Datasets:

common-pile
/

arxiv_papers

Dataset card Data Studio Files Files and versions Community

Split (1)

train · ~317k rows (showing the first 75.7k)

id string	text string	source string	created timestamp[s]	added string	metadata dict
1111.1671	# Construction of wedge-local nets of observables through Longo-Witten endomorphisms. II Marcel Bischoff 111Supported in part by the ERC Advanced Grant 227458 OACFT “Operator Algebras and Conformal Field Theory”. and Yoh Tanimoto∗ e-mail: bischoff@mat.uniroma2.it, tanimoto@mat.uniroma2.it Dipartimento di Matematica, Università di Roma “Tor Vergata” Via della Ricerca Scientifica, 1 - I–00133 Roma, Italy. ###### Abstract In the first part, we have constructed several families of interacting wedge- local nets of von Neumann algebras. In particular, there has been discovered a family of models based on the endomorphisms of the ${\rm U(1)}$-current algebra ${{\mathcal{A}}^{(0)}}$ of Longo-Witten. In this second part, we further investigate endomorphisms and interacting models. The key ingredient is the free massless fermionic net, which contains the ${\rm U(1)}$-current net as the fixed point subnet with respect to the ${\rm U(1)}$ gauge action. Through the restriction to the subnet, we construct a new family of Longo-Witten endomorphisms on ${{\mathcal{A}}^{(0)}}$ and accordingly interacting wedge-local nets in two-dimensional spacetime. The ${\rm U(1)}$-current net admits the structure of particle numbers and the S-matrices of the models constructed here do mix the spaces with different particle numbers of the bosonic Fock space. Dedicated to Roberto Longo on the occasion of his 60th birthday ###### Contents 1. 1 Introduction 2. 2 Preliminaries 1. 2.1 Fermi nets on $S^{1}$ 2. 2.2 Subnets and the character argument 3. 2.3 Scattering theory of waves in ${\mathbb{R}}^{2}$ (revisited) 4. 2.4 Restriction of wedge-local nets 3. 3 Examples of fermi nets 1. 3.1 ${\rm U(1)}$-current net ${{\mathcal{A}}^{(0)}}$ 2. 3.2 The free complex fermion net $\mathrm{Fer}_{\mathbb{C}}$ 3. 3.3 ${\rm U(1)}$-current net as a subnet of $\mathrm{Fer}_{\mathbb{C}}$ 4. 4 A new family of Longo-Witten endomorphisms on ${\rm U(1)}$-current net 5. 5 Interacting wedge-local net with particle production 1. 5.1 Construction of scattering operators 2. 5.2 Action of the S-matrix on the 1+1 particle space 6. 6 Conclusion and outlook ## 1 Introduction As already explained in Part I [Tan11a], construction of interacting models of Quantum Field Theory in (physical) four spacetime dimensions has been a long- standing open problem, and recently the algebraic approach had several progress [Lec08, GL07, GL08, BS08, BLS11, Lec11] and two dimensional cases work particularly well: these works constructed models of QFT with weaker localization property, and in some case such models turned out to be strictly local and fully interacting [Lec08]. One should recall, however, that the models in [Lec08] allow a complete interpretation in terms of particles (asymptotic completeness) and the particle number is preserved under the scattering operator. On the other hand, it is known that in four dimensions an interacting model inevitably involves particle production [Aks65]. In the present paper, we construct a further new family of interacting wedge-local two-dimensional massless models and find that their S-matrices mix the spaces with different particle numbers. In fact, the requirement to involve particle production non-perturbatively is already not simple. On the one hand, an asymptotically complete model must behave like the free theory and hence must be compatible with the Fock space structure at asymptotic time. On the other hand, a particle production process properly means a violation of the Fock structure at physical time. To overcome this difficulty, one would have to “deform” the free theory in a somewhat involved way (cf. [Lec11]) or should rely on a nice trick. Here we take the second way. Standard examples and techniques from Conformal Field Theory provide such a trick. Conformal Field Theory has been well studied particularly on the circle, which can be seen as a chiral part of 1+1 dimensional theory. There are many important examples of such models, or nets in operator-algebraic terms, and both field-theoretic and operator-algebraic techniques allow one to analyze their interrelationships. Our trick can be briefly summarized as follows: we consider the free complex fermionic field $\psi$ on the circle; the field $\psi$ admits a gauge group action by ${\rm U(1)}$, and the fixed point with respect to this action is known to be isomorphic to the algebra of the ${\rm U(1)}$-current $J$. Both fields are free fields acting naturally on the Fock space (fermionic and bosonic, respectively) but the correspondence between the spaces is quite involved. The passage to 1+1 dimensional models is simply the tensor product of two such chiral parts. Now, we can easily “deform” the two- dimensional Dirac field (built up from the chiral parts $\psi\otimes{\mathbbm{1}}$ and ${\mathbbm{1}}\otimes\psi$) in such a way that it commutes with the product action of the gauge group ${\rm U(1)}\times{\rm U(1)}$. Hence the deformation restricts to the algebra of the conserved current $J^{\mu}=(J^{0},J^{1})=(J\otimes{\mathbbm{1}}+{\mathbbm{1}}\otimes J,{\mathbbm{1}}\otimes J-J\otimes{\mathbbm{1}})$, and this deformation is sufficiently complicated so that the resulting S-matrix does not preserve the bosonic Fock structure, thanks to the involved fermion-boson correspondence. In Part I, we have constructed a family of two-dimensional massless models based on the free current $J^{\mu}$ or more precisely its net ${{\mathcal{A}}^{(0)}}\otimes{{\mathcal{A}}^{(0)}}$ of von Neumann algebras of observables. The main ingredient was endomorphisms of the algebra ${{\mathcal{A}}^{(0)}}({\mathbb{R}}_{+})$ of observables localized in the positive half-line ${\mathbb{R}}_{+}$ commuting with the translations. A family of such endomorphisms has been studied first by Longo and Witten [LW11] in order to construct Quantum Field Theory with boundary. We used those endomorphisms to construct two-dimensional models without boundary. In the present article, we study the fermi net $\mathrm{Fer}_{\mathbb{C}}$ generated by the free complex fermionic field $\psi$ and its Longo-Witten endomorphisms. We construct endomorphisms of $\mathrm{Fer}_{\mathbb{C}}$ which commute with the gauge action of ${\rm U(1)}$, hence restrict to the fixed point subnet of ${{\mathcal{A}}^{(0)}}$. It turns out that the restricted endomorphisms cannot be implemented by second quantization operators, hence are different from the ones considered in [LW11]. We again knit them up to construct S-matrices and wedge-local nets. Then the fixed point with respect to the action of ${\rm U(1)}\times{\rm U(1)}$ is considered. We find that its asymptotic behaviour is the same as the free (bosonic) current $J^{\mu}$ and the S-matrix does not preserve the space of $1+1$ particles ($1$ left and $1$ right moving particle) in the sense of the Fock space structure. We stress that the Fock space particle number has no intrinsic meaning as particles, because we are in a massless case where just a scattering between two waves is considered. One has to pass to the massive case to talk about particle production. We will discuss in more detail the implication of this phenomenon at the end of Section 5. This paper is organized as follows. In Section 2 we recall the standard notions of algebraic QFT and the scattering theory of two-dimensional massless models [Buc75, DT11]. Some simple observations are given about subtheories and inner symmetries. Main examples of nets, the free complex fermionic net $\mathrm{Fer}_{\mathbb{C}}$ and the ${\rm U(1)}$-current net ${{\mathcal{A}}^{(0)}}$, are introduced in Section 3. Although it is well- known [KR87, Kac98, Reh98] that the fixed point subnet $\mathrm{Fer}_{\mathbb{C}}$ with respect to ${\rm U(1)}$ is ${{\mathcal{A}}^{(0)}}$ at the field-theoretic level, we prove it in the framework of algebraic approach. Section 4 is devoted to the construction of new Longo-Witten endomorphisms on ${{\mathcal{A}}^{(0)}}$. They are used in Section 5 to construct new interacting wedge-local nets. Outlook and open problems are summarized in Section 6. ## 2 Preliminaries ### 2.1 Fermi nets on $S^{1}$ Here we give a summary of one-dimensional nets, since they will be our building blocks of the construction of two-dimensional interacting models. In the first part, we considered local nets of von Neumann algebras on $S^{1}$. Since we need to exploit the free fermionic field in this second part, a generalized concept of nets is recalled. We follow the definition in [CKL08] and denote by ${\rm M\ddot{o}b}^{(2)}(\cong{\rm SL}(2,{\mathbb{R}})\cong{\rm SU}(1,1))$ the double cover of the Möbius group ${\rm PSL}(2,{\mathbb{R}})$. We denote by $\mathcal{I}$ the set of proper intervals $I\subset S^{1}$, where proper means that $I$ is open and connected and neither dense nor the empty set. A (Möbius covariant) fermi net is an assignment of von Neumann algebras ${\mathcal{F}}_{0}(I)$ on ${\mathcal{H}}_{{\mathcal{F}}_{0}}$ to intervals $I\in\mathcal{I}$ on $S^{1}$ satisfying the following conditions: 1. (1) Isotony. If $I_{1}\subset I_{2}$, then ${\mathcal{F}}_{0}(I_{1})\subset{\mathcal{F}}_{0}(I_{2})$. 2. (2) Möbius covariance. There exists a strongly continuous unitary representation $U_{0}$ of the group ${\rm M\ddot{o}b}^{(2)}$ such that for any interval $I\in\mathcal{I}$ it holds that $U_{0}(g){\mathcal{F}}_{0}(I)U_{0}(g)^{}={\mathcal{F}}_{0}(gI),\mbox{ for }g\in{\rm M\ddot{o}b}^{(2)},$ where the action of ${\rm M\ddot{o}b}^{(2)}\cong{\rm SU}(1,1)$ on $S^{1}$ is defined through linear fractional transformation. 3. (3) Positivity of energy. The generator of the one-parameter subgroup of the lift of rotations in $\mathrm{M\ddot{o}b}$ in the representation $U_{0}$ is positive. 4. (4) Existence of the vacuum. There is a unique (up to a phase) unit vector $\Omega_{0}$ in ${\mathcal{H}}_{{\mathcal{F}}_{0}}$ which is invariant under the action of $U_{0}$, and cyclic for $\bigvee_{I\Subset S^{1}}{\mathcal{F}}_{0}(I)$. 5. (5) ${\mathbb{Z}}_{2}$-grading. There is a unitary operator $\Gamma_{0}$ with $\Gamma_{0}^{2}={\mathbbm{1}}$ such that $\Gamma_{0}\Omega_{0}=\Omega_{0}$ and ${\hbox{\rm Ad\,}}\Gamma_{0}({\mathcal{F}}_{0}(I))={\mathcal{F}}_{0}(I)$. 6. (6) Graded locality. If $I_{1}\cap I_{2}=\emptyset$, then $[{\mathcal{F}}_{0}(I_{1}),{\hbox{\rm Ad\,}}Z_{0}({\mathcal{F}}_{0}(I_{2}))]=0$, where $Z_{0}:=\frac{{\mathbbm{1}}-\mathrm{i}\Gamma_{0}}{1-\mathrm{i}}$. If the grading operator is trivial: $Z_{0}={\mathbbm{1}}$, then the net ${\mathcal{F}}_{0}$ is said to be local. Among the consequences of these conditions are (see [CKL08]): 1. (7) Reeh-Schlieder property. The vector $\Omega_{0}$ is cyclic and separating for each ${\mathcal{F}}_{0}(I)$. 2. (8) Additivity. If $I=\bigcup_{i}I_{i}$, then ${\mathcal{F}}_{0}(I)=\bigvee_{i}{\mathcal{F}}_{0}(I_{i})$. 3. (9) Twisted Haag duality on $S^{1}$. For an interval $I\in\mathcal{I}$, it holds that ${\mathcal{F}}_{0}(I)^{\prime}={\hbox{\rm Ad\,}}Z_{0}({\mathcal{F}}_{0}(I^{\prime}))$, where $I^{\prime}$ is the interior of the complement of $I$ in $S^{1}$. 4. (10) Bisognano-Wichmann property. The modular group $\Delta_{{\mathcal{F}}_{0}({\mathbb{R}}_{+})}^{\mathrm{i}t}$ of ${\mathcal{F}}_{0}({\mathbb{R}}_{+})$ with respect to $\Omega_{0}$ is equal to $U_{0}(\delta(-2\pi t))$, where $S^{1}$ is identified as the one-point compactification of ${\mathbb{R}}$ as below and $\delta$ is the one-parameter group of dilations. 5. (11) Irreducibility. It holds that $\bigvee_{I\in\mathcal{I}}{\mathcal{F}}_{0}(I)=B({\mathcal{H}}_{{\mathcal{F}}_{0}})$. Each algebra ${\mathcal{F}}_{0}(I)$ is referred to as a local algebra (even for a fermi net). Note that if the grading operator $\Gamma_{0}$ is trivial, then the definition of fermi net coincides with the one of local Möbius- covariant net. We identify the circle $S^{1}$ and the compactified real line ${\mathbb{R}}\cup\\{\infty\\}$ through the Cayley transform $t=-\mathrm{i}\frac{z-1}{z+1}\Longleftrightarrow z=-\frac{t-\mathrm{i}}{t+\mathrm{i}},\phantom{...}t\in{\mathbb{R}},\phantom{..}z\in S^{1}\subset{\mathbb{C}}$ and refer to the algebra ${\mathcal{F}}_{0}(I)$ for an interval $I\subset{\mathbb{R}}$. The representation $U_{0}$ of ${\rm M\ddot{o}b}^{(2)}\cong\mathrm{SL}(2,{\mathbb{R}})$ restricts indeed to a projective unitary representation of ${\rm PSL}(2,{\mathbb{R}})$ [CKL08]. Let $\rho$ be the ($4\pi$ periodic) lift of the rotations in $\mathrm{PSU}(1,1)$ (acting by $\rho(\theta)z=\mathrm{e}^{\mathrm{i}\theta}z$) to $\mathrm{M\ddot{o}b}^{(2)}$ and let us denote $R_{0}(\theta)=U_{0}(\rho(\theta))=\mathrm{e}^{\mathrm{i}\theta L_{0}}$. Under the identification between $S^{1}$ and ${\mathbb{R}}\cup\\{\infty\\}$, one can talk about the translations and dilations of ${\mathbb{R}}$, which are included in $\mathrm{M\ddot{o}b}$. In particular, the representation of translations (which we denote by $\tau$) plays a crucial role. Let us denote $T_{0}(t)=U_{0}(\tau(t))$. A Longo-Witten endomorphism of a fermi net ${\mathcal{F}}_{0}$ is an endomorphism of the algebra ${\mathcal{F}}_{0}({\mathbb{R}}_{+})$ implemented by a unitary $V_{0}$ which commutes with the translation $T_{0}(t)$. A family of Longo-Witten endomorphisms has been found for the ${\rm U(1)}$-current net and the real free fermion net [LW11]. The examples will be explained later in detail. Note that a Longo-Witten endomorphism is uniquely implemented up to scalar. Indeed, since it commutes with translation, ${\hbox{\rm Ad\,}}V_{0}$ is an endomorphism of ${\mathcal{F}}_{0}({\mathbb{R}}_{+}+t)$ for any $t\in{\mathbb{R}}$. If there is another unitary $W_{0}$ which satisfies ${\hbox{\rm Ad\,}}W_{0}(x)={\hbox{\rm Ad\,}}V_{0}(x)$ for any $x\in{\mathcal{F}}_{0}({\mathbb{R}}_{+}+t)$, $t\in{\mathbb{R}}$, then by the irreducibility $W_{0}^{}V_{0}$ must be scalar. ### 2.2 Subnets and the character argument Let ${\mathcal{F}}_{0}$ be a fermi (or local) net on ${\mathcal{H}}_{{\mathcal{F}}_{0}}$. Another assignment ${\mathcal{A}}_{0}$ of von Neumann algebras $\\{{\mathcal{A}}_{0}(I)\\}_{I\in\mathcal{I}}$ on ${\mathcal{H}}_{{\mathcal{F}}_{0}}$ is called a subnet of ${\mathcal{F}}_{0}$ if it satisfies isotony, Möbius covariance with respect to the same $U_{0}$ for ${\mathcal{F}}_{0}$ and it holds that ${\mathcal{A}}_{0}(I)\subset{\mathcal{F}}_{0}(I)$ for every interval $I\in\mathcal{I}$. We simply write ${\mathcal{A}}_{0}\subset{\mathcal{F}}_{0}$. In this case, let us denote ${\mathcal{H}}_{{\mathcal{A}}_{0}}=\overline{\bigvee_{I\in\mathcal{I}}{\mathcal{A}}_{0}(I)\Omega_{0}}$. Then it is immediate to see that ${\mathcal{A}}_{0}(I)$ and $U_{0}$ restrict to ${\mathcal{H}}_{{\mathcal{A}}_{0}}$, and by this restriction ${\mathcal{A}}_{0}\|_{{\mathcal{H}}_{{\mathcal{A}}_{0}}}$ becomes a fermi net with the representation of covariance $U_{0}\|_{{\mathcal{H}}_{{\mathcal{A}}_{0}}}$. This restriction is also said to be a subnet of ${\mathcal{F}}_{0}$ if no confusion arises. For a fermi net ${\mathcal{F}}_{0}$ on $S^{1}$, a gauge automorphism $\alpha_{0}$ is a family of automorphisms $\\{\alpha_{0,I}\\}$ of local algebras which satisfies the consistency condition $\alpha_{0,I_{2}}\|_{{\mathcal{A}}_{0}(I_{1})}=\alpha_{0,I_{1}}\,\,\mbox{ for }I_{1}\subset I_{2}\,.$ If a gauge automorphism $\alpha_{0}$ preserves the vacuum state $\langle\Omega_{0},\cdot\,\Omega_{0}\rangle$, it is said to be an inner symmetry. An inner symmetry $\alpha_{0}$ can be unitarily implemented by the formula $V_{\alpha_{0}}x\Omega_{0}=\alpha_{0}(x)\Omega_{0}$, where $x$ is an element of some local algebra ${\mathcal{F}}_{0}(I)$. We say that a compact group $G$ acts on the net ${\mathcal{F}}_{0}$ when there is automorphisms $\\{\alpha_{0,g}\\}_{g\in G}$ which satisfy the composition law when restricted to local algebras. The fixed point subnet with respect to this action of $G$ is the subnet defined by ${\mathcal{F}}_{0}^{G}(I):={\mathcal{F}}_{0}(I)^{G}$. Let ${\mathcal{F}}_{0}$ be a fermi net and ${\mathcal{A}}_{0}$ be a subnet. Recall that, for a Möbius covariant fermi net, the Bisognano-Wichmann property is automatic. As a consequence, for each interval there is a conditional expectation $E_{0,I}:{\mathcal{F}}_{0}(I)\to{\mathcal{A}}_{0}(I)$ which preserves the vacuum state $\langle\Omega_{0},\cdot\,\Omega_{0}\rangle$ and implemented by the projection $P_{{\mathcal{A}}_{0}}$ onto ${\mathcal{H}}_{{\mathcal{A}}_{0}}$ (see [Tak03, Theorem IX.4.2]). This projection $P_{{\mathcal{A}}_{0}}$ contains much information of ${\mathcal{A}}_{0}$. Consider the case where ${\mathcal{A}}_{0}={\mathcal{F}}_{0}^{G}$ is the fixed point subnet with respect to an action $\alpha_{0}$ of a compact group $G$ by inner symmetry. Then we have a unitary representation $V_{\alpha_{0}}$ of $G$ on ${\mathcal{H}}_{{\mathcal{F}}_{0}}$. If we write the set of invariant vectors with respect to $V_{\alpha_{0}}$ by ${\mathcal{H}}_{{\mathcal{F}}_{0}}^{G}$, it holds that ${\mathcal{H}}_{{\mathcal{F}}_{0}}^{G}={\mathcal{H}}_{{\mathcal{A}}_{0}}$. Indeed, the inclusion ${\mathcal{H}}_{{\mathcal{A}}_{0}}\subset{\mathcal{H}}_{{\mathcal{F}}_{0}}^{G}$ is obvious. On the other hand, for $x\in{\mathcal{F}}_{0}(I)$, we have $\left(\int_{G}\alpha_{0}(x)\,\mathrm{d}g\right)\Omega_{0}=\int_{G}\left(V_{\alpha_{0}}(g)x\Omega_{0}\right)\,\mathrm{d}g,$ which implies that any vector in ${\mathcal{H}}_{{\mathcal{F}}_{0}}^{G}$ can be approximated from ${\mathcal{H}}_{{\mathcal{A}}_{0}}$ by the Reeh-Schlieder property. For the later use, we put here a simple observation. ###### Proposition 2.1. In the situation above, if a Longo-Witten endomorphism is implemented by $W_{0}$ and $W_{0}$ commutes with $V_{\alpha_{0}}$, then ${\hbox{\rm Ad\,}}W_{0}$ restricts to a Longo-Witten endomorphism of the fixed point subnet ${\mathcal{A}}_{0}$. ###### Proof. The unitary $W_{0}$ commutes with the projection $P_{{\mathcal{A}}_{0}}$, hence also with the conditional expectation $E_{0}$ onto ${\mathcal{A}}_{0}$. ∎ Let ${\mathcal{F}}_{0}$ be fermi (or local) net on ${\mathcal{H}}_{{\mathcal{F}}_{0}}$. The Hilbert space ${\mathcal{H}}_{{\mathcal{F}}_{0}}$ is graded by the action of the rotation subgroup $R_{0}(\theta)=\mathrm{e}^{\mathrm{i}\theta L_{0}}$: ${\mathcal{H}}_{{\mathcal{F}}_{0}}={\mathbb{C}}\Omega_{0}\oplus\bigoplus_{r\in\frac{1}{2}{\mathbb{N}}}{\mathcal{H}}_{r}=\bigoplus_{r\in\frac{1}{2}{\mathbb{N}}_{0}}{\mathcal{H}}_{r}$ with ${\mathcal{H}}_{r}=\\{\xi\in{\mathcal{H}}_{{\mathcal{F}}_{0}}:R_{0}(\theta)\xi=\mathrm{e}^{\mathrm{i}r\theta}\xi\\}$ and the sum only going over ${\mathbb{N}}_{0}$ for a local net. The conformal character of the net ${\mathcal{F}}_{0}$ is given as a formal power series of $t=\mathrm{e}^{-\beta}$: $\operatorname{tr}_{{\mathcal{H}}_{{\mathcal{F}}_{0}}}(\mathrm{e}^{-\beta L_{0}})=\sum_{r\in\frac{1}{2}{\mathbb{N}}_{0}}^{\infty}{\hbox{dim}\,}{\mathcal{H}}_{r}\cdot t^{r}\,.$ Let us assume that there is an action of $G={\rm U(1)}$ by inner symmetry. We denote by $V_{0}(\theta)$ the implementing unitary. Then $V_{0}$ and $U_{0}$ commute and ${\mathcal{H}}_{F_{0}}$ is graded also by the gauge action $V_{0}(\theta)=\mathrm{e}^{\mathrm{i}\theta Q_{0}}$: ${\mathcal{H}}_{{\mathcal{F}}_{0}}={\mathbb{C}}\Omega_{0}\oplus\bigoplus_{r\in\frac{1}{2}{\mathbb{N}},q\in{\mathbb{Z}}}{\mathcal{H}}_{r,q}=\bigoplus_{q\in{\mathbb{Z}}}{\mathcal{H}}_{\,\cdot\,,q},\qquad\text{with}\qquad{\mathcal{H}}_{\,\cdot\,,q}:=\bigoplus_{r\in\frac{1}{2}{\mathbb{N}}_{0}}{\mathcal{H}}_{r,q}$ and the character is given as a formal power series in $t=\mathrm{e}^{-\beta}$ and $z=\mathrm{e}^{-E}$: $\operatorname{tr}_{{\mathcal{H}}_{{\mathcal{F}}_{0}}}(\mathrm{e}^{-\beta L_{0}-EQ_{0}})=\sum_{r\in\frac{1}{2}{\mathbb{N}}_{0},q\in{\mathbb{Z}}}{\hbox{dim}\,}{\mathcal{H}}_{r,q}\cdot t^{r}z^{q}\,.$ Recall that it holds that ${\mathcal{H}}_{{\mathcal{F}}_{0}}^{G}={\mathcal{H}}_{{\mathcal{A}}_{0}}$. The operator $Q_{0}$ acts by $0$ on ${\mathcal{H}}_{{\mathcal{F}}_{0}}^{G}$, hence we can obtain the conformal character of ${\mathcal{A}}_{0}$ just by taking the coefficient of $z^{0}$ in $\operatorname{tr}_{{\mathcal{H}}_{{\mathcal{F}}_{0}}}(\mathrm{e}^{-\beta L_{0}-EQ_{0}})$. Later in this paper we need to compare the size of two subnets. Let ${\mathcal{A}}_{0}\subset{\mathcal{B}}_{0}\subset{\mathcal{F}}_{0}$ be an inclusion of three fermi nets. If the conformal characters of ${\mathcal{A}}_{0}$ and ${\mathcal{B}}_{0}$ coincide, then this means that the subspaces ${\mathcal{H}}_{{\mathcal{A}}_{0}}$ and ${\mathcal{H}}_{{\mathcal{B}}_{0}}$ coincide, since we have already an inclusion ${\mathcal{H}}_{{\mathcal{A}}_{0}}\subset{\mathcal{H}}_{{\mathcal{B}}_{0}}$ and the coefficients of the conformal character are the dimensions of eigenspaces of $L_{0}$. This in turn implies that two subnets ${\mathcal{A}}_{0}$ and ${\mathcal{B}}_{0}$ are the same since the conditional expectations which are implemented by $P_{{\mathcal{A}}_{0}},P_{{\mathcal{B}}_{0}}$ are the same. We will see such an argument in an example. ### 2.3 Scattering theory of waves in ${\mathbb{R}}^{2}$ (revisited) Here we just collect some basic notions regarding scattering theory of two- dimensional massless models. As recalled in Part I [Tan11a], this theory has been established by Buchholz [Buc75] and extended to the wedge-local case [DT11]. A Borchers triple on a Hilbert space ${\mathcal{H}}$ is a triple $({\mathcal{M}},T,\Omega)$ of a von Neumann algebra ${\mathcal{M}}\subset B({\mathcal{H}})$, a unitary representation $T$ of ${\mathbb{R}}^{2}$ on ${\mathcal{H}}$ and a vector $\Omega\in{\mathcal{H}}$ such that * • ${\hbox{\rm Ad\,}}T(t_{0},t_{1})({\mathcal{M}})\subset{\mathcal{M}}$ for $(t_{0},t_{1})\in W_{\mathrm{R}}=\\{(x_{0},x_{1})\in{\mathbb{R}}^{2}:x_{1}>\|x_{0}\|\\}$, the standard right wedge. * • The joint spectrum ${\rm sp}\,T$ is contained in the closed forward lightcone $\overline{V}_{+}=\\{(p_{0},p_{1})\in{\mathbb{R}}_{2}:p_{0}\geq\|p_{1}\|\\}$. * • $\Omega$ is a unique (up to scalar) invariant vector under $T$, and cyclic and separating for ${\mathcal{M}}$. We recall that one interprets ${\mathcal{M}}$ as the algebra assigned to the wedge $W_{\mathrm{R}}$. Let ${\mathcal{W}}$ be the set of wedges, i.e. the set of all $W=gW_{\mathrm{R}}$ where $g$ is a Poincaré transformation, then we define the wedge-local net ${\mathcal{W}}\ni W\mapsto{\mathcal{M}}(W)$ associated with the Borchers triple $({\mathcal{M}},T,\Omega)$ by ${\mathcal{M}}(W_{R}+a)=T(a){\mathcal{M}}T(a)^{\ast}$ and ${\mathcal{M}}(W_{R}^{\prime}+a)=T(a){\mathcal{M}}^{\prime}T(a)^{\ast}$. With the help of the modular objects one can define a representation of the Poincaré group extending the one of translations $T$ [Bor92]. For details we refer to the first part. Take a Borchers triple $({\mathcal{M}},T,\Omega)$ and $x\in B({\mathcal{H}})$. We write $x(a)={\hbox{\rm Ad\,}}T(a)(x)$ for $a\in{\mathbb{R}}^{2}$ and consider observables sent to lightlike directions with parameter ${\mathcal{T}}$: $x_{\pm}(h_{\mathcal{T}}):=\int h_{\mathcal{T}}(t)x(t,\pm t)\,\mathrm{d}t,$ where $h_{\mathcal{T}}(t)=\|{\mathcal{T}}\|^{-\varepsilon}h(\|{\mathcal{T}}\|^{-\varepsilon}(t-{\mathcal{T}}))$, $0<\varepsilon<1$ is a constant, ${\mathcal{T}}\in{\mathbb{R}}$ and $h$ is a nonnegative symmetric smooth function on ${\mathbb{R}}$ such that $\int h(t)\,\mathrm{d}t=1$. Then for $x\in{\mathcal{M}}$, the limits $\Phi^{\mathrm{out}}_{+}(x):=\underset{{\mathcal{T}}\to+\infty}{{{\mathrm{s}\textrm{-}\lim}}}\,x_{+}(h_{\mathcal{T}})$ and $\Phi^{\mathrm{in}}_{-}(x):=\underset{{\mathcal{T}}\to-\infty}{{{\mathrm{s}\textrm{-}\lim}}}\,x_{-}(h_{\mathcal{T}})$ exist. Furthermore we set $\Phi_{+}^{\mathrm{in}}(y^{\prime}):=J_{\mathcal{M}}\Phi_{+}^{\mathrm{out}}(J_{\mathcal{M}}y^{\prime}J_{\mathcal{M}})J_{\mathcal{M}},\,\,\Phi_{-}^{\mathrm{out}}(y^{\prime}):=J_{\mathcal{M}}\Phi_{-}^{\mathrm{in}}(J_{\mathcal{M}}y^{\prime}J_{\mathcal{M}})J_{\mathcal{M}}$ for $y^{\prime}\in{\mathcal{M}}^{\prime}$, where $J_{\mathcal{M}}$ is the modular conjugation of ${\mathcal{M}}$ with respect to $\Omega$. The properties of these asymptotic fields are summarized in [DT11, Tan11a]. For example, it holds that $\Phi_{+}^{\mathrm{in}}(y^{\prime})=\underset{{\mathcal{T}}\to-\infty}{{{\mathrm{s}\textrm{-}\lim}}}\,y^{\prime}_{+}(h_{\mathcal{T}})$ and $\Phi_{-}^{\mathrm{out}}(y^{\prime})=\underset{{\mathcal{T}}\to+\infty}{{{\mathrm{s}\textrm{-}\lim}}}\,y^{\prime}_{-}(h_{\mathcal{T}})$. Let ${\mathcal{H}}_{+}$ (respectively by ${\mathcal{H}}_{-}$) be the space of the single excitations with positive momentum, (respectively with negative momentum), i.e. ${\mathcal{H}}_{+}=\\{\xi\in{\mathcal{H}}:T(t,t)\xi=\xi\mbox{ for }t\in{\mathbb{R}}\\}$ (respectively ${\mathcal{H}}_{-}=\\{\xi\in{\mathcal{H}}:T(t,-t)\xi=\xi\mbox{ for }t\in{\mathbb{R}}\\}$). For $\xi_{+}\in{\mathcal{H}}_{+}$, $\xi_{-}\in{\mathcal{H}}_{-}$, there are sequences of local operators $\\{x_{n}\\},\\{y_{n}\\}\subset{\mathcal{M}}$ and $\\{x^{\prime}_{n}\\},\\{y^{\prime}_{n}\\}\subset{\mathcal{M}}^{\prime}$ such that $\underset{n\to\infty}{{{\mathrm{s}\textrm{-}\lim}}}\,P_{+}x_{n}\Omega=\underset{n\to\infty}{{{\mathrm{s}\textrm{-}\lim}}}\,P_{+}x^{\prime}_{n}\Omega=\xi_{+}$ and $\underset{n\to\infty}{{{\mathrm{s}\textrm{-}\lim}}}\,P_{-}y_{n}\Omega=\underset{n\to\infty}{{{\mathrm{s}\textrm{-}\lim}}}\,P_{+}y^{\prime}_{n}\Omega=\xi_{-}$. We define collision states as in [DT11]: $\xi_{+}{\overset{{\mathrm{in}}}{\times}}\xi_{-}=\underset{n\to\infty}{{{\mathrm{s}\textrm{-}\lim}}}\,\Phi^{\mathrm{in}}_{+}(x^{\prime}_{n})\Phi^{\mathrm{in}}_{-}(y_{n})\Omega,\quad\xi_{+}{\overset{{\mathrm{out}}}{\times}}\xi_{-}=\underset{n\to\infty}{{{\mathrm{s}\textrm{-}\lim}}}\,\Phi^{\mathrm{out}}_{+}(x_{n})\Phi^{\mathrm{out}}_{-}(y^{\prime}_{n})\Omega\,.$ We denote by ${\mathcal{H}}^{\mathrm{in}}$ (respectively ${\mathcal{H}}^{\mathrm{out}}$) the subspace generated by $\xi_{+}{\overset{{\mathrm{in}}}{\times}}\xi_{-}$ (respectively $\xi_{+}{\overset{{\mathrm{out}}}{\times}}\xi_{-}$). The isometry $S:{\mathcal{H}}^{\mathrm{out}}\ni\xi_{+}{\overset{{\mathrm{out}}}{\times}}\xi_{-}\longmapsto\xi_{+}{\overset{{\mathrm{in}}}{\times}}\xi_{-}\in{\mathcal{H}}^{\mathrm{in}}$ is called the scattering operator or the S-matrix of the Borchers triple $({\mathcal{M}},T,\Omega)$. We say that the Borchers triple $({\mathcal{M}},T,\Omega)$ is interacting if $S$ is not equal to the identity operator on ${\mathcal{H}}^{\mathrm{out}}$ and asymptotically complete (with respect to waves) if it holds that ${\mathcal{H}}^{\mathrm{in}}={\mathcal{H}}^{\mathrm{out}}={\mathcal{H}}$. We have studied the general structure of asymptotically complete local and wedge-local nets (using Borchers triple) in [Tan11a, Section 3]. The point was that for a given (strictly local) $({\mathcal{M}},T,\Omega)$ we can construct the chiral net, and the original object ${\mathcal{M}}$ can be recovered from the chiral net and a single operator $S$. Here we rephrase this observation from the point of view of constructing examples based on chiral components. See also the general structure of asymptotically complete strictly local nets [Tan11a, Section 3] $x$$t$$W_{\mathrm{R}}$$\scriptstyle\mathrm{Ad\,}S({\mathbbm{1}}\otimes{\mathcal{F}}_{-}({\mathbb{R}}_{+}))$$\scriptstyle{\mathcal{F}}_{+}({\mathbb{R}}_{-})\otimes{\mathbbm{1}}$$W_{\mathrm{R}}+(t_{1},x_{1})$$\scriptstyle\mathrm{Ad\,}S({\mathbbm{1}}\otimes{\mathcal{F}}_{-}({\mathbb{R}}_{+}+\frac{t_{1}+x_{1}}{\sqrt{2}}))$$\scriptstyle{\mathcal{F}}_{+}({\mathbb{R}}_{-}+\frac{t_{1}-x_{1}}{\sqrt{2}})\otimes{\mathbbm{1}}$$x$$t$ Figure 1: On the definition of the wedge-local net ###### Proposition 2.2. Let ${\mathcal{F}}_{\pm}$ be two fermi nets on $S^{1}$ defined on ${\mathcal{H}}_{\pm}$ and assume that there is a unitary operator $S$ on ${\mathcal{H}}_{+}\otimes{\mathcal{H}}_{-}$ commuting with $T_{+}\otimes T_{-}$, leaving ${\mathcal{H}}_{+}\otimes\Omega_{-}$ and $\Omega_{+}\otimes{\mathcal{H}}_{-}$ pointwise invariant, such that $x\otimes{\mathbbm{1}}$ commutes with ${\hbox{\rm Ad\,}}S(x^{\prime}\otimes{\mathbbm{1}})$ where $x\in{\mathcal{F}}_{+}({\mathbb{R}}_{-})$ and $x^{\prime}\in{\hbox{\rm Ad\,}}Z_{+}({\mathcal{F}}_{+}({\mathbb{R}}_{+}))$, and ${\hbox{\rm Ad\,}}S({\mathbbm{1}}\otimes y)$ commutes with ${\mathbbm{1}}\otimes y^{\prime}$ where $y\in{\mathcal{F}}_{-}({\mathbb{R}}_{+})$ and $y^{\prime}\in{\hbox{\rm Ad\,}}Z_{-}({\mathcal{F}}_{-}({\mathbb{R}}_{-}))$. Then the triple * • ${\mathcal{M}}_{S}:=\\{x\otimes{\mathbbm{1}},{\hbox{\rm Ad\,}}S({\mathbbm{1}}\otimes y):x\in{\mathcal{F}}_{+}({\mathbb{R}}_{-}),y\in{\mathcal{F}}_{-}({\mathbb{R}}_{+})\\}^{\prime\prime}$, * • $T(t,x):=T_{+}(\frac{t-x}{\sqrt{2}})\otimes T_{-}(\frac{t+x}{\sqrt{2}})$, * • $\Omega:=\Omega_{+}\otimes\Omega_{-}$ is an asymptotically complete Borchers triple with the S-matrix $S$. ###### Proof. As in Part I [Tan11a], the conditions on $T$ and $\Omega$ are automatic because they are just tensor products of objects for fermi nets. Similarly, the condition that ${\hbox{\rm Ad\,}}T(a){\mathcal{M}}_{S}\subset{\mathcal{M}}_{S}$ for $a\in W_{\mathrm{R}}$ is easily seen from the assumption that $T$ commutes with $S$ and the covariance of fermi nets. What remains is the cyclicity and separating property of $\Omega$ for ${\mathcal{M}}_{S}$. Cyclicity is immediate because we have $\displaystyle{\mathcal{M}}_{S}\Omega$ $\displaystyle\supset$ $\displaystyle\\{x\otimes{\mathbbm{1}}\cdot S({\mathbbm{1}}\otimes y)S^{}\cdot\Omega:x\in{\mathcal{F}}_{+}({\mathbb{R}}_{-}),y\in{\mathcal{F}}_{-}({\mathbb{R}}_{+})\\}$ $\displaystyle=$ $\displaystyle\\{x\otimes y\cdot\Omega:x\in{\mathcal{F}}_{+}({\mathbb{R}}_{-}),y\in{\mathcal{F}}_{-}({\mathbb{R}}_{+})\\}$ by the assumed property of $S$, and the latter set is total in ${\mathcal{H}}_{+}\otimes{\mathcal{H}}_{-}$ by the Reeh-Schlieder property for fermi nets. As for the separating property, we define: ${\mathcal{M}}_{S}^{1}:=\\{{\hbox{\rm Ad\,}}S(x^{\prime}\otimes{\mathbbm{1}}),{\mathbbm{1}}\otimes y^{\prime}:x^{\prime}\in{\hbox{\rm Ad\,}}Z_{+}({\mathcal{F}}_{+}({\mathbb{R}}_{+})),y^{\prime}\in{\hbox{\rm Ad\,}}Z_{-}({\mathcal{F}}_{-}({\mathbb{R}}_{-}))\\}^{\prime\prime}.$ By an analogous proof, one sees that $\Omega$ is cyclic for ${\mathcal{M}}_{S}^{1}$. Furthermore, ${\mathcal{M}}_{S}$ and ${\mathcal{M}}_{S}^{1}$ commute by assumption. Hence $\Omega$ is separating for ${\mathcal{M}}_{S}$. In other words, $({\mathcal{M}}_{S},T,\Omega)$ is a Borchers-triple. It is immediate that $\Phi^{\mathrm{out}}_{+}(x\otimes{\mathbbm{1}})=x\otimes{\mathbbm{1}}$ and $\Phi^{\mathrm{in}}_{-}({\hbox{\rm Ad\,}}S({\mathbbm{1}}\otimes y))={\hbox{\rm Ad\,}}S({\mathbbm{1}}\otimes y)$ (the latter follows since $S$ commutes with $T$). Similarly, we have $\Phi^{\mathrm{in}}_{+}({\hbox{\rm Ad\,}}S(x^{\prime}\otimes{\mathbbm{1}}))={\hbox{\rm Ad\,}}S(x^{\prime}\otimes{\mathbbm{1}})$ and $\Phi^{\mathrm{out}}_{-}({\mathbbm{1}}\otimes y^{\prime})={\mathbbm{1}}\otimes y^{\prime}$. From this, one concludes that the Borchers triple $({\mathcal{M}}_{S},T,\Omega)$ is asymptotically complete and its S-matrix is $S$. ∎ We remark that we see $({\mathcal{M}}_{S},T,\Omega)$ as a fermi (i.e. twisted local) net defined by ${\mathcal{M}}(W_{\mathrm{R}}^{\prime})={\hbox{\rm Ad\,}}Z_{+}\otimes Z_{-}({\mathcal{M}})$ and that the scattering theory of waves [Buc75] is considered to be an analogue of the Haag-Ruelle scattering theory and it is not intended to be applied to fermionic nets. But we will not pay much attention to this restriction, since our result is a construction of wedge-local nets with a free massless bosonic net as the asymptotic net, and fermionic nets appear as auxiliary objects. ### 2.4 Restriction of wedge-local nets We consider a Borchers triple $({\mathcal{M}},T,\Omega)$. It is in some cases interesting to consider a subalgebra ${\mathcal{N}}$ of ${\mathcal{M}}$. Let us denote ${\mathcal{H}}_{\mathcal{N}}:=\overline{{\mathcal{N}}\Omega}$. ###### Proposition 2.3. If the subspace ${\mathcal{H}}_{\mathcal{N}}$ is invariant under $T$ and ${\hbox{\rm Ad\,}}T(a)({\mathcal{N}})\subset{\mathcal{N}}$ for $a\in W_{\mathrm{R}}$. Then $({\mathcal{N}}\|_{{\mathcal{H}}_{\mathcal{N}}},T\|_{{\mathcal{H}}_{\mathcal{N}}},\Omega)$ is a Borchers triple on ${\mathcal{H}}_{\mathcal{N}}$. ###### Proof. The components ${\mathcal{N}},T$ and $\Omega$ naturally restricts to ${\mathcal{H}}_{\mathcal{N}}$. The conditions on $T$ and $\Omega$ are trivial, even restricted to ${\mathcal{H}}_{\mathcal{N}}$. The cyclicity of $\Omega$ is immediate from the definition of ${\mathcal{H}}_{\mathcal{N}}$. Since $\Omega$ is already separating for ${\mathcal{M}}$, so is also for ${\mathcal{N}}$. Endomorphic action of $T$ on ${\mathcal{N}}$ is in the hypothesis. ∎ We call a triple $({\mathcal{N}},T,\Omega)$ a (Borchers) subtriple of $({\mathcal{M}},T,\Omega)$ if ${\mathcal{N}}$ is a subalgebra of ${\mathcal{M}}$, ${\mathcal{H}}_{\mathcal{N}}$ is invariant under $T(a)$, ${\hbox{\rm Ad\,}}T(a)({\mathcal{N}})\subset{\mathcal{N}}$ for $a\in W_{\mathrm{R}}$, and ${\mathcal{N}}$ is invariant under ${\hbox{\rm Ad\,}}\Delta_{\mathcal{M}}^{\mathrm{i}t}$, where $\Delta_{\mathcal{M}}$ is the modular operator of ${\mathcal{M}}$ with respect to $\Omega$. Recall that a Borchers triple $({\mathcal{M}},T,\Omega)$ gives rise to a strictly local net if $\Omega$ is cyclic for ${\mathcal{M}}\cap{\hbox{\rm Ad\,}}T(a)({\mathcal{M}})^{\prime}$ for any $a\in W_{\mathrm{R}}$. We call such a triple therefore strictly local. The following proposition shows that the concept of Borchers subtriple corresponds to the one of a local subnet. ###### Proposition 2.4. If a Borchers triple $({\mathcal{M}},T,\Omega)$ is strictly local, any subtriple $({\mathcal{N}},T,\Omega)$ is again strictly local when restricted on $\overline{{\mathcal{N}}\Omega}$. ###### Proof. Since ${\mathcal{N}}$ is invariant under the modular automorphism ${\hbox{\rm Ad\,}}\Delta_{\mathcal{M}}^{\mathrm{i}t}$, there is a conditional expectation $E$ from ${\mathcal{M}}$ onto ${\mathcal{N}}$ which preserves the state $\langle\Omega,\,\cdot\,\Omega\rangle$ and is implemented by the projection $P_{\mathcal{N}}$ (see [Tak03, Theorem IX.4.2] for the original reference and [Tan11b, Appendix A] for an application to nets). We have to show that $\Omega$ is cyclic for the relative commutant ${\mathcal{N}}\cap{\hbox{\rm Ad\,}}T(a)({\mathcal{N}})^{\prime}$ on the subspace ${\mathcal{H}}_{\mathcal{N}}$. Let us denote ${\mathcal{M}}_{0,a}:={\mathcal{M}}\cap{\hbox{\rm Ad\,}}T(a)({\mathcal{M}})^{\prime}$. We claim that $E({\mathcal{M}}_{0,a})$ is contained in ${\mathcal{N}}\cap{\hbox{\rm Ad\,}}T(a)({\mathcal{N}})^{\prime}$. Indeed, by the definition of $E$, the image of $E$ is contained in ${\mathcal{N}}$. Furthermore, if $x\in{\mathcal{M}}_{0,a}$, $y\in{\hbox{\rm Ad\,}}T(a)({\mathcal{N}})\subset{\hbox{\rm Ad\,}}T(a)({\mathcal{M}})$, then $E(x)y=E(xy)=E(yx)=yE(x),$ hence they commute and the image $E({\mathcal{M}}_{0,a})$ lies in the relative commutant. Now we have $\overline{\left({\mathcal{N}}\cap{\hbox{\rm Ad\,}}T(a)({\mathcal{N}})^{\prime}\right)\Omega}\supset\overline{E({\mathcal{M}}_{0,a})\Omega}\supset\overline{P_{\mathcal{N}}{\mathcal{M}}_{0,a}\Omega}={\mathcal{H}}_{\mathcal{N}}$ by the assumed strict locality of $({\mathcal{M}},T,\Omega)$. ∎ Let $({\mathcal{B}},U,\Omega)$ be an asymptotically complete local Poincaré covariant net on ${\mathbb{R}}^{2}$ fulfilling the Bisognano-Wichmann property (see [Tan11a] for related definitions). We recall that one can define the (out-) asymptotic algebras ${\mathcal{B}}_{+}\otimes{\mathcal{B}}_{-}$ and the scattering operator $S$ which is a unitary operator, and that it is possible to recover the original net by the formula ${\mathcal{B}}(W_{\mathrm{R}})=\\{x\otimes{\mathbbm{1}},{\hbox{\rm Ad\,}}S({\mathbbm{1}}\otimes y):x\in{\mathcal{B}}_{+}({\mathbb{R}}_{-}),y\in{\mathcal{B}}_{-}({\mathbb{R}}_{+})\\}^{\prime\prime}.$ Note that $({\mathcal{B}}(W_{\mathrm{R}}),U\|_{{\mathbb{R}}^{2}},\Omega)$ is an asymptotically complete, strictly local Borchers triple. Here we exhibit a simple way to construct subtriples. Let ${\mathcal{A}}_{+},{\mathcal{A}}_{-}$ be (Möbius covariant) subnets of ${\mathcal{B}}_{+},{\mathcal{B}}_{-}$, respectively. If we define ${\mathcal{N}}=\\{x\otimes{\mathbbm{1}},{\hbox{\rm Ad\,}}S({\mathbbm{1}}\otimes y):x\in{\mathcal{A}}_{+}({\mathbb{R}}_{-}),y\in{\mathcal{A}}_{-}({\mathbb{R}}_{+})\\}^{\prime\prime},$ then $({\mathcal{N}},U\|_{{\mathbb{R}}^{2}},\Omega)$ is a Borchers subtriple of $({\mathcal{B}}(W_{\mathrm{R}}),U\|_{{\mathbb{R}}^{2}},\Omega)$. Indeed, conditions regarding ${\mathcal{N}},U\|_{{\mathbb{R}}^{2}},\Omega$ are immediate. As for the invariance of ${\mathcal{N}}$ under ${\hbox{\rm Ad\,}}\Delta_{\mathcal{M}}^{\mathrm{i}t}$, it suffices to note that $S$ and $\Delta_{\mathcal{M}}^{\mathrm{i}t}$ commute ([Tan11a, Lemma 2.4], cf. [Buc75]) and that ${\mathcal{A}}_{+}({\mathbb{R}}_{-})$ and ${\mathcal{A}}_{-}({\mathbb{R}}_{+})$ are preserved by ${\hbox{\rm Ad\,}}\Delta_{\mathcal{M}}^{\mathrm{i}t}$ because of Bisognano-Wichmann property. The trouble is, however, that such Borchers triples constructed as above are not necessarily asymptotically complete in general. Indeed, the out-asymptotic states span the subspace $\overline{{\mathcal{A}}_{+}({\mathbb{R}}_{-})\Omega}\otimes\overline{{\mathcal{A}}_{-}({\mathbb{R}}_{+})\Omega}$. It is easy to see that this coincides with the full space $\overline{{\mathcal{N}}\Omega}$ if and only if it is invariant under $S$. Since a clear-cut scattering theory is so far available only for asymptotically complete nets, it is worthwhile to give a general condition to assure that subnets are asymptotically complete. For simplicity, we consider the following situation: let ${\mathcal{A}}_{0}$ be a fermi net on ${\mathcal{H}}_{0}$ with an action of a compact group $G$ by inner symmetry implemented by $V_{g}$. Suppose that there is a unitary operator $S$ on ${\mathcal{H}}_{0}\otimes{\mathcal{H}}_{0}$ such that $({\mathcal{M}}_{S},T,\Omega)$ is a Borchers triple where • ${\mathcal{M}}_{S}:=\\{x\otimes{\mathbbm{1}},{\hbox{\rm Ad\,}}S({\mathbbm{1}}\otimes y):x\in{\mathcal{A}}_{0}({\mathbb{R}}_{-}),y\in{\mathcal{A}}_{0}({\mathbb{R}}_{+})\\}^{\prime\prime}$, * • $T(t,x):=T_{0}(\frac{t-x}{\sqrt{2}})\otimes T_{0}(\frac{t+x}{\sqrt{2}})$, * • $\Omega:=\Omega_{0}\otimes\Omega_{0}$, as in Proposition 2.2. ###### Proposition 2.5. If $S$ commutes with $V_{g}\otimes V_{g^{\prime}}$, $g,g^{\prime}\in G$, then the triple * • ${\mathcal{N}}_{S}:=\\{x\otimes{\mathbbm{1}},{\hbox{\rm Ad\,}}S({\mathbbm{1}}\otimes y):x\in{\mathcal{A}}_{0}^{G}({\mathbb{R}}_{-}),y\in{\mathcal{A}}_{0}^{G}({\mathbb{R}}_{+})\\}^{\prime\prime}$ (restricted to $\overline{{\mathcal{N}}_{S}\Omega}$), * • $T(t,x):=T_{0}(\frac{t-x}{\sqrt{2}})\otimes T_{0}(\frac{t+x}{\sqrt{2}})$, (restricted to $\overline{{\mathcal{N}}_{S}\Omega}$) * • $\Omega:=\Omega_{0}\otimes\Omega_{0}$ is an asymptotically complete Borchers triple with asymptotic algebra ${\mathcal{A}}_{0}^{G}\otimes{\mathcal{A}}_{0}^{G}$ and scattering operator $S\|_{\overline{{\mathcal{N}}_{s}\Omega}}$. ###### Proof. As remarked above, $({\mathcal{N}}_{S},T,\Omega)$ is a Borchers triple on ${\mathcal{H}}_{{\mathcal{N}}_{S}}$, hence the only thing to be proven is asymptotic completeness. We show that the subspace $\overline{{\mathcal{A}}_{0}^{G}({\mathbb{R}}_{-})\otimes{\mathcal{A}}_{0}^{G}({\mathbb{R}}_{+})\Omega}$ is invariant under $S$. We claim that $\overline{{\mathcal{A}}_{0}^{G}({\mathbb{R}}_{-})\Omega_{0}}$ coincides with the subspace ${\mathcal{H}}_{0}^{G}$ of invariant vectors under $\\{V_{g}\\}_{g\in G}$. Indeed, for any $x\in{\mathcal{A}}_{0}$, the averaging $\int_{g}V_{g}x\Omega_{0}\,\mathrm{d}g=\left(\int_{g}\alpha_{g}(x)\,\mathrm{d}g\right)\Omega_{0}$ gives a projection onto ${\mathcal{H}}_{0}^{G}$. By the Reeh-Schlieder property, any vector in ${\mathcal{H}}_{0}^{G}$ can be approximated by vectors in ${\mathcal{A}}_{0}^{G}({\mathbb{R}}_{-})\Omega_{0}$. The converse inclusion is obvious. Now it is easy to see that $\overline{{\mathcal{A}}_{0}^{G}({\mathbb{R}}_{-})\otimes{\mathcal{A}}_{0}^{G}({\mathbb{R}}_{+})\Omega}={\mathcal{H}}_{0}^{G}\otimes{\mathcal{H}}_{0}^{G}$. This is the space of invariant vectors under the action $\\{V_{g}\otimes V_{g^{\prime}}:g,g^{\prime}\in G\\}$. Since $S$ commutes with $V_{g}\otimes V_{g^{\prime}}$ by assumption, this subspace is preserved under $S$. Then, as remarked before, $\overline{{\mathcal{N}}_{S}\Omega}$ coincides with $\overline{{\mathcal{A}}_{0}^{G}({\mathbb{R}}_{-})\otimes{\mathcal{A}}_{0}^{G}({\mathbb{R}}_{+})\Omega}$ and we obtain the asymptotic completeness. The statement on S-matrix is immediate from the definition and by Proposition 2.2. ∎ ## 3 Examples of fermi nets ### 3.1 ${\rm U(1)}$-current net ${{\mathcal{A}}^{(0)}}$ Let $U_{1}$ be the irreducible unitary positive-energy representation of $\operatorname{M\ddot{o}b}$ with lowest weight $1$ on a Hilbert space denoted by ${\mathcal{H}}_{\mathrm{{{\mathcal{A}}^{(0)}}}}^{1}$, which can be identified with the one-particle space of the ${\rm U(1)}$-current. This has the following concrete realization: consider $C^{\infty}(S^{1},{\mathbb{R}})$, where we write the periodic function $f\in C^{\infty}(S^{1},{\mathbb{R}})$ as a Fourier series $f(\theta)=\sum_{k\in{\mathbb{Z}}}\hat{f}_{k}\mathrm{e}^{\mathrm{i}k\theta},\quad\hat{f}_{k}=\int_{0}^{2\pi}\mathrm{e}^{-\mathrm{i}k\theta}f(\theta)\frac{\,\mathrm{d}\theta}{2\pi}=\overline{\hat{f}_{-k}}\,.$ We introduce a semi-norm $\\|f\\|^{2}=\sum_{k=1}^{\infty}k\cdot\|\hat{f}_{k}\|^{2}$ and a complex structure, i.e. an isometry $\mathcal{J}$ w.r.t. $\\|\,\cdot\,\\|$ satisfying $\mathcal{J}^{2}=-1$, by $\mathcal{J}:\hat{f}_{k}\mapsto-\mathrm{i}\operatorname{sign}(k)\hat{f}_{k}$ and finally we get the Hilbert space ${\mathcal{H}}_{\mathrm{{{\mathcal{A}}^{(0)}}}}^{1}=\overline{C^{\infty}(S^{1},{\mathbb{R}})/{\mathbb{R}}}^{\\|\,\cdot\,\\|}$ by completion with respect to the norm $\\|\,\cdot\,\\|$, where ${\mathbb{R}}$ is identified with the constant functions. By abuse of notation we denote also the image of $f\in C^{\infty}(S^{1},{\mathbb{R}})$ in ${\mathcal{H}}_{\mathrm{{{\mathcal{A}}^{(0)}}}}^{1}$ by $f$. The scalar product (linear in the second component) and the sesquilinear form $\omega(\,\cdot\,,\,\cdot\,)\equiv\Im\langle\,\cdot\,,\,\cdot\,\rangle$ are given by $\displaystyle\langle f,g\rangle$ $\displaystyle=\sum_{k=1}^{\infty}k\hat{f}_{k}\hat{g}_{-k}\,,$ $\displaystyle\omega(f,g)$ $\displaystyle=\frac{-\mathrm{i}}{2}\sum_{k\in{\mathbb{Z}}}k\hat{f}_{k}\hat{g}_{-k}=\frac{1}{2}\int_{0}^{2\pi}f(\theta)g^{\prime}(\theta)\frac{\,\mathrm{d}\theta}{2\pi}=\frac{1}{4\pi}\int f\,\mathrm{d}g\,,$ respectively. The unitary action $U_{1}$ of $\operatorname{M\ddot{o}b}$ on ${\mathcal{H}}_{\mathrm{{{\mathcal{A}}^{(0)}}}}^{1}$ is induced by the action on $C^{\infty}(S^{1},{\mathbb{R}})$ $(U_{1}(g)f)=(g_{\ast}f)(\theta):=f(g^{-1}(\theta))$. For $I\in\mathcal{I}$ we denote by $H(I)$ the closure of the subspace of real functions with support in $I$. This space is standard (i.e. $H(I)+\mathrm{i}H(I)$ is dense in ${\mathcal{H}}_{\mathrm{{{\mathcal{A}}^{(0)}}}}^{1}$ and $H(I)\cap\mathrm{i}H(I)=\\{0\\}$) and the family $\\{H(I)\\}_{I\in\mathcal{I}}$ is a local Möbius covariant net of standard subspaces [Lon08, LW11]. We explain briefly the bosonic second quantization procedure in general. Let ${\mathcal{H}}^{1}$ be a separable Hilbert space, the one-particle space, and $\omega(\,\cdot\,,\,\cdot\,)=\Im\langle\,\cdot\,,\,\cdot\,\rangle$ the sesquilinear form. There are unitaries $W(f)$ for $f\in{\mathcal{H}}^{1}$ fulfilling $W(f)W(g)=\mathrm{e}^{-\mathrm{i}\omega(f,g)}W(f+g)=\mathrm{e}^{-2\mathrm{i}\omega(f,g)}W(g)W(f)$ and acting naturally on the bosonic Fock space $\mathrm{e}^{{\mathcal{H}}^{1}}$ over ${\mathcal{H}}^{1}$. This space is given by $\mathrm{e}^{{\mathcal{H}}^{1}}=\oplus_{n=0}^{\infty}P_{n}({\mathcal{H}}^{1})^{\otimes n}$, where $P_{n}$ is the projection $P_{n}(\xi_{1}\otimes\cdots\otimes\xi_{n})=1/n!\sum_{\sigma}\xi_{\sigma(1)}\otimes\cdots\otimes\xi_{\sigma(n)}$ and the sum goes over all permutations. The set of coherent vectors $\mathrm{e}^{h}:=\oplus_{n=0}^{\infty}h^{\otimes n}/\sqrt{n!}$ with $h\in{\mathcal{H}}^{1}$ is total in $\mathrm{e}^{{\mathcal{H}}^{1}}$ and it holds $\langle\mathrm{e}^{f},\mathrm{e}^{h}\rangle=\mathrm{e}^{\langle f,h\rangle}$. The vacuum is given by $\Omega=\mathrm{e}^{0}$ and the action of $W(f)$ is given by $W(f)\mathrm{e}^{0}=\mathrm{e}^{-\frac{1}{2}\\|f\\|^{2}}\mathrm{e}^{f}$, in other words the vacuum representation is characterized by $\phi(W(f))=\mathrm{e}^{-\frac{1}{2}\\|f\\|^{2}}$, where $\phi(\,\cdot\,)=\langle\Omega,\,\cdot\,\Omega\rangle$. For a real subspace $H\subset{\mathcal{H}}^{1}$, we define the von Neumann algebra $\displaystyle R(H)=\\{W(f):f\in H\\}^{\prime\prime}\subset B(\mathrm{e}^{{\mathcal{H}}^{1}})\,.$ Let $U$ be a unitary on the one-particle space ${\mathcal{H}}^{1}$ then $\mathrm{e}^{U}:=\oplus_{n=0}^{\infty}U^{\otimes n}$ acts on coherent states by $\mathrm{e}^{U}\mathrm{e}^{h}=\mathrm{e}^{Uh}$ and is therefore a unitary on $\mathrm{e}^{{\mathcal{H}}^{1}}$, the second quantization unitary. We obtain the ${\rm U(1)}$-current net ${{\mathcal{A}}^{(0)}}$ on ${\mathcal{H}}_{\mathrm{{{\mathcal{A}}^{(0)}}}}:=\mathrm{e}^{{\mathcal{H}}_{\mathrm{{{\mathcal{A}}^{(0)}}}}^{1}}$ with $\Omega_{0}=\mathrm{e}^{0}$ by defining ${{\mathcal{A}}^{(0)}}(I):=R(H(I))$ which is covariant with respect to $U(g):=\mathrm{e}^{U_{1}(g)}$. For $f\in C^{\infty}(S^{1},{\mathbb{R}})$ we consider a self-adjoint operator $J(f)$ given by the generator of the unitary one-parameter group $W(t\cdot f)=\mathrm{e}^{\mathrm{i}t\cdot J(f)}$ with $t\in{\mathbb{R}}$. This defines the usual current (field operator) smeared with the real test function $f$, which fulfills $J(f)\Omega_{0}=f\in{\mathcal{H}}_{\mathrm{{{\mathcal{A}}^{(0)}}}}^{1}$ and $\displaystyle[J(f),J(g)]$ $\displaystyle=2\mathrm{i}\omega(f,g)=\sum_{k}k\hat{f}_{k}\hat{g}_{-k}=\frac{\mathrm{i}}{2\pi}\int f\,\mathrm{d}g\,.$ It can be extended to complex test functions via $J(f+\mathrm{i}g)=J(f)+\mathrm{i}J(g)$, and one obtains the usual operator valued ($z$-picture) distribution $J(z)$ with the relations $\displaystyle J(f)$ $\displaystyle=\sum_{n\in{\mathbb{Z}}}\hat{f}_{n}J_{n}=\oint_{S^{1}}f(z)J(z)\frac{\,\mathrm{d}z}{2\pi\mathrm{i}},$ $\displaystyle J(z)=\sum_{n}J_{n}z^{-n-1}$ $\displaystyle[J_{m},J_{n}]$ $\displaystyle=m\delta_{m+n,0}\,,$ where the modes $J_{n}=J(e_{n})$ with $e_{n}(\theta)=\mathrm{e}^{\mathrm{i}n\theta}$ satisfy $J_{n}\Omega_{0}=0$ for $n\geq 0$. The space ${\mathcal{H}}_{\mathrm{{{\mathcal{A}}^{(0)}}}}$ is spanned by vectors of the form $\xi=J_{-n_{1}}\cdots J_{-n_{k}}\Omega_{0}$ with $0<n_{1}\leq\cdots\leq n_{k}$ with “energy” $N=\sum_{m}n_{m}$, i.e. $R(\theta)\xi=\mathrm{e}^{\mathrm{i}N\theta}\xi$. Therefore it is graded with respect to the rotations $\displaystyle{\mathcal{H}}_{\mathrm{{{\mathcal{A}}^{(0)}}}}$ $\displaystyle={\mathbb{C}}\Omega_{0}\oplus\bigoplus_{n\in{\mathbb{N}}}{\mathcal{H}}_{\mathrm{{{\mathcal{A}}^{(0)}}},n}$ $\displaystyle{\mathcal{H}}_{\mathrm{{{\mathcal{A}}^{(0)}}},n}$ $\displaystyle=\bigoplus_{k=1}^{n}\bigoplus_{\begin{subarray}{c}0<n_{1}\leq\cdots\leq n_{k}\\\ n_{1}+\cdots+n_{k}=n\end{subarray}}{\mathbb{C}}J_{-n_{1}}\cdots J_{-n_{k}}\Omega_{0}$ and ${\hbox{dim}\,}{\mathcal{H}}_{\mathrm{{{\mathcal{A}}^{(0)}}},n}$ is the number of partitions of $n$ elements, whose generating function $p(t)$ is the inverse of Euler’s function $\phi(t)=\prod_{k=1}^{\infty}(1-t^{k})$ and therefore the conformal character of the ${\rm U(1)}$-current net is given by ($t=\mathrm{e}^{-\beta}$): $\operatorname{tr}_{{\mathcal{H}}_{\mathrm{{{\mathcal{A}}^{(0)}}}}}(\mathrm{e}^{-\beta L_{0}})=\sum_{n=0}^{\infty}{\hbox{dim}\,}{\mathcal{H}}_{\mathrm{{{\mathcal{A}}^{(0)}}},n}\cdot t^{n}=\prod_{n\in{\mathbb{N}}}(1-t^{n})^{-1}$ (a conformal character is defined as a formal power series, but it is often convergent for $\|t\|<1$ and here we used the formula $(1-z)^{-1}=1+z+z^{2}\cdots$). It will be convenient to use the real parametrization $x\in{\mathbb{R}}\cong S^{1}\setminus\\{-1\\}$ of the cut circle and use the conventions $\displaystyle f(s)=\int_{\mathbb{R}}\mathrm{e}^{-\mathrm{i}sp}\hat{f}(p)\,\mathrm{d}p.$ By writing $f(s)=f_{0}(\theta(s))$ for $f_{0}\in C^{\infty}(S^{1},{\mathbb{R}})$ where $\theta(s)=2\arctan(s)$, the space ${\mathcal{H}}_{\mathrm{{{\mathcal{A}}^{(0)}}}}^{1}$ above can be identified with the space $L^{2}({\mathbb{R}}_{+},p\,\mathrm{d}p)$ in which the space ${\mathcal{S}}({\mathbb{R}},{\mathbb{R}})$ embeds by restriction of the Fourier transformation to ${\mathbb{R}}_{+}$. In other words ${\mathcal{H}}_{\mathrm{{{\mathcal{A}}^{(0)}}}}^{1}$ can be seen as the closure of the space ${\mathcal{S}}({\mathbb{R}},{\mathbb{R}})$ with complex structure $\mathcal{J}\hat{f}(p)=\mathrm{i}\operatorname{sign}(p)\hat{f}(p)$ and the scalar product and sesquilinear form given by: $\displaystyle\langle f,g\rangle$ $\displaystyle=\int_{{\mathbb{R}}_{+}}\hat{f}(-p)\hat{g}(p)p\,\mathrm{d}p\,,$ $\displaystyle\omega(f,g)$ $\displaystyle=\frac{-\mathrm{i}}{2}\int_{{\mathbb{R}}}\hat{f}(-p)\hat{g}(p)p\,\mathrm{d}p=\frac{1}{4\pi}\int_{\mathbb{R}}f(x)g^{\prime}(x)\,\mathrm{d}x\,.$ Using the above identification we denote for $f\in{\mathcal{S}}({\mathbb{R}},{\mathbb{R}})$ by $J(f)$ the smeared current with $J(f)\Omega_{0}=f\in{\mathcal{H}}_{\mathrm{{{\mathcal{A}}^{(0)}}}}^{1}$. In this parametrization commutation relations read: $\displaystyle[J(f),J(g)]$ $\displaystyle=\frac{\mathrm{i}}{2\pi}\int_{\mathbb{R}}f(x)g^{\prime}(x)\,\mathrm{d}x=\int_{{\mathbb{R}}}\hat{f}(-p)\hat{g}(p)p\,\mathrm{d}p\,.$ ### 3.2 The free complex fermion net $\mathrm{Fer}_{\mathbb{C}}$ We construct the net of the free complex fermion on the circle, which can be seen as the chiral part of the net of the free massless Dirac (or complex) fermion on two dimensional Minkowski space. The notations of this section are basically in accordance with [Was98], but we use a different convention of positive-energy, which leads to the conjugated complex structure. For giving a simple description of the one-particle space, we consider first the Hilbert space $L^{2}(S^{1})$ and the Hardy space $H^{2}(S^{1})$, namely $H^{2}(S^{1}):=\left\\{f:\mbox{ analytic on the unit disk }D,\sup_{0\leq r<1}\int_{0}^{2\pi}\|f(r\mathrm{e}^{\mathrm{i}\theta})\|^{2}\,\mathrm{d}\theta<\infty\right\\}\,.$ Any function in $H^{2}(S^{1})$ has a $L^{2}$-boundary value and can be considered as an element of $L^{2}(S^{1})$. In this sense, $H^{2}(S^{1})$ is a subspace of $L^{2}(S^{1})$. Furthermore, it holds that $H^{2}(S^{1})=\\{f\in L^{2}(S^{1}):\hat{f}_{n}=0\mbox{ for }n<0\\},$ where $\hat{f}_{n}$ is the $n$-th Fourier component of $f$. We denote the orthogonal projection onto $H^{2}(S^{1})$ by $P$. The group $\mathrm{SU}(1,1):=\left\\{\left(\begin{matrix}\alpha&\beta\\\ \overline{\beta}&\overline{\alpha}\end{matrix}\right)\in M_{2}({\mathbb{C}}):\|\alpha\|^{2}-\|\beta\|^{2}=1\right\\}$ acts on the circle $S^{1}$ by $g\cdot z=\frac{\alpha z+\beta}{\overline{\beta}z+\overline{\alpha}}$ and there is a unitary action of $\mathrm{SU}(1,1)$ on $L^{2}(S^{1})$ by $(U(g)f)(z):=(V_{g}f)(z)=\frac{1}{-\overline{\beta}z+\alpha}f(g^{-1}\cdot z)\,.$ One sees that the projection $P$ commutes with $V_{g}$ , since $V_{g}f$ is still an analytic function for $\|\alpha\|>\|\beta\|$. Then one defines a new Hilbert space ${\mathcal{H}}_{\mathrm{Fer}_{{\mathbb{C}}}}^{1}=\overline{PL^{2}(S^{1})}\oplus({\mathbbm{1}}-P)L^{2}(S^{1})$: namely, ${\mathcal{H}}_{\mathrm{Fer}_{{\mathbb{C}}}}^{1}$ is identical with $L^{2}(S^{1})$ as a real linear space and the multiplication by $\mathrm{i}$ is given by $-\mathrm{i}(2P-{\mathbbm{1}})$, or in other words, by $-\mathrm{i}$ on $PL^{2}(S^{1})$ and $\mathrm{i}$ on $({\mathbbm{1}}-P)L^{2}(S^{1})$. Because $P$ and $U(g)$ commute, the action of $\mathrm{SU}(1,1)$ remains unitary on ${\mathcal{H}}_{\mathrm{Fer}_{{\mathbb{C}}}}^{1}$. Then for $I\in\mathcal{I}$ one takes real Hilbert subspaces $K(I):=L^{2}(I)$ of ${\mathcal{H}}_{\mathrm{Fer}_{{\mathbb{C}}}}^{1}$. This subspaces turn out to be standard [Was98, Theorem (p. 497)]. If $I_{1}$ and $I_{2}$ are disjoint intervals, $K(I_{1})$ are real orthogonal to $K(I_{2})$, in other words $K(I_{1})\subset K(I_{2})^{\perp}$, where $K^{\perp}=\\{\xi\in{\mathcal{H}}:\Re\langle\xi,K\rangle=0\\}$. It turns out that $I\mapsto K(I)$ is a twisted-local Möbius covariant net of standard subspaces. We briefly explain the fermionic second quantization in general. Let ${\mathcal{H}}^{1}$ be a complex Hilbert space and ${\mathcal{H}}=\Lambda({\mathcal{H}}^{1})$ the antisymmetric (fermionic) Fock space obtained by completing the exterior algebra with the inner product. For $A\in B({\mathcal{H}}^{1})$ with $\\|A\\|\leq 1$ we define $\Lambda(A)$ to be $A^{\otimes k}$ on ${\mathcal{H}}^{k}:=\Lambda^{k}({\mathcal{H}}^{1})\subset({\mathcal{H}}^{1})^{\otimes k}$. The space is ${\mathbb{Z}}_{2}$ graded by $\Gamma:=\Lambda(-{\mathbbm{1}})$. We define $Z=\frac{{\mathbbm{1}}-\mathrm{i}\Gamma}{1-\mathrm{i}}$ and note that $Z^{2}=\Gamma$. For $f\in{\mathcal{H}}^{1}$ let $a(f)$ be the bounded operator obtained by continuing the exterior multiplication $f\wedge\cdot$. The operators fulfill the complex Clifford relations $a(f)^{\ast}a(g)+a(g)a(f)^{\ast}=\langle f,g\rangle$ and $\\{a(f),a(g)\\}=\\{a(f)^{\ast},a(g)^{\ast}\\}=0$ for all $f,g\in{\mathcal{H}}^{1}$. For a standard subspace $K\subset{\mathcal{H}}^{1}$ we define the von Neumann algebra $C(K)=\\{c(f):f\in K\\}^{\prime\prime}\subset B(\Lambda{\mathcal{H}}^{1})$ where $c(f)=a(f)+a(f)^{\ast}$, which fulfills the real Clifford relations $c(f)c(g)+c(g)c(f)=2\Re\langle f,g\rangle$. By $\Omega=1\in\Lambda^{0}$ we denote the vacuum which is cyclic and separating for $C(K)$ for every standard subspace $K\subset{\mathcal{H}}^{1}$. Further it holds Haag-Araki duality, i.e. $C(K^{\perp})$ equals $C(K)^{\perp}:=ZC(K)^{\prime}Z^{\ast}$, the twisted commutant of $C(K)$. For a unitary $U$ on ${\mathcal{H}}^{1}$ it holds $\Lambda(U)c(f)\Lambda(U^{\ast})=c(Uf)$, which implies that $C$ is covariant with respect to the unitaries $U({\mathcal{H}}^{1})$, i.e. $\Lambda(U)C(K)\Lambda(U)^{\ast}=C(UK)$. We note that in the case like the complex fermion the one-particle space is obtained from a Hilbert space ${\mathcal{H}}^{1}$ (the space of test functions) and a projection $P$ by ${\mathcal{H}}^{1}_{P}=P{\mathcal{H}}^{1}\oplus\overline{P^{\perp}{\mathcal{H}}^{1}}$ and one gets a new representation of the complex Clifford algebra on $\Lambda({\mathcal{H}}^{1}_{P})$ by $a_{P}(f)=a(Pf)+a(\overline{P^{\perp}f})^{\ast}$ where $a(f)$ is the creation operator. For a standard subspace $K\subset{\mathcal{H}}^{1}_{P}$ which is invariant under the multiplication of $\mathrm{i}_{{\mathcal{H}}^{1}}$ in ${\mathcal{H}}^{1}$, the von Neumann algebra $C(K)$ on $\Lambda({\mathcal{H}}_{P}^{1})$ coincides with the von Neumann algebra $\\{a_{P}(f),a_{P}(f)^{\ast}:f\in K\\}^{\prime\prime}$. Indeed, the one inclusion follows from $c(f)=a_{P}(f)+a_{P}(f)^{\ast}$ and the other follows from Araki-Haag duality and $\\{a_{P}(f),c(g)\\}=\langle g,f\rangle_{{\mathcal{H}}^{1}}=\Re\langle g,f\rangle_{{\mathcal{H}}^{1}_{P}}-\mathrm{i}\Re\langle g,\mathrm{i}_{{\mathcal{H}}^{1}}f\rangle_{{\mathcal{H}}^{1}_{P}}=0$ for $f\in K$ and $g\in K^{\perp}$. We further note that the space $\Lambda({\mathcal{H}}^{1}_{P})$ is as a real Hilbert space the same as $\Lambda({\mathcal{H}}^{1})$ and can be identified canonically with $\Lambda(P{\mathcal{H}}^{1})\otimes\Lambda(\overline{P^{\perp}{\mathcal{H}}^{1}})$. We turn to the concrete case where ${\mathcal{H}}^{1}=L^{2}(S^{1})$ and define the net $\mathrm{Fer}_{\mathbb{C}}(I):=C(K(I))=\\{a_{P}(f),a_{P}(f)^{\ast}:f\in L^{2}(I)\\}^{\prime\prime}$ (where here $a_{P}(f):=a(\overline{Pf})+a(P^{\perp}f)^{\ast}$) on ${\mathcal{H}}_{\mathrm{Fer}_{{\mathbb{C}}}}=\Lambda({\mathcal{H}}_{\mathrm{Fer}_{{\mathbb{C}}}}^{1})\cong\Lambda(\overline{PL^{2}(S^{1})})\otimes\Lambda(P^{\perp}L^{2}(S^{1}))$ which is isotonic by definition and fulfills twisted duality, namely by Haag- Araki duality $\mathrm{Fer}_{\mathbb{C}}(I^{\prime})=C(K(I)^{\perp})=C(K(I))^{\perp}=\mathrm{Fer}_{\mathbb{C}}(I)^{\perp}$. In addition, the net $\mathrm{Fer}_{\mathbb{C}}$ is Möbius covariant. Indeed, we can take the representation $\Lambda U(\cdot)$ by promoting the one- particle representation $U$ to the second quantization operator. It is easy to see that the covariance of this net $\mathrm{Fer}_{\mathbb{C}}$ follows from the covariance of the net of standard spaces $K$. The representation $\Lambda U$ has positive energy since so does the representation $U$, and leaves invariant the vacuum vector $\Omega_{0}$ of the Fock space. Summing up, the net $\mathrm{Fer}_{\mathbb{C}}$ is a fermi net (cf. [Was98]). This net is referred to as the free complex fermi net on $S^{1}$. The scalar multiplication by a constant phase $\mathrm{e}^{-\mathrm{i}\vartheta}$ in the original structure of the one-particle space is still a unitary operator in the new structure. Its promotion by the second quantization $V(\theta)$ implements an action of ${\rm U(1)}$ on $\mathrm{Fer}_{\mathbb{C}}$ by inner symmetry. This will be referred to as the ${\rm U(1)}$-gauge action. For $r\in\frac{1}{2}+{\mathbb{Z}}$ let $\psi_{r}=a_{P}(e_{-r-\frac{1}{2}})$ and $\bar{\psi}_{r}=a_{P}(e_{r-\frac{1}{2}})^{\ast}$ where $e_{r}\in L^{2}(S^{1})$ with $e_{r}(\theta)=\mathrm{e}^{\mathrm{i}\theta r}$. The $\psi_{r},\bar{\psi}_{r}$ are the modes of the free complex fermion, namely $\displaystyle\\{\psi_{n},\psi_{m}\\}$ $\displaystyle=\\{\bar{\psi}_{m},\bar{\psi}_{n}\\}=0$ $\displaystyle\\{\bar{\psi}_{n},\psi_{m}\\}$ $\displaystyle=\delta_{m+n,0}$ $\displaystyle\psi_{n}^{\ast}$ $\displaystyle=\bar{\psi}_{-n}$ and it holds that $\psi_{r}\Omega_{0}=\bar{\psi}_{r}\Omega_{0}=0$ for $r\in\frac{1}{2}+{\mathbb{N}}_{0}$. Each of $\psi_{r}$ or $\bar{\psi}_{r}$ has norm $1$ following from the commutation relation. We can introduce the usual fields ($f,g\in L^{2}(S^{1})$) and operator valued distributions in the $z$-picture: $\displaystyle\Psi(f)$ $\displaystyle=\sum_{r\in\frac{1}{2}+{\mathbb{Z}}}\hat{f}_{r}\Psi_{r}=\oint_{S^{1}}f(z)z^{-\frac{1}{2}}\Psi(z)\frac{\,\mathrm{d}z}{2\pi\mathrm{i}}\,,$ $\displaystyle\Psi(z)$ $\displaystyle=\sum_{r\in\frac{1}{2}+{\mathbb{Z}}}\Psi_{r}z^{-r-\frac{1}{2}}\,,$ $\displaystyle\bar{\psi}(f)$ $\displaystyle=\psi(\overline{f})^{\ast}=a_{P}(e_{-\frac{1}{2}}f)^{\ast}\,,$ $\displaystyle\\{\bar{\psi}(f),\psi(g)\\}$ $\displaystyle=\oint f(z)g(z)\frac{\,\mathrm{d}z}{2\pi\mathrm{i}z}\,,$ where $\Psi$ is either $\psi$ or $\bar{\psi}$. The fields $\psi,\bar{\psi}$ are covariant, e.g. $U(g)\psi(f)U(g)^{\ast}=\psi(f_{g})$ with $f_{g}(z)=\frac{1}{\|\alpha-\overline{\beta}z\|}f(g^{-1}z)$ for $g=\left(\begin{matrix}\alpha&\beta\\\ \overline{\beta}&\overline{\alpha}\end{matrix}\right)\in\mathrm{SU(1,1)}$. We note that vectors of the form $\xi=\psi_{-r_{1}}\cdots\psi_{-r_{k}}\bar{\psi}_{-s_{1}}\cdots\bar{\psi}_{-s_{\ell}}\Omega_{0}$ with $0<r_{1}<\cdots<r_{k}$ and $0<s_{1}<\cdots<s_{\ell}$ form a basis of ${\mathcal{H}}_{\mathrm{Fer}_{{\mathbb{C}}}}=\Lambda({\mathcal{H}}_{\mathrm{Fer}_{{\mathbb{C}}}}^{1})$ and that such a $\xi$ is an eigenvector for the rotations, $R(\theta)\xi\equiv\mathrm{e}^{\mathrm{i}\theta L_{0}}\xi=\mathrm{e}^{\mathrm{i}N\theta}\xi$ with $N=\sum_{j=1}^{k}r_{j}+\sum_{j=1}^{\ell}s_{j}$ and of the gauge action $V(\theta)\xi\equiv\mathrm{e}^{\mathrm{i}\theta Q}\xi=\mathrm{e}^{\mathrm{i}(k-\ell)\theta}\xi$. In each vector of this basis the $r$-th energy level can either be empty, be occupied by $\psi_{-r}$ or $\bar{\psi}_{-r}$ or occupied by both. The contribution of this level to the character $\operatorname{tr}_{{\mathcal{H}}_{\mathrm{Fer}_{{\mathbb{C}}}}}(\mathrm{e}^{-\beta L_{0}-EQ})$ is then $1$, $zt^{r}$, $z^{-1}t^{r}$ or $t^{2r}$, respectively, where $t=\mathrm{e}^{-\beta}$ and $z=\mathrm{e}^{-E}$. By summing over all possibilities one gets that the character of $\mathrm{Fer}_{\mathbb{C}}$ is given by (cf. [Kac98, Reh98]): $\displaystyle\operatorname{tr}_{{\mathcal{H}}_{\mathrm{Fer}_{{\mathbb{C}}}}}(\mathrm{e}^{-\beta L_{0}-EQ})=\operatorname{tr}_{{\mathcal{H}}_{\mathrm{Fer}_{{\mathbb{C}}}}}(t^{L_{0}}z^{Q})$ $\displaystyle=\prod_{r\in{\mathbb{N}}_{0}+\frac{1}{2}}(1+zt^{r}+z^{-1}t^{r}+t^{2r})$ $\displaystyle=\prod_{r\in{\mathbb{N}}_{0}+\frac{1}{2}}(1+zt^{r})(1+z^{-1}t^{r})$ $\displaystyle=p(t)\sum_{q\in{\mathbb{Z}}}z^{q}t^{\frac{q^{2}}{2}}\,,$ where the last equality follows directly from the Jacobi triple product formula (see [Apo76, Theorem 14.6]) $\displaystyle\prod_{r\in{\mathbb{N}}}(1+zw^{2r-1})(1+z^{-1}w^{2r-1})(1-w^{2r})=\sum_{q\in{\mathbb{Z}}}z^{q}w^{q}$ by setting $2r-1=2n$ and $t=w^{2}$. In particular, for the local net $\mathrm{Fer}_{\mathbb{C}}^{{\rm U(1)}}$ the character is given by $\operatorname{tr}_{{\mathcal{H}}_{\mathrm{Fer}_{{\mathbb{C}}}}^{{\rm U(1)}}}(\mathrm{e}^{-\beta L_{0}})=p(t)$, since it is the fixed point with respect to the ${\rm U(1)}$-gauge action and the conformal character is the coefficient of $z^{0}$. ### 3.3 ${\rm U(1)}$-current net as a subnet of $\mathrm{Fer}_{\mathbb{C}}$ In this section we use the well-known fact that the Wick product $:\\!\bar{\psi}\psi\\!:$ of the complex fermion $\psi$ equals the ${\rm U(1)}$-current and give an analogue of the boson-fermion correspondence (see e.g. [Kac98, 5.2]) in the operator algebraic setting. Let us denote by ${\mathcal{D}}_{0}$ the subspace of $\Lambda({\mathcal{H}}_{P}^{1})$ of vectors with finite energy: ${\mathcal{D}}_{0}:=\mathrm{span}\left\\{\psi_{-r_{1}}\cdots\psi_{-r_{k}}\bar{\psi}_{-s_{1}}\cdots\bar{\psi}_{-s_{l}}\Omega_{0}:k,l\in{\mathbb{N}}_{0},r_{i},s_{j}\in{\mathbb{N}}+\frac{1}{2}\right\\}\,.$ Then we define the unbounded operators on the domain ${\mathcal{D}}_{0}$: $\displaystyle J_{n}=\sum_{r+s=n}{:\\!\bar{\psi}_{r}\psi_{s}\\!:}$ $\displaystyle=$ $\displaystyle\sum_{r<0}\bar{\psi}_{r}\psi_{n-r}-\sum_{r>0}\psi_{n-r}\bar{\psi}_{r}$ $\displaystyle=$ $\displaystyle\sum_{r}\left(\bar{\psi}_{r}\psi_{n-r}-\langle\Omega_{0},\bar{\psi}_{r}\psi_{n-r}\Omega_{0}\rangle\right)$ with $r,s\in\frac{1}{2}+{\mathbb{Z}}$. Note that any vector in ${\mathcal{D}}_{0}$ is annihilated by $\psi_{r}$ for sufficiently large $r$, thus the action of $J_{n}$ on such a vector can be defined and remains in ${\mathcal{D}}_{0}$. In particular, we have $J_{n}\Omega_{0}=0$ for $n\in{\mathbb{N}}_{0}$. ###### Lemma 3.1. On ${\mathcal{D}}_{0}$ it holds that 1. 1. $[J_{n},\psi_{k}]=-\psi_{n+k}$ and $[J_{n},\bar{\psi}_{k}]=\bar{\psi}_{n+k}$ 2. 2. $[J_{m},J_{n}]=m\delta_{m+n,0}$ ###### Proof. Using $[ab,c]=a\\{b,c\\}-\\{a,c\\}b$, one obtains $[\bar{\psi}_{r}\psi_{n},\psi_{k}]=-\delta_{r+k,0}\psi_{n}$ and $[\psi_{n}\bar{\psi}_{r},\psi_{k}]=\delta_{r+k,0}\psi_{n}$ from which directly follows $[J_{n},\psi_{k}]=\sum_{r<0}[\bar{\psi}_{r}\psi_{n-r},\psi_{k}]-\sum_{r>0}[\psi_{n-m}\bar{\psi}_{r},\psi_{k}]=-\psi_{n+k}$. Analogously one shows $[J_{n},\bar{\psi}_{k}]=\bar{\psi}_{n+k}$. From the Jacobi identity, it follows immediately that $[J_{n},J_{m}]$ commutes with all $\psi_{k}$ and $\bar{\psi}_{k}$ and hence $[J_{n},J_{m}]$ is a multiple of the identity, therefore $[J_{n},J_{m}]=\langle\Omega_{0},[J_{n},J_{m}]\Omega_{0}\rangle{\mathbbm{1}}$. It is $\displaystyle[J_{n},J_{p}]$ $\displaystyle=\sum_{r<0}[J_{n},\bar{\psi}_{r}\psi_{p-r}]-\sum_{r>0}[J_{n},\psi_{p-r}\bar{\psi}_{r}]$ $\displaystyle=-\sum_{r<0}\left(\bar{\psi}_{r}\psi_{p-r+n}-\bar{\psi}_{r+n}\psi_{p-r}\right)-\sum_{r>0}\left(\psi_{p-r}\bar{\psi}_{r+n}-\psi_{p-r+n}\bar{\psi}_{r}\right)$ and in the case $p\neq-n$ we get $\langle\Omega_{0},[J_{n},J_{p}]\Omega_{0}\rangle=0$, and otherwise $\displaystyle\langle\Omega_{0},[J_{n},J_{-n}]\Omega_{0}\rangle$ $\displaystyle=\begin{cases}\sum_{r<0}\langle\Omega_{0},\bar{\psi}_{r+n}\psi_{-r-n}\Omega_{0}\rangle=\sum_{r=\frac{1}{2}}^{n-\frac{1}{2}}\langle\Omega_{0},\\{\bar{\psi}_{r},\psi_{-r}\\}\Omega_{0}\rangle&n>0\\\ -\sum_{r>0}\langle\Omega_{0},\psi_{-r-n}\bar{\psi}_{r+n}\Omega_{0}\rangle=-\sum_{r=\frac{1}{2}}^{-n-\frac{1}{2}}\langle\Omega_{0},\\{\psi_{r},\bar{\psi}_{-r}\\}\Omega_{0}\rangle&n<0\end{cases}$ $\displaystyle=n\,,$ which completes the proof. ∎ Let $L_{0}$ be the generator of the rotation: $R(\theta)=\mathrm{e}^{\mathrm{i}\theta L_{0}}$. From its action (see the end of Section 3.2), one verifies that ${\mathcal{D}}_{0}$ is a core for $L_{0}$. ###### Lemma 3.2 (Linear energy bounds). It holds that $[L_{0},J_{n}]=-nJ_{n}$ on ${\mathcal{D}}_{0}$. For a trigonometric polynomial $f=\sum_{n}\hat{f}_{n}e_{n}$ where the sum is finite and $\xi\in{\mathcal{D}}_{0}$, we have $\displaystyle\\|J(f)\xi\\|$ $\displaystyle\leq c_{f}\\|(L_{0}+1)\xi\\|$ $\displaystyle\\|[L_{0},J(f)]\xi\\|$ $\displaystyle\leq c_{\partial_{\theta}f}\\|(L_{0}+1)\xi\\|,$ where $c_{f}$ depends only on $f$. ###### Proof. For the commutation relation, it is enough to choose an energy eigenvector $\xi\in{\mathcal{D}}_{0}$, i.e. $L_{0}\xi=N\xi$. It is $J_{n}L_{0}\xi=NJ_{n}\xi$ and $L_{0}J_{n}\xi=L_{0}\left(\sum_{r<0}\bar{\psi}_{r}\psi_{n-r}\xi-\sum_{r>0}\psi_{n-r}\bar{\psi}_{r}\xi\right)=(N-n)J_{n}\xi,$ and the first statement follows. We have seen that $\psi_{r}$ and $\bar{\psi}_{r}$ have norm $1$ in Section 3.2. First we claim that $\\|J_{n}\xi\\|\leq\\|(2(L_{0}+1)+\|n\|)\xi\\|$. Let $\xi$ be again an eigenvector of $L_{0}$, i.e. $L_{0}\xi=N\xi$. From the defining sum of $J_{n}$, one sees that only $2N+\|n\|+2$ terms contribute to $J_{n}\xi$. Hence we have $\\|J_{n}\xi\\|\leq(2N+\|n\|+2)\\|\xi\\|=\\|2(L_{0}+1)+\|n\|\xi\\|$. If the inequality holds for eigenvectors, then for $\\{\xi_{r}\\}$ with different eigenvalues, we have $\xi_{r}\perp\xi_{s}$ and $J_{n}\xi_{r}\perp J_{n}\xi_{s}$, and hence $\displaystyle\left\\|J_{n}\sum_{r}\xi_{r}\right\\|^{2}$ $\displaystyle=$ $\displaystyle\sum_{r}\\|J_{n}\xi_{r}\\|^{2}$ $\displaystyle\leq$ $\displaystyle\sum_{r}\\|(2(L_{0}+1)+\|n\|)\xi_{r}\\|^{2}$ $\displaystyle=$ $\displaystyle\left\\|(2(L_{0}+1)+\|n\|)\sum_{r}\xi_{r}\right\\|^{2}$ and the general case follows. For a smeared field, we have $\\|J(f)\xi\\|=\left\\|\sum_{n}\hat{f}_{n}J_{n}\xi\right\\|\leq 2\tilde{c}_{f}\\|(L_{0}+1)\xi\\|+\tilde{c}_{\partial_{\theta}f}\\|\xi\\|\leq(2\tilde{c}_{f}+\tilde{c}_{\partial_{\theta}f})\\|(L_{0}+1)\xi\\|,$ where $\tilde{c}_{f}=\sum_{n}\|\hat{f}_{n}\|$. By defining $c_{f}=2\tilde{c}_{f}+\tilde{c}_{\partial_{\theta}f}$, we obtain the first inequality of the statement. The rest follows by noting that $[L_{0},J(f)]=J(\mathrm{i}\partial_{\theta}f)$. ∎ For a smooth function $f=\sum_{n\in{\mathbb{Z}}}\hat{f}_{n}e_{n}\in C^{\infty}(S^{1})$, its Fourier coefficients $\hat{f}_{n}$ are strongly decreasing and, in particular, it is summable: $\sum_{n}\|\hat{f}_{n}\|=\tilde{c}_{f}<\infty$. Hence we can naturally extend the definition of the smeared current to smooth functions using the above estimate by $J(f)=\sum_{n\in{\mathbb{Z}}}f_{n}J_{n}=\sum_{r,s\in\frac{1}{2}+{\mathbb{Z}}}f_{r+s}:\\!\psi_{r}\bar{\psi}_{s}\\!:,$ and the same inequality in Lemma 3.2 holds. The operator is closable since we have $J(f)\subset J(\overline{f})^{}$ and we still denote the closure by $J(f)$. We note that from the above definition it follows that $J(f)$ is obtained by a limit $\sum_{n}:\\!\psi(h_{n})\bar{\psi}(k_{n})\\!:$ with suitable functions such that $\sum_{n}h_{n}(\theta)k_{n}(\vartheta)\to 2\pi f(\theta)\delta(\theta-\vartheta)$. This implies covariance of the “field”, i.e. $U(g)J(f)U(g)^{\ast}=J(f\circ g^{-1})$. Recall that $\\|\psi_{r}\\|=1$, hence the smeared field is still bounded: $\\|\psi(g)\\|\leq\tilde{c}_{g}$. We claim that, for $f,g\in C^{\infty}(S^{1})$ and $\xi\in{\mathcal{D}}_{0}$, $\psi(g)\xi$ is in the domain of $J(f)$. Indeed, for a trigonometric polynomial $g$, we have the estimate $\displaystyle\\|J(f)\psi(g)\xi\\|$ $\displaystyle\leq$ $\displaystyle c_{f}\\|(L_{0}+1)\psi(g)\xi\\|$ $\displaystyle\leq$ $\displaystyle c_{f}(\tilde{c}_{g}\\|\xi\\|+\\|[L_{0},\psi(g)]\xi+\psi(g)L_{0}\xi\\|)$ $\displaystyle\leq$ $\displaystyle c_{f}(\tilde{c}_{g}(\\|\xi\\|+\\|L_{0}\xi\\|)+\tilde{c}_{\partial_{\theta}g}\\|\xi\\|)\,.$ Then if we have a sequence of trigonometric polynomial $g_{n}$ converging to a smooth function $g\in C^{\infty}(S^{1})$, the sequence $\\{J(f)\psi(g_{n})\xi\\}$ is also converging. ###### Lemma 3.3. For $\xi,\eta\in{\mathcal{D}}_{0}$, it holds that $\displaystyle[J(f),\psi(g)]\xi$ $\displaystyle=-\psi(f\cdot g)\xi$ $\displaystyle[J(f),\bar{\psi}(g)]\xi$ $\displaystyle=\bar{\psi}(f\cdot g)\xi$ $\displaystyle\langle J(\bar{f})\xi,J(g)\eta\rangle$ $\displaystyle=\langle J(\bar{g})\xi,J(f)\eta\rangle+2\mathrm{i}\omega(f,g)\langle\xi,\eta\rangle.$ ###### Proof. For trigonometric polynomials $f,g$, the statements can be proved easily from Lemma 3.1. The general case is shown by approximating first $f$ by polynomials, then $g$, according to the convergence considered above (as for the third statement, obviously the order of limits does not matter). ∎ We need the following well-known result [DF77, Theorem 3.1]: ###### Theorem 3.4 (The commutator theorem). Let $H$ be a positive self-adjoint operator and $A,B$ symmetric operators defined on a core ${\mathcal{D}}_{0}$ for $(H+{\mathbbm{1}})^{2}$. Assume that there is a constant $C$ such that $\displaystyle\\|A\xi\\|\leq C\\|(H+{\mathbbm{1}})\xi\\|,\,\,\,\\|B\xi\\|\leq C\\|(H+{\mathbbm{1}})\xi\\|,$ $\displaystyle\\|[H,A]\xi\\|\leq C\\|(H+{\mathbbm{1}})\xi\\|,\,\,\,\\|[H,B]\xi\\|\leq C\\|(H+{\mathbbm{1}})\xi\\|,$ $\displaystyle\langle A\xi,B\eta\rangle=\langle B\xi,A\eta\rangle\mbox{ for any }\xi,\eta\in{\mathcal{D}}_{0}.$ Then $A$ and $B$ are essentially self-adjoint on any core of $H$ and any bounded functional calculus of $A$ and $B$ commute. ###### Remark 3.5. In the original literature [DF77], this Theorem is proved under the assumption of certain operator inequalities. In fact, what is really used in the proof of commutativity of bounded functions is the norm estimates $\\|A(H+{\mathbbm{1}})^{-1}\\|<C,\\|[H,A](H+{\mathbbm{1}})^{-1}\\|<C$ etc. and they follow from the assumptions here. The essential self-adjointness of $A$ and $B$ can be proved by [RS75, Theorem X.37]. An analogous application of this theorem with norm estimates can be found in [BSM90]. By the commutator theorem, we get that $J(f)$ is self-adjoint for $f\in C^{\infty}(S^{1},{\mathbb{R}})$ and that all bounded functions of $J(f)$ commute with all bounded functions of $J(g)$ for $f,g\in C^{\infty}(S^{1},{\mathbb{R}})$ with disjoint support. Let $I$ be a proper interval and let us define the von Neumann algebra ${\mathcal{B}}(I)=\\{\mathrm{e}^{\mathrm{i}J(f)}:{\rm supp}f\subset I\\}^{\prime\prime}.$ The local net ${\mathcal{B}}(I)$ restricted to $\overline{{\mathcal{B}}(I)\Omega_{0}}$ can be identified with the ${\rm U(1)}$-current net ${{\mathcal{A}}^{(0)}}$ on ${\mathcal{H}}_{\mathrm{{{\mathcal{A}}^{(0)}}}}$, in particular we can identify $\overline{{\mathcal{B}}(I)\Omega_{0}}\cong{\mathcal{H}}_{\mathrm{{{\mathcal{A}}^{(0)}}}}$. ###### Proposition 3.6. Let $I$ be a proper interval, then ${\mathcal{B}}(I)\subset\mathrm{Fer}_{\mathbb{C}}^{{\rm U(1)}}(I)$. ###### Proof. We see that ${\mathcal{B}}(I)$ commutes with $\mathrm{Fer}_{\mathbb{C}}(I^{\prime})=\\{c(g):g\in L^{2}(I^{\prime})\\}^{\prime\prime}$ because, for $f,g$ with disjoint supports, $c(g)$ commutes with $J(f)$ on a core by Lemma 3.3 and therefore any spectral projection of $c(g)$ commutes with $J(f)$, and hence with any bounded functions of $J(f)$. Further because $J(f)$ commutes by construction with the gauge action $V(t)$ and is in particular even because $V(\pi)=\Gamma$, it follows that ${\mathcal{B}}(I)$ lies in the twisted commutant $\mathrm{Fer}_{\mathbb{C}}(I^{\prime})^{\perp}$. By twisted Haag duality it is ${\mathcal{B}}(I)\subset\mathrm{Fer}_{\mathbb{C}}(I^{\prime})^{\perp}=\mathrm{Fer}_{\mathbb{C}}(I)$ and therefore ${\mathcal{B}}(I)={\mathcal{B}}(I)^{{\rm U(1)}}\subset\mathrm{Fer}_{\mathbb{C}}^{{\rm U(1)}}(I)$. ∎ Since the covariance has been seen, we have the following. ###### Corollary 3.7. ${\mathcal{B}}$ is a subnet of $\mathrm{Fer}_{\mathbb{C}}^{{\rm U(1)}}$. Now the following is straightforward. ###### Proposition 3.8. The ${\rm U(1)}$-fixed point subnet of the complex free fermion net $\mathrm{Fer}_{\mathbb{C}}$ is the ${\rm U(1)}$-current net, i.e. $\mathrm{Fer}_{\mathbb{C}}^{{\rm U(1)}}={\mathcal{B}}\cong{{\mathcal{A}}^{(0)}}$. ###### Proof. Let us see ${\mathcal{B}}$ as a subnet of the fermi net $\mathrm{Fer}_{\mathbb{C}}^{{\rm U(1)}}$ on ${\mathcal{H}}_{\mathrm{Fer}_{{\mathbb{C}}}}^{{\rm U(1)}}\equiv{\mathcal{H}}_{\,\cdot\,,0}$. Further $\overline{{\mathcal{B}}(I)\Omega}$ does not depend on $I$ by the same proof of the Reeh-Schlieder property and is clearly a subspace of ${\mathcal{H}}_{\mathrm{Fer}_{{\mathbb{C}}}}^{{\rm U(1)}}\equiv{\mathcal{H}}_{\,\cdot\,,0}$. In fact they coincide, since we have confirmed that $\operatorname{tr}_{{\mathcal{H}}_{\mathrm{{{\mathcal{A}}^{(0)}}}}}(\mathrm{e}^{-\beta L_{0}})=\operatorname{tr}_{{\mathcal{H}}_{\,\cdot\,,0}}(\mathrm{e}^{-\beta L_{0}})=p(t)$, where $\mathrm{e}^{-\beta}=t$, namely, their conformal characters coincide (see also Section 2.2). ∎ We finish this section by giving the parametrization in $x$-picture, where the action of the translation is more natural. With $f(x)=\frac{1}{\sqrt{2\pi}}\sqrt{\left\|\frac{\partial\theta(x)}{\partial x}\right\|}\mathrm{e}^{\mathrm{i}\theta(x)/2}f_{0}(\theta(x))$ we identify $L^{2}({\mathbb{R}})=L^{2}({\mathbb{R}},\,\mathrm{d}x)$ with $L^{2}(S^{1})=L^{2}([0,2\pi],\,\mathrm{d}\theta/(2\pi))$ and therefore the space ${\mathcal{H}}_{\mathrm{Fer}_{{\mathbb{C}}}}^{1}$ is given by $PL^{2}({\mathbb{R}})\oplus\overline{P^{\perp}L^{2}({\mathbb{R}})}$ with $P:\hat{f}(p)\mapsto\Theta(p)\hat{f}(p)$ and it can be identified in “momentum space” with $L^{2}({\mathbb{R}}_{+},2\pi\,\mathrm{d}p)\oplus L^{2}({\mathbb{R}}_{+},2\pi\,\mathrm{d}q)$ by $\displaystyle f(x)$ $\displaystyle\longmapsto\widehat{Pf}(p)\oplus\overline{\widehat{P^{\perp}f}(-q)}$ $\displaystyle p,q>0\,.$ The field operators are defined for $f\in L^{2}({\mathbb{R}})$ by $\psi(f)=a_{P}(f)$ and $\bar{\psi}(f)=a_{P}(\overline{f})^{\ast}$. For $\Psi\in{\mathcal{H}}_{\mathrm{Fer}_{{\mathbb{C}}}}$ we write its components $\Psi_{m,n}\in{\mathcal{H}}_{m,n}:=L^{2}({\mathbb{R}}_{+}^{m+n},(2\pi)^{m+n}\,\mathrm{d}p_{1}\cdots\,\mathrm{d}p_{m}\,\mathrm{d}q_{1}\cdots q_{n})_{-},$ (1) where $-$ means the antisymmetrization within $p_{1},\ldots,p_{m}$ and $q_{1},\ldots,q_{n}$. By this notation $(\psi(f)\Omega_{0})_{1,0}(p)=\hat{f}(p)$ and $(\bar{\psi}(f)\Omega_{0})_{0,1}(q)=\hat{f}(q)$. Further the bi-field ${:\\!\bar{\psi}(f)\psi(g)\\!:}=\bar{\psi}(f)\psi(g)-\langle\Omega_{0},\bar{\psi}(f)\psi(g)\Omega_{0}\rangle{\mathbbm{1}}$ creates from the vacuum $\Omega_{0}$ a fermionic 1+1 particle state $\Psi_{f,g}:={:\\!\bar{\psi}(f)\psi(g)\\!:}\Omega_{0}$ with $(\Psi_{f,g})_{1,1}(p,q)=-\hat{f}(q)\hat{g}(p)$ and it follows for $h\in C^{\infty}({\mathbb{R}},{\mathbb{R}})$ that for the ${\rm U(1)}$-current $J$, it holds $(J(h)\Omega_{0})_{1,1}(p,q)=-\frac{1}{2\pi}\hat{h}(p+q)$ which is obtained by taking a limit $\sum_{n}\Psi_{f_{n},g_{n}}$ with test functions $\sum_{n}f_{n}(x)g_{n}(y)\to h(x)\delta(x-y)$. We make the important observation that the $J(f)\Omega_{0}$ generate the one-particle space which we can identify with ${\mathcal{H}}_{\mathrm{{{\mathcal{A}}^{(0)}}}}^{1}$ and this is obviously a proper subspace of the fermionic 1+1-particle space ${\mathcal{H}}_{\mathrm{Fer}_{{\mathbb{C}}}}^{1,1}$. ## 4 A new family of Longo-Witten endomorphisms on ${\rm U(1)}$-current net We use the description of ${\mathcal{H}}_{\mathrm{Fer}_{{\mathbb{C}}}}^{1}=\overline{PL^{2}(S^{1})}+P^{\perp}L^{2}(S^{1})$ which equals $L^{2}(S^{1})$ as a real Hilbert space and is described in the beginning of Section 3.2. First we decompose ${\mathcal{H}}_{\mathrm{Fer}_{{\mathbb{C}}}}^{1}$ into irreducible representations of $\mathrm{SU}(1,1)$ in a compatible way with $K(I)$. Let us define $\displaystyle{\mathcal{H}}_{\Re}$ $\displaystyle:=$ $\displaystyle\\{f\in{\mathcal{H}}^{1}_{\mathrm{Fer}_{\mathbb{C}}}:z^{\frac{1}{2}}f(z)\mbox{ is real}\\},$ $\displaystyle{\mathcal{H}}_{\Im}$ $\displaystyle:=$ $\displaystyle\\{f\in{\mathcal{H}}^{1}_{\mathrm{Fer}_{\mathbb{C}}}:z^{\frac{1}{2}}f(z)\mbox{ is pure imaginary}\\}.$ By their definition, it is clear that ${\mathcal{H}}_{\Re}$ and ${\mathcal{H}}_{\Im}$ are real Hilbert subspaces of $L^{2}(S^{1})$. In fact, they are complex subspaces with respect to the new complex structure. To see this, we take another description of ${\mathcal{H}}_{\Re}$: in terms of Fourier components, it holds that $f\in{\mathcal{H}}_{\Re}$ if and only if $f_{n}=\overline{f_{-n-1}}$. Recall that, on $L^{2}(S^{1})$, the new scalar multiplication by $\mathrm{i}$ is given by $\mathrm{i}\cdot f_{n}=-\mathrm{i}f_{n}$, $\mathrm{i}\cdot f_{-n-1}=\mathrm{i}f_{-n-1}$ for $n\geq 0$. Hence this condition is preserved under the multiplication by $\mathrm{i}$ and ${\mathcal{H}}_{\Re}$ is a complex subspace. An analogous argument holds for ${\mathcal{H}}_{\Im}$. Next we see that ${\mathcal{H}}_{\Re}$ and ${\mathcal{H}}_{\Im}$ are orthogonal. Note that because of the change of the complex structure, for $f(z)=\sum_{n}f_{n}z^{n}$ and $h(z)=\sum_{n}h_{n}z^{n}$ the inner product is written as follows: $\langle f,h\rangle=\sum_{n\geq 0}f_{n}\overline{h_{n}}+\sum_{n<0}\overline{f_{n}}h_{n}.$ Now $f\in{\mathcal{H}}_{\Re}$ implies $f_{n}=\overline{f_{-n-1}}$ and $h\in{\mathcal{H}}_{\Im}$ implies $h_{n}=-\overline{h_{-n-1}}$ for non- negative $n$, hence it is easy to see that $\langle f,h\rangle=\sum_{n\geq 0}f_{n}\overline{h_{n}}+\sum_{n<0}\overline{f_{n}}h_{n}=-\sum_{n\geq 0}\overline{f_{-n-1}}{h_{-n-1}}+\sum_{n<0}\overline{f_{n}}{h_{n}}=0.$ In other words, these two complex subspaces are mutually orthogonal. Furthermore, ${\mathcal{H}}_{\Re}$ and ${\mathcal{H}}_{\Im}$ are invariant under the action of $\mathrm{SU}(1,1)$. We recall that the action is given by $(V_{g}f)(z)=\frac{1}{-\overline{\beta}z+\alpha}f\left(\frac{\overline{\alpha}z-\beta}{-\overline{\beta}z+\alpha}\right)$. Then if $z^{\frac{1}{2}}f(z)$ is real then it holds that $\displaystyle z^{\frac{1}{2}}(V_{g}f)(z)$ $\displaystyle=$ $\displaystyle\frac{1}{{\bar{z}}^{\frac{1}{2}}(-\overline{\beta}z+\alpha)}\left(\frac{\overline{\alpha}z-\beta}{-\overline{\beta}z+\alpha}\right)^{-\frac{1}{2}}\cdot\left(\frac{\overline{\alpha}z-\beta}{-\overline{\beta}z+\alpha}\right)^{\frac{1}{2}}f\left(\frac{\overline{\alpha}z-\beta}{-\overline{\beta}z+\alpha}\right)$ $\displaystyle=$ $\displaystyle\frac{1}{(-\overline{\beta}+\alpha\bar{z})^{\frac{1}{2}}(\overline{\alpha}z-\beta)^{\frac{1}{2}}}\cdot\left(\frac{\overline{\alpha}z-\beta}{-\overline{\beta}z+\alpha}\right)^{\frac{1}{2}}f\left(\frac{\overline{\alpha}z-\beta}{-\overline{\beta}z+\alpha}\right)$ and both factors are real. Similarly one shows that ${\mathcal{H}}_{\Im}$ is preserved under $V_{g}$. It is obvious that these two representations are intertwined by the multiplication by $\mathrm{i}$ in the old complex structure. This is still a unitary map, thus they are unitarily equivalent. One can see that each representation is indeed irreducible, and when restricted to $\mathrm{PSU}(1,1)={\rm PSL}(2,{\mathbb{R}})$, it is the projective positive energy representation with lowest weight $\frac{1}{2}$. It is easy to see that $e^{\Re}_{n}:=\\{e_{n}+e_{-n+1},n\geq 0\\}$ and $e^{\Im}_{n}:=\\{\mathrm{i}(e_{n}+e_{-n+1}),n\geq 0\\}$ form bases of ${\mathcal{H}}_{\Re}$ and ${\mathcal{H}}_{\Im}$, respectively, where $e_{n}(z)=z^{n}$ and the multiplication by $\mathrm{i}$ is given in the old structure. Now we describe the gauge action in terms of this basis. By the definition, for a given complex number $\alpha$ with modulus $1$, the action is given by the multiplication in the old structure. Hence if $\alpha=\cos\theta+\mathrm{i}\sin\theta$, we have $U_{\alpha}e^{\Re}_{n}=\cos\theta e^{\Re}_{n}+\sin\theta e^{\Im}_{n}$ and $U_{\alpha}e^{\Im}_{n}=-\sin\theta e^{\Re}_{n}+\cos\theta e^{\Im}_{n}$. This means that $U_{\alpha}$ acts as the real rotation by $\theta$ in this basis. #### Construction of endomorphisms We construct Longo-Witten endomorphisms on the free fermion net $\mathrm{Fer}_{\mathbb{C}}$ commuting with the gauge action. The key is the following theorem. We remind that a standard pair $(\tilde{H},\tilde{T})$ is a standard subspace $\tilde{H}\subset\tilde{\mathcal{H}}$ of a Hilbert space $\tilde{\mathcal{H}}$ and a positive energy representation $\tilde{T}$ of ${\mathbb{R}}$ on $\tilde{\mathcal{H}}$, such that $\tilde{T}(a)\tilde{H}\subset\tilde{H}$ for $a\geq 0$. If $\tilde{T}$ is maximally abelian, the standard pair is said to be irreducible and there is a (up to unitary equivalence) unique irreducible standard pair. ###### Theorem 4.1 ([LW11, Theorem 2.6]). Let $(\tilde{H},\tilde{T})$ be a standard pair with multiplicity $n$, i.e. it decomposes into $n$-fold direct sum of irreducible standard pairs, each unitarily equivalent to the unique standard pair $(H,T)$ and $T(t)=\mathrm{e}^{\mathrm{i}tP}$. Then a unitary $\tilde{V}$ commuting with the translation $\tilde{T}$ preserves $\tilde{H}$ if and only if $\tilde{V}$ is a $n\times n$ matrix $(V_{hk})$ (with respect to the decomposition of $\tilde{\mathcal{H}}$ into $n$ direct sum as above) such that $V_{hk}=\varphi_{hk}(P)$, where $\varphi_{hk}:{\mathbb{R}}\to{\mathbb{C}}$ are complex Borel functions such that $(\varphi_{hk})$ is a unitary matrix for almost every $p>0$, each $\varphi_{hk}$ is the boundary value of a function in ${\mathbb{H}}(\SS_{\infty})$ and is symmetric, i.e. $\varphi_{hk}(-p)=\overline{\varphi_{hk}(p)}$. Consider the one-particle space ${\mathcal{H}}_{\mathrm{Fer}_{{\mathbb{C}}}}^{1}$ for $\mathrm{Fer}_{\mathbb{C}}$. The pair of the standard space $K({\mathbb{R}}_{+})=L^{2}({\mathbb{R}}_{+})$ defined in Section 3.2 (under the identification of $S^{1}$ and ${\mathbb{R}}\cup\\{\infty\\})$ and the natural translation has multiplicity 2. If we take a matrix-valued function $(\varphi_{hk})$ as above and take the second quantization operator $\Lambda(V)$ of the (matrix-valued) operator $(V_{kh})=(\varphi_{hk}(P))$, then it implements a Longo-Witten endomorphism of $\mathrm{Fer}_{\mathbb{C}}$ (see [LW11]). As the gauge group acts by real rotation $\left(\begin{matrix}\cos\theta&-\sin\theta\\\ \sin\theta&\cos\theta\end{matrix}\right)$, any matrix-valued function of $p$ which commute with them must have the form $\left(\begin{matrix}a(p)&\mathrm{i}b(p)\\\ -\mathrm{i}b(p)&a(p)\end{matrix}\right)$. If each component is symmetric, then $a$ is symmetric and $b$ is antisymmetric. Such a matrix-valued function can be diagonalized by the matrix $\left(\begin{matrix}1&\mathrm{i}\\\ \mathrm{i}&1\end{matrix}\right)$ and becomes $\left(\begin{matrix}a(p)+b(p)&0\\\ 0&a(p)-b(p)\end{matrix}\right)$. We claim that such $a$ and $b$ exist. Indeed, let $\varphi$ be a inner function (not necessarily symmetric), namely the boundary value with modulus $1$ of a bounded analytic function on the upper half-plane ${\mathbb{H}}$, and define $a(p)=\frac{1}{2}(\varphi(p)+\overline{\varphi(-p)})$, $b(p)=\frac{1}{2}(\varphi(p)-\overline{\varphi(-p)})$. Then it is obvious that $a$ is symmetric and $b$ is antisymmetric. In addition, $a(p)+b(p)=\varphi(p)$ and $a(p)-b(p)=\overline{\varphi(-p)}$, hence the diagonalized matrix is unitary for almost every $p$. By the theorem of Longo-Witten, the operator $\varphi(P_{1}):=\left(\begin{matrix}a(P)&\mathrm{i}b(P)\\\ -\mathrm{i}b(P)&a(P)\end{matrix}\right)$ preserves the real Hilbert space $\widetilde{H}:=K({\mathbb{R}}_{+})$, where $P_{1}$ is the generator of the translation in ${\mathcal{H}}_{\mathrm{Fer}_{\mathbb{C}}}^{1}$ which has multiplicity $2$ and $P$ is the generator of $T$ of the irreducible standard pair $(H,T)$. It is easy to see that the above diagonalization is given exactly by the decomposition ${\mathcal{H}}_{\mathrm{Fer}_{{\mathbb{C}}}}^{1}=\overline{PL^{2}(S^{1})}\oplus({\mathbbm{1}}-P)L^{2}(S^{1})$. We remind that ${\mathcal{H}}_{\mathrm{Fer}_{{\mathbb{C}}}}^{1}$ can be identified with $L^{2}({\mathbb{R}})$ as a real space. In $L^{2}({\mathbb{R}})$ the function $\varphi(P_{1})f$ is the function with Fourier transform $\varphi(p)\hat{f}(p)$ and we remark that it also follows directly from the Paley-Wiener theorem that $\varphi(P_{1})$ leaves $L^{2}({\mathbb{R}}_{+})\subset L^{2}({\mathbb{R}})$ invariant for $\varphi$ inner. Further using that the space ${\mathcal{H}}_{\mathrm{Fer}_{{\mathbb{C}}}}$ decomposes in ${\mathcal{H}}_{\mathrm{Fer}_{{\mathbb{C}}}}=\bigoplus_{m,n\in{\mathbb{N}}_{0}}{\mathcal{H}}_{m,n}$ like in (1) with the gauge action given by $V(\theta)\Psi_{m,n}=\mathrm{e}^{\mathrm{i}(m-n)\theta}\Psi_{m,n}$, the action of the Longo-Witten unitary $V_{\varphi}=\Lambda(\varphi(P_{1}))$ is given by $\displaystyle(V_{\varphi}\Psi)_{m,n}(p_{1},\cdots,p_{m},q_{1},\cdots,q_{n})$ $\displaystyle\quad=\varphi(p_{1})\cdots\varphi(p_{m})\overline{\varphi(-q_{1})}\cdots\overline{\varphi(-q_{n})}\Psi_{m,n}(p_{1},\cdots,p_{m},q_{1},\cdots,q_{n})\,.$ ###### Lemma 4.2. Let $\iota:\Psi\in L^{2}({\mathbb{R}}_{+},p\,\mathrm{d}p)\equiv{\mathcal{H}}_{\mathrm{{{\mathcal{A}}^{(0)}}}}^{1}\hookrightarrow{\mathcal{H}}_{1,1}\subset{\mathcal{H}}_{\mathrm{Fer}_{{\mathbb{C}}}}$ be the embedding given by $\iota(\Psi)_{1,1}(p,q)=-\frac{1}{2\pi}\Psi(p+q)$. A second quantization Longo-Witten unitary $V_{\varphi}$ commuting with the gauge action $V(\,\cdot\,)$ satisfies $V_{\varphi}\iota{\mathcal{H}}_{\mathrm{{{\mathcal{A}}^{(0)}}}}^{1}\subset\iota{\mathcal{H}}_{\mathrm{{{\mathcal{A}}^{(0)}}}}^{1}$ if and only if $V_{\varphi}=V(\theta)T(t)$ with $t\geq 0$. ###### Proof. The translations commute with the gauge action and it follows immediately that they leave ${\mathcal{H}}_{\mathrm{{{\mathcal{A}}^{(0)}}}}^{1}$ invariant. We note that $\varphi(p)\overline{\varphi(-q)}\Psi(p+q)$ belongs to ${\mathcal{H}}_{\mathrm{{{\mathcal{A}}^{(0)}}}}^{1}$ only if it can be written as a function of $g(p+q)$. This means that $\varphi(p)\overline{\varphi(-q)}=\widetilde{\varphi}(p+q)$ for $p,q\geq 0$, where $\widetilde{\varphi}$ is another function. Then, putting $q=0$ and $p=0$ respectively, we see that $\varphi(p)\overline{\varphi(0)}=\widetilde{\varphi}(p)$ for $p\geq 0$ and $\varphi(0)\overline{\varphi(-q)}=\widetilde{\varphi}(q)$ for $q\geq 0$, in particular $\widetilde{\varphi}(0)=1$. Multiplying the each side of these equations, one sees that $\widetilde{\varphi}(p+q)=\widetilde{\varphi}(p)\widetilde{\varphi}(q)$ because $\|\varphi(0)\|=1$. Then it follows that $\widetilde{\varphi}(p)=\mathrm{e}^{\mathrm{i}\kappa p}$ for some $\kappa\geq 0$, and $\varphi(p)=\mathrm{e}^{\mathrm{i}(\kappa p+\theta)}$ for some $\theta\in{\mathbb{R}}$ (in fact, the arguments here should be treated with care because the relation is given only almost everywhere, but both $\varphi$ and $\check{\varphi}$ analytically continue and in the domain of analyticity it holds everywhere). Such a $\varphi$ is a Longo-Witten unitary only for $\kappa\geq 0$. The constant factor $\mathrm{e}^{\mathrm{i}\theta}$ corresponds to the factor $V(\theta)$. ∎ ###### Theorem 4.3. Let $\varphi$ be an inner function as above. The endomorphism implemented by the second quantization $V_{\varphi}$ of the operator constructed above restricts to the ${\rm U(1)}$-current subnet. The restriction cannot be implemented by any second quantization operator if $\varphi(p)\neq\mathrm{e}^{\mathrm{i}(\kappa p+\theta)}$. ###### Proof. The operator $V_{\varphi}$ restricts to the subnet ${{\mathcal{A}}^{(0)}}$ by the general argument in Proposition 2.1. It cannot be implemented by a second quantization operator, since any second quantization operator preserves the particle number, while $V_{\varphi}$ does not for non-exponential $\varphi$ as we saw above, and a Longo-Witten endomorphism is uniquely implemented up to scalar (see Section 2.1). ∎ ###### Remark 4.4. By the construction in [LW11], each unitary $V=V_{\varphi}\|_{{\mathcal{H}}_{\mathrm{{{\mathcal{A}}^{(0)}}}}}$ related to an inner function $\varphi$ from above gives rise to a local, time-translation covariant net of von Neumann algebras on the Minkowski half-space $M_{+}=\\{(t,x)\in{\mathbb{R}}^{2}:x>0\\}$. This net is associated with the ${\rm U(1)}$-current net ${{\mathcal{A}}^{(0)}}$ and defined by ${{\mathcal{A}}^{(0)}}_{V}(O)={{\mathcal{A}}^{(0)}}(I_{1})\vee V{{\mathcal{A}}^{(0)}}(I_{2})V^{\ast}$, where $O=I_{1}\times I_{2}=\\{(t,x)\in{\mathbb{R}}^{2}:t-x\in I_{1},t+x\in I_{2}\\}$ is a double cone with $\overline{O}\subset M_{+}$ corresponding uniquely to the two intervals $I_{1}$ and $I_{2}$ with disjoint closures. In the case where $\varphi$ is not exponential $V_{\varphi}$ does not come from second quantization—in contrast to the unitaries constructed by Longo and Witten in [LW11]—and therefore gives new examples. ## 5 Interacting wedge-local net with particle production ### 5.1 Construction of scattering operators In the previous section we saw that, in the basis $\\{e_{n}+e_{-n},e_{n}-e_{-n}\\}$ the matrix operator $\left(\begin{matrix}a(P)&\mathrm{i}b(P)\\\ -\mathrm{i}b(P)&a(P)\end{matrix}\right)$ implements a Longo-Witten endomorphism if $a$ is symmetric and $b$ is antisymmetric, and after the simultaneous diagonalization it becomes $\left(\begin{matrix}\varphi(P)&0\\\ 0&\check{\varphi}(P)\end{matrix}\right)$ where $\varphi$ is an inner function and $\check{\varphi}(p)=\overline{\varphi(-p)}$ (note that if $\varphi$ extends to an analytic function $\varphi(z)$ on ${\mathbb{H}}$, then $\check{\varphi}(z)=\overline{\varphi(-\overline{z})}$ also extends to ${\mathbb{H}}$, hence $\check{\varphi}$ is again an inner function). By the same argument one sees that $\left(\begin{matrix}\check{\varphi}(P)&0\\\ 0&\varphi(P)\end{matrix}\right)$ implements an endomorphism since $\check{\check{\varphi}}=\varphi$. With respect to the basis after diagonalization, we split the Hilbert space ${\mathcal{H}}_{\mathrm{Fer}_{\mathbb{C}}}^{1}=:{\mathcal{H}}_{+}\oplus{\mathcal{H}}_{-}$ and the generator of translation $P_{1}=:P_{+}\oplus P_{-}$. Then the tensor product space can be written as follows: ${\mathcal{H}}_{\mathrm{Fer}_{\mathbb{C}}}^{1}\otimes{\mathcal{H}}_{\mathrm{Fer}_{\mathbb{C}}}^{1}=\left({\mathcal{H}}_{+}\otimes{\mathcal{H}}_{+}\right)\oplus\left({\mathcal{H}}_{+}\otimes{\mathcal{H}}_{-}\right)\oplus\left({\mathcal{H}}_{-}\otimes{\mathcal{H}}_{+}\right)\oplus\left({\mathcal{H}}_{-}\otimes{\mathcal{H}}_{-}\right)\,.$ According to this decomposition into a direct sum of four subspaces, we define an operator $M_{\varphi}:=\varphi(P_{+}\otimes P_{+})\oplus\check{\varphi}(P_{+}\otimes P_{-})\oplus\check{\varphi}(P_{-}\otimes P_{+})\oplus\varphi(P_{-}\otimes P_{-})\,.$ Then this restricts to the subspace ${\mathcal{H}}_{\mathrm{Fer}_{\mathbb{C}}}^{1}\otimes{\mathcal{H}}_{+}=\left({\mathcal{H}}_{+}\oplus{\mathcal{H}}_{-}\right)\otimes{\mathcal{H}}_{+}$ and it is $\varphi(P_{+}\otimes P_{+})\oplus\check{\varphi}(P_{-}\otimes P_{+})$, or we can decompose it with respect to the spectral measure of $P_{+}$: $\int_{{\mathbb{R}}_{+}}\left(\begin{matrix}\varphi(pP_{+})&0\\\ 0&\check{\varphi}(pP_{-})\end{matrix}\right)\otimes\,\mathrm{d}E_{+}(p).$ Similarly, the restriction to ${\mathcal{H}}_{\mathrm{Fer}_{\mathbb{C}}}^{1}\otimes{\mathcal{H}}_{-}$ is written as $\int_{{\mathbb{R}}_{+}}\left(\begin{matrix}\check{\varphi}(pP_{+})&0\\\ 0&\varphi(pP_{-})\end{matrix}\right)\otimes\,\mathrm{d}E_{-}(p).$ Using the two-point set ${\mathbb{Z}}_{2}=\\{+,-\\}$ we define $\varphi_{+}(p,+):=\varphi(p),\,\,\,\,\varphi_{+}(p,-)=\check{\varphi}(p),\,\,\,\,\varphi_{-}(p,+)=\check{\varphi}(p),\,\,\,\,\varphi_{-}(p,-)=\varphi(p).$ By defining the spectral measure $E_{1}=E_{+}\oplus E_{-}$ on ${\mathcal{H}}^{1}$, $M_{\varphi}$ can be simply written as $M_{\varphi}=\int_{{\mathbb{R}}_{+}\times{\mathbb{Z}}_{2}}\left(\begin{matrix}\varphi_{+}(pP_{+},\iota)&0\\\ 0&\varphi_{-}(pP_{-},\iota)\end{matrix}\right)\otimes\,\mathrm{d}E_{1}(p,\iota),$ where $\iota=\pm$. As in [Tan11a], we construct the scattering matrix first on the unsymmetrized Fock space, then restrict it to the antisymmetric space. For an operator $A$ on ${\mathcal{H}}_{\mathrm{Fer}_{\mathbb{C}}}^{1}\otimes{\mathcal{H}}_{\mathrm{Fer}_{\mathbb{C}}}^{1}$, we denote by $A^{m,n}_{i,j}$ on $({\mathcal{H}}_{\mathrm{Fer}_{\mathbb{C}}}^{1})^{\otimes m}\otimes({\mathcal{H}}_{\mathrm{Fer}_{\mathbb{C}}}^{1})^{\otimes n}$ the operator which acts only on the $i$-th factor in $({\mathcal{H}}_{\mathrm{Fer}_{\mathbb{C}}}^{1})^{\otimes m}$ and $j$-th factor in $({\mathcal{H}}_{\mathrm{Fer}_{\mathbb{C}}}^{1})^{\otimes n}$ as $A$. As a convention, $A^{m,n}_{i,j}$ equals to the identity operator if $m$ or $n$ is $0$. Let us denote simply $\widetilde{\varphi}(p,\iota):=\left(\begin{matrix}\varphi_{+}(p,\iota)&0\\\ 0&\varphi_{-}(p,\iota)\end{matrix}\right)$ and $\widetilde{\varphi}(P_{1},\iota):=\left(\begin{matrix}\varphi_{+}(P_{+},\iota)&0\\\ 0&\varphi_{-}(P_{-},\iota)\end{matrix}\right)$. From the observation above, it is straightforward to see that $(M_{\varphi})^{m,n}_{i,j}=\int\left({\mathbbm{1}}\otimes\cdots\otimes\underset{i\mbox{-th}}{\widetilde{\varphi}(p_{j}P_{1},\iota_{j})}\otimes\cdots\otimes{\mathbbm{1}}\right)\otimes\,\mathrm{d}E_{1}(p_{1},\iota_{1})\otimes\cdots\otimes\,\mathrm{d}E_{1}(p_{n},\iota_{n})$ (the case where $m$ or $n$ is $0$ is treated separately). Then we define, as in [Tan11a], $\displaystyle S_{\varphi}^{m,n}$ $\displaystyle:=$ $\displaystyle\prod_{i,j}(M_{\varphi})^{m,n}_{i,j}$ $\displaystyle S_{\varphi}$ $\displaystyle:=$ $\displaystyle\bigoplus_{m,n}S_{\varphi}^{m,n}\,.$ Let ${\mathcal{H}}_{\mathrm{Fer}_{\mathbb{C}}}^{\Sigma}$ be the unsymmetrized Fock space based on ${\mathcal{H}}_{\mathrm{Fer}_{\mathbb{C}}}^{1}$. Note that $S_{\varphi}$ is defined on ${\mathcal{H}}_{\mathrm{Fer}_{\mathbb{C}}}^{\Sigma}\otimes{\mathcal{H}}_{\mathrm{Fer}_{\mathbb{C}}}^{\Sigma}$, and it naturally restricts to ${\mathcal{H}}_{\mathrm{Fer}_{\mathbb{C}}}\otimes{\mathcal{H}}_{\mathrm{Fer}_{\mathbb{C}}}^{\Sigma}$, ${\mathcal{H}}_{\mathrm{Fer}_{\mathbb{C}}}^{\Sigma}\otimes{\mathcal{H}}_{\mathrm{Fer}_{\mathbb{C}}}$ and ${\mathcal{H}}_{\mathrm{Fer}_{\mathbb{C}}}\otimes{\mathcal{H}}_{\mathrm{Fer}_{\mathbb{C}}}$. This $S_{\varphi}$ will be interpreted as the scattering matrix. In order to confirm this, we have to take the spectral decomposition of $S_{\varphi}$ only with respect to the right or left component. Namely, $\displaystyle S_{\varphi}$ $\displaystyle:=$ $\displaystyle\bigoplus_{m,n}\prod_{i,j}(M_{\varphi})^{m,n}_{i,j}$ $\displaystyle=$ $\displaystyle\bigoplus_{m,n}\prod_{i,j}\int\left({\mathbbm{1}}\otimes\cdots\otimes\underset{i\mbox{-th}}{\widetilde{\varphi}(p_{j}P_{1},\iota_{j})}\otimes\cdots\otimes{\mathbbm{1}}\right)\otimes\,\mathrm{d}E_{1}(p_{1},\iota_{1})\otimes\cdots\otimes\,\mathrm{d}E_{1}(p_{n},\iota_{n})$ $\displaystyle=$ $\displaystyle\bigoplus_{m,n}\int\prod_{i,j}\left({\mathbbm{1}}\otimes\cdots\otimes\underset{i\mbox{-th}}{\widetilde{\varphi}(p_{j}P_{1},\iota_{j})}\otimes\cdots\otimes{\mathbbm{1}}\right)\otimes\,\mathrm{d}E_{1}(p_{1},\iota_{1})\otimes\cdots\otimes\,\mathrm{d}E_{1}(p_{n},\iota_{n})$ $\displaystyle=$ $\displaystyle\bigoplus_{n}\int\bigoplus_{m}\prod_{j}\left(\widetilde{\varphi}(p_{j}P_{1},\iota_{j})\right)^{\otimes m}\otimes\,\mathrm{d}E_{1}(p_{1},\iota_{1})\otimes\cdots\otimes\,\mathrm{d}E_{1}(p_{n},\iota_{n})$ $\displaystyle=$ $\displaystyle\bigoplus_{n}\int\prod_{j}\bigoplus_{m}\left(\widetilde{\varphi}(p_{j}P_{1},\iota_{j})\right)^{\otimes m}\otimes\,\mathrm{d}E_{1}(p_{1},\iota_{1})\otimes\cdots\otimes\,\mathrm{d}E_{1}(p_{n},\iota_{n})$ $\displaystyle=$ $\displaystyle\bigoplus_{n}\int\prod_{j}\Lambda(\widetilde{\varphi}(p_{j}P_{1},\iota_{j}))\otimes\,\mathrm{d}E_{1}(p_{1},\iota_{1})\otimes\cdots\otimes\,\mathrm{d}E_{1}(p_{n},\iota_{n})\,,$ where the integral and the product commute in the third equality since the spectral measure is disjoint for different values of $p$’s and $\iota$’s, and the sum and the product commute in the fifth equality since the operators in the integrand act on mutually disjoint spaces, namely on $({\mathcal{H}}_{\mathrm{Fer}_{\mathbb{C}}}^{1})^{\otimes m}\otimes{\mathcal{H}}_{\mathrm{Fer}_{\mathbb{C}}}^{\Sigma}$ for different $m$. In the final expression, all operators appearing in the integrand are the second quantization operators, thus this formula naturally restricts to the partially antisymmetrized space ${\mathcal{H}}_{\mathrm{Fer}_{\mathbb{C}}}\otimes{\mathcal{H}}_{\mathrm{Fer}_{\mathbb{C}}}^{\Sigma}$. Now we define • ${\mathcal{M}}_{\varphi}:=\\{x\otimes{\mathbbm{1}},{\hbox{\rm Ad\,}}S_{\varphi}({\mathbbm{1}}\otimes y):x\in\mathrm{Fer}_{\mathbb{C}}({\mathbb{R}}_{-}),y\in\mathrm{Fer}_{\mathbb{C}}({\mathbb{R}}_{+})\\}^{\prime\prime}$, * • $T(t,x):=T_{0}(\frac{t-x}{\sqrt{2}})\otimes T_{0}(\frac{t+x}{\sqrt{2}})$, * • $\Omega:=\Omega_{0}\otimes\Omega_{0}$. As the net $\mathrm{Fer}_{\mathbb{C}}$ is fermionic by nature, the interpretation of the scattering theory of [Buc75] is not clear. Nevertheless, we can show the following by an almost same proof as in [Tan11a, Lemma 5.2, Theorem 5.3]. ###### Lemma 5.1. The triple $({\mathcal{M}}_{\varphi},T,\Omega)$ is a Borchers triple. ###### Proof. To apply Proposition 2.2, it is immediate that $S_{\varphi}$ commutes with translation since it is defined through the spectral measure as above. It preserves ${\mathcal{H}}_{\mathrm{Fer}_{\mathbb{C}}}\otimes\Omega_{0}$ and $\Omega_{0}\otimes{\mathcal{H}}_{\mathrm{Fer}_{\mathbb{C}}}$ pointwise, since these subspaces correspond to the case where $m$ or $n$ is $0$ in the above decomposition and $S_{\varphi}$ acts as the identity operator by definition. What remains to show is the commutation property. As we saw above, the operator $S_{\varphi}$ can be written as $S_{\varphi}=\bigoplus_{n}\int\prod_{j}\Lambda(\widetilde{\varphi}(p_{j}P_{1},\iota_{j}))\otimes\,\mathrm{d}E_{1}(p_{1},\iota_{1})\otimes\cdots\otimes\,\mathrm{d}E_{1}(p_{n},\iota_{n})\,.$ The point is that the operators which appear in the integrand implement Longo- Witten endomorphisms as we saw above since $p_{j}\geq 0$ in the support of the integration. Let $x^{\prime}\in\mathrm{Fer}_{\mathbb{C}}({\mathbb{R}}_{+})$ and consider $x^{\prime}\otimes{\mathbbm{1}}$ as an operator on ${\mathcal{H}}_{\mathrm{Fer}_{\mathbb{C}}}\otimes{\mathcal{H}}_{\mathrm{Fer}_{\mathbb{C}}}^{\Sigma}$. We have ${\hbox{\rm Ad\,}}S_{\varphi}(x^{\prime}\otimes{\mathbbm{1}})=\bigoplus_{n}\int{\hbox{\rm Ad\,}}\left(\prod_{j}\Lambda(\widetilde{\varphi}(p_{j}P_{1},\iota_{j}))\right)(x^{\prime})\otimes\,\mathrm{d}E_{1}(p_{1},\iota_{1})\otimes\cdots\otimes\,\mathrm{d}E_{1}(p_{n},\iota_{n})\,.$ Although this formula is not closed on ${\mathcal{H}}_{\mathrm{Fer}_{\mathbb{C}}}\otimes{\mathcal{H}}_{\mathrm{Fer}_{\mathbb{C}}}$, the left hand side obviously restricts there. One sees that the integrand remains in $\mathrm{Fer}_{\mathbb{C}}({\mathbb{R}}_{+})$. Recall the operator $Z_{0}$ which gives the graded locality of $\mathrm{Fer}_{\mathbb{C}}$. One has to remind that $Z_{0}=\frac{{\mathbbm{1}}-\mathrm{i}\Gamma_{0}}{1-\mathrm{i}}$ where $\Gamma_{0}=\Lambda(-{\mathbbm{1}})$, hence $Z_{0}$ commutes with any second quantization operator. Then by the disintegration above (and the corresponding disintegration with respect to the left component), it is easy to see that $Z_{0}\otimes{\mathbbm{1}}$ commutes with $S_{\varphi}$. Let us check the commutation property of the assumptions in Proposition 2.2. Note that ${\hbox{\rm Ad\,}}Z_{0}(x)\otimes{\mathbbm{1}}$ and ${\hbox{\rm Ad\,}}Z_{0}(x)\in\mathrm{Fer}_{\mathbb{C}}({\mathbb{R}}_{+})^{\prime}$ for $x\in\mathrm{Fer}_{\mathbb{C}}({\mathbb{R}}_{-})$. Since $Z_{0}\otimes{\mathbbm{1}}$ and $S_{\varphi}$ commute as we saw above, to prove the first commutation relation, it is enough to show that $[{\hbox{\rm Ad\,}}Z_{0}(x)\otimes{\mathbbm{1}},{\hbox{\rm Ad\,}}S_{\varphi}(x^{\prime}\otimes{\mathbbm{1}})]=0$ for $x\in\mathrm{Fer}_{\mathbb{C}}({\mathbb{R}}_{-})$ and $x^{\prime}\in\mathrm{Fer}_{\mathbb{C}}({\mathbb{R}}_{+})$. As operators acting on ${\mathcal{H}}_{\mathrm{Fer}_{\mathbb{C}}}\otimes{\mathcal{H}}_{\mathrm{Fer}_{\mathbb{C}}}^{\Sigma}$, this is done by the above disintegration of ${\hbox{\rm Ad\,}}S_{\varphi}(x^{\prime}\otimes{\mathbbm{1}})$. Then both operators naturally restrict to ${\mathcal{H}}_{\mathrm{Fer}_{\mathbb{C}}}\otimes{\mathcal{H}}_{\mathrm{Fer}_{\mathbb{C}}}$, and we obtain the claim (cf. [Tan11a, Lemma 5.2, Theorem 5.3]). The second commutation relation for Proposition 2.2 can be proven analogously. ∎ Finally we arrive at a new family of interacting Borchers triples with asymptotic algebra ${{\mathcal{A}}^{(0)}}\otimes{{\mathcal{A}}^{(0)}}$. ###### Theorem 5.2. Let us define * • ${\mathcal{N}}_{\varphi}:=\\{x\otimes{\mathbbm{1}},{\hbox{\rm Ad\,}}S_{\varphi}({\mathbbm{1}}\otimes y):x\in\mathrm{Fer}_{\mathbb{C}}^{{\rm U(1)}}({\mathbb{R}}_{-}),y\in\mathrm{Fer}_{\mathbb{C}}^{{\rm U(1)}}({\mathbb{R}}_{+})\\}^{\prime\prime}$, * • $T(t,x):=T_{0}(\frac{t-x}{\sqrt{2}})\otimes T_{0}(\frac{t+x}{\sqrt{2}})$, * • $\Omega:=\Omega_{0}\otimes\Omega_{0}$. Then the triple $({\mathcal{N}}_{\varphi},T,\Omega)$, restricted to $\overline{{\mathcal{N}}_{\varphi}\Omega}$, is an asymptotically complete, interacting Borchers triple with the asymptotic algebra ${{\mathcal{A}}^{(0)}}\otimes{{\mathcal{A}}^{(0)}}$ and scattering operator $S_{\varphi}\|_{\overline{{\mathcal{N}}_{\varphi}\Omega}}$. It also holds that $\overline{{\mathcal{N}}_{\varphi}\Omega}=\overline{{{\mathcal{A}}^{(0)}}(I_{+})\otimes{{\mathcal{A}}^{(0)}}(I_{-})\Omega}$ for arbitrary intervals $I_{+},I_{-}$. ###### Proof. Substantial arguments are already done: In Lemma 5.1 we constructed Borchers triples with $\mathrm{Fer}_{\mathbb{C}}\otimes\mathrm{Fer}_{\mathbb{C}}$ as the asymptotic algebra. We have seen in Section 3.3 the ${\rm U(1)}$-current net ${{\mathcal{A}}^{(0)}}$ is the fixed point subnet of $\mathrm{Fer}_{\mathbb{C}}$ with respect to the action of ${\rm U(1)}$. From the construction in Section 5.1 and Theorem 4.3, it is easy to see that $S_{\varphi}$ commutes with the product action of the inner symmetries. Then all the statements of the Theorem follow from the general consideration of Proposition 2.5. ∎ ### 5.2 Action of the S-matrix on the 1+1 particle space In this Section we want to analyze the action of the S-matrix of the models constructed in Section 5.1 on the 1+1 particle space ${\mathcal{H}}_{\mathrm{{{\mathcal{A}}^{(0)}}}}^{1}\otimes{\mathcal{H}}_{\mathrm{{{\mathcal{A}}^{(0)}}}}^{1}$, i.e. one left and one right moving particle, where we use the word particle in the sense of Fock space excitations. We note that on the $n$+0 and 0+$n$ particle spaces ${\mathcal{H}}_{n}\otimes{\mathbb{C}}\Omega_{0}$ and ${\mathbb{C}}\Omega_{0}\otimes{\mathcal{H}}_{n}$, respectively, the S-matrix $S$ acts trivially. A typical vector in ${\mathcal{H}}_{\mathrm{{{\mathcal{A}}^{(0)}}}}^{1}\otimes{\mathcal{H}}_{\mathrm{{{\mathcal{A}}^{(0)}}}}^{1}$ is of the form $\Psi:=J(f)\Omega_{0}\otimes J(g)\Omega_{0}$ which we express as the function $\Psi(p,\bar{p})=\hat{f}(p)\hat{g}(\bar{p})$. The embedding $\iota:L^{2}({\mathbb{R}}_{+},p\,\mathrm{d}p)\otimes L^{2}({\mathbb{R}}_{+},\bar{p}\,\mathrm{d}\bar{p})\cong{\mathcal{H}}_{\mathrm{{{\mathcal{A}}^{(0)}}}}^{1}\otimes{\mathcal{H}}_{\mathrm{{{\mathcal{A}}^{(0)}}}}^{1}\hookrightarrow{\mathcal{H}}_{1,1}\otimes{\mathcal{H}}_{1,1}\subset{\mathcal{H}}_{\mathrm{Fer}_{{\mathbb{C}}}}\otimes{\mathcal{H}}_{\mathrm{Fer}_{{\mathbb{C}}}}$ is given by $\iota(\Psi)_{1,1;1,1}(p,q,\bar{p},\bar{q})=\frac{1}{(2\pi)^{2}}\Psi(p+q,\bar{p}+\bar{q})$. We have an analogue of Lemma 4.2 ###### Proposition 5.3. Let $\varphi$ be some inner function. The unitary $S_{\varphi}$ satisfies $S_{\varphi}({\mathcal{H}}_{\mathrm{{{\mathcal{A}}^{(0)}}}}^{1}\otimes{\mathcal{H}}_{\mathrm{{{\mathcal{A}}^{(0)}}}}^{1})\subset{\mathcal{H}}_{\mathrm{{{\mathcal{A}}^{(0)}}}}^{1}\otimes{\mathcal{H}}_{\mathrm{{{\mathcal{A}}^{(0)}}}}^{1}$ if and only if $\varphi(p)=\mathrm{e}^{\mathrm{i}(kp+\theta)}$. ###### Proof. The action of $S_{\varphi}$ on $\Psi\in{\mathcal{H}}_{1}\otimes{\mathcal{H}}_{1}$ is given by $S_{\varphi}\Psi(p+q,\bar{p}+\bar{q})=\varphi(p\cdot\bar{p})\check{\varphi}(q\cdot\bar{p})\check{\varphi}(p\cdot\bar{q})\varphi(q\cdot\bar{q})\Psi(p+q,\bar{p}+\bar{q})$ which is again in ${\mathcal{H}}_{\mathrm{{{\mathcal{A}}^{(0)}}}}^{1}\otimes{\mathcal{H}}_{\mathrm{{{\mathcal{A}}^{(0)}}}}^{1}$ if it can be written as a function $\tilde{\Psi}(p+q,\bar{p}+\bar{q})$, in particular if $\varphi(p\cdot\bar{p})\check{\varphi}(q\cdot\bar{p})\check{\varphi}(p\cdot\bar{q})\varphi(q\cdot\bar{q})=\widetilde{\varphi}(p+q,\bar{p}+\bar{q})$. Setting $\bar{p}=1$ and $\bar{q}=0$, we have $\varphi(p)\check{\varphi}(q)=\widetilde{\varphi}(p+q,1)$. The rest follows as Lemma 4.2. ∎ ###### Remark 5.4. In the case $\varphi(p)=\mathrm{e}^{\mathrm{i}\kappa p}$, one gets the models obtained in [DT11] using warped convolution. ###### Proposition 5.5. Let $e$ be the projection on ${\mathcal{H}}_{\mathrm{{{\mathcal{A}}^{(0)}}}}^{1}\otimes{\mathcal{H}}_{\mathrm{{{\mathcal{A}}^{(0)}}}}^{1}$, then $eS_{\varphi}e=\tilde{\varphi}(P\otimes P)$, where $\tilde{\varphi}$ is boundary value of an analytic function in $\mathbb{H}$ with $\|\tilde{\varphi}(p)\|\leq 1$ and $P$ is the generator of translation restricted to the one-particle space (which gives rise the irreducible standard pair). ###### Proof. It can be checked that $(e_{0}f)(p,q)=\frac{1}{p+q}\int_{0}^{p+q}f(p+q-x,x)\,\mathrm{d}x$ is the projection on ${\mathcal{H}}_{\mathrm{{{\mathcal{A}}^{(0)}}}}^{1}\subset{\mathcal{H}}_{1,1}$. Then the action of $eS_{\varphi}$ on a $f\in{\mathcal{H}}_{\mathrm{{{\mathcal{A}}^{(0)}}}}^{1}\otimes{\mathcal{H}}_{\mathrm{{{\mathcal{A}}^{(0)}}}}^{1}$ can be calculated to be $\varphi^{\prime}(P\otimes 1,1\otimes P)$ with $\varphi^{\prime}(p,q)=\frac{1}{p\cdot q}\int_{0}^{p}\int_{0}^{q}\varphi((p-x)\cdot(q-y))\varphi(x\cdot y)\check{\varphi}((p-x)\cdot y)\check{\varphi}(x\cdot(q-y))\,\mathrm{d}y\,\mathrm{d}x$ and it is easy to check that with $\tilde{\varphi}(p):=\varphi^{\prime}(p,1)$ it holds $\varphi^{\prime}(p,q)=\tilde{\varphi}(p\cdot q)$ for all $p,q>0$. That $\|\tilde{\varphi}(p)\|\leq 1$ can be checked directly or follows from the fact that $S_{\varphi}$ is unitary. ∎ ###### Remark 5.6. It is a general feature of asymptotically complete Borchers triples with asymptotic algebra ${{\mathcal{A}}^{(0)}}\otimes{{\mathcal{A}}^{(0)}}$ that the restriction of the scattering matrix $S$ to $eSe$ is a functional calculus of $P\otimes P$. Indeed, both $e$ and $S$ commute with the translation $T$, but $T$ is maximally abelian when restricted to ${\mathcal{H}}_{\mathrm{{{\mathcal{A}}^{(0)}}}}^{1}\otimes{\mathcal{H}}_{\mathrm{{{\mathcal{A}}^{(0)}}}}^{1}$, hence there is a function $\varphi_{S}$ such that $eSe=\varphi_{S}(P\otimes{\mathbbm{1}},{\mathbbm{1}}\otimes P)$. Furthermore, both $e$ and $S$ commute with boosts, so does $\varphi_{S}$ and one obtains the form $eSe=\varphi_{S}^{\prime}(P\otimes P)$. We note that the proof above shows that $\|\tilde{\varphi}(M^{2}/2)\|$ is the probability that an improper state in ${\mathcal{H}}^{1}_{{\mathcal{A}}^{(0)}}\otimes{\mathcal{H}}^{1}_{{\mathcal{A}}^{(0)}}$ with mass $M^{2}$ is scattered elastically in the sense of Fock space particles, where $\tilde{\varphi}(p)=\frac{1}{p}\int_{0}^{p}\int_{0}^{1}\varphi((p-x)(1-y))\varphi(xy)\check{\varphi}((p-x)y)\check{\varphi}(x(1-y))\,\mathrm{d}y\,\mathrm{d}x\,.$ As we discussed in 3.1, the Hilbert space of the ${\rm U(1)}$-current net, and hence the tensor product of two copies of it, admit the bosonic Fock space structure, hence we can consider the particle number. Although we admit that this concept does not have an intrinsic meaning, we claim that it is possible to interpret this as the number of massless particles. An evidence comes from the comparison with massive cases. In [Lec08] Lechner has constructed a family of massive interacting models parametrized by so- called scattering functions, and later he reinterpreted them as deformations of the massive free field [Lec11]. If one applies the same deformation procedure to the derivative of the massless free field whose net is ${{\mathcal{A}}^{(0)}}\otimes{{\mathcal{A}}^{(0)}}$ (with scattering functions satisfying $S_{2}(0)=1$), he obtains the Borchers triples with ${{\mathcal{A}}^{(0)}}\otimes{{\mathcal{A}}^{(0)}}$ as the asymptotic net constructed in [Tan11a] 222Private communication with Gandalf Lechner and Jan Schlemmer. This will be presented elsewhere.. Hence the models in [Tan11a] should be considered as the massless versions of the models in [Lec08]. Likewise, it can be said that the models constructed in the present paper are the deformed (in an appropriate sense) version of the massless free field. In massive case, there is a mass gap in the spectrum of the spacetime translation and the one-particle space of the Fock space has an intrinsic meaning. In massless case, such an intrinsic interpretation is lost but there is still the Fock space structure. Thus we think that, if the two-particle space in the Fock structure is not preserved by the S-matrix, as in the case where $\varphi$ is not exponential (see Proposition 5.3), then it represents massless particle production. ## 6 Conclusion and outlook In this paper we have constructed a new family of Longo-Witten endomorphisms on ${{\mathcal{A}}^{(0)}}$ through the inclusion ${{\mathcal{A}}^{(0)}}=\mathrm{Fer}_{\mathbb{C}}^{\rm U(1)}\subset\mathrm{Fer}_{\mathbb{C}}$. We combined them to construct interacting wedge-local nets with ${{\mathcal{A}}^{(0)}}\otimes{{\mathcal{A}}^{(0)}}$ as the asymptotic algebra and showed that their S-matrices do not preserve the $n$-particle space of the bosonic Fock space. Particle production is a necessary feature of interacting models in higher dimensions [Aks65], thus this result opens up some hope for algebraic construction of higher dimensional interacting models. However, there are at least two shortcomings with the present method. The first is that we proved only wedge-locality of the models. As already shown in [Tan11a], a wedge-local net can be dilation-covariant and at the same time interacting. On the other hand, a strictly local dilation-covariant (asymptotically complete) net is necessarily not interacting [Tan11b]. Hence, interaction of wedge-local nets could be just a false-positive and strict locality is desired. The second is the fact that the concept of particle in massless case is not intrinsically defined. Although the Fock space structure is easily understood, its interpretations should be treated with care. These issues could be overcome by considering massive cases. As for strict locality, it has been shown that the deformation of the massive free field by a suitably regular function is again strictly local [Lec08, Lec11]. On the other hand, in massless situation, even the simplest case $\varphi(p)=-1$ (where $\varphi$ is an inner symmetric function used in [Tan11a] to deform directly ${{\mathcal{A}}^{(0)}}\otimes{{\mathcal{A}}^{(0)}}$) is already not strictly local [Tan11a]. Hence we believe that strict locality should be addressed in massive models. Furthermore, for a massive asymptotically complete model, the notion of particle production is intrinsic. Fortunately, it is known that the construction in [Tan11a] coincides with the deformation of the massive free field as we remarked in the last section, hence a further correspondence between massive and massless cases are expected. We hope to investigate this problem in a future publication. Of course, interacting models in higher dimensions are always one of the most important issues. Although conformal nets themselves are not interacting [BF77], some new constructions based on CFT could be possible and ideas from the present article could be useful. #### Acknowledgment. We thank our supervisor Roberto Longo for his constant support and useful suggestions. Y. T. thanks Gandalf Lechner and Jan Schlemmer for discussions on the relation between the present construction and the deformation of [Lec11]. ## References * [Aks65] S. O. Aks. Proof that scattering implies production in quantum field theory. J. Mathematical Phys., 6:516–532, 1965. * [Apo76] T. M. Apostol. Introduction to analytic number theory. Springer-Verlag, New York, 1976. * [BF77] D. Buchholz and K. Fredenhagen. Dilations and interaction. J. Math. Phys., 18(5):1107–1111, 1977. * [BLS11] D. Buchholz, G. Lechner, and S.J. Summers. Warped convolutions, Rieffel deformations and the construction of quantum field theories. Commun. Math. Phys., 304:95–123, 2011. * [Bor92] H.-J. Borchers. The CPT-theorem in two-dimensional theories of local observables. Comm. Math. Phys., 143(2):315–332, 1992. * [BS08] D. Buchholz and S.J. Summers. Warped convolutions: a novel tool in the construction of quantum field theories. In Quantum field theory and beyond, pages 107–121. World Sci. Publ., Hackensack, NJ, 2008. * [BSM90] D. Buchholz and H. Schulz-Mirbach. Haag duality in conformal quantum field theory. Rev. Math. Phys., 2(1):105–125, 1990. * [Buc75] D. Buchholz. Collision theory for waves in two dimensions and a characterization of models with trivial $S$-matrix. Comm. Math. Phys., 45(1):1–8, 1975. * [CKL08] S. Carpi, Y. Kawahigashi, and R. Longo. Structure and classification of superconformal nets. Ann. Henri Poincaré, 9(6):1069–1121, 2008. * [DF77] W. Driessler and J. Fröhlich. The reconstruction of local observable algebras from the euclidean green’s functions of relativistic quantum field theory. Annales de L’Institut Henri Poincare Section Physique Theorique, 27:221–236, 1977. * [DT11] W. Dybalski and Y. Tanimoto. Asymptotic completeness in a class of massless relativistic quantum field theories. Comm. Math. Phys., 305:427–440, 2011. * [GL07] H. Grosse and G. Lechner. Wedge-local quantum fields and noncommutative Minkowski space. J. High Energy Phys., (11):012, 26, 2007. * [GL08] H. Grosse and G. Lechner. Noncommutative deformations of Wightman quantum field theories. J. High Energy Phys., (9):131, 29, 2008. * [Kac98] V. G. Kac. Vertex algebras for beginners. American Mathematical Society, 1998. * [KR87] V. G. Kac and A. K. Raina. Bombay lectures on highest weight representations of infinite-dimensional Lie algebras, volume 2 of Advanced Series in Mathematical Physics. World Scientific Publishing Co. Inc., Teaneck, NJ, 1987. * [Lec08] G. Lechner. Construction of quantum field theories with factorizing $S$-matrices. Comm. Math. Phys., 277(3):821–860, 2008. * [Lec11] G. Lechner. Deformations of quantum field theories and integrable models. Commun. Math. Phys., 212:265–302, 2012. * [Lon08] R. Longo. Real Hilbert subspaces, modular theory, ${\rm SL}(2,{\bf R})$ and CFT. In Von Neumann algebas in Sibiu: Conference Proceedings, pages 33–91. Theta, Bucharest, 2008. * [LW11] R. Longo and E. Witten. An algebraic construction of boundary quantum field theory. Commun. Math. Phys., 303:213–232, 2011. * [Reh98] K.-H. Rehren. Konforme quantenfeldtheorie. Lecture note available at http://www.theorie.physik.uni-goettingen.de/~rehren/ps/cqft.pdf * [RS75] M. Reed and B. Simon. Methods of modern mathematical physics. II. Fourier analysis, self-adjointness. Academic Press [Harcourt Brace Jovanovich Publishers], New York, 1975\. * [Tak03] M. Takesaki. Theory of operator algebras. II, volume 125 of Encyclopaedia of Mathematical Sciences. Springer-Verlag, Berlin, 2003. * [Tan11a] Y. Tanimoto. Construction of wedge-local nets of observables through longo-witten endomorphisms. Commun. Math. Phys., 314(2):443–469, 2012. * [Tan11b] Y. Tanimoto. Noninteraction of waves in two-dimensional conformal field theory. Commun. Math. Phys., 314(2):419–441, 2012. * [Was98] A. Wassermann. Operator algebras and conformal field theory. III. Fusion of positive energy representations of ${\rm LSU}(N)$ using bounded operators. Invent. Math., 133(3):467–538, 1998.	arxiv-papers	2011-11-07T18:35:11	2024-09-04T02:49:24.082565	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/", "authors": "Marcel Bischoff and Yoh Tanimoto", "submitter": "Yoh Tanimoto", "url": "https://arxiv.org/abs/1111.1671" }
1111.1805	# Repulsive and attractive Casimir interactions in liquids Anh D. Phan University of South Florida, Tampa, USA anhphan@mail.usf.edu N. A. Viet Institute of Physics, Hanoi, Vietnam ###### Abstract The Casimir interactions in the solid-liquid-solid systems as a function of separation distance have been studied by the Lifshitz theory. The dielectric permittivity functions for a wide range of materials are described by Drude, Drude-Lorentz and oscillator models. We find that the Casimir forces between gold and silica or MgO materials are both the repulsive and attractive. We also find the stable forms for the systems. Our studies would provide a good guidance for the future experimental studies on the dispersion interactions. ###### pacs: Valid PACS appear here ## I Introduction The dispersion interactions, the Casimir force, between neutral objects have brought attraction for many years. There are a lot of factors affecting on the value of force, such as geometry and material properties. Each of them gives rise to hot subjects of ongoing investigation. Some experiments have examined the influence of the dielectric properties of objects on the Casimir force 1 ; 2 ; 3 ; 6 . A number of settings used to study the interaction in terms of theory are ideal metals, real metals and semiconductors 3 ; 4 ; 6 , metamaterials, and two objects placed in liquids 1 ; 2 ; 5 . These studies have significantly advanced our understanding of the subtle effect of geometry and material on the Casimir-Lifshitz interactions, especially for designing nanodevices and nanotechnologies. In the Lifshitz theory, the dispersion interactions primarily depend on dielectric permittivity functions of materials. Changing dielectric function alters the Casimir interactions. There are some ways to modify dielectric functions, including illuminating a light on the silicon 7 ; 8 , which make drifting carriers on semiconductor materials. In principle, there are some models to describe dielectric response functions of real materials, for example, plasma and Drude models for metals 6 ; 9 ; 11 , Drude-Lorentz and oscillator models for liquids 2 ; 11 , oxides and others 10 ; 11 ; 12 . Based on these models, the Casimir forces were obtained by numerical integrations and series expansion methods 19 . It has been theoretically shown that the attractive Casimir interaction always occurs between two (non-magnetic) dielectric bodies related by reflection. Therefore, the repulsive force is a striking feature creating inspiration for scientists to make accurate measurements of nano electromechanical machines where the repulsive force plays an important role and might resolve the stiction problems. The repulsive Casimir forces can be observed in systems which have the presence of liquids 2 , metamaterials and metallic geometries 16 . Recent experiments have pointed out that there a repulsive force exists between a gold sphere and a silica plate, separated by bromobenzen 2 . As a matter of fact, the repulsive Casimir forces between solids arise when the dielectric of material surfaces 1 and 2 and an intervening liquid obey the relation $\varepsilon_{1}(i\xi)>\varepsilon_{liquid}(i\xi)>\varepsilon_{2}(i\xi)$ over a wide imaginary frequency range $\xi$. A previous theoretical 14 study has noticed that it is difficult to establish an equilibrium configuration of sytems in a vacuum medium. In the reference 15 , the authors showed that they were able to form some stable configurations of Teflon-Si and Silica-Si immersed in ethanol. The equilibrium is explicitly explained by dispersion properties. In the present work, our theoretical studies have shown that the equilibra can be obtained by placing Au-MgO, Silica-MgO and Au-Silica sytems in bromobenzen. In this paper, the Casimir-Lifshitz forces between material plate systems made in oxides and metals immersed in bromobenzen are calculated. The combination between these results and the proximity force approximation (PFA) method allow us to compute the Casimir interactions in different configurations. We find that the magnitude of the Casimir force between two dielectric bodies depends on the configuration and distance between two bodies. The shape usually used in experiments is a combination of a sphere and a plate because one can avoid the problem of alignment and easily control the distance between them. The energy interactions between a plate-plate system per unit area can be obtained by using the relationship between the Casimir energy of two plannar objects and the dispersion force of a sphere-plate system. The rest of the paper is organized as follows: In Sec. II the theoretical formulations of Casimir-Lifshitz force interaction are introduced. In section III, the numerical results for the Casimir force between two bodies are presented. Important conclusions and discussions are finally given in section IV. ## II Lifshitz theory for force calculations For the force calculations, we used Lifshitz theory without considering effect of temperature. The separations used here were less than 1 $\mu$m, therefore thermal corrections at $T=300K$ are not significant. As previously noted in 2 ; 3 ; 20 ; 21 , the Lifshitz formula at zero temperature for the Casimir force acting between between two parallel flat bodies per unit area, separated by a distance $d$ are given by $\displaystyle F(d)$ $\displaystyle=$ $\displaystyle-\dfrac{\hbar}{2\pi^{2}}\int_{0}^{\infty}qk_{\perp}dk_{\perp}\int_{0}^{\infty}d\xi$ (1) $\displaystyle\times\left(\dfrac{r_{TM}^{(1)}r_{TM}^{(2)}}{e^{2qd}-r_{TM}^{(1)}r_{TM}^{(2)}}+\dfrac{r_{TE}^{(1)}r_{TE}^{(2)}}{e^{2qd}-r_{TE}^{(1)}r_{TE}^{(2)}}\right).$ Here the reflection coefficients $r_{TM,TE}^{(1)}$ and $r_{TM,TE}^{(2)}$ for two independent polarizations of the electromagnetic field (transverse magnetic and transverse electric fields) are $\displaystyle r_{TM}^{(p)}=r_{TM}^{(p)}\left({\xi,k_{\bot}}\right)=\frac{{\varepsilon^{(p)}\left({i\xi}\right)q-\varepsilon^{(2)}\left({i\xi}\right)k^{(p)}}}{{\varepsilon^{(p)}\left({i\xi}\right)q+\varepsilon^{(2)}\left({i\xi}\right)k^{(p)}}},$ (2) $\displaystyle r_{TE}^{(p)}=r_{TE}^{(p)}\left({\xi,k_{\bot}}\right)=\frac{{\mu^{(2)}(i\xi)k^{(p)}-\mu^{(p)}(i\xi)q}}{{\mu^{(2)}(i\xi)k^{(p)}+\mu^{(p)}(i\xi)q}},$ (3) where $\displaystyle q=\sqrt{k_{\bot}^{2}+\varepsilon^{(2)}\left({i\xi}\right)\mu^{(2)}\left({i\xi}\right)\frac{{\xi^{2}}}{{c^{2}}}},$ (4) $\displaystyle k^{(p)}=\sqrt{k_{\bot}^{2}+\varepsilon^{(p)}\left({i\xi}\right)\mu^{(p)}\left({i\xi}\right)\frac{{\xi^{2}}}{{c^{2}}}}.$ (5) in which $\varepsilon^{(p)}\left({i\xi}\right)$ and $\mu^{(p)}\left({i\xi}\right)$ are the dielectric permittivity and the magnetic permeability of the first body (p = 1) and the second body (p=3), respectively. $\varepsilon^{(2)}\left(\omega\right)$ and $\mu^{(2)}\left({i\xi}\right)$ are the dielectric function and the permeability of a liquid filled between two bodies. Here, medium ‘2’ selected is a bromobenzen so $\mu^{(2)}\left({i\xi}\right)=1$. Moreover, in this paper, the non-magnetic materials used such as germanium, gold and oxides have also $\mu^{(p)}\left({i\xi}\right)=1$. $k_{\bot}$ magnitude of the wave vector component perpendicular on the plate, is frequency variable along the imaginary axis ($\omega=i\xi$). We recall that Lifshitz formula, routinely used to interpret current experiments, express the Casimir force between two parallel plates as an integral over imaginary frequencies $i\xi$ of a quantity involving the dielectric permittivities of the plates $\omega=i\xi$. It is important to note that, in principle, recourse to imaginary frequencies is not mandatory because it is possible to rewrite Lifshitz formula in a mathematically equivalent form, involving an integral over the real frequency axis. In this case, however, the integrand becomes a rapidly oscillating function of the frequency, which hampers any possibility of numerical evaluation. Another remarkable point is that occurrence of imaginary frequencies in the expression of the Casimir force is a general feature of all recent formalisms hence extending Lifshitz theory to non-planar geometries 17 ; 18 . The problem is that the electric permittivity $\varepsilon(i\xi)$ at imaginary frequencies cannot be measured directly by any experiment. The only way to determine it by means of dispersion relations, which allow the expression of $\varepsilon(i\xi)$ in terms of the observable real-frequency electric permittivity $\varepsilon(i\xi)$. In the standard works on the Casimir effect, $\varepsilon(i\xi)$ is expressed with the Kramers-Kronig relation in terms of an integral of a quantity involving the imaginary part of the electric permittivity 13 $\displaystyle\varepsilon(i\xi)=1+\frac{2}{\pi}\int\limits_{0}^{\infty}{d\omega\frac{{\omega{\mathop{\rm Im}\nolimits}\varepsilon(\omega)}}{{\omega^{2}+\xi^{2}}}},$ (6) where ${\mathop{\rm Im}\nolimits}\varepsilon(\omega)$ is calculated using the tabulated optical data for the complex index of refraction. The well-known dielectric function described for gold is the Drude model 13 $\displaystyle\varepsilon(i\xi)=1+\frac{{\omega_{p}^{2}}}{{\xi(\xi+\gamma)}},$ (7) where $\omega_{p}=9.0$ eV, $\gamma=0.035$ eV are the plasma frequency and the relaxation parameter of Au, respectively. The imaginary part of the resulting dielectric function at 6 and 295 K of pure MgO are shown in 12 . The optical features have been fitted to a classical oscillator model using the complex dielectric function $\displaystyle\varepsilon(\omega)=\varepsilon_{\infty}+\sum\limits_{j}{\frac{{\omega_{p,j}^{2}}}{{\omega_{TO,j}^{2}-\omega^{2}-i2\omega\gamma_{i}}}},$ (8) where $\varepsilon_{\infty}$ is a high-frequency contribution, and $\omega_{TO,j},{2\gamma_{i}}$ and $\omega_{p,j}$ are the frequency, full width and effective plasma frequency of the $j$th vibration. The values of these parameters can be found in 12 . Of course with such simple model for the permittivity of MgO, there is no need to use dispersion relations to obtain the expression of $\varepsilon(i\xi)$, for this can be simply done by the substitution $\omega\to i\xi$ in the r.h.s of Eq.(8) 24 $\displaystyle\varepsilon(i\xi)=\varepsilon_{\infty}+\sum\limits_{j}{\frac{{\omega_{p,j}^{2}}}{{\omega_{TO,j}^{2}+\xi^{2}+2\xi\gamma_{j}}}}.$ (9) In the case of bromobenzen and silica, it has recently been used for measurement of repulsive forces between gold and silica surfaces. Extremely weak repulsion was measured, indicating that the dielectric functions of bromobenzen and silica are very similar in magnitude. In fact, oscillator models are constructed to represent the dielectric function at imaginary frequencies. The form of the oscillator model is given by $\displaystyle\varepsilon(i\xi)=1+\sum_{i}{\frac{{C_{i}}}{{1+\xi^{2}/\omega_{i}^{2}}}},$ (10) where the coefficients $C_{i}$ are the oscillator’s strengths corresponding to (resonance) frequencies $\omega_{i}$ 1 ; 2 ; 25 . The dielectric data was fitted in a wide frequency range 2 . They are more accurate in comparison with other simple oscillator models. Moreover, many older references used limited dielectric data, so the oscillator models with second or third order may lead to the difference in Casimir force calculations. The parameters we used in the present paper for bromobenzen and also silica come from 2 . ## III Numerical results and discussions The Casimir attractive force usually occurs in experiments and theoretical calculations. When bromobenzen is filled in the gap between two bodies, the Casimir force is attractive if the dielectric functions do not satisfy one condition $\varepsilon_{1}(i\xi)>\varepsilon_{liquid}(i\xi)>\varepsilon_{2}(i\xi)$ for all frequencies $\xi$. Therefore, by describing Fig. 1 as the dielectric response function as a function of the frequency gives us some predictions of repulsive and attractive forces. Figure 1: (Color online) The dielectric function of various materials plotted at imaginary frequencies $\xi$. This graph shows that $\varepsilon_{Au}(i\xi)>\varepsilon_{MgO}(i\xi)>\varepsilon_{liquid}(i\xi)$ and $\varepsilon_{MgO}(i\xi)>\varepsilon_{Au}(i\xi)>\varepsilon_{liquid}(i\xi)$ at $\xi<6.5$ eV, thus the interactions between Au and MgO body immersed in bromobenzen liquid and in vacuum are attractive in this range. In the range of $\xi>6.5$ eV, $\varepsilon_{MgO}(i\xi)>\varepsilon_{liquid}(i\xi)>\varepsilon_{Au}(i\xi)$, it causes the repulsive interaction. Similarly, in the gold-bromobenzen-silica system, at extremely small frequencies , the forces are attractive. In the larger frequency region, the Casimir forces are repulsive. Besides, the similar explainations are applied to understand the interaction in the MgO- bromobenzen-Au system. The numerical calculations of the normalized Casimir force are provided in Fig. 2 Figure 2: (Color online) Relative Casimir force between two semi-infinite plates normalized by the perfect metal force $F_{o}(d)=-\pi^{2}\hbar c/240d^{4}$. The liquid used in this calculation is bromobenzen. In the MgO-bromobenzen-Silica system, it can be clearly seen that there are two positions in each curve where the Casimir force is equal to zero. The first points are corresponding to unstable equilibria $d_{us}^{(1)}\approx 13$ nm because the interaction force changes from the attactive force to the repulsive force, the second point $d_{s}^{(1)}\approx 110$ nm is a stable position. There is only one position in the Au-bromobenzen-Silica system and the Au-bromobenzen-MgO sytem, The interaction forces disappear at $d_{s}^{(2)}\approx 275$ nm and $d_{s}^{(3)}\approx 5.5$ nm, stable position of each sytem, respectively. Figure 3: (Color online) Schematic picture of the setting considered in our calculations. A sphere is located in bromobenzen at a distance $d$ away from a material plate. In order to consider the Casimir interactions between a spherical body and a plate at a distance of close approach $d$ at a temperature T = 300 K, it is very useful to utilize the PFA method to calculate. Experimental results for the Casimir force in the plane-sphere geometry are usually compared with PFA- based theoretical models. The spherical surface is assumed to be nearly flat over the scale of $d$. Although the Casimir force is not additive, PFA is often expected to provide an accurate description when $R\gg d$. Here, the radius of Au sphere that is used in configurations is $R=40$ $\mu$m in order to calculate Casimir interaction by the proximity force approximation (PFA) method because the ratio of $d$ to $R$ is small enough to PFA results becoming enormously accurate. It can be described by Fig. 3. In this approach, the surfaces of the bodies are treated as a superposition of infinitesimal parallel plates 22 . $\displaystyle F_{sp}^{PFA}(d)=\int\limits_{0}^{R}{F_{pp}(d+R-\sqrt{R^{2}-r^{2}})}2\pi rdr.$ (11) here ${F_{pp}}$ is the Casimir force for two parallel plates of unit area. When using the PFA method, one important point is that the interactions between a gold sphere or a magnesium oxide sphere and a silica plate are equal to the interactions between a magnesium oxide plate, which has the same radius, and a gold plate or a silica plate. There is no difference in calculations and results as well because the PFA method does not consider a structure of bodies when their shape is modified or is spherical or cylindrical shape. The equivalent situations occur in other materials. The resulting Casimir forces are shown in Fig. 4. Figure 4: (Color online) The Casimir forces of various sphere-bromobenzen- plate systems are estimated as a function of separation, described in the text with the spherical redius $R=40$ $\mu$m. In the reference 2 , the authors experimentally measured and theoretically calculated the Casimir interaction between a gold sphere and a silica plate immersed in bromobenzen in the range from $20$ nm to $60$ nm. Our results in this range are the same for this range. But when we extend the considered range of distance, the attractive-repulsive transition occurs at approximately $190$ nm. This position makes this system stable. Another consequence of Fig. 4 demonstrates that stable position of the Au-bromobenzen-MgO sytem moves to $3.5$ nm to balance between the attractive and repulsive forces. It can be explained that increasing the separation distance of infinitesimal parallel plates causes the fast reduction of the dispersion interaction. At the same minimal separation distance $d$, the attractive force acting on a sphere is less than that of a plate in the same effective area. Finally, in the system of a MgO sphere and a silica plate embeded in bromobenzen, there is only one presence of non-interaction posion at nearly $10$ nm. It is an unstable position. Figure 5: (Color online) The Casimir energy is calculated as a function of separation for different materials . In addition, PFA formula and Eq.(11) can allow us to estimate the Casimir energy per unit area between two plate bodies illustrated in Fig. 5. The Casimir energy is approximated by 22 $\displaystyle F_{sp}^{PFA}(d)=2\pi RE(d),$ (12) where $E(d)$ is the Casimir energy per a unit area for planar bodies. We have also applied the PFA method to calculate the Casimir force in sphere- sphere systems, we continue to calculate by the PFA method. The formula for this calculation is given $\displaystyle F_{ss}^{PFA}(d)=2\pi\int\limits_{0}^{R_{2}}rdr\times$ $\displaystyle F_{pp}(d+R_{1}-\sqrt{R_{1}^{2}-r^{2}}+R_{2}-\sqrt{R_{2}^{2}-r^{2}}),$ (13) where the radii of two spherical objects are ${R_{1}}$ and ${R_{2}}$, respectively. It is assumed that $R_{2}<R_{1}$. In this study, we consider $R_{1}=40$ $\mu$m and the case of $R_{2}=R_{1}$, $R_{1}=2R_{2}$ and $R_{1}=2R_{2}$. Here, having calculated $F_{ss}^{PFA}(d)$ in a sphere-sphere system using Eq.(13) and $F_{sp}^{PFA}(d)$ in a sphere-plate system using Eq.(11). These results obtained show that, when increasing $d$, the ratio $F_{ss}^{PFA}(d)/F_{sp}^{PFA}(d)$ does not depend on the distance $d$. It is a constant with its magnitude as a function of the radius of two spheres, $F_{ss}^{PFA}(d)/F_{sp}^{PFA}(d)=1/2$ when $R_{1}=R_{2}$ and $F_{ss}^{PFA}(d)/F_{sp}^{PFA}(d)=1/3$ when $R_{1}=2R_{2}$. Generalizing this ratio, if $R_{1}=nR_{2}$, we have $F_{ss}^{PFA}(d)/F_{sp}^{PFA}(d)=1/(n+1)$. This character is likely to be explained by the results in 22 . When the second sphere is extremely small in comparison with the first one, the interaction force goes to zero. In this case, the Lifshitz formula used to calculate the Casimir force should be transformed to the Casimir-Polder formula describing the interaction between an atom and a microscopic object. Moreover, the PFA method is not accurate because this approach is useful if the size of objects is much larger than the separation between them. On the other hand, we have $\displaystyle F_{ss}^{PFA}(d)=\frac{1}{n+1}F_{sp}^{PFA},$ (14) where $R_{1}=nR_{2}$. If $F_{sp}^{PFA}=0$, $F_{ss}^{PFA}$ must be zero. Therefore, the unstable and stable positions are constant and unchanged when a radius of a second sphere varies. One demonstrated that the Casimir force between two objects embedded in liquids can be derived from the well-known Lifshitz formula at least if the object is not made of nonabsorbing materials 5 . That explains why the Lifshitz expressions is used in order to calculate the Casimir force and compare with experimental data. Nothing changes in the dielectric functions of bodies immersed in liquids. On the other hand, several experiments verified that when metals are placed in liquids, there is a variation of Drude parameters in the metal 23 . The discrepacy of the interaction between “dry” and “wet” can reach to 15 $\%$ in this case. But they measured the Casimir force between two metal plates and got the error. Besides, maybe the dielectric of liquids and low index materials play much more important role in Casimir force. In the reference 2 , the change of Drude parameters are not taken into account but the theoretical calculations are close to the experimental data curves when we have liquids and the low index materials. ## IV Conclusions In this work, we have extended the Lifshitz theory to calculate the Casimir force. Liquid, silica and magnesium oxide are represented by oscillator models. Although further studies are required to determine the repulsive Casimir force accurately, our results show that MgO and silica is a good candidate for the demonstration of quantum levitation. The contribution of bromobenzen is important because it is an important factor making the purely repulsive force or the repulsive-attractive transition. After calculating the Casimir force between two bodies per unit area and associating proximity force approximation method, it is easy to compute the interaction of different material plates with a material sphere. Based on the formula of the Casimir force between a sphere and a plate, it is convenient to estimate the free energy interaction of bodies. The result is a prediction for further experimental studies. ###### Acknowledgements. This work was supported by the Nafosted Grant No. 103.02.57.09. We thank Prof. Lilia M. Woods and Prof. P. J. van Zwol for helpful discussions and comments. ## References * (1) J. N. Munday, F. Capasso, and V. A. Parsegian, Nature 457, 170 (2009). * (2) P. J. van Zwol, and G. Palasantzas, Phys. Rev. A 81, 062502 (2010). * (3) F. Chen, U. Mohideen, G. L. Klimchitskaya, and V. M. Mostepanenko, Phys. Rev. A 74, 022103 (2006). * (4) I. Pirozhenko, and A. Lambrecht, Phys. Rev. A 77, 013811 (2008). * (5) Christian Raabe, and Drik-Gunnar Welsch, Phys. Rev. A 71, 013811 (2005). * (6) V.B. Svetovoy, P.J. van Zwol, G. Palasantzas, and J. Th. M. De Hosson, Phys. Rev. B 77, 035439 (2008). * (7) Norio INUI, J. Phys. Soc. Jpn. 75, 024004 (2006). * (8) F. Chen, G.L. Klimchitskaya, V.M. Mostepanenko, and U. Mohideen, Phys. Rev. B 76, 035338 (2007). * (9) R. Esquivel-Sirvent, Phys. Rev. A 77, 042107 (2008). * (10) R. Castillo-Garza,C.-C. Chang, D. Jimenez, G. L. Klimchitskaya, V. M. Mostepanenko, and U. Mohideen, Phys. Rev. A 75, 062114 (2007). * (11) I. Pirozhenko, and A. Lambrecht, Phys. Rev. A 77, 013811 (2008). * (12) Tao Sun, Philip B. Allen, David G. Stahnke, Steven D. Jacobsen, and Christopher C. Homes, Phys. Rev. B 77, 134303 (2008). * (13) Giuseppe Bimonte, Phys. Rev. A 83, 042109 (2011). * (14) S. J. Rahi, M. Kardar, and T. Emig, Phys. Rev. Lett. 105, 070404 (2010). * (15) Alejandro W. Rodriguez, Alexander P. McCauley, David Woolf, F. Capasso, J. D. Joannopoulos, and Steven G. Johnson, Phys. Rev. Lett. 104, 160402 (2010). * (16) R. Zhao, J. Zhou1, Th. Koschny, E. N. Economou, and C. M. Soukoulis, Phys. Rev. Lett. 103, 103602 (2009). * (17) T. Emig, N. Graham, R. L. Jaffe, and M. Kardar, Phys. Rev. Lett. 99, 170403 (2007); Phys. Rev. D 77, 025005, (2008). * (18) O. Kenneth, and I. Klich, Phys. Rev. B 78, 014103 (2008). * (19) R. S. Decca, E. Fischbach, G. L. Klimchitskaya, D. E. Krause, D. L ̵́ pez, and V. M. Mostepanenko, Phys. Rev. A 84, 042502 (2011). * (20) J. N. Munday, F. Capasso, V. A. Parsegian, and S. M. Bezrukov, Phys. Rev. A 78, 032109 (2008). * (21) P. J. van Zwol, G. Palasantzas, and J. Th. M. De Hosson, Phys. Rev. B 79, 195428 (2009). * (22) Bo. E. Sernelius, and C. E. Romn-Velzquez, Phys. Rev. A 78, 032111 (2008). * (23) R. Esquivel-Sirvent, J. Chem. Phys 132, 194707 (2010). * (24) Giuseppe Bimonte, Phys. Rev. A 81, 062501 (2010). * (25) I. Pirozhenko, and A. Lambrecht, Phys. Rev. A 80, 042510 (2009).	arxiv-papers	2011-11-08T05:11:39	2024-09-04T02:49:24.100820	{ "license": "Public Domain", "authors": "Anh D. Phan, N. A. Viet", "submitter": "Anh Phan", "url": "https://arxiv.org/abs/1111.1805" }
1111.1810	Harmonic analysis of the functions $\tilde{\Delta}(x)$ and $N(T)$ Jining Gao Department of Mathematics, Shanghai Jiaotong University, Shanghai ,P. R. China In this paper, under the Riemann hypothesis, we study the Fourier analysis about the functions $\tilde{\Delta}(x)$ and $N(T)$ . ## 1 INTRODUCTION Riemann hypothesis has been studied in many different ways, in this paper, we will try to use somewhat new angles to study RH. Most results of this paper are obtained under the RH. As we know, Guinand formula is a representation of $N(T)$ which is the distribution function of Riemann zeros in term of series of the prime number powers, although Guinand formula [2]is a result under the assumption of RH,it provides an explicit method to figure out all non trivial Riemann zeros. Actually, this fact is far from trivial because once we have prime number representation of $N(T)$ (Guinand formula) at hand, we can immediately restore a function via the distribution of it’s zeros,so Guinand formula is equivalent to RH and we will prove it in the first section. Since Guinand formula is very important in this paper and Guinand original proof is complicated and full of the favor of harmonic analysis, we will first of all give another simple and elementary proof based on the lemma, which is the ground stone of this paper, besides ,our new proof gives out a stronger conclusion than the original statement of Guinand formula. This stronger result will help us to check the truth of RH much more efficiently. In the second section we rewrite Guinand formula and Riemann-Mangoldt formula as two integral equations of two ”functional variables” $\tilde{\Delta}(x)$ and $S(T)$, which seems to imply Guinand formula and Riemann-Mangoldt formula are reciprocal to each other and such integral representation will be used in the 4th section. In the third section ,first of all, we derive an elementary formula based on functional equation of Riemann zeta function and lemma. This formula provides infinitely many non trivial integral equations of $N(T)$, also, we use the elementary formula to prove a theorem which claims $\mid\tilde{\Delta}(x)\mid$ has a non-zero measurement of a positive lower bound. ## 2 Guinand formula with an error term and it’s inverse theorem In this section, we will give out a proof of Guinand formula with the uniformly convergent error term, besides, we also give out an inverse theorem of Guinand formula. First of all we need following notations and formulas which will be used throughout this paper[1],[3]. Chebyshev function $\displaystyle\psi(x)=\sum_{n<x}\Lambda(n)$ Where the Von Mangoldt function $\Lambda(n)=logp$ if $n=p^{k}$ for some $k$ and some prime number $p$ ,$\Lambda(n)=0$ otherwise. ###### Theorem 1 ( Mangoldt and Riemann explicit formula) $\displaystyle\psi(x)=x-lim_{T\rightarrow\infty}\sum_{\left\|Im\rho\right\|<T}\frac{x^{\rho}}{\rho}-log(2\pi)+\sum_{n=1}^{\infty}\frac{x^{-2n}}{2n}$ Where $\rho$ runs through all non-trivial Riemann zeros. ###### Theorem 2 [3] $\displaystyle\psi(x)=x-\sum_{\left\|Im\rho\right\|<T}\frac{x^{\rho}}{\rho}+O(\frac{xlog^{2}x}{T})$ we set $\displaystyle\tilde{\psi}(x)=\frac{x^{2}}{2}-\sum_{\rho}\frac{x^{\rho+1}}{\rho(\rho+1)}-xln(2\pi)-\sum_{n=1}^{\infty}\frac{x^{-2n+1}}{2n(2n-1)}$ (1) It’s easy to prove that when $x$ is not equal to any integer,$\tilde{\psi}(x)$ is differentialble and it’s derivative is just $\psi(x)$ and it’s continuous when $x>0$ [1]. We set $\Delta(x)=\psi(x)-x$ and $\tilde{\Delta}(x)=\tilde{\psi}(x)-\frac{x^{2}}{2}$ . The following lemma will is important for deducing the Guinand formula with the error term. ###### Lemma 3 when $s\neq\rho$, $\displaystyle\frac{\zeta^{\prime}}{\zeta}(s)=-\sum_{n<X}\frac{\Lambda(n)}{n^{s}}+\psi(X)X^{-s}+s\tilde{\psi}(X)X^{-s-1}-$ $\displaystyle\frac{s(s+1)}{2(s-1)}X^{1-s}+\sum_{\rho}\frac{s(s+1)X^{\rho-s}}{\rho(\rho+1)(s-\rho)}+\sum_{n\geq 1}\frac{s(s+1)X^{-2n-2s}}{2n(2n-1)(s+2n)}$ (2) Proof. Let $f_{X}(s)=\sum_{n<X}\frac{\Lambda(n)}{n^{s}}$ Using integration by parts twicely, we have that $\displaystyle f_{X}(s)=\int_{1}^{X}x^{-s}d\psi(x)=\psi(X)X^{-s}-\int_{1}^{X}\psi(x)dx^{-s}=\psi(X)X^{-s}+s\int_{1}^{X}\psi(x)x^{-s-1}dx$ $\displaystyle=\psi(X)X^{-s}+s\int_{1}^{X}x^{-s-1}d\tilde{\psi}(x)$ $\displaystyle=\psi(X)X^{-s}+s\tilde{\psi}(X)X^{-s-1}+s(s+1)\int_{1}^{X}\tilde{\psi}(x)x^{-s-2}dx$ (3) and by formula 1,we can further get $\displaystyle\int_{1}^{X}\tilde{\psi}(x)x^{-s-2}dx=\int_{1}^{X}\frac{\frac{x^{2}}{2}-\sum_{\rho}\frac{x^{\rho+1}}{\rho(\rho+1)}-ln(2\pi)x-\sum_{n=1}^{\infty}\frac{x^{-2n+1}}{2n(2n-1)}}{x^{s+2}}dx$ $\displaystyle=\frac{1}{2}(\frac{X^{1-s}}{1-s}-\frac{1}{1-s})-\sum_{\rho}\frac{1}{\rho(\rho+1)}(\frac{X^{\rho-s}}{\rho-s}-\frac{1}{\rho-s})$ $\displaystyle+ln(2\pi)(\frac{X^{-s}}{s}-\frac{1}{s})+\sum_{n\geq 1}\frac{1}{2n(2n-1)}(\frac{X^{-s-2n}}{s+2n}-\frac{1}{s+2n})$ We collect all above terms as two groups $J_{X}(s)$ and $I(s)$,obviously, $I(s)=\frac{1}{2}\frac{1}{s-1}-\sum_{\rho}\frac{1}{\rho(\rho+1)}\frac{1}{s-\rho}-\frac{ln(2\pi)}{s}-\sum_{n\geq 1}\frac{1}{2n(2n-1)}\frac{1}{s+2n}$ By formula 3 and using the notations $J_{X}(s)$ and $I(s)$, we get $\displaystyle f_{X}(s)=\psi(X)X^{-s}+s\tilde{\psi}(X)X^{-s-1}+s(s+1)J_{X}(s)+s(s+1)I(s)$ (4) and $s(s+1)I(s)=\frac{s(s+1)}{2(s-1)}-\sum_{\rho}\frac{s(s+1)}{\rho(\rho+1)(s-\rho)}-(s+1)ln(2\pi)-\sum_{n\geq 1}\frac{s(s+1)}{2n(2n-1)(s+2n)}$ By the following identity, $\displaystyle\frac{s(s+1)}{z(z+1)(s-z)}=\frac{s}{z(z+1)}+\frac{1}{s-z}+\frac{1}{z}$ we have that $\displaystyle s(s+1)I(s)=-\frac{\zeta^{\prime}}{\zeta}(s)+as+b$ (5) where $a,b$ are some constants which can be determined immediately. According 4 and 5 ,we get a new representation of $\frac{\zeta^{\prime}}{\zeta}(s)$ when $s\neq\rho$ as follows: $\displaystyle\frac{\zeta^{\prime}}{\zeta}(s)=-\sum_{n<X}\frac{\Lambda(n)}{n^{s}}+\psi(X)X^{-s}+s\tilde{\psi}(X)X^{-s-1}$ $\displaystyle-\frac{s(s+1)}{2(s-1)}X^{1-s}+\sum_{\rho}\frac{s(s+1)X^{\rho-s}}{\rho(\rho+1)(s-\rho)}+\sum_{n\geq 1}\frac{s(s+1)X^{-2n-s}}{2n(2n-1)(s+2n)}+as+b$ (6) Since when $Res>1$, $\frac{\zeta^{\prime}}{\zeta}(s)=-\sum_{n=1}^{\infty}\frac{\Lambda(n)}{n^{s}}$ Let $X\rightarrow\infty$ on the right side of 6 when $Res>1$, we immediately get $a=b=0$, that follows our theorem. As we know, $log\zeta(s_{0})=\int_{2}^{s_{0}}\frac{\zeta^{\prime}}{\zeta}(s)ds$ ,where the integral path is a positive orient half rectangle with vertices $2,2+iT,\sigma+iT$ and $s_{0}=\sigma+iT$ $s_{0}\neq\rho$. Taking this complex integral on both sides of 6, we directly get following theorem: ###### Theorem 4 When $s_{0}\neq\rho$, we have that $\displaystyle log\zeta(s_{0})=\sum_{n<X}\frac{\Lambda(n)}{(logn)n^{s_{0}}}-\frac{\Delta(X)}{(logX)X^{s_{0}}}-\frac{\tilde{\Delta}(X)}{(logX)X^{s_{0}+1}}(s_{0}+\frac{1}{lnX})$ $\displaystyle+\int_{2}^{s_{0}}\frac{X^{1-s}}{1-s}ds+\tilde{J}_{X}(s_{0})+C_{0}$ (7) Where $\displaystyle\tilde{J}_{X}(s_{0})=\int_{2}^{s_{0}}s(s+1)J(X)ds=-\frac{1}{lnX}\sum_{\rho}\frac{1}{\rho(\rho+1)}[\frac{s_{0}(s_{0}+1)X^{\rho- s_{0}}}{s_{0}-\rho}$ $\displaystyle-\int_{2}^{s_{0}}X^{\rho-s}(\frac{2s+1}{s-\rho}-\frac{s^{2}+s}{(s-\rho)^{2}})ds]$ $\displaystyle-\frac{1}{lnX}\sum_{n\geq 1}\frac{1}{2n(2n-1)}[\frac{s_{0}(s_{0}+1)X^{-2n-s}}{s+2n}-\int_{2}^{s_{0}}X^{-2n-s}(\frac{2s+1}{s+2n}-\frac{s^{2}+s}{(s+2n)^{2}})ds]$ and $C_{0}$ is a real constant. Proof. Using integration by parts and collecting all terms containing $X^{1-s}$, we immediately get above results. setting $s_{0}=\frac{1}{2}+iT$ in the formula 7 and taking imaginary parts on both sides, we have that ###### Theorem 5 If the Riemann hypothesis is true, and $\delta$ is the distance between $T$ and the coordinate of the nearest Riemann zero,we have $\displaystyle\pi S(T)=-\sum_{n<X}\frac{\Lambda(n)sin(Tlogn)}{\sqrt{n}}+\frac{\Delta(X)sin(TlogX)}{\sqrt{X}(logX)}+Im(\int_{2}^{\frac{1}{2}+iT}\frac{X^{1-s}}{1-s}ds)+O(\frac{T^{3}}{\delta^{2}lnX})$ (8) in the limit language, we have $\displaystyle\pi S(T)=-lim_{X\rightarrow\infty}[\sum_{n<X}\frac{\Lambda(n)sin(Tlogn)}{\sqrt{n}(logn)}$ $\displaystyle-\frac{\Delta(X)sin(TlogX)}{\sqrt{X}(logX)}-Im(\int_{2}^{\frac{1}{2}+iT}\frac{X^{1-s}}{1-s}ds)]$ (9) From now on ,we will prove formula 9 is the same as Guinand formula. To achieve it, we need to make some simplification as follows: Let’s first simplify the term $Im(\int_{2}^{\frac{1}{2}+iT}\frac{X^{1-s}}{1-s}ds)$ ,Let’s transform the original integral path which is half rectangle with vertices $2,2+iT,\frac{1}{2}+iT$ to another half rectangle with vertices $2,\frac{1}{2},\frac{1}{2}+iT$ and orient is clockwise, we get $\displaystyle\int_{2}^{\frac{1}{2}+iT}\frac{X^{1-s}}{1-s}ds=i\int_{0}^{T}\frac{X^{\frac{1}{2}-it}}{\frac{1}{2}-it}dt+\int_{\Gamma_{r}}\frac{X^{1-s}}{1-s}ds$ (10) Where $\gamma_{r}=[\frac{1}{2},1-r]\cup S_{r}\cup[1+r,2]$ and $S_{r}$ is upper half semi-circle with radius $r$ and centered at $z=1$ Taking imaginary part on both side of 10,we get the first term of right hand side is equal to $\sqrt{X}\int_{0}^{T}\frac{2cos(tlogX)+4tsin(tlogX)}{1+4t^{2}}dt$ which is set to be $f_{1}(T,X)$ For the second term of right side of 10, we have that $\displaystyle Im(\int_{\Gamma_{r}}\frac{X^{1-s}}{1-s}ds)=Im(\int_{J_{r}}\frac{X^{1-s}}{1-s}ds)$ $\displaystyle=lim_{r\rightarrow 0}Im(\int_{J_{r}}\frac{X^{1-s}}{1-s}ds)=-\pi$ (11) Let’s pick up the second term of righ side of 47 i.e. $\int_{1}^{X}\frac{sin(Tlogt)}{\sqrt{t}logt}dt$ and set it to be $f_{2}(T,X)$, Since $\frac{sin(Tlogt)}{\sqrt{t}logt}$ is continously differentiable with respect to $T$, we have that $\displaystyle\frac{\partial f_{2}}{\partial T}=\int_{1}^{X}\frac{cos(Tlogt)}{\sqrt{t}}dt$ $\displaystyle=\int_{0}^{lnX}e^{\frac{1}{2}u}cos(Tu)du=\frac{\sqrt{X}}{1+4T^{2}}[2cos(TlogX)+4Tsin(TlogX]-\frac{2}{1+4T^{2}}$ (12) in which we have used the substituation $u=logt$ We can also notice that $\frac{\partial f_{1}}{\partial T}=\frac{\sqrt{X}}{1+4T^{2}}[2cos(TlogX)+4Tsin(TlogX]$ By 12, we get $\displaystyle\frac{\partial f_{1}}{\partial T}-\frac{\partial f_{2}}{\partial T}=\frac{2}{1+4T^{2}}$ Thus, $\displaystyle f_{1}(T,X)-f_{2}(T,X)=\int_{0}^{T}\frac{2}{1+4t^{2}}dt$ (13) $\displaystyle=arctan2T$ Consequently,by 10, 11,13 $\displaystyle Im(\int^{\frac{1}{2}+iT}\frac{X^{1-s}}{1-s}ds)-\int_{1}^{X}\frac{sin(Tlogt)}{\sqrt{t}logt}dt=arctan2T-\pi$ (14) Let’s single out the term $\frac{sin(TlogX)}{logX}\left\\{\sum_{n<X}\Lambda(n)n^{-\frac{1}{2}}-2X^{\frac{1}{2}}\right\\}$ in right hand of 47 and get it simplified as follows: $\displaystyle\frac{sin(TlogX)}{logX}\left\\{\sum_{n<X}\Lambda(n)n^{-\frac{1}{2}}-2X^{\frac{1}{2}}\right\\}$ $\displaystyle=\frac{sin(TlogX)}{logX}[\int_{1}^{X}x^{-\frac{1}{2}}d\psi(x)-2\sqrt{X}]$ $\displaystyle=\frac{sin(TlogX)}{logX}[\psi(X)X^{-\frac{1}{2}}+\frac{1}{2}\int_{1}^{X}\psi(x)x^{-\frac{3}{2}}dx-2\sqrt{X}]$ $\displaystyle=\frac{\Delta(X)sin(TlogX)}{\sqrt{X}(logX)}+\frac{sin(TlogX)}{2logX}\int_{1}^{X}\Delta(x)x^{-\frac{3}{2}}dx$ (15) and by the theorem 2 , we have that $\displaystyle\int_{1}^{X}\Delta(x)x^{-\frac{3}{2}}dx=-\sum_{\rho}\frac{X^{\rho-\frac{1}{2}}-1}{\rho(\rho-\frac{1}{2})}+2ln(2\pi)(X^{-\frac{1}{2}}-1)$ $\displaystyle-\sum_{n\geq 1}\frac{X^{-2n-\frac{1}{2}}-1}{2n(2n+\frac{1}{2})}=O(1)$ With formula 15, we have $\displaystyle\frac{sin(TlogX)}{logX}[\sum_{n\leq X}\Lambda(n)n^{-\frac{1}{2}}-2X^{\frac{1}{2}}]=\frac{\Delta(X)sin(TlogX)}{\sqrt{X}(logX)}+O(\frac{1}{logX})$ (16) Since $\displaystyle N(T)=\frac{1}{\pi}arg\xi(\frac{1}{2}+iT)$ $\displaystyle=\frac{1}{\pi}args(s-1)\pi^{-\frac{s}{2}}\Gamma(\frac{s}{2})\zeta(s)\|_{s=\frac{1}{2}+iT}$ $\displaystyle=\frac{1}{\pi}arg(-\frac{1}{4}-T^{2})-\frac{Tln\pi}{2\pi}+\frac{1}{\pi}arg\Gamma(\frac{1}{4}+\frac{iT}{2})+S(T)$ $\displaystyle=1-\frac{Tln\pi}{2\pi}+\frac{1}{\pi}arg\Gamma(\frac{1}{4}+\frac{iT}{2})+S(T)$ (17) Whenever $T$ is not equal to any cordinates of some Riemann zeros,we can rewrite Guinand formula 47 as follows $\displaystyle\pi S(T)=F_{X}(T)+arctan(2T)-\pi+\frac{1}{2}arg\Gamma(\frac{1}{2}+iT)-arg\Gamma(\frac{1}{4}+\frac{iT}{2})$ $\displaystyle-\frac{Tln2}{2}-\frac{1}{4}arctan(sinh\pi T)$ (18) Where $\displaystyle F_{X}(T)=-lim_{X\rightarrow\infty}[\sum_{n\leq X}\Lambda(n)\frac{sin(Tlogn)}{\sqrt{n}logn}-\int_{1}^{X}\frac{sin(Tlogt)}{\sqrt{t}logt}dt$ $\displaystyle-\frac{sin(TlogX)}{logX}[\sum_{n\leq X}\Lambda(n)n^{-\frac{1}{2}}-2X^{\frac{1}{2}}]]$ Using equation 18(Guinand formula) minus equation 9 and notice 14 and 16, we get that $\displaystyle 0=\frac{1}{2}arg\Gamma(\frac{1}{2}+iT)-arg\Gamma(\frac{1}{4}+\frac{iT}{2})-\frac{Tln2}{2}-\frac{1}{4}arctan(sinh\pi T)$ (19) When $T$ is not cordinates of some Riemann zeros. We set right side of 19 to be $d(T)$,then we just need to prove that $d(T)\equiv 0$ when $T>0$. Let’s show it as follows: By rewriting $\frac{Tln2}{2}=arg2^{\frac{iT}{2}}$ and $arctan(sinh\pi T)=arg(1+isinh\pi T)$ , we have that $\displaystyle 4d(T)=arg\frac{\Gamma^{2}(\frac{1}{2}+iT)}{\Gamma^{4}(\frac{1}{4}+\frac{iT}{2})4^{iT}(1+isinh\pi T)}$ $\displaystyle=Imlog\frac{\Gamma^{2}(\frac{1}{2}+iT)}{\Gamma^{4}(\frac{1}{4}+\frac{iT}{2})4^{iT}(1+isinh\pi T)}$ $\displaystyle=\frac{1}{2i}[log\frac{\Gamma^{2}(\frac{1}{2}+iT)}{\Gamma^{4}(\frac{1}{4}+\frac{iT}{2})4^{iT}(1+isinh\pi T)}-log\frac{\Gamma^{2}(\frac{1}{2}-iT)}{\Gamma^{4}(\frac{1}{4}-\frac{iT}{2})4^{-iT}(1-isinh\pi T)}]$ Let $s=iT$, then $sinh\pi T=-isin\pi s$ and $\displaystyle 4d(T)=\frac{1}{2i}log\frac{\Gamma^{2}(\frac{1}{2}+s)\Gamma^{4}(\frac{1}{4}-\frac{s}{2})(1-sin\pi s)}{\Gamma^{2}(\frac{1}{2}-s)\Gamma^{4}(\frac{1}{4}+\frac{s}{2})4^{2s}(1+sin\pi s)}$ (20) Let $g(s)=\frac{\Gamma^{2}(\frac{1}{2}+s)\Gamma^{4}(\frac{1}{4}-\frac{s}{2})(1-sin\pi s)}{\Gamma^{2}(\frac{1}{2}-s)\Gamma^{4}(\frac{1}{4}+\frac{s}{2})4^{2s}(1+sin\pi s)}$ then $g(s)$ is a meromorphic function in the whole complex number plane and by the formula 20, $g(s)\|_{s=iT}=1$ when $T>0$ . For the convenience of factorizing $g(s)$, let’s set $s=\frac{1}{2}-z$ and reset $f(z)=g(s)$ we have that $\displaystyle f(z)=\frac{\Gamma^{2}(1-z)\Gamma^{4}(\frac{1}{2}z)sin^{2}(\frac{\pi}{2}z)}{\Gamma^{2}(z)\Gamma^{4}(\frac{1}{2}-\frac{1}{2}z)4^{1-2z}cos^{2}(\frac{\pi}{2}z)}$ (21) We just need to prove that $f(z)\equiv 1$ for any $z\in C$ , that can be derived by the formula $\Gamma(z)\Gamma(1-z)=\frac{\pi}{sin(\pi z)}$ With the formulas 16,18,19, we can rewrite the formula 8 as: $\pi S(T)=-\sum_{n<X}\frac{\Lambda(n)sin(Tlogn)}{\sqrt{n}}+\frac{\Delta(X)sin(TlogX)}{\sqrt{X}(logX)}+\int_{1}^{X}\frac{sin(Tlogy)}{\sqrt{y}logy}dy$ $+arctan(2T)-\pi+O(\frac{T^{3}}{\delta^{2}lnX})$ Furthermore, we can rewrite above formula as an integral equation, first of all, $\sum_{n<X}\frac{\Lambda(n)sin(Tlogn)}{\sqrt{n}}=\int_{a}^{X}\frac{sin(Tlogy)}{\sqrt{y}logy}d\psi(y)$ $=\int_{a}^{X}\frac{sin(Tlogy)}{\sqrt{y}logy}dy+\int_{a}^{X}\frac{sin(Tlogy)}{\sqrt{y}logy}d\Delta(y)$ $=\int_{a}^{X}\frac{sin(Tlogy)}{\sqrt{y}logy}dy+\frac{\Delta(X)sin(TlogX)}{\sqrt{X}(logX)}-\frac{\Delta(a)sin(Tloga)}{\sqrt{a}(loga)}$ $-\int_{a}^{X}\frac{Tcos(Tlny)-sin(Tlny)(\frac{lny}{2}+1)}{y\sqrt{y}ln^{2}y}\Delta(y)dy$ Where $1<a<2$. We substitute above formula into 2,we get that $S(T)=-\frac{1}{\pi}\int_{a}^{X}\frac{Tcos(Tlny)-sin(Tlny)(\frac{lny}{2}+1)}{y\sqrt{y}ln^{2}y}\Delta(y)dy$ $-\frac{1}{\pi}[\int_{a}^{1}\frac{sin(Tlogy)}{\sqrt{y}logy}dy-\frac{\Delta(a)sin(Tloga)}{\sqrt{a}(loga)}+arctan(2T)-\pi]$ and $\frac{1}{\pi}\int_{a}^{X}\frac{Tcos(Tlny)-sin(Tlny)(\frac{lny}{2}+1)}{y\sqrt{y}ln^{2}y}\Delta(y)dy$ $=\frac{1}{\pi}\int_{a}^{X}\frac{Tcos(Tlny)-sin(Tlny)(\frac{lny}{2}+1)}{y\sqrt{y}ln^{2}y}d\tilde{\Delta}(y)$ $=\tilde{\Delta}(X)\frac{Tcos(TlnX)-sin(TlnX)(\frac{lnX}{2}+1)}{X\sqrt{X}ln^{2}X}-\tilde{\Delta}(a)\frac{Tcos(Tlna)-sin(Tlna)(\frac{lna}{2}+1)}{a\sqrt{a}ln^{2}a}$ $-\int_{a}^{X}\tilde{\Delta}(y)d\frac{Tcos(Tlny)-sin(Tlny)(\frac{lny}{2}+1)}{y\sqrt{y}ln^{2}y}$ and $\displaystyle\int_{a}^{X}\tilde{\Delta}(y)d\frac{Tcos(Tlny)-sin(Tlny)(\frac{lny}{2}+1)}{y\sqrt{y}ln^{2}y}=\int_{a}^{X}F(T,y)\tilde{\Delta}(y)dy$ (22) Where $F(T,y)=-\frac{T^{2}sin(Tlny)}{y^{\frac{5}{2}}lny}-\frac{2Tcos(Tlny)}{y^{\frac{5}{2}}lny}+\frac{3sin(Tlny)}{4y^{\frac{5}{2}}lny}$ $-\frac{2Tcos(Tlny)}{y^{\frac{5}{2}}ln^{2}y}+\frac{2sin(Tlny)}{y^{\frac{5}{2}}ln^{2}y}+\frac{2sin(Tlny)}{y^{\frac{5}{2}}ln^{3}y}$ By 2,2,22 and when t is not the cordinate of a Riemann zero, let $X\rightarrow\infty$, we have following integral equation $\displaystyle S(t)=-\frac{1}{\pi}\int_{a}^{\infty}F(t,y)\tilde{\Delta}(y)dy+g(a,t)$ (23) Where $g(a,t)=-\frac{1}{\pi}[\int_{a}^{1}\frac{sin(tlogy)}{\sqrt{y}logy}dy-\frac{\Delta(a)sin(tloga)}{\sqrt{a}(loga)}$ $+\tilde{\Delta}(a)\frac{tcos(tlna)-sin(tlna)(\frac{lna}{2}+1)}{a\sqrt{a}ln^{2}a}+arctan(2t)-\pi]$ , $1<a<2$ ## 3 Representing $\tilde{\Delta}(x)$ in term of $S(T)$ In this section, under the RH , we will represent $\tilde{\Delta}(x)$ as an integral of $S(T)$ via Riemann-Von Mangoldt formula. Let $N(T)$ be a function counting the number of non-trivial Riemann zeros whose imaginary is between $0$ and $T$,under the RH,we can rewrite the formula 1in term of $N(T)$ as follows, $\displaystyle\tilde{\Delta}(x)=-\sum_{\rho}\frac{x^{\rho+1}}{\rho(\rho+1)}-xln(2\pi)-\sum_{n=1}^{\infty}\frac{x^{-2n+1}}{2n(2n-1)}$ (24) $\displaystyle=-\int_{0}^{\infty}[\frac{x^{\frac{3}{2}+it}}{(\frac{1}{2}+it)(\frac{3}{2}+it)}+\frac{x^{\frac{3}{2}-it}}{(\frac{1}{2}-it)(\frac{3}{2}-it)}]dN(t)+f(x)$ (25) $\displaystyle=-2x^{\frac{3}{2}}\int_{0}^{\infty}\frac{(\frac{3}{4}-t^{2})cos(tlnx)+2tsin(tlnx)}{4t^{2}+(\frac{3}{4}-t^{2})^{2}}dN(t)+f(x)$ (26) Where $f(x)=-xln(2\pi)-\sum_{n=1}^{\infty}\frac{x^{-2n+1}}{2n(2n-1)}$ Noticing 17,set $g(t)=1-\frac{tln\pi}{2\pi}+\frac{1}{\pi}arg\Gamma(\frac{1}{4}+\frac{it}{2})$ Thus we have $\displaystyle\tilde{\Delta}(x)=-2x^{\frac{3}{2}}\int_{0}^{\infty}\frac{(\frac{3}{4}-t^{2})cos(tlnx)+2tsin(tlnx)}{4t^{2}+(\frac{3}{4}-t^{2})^{2}}dS(t)$ (27) $\displaystyle-2x^{\frac{3}{2}}\int_{0}^{\infty}\frac{(\frac{3}{4}-t^{2})cos(tlnx)+2tsin(tlnx)}{4t^{2}+(\frac{3}{4}-t^{2})^{2}}dg(t)+f(x)$ (28) Putting the last two terms together and setting it to be $\tilde{f}(x)$, we get that $\displaystyle\tilde{\Delta}(x)=-2x^{\frac{3}{2}}\int_{0}^{\infty}\frac{(\frac{3}{4}-t^{2})cos(tlnx)+2tsin(tlnx)}{4t^{2}+(\frac{3}{4}-t^{2})^{2}}dS(t)+\tilde{f}(x)$ (29) Using integration by parts and noticing $S(t)=O(logt)$, we have $\displaystyle\int_{0}^{\infty}\frac{(\frac{3}{4}-t^{2})cos(tlnx)+2tsin(tlnx)}{4t^{2}+(\frac{3}{4}-t^{2})^{2}}dS(t)$ $\displaystyle=-\frac{4S(0)}{3}-\int_{0}^{\infty}S(t)d\frac{(\frac{3}{4}-t^{2})cos(tlnx)+2tsin(tlnx)}{4t^{2}+(\frac{3}{4}-t^{2})^{2}}$ (30) From now on, we are going to evaluate the second term of 30 in detail for the convenience of checking. $\int_{0}^{\infty}S(t)d\frac{(\frac{3}{4}-t^{2})cos(tlnx)+2tsin(tlnx)}{4t^{2}+(\frac{3}{4}-t^{2})^{2}}$ $=\int_{0}^{\infty}S(t)\frac{[(\frac{3}{4}-t^{2})cos(tlnx)+2tsin(tlnx)]^{\prime}[4t^{2}+(\frac{3}{4}-t^{2})^{2}]}{[4t^{2}+(\frac{3}{4}-t^{2})^{2}]^{2}}dt$ $-\int_{0}^{\infty}S(t)\frac{[(\frac{3}{4}-t^{2})cos(tlnx)+2tsin(tlnx)][4t^{2}+(\frac{3}{4}-t^{2})^{2}]^{\prime}}{[4t^{2}+(\frac{3}{4}-t^{2})^{2}]^{2}}dt$ $=\int_{0}^{\infty}S(t)\frac{[-2tcos(tlnx)-(\frac{3}{4}lnx)sin(tlnx)+t^{2}lnxsin(tlnx)+2sin(tlnx)+2tlnxcos(tlnx)][t^{4}+\frac{5}{2}+\frac{9}{16}]}{[4t^{2}+(\frac{3}{4}-t^{2})^{2}]^{2}}dt$ $-\int_{0}^{\infty}S(t)\frac{[\frac{3}{4}cos(tlnx)-t^{2}cos(tlnx)+2tsin(tlnx)][4t^{3}+5t]}{[4t^{2}+(\frac{3}{4}-t^{2})^{2}]^{2}}dt$ $=\int_{0}^{\infty}S(t)\frac{-2t^{5}cos(tlnx)-\frac{3}{4}t^{4}sin(tlnx)+t^{6}lnxsin(tlnx)+2t^{4}sin(tlnx)+2t^{5}lnxcos(tlnx)}{(t^{4}+\frac{5}{2}t^{2}+\frac{9}{16})^{2}}dt\\\ $ $+\int_{0}^{\infty}S(t)\frac{-5t^{3}cos(tlnx)-\frac{15}{8}t^{2}sin(tlnx)+\frac{5}{2}t^{4}lnxsin(tlnx)+5t^{2}sin(tlnx)+5t^{3}lnxcos(tlnx)}{(t^{4}+\frac{5}{2}t^{2}+\frac{9}{16})^{2}}dt$ $+\int_{0}^{\infty}S(t)\frac{-\frac{9}{8}tcos(tlnx)-(\frac{27}{64}lnx)sin(lnx)+\frac{9}{16}t^{2}lnxsin(tlnx)+\frac{9}{8}sin(tlnx)+\frac{9}{8}tlnxcos(tlnx)}{(t^{4}+\frac{5}{2}t^{2}+\frac{9}{16})^{2}}dt$ $-\int_{0}^{\infty}S(t)\frac{3t^{3}-4t^{5}cos(tlnx)+8t^{4}sin(tlnx)+\frac{15}{4}tcos(tlnx)-5t^{3}cos(tlnx)+10t^{2}sin(tlnx)}{(t^{4}+\frac{5}{2}t^{2}+\frac{9}{16})^{2}}dt$ To summarize, we get $\displaystyle\tilde{\Delta}(x)=-2x^{\frac{3}{2}}\int_{0}^{\infty}K(x,t)S(t)dt+f(x)$ (31) Where we still let $f(x)=-2x^{\frac{3}{2}}\int_{0}^{\infty}\frac{(\frac{3}{4}-t^{2})cos(tlnx)+2tsin(tlnx)}{4t^{2}+(\frac{3}{4}-t^{2})^{2}}dg(t)-xln(2\pi)-\sum_{n=1}^{\infty}\frac{x^{-2n+1}}{2n(2n-1)}+\frac{8S(0)}{3}x^{\frac{3}{2}}$ and where $K(x,t)=lnx[t^{6}sin(tlnx)+2t^{5}cos(tlnx)+\frac{7}{4}t^{4}sin(tlnx)+5t^{3}cos(tlnx)-\frac{21}{16}t^{2}sin(tlnx)$ $+\frac{9}{8}tcos(tlnx)-\frac{27}{64}sin(tlnx)]+2t^{5}cos(tlnx)-6t^{4}sin(tlnx)-3t^{3}cos(tlnx)-5t^{2}sin(tlnx)$ $-\frac{39}{8}tcos(tlnx)+\frac{9}{8}sin(tlnx)$ By the equations 23,31, we can get a system of integral equations $\displaystyle\left\\{\begin{array}[]{ccc}\tilde{\Delta}(x)&=&-2x^{\frac{3}{2}}\int_{0}^{\infty}K(x,t)S(t)dt+f(x)\\\ \\\ S(t)&=&-\frac{1}{\pi}\int_{a}^{\infty}F(t,y)\tilde{\Delta}(y)dy+g(a,t)\end{array}\right.$ (35) We shall prove an inverse theorem of the Guinnand formula as follows. ###### Theorem 6 Let $f(t)$ be a function which is continous on the interval $[0,+\infty]$ except some discrete points,which forms a set $E$ ,and $f(t)=lim_{X\rightarrow\infty}f_{X}(t)$, $f_{X}(t)=-\sum_{n<X}\frac{\Lambda(n)sin(Tlogn)}{\sqrt{n}}+\frac{\Delta(X)sin(TlogX)}{\sqrt{X}(logX)}+\int_{1}^{X}\frac{sin(Tlogy)}{\sqrt{y}logy}dy$ and $f_{X}(t)=O(logt)$ on $[0,+\infty]\setminus E$,then the RH holds Proof. Considering the function $F(s)=\int_{0}^{+\infty}\frac{2(s-1)}{(s-\frac{1}{2})^{2}+t^{2}}df(t)$ and $f_{X}(s)=\int_{0}^{+\infty}\frac{2(s-1)}{(s-\frac{1}{2})^{2}+t^{2}}df_{X}(t)$ , where $Res>\frac{1}{2}$ Using integration by parts, $F(s)=-\frac{2f(0)}{s-\frac{1}{2}}+\int_{0}^{+\infty}\frac{4(s-1)t}{[(s-\frac{1}{2})^{2}+t^{2}]^{2}}f(t)dt$ and $f_{X}(s)=-\frac{2f_{X}(0)}{s-\frac{1}{2}}+\int_{0}^{+\infty}\frac{4(s-1)t}{[(s-\frac{1}{2})^{2}+t^{2}]^{2}}f_{X}(t)dt$ Since $f(t)=lim_{X\rightarrow\infty}f_{X}(t)$ and $f_{X}(t)=O(logt)$ on $[0,+\infty]\setminus E$, by the Lebesque CCL, $lim_{X\rightarrow\infty}f_{X}(s)=F(s)$ and $\displaystyle\int_{0}^{+\infty}\frac{4(s-1)t}{[(s-\frac{1}{2})^{2}+t^{2}]^{2}}f_{X}(t)dt=\int_{0}^{+\infty}\frac{4(s-1)t}{[(s-\frac{1}{2})^{2}+t^{2}]^{2}}[-\sum_{n<X}\frac{\Lambda(n)sin(tlogn)}{\sqrt{n}}+\frac{\Delta(X)sin(tlogX)}{\sqrt{X}(logX)}+\int_{1}^{X}\frac{sin(Tlogy)}{\sqrt{y}logy}dy]dt$ (36) Using residue theorem, we have that $\int_{0}^{+\infty}\frac{4(s-1)t}{[(s-\frac{1}{2})^{2}+t^{2}]^{2}}\frac{\Lambda(n)sin(tlogn)}{\sqrt{n}}dt=\frac{\Lambda(n)}{n^{s}}$ Similarly, $\int_{0}^{+\infty}\frac{4(s-1)t}{[(s-\frac{1}{2})^{2}+t^{2}]^{2}}[\frac{\Delta(X)sin(tlogX)}{\sqrt{X}(logX)}]dt=\frac{\Delta(X)}{X^{s+\frac{1}{2}}logX}$ and $\int_{0}^{+\infty}\frac{4(s-1)t}{[(s-\frac{1}{2})^{2}+t^{2}]^{2}}[\int_{1}^{X}\frac{sin(Tlogy)}{\sqrt{y}logy}dy]dt=\frac{1}{1-s}[X^{1-s}-1]$ To summarize, $\displaystyle\int_{0}^{+\infty}\frac{4(s-1)t}{[(s-\frac{1}{2})^{2}+t^{2}]^{2}}f_{X}(t)dt=-\sum_{n<X}\frac{\Lambda(n)}{n^{s}}+\frac{\Delta(X)}{X^{s+\frac{1}{2}}logX}+\frac{1}{1-s}[X^{1-s}-1]$ (37) Since for any $Res>\frac{1}{2}$,$lim_{X\rightarrow\infty}f_{X}(s)=F(s)$, and $F(s)$ is analytical in the half plane $Res>\frac{1}{2}$, we notice that when $Res>1$, we have $\displaystyle lim_{X\rightarrow\infty}f_{X}(s)=-\sum_{n=2}^{\infty}\frac{\Lambda(n)}{n^{s}}+\frac{1}{s-1}$ (38) $\displaystyle=-\frac{\zeta^{\prime}(s)}{\zeta(s)}+\frac{1}{s-1}=F(s)$ (39) Which means that RH is true. ## 4 Lower bound of $\tilde{\Delta}(x)$ First of all , Let’s derive a formula based on the functional equation and formula, since $\displaystyle\frac{\zeta^{\prime}(s)}{\zeta(s)}=-s(s+1)\int_{1}^{X}\tilde{\Delta}(x)x^{-s-2}dx$ $\displaystyle-\frac{s(s+1)}{2(s-1)}X^{1-s}+\sum_{\rho}\frac{s(s+1)X^{\rho-s}}{\rho(\rho+1)(s-\rho)}+\sum_{n\geq 1}\frac{s(s+1)X^{-2n-s}}{2n(2n-1)(s+2n)}$ (40) As we know, by the functional equation and 40, we have $\displaystyle Re\frac{\zeta^{\prime}(s)}{\zeta(s)}\|_{s=\frac{1}{2}+it}=\frac{ln\pi}{2}-\frac{1}{2}Re\frac{\Gamma^{\prime}}{\Gamma}(\frac{1}{2}+it)$ (41) Evaluating real part at$s=\frac{1}{2}+it$ on both sides of 42, we get that $\displaystyle\int_{1}^{X}\frac{\tilde{\Delta}(x)}{x^{\frac{5}{2}}}[(\frac{3}{4}-t^{2})cos(tlnx)+2tsin(tlnx)]dx-\sum_{t_{\rho}}\frac{(\frac{3}{4}-t^{2})(\frac{3}{4}-t_{\rho}^{2})+4tt_{\rho}}{(\frac{3}{4}-t_{\rho}^{2})^{2}+4t_{\rho}^{2}}\frac{sin(t_{\rho}-t)lnX}{t_{\rho}-t}$ $\displaystyle-\sum_{t_{\rho}}\frac{(\frac{3}{4}-t^{2})(\frac{3}{4}-t_{\rho}^{2})-4tt_{\rho}}{(\frac{3}{4}-t_{\rho}^{2})^{2}+4t_{\rho}^{2}}\frac{sin(t_{\rho}+t)lnX}{t_{\rho}+t}$ $\displaystyle+\sum_{t_{\rho}}\frac{\frac{3}{2}+2tt_{\rho}}{(\frac{3}{4}-t_{\rho}^{2})^{2}+4t_{\rho}^{2}}cos(t_{\rho}-t)lnX$ $\displaystyle+\sum_{t_{\rho}}\frac{\frac{3}{2}-2tt_{\rho}}{(\frac{3}{4}-t_{\rho}^{2})^{2}+4t_{\rho}^{2}}cos(t_{\rho}+t)lnX=g_{X}(t)$ (42) Where $g_{X}(t)=Re[-\frac{\zeta^{\prime}(s)}{\zeta(s)}+\sum_{n\geq 1}\frac{s(s+1)X^{-2n-s}}{2n(2n-1)(s+2n)}]_{s=\frac{1}{2}+it}$ By above formula, when $t\neq t_{\rho}$, $\displaystyle\int_{1}^{X}\frac{\tilde{\Delta}(x)}{x^{\frac{5}{2}}}[(\frac{3}{4}-t^{2})cos(tlnx)+2tsin(tlnx)]dx=O(1)$ (43) otherwise, $\displaystyle\int_{1}^{X}\frac{\tilde{\Delta}(x)}{x^{\frac{5}{2}}}[(\frac{3}{4}-t_{\rho}^{2})cos(t_{\rho}lnx)+2tsin(t_{\rho}lnx)]dx=lnX+O(1)$ (44) Where $\rho=\frac{1}{2}+it_{\rho}$ are Riemann zeros, to simplify LHS of 44, set $\theta_{\rho}=arctan\frac{\frac{3}{4}-t_{\rho}^{2}}{2t_{\rho}}$,then we have $\displaystyle\int_{1}^{X}\frac{\tilde{\Delta}(x)}{x^{\frac{5}{2}}}sin(t_{\rho}lnx+\theta_{\rho})dx=\frac{1}{\sqrt{(\frac{3}{4}-t_{\rho}^{2})^{2}+4t_{\rho}^{2}}}lnX+O(1)$ (45) Let $u=lnx$, above formula can be reduced to $\int_{1}^{u}\frac{\tilde{\Delta}(e^{u})}{e^{\frac{3}{2}u}}sin(t_{\rho}y+\theta_{\rho})dy=\frac{1}{\sqrt{(\frac{3}{4}-t_{\rho}^{2})^{2}+4t_{\rho}^{2}}}u+R_{\rho}(u)$ Set $f_{\rho}(u)=\frac{\tilde{\Delta}(e^{u})}{e^{\frac{3}{2}u}}sin(t_{\rho}u+\theta_{\rho})$, and $A_{\rho}=\frac{1}{\sqrt{(\frac{3}{4}-t_{\rho}^{2})^{2}+4t_{\rho}^{2}}}$ Thus We have simplified form $\int_{0}^{X}f_{\rho}(t)dt=A_{\rho}X+R_{\rho}(X)$ Let $g(t)=\frac{\tilde{\Delta}(e^{t})}{e^{\frac{3}{2}t}}$ and $max_{0\leq t<+\infty}\mid g(t)\mid=C_{1}$ , $\mu(x)=m\\{t\|\mid g(t)\mid\leq x\\}$, $\overline{\mu}(x)=X-\mu(x)$, and $E_{x}=\\{t\|\mid g(t)\mid\leq x\\}$ By choosing any $x<C_{1}$ , we have following estimate $A_{\rho}X+R_{\rho}(X)=\int_{0}^{X}f_{\rho}(t)dt\leq\int_{0}^{X}\mid g(t)\mid dt=\int_{E_{x}}\mid g(t)\mid dt+\int_{[0,X]\setminus E_{x}}\mid g(t)\mid dt$ $\leq x\mu(x)+C_{1}(X-\mu(x))$ When $X$ is big enough, we have $\mu(x)\leq\frac{C_{1}-A_{\rho}}{C_{1}-x}X+\frac{C_{0\rho}}{C_{1}-x}$ or $\overline{\mu}(x)\geq\frac{A_{\rho}-x}{C_{1}-x}X-\frac{C_{0\rho}}{C_{1}-x}$ Where $C_{0\rho}=max_{0<t<\infty}\mid R_{\rho}(t)\mid$ and set $F_{x}^{X}=\\{u\mid\|\frac{\tilde{\Delta}(u)}{u^{\frac{3}{2}}}\|>x,0<u<X\\}$ Therefore $m(F_{x}^{X})\geq\frac{A_{\rho}-x}{C_{1}-x}X-\frac{C_{0\rho}}{C_{1}-x}$ Where $\tilde{\Delta}(u)=\sum_{n\leq u}(n-\psi(n))\Lambda(n)-\frac{u^{2}}{2}$ Choosing $\rho=\rho_{0}=\frac{1}{2}+14.134....$ which is the first non-trivial Riemann zero, we have following theorem ###### Theorem 7 When $X$ is big enough, $m(F_{x}^{X})\geq\frac{A_{\rho_{0}}-x}{C_{1}-x}X-\frac{C_{0\rho_{0}}}{C_{1}-x}$ Set $F_{X}(t)=\int_{1}^{X}\frac{\tilde{\Delta}(x)}{x^{\frac{5}{2}}}[(\frac{3}{4}-t^{2})cos(tlnx)+2tsin(tlnx)]dx$ , and $G_{X}(t)=\int_{0}^{t}F_{X}(y)dy$, by the formula 43,45,and Guinand formula i.e $lim_{X\rightarrow\infty}G_{X}(t)=N(t)$, we can conjecture that when $X\rightarrow\infty$, $F_{X}(t)$ will behave like a distribution more than an ordinary function. Let’s verify it as follows: First of all, we notice that $\displaystyle\int_{0}^{+\infty}e^{-\epsilon u}[(\frac{3}{4}-k^{2})cos(ku)+2ksin(ku)]\frac{\tilde{\Delta}(e^{u})}{e^{\frac{3}{2}u}}du$ $\displaystyle=Re[s(s+1)\int_{1}^{+\infty}\tilde{\Delta}(x)x^{-s-\epsilon-2}dx]_{s=\frac{1}{2}+ik}$ Using integration by parts couple of times, we get $\displaystyle s(s+1)\int_{1}^{X}\tilde{\Delta}(x)x^{-s-\epsilon-2}dx=s\tilde{\Delta}(1)-\Delta(1)+\int_{1}^{X}x^{-s-\epsilon}d\Delta(x)+\epsilon\int_{1}^{X}x^{-s-\epsilon}\Delta(x)dx$ $\displaystyle-s\epsilon\int_{1}^{X}x^{-s-\epsilon-2}\tilde{\Delta}(x)dx- sx^{-s-\epsilon-1}\tilde{\Delta}(x)$ and $\displaystyle\int_{1}^{X}x^{-s-\epsilon}d\Delta(x)=\int_{1}^{X}x^{-s-\epsilon}d\psi(x)-\int_{1}^{X}x^{-s-\epsilon}dx=\sum_{n<X}\frac{\Lambda(n)}{n^{s+\epsilon}}-\frac{X^{1-s-\epsilon}}{1-s-\epsilon}+1$ By the formula 2, when $Res\geq\frac{1}{2}$we get that $\displaystyle\int_{1}^{+\infty}x^{-s-\epsilon}d\Delta(x)=-\frac{\zeta^{\prime}(s+\epsilon)}{\zeta(s+\epsilon)}+1$ (46) Therefore $\displaystyle\int_{0}^{+\infty}e^{-\epsilon u}[(\frac{3}{4}-k^{2})cos(ku)+2ksin(ku)]\frac{\tilde{\Delta}(e^{u})}{e^{\frac{3}{2}u}}du=Re[\frac{\zeta^{\prime}(s+\epsilon)}{\zeta(s+\epsilon)}]\mid_{s=\frac{1}{2}+ik}+\varphi_{\epsilon}(k)$ (47) Where $\varphi_{\epsilon}(k)=Re[-s\epsilon\int_{1}^{\infty}x^{-s-\epsilon-2}\tilde{\Delta}(x)dx+\epsilon\int_{1}^{\infty}x^{-s-\epsilon-1}\Delta(x)dx]\mid_{s=\frac{1}{2}+ik}$ For the simplicity, we denote LHS of 47 by $J_{\epsilon}(k)$. Choosing any test function $g(k)\in C_{0}^{\infty}(R^{+})$ ,where $C_{0}^{\infty}(R^{+})$ is the set of all smooth functions which have compact supports on $R^{+}$ , let’s compute following inner product,by 47 $\displaystyle lim_{\epsilon\rightarrow 0}\langle J_{\epsilon}(k),g(k)\rangle=Re[\frac{\zeta^{\prime}(s+\epsilon)}{\zeta(s+\epsilon)}]\mid_{s=\frac{1}{2}+ik}+\varphi_{\epsilon}(k),g(k)\rangle$ $\displaystyle=lim_{\epsilon\rightarrow 0}\langle Re[\frac{\zeta^{\prime}(s+\epsilon)}{\zeta(s+\epsilon)}]\mid_{s=\frac{1}{2}+ik},g(k)\rangle+lim_{\epsilon\rightarrow 0}\langle\varphi_{\epsilon}(k),g(k)\rangle$ (48) and it’s not difficult to verify that $lim_{\epsilon\rightarrow 0}\langle\varphi_{\epsilon}(k),g(k)\rangle=0$ Using integration by parts twice, we have that $\displaystyle\langle Re[\frac{\zeta^{\prime}(s+\epsilon)}{\zeta(s+\epsilon)}]\mid_{s=\frac{1}{2}+ik},g(k)\rangle=\langle h(\frac{1}{2}+\epsilon+ik),g^{\prime\prime}(k)\rangle$ (49) Where $h(z)=\int_{1}^{z}ln\zeta(s)ds$ and $Rez>\frac{1}{2}$,the integration path is the conventional contour from $1$ to $z$ Since $h(\frac{1}{2}+\epsilon+ik)=O(klogk)$ uniformly for any small $\epsilon$ and $lim_{\epsilon\rightarrow 0}h(\frac{1}{2}+\epsilon+ik)=S_{1}(k)$ Where $S_{1}(k)=\int_{0}^{k}S(t)dt$ Finally, we have $\displaystyle lim_{\epsilon\rightarrow 0}\langle J_{\epsilon}(k),g(k)\rangle=\langle S_{1}(k),g^{\prime\prime}(k)\rangle$ (50) Before ending this section, let’s take look at the equation 42 again, we can rewrite this equation in term of integral equation as follows: $\displaystyle\int_{0}^{\infty}K_{X}(t,t^{\prime})dN(t^{\prime})=H_{X}(t)$ (51) Where $K_{X}(t)=-\frac{(\frac{3}{4}-t^{2})(\frac{3}{4}-t^{\prime 2})+4tt^{\prime}}{(\frac{3}{4}-t^{\prime 2})^{2}+4t^{\prime 2}}\frac{sin(t^{\prime}-t)lnX}{t^{\prime}-t}-\frac{(\frac{3}{4}-t^{2})(\frac{3}{4}-t^{\prime 2})-4tt^{\prime}}{(\frac{3}{4}-t^{\prime 2})^{2}+4t^{\prime 2}}\frac{sin(t^{\prime}+t)lnX}{t^{\prime}+t}$ $+\frac{\frac{3}{2}+2tt^{\prime}}{(\frac{3}{4}-t^{\prime 2})^{2}+4t^{\prime 2}}cos(t^{\prime}-t)lnX+\frac{\frac{3}{2}-2tt^{\prime}}{(\frac{3}{4}-t^{\prime 2})^{2}+4t^{\prime 2}}cos(t^{\prime}+t)lnX$ and $H_{X}(t)=\int_{1}^{X}\frac{\tilde{\Delta}(x)}{x^{\frac{5}{2}}}[(\frac{3}{4}-t^{2})cos(tlnx)+2tsin(tlnx)]dx+\frac{ln\pi}{2}-\frac{1}{2}Re\frac{\Gamma^{\prime}}{\Gamma}(\frac{1}{2}+it)$ Actually, above equation depends on the parameter $X$,for every fixed $X$,we get a non trivial integral equation of $N(t)$, so we obtain a family of integral equations, noticing that the integral kernel $K_{X}(t,t^{\prime})$ is an explicit function,it’s expected that exploring these integral equations will help us to understand RH further, besides we can consider similar results for $L$ function which satisfies functional equation. ## References * [1] H. M. Edwards Riemann’s zeta function 74-74 * [2] A. P. Guinand A summation formula in the theory of prime numbers Pro. London Math. Soc. (2) 50 (1948) 107-119 * [3] A. A. Karatsuba S. M. Voronin The Riemann zeta function De Gruyter Exposition in Mathmatics. 5 43-56	arxiv-papers	2011-11-08T06:18:07	2024-09-04T02:49:24.107243	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Jining Gao", "submitter": "Jining Gao", "url": "https://arxiv.org/abs/1111.1810" }
1111.1860	# Superconducting proximity effect to the block antiferromagnetism in KyFe2-xSe2 Hong-Min Jiang Department of Physics and Center of Theoretical and Computational Physics, The University of Hong Kong, Hong Kong, China Department of Physics, Hangzhou Normal University, Hangzhou, China Wei-Qiang Chen Department of Physics, South University of Science and Technology of China, Shenzhen, China Department of Physics and Center of Theoretical and Computational Physics, The University of Hong Kong, Hong Kong, China Zi-Jian Yao Department of Physics and Center of Theoretical and Computational Physics, The University of Hong Kong, Hong Kong, China Fu-Chun Zhang Department of Physics and Center of Theoretical and Computational Physics, The University of Hong Kong, Hong Kong, China Department of Physics, Zhejiang University, Hangzhou, China ###### Abstract Recent discovery of superconducting (SC) ternary iron selenides has block antiferromagentic (AFM) long range order. Many experiments show possible mesoscopic phase separation of the superconductivity and antiferromagnetism, while the neutron experiment reveals a sizable suppression of magnetic moment due to the superconductivity indicating a possible phase coexistence. Here we propose that the observed suppression of the magnetic moment may be explained due to the proximity effect within a phase separation scenario. We use a two- orbital model to study the proximity effect on a layer of block AFM state induced by neighboring SC layers via an interlayer tunneling mechanism. We argue that the proximity effect in ternary Fe-selenides should be large because of the large interlayer coupling and weak electron correlation. The result of our mean field theory is compared with the neutron experiments semi- quantitatively. The suppression of the magnetic moment due to the SC proximity effect is found to be more pronounced in the $d$-wave superconductivity and may be enhanced by the frustrated structure of the block AFM state. ###### pacs: 74.20.Mn, 74.25.Ha, 74.62.En, 74.25.nj ## I introduction The recent discovery of high-$T_{c}$ superconductivity in the ternary iron selenides AyFe2-xSe2 (A=K; Rb; Cs;…) JGuo ; AKrzton ; MHFang has triggered a new surge of interest in study of iron-based superconductors (Fe-SC). The fascinating aspect of these material lies in the tunable Fe-vacancies in these materials, which substantially modifies the normal-state metallic behavior and enhances the transition temperature $T_{c}$ to above 30K from 9K for the binary system FeSe at ambient pressure. JGuo ; MHFang ; YZhang1 Particular attention has been focused on the vacancy ordered 245 system, K0.8Fe1.6Se2, as it introduces a novel magnetic structure into the already rich magnetism of Fe-SC. Unlike the collinear Cruz1 ; FMa1 ; XWYan1 or bi-collinear FMa2 ; WBao2 ; SLi1 AFM order observed in the parent compounds of other Fe-SC, the neutron diffraction experiment has clearly shown that these materials have a block AFM (BAFM) order. WBao1 Meanwhile, the AFM order with an unprecedentedly large magnetic moment of $3.31\mu_{B}$/Fe below the Néel temperature is the largest one among all the known parent compounds of Fe-SC. ZShermadini ; WBao1 Moreover, the carrier concentration is extremely low, indicating the parent compound to be a magnetic insulator/semiconductor, yzhou ; RYu in comparison with a metallic spin-density-wave (SDW) state of the parent compound in other Fe-SC. RHYuan ; MHFang The relation between the novel magnetism and superconductivity in ternary Fe selenides is currently an interesting issue under debate. The question is whether the superconductivity and the BAFM order are phase separated or co- exist in certain region of the phase diagram. The neutron experiment shows the suppression of the AFM ordering below SC transition point, WBao1 suggesting the coexistence. Some other experiments, such as two-magnon Raman-scattering, AMZhang and muon-spin rotation and relaxation Shermadini1 are consistent with this picture. On the other hand, the ARPES, FChen1 NMR Torchetti1 and TEM ZWang1 experiments indicate a mesoscopic phase separation between the superconductivity and the insulating AFM state. Most recently, Li et al. showed the superconductivity and the BAFM orders to occur at different layers of the Fe-selenide planes in the STM measurement. wli1 The vacancy in Fe-selenides is an interesting but complicated issue. The vacancy in the Fe-selenide carries a negative charge since the Fe-ion has a valence of 2+. In the equilibrium, we expect the vacancies to repel each other at short distance for the Coulomb interaction and to attract to each other at a long distance for the elastic strain. Such a scenario would be in favor of the phase separation to form a vacancy rich and vacancy poor regions in the compound. The challenge is then to explain the observed suppression of magnetic moment of the BAFM due to the superconductivity. At the phenomenological level, the suppression of magnetism due to superconductivity has been reported previously, aeppli and such phenomenon may be explained by Ginzburg-Landau theory. varma In this paper, we propose that the proximity effect of superconductivity to the BAFM in a mesoscopically phase separated Fe-selenides may be large to account the suppression of the AFM moments observed in neutron experiment. More specifically, we use a microscopic model to study the proximity effect on a layer of the BAFM state induced by adjacent SC layer. The proximity effect in Fe-selenides is expected to be important for the two reasons. One is the weaker correlation effect, and the other is the larger interlayer hopping amplitude, compared with those in cuprates. Both of them may enhance proximity effect on the magnetism from the neighboring SC layer. Our model calculations show the proximity effect in a mesoscopically phase separated state of Fe- selenides may explain various seemingly conflicted experiments. ## II MODEL AND MEAN FIELD THEORY AyFe2-xSe2 is a layered material with FeSe layers separated by alkali atoms, similar to the 122 material in iron pnictides family. To investigate the proximity effect to the BAFM layer, we consider a single BAFM layer next to a SC layer as shown schematically in Fig. 1. The electronic Hamiltonian describing the BAFM layer is given by $\displaystyle H$ $\displaystyle=H_{0}+H_{inter},$ (1) where $H_{0}$ describes the electron motion and spin couplings in the BAFM layer and $H_{inter}$ describes the coupling to the neighboring SC layer. We consider a two-orbital model to describe $H_{0}$, $\displaystyle H_{0}=$ $\displaystyle-\sum_{ij,\alpha\beta,\sigma}t_{ij,\alpha\beta}C^{{\dagger}}_{i,\alpha\sigma}C_{j,\beta\sigma}-\mu\sum_{i,\alpha\sigma}C^{{\dagger}}_{i,\alpha\sigma}C_{i,\alpha\sigma}$ (2) $\displaystyle+J_{1}\sum_{<ij>,\alpha\beta}\textbf{S}_{i,\alpha}\cdot\textbf{S}_{j,\beta}+J_{2}\sum_{<<ij>>,\alpha\beta}\textbf{S}_{i,\alpha}\cdot\textbf{S}_{j,\beta}$ $\displaystyle+J^{\prime}_{1}\sum_{<ij>^{\prime},\alpha\beta}\textbf{S}_{i,\alpha}\cdot\textbf{S}_{j,\beta}+J^{\prime}_{2}\sum_{<<ij>>^{\prime},\alpha\beta}\textbf{S}_{i,\alpha}\cdot\textbf{S}_{j,\beta},$ where $C_{i,\alpha\sigma}$ annihilates an electron at site $i$ with orbital $\alpha$ ($d_{xz}$ and $d_{yz}$) and spin $\sigma$, $\mu$ is the chemical potential. $t_{ij,\alpha\beta}$ are the hopping integrals, and $<ij>$ ($<ij>^{\prime}$) and $<<ij>>$ ($<<ij>>^{\prime}$) denote the intra-block (inter-block) nearest (NN) and next nearest neighbor (NNN) bonds, respectively [see the upper layer in Fig. 1]. $J_{1}$ ($J_{2}$) are the exchange coupling constants for NN (NNN) spins in the same block, and $J^{\prime}_{1}$ ($J^{\prime}_{2}$) are for the two NN (NNN) spins in different blocks. The two-orbital model is a crude approximation for electronic structure. However, it may be a minimal model to capture some of basic physics in examining the proximity effect. The band structure around the obtained Fermi energy from the two-orbital model is shown in Fig. 2, which is very similar to the result obtained in density functional theory. The main shortcoming in using the two- orbital model is that the magnetic moment is $2\mu$B at largest, smaller than the experimentally measured $3.31\mu$B. We consider this to be a quantitative issue, and will not qualitatively change our results. Figure 1: (color online) Schematic diagram of the system in our model to study proximity effect to the block AFM state (upper layer) induced by superconductivity at the lower layer via a pair tunneling process $H_{inter}$ in Eq. (3). We now consider $H_{inter}$, the coupling between SC layer and BAFM layer. Because of the semiconducting gap of the BAFM layer and the SC gap in the SC layer, the leading order of the interlayer coupling are the pairing hopping between the SC and BAFM layers. According to the crystal structures, wli1 such a coupling term can be expressed as mcmillan1 $\displaystyle H_{inter}=$ $\displaystyle\frac{t^{2}_{\tau}}{\omega_{c}}\sum_{ij,\alpha\beta,\alpha^{\prime}\beta^{\prime},\sigma}(C^{{\dagger}}_{i,\alpha,\sigma}C^{B}_{i,\beta,\sigma}C^{{\dagger}}_{j,\alpha^{\prime},\bar{\sigma}}C^{B}_{j,\beta^{\prime},\bar{\sigma}}$ (3) $\displaystyle+H.C.),$ where $\omega_{c}$ is a characteristic energy, and $t_{\tau}$ is the interlayer hopping integral, the superscript $B$ represents the SC layer. With the mean field approximation $\Delta_{ij,\alpha\alpha^{\prime},\sigma\bar{\sigma}}=<C^{B}_{i,\alpha,\sigma}C^{B}_{j,\alpha^{\prime},\bar{\sigma}}>$, we have $\displaystyle H_{inter}=\sum_{ij,\beta\beta^{\prime},\sigma}(V_{\tau,ij}C^{{\dagger}}_{i,\beta,\sigma}C^{{\dagger}}_{j,\beta^{\prime},\bar{\sigma}}+H.C.),$ (4) where $V_{\tau,ij}=\frac{t^{2}_{\tau}}{\omega_{c}}\sum_{\alpha\alpha^{\prime}}\Delta_{ij,\alpha\alpha^{\prime},\sigma\bar{\sigma}}$. The $\sqrt{5}\times\sqrt{5}$ vacancy order and the BAFM order lead to an enlarged unit cell with eight sites per unit cell. We use a mean field theory for the Ising spins in Eq. (2) and obtain the Bogoliubov-de Gennes equations in the enlarged unit cell $\displaystyle\sum_{k}{{}^{\prime}}\sum_{j,\beta}\left(\begin{array}[]{lr}H_{ij,\alpha\beta,\sigma}+\tilde{H}_{ij,\alpha\beta,\sigma}&H_{c,ij,\alpha\beta}\\\ H^{\ast}_{c,ij,\alpha\beta}&-H^{\ast}_{ij,\alpha\beta,\bar{\sigma}}+\tilde{H}_{ij,\alpha\beta,\sigma}\end{array}\right)$ (7) $\displaystyle\times\exp[i\textbf{k}\cdot(\textbf{r}_{j}-\textbf{r}_{i})]\left(\begin{array}[]{lr}u^{k}_{n,j,\beta,\sigma}\\\ v^{k}_{n,j,\beta,\bar{\sigma}}\end{array}\right)=E^{k}_{n}\left(\begin{array}[]{lr}u^{k}_{n,i,\alpha,\sigma}\\\ v^{k}_{n,i,\alpha,\bar{\sigma}}\end{array}\right),$ (12) where, the summation of $k$ are over the reduced Brillouin zone, and $\displaystyle H_{ij,\alpha\beta,\sigma}=$ $\displaystyle- t_{ij,\alpha\beta}-\mu,$ $\displaystyle\tilde{H}_{ij,\alpha\beta,\sigma}=$ $\displaystyle\sum_{\tau}(J_{\tau,intra}+J_{\tau,inter})<S_{i+\tau,\beta}>\delta_{ij}$ $\displaystyle H_{c,ij,\alpha\beta}=$ $\displaystyle V_{\tau,ij}\sum_{\alpha^{\prime}\beta^{\prime}}\Delta_{ij,\alpha^{\prime}\beta^{\prime},\sigma\bar{\sigma}}.$ (13) Here, $J_{\tau,intra}=J_{1}$ ($J_{\tau,inter}=J^{\prime}_{1}$) if $\tau=\pm\hat{x},\pm\hat{y}$ and $J_{\tau,intra}=J_{2}$ ($J_{\tau,inter}=J^{\prime}_{2}$) if $\tau=\pm\hat{x}\pm\hat{y}$. $\hat{x}$ and $\hat{y}$ denote the unit vectors along $x$ and $y$ directions, respectively. $<S_{i+\tau,\beta}>$ is defined as $(n_{i+\tau,\beta,\uparrow}-n_{i+\tau,\beta,\downarrow})/2$. $u^{k}_{n,j,\alpha,\sigma}$ ($u^{k}_{n,j,\beta,\bar{\sigma}}$), $v^{k}_{n,j,\alpha,\sigma}$ ($v^{k}_{n,j,\beta,\bar{\sigma}}$) are the Bogoliubov quasiparticle amplitudes on the $j$-th site with corresponding eigenvalues $E^{k}_{n}$. The self-consistent equations of the mean fields are $\displaystyle n_{i,\beta,\uparrow}=$ $\displaystyle\sum_{k,n}\|u^{k}_{n,i,\beta,\uparrow}\|^{2}f(E^{k}_{n})$ $\displaystyle n_{i,\beta,\downarrow}=$ $\displaystyle\sum_{k,n}\|v^{k}_{n,i,\beta,\downarrow}\|^{2}[1-f(E^{k}_{n})].$ (14) The magnitude of the magnetic order on the $i$-th site and the induced SC pairing correlation in the BAFM layer are defined as, $\displaystyle M(i)=$ $\displaystyle\frac{1}{2}\sum_{\beta}(n_{i,\beta,\uparrow}-n_{i,\beta,\downarrow})$ $\displaystyle\Delta^{A}_{ij,\alpha\beta}=$ $\displaystyle\frac{1}{4}\sum_{k,n}{{}^{\prime}}(u^{k}_{n,i,\alpha,\sigma}v^{k\ast}_{n,j,\beta,\bar{\sigma}}e^{-i\textbf{k}\cdot(\textbf{r}_{j}-\textbf{r}_{i})}$ (15) $\displaystyle+v^{k\ast}_{n,i,\alpha,\bar{\sigma}}u^{k}_{n,j,\beta,\sigma}e^{i\textbf{k}\cdot(\textbf{r}_{j}-\textbf{r}_{i})})\tanh(\frac{E^{k}_{n}}{2k_{B}T}).$ In the calculations, we choose the hopping integrals as follows: wgyin1 Along the $y$ direction, the $d_{xz}-d_{xz}$ NN hopping integral $t_{1}=0.4$ eV and the $d_{yz}-d_{yz}$ NN hopping integral $t_{2}=0.13$ eV; they are exchanged to the $x$ direction; the NN interorbital hoppings are zero; the NNN intraorbital hopping integral $t_{3}=-0.25$ eV for both $d_{xz}$ and $d_{yz}$ orbitals, and the NNN interorbital hopping is $t_{4}=0.07$ eV. The hopping integral $t_{1}$ is taken as the energy unit. We keep $J_{1}:J^{\prime}_{1}:J_{2}:J^{\prime}_{2}=-4:-1:1:2$. CCao1 ; YZYou1 The doping level is given by $\delta=n-2.0$. ## III results To begin with, we present the energy band structure at half filling with $n=2$ in Fig. 2(a), where $J_{1}=2.0$ is so chosen to get a band gap $\sim 500$meV being in agreement with the first principle calculations. CCao1 ; XWYan2 For the electron doping with $n=2.1$, the Fermi level crosses an energy band around the center of the Brillouin zone [$\Gamma$ point in Fig. 2(b)], while it intersects with an energy band around the zone corner at the hole doping with $n=1.9$ [$M$ point in Fig. 2(c)]. Although a simple two-orbital model is adopted here, both the electron and hole doping cases with $\delta=0.1$ are qualitatively consistent with the first principle calculations. XWYan2 In the presence of the ordered vacancies and BAFM order, the original two-band structures are splitting to sixteen subbands as a result of the enlarged unit cell with $8$ sites. At half filling, $8$ lower bands are occupied, i.e., $1/4$ electron per one subband, while another $8$ bands above the Fermi energy are unoccupied, resulting in a band gap in Fig. 2. For the electron and hole doping with $\delta=0.1$, the chemical potential crosses one subband which produce the characteristic features of the Fermi surface and the metallic BAFM state. Figure 2: (color online) Electronic band structures of $H_{0}$ given by Eq. (2). The parameters are given at the end of section II of the text, and $J_{1}=2.0$. (a): at half filling or $n=2.0$; (b): at electron doping $n=2.1$; and (c): at hole doping $n=1.9$. The color scale indicates the relative spectra weight. Motivated by the agreement of the self-consistent mean-field solutions with the mentioned first principle calculations, we consider now the proximity effect in BAFM layer induced by the SC in SC layer. For explicit reason, we choose two possible singlet pairing symmetries in the SC layer, i.e., the NNN $s_{\pm}$-wave and the NN $d$-wave symmetries with their respective gap functions $\Delta_{s_{\pm}}=\Delta_{0}\cos(k_{x})\cos(k_{y})$ and $\Delta_{d}=\Delta_{0}[\cos(k_{x})-\cos(k_{y})]$, where the former results in the NNN bond and the latter the NN bond couplings in the BAFM layer. The interlayer hopping constant $t_{\tau}$ is assumed to be site independent. Fig. 3 displays the moment of the BAFM order as a function of the effective tunneling strength $V_{\tau,ij}$. At the half filling, both symmetries of the SC order in the SC layer introduce the decrease of the BAFM order and simultaneously induce the SC correlation with the same symmetries in the BAFM layer as the tunneling strength increases. A main difference between the $s_{\pm}$\- and the $d$-wave symmetries is the more pronounced proximity effect in reducing the moment of the BAFM order produced by the $d$-wave symmetry as the tunneling strength increase, as shown in Figs. 3(a) and 3(b). In the case of electron doping with $n=2.1$, where the metallic BAFM state results, although the proximity effect is more pronounced, the magnetic and the induced SC correlation remain the qualitatively unchanged, due possibly to the very low total carrier concentration. Figure 3: (color online) Block AFM moment and the SC pairing correlation as functions of the effective tunneling strength $V_{\tau,ij}$. Black curves are for next nearest neighbor $s_{\pm}$-wave pairing, and red for nearest neighbor $d$-wave pairing. Upper panel (a) and (b): $n=2.0$ and lower panel (c) and (d): $n=2.1$. The unique feature of the the effective tunneling in the second order is it’s temperature dependence via the SC pairing $\Delta_{ij,\alpha\alpha^{\prime},\sigma,\sigma^{\prime}}$, which differs from that in one particle tunneling process. mmori1 ; jxzhu1 ; andersen1 The temperature dependence of the SC pairing parameter is modeled by a phenomenological form with $\Delta=\Delta_{0}\sqrt{1-T/T_{c}}$. We present the temperature dependence of the magnetic moment in Fig. 4(a) for the typical choice of the coupling constants $g_{\tau}=2V_{\tau,ij}\Delta_{0}=0.25$. As temperature decrease, the magnetic order increases when temperature is above $T_{c}$, while it decreases when temperature is below $T_{c}$, resulting in a broad peak around $T_{c}$. We note that the temperature dependence of the AFM moment is reminiscent of the neutron diffraction and the two-magnon experiments [Fig. 4(b)]. WBao1 ; AMZhang There is another scenario that the competition between the AFM and the SC orders in the microscopic coexistence of them may also produce the decrease of the AFM moment below $T_{c}$. The study of such possibility is currently under way and the results will be published elsewhere. It is worthwhile to notice that the sizable proximity effect relies on the substantial interlayer hoping constant $t_{\tau}$. Based on the first principle calculation, the interlayer hopping $t_{\tau}$ was estimated to have a comparable magnitude with $t_{1}$ possibly due to the high values of electron mobility from the intercalated alkaline atoms, CCao2 and leads to the highly three dimensional Fermi surface. CCao1 ; XWYan2 Figure 4: (color online) (a) Temperature dependence of the block AFM moment at $n=2.0$. Black and red curves are for next nearest neighbor $s_{\pm}$-wave and nearest neighbor $d$-wave pair couplings, respectively. (b): the re-plotted curve of the neutron data from Ref. WBao1, . ## IV SUMMARY AND DISCUSSIONS In summary, we have proposed that various seemingly conflict experiments on the phase separation or coexistence of superconductivity and BAFM may be explained within a phase separation scenario by taking into account of the proximity effect of superconductivity to the neighboring layer of BAFM. We have theoretically studied the proximity effect to a BAFM layer induced by adjacent SC layers in a simplified two-orbital model for Fe-selenides. The proximity effect in reducing the moment of the BAFM order highly depends on the coupling constant $V_{\tau,ij}$. For realistic parameters of the interlayer tunneling, our calculation shows that the superconductivity proximity effect may result in substantial suppression of the magnetic moment. This is in contrary to that in the cuprate superconductor, where the coupling constant $V_{\tau,ij}$ is very small because of small c-axis hopping integral due to the large anisotropy, and because of the renormalization of $V_{\tau,ij}$ by a factor proportional to hole concentrations due to the no double-occupation condition. maekawa In iron-based superconductor, the anisotropy of iron-based material is much smaller than in the cuprate, which lead to a relative larger $t_{\tau}$. And the moderate correlation effect in iron-based superconductor leads to a moderate renormalization factors. As a consequence, the coupling constant $V_{\tau,ij}$ in iron chalcogenide superconductor should be moderate. We remark that we’d be careful in drawing a concrete conclusion to compare with the experiments. The approximation that only $d_{xz}$ and $d_{yz}$ orbitals are important in the bands close to Fermi energy is good in terms of the band structures. CCao1 ; XWYan2 But the maximum magnetic moment in two- orbital model is only $2\mu$B, smaller than the moment of $3.31\mu$B measured in experiments. ZShermadini ; WBao1 The other effect is that we only calculated the suppression of the BAFM order of the surface layer of the BAFM domain. According to TEM experiment, ZWang1 each BAFM domain has around ten layers. And the suppression of BAFM order of the layers in the middle of domain may be more complicated. In brief, the suppression of the BAFM moment is sizable because of the moderate coupling constant $V_{\tau,ij}$, and our calculation may be viewed as a semi-quantitative result. We also investigated proximity effect for various pairing symmetry of the SC phase. It has shown that the SC pairing with NN $d$-wave symmetry resulted a more pronounced proximity effect in reducing the moment of the BAFM order than the NNN $s_{\pm}$-wave pairing. The second order process induced proximity effect has a temperature dependent as the SC pairing, which may be relevant to the experimental observations. More remarkable proximity effect was found in the BAFM state by comparison with the conventional AFM state, which was the consequence of the frustrated structure and the associated anisotropic exchange interactions. ## V acknowledgement We thank W. Bao, G. Aeppli, Y. Zhou, and T. M. Rice for helpful discussions. This work is supported in part by Hong Kong’s RGC GRF HKU706809 and HKUST3/CRF/09. HMJ is grateful to the NSFC (Grant No. 10904062), Hangzhou Normal University (HSKQ0043, HNUEYT). ## VI appendix In the following, we compare the above proximity effect with that in the single band conventional AFM (CAFM) system. In order to make the comparison more convincing, we choose the dispersion $\varepsilon_{k}=-2t[\cos(k_{x})+\cos(k_{y})]-4t^{\prime}\cos(k_{x})\cos(k_{y})-\mu$ with $t=t_{1}$ and $t^{\prime}=t_{3}$, which gives rise to the similar energy band width with that in the above two-orbital model and is close to the case of the cuprates. The AFM order is introduced by the AFM exchange interaction $J\sum_{\langle ij\rangle}\textbf{S}_{i}\cdot\textbf{S}_{j}$ between the NN sites. At the half filling $n=1$, we find that $J=1.6$ produces the comparable band gap and the electron polarization as in the above BAFM state. In Fig. 5, we present the magnitude of the magnetic order and the induced pairing correlation as a function of the effective tunneling $V_{\tau,ij}$. The upper panel shows the results for the NNN $s_{\pm}$-wave pairing and the lower panel the results for the NN $d$-wave pairing. In the figure, the magnitude of the magnetic order in both cases is renormalized. The proximity effect in reducing the AFM order is more pronounced for the BAFM state as shown in Figs. 5(a) and 5(c). As for the induced pairing correlation, the larger correlation is found in the BAFM state for the $s_{\pm}$-wave paring and in the CAFM state for the $d$-wave pairing, as displayed in Figs. 5(b) and 5(d), respectively. Figure 5: (color online) Comparison of the proximity effect between the block and conventional AFM states. Left column shows the moment of the AFM order, and right column the SC pairing correlation as functions of the effective tunneling strength $V_{\tau,ij}$. Upper panel: for the next nearest neighbor $s_{\pm}$-wave pairing and lower panel for the nearest neighbor $d$-wave pairing. Figure 6: (color online) Comparison of the spin structures and their respective NN and NNN bonds. (a): block AFM state; (b): conventional AFM state. We can understand the above results by considering the different spin configurations of the BAFM and CAFM orders, as shown in Fig. 6. In the BAFM state, when two electrons transfer from the BAFM layer to the SC one, the energy changes due to the bonds breaking for the NN bond coupling are $\Delta E^{\uparrow\uparrow}_{NN}=\|J_{1}\|$ [A1 and A2 bonds in Fig. 6(a)] and $\Delta E^{\uparrow\downarrow}_{NN}=7\|J_{1}\|/4$ [A3 bond in Fig. 6(a)] for the electron pairs with ferromagnetic and antiferromagnetic alignments, respectively. On the other hand, the energy changes due the bonds breaking for the NNN bond are $\Delta E^{\uparrow\uparrow}_{NNN}=9\|J_{1}\|/4$ [B3 bond in Fig. 6(a)] and $\Delta E^{\uparrow\downarrow}_{NNN}=5\|J_{1}\|/2$ [B1 and B2 bonds in Fig. 6(a)]. As a result, the proximity effect in reducing the moment of the AFM order by the $d$-wave pairing is more remarkable than that by the $s_{\pm}$ one. However, the energy changes due to the bonds breaking in the CAFM state are $\Delta E_{NN}=7\|J\|$ [C bond in Fig. 6(b)] and $\Delta E_{NNN}=8\|J\|$ [D bond in Fig. 6(b)] for the NN and NNN band couplings. Therefore, the proximity effect in reducing the moment of the AFM in the CAFM state is rather weak for both the $s_{\pm}$\- and $d$-wave pairing couplings. As for the induced pairing correlation, the extent of the match between the AFM order configuration and the singlet SC pairing largely determines the magnitude of the induced pairing correlation. For example, the CAFM matches well with the NN $d$-wave pairing, so that one can expect a large induced pairing correlation without the severe decrease of the AFM order as displayed in Figs. 5(c) and 5(d). ## References * (1) J. Guo, S. Jin, G. Wang, S. Wang, K. Zhu, T. Zhou, M. He, and X. Chen, Phys. Rev. B 82, 180520(R) (2010). * (2) A. Krzton-Maziopa, Z. Shermadini, E. Pomjakushina, V. Pomjakushin, M. Bendele, A. Amato, R. Khasanov, H. Luetkens, and K. Conder, J. Phys.: Condens. Matter 23, 052203 (2011). * (3) M. H. Fang, H. D. Wang, C. H. Dong, Z. J. Li, C. M. Feng, J. Chen, and H. Q. Yuan, Europhys. Lett. 94, 27009 (2011). * (4) Y. Zhang, L. X. Yang, M. Xu, Z. R. Ye, F. Chen, C. He, H. C. Xu, J. Jiang, B. P. Xie, J. J. Ying, X. F. Wang, X. H. Chen, J. P. Hu, M. Matsunami, S. Kimura, and D. L. Feng, Nature Materials 10, 273 (2011). * (5) C. de la Cruz, Q. Huang, J. W. Lynn, J. Li, W. Ratcliff II, J. L. Zarestky, H. A. Mook, G. F. Chen, J. L. Luo, N. L. Wang, and P. Dai, Nature 453, 899 (2008). * (6) F. Ma, Z. Y. Lu, and T. Xiang, Phys. Rev. B 78, 224517 (2008); Front. Phys. China 5, 150 (2010). * (7) X. W. Yan, M. Gao, Z. Y. Lu, and T. Xiang, Phys. Rev. Lett. 106, 087005 (2011). * (8) F. Ma, W. Ji, J. Hu, Z. Y. Lu, and T. Xiang, Phys. Rev. Lett. 102, 177003 (2009). * (9) W. Bao, Y. Qiu, Q. Huang, M. A. Green, P. Zajdel, M. R. Fitzsimmons, M. Zhernenkov, S. Chang, M. Fang, B. Qian, E. K. Vehstedt, J. Yang, H. M. Pham, L. Spinu, and Z. Q. Mao, Phys. Rev. Lett. 102, 247001 (2009). * (10) S. Li, C. de la Cruz, Q. Huang, Y. Chen, J. W. Lynn, J. Hu, Y. L. Huang, F. C. Hsu, K. W. Yeh, M. K. Wu, and P. Dai, Phys. Rev. B 79, 054503 (2009). * (11) W. Bao, Q. Huang, G. F. Chen,M. A. Green, D. M. Wang, J. B. He, X. Q. Wang, and Y. Qiu, Chinese Phys. Lett. 28, 086104 (2011). * (12) Z. Shermadini, A. Krzton-Maziopa, M. Bendele, R. Khasanov, H. Luetkens, K. Conder, E. Pomjakushina, S. Weyeneth, V. Pomjakushin, O. Bossen, and A. Amato, Phys. Rev. Lett. 106, 117602 (2011). * (13) Y. Zhou, D.-H. Xu, F.-C. Zhang, and W.-Q. Chen, Europhys. Lett. 95, 17003 (2011). * (14) R. Yu, J.-X. Zhu, and Q. Si, Phys. Rev. Lett. 106, 186401 (2011). * (15) R. H. Yuan, T. Dong, G. F. Chen, J. B. He, D. M. Wang, and N. L. Wang, e-print arXiv:1102.1381 (to be published). * (16) A. M. Zhang, J. H. Xiao, Y. S. Li, J. B. He, D. M. Wang, G. F. Chen, B. Normand, Q. M. Zhang, and T. Xiang, e-print arXiv:1106.2706 (unpublished). * (17) Z. Shermadini, A. Krzton-Maziopa, M. Bendele, R. Khasanov, H. Luetkens, K. Conder, E. Pomjakushina, S. Weyeneth, V. Pomjakushin, O. Bossen, and A. Amato, Phys. Rev. Lett. 106, 117602 (2011). * (18) F. Chen, M. Xu, Q. Q. Ge, Y. Zhang, Z. R. Ye, L. X. Yang, Juan Jiang, B. P. Xie, R. C. Che, M. Zhang, A. F. Wang, X. H. Chen, D. W. Shen, X. M. Xie, M. H. Jiang, J. P. Hu, D. L. Feng, e-print arXiv:1106.3026 (unpublished). * (19) D. A. Torchetti, M. Fu, D. C. Christensen, K. J. Nelson, T. Imai, H. C. Lei, and C. Petrovic, Phys. Rev. B 83, 104508 (2011). * (20) Z. Wang, Y. J. Song, H. L. Shi, Z. W. Wang, Z. Chen, H. F. Tian, G. F. Chen, J. G. Guo, H. X. Yang, and J. Q. Li, Phys. Rev. B 83, 140505(R) (2011). * (21) W. Li, H. Ding, P. Deng, K. Chang, C. Song, K. He, L. Wang, X. Ma, J.-P. Hu, X. Chen, and Q.-K. Xue, e-print arXiv:1108.0069 (unpublished). * (22) G. Aeppli, D. Bishop, C. Broholm, E. Bucher, K. Siemensmeyer, M. Steiner, and N. Stusser, Phys. Rev. Lett. 63, 676 (1989). * (23) E. I. Blount, C. M. Varma, G. Aeppli, Phys. Rev. Letts. 64, 3074 (1990). * (24) W. L. McMillan, Phys. Rev. 175, 537 (1968). * (25) W.-G. Yin, C.-C. Lee, and W. Ku, Phys. Rev. Letts. 105, 107004 (2010). * (26) C. Cao, and J. Dai, Phys. Rev. Lett. 107, 056401 (2011). * (27) Y.-Z. You, H. Yao, and D.-H. Lee, Phys. Rev. B 84, 020406(R) (2011). * (28) X.-W. Yan, M. Gao, Z.-Y. Lu, and T. Xiang, Phys. Rev. B 83, 233205 (2011). * (29) M. Mori, and S. Maekawa, Phys. Rev. Lett. 94, 137003 (2005). * (30) J.-X. Zhu, and C. S. Ting, Phys. Rev. B 61, 1456 (2000). * (31) B. M. Andersen, I. V. Bobkova, P. J. Hirschfeld, and Yu. S. Barash, Phys. Rev. B 72, 184510 (2005). * (32) C. Cao, private communications. * (33) M. Mori, T. Tohyama and S. Maekawa, J. Phys. Soc. Jpn. 75, 034708 (2006)	arxiv-papers	2011-11-08T10:33:14	2024-09-04T02:49:24.115189	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Hong-Min Jiang, Wei-Qiang Chen, Zi-Jian Yao, and Fu-Chun Zhang", "submitter": "Hong-Min Jiang", "url": "https://arxiv.org/abs/1111.1860" }
1111.1937	2011 Vol. 0 No. XX, 000–000 11institutetext: School of Physics, Damghan University, Damghan, Iran; kfaghei@du.ac.ir Received [year] [month] [day]; accepted [year] [month] [day] # Numerical study of self-gravitating protoplanetary discs Kazem Faghei ###### Abstract In this paper, the effect of self-gravity on the protoplanetary discs is investigated. The mechanisms of angular momentum transport and energy dissipation are assumed to be the viscosity due to turbulence in the accretion disc. The energy equation is considered in situation that the released energy by viscosity dissipation is balanced with cooling processes. The viscosity is obtained by equality of dissipation and cooling functions, and is used for angular momentum equation. The cooling rate of the flow is calculated by a prescription, $du/dt=-u/\tau_{cool}$, that $u$ and $\tau_{cool}$ are internal energy and cooling timescale, respectively. The ratio of local cooling to dynamical timescales $\Omega\tau_{cool}$ is assumed as a constant and also as a function of local temperature. The solutions for protoplanetary discs show that in situation of $\Omega\tau_{cool}=constant$, the disc does not show any gravitational instability in small radii for a typically mass accretion rate, $\dot{M}=10^{-6}M_{\odot}yr^{-1}$, while by choosing $\Omega\tau_{cool}$ as a function of temperature, the gravitational instability for this amount of mass accretion rate or even less can occur in small radii. Also, by study of the viscous parameter $\alpha$, we find that the strength of turbulence in the inner part of self-gravitating protoplanetary discs is very low. These results are qualitatively consistent with direct numerical simulations of protoplanetary discs. Also, in the case of cooling with temperature dependence, the effect of physical parameters on the structure of the disc is investigated. The solutions represent that disc thickness and Toomre parameter decrease by adding the ratio of disc mass to central object mass. While, the disc thickness and Toomre parameter increase by adding mass accretion rate. Furthermore, for typically input parameters such as mass accretion rate $10^{-6}M_{\odot}yr^{-1}$, the ratio of the specific heats $\gamma=5/3$, and the ratio of disc mass to central object mass $q=0.1$, the gravitational instability can occur in whole radii of the discs excluding very near to the central object. ###### keywords: accretion, accretion discs — planetary systems: protoplanetary discs — planetary systems: formation ## 1 Introduction Accretion discs are important for many astrophysical phenomena, including protoplanetary systems, different types of binary stars, binary X-ray sources, quasars, and Active Galactic Nuclei (AGN). Historically, theory of accretion discs had concentrated in the case of non self-gravitating and occasionally the effect of self-gravity had studied (Paczyński 1978; Kolykhalov & Sunyaev 1979; Lin & Pringle 1987, 1990). On the other hand, in recent years, the importance of study of disc self-gravity has increased, especially in the protostellar discs and Active Galactic Nuclei (AGN) discs. It can be due to increase of computational resources in simulation of self-gravitating accretion discs and the observational evidences that have confirmed the existence of self-gravity on all scale discs, from AGN to protostars (Lodato 2008 and references therein). Also, it appears the development of gravitational instability is important for cool regions of accreting gas that angular momentum transport by magneto-rotational instability (MRI) becomes weak (Fleming et al. 2000; Masada & Sano 2008; Faghei 2011) and angular momentum can transport by gravitational instability. The structure of self-gravitating discs has been studied both through self- similar solutions assuming steady and unsteady state (Mineshige & Umemura 1996, 1997; Tsuribe 1999; Bertin & Lodato 1999, 2001; Shadmehri & Khajenabi 2006; Abbassi et. al. 2006; Shadmehri 2009) and through direct numerical simulations (Gammie 2001; Rice et al. 2003, 2005, 2010; Rice & Armitage 2009; Cossins et al. 2010; Meru & Bate 2011a). Mineshige & Umemura (1996) investigated the role of self-gravity on the classical self-similar solution of advection dominated accretion flows (ADAF, Narayan & Yi 1994) and found global one-dimensional solutions influenced by self-gravity in both the radial and the perpendicular directions of the disc. They extended the previous steady state solutions to the time-dependent case while the effect of the self-gravity of the disc was taken into account. They used an isothermal equation, and so their solutions describe a viscous accretion discs in the slow accretion limit. Tsuribe (1999) studied unsteady viscous accretion in self-gravitating discs. Taking into account the growth of the central point mass, Tsuribe (1999) derived a series of self-similar solutions for rotating isothermal discs. The solutions showed, as a core mass increases, the rotation law changes from flat rotation to Keplerian rotation in the inner disc and in addition to the central point mass, the inner disc grows by mass accumulation due to the differing mass accretion rates in the inner and outer radii. Bertin & Lodato (1999) considered a class of steady- state self-gravitating accretion discs for which efficient cooling mechanisms are assumed to operate so that the disc is self-regulated at a condition of approximate marginal Jeans stability. They investigated the entire parameter space available for such self-regulated accretion discs. In another study, Bertin & Lodato (2001) followed the model that, when the disc is sufficiently cold, the stirring due to Jeans-related instabilities acts as a source of effective heating. The corresponding reformulation of the energy equations, they demonstrated how self-regulation can be established, so that the stability parameter $Q$ is maintained close to a threshold value, with weak dependence on radius. Abbassi et al. (2006) studied the effect of viscosity on the time evolution of axisymmetric, polytropic self-gravitating discs around a new born central object. Thus, they ignored from the gravitational effect of central object and only self-gravity of the disc played an important role. They compared effects of $\alpha$-viscosity prescription (Shakura & Sunyaev) and $\beta$-viscosity prescription (Duschel et al. 2000) on disc structure. They found that accretion rate onto the central object for $\beta$-discs more than $\alpha$-discs at least in the outer regions where $\beta$-discs are more efficient. Also, their results showed gravitational instability can occur everywhere on the $\beta$-discs and thus they suggested that $\beta$-discs can be a good candidate for the origin of planetary systems. Shadmehri & Khejenabi (2006) examined steady self-similar solutions of isothermal self-gravitating discs in the presence of a global magnetic field. Similar to Abbassi et. al. (2006) they neglected from the mass of the central object to the disc mass. By study of Toomre parameter they showed that magnetic field can be important in gravitational stability of the disc. An accretion discs can become gravitationally unstable if Toomre parameter becomes smaller than its critical value, $Q<Q_{crit}$ (Toomre 1964). For axisymmetric instabilities $Q_{crit}\sim 1$, while for non-axisymmetric instabilities $Q_{crit}$ values as high as $1.5\,-\,1.7$ (Durisen et al. 2007). One possible outcome is that unstable discs fragment to produce bound objects and has been suggested as a possible mechanism for forming giant planets (Boss 1998, 2002). However, recently it has been realized that above condition is not sufficient to guarantee fragmentation. Gammie (2001) showed that in addition to the above instability criterion, the disc must cool at a fast enough rate. Let the cooling timescale $\tau_{cool}$ be defined as the gas internal energy divided by the volumetric cooling rate. For power-law equation of state and with $\tau_{cool}$ prescribed to be some value over a annulus of the disc, the thin shearing box simulations of Gammie (2001) show that fragmentation occurs if and only if $\Omega\tau_{cool}\lesssim\beta_{crit}$, where $\beta_{crit}\sim 3$ and $\Omega$ is angular velocity of the disc or inverse of dynamical timescale $\tau_{dyn}=\Omega^{-1}$. The critical value of $\Omega\tau_{cool}$ can be somewhat larger than three for more massive and physically thicker discs (Rice et al. 2003), larger adiabatic index (Rice et al. 2005), and more resolution of simulations (Meru & Bate 2011b). Cossins et al. (2010) by SPH simulation studied the effects of opacity regimes on the stability of self-gravitating protoplanetary discs to fragmentation into bound objects. They showed that $\Omega\tau_{cool}$ has a strong dependence on the local temperature. As, they found that without temperature dependence, for radii $\lesssim 10AU$ a very large accretion rate $~{}10^{-3}M_{\odot}yr^{-1}$ is required for fragmentation, but that this is reduced to $10^{-4}$ with cooling of dependent on temperature. As mentioned, typically semi-analytical studies of self-gravitating discs are regarding polytropic discs (Abbassi et al. 2006), isothermal discs (Mineshige & Umemura 1996, 1997; Tsuribe 1999; Shadmehri & Khajenabi 2006), ADAFs in the extreme limit of no radiative cooling (Shadmehri 2004), and discs without central object (Mineshige & Umemura 1996, 1997; Tsuribe 1999; Shadmehri & Khajenabi 2006; Abbassi et al. 2006). In this paper, it will be interesting to understand under which conditions gravitational instability can occur in accretion discs by a suitable energy equation and assuming a Newtonian potential of a mass point that stands in the disc centre. Thus, to obtain these conditions, we will use a prescription for cooling rate that is introduced by Gammie (2001), $du/dt=-u/\tau_{cool}$, that $u$ and $\tau_{cool}$ are internal energy and cooling timescale, respectively. The ratio of local cooling to dynamical timescales $\Omega\tau_{cool}$ is assumed a power-law function of temperature in adapting Cossins et al. (2010), $\Omega\tau_{cool}=\beta_{0}(T/T_{0})^{\delta}$, where $T_{0}$ and $\delta$ are free parameters, and $\beta_{0}$ is free parameter in Gammie (2001). In $\delta=0$, $\Omega\tau_{cool}$ reduces to Gammie (2001) model that $\Omega\tau_{cool}$ is a constant, while non-zero $\delta$ is qualitatively consistent with results of Cossins et al. (2010). We will examine the effects of $\delta$ parameter on gravitational stability of disc. We will show that the present model is qualitatively consistent with direct numerical simulations (Rice & Armitage 2009; Cossin et al. 2010; Rice et al. 2010) and can provide conditions that gravitational instability occur in whole radii excluding very near to the central object. In section 2, the basic equations of constructing a model for steady self- gravitating disc will be defined. In section 3, we will find asymptotic solutions for outer edge of the disc. In section 4, by exploit of asymptotic solutions as boundary conditions for system equations, we will investigate numerically the effects of physical parameters on structure and stability of the disc. The summary and discussion of the model will appear in section 5. ## 2 Basic Equations We use cylindrical coordinate $(r,\varphi,z)$ centered on the accreting object and make the following standard assumptions: * (i) The flow is assumed to be steady and axisymmetric $\partial_{t}=\partial_{\varphi}=0$, so all flow variables are a function of $r$ and $z$ ; * (ii) The gravitational force of central object on a fluid element is characterized by the Newtonian potential of a point mass, $\Psi=-{GM_{}}/{r}$, with $G$ representing the gravitational constant and $M_{}$ standing for the mass of the central star; * (iii) The equations written in cylindrical coordinates are integrated in the vertical direction, hence all quantities of the flow variables will be expressed in terms of cylindrical radius $r$; The governing equations on the self-gravitating accretion disc for such assumptions is as follows. The continuity equation is $\frac{1}{r}\frac{d}{dr}(r\Sigma v_{r})=0,$ (1) where $v_{r}$ is the radial infall velocity and $\Sigma$ is the surface density, which is defined as $\Sigma=2\rho h$, and $\rho$ and $h$ are density and the disk half-thickness, respectively. The half-thickness of the disc with assume of hydrostatic equilibrium in vertical direction is $h=c_{s}/\Omega$, where $c_{s}$ is sound speed, which is defined as $c_{s}^{2}=p/\rho$, $p$ being the gas pressure and $\Omega$ represents angular velocity of the flow. The equation (1) implies that $\displaystyle\dot{M}=-2\pi r\Sigma v_{r}=constant$ where $\dot{M}$ is the mass accretion rate and is a constant in the present model. The simulation results of protoplanetary discs show that the disc reaches a quasi-steady state in 20000 years or less and might imply that these systems are rarely out of equilibrium. Also, the simulations show that the mass of the disc redistribute itself to produce a state in which the accretion rate, $\dot{M}$, is largely independent of $r$ (Rice & Armitage 2009, Rice et al. 2010). Thus, we can use the mass accretion as a constant and it can not be a limitation for the present model. The momentum equations are $v_{r}\frac{dv_{r}}{dr}=-\frac{1}{\Sigma}\frac{d}{dr}(\Sigma c_{s}^{2})-G\left[\frac{M_{}+M(r)}{r^{2}}\right]+r\Omega^{2},$ (2) $\Sigma v_{r}\frac{d}{dr}(r^{2}\Omega)=\frac{1}{r}\frac{d}{dr}\left[\nu\Sigma r^{3}\frac{d\Omega}{\partial r}\right],$ (3) where $\nu$ is the kinematic viscosity coefficient, and $\gamma$ is the adiabatic index, and $M(r)$ is the mass of a disc within a radius $r$. As in Mineshige & Umemura (1997), we adopt the monopole approximation for the radial gravitational force due to the self-gravity of the disc, which considerably simplifies the calculations and is not expected to introduce any significant error as long as the surface density profile is steeper than $1/r$ (e.g. Li & Shu 1997; Saigo & Hanawa 1998; Tsuribe 1999; Krasnopolsky & Konigl 2002; Shadmehri 2009). Now, we can write $\frac{dM(r)}{dr}=2\pi r\Sigma.$ (4) The energy equation is $\frac{\Sigma v_{r}}{\gamma-1}\frac{dc_{s}^{2}}{dr}+\frac{\Sigma c_{s}^{2}}{r}\frac{d}{dr}\left(rv_{r}\right)=\Gamma-\Lambda,$ (5) where $\Gamma$ is the heating rate of the gas by dissipation processes such as turbulent viscosity and $\Lambda$ represents the energy loss through radiative cooling processes. The forms of the dissipation and cooling functions can be written as $\Gamma=r^{2}\Sigma\nu\|\frac{d\Omega}{dr}\|^{2}$ (6) $\Lambda=\frac{1}{\gamma(\gamma-1)}\frac{\Sigma c_{s}^{2}}{\tau_{cool}}$ (7) where $\tau_{cool}$ is cooling timescale. As noted in the introduction, we are interest to consider the effect of cooling function on the structure of self- gravitating discs. Thus, similar to Rice & Armitage (2009) we will study the effects of it in the case of the heating rate in disc is equal to cooling rate, $\Gamma=\Lambda$. Since fragmentation requires fast cooling, Gammie (2001) suggested the cooling timescale can be parameterized as $\beta=\Omega\tau_{cool}$ , where $\beta$ is a free parameter. Gammie (2001) showed fragmentation requires $\beta~{}\lesssim~{}\beta_{crit}$, where $\beta_{crit}\approx 3$ for the adiabatic index of $\gamma=2$. Rice et al. (2005) performed 3D simulations to show the dependence of $\beta_{crit}$ on $\gamma$: for discs with $\gamma=5/3$ and $7/5$, $\beta_{crit}\approx 6-7$ and $\approx 12-13$, respectively. Recently, Cossins et al. (2010) studied $\beta$ as a function of temperature. They showed that $\beta$ has a strong dependence on the local temperature. They found that without temperature dependence, for radii $\lesssim~{}10au$ a very large accretion rate $~{}10^{-3}~{}M_{\odot}~{}yr^{-1}$ is required for fragmentation, but that this is reduced to $10^{-4}~{}M_{\odot}~{}yr^{-1}$ with cooling of dependent on temperature. So, for simplicity in this paper we will use a cooling timescale with a power-law dependence on temperature for study of the equations (1)-(5) $\displaystyle\tau_{cool}=\frac{\beta_{0}}{\Omega}(\frac{T}{T_{0}})^{\delta}$ $\displaystyle=\frac{\beta_{0}}{\Omega}(\frac{c_{s}}{c_{s_{0}}})^{2\delta}$ (8) that $\delta$ and $\beta_{0}$ are free parameters, and if we select $T_{0}$ as a temperature of the outer part of the disc, then $c_{s_{0}}$ will be sound speed in there. From equation (8) and $\delta=0$, we expect that $\Omega\tau_{cool}$ becomes a constant that is same with Gammie (2001) model. While non-zero $\delta$ is qualitatively consistent with Cossins et al. (2010) model. It is important to stress that the above description for cooling rate is not meant to reproduce any specific cooling law, but is just a convenient way of exploring the role of the cooling timescale in the outcome of the gravitational instability. Here, the kinematic coefficient of viscosity can be obtained by equating of the heating and cooling rates $\nu=\frac{1}{\gamma(\gamma-1)}\frac{\left\|\frac{d\Omega}{dr}\right\|^{-2}}{r^{2}}\frac{c_{s}^{2}}{\tau_{cool}}.$ (9) Thus, by exploit of equation (9) we do not need to use of viscosity descriptions, such as $\alpha$ and $\beta$ prescriptions that are introduced by Shakura & Sunyaev (1973) and Duschel et al. (2000), respectively. Equation (9) implies that the kinematic coefficient of viscosity in the present model depends on physical quantities of the system, specially cooling timescale. The kinematic coefficient of viscosity in $\alpha$-prescription is $\nu=\alpha c_{s}h$, where $\alpha$ is a free parameter and is less than unity (Shakura & Sunyaev 1973). By using equation (9) for $\alpha$ parameter we can write $\displaystyle\alpha=\frac{\nu}{c_{s}h}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$ $\displaystyle=\frac{1}{\gamma(\gamma-1)}\frac{\left\|\frac{d\Omega}{dr}\right\|^{-2}}{r^{2}h}\frac{c_{s}}{\tau_{cool}}.$ (10) The above equation implies that the $\alpha$ parameter is not a constant and varies by position and strongly depends on cooling timescale. We will study the $\alpha$ parameter in section 4 and will show that in the present model it increases by radii. As mentioned in the introduction, the gravitational stability of the disc can be investigated by Toomre parameter (Toomre 1964). The Toomre parameter for an epicyclic motion can be written as $Q=\frac{c_{s}k}{\pi G\Sigma}$ (11) where $k=\Omega\sqrt{4+2\frac{d\log\Omega}{d\log r}}$ (12) is the epicyclic frequency which can be replaced by the angular frequency, $\Omega$. The equations of (1)-(5) and (9) provide a set of ordinary differential equations that describes physical properties of the self-gravitating disc. Since, these equations are nonlinear, we will need suitable boundary conditions to solve it numerically. Thus, in next section we will try to obtain asymptotic solution in outer edge of the disc and then by exploit of this asymptotic solution as a boundary condition, we will able to integrate system equations inward from a point very near to the outer edge of the disc. Before next sections and the numerical study of the model, we shall express all quantities in units with values typical protostellar disc. We will choose astronomical unit ($au$) and the sun mass ($M_{\odot}$) as the units of length and mass, respectively. Thus, the time unit is given by $\sqrt{au^{3}/GM_{\odot}}$ that is equal to a year divided to $2\pi$. ## 3 Outer Limit Here, the asymptotic behavior of the system equations as $r\rightarrow R$ is investigated that $R$ is the outer radius of the disc. The asymptotic solutions are given by $\Sigma(r)\sim\frac{\Sigma_{0}}{R^{1/2}}~{}\left(1+a_{1}\frac{s}{R}+\cdot\cdot\cdot\right)$ (13) $v_{r}(r)\sim- c_{1}\sqrt{\frac{M_{}+M_{disc}}{R}}~{}\left(1+a_{2}\frac{s}{R}+\cdot\cdot\cdot\right)$ (14) $\Omega(r)\sim c_{2}\sqrt{\frac{M_{}+M_{disc}}{R^{3}}}~{}\left(1+a_{3}\frac{s}{R}+\cdot\cdot\cdot\right)$ (15) $c_{s}^{2}(r)\sim c_{3}\frac{M_{}+M_{disc}}{R}~{}\left(1+a_{4}\frac{s}{R}+\cdot\cdot\cdot\right)$ (16) $M(r)\sim M_{disc}-\int^{R}_{r}2\pi r^{\prime}\Sigma(r^{\prime})dr^{\prime}$ (17) where $s=R-r$, $M_{disc}$ is the disc mass, and the coefficients of $c_{i}$, $a_{i}$, and $\Sigma_{0}$ must be determined. Using these solutions, from the continuity, momentum, angular momentum, energy, and viscosity equations [(1)-(5), and (9)], we can obtain the coefficients of $c_{i}$ that have the following forms: $c_{1}=\frac{\dot{M}}{2\pi\Sigma_{0}\sqrt{M_{}+M_{disc}}}$ (18) $\displaystyle c_{2}^{2}+\left[\frac{a_{3}\gamma(\gamma-1)\beta_{0}\dot{M}(a_{3}-2)(a_{1}+a_{4})}{2\pi\Sigma_{0}\sqrt{M_{}+M_{disc}}(a_{1}+a_{3}+a_{4}-1)}\right]c_{2}$ $\displaystyle+\left[\frac{a_{2}\dot{M}^{2}}{4\pi^{2}\Sigma_{0}^{2}(M_{}+M_{disc})}-1\right]=0$ (19) $c_{3}=\left(\frac{a_{3}\gamma\beta_{0}(a_{3}-2)(\gamma-1)\dot{M}}{2\pi\Sigma_{0}(a_{1}+a_{3}+a_{4}-1)\sqrt{M_{}+M_{disc}}}\right)c_{2}$ (20) where $a_{4}=(1+a_{2})(1-\gamma).$ (21) The amount of mass accretion rate can be determined by observational evidences of the protoplanetary discs. Also, $\Sigma_{0}$ approximately can be determined by disc mass, $M_{disc}\sim\pi R^{2}\Sigma$. Thus, knowing the amounts of $\Sigma_{0}$ and $\dot{M}$ from the observations, the value of $c_{3}$ coefficient is only depended on value of $c_{2}$. On the other hand, the value of $c_{2}$ can be obtained by equation (19). Since, we have only one equation for coefficients of $a_{i}$ (equation 21), we will select below values for them in duration of numerical integration of system equations to obtain physical results $a_{1}<-2+\frac{3}{2}\,\gamma,~{}~{}3\,a_{2}=a_{3}=\frac{3}{2},~{}~{}a_{4}=(1+a_{2})(1-\gamma).$ (22) Figure 1: Surface density, thickness, temperature, and Toomre parameter of the disc as a function of radius, for several values of $\delta$. The surface density and the temperature are in $cgs$ system, and the thickness and the distance are in $au$ unit. The solid lines represent $\delta=0$, the dashed lines represent $\delta=0.75$, and the dotted lines represent $\delta=1.5$. The input parameters are set to the disc mass $M_{disc}=0.1M_{\odot}$, the star mass $M_{}=M_{\odot}$, the mass accretion rate $\dot{M}=10^{-6}M_{\odot}yr^{-1}$, the ratio of the specific heats is set to be $\gamma=5/3$, and $\beta_{0}=2$. ## 4 Numerical Results If the value of $R$ is guessed, the equations by Fehlberg-Runge-Kutta fourth- fifth order method can be integrated inwards from a point very near to the outer edge of the disc, using the above expansions. Examples of such solutions for surface density, half-thickness of the disc, temperature, Toomre parameter, and the viscous parameter of $\alpha$ as a function of radius are presented in Figs 1-5. The delineated quantities of $T$ in Figs 1-4 is the mid-plane temperature and can then be determined using $\displaystyle T=\left(\frac{\mu m_{p}}{k_{B}}\right)c_{s}^{2}$ where $\mu=2$ is the mean molecular weight, $m_{p}$ is the proton mass, and $k_{B}$ is Boltzmann’s constant. ### 4.1 The influences of physical parameters on the results The free parameters in the present model are the importance degree of temperature in cooling timescale, $\delta$, the mass accretion rate, $\dot{M}$, the parameter of $\beta_{0}$, the ratio of disc mass to star mass, $q=M_{disc}/M_{}$. #### 4.1.1 $\delta$ parameter The effects of $\delta$ parameter on the physical quantities are presented in Fig. 1. The profiles of surface density and temperature show that they increase by adding $\delta$. But, the increase of surface density is more than temperature. Thus, the Toomre parameter ($Q\propto c_{s}/\Sigma\propto\sqrt{T}/\Sigma$) decreases by adding $\delta$ parameter. The profiles of Toomre parameter represent that for small $\delta$, only outer part of the disc gravitationally is unstable, and the gravitational instability can extend to inner radii by adding $\delta$ parameter. For $\delta_{crit}\sim 1.5$, the Toomre parameter in radii $\gtrsim 5\,au$ becomes smaller than critical Toomre parameter ($Q_{cri}\sim 1$) and the disc becomes gravitationally unstable. In the other words, the profiles of Toomre parameter represent the gravitational instability of the flow strongly depends on cooling timescale with temperature dependence. This result is qualitatively consistent with direct numerical simulations (e. g. Cossins et al. 2010). The disc thickness increases by adding $\delta$ parameter. It can be due to the increase of the temperature ($h\propto c_{s}\propto\sqrt{T}$). Equations 8 and 9 imply that $\frac{\nu_{(\delta\neq 0)}}{\nu_{(\delta=0)}}=\left(\frac{c_{s}}{c_{s_{0}}}\right)^{-2\delta}.$ (23) Since $c_{s}\geq c_{s_{0}}$ the right-hand side of above equation is equal or less than one. On the other hand, non-zero $\delta$ constrains lower viscosity for hotter regions of the disc. The study of gravitational instability shows that it enhances in lower viscosity (Abbassi et al. 2006; Shadmehri & Khajenabi 2006; Khajenabi & Shadmehri 2007). Thus, the gravitational instability can be enhanced by adding the $\delta$ parameter for hotter regions. But there is a limitation for the amount of $\delta$ parameter that we discuss it in next section. Figure 2: Surface density, thickness, temperature, and Toomre parameter of the disc as a function of radius, for several values of $\beta_{0}$. The surface density and the temperature are in $cgs$ system, and the thickness and the distance are in $au$ unit. The solid lines represent $\beta_{0}=1$, the dashed lines represent $\beta_{0}=5.0$, and the dotted lines represent $\beta_{0}=10$. The input parameters are set to the disc mass $M_{disc}=0.1M_{\odot}$, the star mass $M_{}=M_{\odot}$, the mass accretion rate $\dot{M}=10^{-6}M_{\odot}yr^{-1}$, the ratio of the specific heats is set to be $\gamma=5/3$, and $\delta=1.0$. #### 4.1.2 $\beta_{0}$ parameter The influences of parameter of $\beta_{0}$ are shown in Fig. 2. As, we know from the simulations of self-gravitating disc (Gammie 2001; Rice et al. 2003), the reduce of this parameter provides conditions that the disc places on gravitational instability and consequently fragmentation. The profiles of surface density show that it does not change by adding the $\beta_{0}$ parameter and only it shows small deviations in large radii. The disc temperature increases by adding the $\beta_{0}$ parameter. Because, the increase of this parameter reduces the rate of cooling. In large amount of $\beta_{0}$ ($\sim 10$), the disc is gravitationally stable, while by reduce of its value to $5$, the gravitational instability can occur in large radii, and for the small value of it ($\beta_{0}\sim 1$), we can expect gravitational instability in whole of the disc excluding near to the star. These results are qualitatively consistent with direct numerical simulations of protoplanetary disc (Gammie 2001; Rice et al. 2003; Cossins et al. 2010). Also, the solutions show that the disc thickness increases by adding the $\beta_{0}$ parameter. Figure 3: Surface density, thickness, temperature, and Toomre parameter of the disc as a function of radius, for several values of $\dot{M}$. The surface density and the temperature are in $cgs$ system, and the thickness and the distance are in $au$ unit. The solid lines represent $\dot{M}=10^{-7}M_{\odot}yr^{-1}$, the dashed lines represent $\dot{M}=5\times 10^{-7}M_{\odot}yr^{-1}$, and the dotted lines represent $\dot{M}=10^{-6}M_{\odot}yr^{-1}$. The input parameters are set to the disc mass $M_{disc}=0.1M_{\odot}$, the star mass $M_{}=M_{\odot}$, the ratio of the specific heats is set to be $\gamma=5/3$, $\beta_{0}=10$ and $\delta=1.0$. #### 4.1.3 The mass accretion rate Rice & Armitage (2009) showed that beyond of $1\,au$ the disc reaches a quasi- steady state in $20000$ years and mass is redistributing itself to produce a state in which the accretion rate is largely independent of $r$. The mass accretion rate in their simulations finally reached to $10^{-6}-10^{-7}M_{\odot}/yr$ (see Fig 4 in their paper). We will study the behavior of the present model in Fig 3 for several values of the mass accretion rate ($10^{-7}$, $5\times 10^{-7}$, and $10^{-6}M_{\odot}/yr$). The solutions imply that the disc temperature is sensitive to the amount of mass accretion rate and increases by adding the mass accretion rate. While, the surface density is not sensitive to the mass accretion rate and only shows small variations in large radii. Thus, the behavior of the temperature only specifies the behavior of the Toomre parameter ($Q\propto\sqrt{T}/\Sigma$). The profiles of Toomre parameter represent that it increases by adding the mass accretion rate. Also, the solutions show the disc thickness increases by adding mass accretion rate, that is due to increase of the disc temperature. The solutions show that for a low mass accretion rate ($\sim 10^{-7}M_{\odot}/yr$), but cooling timescale with temperature dependence ($\delta\sim 1$), the gravitational instability can occur for radii $\gtrsim 10\,au$. Figure 4: Surface density, thickness, temperature, and Toomre parameter of the disc as a function of radius, for several values of $q=M_{disc}/M_{}$. The surface density and the temperature are in $cgs$ system, and the thickness and the distance are in $au$ unit. The solid lines represent $q=0.05$, the dashed lines represent $q=0.1$, and the dotted lines represent $q=0.15$. The input parameters are set to the star mass $M_{}=M_{\odot}$, the mass accretion rate $\dot{M}=10^{-6}M_{\odot}yr^{-1}$, the ratio of the specific heats is set to be $\gamma=5/3$, $\beta_{0}=2$ and $\delta=1.5$. #### 4.1.4 Mass ratio As noted in the introduction, semi-analytical studies of self-gravitating discs are regarding discs without central object. This simplification is relevant to protostellar discs at the beginning of the accretion phase, during which the mass of the central object is small and only self-gravity of the disk plays an important role. Also, this simplification can correspond to discs at large radii because the effects of the central mass become unimportant in the outer regions of the disc. As, the central object attends in the present model and its effects are not ignored. Thus, the present model does not have limitations of previous studies of semi-analytical self- gravitating discs and can be applied for all region of the disc. Fig. 4 represents the effects of the ratio of the disc mass to the star mass $q=M_{}/M_{disc}$ on the present model. The solutions show the surface density increases and the temperature decreases. Each of the surface density increasing and the temperature decreasing individually can reduce the Toomre parameter. Thus, we expect that Toomre parameter decreases by adding $q$ parameter that the profiles of Toommre parameter confirm it. The disc thickness profiles represent the disc thickness decreases by adding the disc mass. This property is qualitatively consistent with two-dimensional study of self-gravitating disc (e. g. Ghanbari & Abbassi 2004). Figure 5: The viscous parameter of $\alpha$ as a function of radius ($au$). The input parameters are set to the star mass $M_{}=M_{\odot}$, the mass accretion rate $\dot{M}=10^{-6}M_{\odot}yr^{-1}$, the ratio of the specific heats is set to be $\gamma=5/3$. Left panel is for several values of Gammie’s parameter $\beta_{0}$, the solid line represents $\beta_{0}=1$, the dashed line represents $\beta_{0}=5$, and the dotted line represents $\beta_{0}=10$, and $\delta=1.5$. Right panel is for several values of parameter of $\delta$, the solid line represents $\delta=0.5$, the dashed line represents $\delta=1.0$, and the dotted line represents $\delta=1.5$, and $\beta_{0}=1.0$. ### 4.2 The viscous parameter of $\alpha$ In the present model, the viscous parameter of $\alpha$ depends on the physical quantities of the disc (Equation 10), especially the local cooling rate which depends on the local temperature. The profiles of the viscous parameter of $\alpha$ show that it increases by radii that this property is agree with simulation results of Rice & Armitage (2009) and Rice et al. (2010). As, mentioned in the introduction, the minimum cooling timescale depends on the equation of state (Rice et al. 2005) with fragmentation occurring for $\tau_{cool}\leq 3\Omega^{-1}$ when the specific heat ratio $\gamma=5/3$ (Gammie 2001). Rice et al. (2005) showed that fragmentation occurs for $\alpha>0.06$ and this boundary is independent of the specific heat ratio $\gamma$. Left panel of Fig 5 represents the viscous parameter of $\alpha$ as a function of radius for several values of the $\beta_{0}$ parameter. The solutions show the viscous $\alpha$ strongly depends on the $\beta_{0}$ parameter. As, the $\alpha$ parameter decreases by factor of $\beta_{0}$. Also, the solutions for small values of $\beta_{0}$ show the viscous $\alpha$ can reach to its critical value for fragmentation. Right panel of Fig 5 represents the viscous parameter of $\alpha$ as a function of radius for several values of the $\delta$ parameter. The solutions represent the $\alpha$ parameter excluding the outer region of the disc strongly depends on the $\delta$ parameter. In $\delta=0.5$, the value of viscous $\alpha$ in whole of the disc is in the region for fragmentation. However, Rafikov (2005) suggested that it is extremely difficult to see how fragmentation can occur within $10\,au$ even for the relatively massive discs. In $\delta=1.0$ and $\delta=1.5$, the viscous $\alpha$ in the inner disc ($r\lesssim 10$ and $40\,au$, respectively) is well below that required for fragmentation. The requirements for fragmentation are $Q\lesssim 1$ and $\alpha>0.06$ (Rice et al. 2005, 2010; Rice & Armitage 2009). In the present model, apparently the increase of the $\delta$ parameter reduces possibility of fragmentation (Right panel of Fig 5). On the other hand, the increase of $\delta$ parameter can place the disc in gravitational instability (Fig 1). Thus, by a suitable value for the $\delta$ parameter, the disc can obtain two requirements for fragmentation. The Figs 1 and 5 imply that this value for small $\beta_{0}$ can be between $0.5$ and $1.0$. ## 5 Summary and Discussion In this paper, we have studied self-gravitating accretion discs in presence of a Newtonian potential of a point mass. We have used a prescription for cooling that is introduced by Gammie (2001). But, due to recent results of Cossins et al. (2010), we have assumed that cooling timescale in unit of dynamical timescale is a power-law function of temperature. Because of, the system equations are non-linear and there is not self-similar solution for it. First, we have obtained asymptotic solutions for system equations and then by them as boundary conditions, we integrated system equations numerically. The solutions showed that the structure of the disc strongly depends on the present cooling function. As, by adding importance degree of temperature in cooling timescale, gravitational instability extends from outer to inner radii. The solutions showed that in the case of cooling with temperature dependence, the disc thickness increases. But, this change of thickness is important in region with smaller Toomre parameter. In the present model, the effect of physical parameters studied such as mass accretion rate, $\beta_{0}$ parameter, and the ratio of the disc mass to central object mass. The results showed the structure of the disc is sensitive to these parameters. For example, the disc becomes gravitationally stable in larger mass accretion rate. While, the gravitational instability can occur in the larger disc mass. Also, the disc thickness increases by adding the mass accretion rate and decreases by adding the ratio of the disc mass to the star mass. The study of the viscous parameter $\alpha$ in the present model shows that it increases by radii that this result is consistent with direct numerical simulations (e. g. Rice & Armitage 2009; Rice et al. 2010). Also, the solution implies that the viscous $\alpha$ in the outer part of the disc becomes larger than its critical value ($\sim 0.06$) that might mean condition for fragmentation. Here, the solutions represented that the disc thickness is very sensitive to input parameters. Thus, study of the present in a two dimensional approach may be interesting subject for future works. Also, it will be interesting to obtain a suitable $\delta$ value for fragmentation by direct numerical simulations. ## Acknowledgments I would like to acknowledge useful discussions with Alireza Khesali. ## References * [1] * [2] [] Abbassi S., Ghanbari J., Salehi F., 2006, A&A, 460, 357 * [3] * [4] [] Bertin G., Lodato G., 1999, A&A, 350, 694 * [5] * [6] [] Bertin G., Lodato G., 2001, A&A, 370, 34 * [7] * [8] [] Boss A. P., 1998, ApJ, 503, 923 * [9] * [10] [] Boss A. P., 2002, ApJ, 576, 462 * [11] * [12] [] Cossins P., Lodato G., Clarke C., 2010, MNRAS, 401, 2587 * [13] * [14] [] Durisen R. H., Boss A. P., Mayer L., Nelson A. F., Quinn T., Rice W. K. M., 2007, in Reipurth B., Jewitt D., Keil K., eds, Protostars and Planets V. Univ. of Arizona Press, Tucson, p. 701 * [15] * [16] [] Duschl W.J., Stritmatter P. A., Bierman, P. L., 2000, A&A, 357, 1123 * [17] * [18] [] Faghei, K., 2011, J. Astrophys. Astr., accepted * [19] [] Fleming T. P., Stone J. M., Hawley J. F., 2000, ApJ, 530, 464 * [20] * [21] [] Gammie C. F., 2001, ApJ, 553, 174 * [22] * [23] [] Ghanbari J., Abbassi S., 2004, MNRAS, 350, 1437 * [24] * [25] Khajenabi F., Shadmehri M., 2007, MNRAS, 377, 1689 * [26] * [27] [] Kolykhalov P. I., Sunyaev R. A., 1979, Soviet Astronomy Letters, 5, 180 * [28] * [29] [] Krasnopolsky R., Konigl A., 2002, ApJ, 580, 987 * [30] * [31] [] Li Z.-Y., Shu F. H., 1997, ApJ, 475, 237 * [32] * [33] [] Lin D. N. C., Pringle J. E., 1987, MNRAS, 225, 607 * [34] * [35] [] Lin D. N. C., Pringle J. E., 1990, MNRAS, 358, 515 * [36] * [37] [] Lodato G., 2007, La Rivista del Nuovo Cimento, 30, 293 * [38] * [39] [] Masada Y., Sano T, 2008, ApJ, 689, 1234 * [40] * [41] [] Meru F., Bate M. R., 2011a, MNRAS, 410, 559 * [42] * [43] [] Meru F., Bate M. R., 2011b, MNRAS, 411, L1 * [44] * [45] [] Mineshige, S., Umemura, M., 1996, ApJ, 469, L49 * [46] * [47] [] Mineshige, S., Umemura, M., 1997, ApJ, 480, 167 * [48] * [49] [] Narayan, R., Yi, I. 1994, ApJ, 428, L13 * [50] * [51] [] Paczyński B., 1978, Acta Astronomica, 28, 91 * [52] * [53] [] Rafikov R. R., 2005, ApJ, 621, L69 * [54] * [55] [] Rice W. K. M., Armitage P. J., Bate M. R., Bonnell I. A., 2003, MNRAS, 339, 1025 * [56] * [57] [] Rice W. K. M., Lodato G., Armitage P. J., 2005, MNRAS, 364, L56 339, 1025 * [58] * [59] [] Rice W. K. M., Armitage P. J., 2009, MNRAS, 396, 2228 * [60] * [61] [] Rice W. K. M., Mayo J. H., Armitage P. J., 2010, MNRAS, 402, 1740 * [62] * [63] [] Saigo K., Hanawa T., 1998, ApJ, 493, 342 * [64] * [65] [] Shadmehri M., 2004, ApJ, 612, 1000 * [66] * [67] [] Shadmehri M., Khajenabi F., 2006, ApJ, 637, 439 * [68] * [69] Shadmehri M., 2009, MNRAS, 395, 877 * [70] * [71] [] Shakura, N.I., Sunyaev, R.A., 1973, A&A, 24, 337 * [72] * [73] [] Toomre A., 1964, ApJ, 139, 1217 * [74] * [75] [] Tsuribe T., 1999, ApJ, 527, 102	arxiv-papers	2011-11-08T15:27:44	2024-09-04T02:49:24.123875	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Kazem Faghei", "submitter": "Kazem Faghei", "url": "https://arxiv.org/abs/1111.1937" }
1111.1962	# Quantum solution to a three player Kolkata restaurant problem using entangled qutrits Puya Sharif† and Hoshang Heydari Department of physics, Stockholm university 10691 Stockholm Sweden $\dagger$ Email: ps@puyasharif.net ###### Abstract Three player quantum Kolkata restaurant problem is modeled using three entangled qutrits. This first use of three level quantum states in this context is a step towards a $N$-choice generalization of the $N$-player quantum minority game. It is shown that a better than classical payoff is achieved by a Nash equilibrium solution where the space of available strategies is spanned by subsets of SU(3) and the players share a tripartite entangled initial state. Keywords: Quantum information theory, Quantum game theory, Quantum minority games, Qutrits, Three level systems, Multipartite entangled states. ## 1 Introduction Quantum game theory is a fairly recent extension of game theoretical analysis to situations formulated in the framework of quantum information theory. The first papers appeared in 1999. Meyer showed with a model of a penny-flip game that a player making a _quantum move_ always comes out as a winner against a player making a _classical_ move regardless of the classical players choice [1]. The same year Eisert et al. published a quantum protocol in which they overcame the dilemma in Prisoners dilemma [2]. In 2003 Benjamin and Hayden generalized Eisert’s protocol to handle multiplayer quantum games and introduced the quantum minority game together with a solution for the four player case which outperformed the classical randomization strategy [3]. This result was later generalized to the $n$-players by Chen et al. in 2004 [4]. Multiplayer minority games has since then been extensively investigated by Flitney et al. [5, 6, 7]. We will here extend quantum minority games to situations where there are not only multiple players, but also multiple choices. A quantum version of the Kolkata restaurant problem, which is a generalized minority game will be presented. The players uses maximally entangled qutrits as a quantum resource and selects their strategy by locally acting with a general SU(3) operator on the qutrit in their possession. ### 1.1 Kolkata restaurant problem The Kolkata restaurant problem is a minority-type game [8, 9, 10, 11, 12]. In its most general form $N$ non-communicating agents (players), have to choose between $n$ choices. The agents receive a gain in their utility if their choice is not too crowded, i.e the number of agents that made the same choice is under some threshold limit. The choices can also have different values of utility associated with them, accounting for a preference profile over the set of choices. The original formulation comes with a story of workers in Kolkata that during lunch hours has to choose between a fixed number of cheap restaurants. Each restaurant can only serve a finite number of customers, so workers arriving to a crowded restaurant will simply miss the opportunity of having lunch. Often is the number of agents taken to be equal to the number of restaurants, and the maximum number of costumers per restaurant limited to one. The problem is usually modeled as an iterative game where agents ought to base their decision on information about the distribution of agents over choices in the previous iterations. The Kolkata restaurant problem offers therefore a method for modeling heard behavior and market dynamics, where visiting a restaurant translates to buying a security, in which case an agent wishes to be the only bidder. ### 1.2 The model In our simplified model there are just three agents, Alice, Bob and Charlie. They have three possible choices: security 0, security 1 and security 2. They receive a payoff $\$$ of one unit if their choice is unique, i.e that nobody else has made the same choice, otherwise they receive $\$=0$. The game is so called _one shoot_ , which means that it is non-iterative, and the agents have no information from previous rounds to base their decisions on. Under the constraint that they cannot communicate, there is nothing left to do other than randomizing between the choices. Given the symmetric nature of the problem, any deterministic strategy would lead all three agents to the same strategy, which in turn would mean that all three would leave empty handed. There are $27$ different strategy profiles possible, i.e combinations of choices. $12$ of which gives a payoff of $\$=1$ to each one of them. Randomization gives therefore agent $i$ an expected payoff of $E^{c}(\$)=\frac{4}{9}$, where the superscript denotes that the result is due to the best _classical_ strategy (as opposed to _quantum_ strategy). In the framework of quantum game theory [13, 14, 15, 16], Alice, Bob and Charlie shares a quantum resource. Each has a part of a multipartite quantum state. They play their strategy by manipulating their own part of the combined system, before measuring their subsystems and choosing accordingly. Whereas classically the players would be allowed randomizing over a discrete set of choices, in the quantum version each subsystem is allowed to be transformed with the full machinery of quantum operations. A strategy, or choice therefore translates to choosing a unitary operator $U$. In the absence of entanglement, quantum games of this type usually yield the same payoffs as their classical counterparts, whereas the combination of unitary operators (or a subset therein) and entanglement, sometimes strongly outperforms classical games and decision theoretic models. We will here present such a case. ## 2 Qutrits and parametrization of SU(3) A qutrit is a 3-level quantum system on 3-dimensional Hilbert space $\mathcal{H}=\mathbb{C}^{3}$ , written in the computational basis as: $\|\psi\rangle=a_{0}\|0\rangle+a_{1}\|1\rangle+a_{2}\|2\rangle\in\mathbb{C}^{3},$ (2.0.1) with $a_{0},a_{1},a_{2}\in\mathbb{C}$ and $\|a_{0}\|^{2}+\|a_{1}\|^{2}+\|a_{2}\|^{2}=1$. A general $N$-qutrit system $\left\|\Psi\right\rangle$ is a vector on $3^{N}$-dimensional Hilbert space, and is written as a linear combination of $3^{N}$ orthonormal basis vectors. $\left\|\Psi\right\rangle=\sum_{x_{N},..,x_{1}=0}^{2}a_{x_{N}...x_{1}}\left\|x_{N}\cdots x_{1}\right\rangle,$ (2.0.2) where $\left\|x_{N}\cdots x_{1}\right\rangle=\left\|x_{N}\right\rangle\otimes\left\|x_{N-1}\right\rangle\otimes\cdots\otimes\left\|x_{1}\right\rangle\in\mathcal{H}=\overbrace{\mathbb{C}^{3}\otimes...\otimes\mathbb{C}^{3}}^{\text{$N$-times}},$ (2.0.3) with $x_{i}\in\\{0,1,2\\}$ and complex coefficients $a_{x_{i}}$, obeying $\sum\|a_{x_{N}...x_{1}}\|^{2}=1$. Single qutrits are transformed with unitary operators $U\in$ SU(3), i.e operators from the special unitary group of dimension 3, acting on $\mathcal{H}$ as $U:\mathcal{H}\rightarrow\mathcal{H}$. In a multi-qutrit system, operations on single qutrits are said to be local. They affect the state-space of the corresponding qutrit only. The transformation of a multi- qutrit state vector under local operations is given by the tensor products of the individual operators: $\left\|\Psi_{fin}\right\rangle=U_{N}\otimes U_{N-1}\otimes\cdots\otimes U_{1}\left\|\Psi_{in}\right\rangle,$ (2.0.4) where $\left\|\Psi_{in}\right\rangle$ and $\left\|\Psi_{fin}\right\rangle$ denotes the initial and final state of the system respectively. There are a number of ways you can parameterize SU(3) [18, 19]. One common approach is through the Lie algebra of the group, the eight traceless $3\times 3$ Gell-Mann matrices. We are using a different and maybe slightly more intuitive parametrization [17]. Let $\bar{x},\bar{y},\bar{z}$ be three general, mutually orthogonal complex unit vectors, such that $\bar{x}\cdot\bar{y}=0$ and $\bar{x}^{}\times\bar{y}=\bar{z}$. We construct a SU(3) matrix by placing $\bar{x},\bar{y}^{}$ and $\bar{z}$ as its columns. Now a general complex unit vector is given by: $\bar{x}=\left(\begin{array}[]{c}\sin\theta\cos\phi e^{i\alpha_{1}}\\\ \sin\theta\sin\phi e^{i\alpha_{2}}\\\ \cos\theta e^{i\alpha_{3}}\end{array}\right),$ (2.0.5) and one complex unit vector orthogonal to $\bar{x}$ is given by: $\bar{y}=\left(\begin{array}[]{c}\cos\chi\cos\theta\cos\phi e^{i(\beta_{1}-\alpha_{1})}+\sin\chi\sin\phi e^{i(\beta_{2}-\alpha_{1})}\\\ \cos\chi\cos\theta\sin\phi e^{i(\beta_{1}-\alpha_{2})}-\sin\chi\cos\phi e^{i(\beta_{2}-\alpha_{2})}\\\ -\cos\chi\sin\theta e^{i(\beta_{1}-\alpha_{3})}\end{array}\right),$ (2.0.6) where $0\leq\phi,\theta,\chi,\leq\pi/2$ and $0\leq\alpha_{1},\alpha_{2},\alpha_{3},\beta_{1},\beta_{2}\leq 2\pi$. We have a general SU(3) matrix $U$, given by: $U=\left(\begin{array}[]{ccc}x_{1}&y_{1}^{}&x_{2}^{}y_{3}-y_{3}^{}x_{2}\\\ x_{2}&y_{2}^{}&x_{3}^{}y_{1}-y_{1}^{}x_{3}\\\ x_{3}&y_{3}^{}&x_{1}^{}y_{2}-y_{2}^{}x_{1}\end{array}\right),$ (2.0.7) and it is controlled by eight real parameters ${\phi,\theta,\chi,\alpha_{1},\alpha_{2},\alpha_{3},\beta_{1},\beta_{2}}$. ## 3 The scheme The scheme under study is a development of one first introduced by Eisert et al. [2], and later generalized by Benjamin and Hayden [3]. It starts out with Alice, Bob and Charlie, $A,B$ and $C$ respectively, sharing a quantum resource, an entangled tripartite 3-level quantum state. We need to allow the quantum states to have a common origin, since creating entanglement is a global operation, and can t be done by acting locally on the subsystems. We assume that there exists an unbiased referee that prepares the state and distributes the subsystems among the players. From that point on, no communication is allowed between the players and the referee. Each qutrit is due to be measured by the player owning it, at the end of the protocol in the $\\{\|0\rangle,\|1\rangle,\|2\rangle\\}$ -basis, where basis vector corresponds to one of the three choices: security 0, security 1, and security 2. The players plays their strategy by applying an operator from the set of allowed strategies $S$, followed by a local measurement which determines their final choice. The unitary operations done by $A,B,C$ are done locally, which means that the operator is applied on the subsystem held by the player. As mentioned, this translates to the transformation of $\|\psi_{in}\rangle$ by the tensor product of the unitary operators applied by the players. We want to create a quantization of the classical game in which we expand the set of available strategies to include quantum moves. While we are proposing a quantum game which in some sense is fundamentally different from the classical version, we require it to be an extension, not an addition to the classical Kolkata restaurant problem. Tracing the steps of the predecessors of this protocol, we restrict our formulation to have the classical game fully present at all times, accessible in the form of restrictions on the set of allowed local operations. We simply require that there exists a set of operators that when applied locally on an entangled initial state gives the same outcomes as in the classical non-quantum version. Lets first look at the classical game presented with quantum formalism. Note that there is nothing quantum mechanical happening at this point. The initial state $\|\psi_{in}\rangle=\|000\rangle=\|0\rangle_{C}\otimes\|0\rangle_{B}\otimes\|0\rangle_{A}$ corresponds to the case where the three players chooses security 0, by default. The individual choices are made by applying operators $s_{i},s_{j},s_{k}\in S=\\{s_{0},s_{1},s_{2}\\}$ to each subsystem. The exact form of these operators can be left to discuss later. The only restriction at the moment is that they obey: $s_{0}\|0\rangle=\|0\rangle,\,s_{1}\|0\rangle=\|1\rangle,\,s_{2}\|0\rangle=\|2\rangle$, resulting in fully deterministic outcomes: $s_{i}\otimes s_{j}\otimes s_{k}\|000\rangle=\|i\,j\,k\rangle.$ (3.0.1) As mentioned earlier there are 27 different such outcomes, each linked to different combinations of the operators $\\{s_{0},s_{1},s_{2}\\}$, 12 of which gives a player a payoff $\$=1$, and the rest $\$=0$. Clearly there are no operators available corresponding to mixed strategies, so randomization processes leading to classical mixed strategies are here lifted outside the protocol and is done by the players before the choices are made. Having finished the first step in the quantization process our task is now to keep the classical game present throughout the coming steps while we add quantum structure by choosing an entangled initial state and expanding the set of strategy operators to include any $U\in$ $S$ = SU(3). Since the game is symmetric and unbiased in regards to permutation of player positions, then this is a property that has to be true of the initial state $\|\psi_{in}\rangle$, to assure that the payoff functions $\$_{i}(\|\psi_{in}\rangle,U_{A},U_{B},U_{C})$ of all three players $i$ are identical up to some permutation of $U_{A},U_{B},U_{C}$. Note that when dealing with mixed classical and quantum strategies the payoff function becomes an expectation value $E(\$)$ of a probability distribution over the different outcomes. We summarize the criteria for choosing an initial state: 1. 1. The state ought to be entangled, to accommodate for correlated randomization among the players. 2. 2. It should be symmetric and unbiased in regards to player positions. 3. 3. It must allow for classical game to be accessed by restrictions on the space of available strategy operators. The three qutrit GHZ-type-state: $\mid\psi_{in}\rangle=\frac{1}{\sqrt{3}}\left(\|000\rangle+\|111\rangle+\|222\rangle\right),$ (3.0.2) not only fulfills the above criteria, it is also a _maximally_ entangled state on $\mathcal{H}=\mathbb{C}^{3}\otimes\mathbb{C}^{3}\otimes\mathbb{C}^{3}$. It has the additional property of initializing the game in an maximally undesired state. i.e. one in which none of the players receives any payoff. In order to change their situation, they will have to make an active choice. It is left to show that we can define a set of operators corresponding to classical pure strategies that gives raise to deterministic classical payoffs when applied to the entangled initial state. This problem was addressed by Eisert et. al. [2], and further developed by Benjamin et.al. [3] for cases of $n$ players and two choices, by defining an entangling operator $J$ and its inverse $J^{\dagger}$, acting on a $n$-_qubit_ product state $\|00\cdots 0\rangle$ with Hermitian strategy operators $\hat{s_{0}},\hat{s_{1}}$, sandwiched in between. By showing that any combination of the classical strategies $\hat{s}_{x_{1}}\otimes\hat{s}_{x_{2}}\otimes\cdots\otimes\hat{s}_{x_{n}},x_{i}\in\\{0,1\\}$ commutes with $J$, one guarantees that the classical game is embedded in the quantum version. That route is not possible when formulating a game with aid of higher dimensional quantum states like qutrits, since at least _two different_ Lie-algebra elements of su(3) must appear in the Hamiltonian of $J$ (For the GHZ-type-state), whereby commutation is no longer a fact in the general case. We need a set of operators that replicates the classical payoffs when applied directly on our entangled initial state $\mid\psi_{in}\rangle$. The cyclic group of order three, $C_{3}$, generated by the matrix: $s=\ \left(\begin{array}[]{ccc}0&0&1\\\ 1&0&0\\\ 0&1&0\end{array}\right)\ ,$ (3.0.3) where $s^{3}=s^{0}=I$ and $s^{2}=s^{T}$, has the properties we are after. The set of classical strategies $S=\\{s^{0},s^{1},s^{2}\\}$ with $s^{i}\otimes s^{j}\otimes s^{k}\|000\rangle=\|i\,j\,k\rangle$ acts on the GHZ-state as: $s^{i}\otimes s^{j}\otimes s^{k}\frac{1}{\sqrt{3}}\left(\|000\rangle+\|111\rangle+\|222\rangle\right)=\\\ =\frac{1}{\sqrt{3}}\left(\|0+i\;0+j\;0+k\rangle+\|1+i\;1+j\;1+k\rangle+\|2+i\;2+j\;2+k\rangle\right).$ (3.0.4) Note that the superscripts denotes powers of the generator and that the addition is modulo 3. In the case under study, where there is no preference profile over the different choices, any combination of the operators in $S=\\{s^{0},s^{1},s^{2}\\}$ leads to the same payoffs when applied to the GHZ- state as to $\|000\rangle$. Now that an entangled initial state $\mid\psi_{in}\rangle$ is chosen, the scheme for the quantum game proceeds as follows. We form a density matrix $\rho_{in}$ out of the initial state $\mid\psi_{in}\rangle$ and add noise that can be controlled by the parameter $f$ [7]. We get: $\rho_{in}=f\mid\psi_{in}\rangle\langle\psi_{in}\mid+\frac{1-f}{27}\mathbb{I_{\mathrm{27}}},$ (3.0.5) where $\mathbb{I_{\mathrm{27}}}$ is the $27\times 27$ identity matrix. Alice, Bob and Charlie now applies a unitary operator $U$ that maximizes their chances of receiving a payoff $\$=1$, and thereby the initial state $\rho_{in}$ is transformed into the final state $\rho_{fin}$. $\rho_{fin}=U^{\dagger}\otimes U^{\dagger}\otimes U^{\dagger}\rho_{in}U\otimes U\otimes U.$ (3.0.6) Note that they are all applying the same operator $U$ since in the absence of communication, coordination of which operator to be applied by whom, would be impossible. We define for each player $i$ a payoff-operator $P_{i}$ , which contains the sum of orthogonal projectors associated with the states for which player $i$ receives a payoff $\$=1$ . For Alice this would correspond to $P_{A}=\left(\sum_{x_{3},x_{2},x_{1}=0}^{2}\|x_{3}x_{2}x_{1}\rangle\langle x_{3}x_{2}x_{1}\|,\,x_{3}\neq x_{2},x_{3}\neq x_{1},x_{2}\neq x_{1}\right)+\\\ +\left(\sum_{x_{3},x_{2},x_{1}=0}^{2}\|x_{3}x_{2}x_{1}\rangle\langle x_{3}x_{2}x_{1}\|,\,x_{3}=x_{2}\neq x_{1}\right).$ (3.0.7) The expected payoff $E_{i}(\$)$ of player $i$ is calculated by taking the trace of the product of the final state $\rho_{fin}$ and the payoff-operator $P_{i}$: $E_{i}(\$)=\mathrm{Tr\left(\mathit{P}_{i}\rho_{fin}\right)}.$ (3.0.8) ## 4 Optimal strategy The problem now is to find the unitary operator $U(\phi,\theta,\chi,\alpha_{1},\alpha_{2},\alpha_{3},\beta_{1},\beta_{2})$ that maximizes the expected payoff. Due to the symmetry of the problem, optimization can be done with respect to the $P_{i}$ of any of the three players. Doing so one arrives at a maximum expected payoff of $E(\$)=\frac{6}{9}$, assuming ($f=1$), compared to the classical $E^{c}(\$)=\frac{4}{9}$. Which is an $50\%$ increase. This occurs when the players applies the optimal unitary operator $U^{opt}$, whose parameters are listed in table 1. Parameter \| $\phi$ \| $\theta$ \| $\chi$ \| $\alpha_{1}$ \| $\alpha_{2}$ \| $\alpha_{3}$ \| $\beta_{1}$ \| $\beta_{2}$ ---\|---\|---\|---\|---\|---\|---\|---\|--- Value \| $\frac{\pi}{4}$ \| $\cos^{-1}\left(\frac{1}{\sqrt{3}}\right)$ \| $\frac{\pi}{4}$ \| $\frac{5\pi}{18}$ \| $\frac{5\pi}{18}$ \| $\frac{5\pi}{18}$ \| $\frac{\pi}{3}$ \| $\frac{11\pi}{6}$ Table 1: $U^{opt}$ in the given parametrization. Because of the periodic nature of the solution, there could be more than one unique choice for some of the parameters within the allowed ranges $0\leq\phi,\theta,\chi\leq\pi/2$ and $0\leq,\alpha_{1},\alpha_{2},\alpha_{3},\beta_{1},\beta_{2}\leq 2\pi$. This is the case for $\alpha_{1},\alpha_{2},\alpha_{3}$, where maximum expected payoff is achieved for $\left(\frac{5+12n}{18}\right)\pi$, $n\in\\{0,1,2\\}$. Noting that the the center of SU(3), Z(3) =$\\{I,e^{\pm\frac{i2\pi}{3}}I\\}$ only adds a global phase and leaves the density matrix invariant, one concludes that the transformation belongs to SU(3)/Z(3). This removes the above ambiguity, ending up with $\alpha_{1},\alpha_{2},\alpha_{3}=\frac{5\pi}{18}$. The final state arrived at by playing $U^{opt}$ is given by: $\mid\psi_{fin}\rangle=\frac{1}{3}\left(\|000\rangle+\|012\rangle+\|021\rangle+\|102\rangle\right.+\\\ \left.\|111\rangle+\|120\rangle+\|201\rangle+\|210\rangle+\|222\rangle\right).$ (4.0.1) This is an even distribution of all the states that leads to payoff to all three players and the states which gives payoff to none and shows that the $U^{opt}\otimes U^{opt}\otimes U^{opt}$-operation fails to make the state fully depart from the space spanned by $\|000\rangle,\|111\rangle,\|222\rangle$. This failure accounts for the expected payoff not reaching unity. Now by setting $\alpha_{1},\alpha_{2}=0$ and $\alpha_{3}=\alpha$, in the parametrization, one arrives at a six parameter subset of SU(3), given by operators $U(\phi,\theta,\chi,\alpha,\beta_{1},\beta_{2})$. The optimum is at the same value as with the transformation belonging to its domain. There is thereby a $V^{opt}$ in this subset, given in table 2 below, that gives each player an expected payoff of $E(\$)=\frac{6}{9}=\frac{2}{3}$. Parameter \| $\phi$ \| $\theta$ \| $\chi$ \| $\alpha$ \| $\beta_{1}$ \| $\beta_{2}$ ---\|---\|---\|---\|---\|---\|--- Value \| $\frac{\pi}{4}$ \| $\cos^{-1}\left(\frac{1}{\sqrt{3}}\right)$ \| $\frac{\pi}{4}$ \| $\frac{\pi}{2}$ \| $\frac{\pi}{3}$ \| $\frac{5\pi}{6}$ Table 2: $V^{opt}$ in the reduced parametrization. $U^{opt}$ and $V^{opt}$ differs only by a constant phase factor, so for our purposes, what’s true of one is true of the other. We will therefore regard the reduced parametrization when showing that the solution is a Nash equilibrium in the next section. If we further reduce the parametrization by letting all phase parameters $\alpha_{1},\alpha_{2},\alpha_{3},\beta_{1},\beta_{2}=0$, we end up with an operator $O(\phi,\theta,\chi)\in$ SO(3), i.e. the elements of the special orthogonal group of dimension 3. These operators corresponds to rotations in $\mathbb{R}^{3}$. In quantum games with two choices, like quantum prisoners dilemma and in minority games, local _orthogonal_ operations merely achieves to replicate the results of classical mixed strategies and offers no improvement in the expected payoff, even with a maximally entangled initial state. In this case though, there exists an $O^{opt}\in$ SO(3), given in table 3 that outperforms the classical expected payoff by a small margin. Each player would in this case receive a payoff of $E(\$)=\frac{40}{81}$, compared to the classical $E(\$)=\frac{4}{9}=\frac{36}{81}$. This result might open up the possibility of a new classification of quantum games, where there could exist a category of quantum games with classical strategies that are fundamentally different than classical games with classical strategies. Parameter \| $\phi$ \| $\theta$ \| $\chi$ ---\|---\|---\|--- Value \| $\frac{\pi}{6}$ \| $\cos^{-1}\left(\frac{1}{3}\right)$ \| $\frac{\pi}{6}$ Table 3: $O^{opt}$ in the given parametrization. ### 4.1 Nash equilibrium To show that this solution is valid from a game-theoretical point of view, we need to show that $V_{opt}$ is a Nash equilibrium, i.e. that none of the players gains by unilaterally changing strategy from $V^{opt}$ to any other strategy $U(\phi,\theta,\chi,\alpha,\beta_{1},\beta_{2})$. Without loss of generality, we show for the expected payoff $E_{A}(\$)$ of Alice that the following inequality holds for any $V$: $E_{A}(\$)(V_{C}^{opt}\otimes V_{B}^{opt}\otimes V_{A}^{opt})\geq E_{A}(\$)(V_{C}^{opt}\otimes V_{B}^{opt}\otimes V_{A}).$ (4.1.1) We show that this is the case by letting Alice act with a general unitary operator $V(\phi,\theta,\chi,\alpha,\beta_{1},\beta_{2})\in$ SU(3), while Bob and Charlie acts with $V^{opt}\;(U^{opt})$. Then we take the partial derivatives of $E_{A}(\$)$ with respect of each of the parameters while keeping the rest at optimal value. Vanishing partial derivatives together with a negative definite Hessian matrix at the values of $V^{opt}$ proves that $V^{opt}$ is Alice’s dominant strategy and because of the symmetry of the protocol, thereby a Nash equilibrium. $\left.\frac{\partial E_{A}(\$)}{\partial\phi}\right\|_{\phi=\phi^{{}^{\prime}}}=\left.\frac{2}{9}\cos(2\phi)\right\|_{\phi=\phi^{{}^{\prime}}}=0,\qquad\left.\frac{\partial^{2}E_{A}(\$)}{\partial\phi^{2}}\right\|_{\phi=\phi^{{}^{\prime}}}<0,$ (4.1.2) $\left.\frac{\partial E_{A}(\$)}{\partial\theta}\right\|_{\theta=\theta^{{}^{\prime}}}=\left.\frac{1}{27}\left(-\sqrt{3}\sin(\theta)+3\sqrt{2}\cos(2\theta)+\right.\right.\\\ \left.\left.\left(3\sin(\theta)+\sqrt{6}\right)\cos(\theta)\right)\right\|_{\theta=\theta^{{}^{\prime}}}=0,\qquad\left.\frac{\partial^{2}E_{A}(\$)}{\partial\theta^{2}}\right\|_{\theta=\theta^{{}^{\prime}}}<0,$ (4.1.3) $\left.\frac{\partial E_{A}(\$)}{\partial\chi}\right\|_{\chi=\chi^{{}^{\prime}}}=\left.\frac{2}{27}(\cos(\chi)-\sin(\chi))\left(\sin(\chi)+\cos(\chi)+\sqrt{2}\right)\right\|_{\chi=\chi^{{}^{\prime}}}=0,\\\ \left.\frac{\partial^{2}E_{A}(\$)}{\partial\chi^{2}}\right\|_{\chi=\chi^{{}^{\prime}}}<0,$ (4.1.4) $\left.\frac{\partial E_{A}(\$)}{\partial\alpha}\right\|_{\alpha=\alpha^{{}^{\prime}}}=\left.\frac{4\cos(\text{$\alpha$})}{27}\right\|_{\alpha=\alpha^{{}^{\prime}}}=0,\qquad\left.\frac{\partial^{2}E_{A}(\$)}{\partial\alpha^{2}}\right\|_{\alpha=\alpha^{{}^{\prime}}}<0,$ (4.1.5) $\left.\frac{\partial E_{A}(\$)}{\partial\beta_{1}}\right\|_{\beta_{1}=\beta_{1}^{{}^{\prime}}}=\left.\frac{1}{54}\left(-3\sin(\text{$\beta_{1}$})+\sin(2\text{$\beta_{1}$})+\right.\right.\\\ \left.\left.+3\sqrt{3}\cos(\text{$\beta_{1}$})+\sqrt{3}\cos(2\text{$\beta_{1}$})\right)\right\|_{\beta_{1}=\beta_{1}^{{}^{\prime}}}=0,\qquad\left.\frac{\partial^{2}E_{A}(\$)}{\partial\beta_{1}^{2}}\right\|_{\beta_{1}=\beta_{1}^{{}^{\prime}}}<0,$ (4.1.6) $\left.\frac{\partial E_{A}(\$)}{\partial\beta_{2}}\right\|_{\beta_{2}=\beta_{2}^{{}^{\prime}}}=\left.\frac{1}{54}\left((2\sin(\text{$\beta_{2}$})+3)(-\cos(\text{$\beta_{2}$}))-\right.\right.\\\ \left.\left.-\sqrt{3}(3\sin(\text{$\beta_{2}$})+\cos(2\text{$\beta_{2}$}))\right)\right\|_{\beta_{2}=\beta_{2}^{{}^{\prime}}}=0,\qquad\left.\frac{\partial^{2}E_{A}(\$)}{\partial\beta_{2}^{2}}\right\|_{\beta_{2}=\beta_{2}^{{}^{\prime}}}<0.$ (4.1.7) By calculating the Hessian $H$ with $\left.H_{ij}=\frac{\partial^{2}}{\partial a_{i}\partial a_{j}}E_{A}(\$)\right\|_{a_{i}=a_{i}^{opt},a_{j}=a_{j}^{opt}},$ (4.1.8) where $a_{i},a_{j}\in\\{\phi,\theta,\chi,\alpha,\beta_{1},\beta_{2}\\}$, and confirming that all eigenvalues are negative, we conclude that $V^{opt}\;(U^{opt})$ is indeed a Nash equilibrium. ### 4.2 Adjusting entanglement and fidelity We have included a simple model of noise, to show the behavior of the expected payoff, when the initial state was adjusted towards a completely mixed state. This was done by controlling the fidelity $f$ of the initial state, by mixing it with an even distribution of all basis states in $\mathcal{H}=\mathbb{C}^{3}\otimes\mathbb{C}^{3}\otimes\mathbb{C}^{3}$. Clearly as $f\rightarrow 0$, we should expect the entanglement as a resource in the initial state to vanish. This is of course the case and we have $E(\$)(U^{opt},f)=(2(2+f))/9$. For $f=0$ we simply end up with the classical result. A way of directly adjusting the strength of entanglement in the initial state, while keeping the state pure is to start with $\mid\psi_{in}\rangle=\sin\vartheta\cos\varphi\|000\rangle+\sin\vartheta\sin\varphi\|111\rangle+\cos\vartheta\|222\rangle,$ (4.2.1) where $0\leq\vartheta\leq\pi$ and $0\leq\varphi\leq 2\pi$. We retrieve the maximally entangled state (3.0.2) for $\varphi=\frac{\pi}{4},\frac{3\pi}{4}$ and $\vartheta=\pm\cos^{-1}(1/\sqrt{3})$. The expected payoff is given by: $E(\$)(U^{opt},\vartheta,\varphi)=\frac{1}{9}\left(\sin(\varphi)\sin(2\vartheta)+\cos(\varphi)\left(2\sin(\varphi)\sin^{2}(\vartheta)+\sin(2\vartheta)\right)+4\right),$ (4.2.2) which shows that any deviation from maximal entanglement reduces the expected payoff towards the classical $E^{c}(\$)$, graphically shown in figure 1. A point to note here is that the maximum expected payoff strongly depends on the choice of initial state $\|\psi_{in}\rangle$, and that there can exist more or less suitable initial states depending on the task. We chose the GHZ-state for this protocol because it is an unbiased maximally entangled state, which lets the classical game be present and accessible trough restrictions on $S$. Would our preferences been different and we had chosen for example the antisymmetric Aharonov state instead: $\|\mathcal{A_{-}}\rangle=\frac{1}{\sqrt{6}}\sum_{x_{3},x_{2},x_{1}=0}^{2}\epsilon_{x_{3}x_{2}x_{1}}\|x_{3}x_{2}x_{1}\rangle,$ (4.2.3) where $\epsilon_{x_{3}x_{2}x_{1}}$ is the completely antisymmetric tensor, then the expected payoff would have been $E(\$)=1$, just by letting the players apply the identity operator. This state would guarantee that everybody ends up with a unique choice every time. But that wouldn’t be of any interest from a game theoretical perspective since outcomes would have resembled a classical game with unrestricted communication. However, due to the the invariance of $\|\mathcal{A_{-}}\rangle$ under local unitary transformations of the form $U\otimes U\otimes U$, superpositions of $\|\mathcal{A_{-}}\rangle$ and $\|000\rangle$ under some restricted set of operators resembling the set of mixed classical strategies, could model a classical game under different amounts of communication. Figure 1: Expected payoff $E(\$)(U^{opt},\vartheta,\varphi)$ as a function of $\vartheta$ and $\varphi$ at the Nash equilibrium strategy. ## 5 Conclusions We have created the first quantum model for a three player, three restaurant Kolkata restaurant problem. We have shown that when the players share an initial tripartite entangled state, there exists a local unitary operation for which the players can increase their expected payoff $E(\$)$ by 50% compared with classical randomization. This solution is a Nash equilibrium and therefore a natural attractor in the space of available strategies. The achievement of this performance is highly dependent on the strength of entanglement and the fidelity of the initial state. Acknowledgments: We wish to thank Ole Andersson for valuable inputs and fruitful discussions. This study was supported by the Swedish Research Council (VR). ## References [1] D. Meyer, ”Quantum strategies”, Physical Review Letters 82, (1999), 1052 1055. * [2] J. Eisert, M. Wilkens, M. Lewenstein, ”Quantum games and quantum strategies” Physical Review Letters 83,(1999) 3077 3080. * [3] S. Benjamin, P. Hayden, ”Multiplayer quantum games”, Physical Review A 64,(2001) 030301. * [4] Q. Chen, Y. Wang, ”N-player quantum minority game”, Physics Letters A, 327 (2004), 98, 102. * [5] A. Flitney, L.C.L. Hollenberg, ”Multiplayer quantum minority game with decoherence”, Quant. Inform. Comput. 7 (2007) 111-126. * [6] A. Flitney, A. Greentree, ”Coalitions in the quantum minority game: classical cheats and quantum bullies”, Physics Letters A 362 (2007) 132 137. * [7] C. Schmid, A. P. Flitney, W. Wieczorek, N. Kiesel, H. Weinfurter, L. C. L. Hollenberg, ”Experimental implementation of a four-player quantum game”, New J. Phys. 12 (2010) 063031. * [8] A. Chakrabarti, New Economic Windows Series, Springer, Milan, (2007), pp. 220-227. * [9] A. S. Chakrabarti, B. K. Chakrabarti, A. Chatterjee, M. Mitra, ”The Kolkata Paise Restaurant problem and resource utilization”, Physica A 388,(2009) 2420-2426. * [10] A. Ghosh, A. S. Chakrabarti, B. K. Chakrabarti, 2010, ”Kolkata Paise Restaurant problem in some uniform learning strategy limits”, in Econophysics & Economis of Games, Social Choices & Quantitative Techniques, New Economic Windows, Eds. B. Basu, B. K. Chakrabarti, S. R. Chakravarty, K. Gangopadhyay, Springer, Milan, pages 3-9. * [11] A. Ghosh, A. Chatterjee, M. Mitra, B. K. Chakrabarti, New J Phys 12 (2010) 075033. * [12] Arthur, W. B., 1994, ”Inductive reasoning and bounded rationality: El Farol problem”, Am. Eco. Assoc. Papers & Proc. 84, 406. * [13] A. P. Flitney, ”Review of quantum games”, in Game Theory Strategies, Equilibria, and Theorems, I. N. Haugen, A. S. Nilsen, eds.,(2008) 1 40. * [14] E. W. Piotrowski, J. Sladkowski, ”An invitation to quantum game theory”, International Journal of Theoretical Physics 42 (2003) 1089. * [15] F. S. Khan, S. J.D. Phoenix, ”Nash equilibrium in quantum superpositions”, Proceedings of SPIE, Vol. 8057 80570K-1. * [16] S. E. Landsburg, ”Quantum Game Theory”, Wiley Encyclopedia of Operations Research and Management Science, (2011). * [17] M. Mathur and D. Sen, ”Coherent states for SU(3)”, J.Math.Phys. 42 (2001) 4181-4196. * [18] A. T. Bolukbasi, T. Dereli, ”On the SU(3) parametrization of qutrits”,12th Central European Workshop on Quantum Optics, 6-9 (June 2005), Bilkent Univ., Ankara, Turkey. * [19] D. E. Burlankov,”The SU3 space and its quotient spaces”, Theoretical and Mathematical Physics, 138(1): 78 87 (2004).	arxiv-papers	2011-11-08T16:34:40	2024-09-04T02:49:24.132141	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Puya Sharif and Hoshang Heydari", "submitter": "Puya Sharif", "url": "https://arxiv.org/abs/1111.1962" }
1111.2083	# Energy spectrum and effective mass using a non-local 3-body interaction Alexandros Gezerlis1 and G. F. Bertsch1,2 1Department of Physics, University of Washington, Seattle, WA 98195–1560 USA 2Institute for Nuclear Theory, University of Washington, Seattle, WA 98195–1560 USA ###### Abstract We recently proposed a nonlocal form for the 3-body induced interaction that is consistent with the Fock space representation of interaction operators but leads to a fractional power dependence on the density. Here we examine the implications of the nonlocality for the excitation spectrum. In the two- component weakly interacting Fermi gas, we find that it gives an effective mass that is comparable to the one in many-body perturbation theory. Applying the interaction to nuclear matter, it predicts a huge enhancement to the effective mass. Since the saturation of nuclear matter is partly due to the induced 3-body interaction, fitted functionals should treat the effective mass as a free parameter, unless the two- and three-body contributions are determined from basic theory. Zero- and finite-range nuclear energy-density functionals have a long history and a successful track record, allowing the description of heavy nuclei without region-specific parametrizations.Bender:2003 The most popular functionals use interactions that depend on fractional powers of density, which causes serious problems when one tries to extend the theory to include correlations Duguet:2003 ; Robledo:2007 ; Duguet:2009 . Ideally, to avoid these problems the effective theory should be based on a Fock-space Hamiltonian operator. As a partial solution, one can consider energy functionals of integral powers of the density; there have been a number of attempts to construct functionals of this kind Baldo:2010 ; Erler:2010 . With this in mind, we recently proposed a nonlocal effective three-body interaction that achieves a fractional dependence on density using only integral powers of the density matrix Gezerlis:2010a . This was derived using the many-body perturbation theory of the dilute, weakly interacting Fermi gas. By construction, the interaction gives the correct Lee-Yang contribution Lee:1957 to the Fermi-gas energy to order $\rho^{7/3}$. The interaction was validated for finite systems in a harmonic trap by comparing with numerically accurate calculations performed by the Green’s Function Monte Carlo method. At very weak coupling, the new operator led to results that are identical with the Lee-Yang dependence, while for stronger coupling the contribution of the new 3-body operator turned out to be more repulsive than in Lee-Yang (though with the same power-law behavior), thus providing a more accurate description of the microscopic simulation. Using the new interaction, the internal energy of the dilute Fermi gas can be expressed in terms of the one-body density matrix as: $\begin{split}E&=\frac{\hbar^{2}k_{F}^{2}}{m}\int d^{3}r_{1}\,\left(\frac{\nabla_{r_{1}}\cdot\nabla_{r_{2}}}{2}\rho({\bf r_{1},r_{2}})\|_{{\bf r_{1}=r_{2}}}+4\pi a\rho_{\downarrow}({\bf r_{1}},{\bf r_{1}})\rho_{\uparrow}({\bf r_{1}},{\bf r_{1}})\right)\\\ &+C\int d^{3}r_{1}d^{3}r_{2}\,\frac{\rho_{\uparrow}({\bf r_{1},r_{2}})\rho_{\downarrow}({\bf r_{1},r_{2}})\rho({\bf r_{1},r_{2}})}{\|{\mathbf{r}_{1}}-{\mathbf{r}_{2}}\|}.\end{split}$ (1) The subscript $i$ on $\rho_{i}$ denotes the spin state, $\rho$ without a subscript is the total density. Also, if $a$ is the scattering length associated with the two-body interaction, $C$ is a constant proportional to $a^{2}$. The value of $C$ was derived in Ref. Gezerlis:2010a by demanding that the formula reproduce the Lee-Yang energy in the uniform Fermi gas. The energy $E$ or energy density $\cal E$ is given by $\frac{E}{A}=\frac{\cal E}{\cal\rho}=\frac{\hbar^{2}k_{F}^{2}}{2m}\left(\frac{3}{5}+\frac{2}{3\pi}ak_{F}+\frac{4}{35\pi^{2}}\left(11-2\ln 2\right)\left(ak_{F}\right)^{2}\right)~{}.$ (2) It is convenient for later use to rederive from Eq. (1) the formula for $C$, which was originally derived from the perturbation theory in a momentum space representation. We insert in Eq. (1) the free Fermi gas density matrix and drop one of the integrals to get the energy density. The Fermi gas density matrix only depends on the relative coordinate ${\bf r}={\bf r_{1}}-{\bf r_{2}}$ and can be written $\rho_{0i}({\bf r})=\int_{0}^{k_{F}}\frac{d^{3}k}{(2\pi)^{3}}e^{i{\bf k}\cdot{\bf r}}=\rho_{0i}(0)F(k_{F}r)$ (3) where $F(x)=3j_{1}(x)/x$. The integral to be evaluated may be expressed $\frac{E_{3}}{A}=C\frac{24\pi^{3}}{k_{F}^{5}}\rho_{0i}^{3}(0)\int_{0}^{\infty}xdxF^{3}(x)$ (4) The integration can be performed analytically; the final result for the strength parameter $C$ is $C={\hbar^{2}a^{2}\over m}{64\pi(11-\ln 2)\over 3(92-27\ln 3)}$ (5) We now calculate the single-particle energy with functional Eq. (1) and the value of $C$ fixed by Eq. (5). The density matrix with a particle added to the Fermi sea is $\rho_{i}({\bf r)}=\rho_{0i}({\bf r})+\rho_{ki}e^{i{\bf k}\cdot{\bf r}}$ (6) The second term represents a particle of momentum $k$ in spin state $i$; the coefficient $\rho_{ki}$ has dimensions of density. With this definition the single-particle energy may be computed as $\varepsilon_{i}(k)=\frac{d{\cal E}}{d\rho_{ki}}\Big{\|}_{\rho_{ki}=0}.$ (7) Carrying out the differentiation on the energy expression Eq. (1), the first term gives the usual kinetic energy and the second term is independent of $k$. The third term is rather complicated. Assuming equal populations of spin up and spin down in $\rho_{0}$, the derivative is given by the integral: $\varepsilon_{3}(k)=3C\int d^{3}r\,e^{i{\bf k}\cdot({\bf r})}\frac{\rho_{0i}^{2}({\bf r})}{r}=3C{4\pi\over k_{F}^{2}}\rho_{0i}^{2}(0)\int_{0}^{\infty}dx~{}x\,\,j_{0}\Big{(}{k\over k_{F}}x\Big{)}F^{2}(x).$ (8) The factor of 3 is a direct consequence of the spin structure of the numerator in the third term of Eq. (1). The integral can also be expressed analytically: $\begin{split}\int_{0}^{\infty}dx~{}x\,j_{0}(yx)F^{2}(x)&=\frac{3}{160}\Big{[}4(22+y^{2})+\frac{(y-2)^{3}}{y}(y^{2}+6y+4)\log(2-y)\\\ &-2y^{2}(y^{2}-20)\log y+\frac{(y+2)^{3}}{y}(y^{2}-6y+4)\log(2+y)\Big{]},\end{split}$ (9) where $y=k/k_{F}$. The 3-body contribution to the single-particle energy $\varepsilon_{3}$ is plotted in Fig. 1, with the dimensionful factors divided out. Figure 1: Single-particle energy in the dilute Fermi gas, normalized to $\hbar^{2}a^{2}\rho_{0i}^{2}(0)/mk_{F}^{2}$. The solid line shows result using the effective 3-body interaction, Eq. (8). The dashed line shows the contribution to the quasiparticle energy obtained by Galitskii. The dotted line is the slope of the Galitskii expression. Galitskii’s expression for the real part of the quasiparticle energy (Galitskii:1958, , Eq. (34)) is plotted with the same normalization on the graph. The perfect agreement of the two at the Fermi momentum is not accidental: the single-particle energy at the Fermi surface is identical to the chemical potential $\mu$, which can be extracted from the interaction energy by the formula $\mu=\partial{\cal E}/\partial\rho$. Since we fit the total 3-body interaction energy to the dilute Fermi gas, the chemical potential must agree as well. The momentum-dependence of the single-particle energy gives rise to an effective mass $m^{}$ for the quasiparticle spectrum, $\frac{m^{}}{m}=\left(1+{m\over\hbar^{2}k_{F}}{\partial\epsilon_{3}(k)\over\partial k}\Big{\|}_{k=k_{F}}\right)^{-1}$ (10) The derivative in this expression is negative, implying that the effective mass will be larger than $m$. Fig. 1 also shows the derivative for Galitskii’s quasiparticle energy, as the straight line (see also Ref. Fetter:1971 ). We note that the slope for the 3-body single-particle energy is smaller, implying less of an effective-mass enhancement. Even so, the two results are close enough in magnitude to motivate the application of the new operator to a nuclear energy functional. As stated in the introduction, our main interest is to find an improved effective Hamiltonian for nuclear structure theory. There is no reliable low- density expansion in the nuclear many-body problem, and in fact one must impose some length scale in the interactions to avoid collapse. Nevertheless, in some formulations there will be a contribution to saturation coming from the Pauli effects that we are concerned with here. To assess the importance of the nonlocality, we take $C$ as an adjustable parameter to be fitted in the functional, similar to the parameter $t_{3}$ of the Skyrme interaction. The counting of the contributing graphs is different in the four-component Fermi system than in the two-component case treated by Galitskii, but the scaling between the total energy and the single-particle energy remains the same under plausible assumptions about the spin-isospin character of the interaction. Thus we may use the same formulas, only remembering that in the nuclear context $\rho_{0i}$ is the density associated with a specific spin-isospin projection, e.g. neutrons with spin up. Table 1: Contributions to the energy of 208Pb in density functional theory. The numbers for the Skyrme Ska and Gogny D1S functionals were obtained with the ev8 code Bonche:2005 and the HFBaxial code robledo , respectively. \| Ska \| D1S ---\|---\|--- Kinetic \| 3863 \| 3920 Coulomb direct/exchange \| 831/-31 \| 832/-31 Spin-orbit \| -97 \| -105 Central 2B \| -12480 \| -12783 $t_{3}$ \| 6274 \| 6530 Total \| -1640 \| -1637 While we cannot calculate $C$, we can at least put a bound on its value using the magnitude of the 3-body interaction energy that is obtained from phenomenological energy functionals. With our form for the interaction, the relation between the 3-body energy and the effective mass is $\frac{m^{}}{m}=\left(1+d\frac{E_{3}/A}{\hbar^{2}k_{F}^{2}/(2m)}\right)^{-1}$ (11) where $d\approx-1.32$, and the two-body contribution has been omitted. To see what the scale of the effect would be, we show in Table 1 the various contributions to the energy of 208Pb found using the Ska Skyrme functional and the D1S Gogny functional. Both these functionals have the same $\rho^{1/3}$ density-dependent interaction as in the Lee-Yang expansion. One sees that the decomposition into the two-body and three-body contributions is quite similar, although the two-body interactions have a very different construction. Eq. (11) gives a negative effective mass for both functionals, which is of course unphysical. The two-body nonlocality gives a contribution of the opposite sign, but not enough to produce an effective mass in the physical range ($m^{}/m\sim 1$). As mentioned earlier, there must be other 3-body contributions containing intrinsic length scales in order to achieve nuclear saturation. However, unless the nonlocalities can be calculated in detail, it does not seem feasible to derive a theoretical effective mass to be used with an effective Hamiltonian. The extreme sensitivity to the induced 3-body interaction suggests that the effective mass may need to be an unconstrained free parameter when constructing an effective Hamiltonian for mean-field theory and its extensions. In summary, we have applied our newly proposed non-local effective 3-body operator to the study of the single-particle excitation spectrum, both at weak coupling and at strong coupling. At weak coupling we see that the new operator has similar behavior to that found by Galitskii. We also applied the new operator to the nuclear case. The effects pointed to are very large, implying that the effective mass cannot be simply taken to be reduced from the bare mass based on mean-field theory: as long as no dependable ab initio results are available, the effective mass should also be treated like an undetermined parameter. We would like to thank L. Robledo for providing us with the energies of 208Pb for the Gogny D1S interaction. This work was supported by DOE Grant Nos. DE- FG02-97ER41014 and DE-FG02-00ER41132. ## References * (1) M. Bender, P.-H. Heenen, and P.-G. Reinhard, Rev. Mod. Phys. 75, 121 (2003). * (2) T. Duguet and P. Bonche, Phys. Rev. C 67, 054308 (2003). * (3) L. M. Robledo, Int. J. Mod. Phys. E 16, 337 (2007). * (4) T. Duguet, M. Bender, K. Bennaceur, D. Lacroix, and T. Lesinski, Phys. Rev. C 79, 044320 (2009). * (5) M. Baldo, L. M. Robledo, P. Schuck, X. Viñas, J. Phys. G 37, 064015 (2010). * (6) J. Erler, P. Klüpfel, P.-G. Reinhard, Phys. Rev. C 82, 044307 (2010). * (7) A. Gezerlis and G. F. Bertsch, Phys. Rev. Lett. 105, 212501 (2010). * (8) T. D. Lee and C. N. Yang, Phys. Rev. 105, 1119 (1957). * (9) V. M. Galitskii, Sov. Phys. (JETP) 34, 151 (1958). * (10) A. L. Fetter and J. D. Walecka, Quantum Theory of Many-Particle Systems (McGraw-Hill, New York, 1971). * (11) P. Bonche, H. Flocard, and P.H. Heenen, Comp. Phys. Comm. 171, 49 (2005). * (12) ??	arxiv-papers	2011-11-09T00:20:35	2024-09-04T02:49:24.141614	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Alexandros Gezerlis, G. F. Bertsch", "submitter": "George F. Bertsch", "url": "https://arxiv.org/abs/1111.2083" }
1111.2157	\+ daix@aphy.iphy.ac.cn # Implementation of LDA+DMFT with pseudo-potential-plane-wave method Jian-Zhou Zhao1,2, Jia-Ning Zhuang1, Xiao-Yu Deng1, Yan Bi2, Ling-Cang Cai2, Zhong Fang1, and Xi Dai1 1Institute of Physics, Chinese Academy of Sciences, Beijing 100190, People’s Republic of China 2National key Laboratory of Shock Wave and Detonation Physics, Institute of Fluid Physics, China Academy of Engineering Physics, Mianyang, 621900, China ###### Abstract In this paper, we propose an efficient implementation of combining Dynamical Mean field theory (DMFT) with electronic structure calculation based on the local density approximation (LDA). The pseudo-potential-plane-wave method is used in the LDA part, which makes it possible to be applied to large systems. The full loop self consistency of the charge density has been reached in our implementation which allows us to compute the total energy related properties. The procedure of LDA+DMFT is introduced in detail with a complete flow chart. We have also applied our code to study the electronic structure of several typical strong correlated materials, including Cerium, Americium and NiO. Our results fit quite well with both the experimental data and previous studies. ###### pacs: 71.27.+a, 71.15.Mb, 71.15.Nc, 71.20.-b ## 1 Introduction The first principle calculation based on the density functional theory (DFT) with the local density approximation (LDA) and its generalization generalized gradient approximation (GGA) is very successful in predicting ground-state properties and band structures of a wide range of real materials. However it is known for a long time that it is not sufficient to calculate the electronic structure of those strongly correlated materials (i.e. transition metal oxides, actinide and lanthanide-based materials, high $T_{c}$ superconductors) by these methods alone even on the qualitative level. In order to overcome these shortcomings of the traditional DFT-LDA scheme, remedies such as LDA+$U$ have been proposed, which can describe some of the strong correlated materials with long range order in their ground states. While even LDA+$U$ can not be applied to many of the strongly correlated materials, for example, the paramagnetic Mott insulator phase and the materials containing unfilled f-shell with strong multiplet effects. Therefore it is important to have a first principle method which can be applied to the strongly correlated materials with the ability to capture the dynamical properties, finite temperature properties and multiplet effects. In the last two decades, the dynamical mean field theory (DMFT) has been developed very fast to be the standard tool to deal with the on-site correlation effect in the limit of large dimension [1]. After being successfully applied to many model systems, DMFT has been considered as a powerful tool to capture the on-site correlation effects based on the Hubbard like Hamiltonians containing both the local interaction terms and the single particle Hamiltonians extracted from LDA. Therefore the LDA+DMFT method which combines the DFT-based band structure techniques with DMFT has been proposed and developed quickly in the last decade. By DMFT the local correlation effect can be well described by the self energy, which has frequency dependence and in general takes the matrix form within the subspace spanned by the correlation orbitals. Using the Green’s function containing self energy, many of the physical properties of strongly correlated materials can be calculated, i.e. the electronic spectral function, the total energy, the optical conductivity and the local spin susceptibility. Most of the LDA+DMFT calculation till now have been performed by the partially self-consistent scheme, where the local self-energy is obtained by the DMFT calculation with a fixed LDA Hamiltonian generated from the fixed LDA charge density. Therefore, in this simplified scheme one neglects the inference of the strong on-site Coulomb interaction on the charge density. The above mentioned “one-shot” DMFT calculation works quite well for the electronic structure. While in order to obtain reliable total energy related properties, i.e. the equilibrium volume, the elastic constants and the phonon frequencies, the LDA+DMFT scheme with full charge density self consistency is needed. Up to date, there have been several fully self-consistent LDA+DMFT schemes [2, 3] as well as actual implementations [4, 5, 6, 7]. As mentioned in [7], there are three major issues that have to be addressed in DFT+DMFT implementations: i) quality of the basis set, ii) quality of the impurity solvers, iii) choice of correlated orbitals onto which the full Green’s function is projected. In the implementations of LDA+DMFT mentioned above with full charge self-consistency , the basis set of the linear muffin-tin orbitals (LMTO) are used, and the local correlated orbitals are naturally chosen based on the muffin-tin orbitals . While on the other hand, there are very few implementations of LDA+DMFT with full charge self-consistency are based on the plane wave methods. In the present paper, we implement a full self-consistent LDA+DMFT scheme using pseudo-potential-plane-wave as the basis set, and we use either atomic orbitals or Wannier orbitals depending on different physics systems. The Hubbard-I approximation is used as the impurity solver in the present paper, but it can be replace by more precise solver like continuous time quantum Monte Carlo (CTQMC) for metallic systems. The present paper is organized as follows. In section 2, we describe the way to choose and construct the correlated orbitals onto which the local interactions exert. In section 3, we show the full-loop flow and LDA+DMFT(Hubbard-I) formalism in detail. We apply this LDA+DMFT approach to several correlated materials such as Ce, Am, and NiO in section 4. Finally we conclude our work in section 5. ## 2 Projection onto localized orbitals In the LDA+DMFT calculation, all the energy bands are divided into two groups. And only the on-site interactions among those local orbitals are treated more precisely by DMFT. Therefore the choice of localized orbitals does affect the results obtained by LDA+DMFT method and becomes one of the important issues in LDA+DMFT. A natural choice for the local orbitals is a set of atomic like wave functions with corresponding $d$ or $f$ characters. Typical atomic like local orbitals is the linear muffin-tin orbitals (LMTOs)[8] adopted in early LDA+DMFT implementations[9, 10, 11]. However a more physical choice should be wannier functions, since the shape of the local orbitals will be altered in crystals especially when there is strong hybridization between localized orbital and delocalized $p$-type or $s$-type orbitals. The wannier functions are not uniquely determined by the Bloch wave functions, so different choice can be made, for example, the Nth-order muffin-tin orbitals[12], the projected wannier functions[13] and the Maximally localized Wannier functions (MLWFs)[14, 15]. Comparisons of these choices have been made in previous literature. In this paper, two kinds of local orbitals, atomic like functions and the projected wannier functions are adopted according to the different systems. This implementation of LDA+DMFT method reported in this paper is developed on an existing DFT package BSTATE (Beijing Simulation Tool of Atomic TEchnique), which is based on the pseudo-potentials and plane waves. Unlike the LMTO methods, the local orbitals do not enter the basis set of plane waves and should be constructed in a suitable way. Projected wannier functions or MLWFs as local orbitals have been used by previous reported LDA+DMFT implementations based on pseudo-potentials (or projector augmented wave (PAW)) method) and plane wave scheme[16, 17, 18]. In this report, two kinds of orbitals are used according to the need, one is directly derived from the atomic wave function of an isolated atom and the other is projected wannier functions constructed from these atomic wave functions. In order to be self-contained the construction procedure is presented below in detail. Since all our calculations are based on plane wave set, the local orbitals will be projected onto these plane waves. First we consider the atomic like local orbitals. The localized nature of the correlated bands with $d$ or $f$ characters in crystals ensures that these local orbitals are very similar to the corresponding atomic wave functions of an isolated atom. The atomic wave functions can be picked as the local orbitals directly if the hybridization in crystal could be neglected. All atomic wave functions for the isolated atom could be obtained by the solving the all-electron radical Schrödinger equation $(T+V(r))\psi_{nl}^{ALL}(r)=E_{nl}\psi_{nl}^{ALL}(r)$ (1) In this manner, all orbitals are well defined by the primer quantum number $n$ and angular quantum number $l$. In realistic systems, the local orbitals usually come from the partially filled $d$ or $f$ shell and could be picked according to its character. Often there is only one type of local orbitals, so the local subspace can be labeled by $l$. Of course, considering the spherical symmetry of the isolated atom, the spherical harmonics should be multiplied the radical wave functions to form a complete set. The local orbitals are $\|\phi_{lm,I}\rangle=\|\psi_{l,I}^{ALL}Y_{lm}\rangle,$ (2) and in which $m=1\dots 2l+1$ denote different angular components. In methods based on plane waves, it is not desirable to use the all-electron wave function directly since it requires a large amount of plane waves to expand in momentum space. To avoid this problem, in PAW method all-electron atomic partial waves can be used, while in pseudo-potentials method, pseudo atomic wave functions can be chosen. The latter is used in this paper. During the generation of pseudo-potentials, the Schrödinger equation obeyed by the pseudo wave functions is as below $(T+V^{PS}(r))\psi_{nl}^{PS}(r)=E_{nl}^{PS}\psi_{nl}^{PS}(r)$ (3) The details on the generation of pseudo-potentials can be referred to previous literature[19, 20]. The spirit of pseudo-potentials is quite straight forward. Beyond a core radius $r_{c}$, the pseudo-potential $V^{PS}$ play the role of a scattering center just as a real atomic potential $V^{ALL}$ do. Thus, the pseudo eigen energy $E_{nl}^{PS}$ should be the same as the realistic one $E_{nl}^{ALL}$, and both the pseudo wave functions $\|\psi_{n,I}^{PS}\rangle$ and the pseudo-potential $V^{PS}$ coincide with the exact wave functions $\|\psi_{n,I}^{ALL}\rangle$ and the realistic potential $V^{ALL}$ of an isolated atoms beyond this core radius $r_{c}$, respectively. $\eqalign{E_{nl}^{PS}=E_{nl};\cr V^{PS}(r)=V(r),\qquad\psi^{PS}(r)=\psi^{ALL}(r)\qquad r>r_{c}}$ (4) The quality of pseudo wave functions is the same as the quality of the pseudo- potentials, which could be justified by comparing simple DFT calculations with accepted results. The pseudo wave functions bear the same atomic features as the exact atomic wave functions, and can be picked as the local orbitals as above (angular quantum number $l$ ignored since usually only one kind of local orbital are considered). $\|\phi_{m,I}\rangle=\|\psi_{l,I}^{PS}Y_{lm}\rangle,$ (5) It is convenient to transform the local orbitals from real space to momentum space in crystal calculations $\|\phi_{m,{\mathbf{k}}}\rangle=\sum_{I}e^{i{\mathbf{k}}\cdot{\mathbf{R}}_{I}}\|\phi_{m,I}\rangle.$ (6) Since the atomic wave functions at different site $I$ are not orthogonal, an orthogonalization procedure is needed for the above local orbitals. The DFT calculations give a set of Bloch waves $\|\Psi_{n\mathbf{k}}\rangle$ spanning the total Hilbert space in which lies the local subspace spanning by the local orbitals. Thus, the physical choice of local orbitals is Wannier functions derived from the Bloch waves. The Wannier functions are localized in real space and could be constructed from the Bloch waves via a unitary transformation. However, this unitary transformation is not unique since the phase factors of Bloch waves are uncertain, which results in that there are many kinds of Wannier functions in use, e.g., the projected Wannier functions, NMTOs and MLWFs. Also it is not necessary to include all the Bloch waves but the relevant bands lying in a certain energy range to construct the Wannier functions. The local orbitals could be picked as a subset of these Wannier functions with specified $d$ or $f$ localized character. The projecting Wannier functions method is a simple way to construct wannier functions, in which trial wave functions are projecting onto physical relevant Bloch bands. The atomic like local orbitals above are used as trial functions here. The Bloch bands can be selected by band indices ($N_{i},...N_{j}$) or by an energy interval ($E_{i},E_{j}$) enclosing them. $\displaystyle\|W_{m,{\mathbf{k}}}\rangle$ $\displaystyle=$ $\displaystyle\sum_{n=N_{i}}^{N_{j}}\|\psi_{n{\mathbf{k}}}\rangle\langle\psi_{n{\mathbf{k}}}\|\phi_{lm,{\mathbf{k}}}\rangle$ (7) $\displaystyle=$ $\displaystyle\sum_{n(E_{i}<\epsilon_{n{\mathbf{k}}}<E_{j})}\|\psi_{n{\mathbf{k}}}\rangle\langle\psi_{n{\mathbf{k}}}\|\phi_{m,{\mathbf{k}}}\rangle.$ These orbitals need normalization and orthogonalization to be true Wannier functions. $\|\tilde{W}_{m,{\mathbf{k}}}\rangle=\sum_{m^{\prime}}O_{m,m^{\prime}}({\mathbf{k}})^{-1/2}\|{W}_{m,{\mathbf{k}}}\rangle$ (8) in which $O$ is the overlap matrix between different $\|W_{m,{\mathbf{k}}}\rangle$s $O_{m,m^{\prime}}({\mathbf{k}})=\langle W_{m,{\mathbf{k}}}\|W_{m^{\prime},{\mathbf{k}}}\rangle$ (9) Although the construction method can not give the most localized Wannier orbitals, the basic features of the localized bands are captured to a very good extend as indicated by a few reports[13, 16, 17, 18], and also proved by our calculations in this paper. ## 3 LDA+DMFT(Hubbard-I) Formalism We will introduce the detail procedge of LDA+DMFT with full loop charge density self consistency in this section. First we plot the general flow chart in figure 1, which contains a inner DMFT loop and a outer charge density loop. However, if we use Hubbard-I approximation as the impurity solver, the inner DMFT loop is thus neglected. Figure 1: The most general flow chart for LDA+DMFT scheme. The first LDA calculation gives out the band structure on LDA level, and constructs local orbitals, then the full Hamiltonian is established and solved by a DMFT loop, after which the charge density can be recalculated by Fourier transforming the k-space density matrix to real space. The “non-interacting” Hamiltonian is thus regenerated based on the new charge density profile and the calculation will be completed when the self consistency has been reached for the full loop. In the following subsections, we are going to describe the whole process in detail. ### 3.1 First LDA Calculation The first step of the full loop LDA+DMFT is an LDA self-consistent calculation, whose main purpose is to generate an effective single particle Hamiltonian $\hat{H}_{LDA}$ and construct the correlated orbitals. Generally the effective LDA Hamiltonian can be expressed as the following: $\displaystyle\hat{H}_{LDA}=\sum_{n\mathbf{k}}E_{n\mathbf{k}}\hat{c}^{\dagger}_{n\mathbf{k}}\hat{c}_{n\mathbf{k}}$ (10) In the above equation, $n=1\sim N_{band}$ are the joint indices of band and spin; $E_{n\mathbf{k}}$ is the eigen energy of the Kohn-Sham equation determined by $\hat{H}_{LDA}\|\varphi_{n\mathbf{k}}\rangle=E_{n\mathbf{k}}\|\varphi_{n\mathbf{k}}\rangle$, where $\|\varphi_{n\mathbf{k}}\rangle$ represents a set of orthonormal Bloch functions. The LDA calculation also complement the construction of local correlated orbitals. In the present implementation the atomic orbitals and the Wannier functions are two types of commonly used local basis. In LDA+DMFT, the local interactions have been considered on the DMFT level only within the local orbitals and in order to set up the DMFT self consistent equation we need to obtain the overlap matrix between local orbitals and Bloch wave functions, which takes the form of $S^{\mathbf{k}}_{\alpha,n}=\langle\alpha_{\mathbf{k}}\|\varphi_{n\mathbf{k}}\rangle$, with Greek letter $\alpha$ denote the index of local orbitals. The completeness of the Bloch basis set gives $\displaystyle\sum_{n}\|S^{\mathbf{k}}_{\alpha,n}\|^{2}=1$ (11) for any local orbital $\alpha$ and any k-point. ### 3.2 DMFT Loop The purpose of the DMFT loop is to calculate the self-energy caused by the local interactions through the DMFT self consistent loop. As we have mentioned before, the DMFT loop is not necessity if we use Hubbard-I approximation as the impurity solver. However, here we first introduce the algorithm with more general impurity solvers. By adding the local interaction terms to LDA, we get the total Hamiltonian of the system which is to be solved by DMFT, $\displaystyle\hat{H}_{LDA+DMFT}=\hat{H}_{LDA}+\sum_{i}(\hat{H}^{i}_{U}-\hat{V}^{i}_{DC})-\mu\hat{N}$ (12) where $i$ is the site index. For a specified site, we remove the $i$ index and $\displaystyle\hat{H}_{U}=\frac{1}{2}\sum_{\alpha\beta\gamma\delta}U_{\alpha\beta\gamma\delta}\hat{f}^{\dagger}_{\alpha}\hat{f}^{\dagger}_{\beta}\hat{f}_{\gamma}\hat{f}_{\delta}$ (13) is written as a general form of two-body local interactions, in which the creation and annihilation operators of correlated orbitals $\hat{f}^{\dagger}_{\alpha}$ and $\hat{f}_{\alpha}$ are associated with Bloch operators in term of $\displaystyle\hat{f}^{\dagger}_{\alpha}=\sum_{n\mathbf{k}}\hat{c}^{\dagger}_{n\mathbf{k}}\langle\varphi_{n\mathbf{k}}\|\alpha_{\mathbf{k}}\rangle$ (14) $\displaystyle\hat{f}_{\alpha}=\sum_{n\mathbf{k}}\hat{c}_{n\mathbf{k}}\langle\alpha_{\mathbf{k}}\|\varphi_{n\mathbf{k}}\rangle$ (15) The third term of (12) is the double-counting term correspond to the correlated energy that has already been considered in LDA calculation at a Hartree-Fock mean field level, $\hat{V}_{DC}=\sum_{\alpha}V_{DC}^{\alpha}\hat{f}^{\dagger}_{\alpha}\hat{f}_{\alpha}$ (16) and this term will be discussed in section 3.2.3. The last term of (12) $\hat{N}=\sum_{n\mathbf{k}}\hat{c}^{\dagger}_{n\mathbf{k}}\hat{c}_{n\mathbf{k}}$ (17) is the total number of particle operator, while the chemical potential $\mu$ controls the occupation number of the unit cell. #### 3.2.1 Quantum Impurity Hamiltonian In DMFT, the correlation problem on the lattice can be mapped to a quantum impurity model, which contains the same on-site interaction and reads, $\displaystyle\hat{H}_{imp}=\sum_{q\alpha}\epsilon_{q\alpha}\hat{c}^{\dagger}_{q\alpha}\hat{c}_{q\alpha}+\sum_{q\alpha}V_{q\alpha}(\hat{c}^{\dagger}_{q\alpha}\hat{f}_{\alpha}+h.c.)+\sum_{\alpha}E^{imp}_{\alpha}\hat{f}^{\dagger}_{\alpha}\hat{f}_{\alpha}+\hat{H}_{U}$ (18) The inference from the rest of the lattice site besides the one considered in the impurity model is simulated by a non-interacting “heat bath”, which is described by the first term in (18). The second term describes the coupling between the impurity site and the heat bath and the rest two terms describe the local interactions. #### 3.2.2 Hybridization function and Weiss field The above quantum impurity model can be solved by the impurity solver, i.e. Hubbard-I, and the self energy $\hat{\Sigma}(i\omega)$ is then obtained, with which we can construct the lattice Green’s function by applying the same self energy term as, $[\hat{G}^{\mathbf{k}}_{lattice}]^{-1}=\rmi\omega-\hat{H}_{\mathbf{k}}-\hat{\Sigma}_{\mathbf{k}}(\rmi\omega)+\mu$ (19) The hybridization function, which characterizes the dynamics of the “heat bath” is defined as $\Delta(\rmi\omega)_{\alpha\beta}=\delta_{\alpha\beta}\sum\limits_{q}\frac{\|V_{q\alpha}\|^{2}}{\rmi\omega-\epsilon_{q\alpha}}$ and can be obtained iteratively by the following DMFT self consistent equation. $\hat{\Delta}(\rmi\omega)=\rmi\omega-\hat{E}_{imp}-\hat{\Sigma}+\mu-\left[\sum_{k}G^{\mathbf{k}}_{lattice}(\rmi\omega)\right]^{-1}$ (20) The above equation is obtained by requiring that the local Green’s function on the lattice should equal to the Green’s function of the quantum impurity problem. The equations (19) and (20) form a closed self consistent loop, which determines both the self energy and the hybridization function iteratively. #### 3.2.3 Double Counting Term In this section we discuss the explicit expression of $E^{imp}_{\alpha}$ as well as the double-counting term in (12). For 3d system under cubic symmetry, the local interaction can be simplified as coulomb interaction $U$ and Hund’s rule coupling $J$, which can be written as, $\hat{H}_{U}=U\sum_{b}\hat{n}_{b,\uparrow}\hat{n}_{b,\downarrow}+(U-2J)\sum_{b<b^{\prime}\atop\sigma\sigma}\hat{n}_{b,\sigma}\hat{n}_{b^{\prime},\sigma}-J\sum_{b<b^{\prime}\atop\sigma}\hat{n}_{b\sigma}\hat{n}_{b^{\prime}\sigma}$ (21) And the double counting energy in this case has been studied by Held et al[21]. and can be approximately chosen as $\displaystyle E_{DC}=\frac{1}{2}\bar{U}n_{\mbox{\tiny LDA+DMFT}}(n_{\mbox{\tiny LDA+DMFT}}-1)$ (22) where $\displaystyle\bar{U}$ $\displaystyle=$ $\displaystyle\frac{U+2(M-1)(U-2J)-(M-1)J}{2M-1}$ (23) $\displaystyle M$ $\displaystyle=$ $\displaystyle 2l+1$ (24) and $n_{\mbox{\tiny LDA+DMFT}}$ is the total number of electrons on correlated orbitals for a specific atom, hence in (10) becomes $\displaystyle\hat{V}_{DC}$ $\displaystyle=$ $\displaystyle\sum_{\alpha}\frac{\partial E_{DC}}{\partial n_{\alpha}}\hat{n}_{\alpha}$ (25) $\displaystyle=$ $\displaystyle\sum_{\alpha}\bar{U}(n_{\mbox{\tiny LDA+DMFT}}-\frac{1}{2})\hat{n}_{\alpha}$ The final expression of $E^{imp}$ is $\displaystyle E^{imp}_{\alpha}$ $\displaystyle=$ $\displaystyle\sum_{k}\langle\alpha_{k}\|\hat{H}_{LDA}\|\alpha_{k}\rangle- V_{DC}^{\alpha}$ (26) where $\displaystyle V_{DC}^{\alpha}=\bar{U}(n_{\mbox{\tiny LDA+DMFT}}-\frac{1}{2})$ (27) The way to remove the form of double-counting energy is not unique, and in fact this process needs intuition of physics. The double-counting discussed in earlier sections is a very usual way which is also used in LDA+$U$ method. Besides, the following two ways are also commonly used * • to remove self-energy at zero-frequency $\hat{\Sigma}(\rmi\omega)\rightarrow\hat{\Sigma}(\rmi\omega)-\hat{\Sigma}(0)$ * • to remove self-energy at infinity-frequency $\hat{\Sigma}(\rmi\omega)\rightarrow\hat{\Sigma}(\rmi\omega)-\hat{\Sigma}(\infty)$ Furthermore, the double-counting energy is also regarded as “impurity solver dependent”. For example, it is reasonable to use an integer to replace $n_{\mbox{\tiny LDA+DMFT}}$ in (22) for the Hubbard-I solver[6], and in this paper we also follow their suggestion. #### 3.2.4 Hubbard-I solver The essential point of Hubbard-I solver is to neglect the effect of the heat bath and take the atomic self energy as the zeroth order approximation for the quantum impurity problem, which can be written as $\displaystyle\hat{\Sigma}^{atom}=[\rmi\omega_{n}+\mu_{atom}-\hat{E}^{imp}]^{-1}-[\hat{G}^{atom}]^{-1}$ (28) In the above equation $\hat{G}^{atom}$ is the Green’s function for a single atom, which can be expressed as $\displaystyle G^{atom}_{\alpha\alpha^{\prime}}(\rmi\omega)=\sum_{\Gamma\Gamma^{\prime}}\frac{(F^{\alpha})_{\Gamma\Gamma^{\prime}}(F^{\alpha^{\prime}\dagger})_{\Gamma^{\prime}\Gamma}(X_{\Gamma}+X_{\Gamma^{\prime}})}{\rmi\omega- E_{\Gamma^{\prime}}+E_{\Gamma}}$ (29) where $\|\Gamma\rangle$ and $\|\Gamma^{\prime}\rangle$ are the atomic eigenstates obtained by exact diagonalization of a single atom problem, $X_{\Gamma^{\prime}}={e^{-\beta E_{\Gamma^{\prime}}}}/({\sum_{\Gamma}e^{-\beta E_{\Gamma}}})$ represents the occupation probability of the local configuration $\|\Gamma\rangle$, and $F^{\alpha}_{\Gamma\Gamma^{\prime}}=\langle\Gamma\|f_{\alpha}\|\Gamma^{\prime}\rangle$. Hubbard-I approximation is $\displaystyle\hat{\Sigma}\approx\hat{\Sigma}^{atom}$ (30) In reference [22] , Savrasov et alpropose that the above self energy can be written by the summation of a set of poles, which greatly simplifies the DMFT calculation, because it is not necessary to handle the full frequency dependence of the Green’s function. In Hubbard-I approximation, it can be proved that both the atomic Green’s function and self energy are diagonal and can be written in the form of pole expansion if the following two necessary conditions are satisfied. 1) The single particle basis used here should be the one which diagonalize the local density matrix; 2) The atomic Hamiltonian only contains the two-body interaction terms. Obviously by chosen the proper single particle basis the atomic Hamiltonian (18) in general will satisfy the above two condition and thus can be always written in terms of pole expansion. Unlike the reference[22], where they only use a few poles to capture the main features of the self energy, in the present implementation, we keep all the poles in the self energy, which makes it more accurate. From (29), it is very clear that the atomic green’s function has already been expressed in terms of poles and the pole expansion of the corresponding self energy can be obtained using (28), which is introduced in details in the Appendix. Therefore in general the above atomic self energy can be written as $\hat{\Sigma}(\rmi\omega)=\hat{\Sigma}({\infty})+\sum^{np}_{i=1}\hat{V}^{\dagger}_{i}(\rmi\omega- p_{i})^{-1}\hat{V}_{i}$ (31) where $i$ labels the number of poles, $np$ is the total number of poles in the self energy and $\hat{V}_{i}$ is a vector defined in the orbital space describing the distribution of the $ith$ “pole states” among the local orbitals. ### 3.3 Correction of the Density Matrices and Pole Expansion of the Self- energy Once we obtain the converged local self-energy, we are at the point to correct the k-dependent lattice Green’s functions as well as the density matrices. In general we need to do the summation over the Matsubara frequencies, which is time consuming for realistic materials contains lots of bands. But it can be greatly simplified when the self energy can be expressed as the summation of poles, which can be written as $\displaystyle\langle\varphi_{n_{1}\mathbf{k}}\|\hat{\rho}_{LDA+DMFT}\|\varphi_{n_{2}\mathbf{k}}\rangle$ (32) $\displaystyle=$ $\displaystyle\frac{1}{\beta}\sum_{i\omega_{n}}\langle\varphi_{n_{1}\mathbf{k}}\|\frac{1}{\rmi\omega_{n}+\mu-\hat{H}_{LDA}+\hat{V}_{DC}-\hat{\Sigma}(i\omega_{n})}\|\varphi_{n_{2}\mathbf{k}}\rangle$ As pointed out firstly in reference [22], when the self energy can be written in terms of poles, the full green’s function can be expressed as the “physical” part of an enlarged “pseudo Hamiltonian”, defined as $\displaystyle\hat{H}_{ps}(\mathbf{k})=\left(\begin{array}[]{cc}\hat{H}_{LDA}(\mathbf{k})-\hat{V}_{DC}+\hat{\Sigma}(\infty)&\qquad-\hat{V}^{\dagger}\\\ -\hat{V}&\qquad\hat{P}\end{array}\right)$ (35) where $\hat{H}_{LDA}(\mathbf{k})=E_{n\mathbf{k}}\hat{c}^{\dagger}_{n\mathbf{k}}\hat{c}_{n\mathbf{k}}$ is the physical Hilbert space, $\hat{V}_{i}$ is defined in (31) and $\hat{P}=Diag(p_{1},p_{2},....,p_{np})$. Therefore the physical green’s function can be expressed in terms of eigenstate of the pseudo Hamiltonian simply as $\displaystyle\langle\varphi_{n_{1}\mathbf{k}}\bigg{\|}\frac{1}{\rmi\omega_{n}-\hat{H}_{LDA}(\mathbf{k})+\hat{V}_{DC}-\hat{\Sigma}(i\omega_{n})}\bigg{\|}\varphi_{n_{2}\mathbf{k}}\rangle$ (36) $\displaystyle=$ $\displaystyle\langle\varphi_{n_{1}\mathbf{k}}\|\psi^{ps}_{l\mathbf{k}}\rangle\frac{1}{\rmi\omega_{n}-E^{ps}_{l}(\mathbf{k})}\langle\psi^{ps}_{l\mathbf{k}}\|\varphi_{n_{2}\mathbf{k}}\rangle$ where $\|\psi^{ps}_{l\mathbf{k}}\rangle$ and $E^{ps}_{l}(\mathbf{k})$ are the eigenstate and eigenvalue of pseudo Hamiltonian respectively. Then the sum of frequencies in (32) can be performed directly $\displaystyle\langle\varphi_{n_{1}\mathbf{k}}\|\hat{\rho}_{LDA+DMFT}\|\varphi_{n_{2}\mathbf{k}}\rangle$ (37) $\displaystyle=$ $\displaystyle\frac{1}{\beta}\sum_{i\omega_{n}}\sum_{l\mathbf{k}}\langle\varphi_{n_{1}\mathbf{k}}\|\psi^{ps}_{l\mathbf{k}}\rangle\frac{1}{\rmi\omega_{n}-E_{l}^{ps}(\mathbf{k})}\langle\psi^{ps}_{l\mathbf{k}}\|\varphi_{n_{2}\mathbf{k}}\rangle$ $\displaystyle=$ $\displaystyle\sum_{l\mathbf{k}}\langle\varphi_{n_{1}\mathbf{k}}\|\psi^{ps}_{l\mathbf{k}}\rangle\langle\psi^{ps}_{l\mathbf{k}}\|\varphi_{n_{2}\mathbf{k}}\rangle n_{F}[E_{l}^{ps}(\mathbf{k})-\mu]$ in which $\mu$ is exactly the chemical potential in (12), determined by the occupation number of electrons $N_{tot}$ in the unit cell $\displaystyle N_{tot}$ $\displaystyle=$ $\displaystyle\sum_{n\mathbf{k}}\langle\varphi_{n\mathbf{k}}\|\hat{\rho}\|\varphi_{n\mathbf{k}}\rangle$ (38) $\displaystyle=$ $\displaystyle\sum_{ln\mathbf{k}}\|\langle\varphi_{n\mathbf{k}}\|\psi^{ps}_{l\mathbf{k}}\rangle\|^{2}n_{F}[E_{l}^{ps}(\mathbf{k})-\mu]$ ### 3.4 Evaluation of Real Space Charge Density and Complete of the Full Loop After DMFT obtains the corrected density matrix $\hat{\rho}$, the real space charge density will be generated based on it by Fourier transformation, and then, the new LDA Hamiltonian as well as the new overlap matrices between Bloch states and local orbitals will be recalculated, which closes the full iteration loop for the charge density self consistency. The occupation number of the Bloch basis can also be obtained by, $\displaystyle n_{\mbox{\tiny LDA+DMFT}}=\sum_{\mathbf{k},n_{1}n_{2},\alpha}\langle\alpha_{\mathbf{k}}\|\varphi_{n_{1}\mathbf{k}}\rangle\langle\varphi_{n_{1}\mathbf{k}}\|\hat{\rho}\|\varphi_{n_{2}\mathbf{k}}\rangle\langle\varphi_{n_{2}\mathbf{k}}\|\alpha_{\mathbf{k}}\rangle$ (39) ### 3.5 Calculation of Physical Quantities #### 3.5.1 Energy Functional Most of the physical quantities we are interested in can be calculated once we have reached the charge density self consistency. First we present the energy functional of the full loop LDA+DMFT, which can be used to calculate many quantities, i.e. the total energy, force and so on. It can be written as, [6] $\displaystyle E=E_{LDA}[\rho]-\langle H_{KS}\rangle_{LDA}+\langle H_{KS}\rangle+\langle H_{U}\rangle-E_{DC}$ (40) in which $E_{LDA}[\rho]$ is the expression of the energy within density- functional theory; $\langle H_{KS}\rangle_{LDA}=\tr[\hat{H}_{LDA}\hat{G}_{LDA}]$ is the non-interacting energy at LDA level; $\langle H_{KS}\rangle=\tr[\hat{H}_{LDA}\hat{G}]$ is the non-interacting energy at DMFT level; $\langle H_{U}\rangle=\frac{1}{2}\tr[\hat{\Sigma}\hat{G}]$ is the interaction energy caused by local correlation interactions; the last term $E_{DC}$ is the double counting energy given before in (22). In above the meaning for “trace” is defined as $\displaystyle\tr[\mathcal{A}]=\frac{1}{\beta}\sum_{n\mathbf{k}}\sum_{\rmi\omega_{n}}\langle\varphi_{n\mathbf{k}}\|\mathcal{A}(\rmi\omega_{n})\|\varphi_{n\mathbf{k}}\rangle$ (41) and the integral path surrounds all the energies of occupation states. The first and the second term is easily calculated in the LDA framework, and the three terms remaining must be evaluated in the DMFT process, explicitly $\displaystyle\langle H_{KS}\rangle=\sum_{n\mathbf{k}}E_{n\mathbf{k}}\langle\varphi_{n\mathbf{k}}\|\hat{\rho}\|\varphi_{n\mathbf{k}}\rangle$ (42) and $\displaystyle\langle H_{U}\rangle$ $\displaystyle=$ $\displaystyle\frac{1}{2}\tr[\hat{\Sigma}\hat{G}]$ (43) $\displaystyle=$ $\displaystyle\frac{1}{2}\tr[(\hat{G}^{-1}_{0}-\hat{G}^{-1})\hat{G}]$ $\displaystyle=$ $\displaystyle\frac{1}{2}\left[\sum_{m,n\mathbf{k}}(E^{ps}_{m\mathbf{k}}-\mu)\|\langle\varphi_{n\mathbf{k}}\|\psi_{m\mathbf{k}}\rangle\|^{2}n_{F}(E^{ps}_{m\mathbf{k}}-\mu)\right.$ $\displaystyle\left.-\langle H_{KS}\rangle+\langle\hat{V}_{DC}\rangle\right]$ while $\langle H_{KS}\rangle$ is expressed in (42), and $\langle\hat{V}_{DC}\rangle$ is the expected value of $\hat{V}_{DC}$ at DMFT level $\displaystyle\langle\hat{V}_{DC}\rangle$ $\displaystyle=$ $\displaystyle\sum_{i}\bar{U}_{i}(n^{i}_{\mbox{\tiny LDA+DMFT}}-\frac{1}{2})\sum_{\alpha}\langle\hat{n}^{i}_{\alpha}\rangle$ (44) $\displaystyle=$ $\displaystyle\bar{U}_{i}(n^{i}_{\mbox{\tiny LDA+DMFT}}-\frac{1}{2})n^{i}_{\mbox{\tiny LDA+DMFT}}$ (45) Notice that here emerges the site index $i$. We should be very careful in the case that the number of correlated atoms is larger than one. The final expression of the total energy is $\displaystyle E=E_{LDA}[\rho]-\langle H_{KS}\rangle_{LDA}+\langle H_{KS}\rangle$ $\displaystyle+\frac{1}{2}\left[\sum_{m,nk}(E^{ps}_{mk}-\mu)\|\langle\varphi_{nk}\|\psi_{mk}\rangle\|^{2}n_{F}(E^{ps}_{mk}-\mu)-\langle H_{KS}\rangle\right]$ $\displaystyle+\frac{1}{4}\sum_{i}\bar{U}_{i}n^{i}_{\mbox{\tiny LDA+DMFT}}$ (46) However, if we use Hubbard-I as the impurity solver, and remove double- counting term by using an integer, the energy functional is given by Haule[7] $\displaystyle E=E_{LDA}[\rho]-\langle H_{KS}\rangle_{LDA}+\langle H_{KS}\rangle$ $\displaystyle+\frac{1}{2}\left[\sum_{m,n\mathbf{k}}(E^{ps}_{m\mathbf{k}}-\mu)\|\langle\varphi_{n\mathbf{k}}\|\psi_{m\mathbf{k}}\rangle\|^{2}n_{F}(E^{ps}_{m\mathbf{k}}-\mu)-\langle H_{KS}\rangle\right]$ $\displaystyle+\frac{1}{4}\sum_{i}\bar{U}_{i}n^{i}_{0}$ (47) where $n^{i}_{0}$ refers to the integer used to remove double counting for the $i$th correlated atom. #### 3.5.2 Expressions of DOS and PDOS In order to calculate the density of state (DOS) or the partial density of state (PDOS) on correlated orbitals, it is necessary to calculate the retarded form of the Green’s function. By using the virtue of pole expansion method, we write directly $\displaystyle D(\epsilon)=(-\frac{1}{\pi})\mathrm{Im}\bigg{[}\sum_{\mathbf{k},m,n}\frac{\langle\varphi_{n\mathbf{k}}\|\psi_{m\mathbf{k}}\rangle\langle\psi_{m\mathbf{k}}\|\varphi_{n\mathbf{k}}\rangle}{\epsilon+\mu-E^{ps}_{m\mathbf{k}}+\rmi\eta}\bigg{]}\ (\eta\rightarrow 0^{+})$ (48) and the PDOS of orbital $\alpha$ $\displaystyle D_{\alpha}(\epsilon)$ $\displaystyle=$ $\displaystyle(-\frac{1}{\pi})\mathrm{Im}\left[\sum_{\mathbf{k},m}\frac{\langle\alpha_{\mathbf{k}}\|\psi_{m\mathbf{k}}\rangle\langle\psi_{m\mathbf{k}}\|\alpha_{\mathbf{k}}\rangle}{\epsilon+\mu-E^{ps}_{m\mathbf{k}}+\rmi\eta}\right]\ (\eta\rightarrow 0^{+})$ (49) where $\displaystyle\langle\alpha_{\mathbf{k}}\|\psi_{m\mathbf{k}}\rangle=\sum_{n}\langle\alpha_{\mathbf{k}}\|\varphi_{n\mathbf{k}}\rangle\langle\varphi_{n\mathbf{k}}\|\psi_{m\mathbf{k}}\rangle$ (50) ## 4 Benchmark To validate the LDA+DMFT framework based on pseudo-potentials-planewave package and pole expansion of self energy, we benchmark our implementation by applying to $\gamma$-cerium, americium and paramagnetic NiO. These three are typical strongly correlated systems in which the valence electrons are believed to be on the localized side as reported in previous literatures. We use the Hubbard-I method as impurity solver, which is good enough to capture the atomic-like features in the Mott insulators, so we expect that these systems could be well described by our method. We would emphasize here that in our implementation we can replace the Hubbard-I solver by any of the solvers as long as the self energy can be written in pole expansion form. ### 4.1 Cerium The cerium metal attracts lots of research interests for its isostructural volume-collapse transition from $\gamma$ phase to $\alpha$ phase. The volume change is about $15\%$ during the transition, which is possibly driven by the entropy change[23]. Normal LDA calculations could not give a correct description of cerium, as show in table 1, the equillibrium volume of cerium given by LDA calculations is smaller than the experimental volume of $\alpha$ phase. This is due to the fact that in LDA the $f$-electrons are treated as itinerant and the strong correlation effects among them can not be well captured. When $f$-electron is treated as core electrons, we see that the equilibrium volume is larger but close to $\gamma$ phase. These simple LDA results implies that in $\gamma$ phase, the $f$-electrons are more localized than in $\alpha$-phase. Therefore $f$-electrons in $\gamma$ phase is quite close to the localized picture like the situation in Mott insulators, which makes it suitable to applying the Hubbard-I solver. The $\alpha$-Ce which is stable in low temperature is a correlated metal, which is confirmed by many other LDA+DMFT calculations[23], and also by our newly development LDA+Gutzwiller method[24]. In our calculation for $\gamma$-Ce, we used ultrasoft pseudo-potentional in which 4 _f_ , 5 _s_ , 5 _d_ , 6 _s_ orbitals are taken as valence orbitals. The energy cutoff for plane-wave expansion is 12.5 Ha for convergence. Calculations are performed with $10\times 10\times 10$ k-points. For DMFT calculation, the on-site Coulomb interaction $U$ is chosen to be 6.0 eV as suggested in previous reports.[25, 23, 26] In figure 2 the total energy versus volume for $\gamma$-cerium obtained by LDA, LDA+$U$ and LDA+DMFT are plotted. The equilibrium volume and bulk modulus are also shown in table 1. We can find in figure 2 that the equilibrium volume obtained by our LDA+DMFT calculation is very close to the experimental data, which shows great improvements over LDA . The bulk modulus is also improved but still larger than the experimental data, which may imply the contribution for lattice vibration is inneglectable as discussed in reference [6]. Table 1: Lattice parameter and bulk modulus of $\gamma$-cerium according to both experiment and calculations * \| Volume($\mathrm{\AA}^{3}$) \| Bulk modulus(GPa) ---\|---\|--- Experiments \| 34.35[27] \| 21[28] LDA \| 22.36 \| 62.09 LDA(fcore)[29] \| 36.39 \| 30.2 LDA+$U$/PAW[30] \| 32.0 \| 34 LDA+DMFT[6] \| 30.10 \| 48.49 LDA+DMFT \| 33.29 \| 38.27 Figure 2: Total energy versus volume curves obtained by LDA, LDA+$U$[6], and LDA+DMFT. The dashed lines are fitted by Birch EOS. The energy were shifted for different curves. The experimental equilibrium volume is from reference [27] Figure 3 shows the 4 _f_ partial DOS for $\gamma$ cerium at the equilibrium volume, the position of the two Hubbard bands agree well with the experimental result. Figure 3: 4 _f_ pdos for $\gamma$ cerium in LDA+DMFT (Hubbard I), XPS and BIS experimental data are from reference [31] and [32] ### 4.2 Americium Americium, which is widely used in smoke detectors, can be viewed as the ”changing point” of the actinide series, where the 5 _f_ electrons changes from delocalized to localized states[33]. Valence-band ultraviolet photoemission experiment by Naegele et al[34] shows that the 5 _f_ states are strongly localized. Am undergoes a series of structure phase transitions with the increment of pressure, from double hexagonal close packed(dhcp, P63/mmc), to face centered cubic(fcc, Fm3m, 6.1 GPa), face centered orthorhombic(Fddd, 10.0 GPa), and to primitive orthorhombic structure(Pnma, 16 PGa)[35]. Therefore Am provides an interesting playfield to investigate the relation between $f$-electron localization and structure transition. Here we focus on the fcc phase and dhcp phase. In the calculation, we used ultrasoft pseudo-potentional contains 5 _f_ , 6 _p_ , 6 _d_ , 7 _s_ orbitals as valence orbitals, the plane-wave expansion kinetic energy cutoff of 12.5 Ha. The LDA self-consistent calculations were performed with $10\times 10\times 10$ k-points grid for fcc phase, $9\times 9\times 3$ k-points grid for dhcp phase. For the volume calculation, we only considered density-density interaction, and on-site Coulomb interaction $F^{(0)}=4.5$ eV, for the partial DOS calculation, we considered general interaction, and take the atomic values $F^{(2)}=7.2$ eV, $F^{(4)}=4.8$ eV, $F^{(6)}=3.6$ eV[36]. We present the optimized volume and bulk modulus for both dhcp and fcc phase in table 2 and figure 4. Our LDA+DMFT results are quite close to the experimental data and show great improvement over LDA. Also our results are quite consistent with the reference [22], in which the full potential LMTO methods are used in the LDA part. The equation of state of Am obtained by our LDA+DMFT calculation is plotted in figure 5, which again shows very good agreement with both the experimental data and previous LDA+DMFT calculation based on LMTO method[22]. Table 2: Am equilibrium volume and bulk modulus for dhcp and fcc phase by different methods, together with the experimental results[35, 37]. * \| Volume($\mathrm{\AA}^{3}$) \| Bulk modulus(GPa) ---\|---\|--- experiment \| 29.250 \| 29.9 GGA(dhcp)[38] \| 19.916 \| 70.0 LDA(dhcp) \| 18.55 \| 118.99 LDA(fcc) \| 17.73 \| 169.40 LDA+DMFT(dhcp) \| 27.04 \| 52.70 LDA+DMFT(fcc) \| 26.99 \| 50.75 LDA+DMFT[22] \| 27.4 \| 45 Figure 4: Total energy per atom of Am obtained by LDA and LDA+DMFT. The LDA+DMFT results of Am for both phase are very close, and the experimental equilibrium volume is denoted by dot line. Figure 5: Calculated P-V relation of dhcp and fcc Am. Here, $V_{0}$ is experimental equilibrium volume, The experimental P-V results[37] are shown by black dots. Figure 6 shows the 5 _f_ partial DOS of fcc phase Am obtained both by our LDA+DMFT calculation and LDA+DMFT(OCA) by Savrasov[22],together with the photoemission data from ref.[34]. Our results is quite consistent with the previous calculations and the experimental results. Figure 6: The spectral function of Am evaluated by LDA+DMFT. The experimental photoemission results is from reference [34] are denoted by solid dots. ### 4.3 Paramagnetic phase of NiO NiO has been heavily studied as a prototype of Mott insulators. It has been studied within the frame of LDA+DMFT by many groups[39, 40]. Therefore NiO can be used as a good benchmark material for the implementation of LDA+DMFT. In this paper, we apply our code to study the paramagnetic phase of NiO. First we plot the partial density of states (PDOS) obtained by LDA in figure 7, which clearly shows metallic behavior and is not consistent with the experiments. In our LDA+DMFT(Hub1) calculation, we use the wannier functions as the local basis and choose on-site interaction $U=8.0eV$, Hund’s coupling $J=1.0eV$. The Mott insulating features of NiO then can be well captured by our LDA+DMFT method, as shown by the PDOS in figure 8). Figure 7: The partial DOS of NiO obtained by LDA. The main contribution to the Fermi surface is attributed to eg-like Wannier orbitals, and the t2g-like orbitals are almost fully occupied. Figure 8: The spectral function of NiO obtained by LDA+DMFT calculation. As shown in the figure, compared with LDA results, the t2g-like orbitals are still fully occupied while the eg-like orbitals split into upper and lower Hubbard bands. The energy gap obtained by our calculation is around 4.0eV, which is also quite consistent with the experiments. The photo emission and BIS data obtained by Sawatzky and Allen[41] are also plotted in figure 9. We can find that our results fit the photo emission and BIS data very well. Figure 9: The spectral function of NiO evaluated by LDA+DMFT, with the comparison of experimental data[41]. The behavior of the density of state near the fermi surface fits well with the experimental data. ## 5 Conclusions The new implementation of LDA+DMFT based on the pseudo-potential plane-wave method is introduced in detail in this paper. We choose the Hubbard-I method as the impurity solver to solve the quantum impurity problem generated by DMFT, which is quite suitable for the Mott insulator materials. The most important advantage of Hubbard-I method is that it is simple enough, which makes the full loop charge self consistent calculation accessible. We also point out in the paper that the difficulty of handling frequency dependent Green’s function can be completely removed by expressing the self energy in terms of pole expansion, which greatly raise the efficiency of the method. Finally we benchmark our implementation of LDA+DMFT on several important correlation materials including $\gamma$-Ce,Am and NiO. Our results for all these materials fit very well with both the experimental data and the previous LDA+DMFT results. ## Appendix A Self-energy in pole expansion form In this appendix, we will introduce how to evaluate self-energy from Green’s function in the pole expansion form. Green’s function can be expressed as $G(i\omega)=\sum^{N_{G}}_{i=1}\frac{V_{i}}{i\omega-P_{i}}$ (51) where $P_{i}$ is the $i$th pole of the Green’s function, $V_{i}$ is the weight of the pole, and $N_{G}$ is the number of Green’s function poles. Besides the self-energy can be expressed as $\Sigma(i\omega)=\sum^{N_{S}}_{i=1}\frac{W_{i}}{i\omega-Q_{i}}+\Sigma(\infty)$ (52) where $Q_{i}$ is the $i$th pole of the self-energy, $W_{i}$ is the weight of the pole, and $N_{S}$ is the number of self-energy poles. Then the Dyson’s equation (28) becomes $\sum^{N_{S}}_{i=1}\frac{W_{i}}{i\omega- Q_{i}}+\Sigma(\infty)=i\omega+\mu-\epsilon_{imp}-\left[\sum^{N_{G}}_{i=1}\frac{V_{i}}{i\omega- P_{i}}\right]^{-1}$ (53) First consider the $\Sigma(\infty)$, when $\omega\rightarrow\infty$ $\displaystyle\Sigma(\omega\rightarrow\infty)$ $\displaystyle=i\omega+\mu-\epsilon_{imp}-\left[\sum^{N_{G}}_{i=1}\frac{V_{i}}{i\omega- P_{i}}\right]^{-1}$ (54) $\displaystyle=i\omega+\mu-\epsilon_{imp}-\left[\frac{1}{i\omega}\sum^{N_{G}}_{i=1}\frac{V_{i}}{1-\frac{P_{i}}{i\omega}}\right]^{-1}$ $\displaystyle=i\omega+\mu-\epsilon_{imp}-i\omega\left[\sum^{N_{G}}_{i=1}V_{i}-\frac{1}{i\omega}\sum^{N_{G}}_{i=1}V_{i}P_{i}\right]^{-1}$ $\displaystyle=i\omega+\mu-\epsilon_{imp}-i\omega\left(1-\frac{\sum^{N_{G}}_{i=1}V_{i}P_{i}}{i\omega}\right)$ $\displaystyle=\mu-\epsilon_{imp}+\sum^{N_{G}}_{i=1}V_{i}P_{i}$ For the the self energy poles in (53), it corresponding to the zero in the square bracket of the right side of (53), that is for $Q\in\\{Q_{i}\\}$, we have $\sum^{N_{G}}_{i=1}\frac{V_{i}}{Q-P_{i}}=0$ (55) The left hand side is monotonically decreasing between two adjacent Green’s function poles, so we could use bisection method to find the self energy poles. We rewrite (53) as $\sum^{N_{s}}_{i=1}\frac{W_{i}}{i\omega- Q_{i}}=i\omega+\mu-\epsilon-\Sigma(\infty)-\left[\sum^{N_{g}}_{i=1}\frac{V_{i}}{i\omega- P_{i}}\right]^{-1}$ (56) For the left hand side $\displaystyle W_{i}$ $\displaystyle=\lim_{i\omega\rightarrow Q_{i}}\left(\sum^{N_{S}}_{j=1}\frac{W_{j}}{i\omega-Q_{j}}\right)(i\omega- Q_{i})$ (57) $\displaystyle=\lim_{i\omega\rightarrow Q_{i}}\left[i\omega+\mu-\epsilon_{imp}-\Sigma(\infty)-\left(\sum^{N_{g}}_{j=1}\frac{V_{j}}{i\omega- P_{j}}\right)^{-1}\right](i\omega-Q_{i})$ $\displaystyle=\lim_{i\omega\rightarrow Q_{i}}-(i\omega- Q_{i})\left(\sum^{N_{G}}_{j=1}\frac{V_{j}}{i\omega-P_{j}}\right)^{-1}$ here we define $\delta=i\omega-Q_{i}$ and $h(\delta)=\sum^{N_{G}}_{j=1}\frac{V_{j}}{Q_{i}+\delta-P_{j}}$, expand $h(\delta)$ as $h(\delta)\approx\delta\cdot h^{\prime}(0)=-\sum^{N_{G}}_{j=1}\frac{V_{j}}{(Q_{i}-P_{j})^{2}}\cdot\delta$ (58) so $W_{i}=\left[\sum^{N_{G}}_{j=1}\frac{V_{j}}{(Q_{i}-P_{j})^{2}}\right]^{-1}$ (59) here we have got both self energy poles and its weight. The authors thank L.Wang, G. Xu, and F. Lu for their helpful discussions. We acknowledge the support from the 973 Program of China (Grant No. 2011CBA00108), from the NSF of China (Grants No. NSFC 10876042 and No. NSFC 10874158), and that from the Fund of Laboratory of Shock Wave and Detonation Physics, Institute of Fluid Physics, CAEP, Contract No. 2011-056000-0833F. ## References ## References * [1] Georges A, Kotliar G, Krauth W, and Rozenberg M J. Dynamical mean-field theory of strongly correlated fermion systems and the limit of infinite dimensions. Rev. Mod. Phys., 68(1):13–125, January 1996. * [2] Lechermann F, Georges A, Poteryaev A, Biermann S, Posternak M, Yamasaki A, and Andersen O K. Dynamical mean-field theory using Wannier functions: A flexible route to electronic structure calculations of strongly correlated materials. Phys. Rev.B, 74(12):125120, September 2006. * [3] Anisimov V I, Kozhevnikov A V, Korotin M A, Lukoyanov A V, and Khafizullin D A. Orbital density functional as a means to restore the discontinuities in the total-energy derivative and the exchange–correlation potential. J. Phys.: Condens. Matter, 19(10):106206, March 2007. * [4] Savrasov S Y and Kotliar G. Spectral density functionals for electronic structure calculations. Phys. Rev.B, 69(24):245101, June 2004. * [5] Minár J, Chioncel L, Perlov A, Ebert H, Katsnelson M, and Lichtenstein A. Multiple-scattering formalism for correlated systems: A KKR-DMFT approach. Phys. Rev.B, 72(4):045125, July 2005. * [6] Pourovskii L V, Amadon B, Biermann S, and Georges A. Self-consistency over the charge density in dynamical mean-field theory: A linear muffin-tin implementation and some physical implications. Phys. Rev.B, 76(23):235101, December 2007. * [7] Haule K, Yee Chuck-Hou, and Kim Kyoo. Dynamical mean-field theory within the full-potential methods: Electronic structure of CeIrIn5 , CeCoIn5 , and CeRhIn5. Phys. Rev.B, 81(19):195107, May 2010. * [8] Andersen O K. Linear methods in band theory. Phys. Rev.B, 12(8):3060–3083, October 1975. * [9] Anisimov V I, Aryasetiawan F, and Lichtenstein A I. First-principles calculations of the electronic structure and spectra of strongly correlated systems: the LDA + U method. J. Phys.: Condens. Matter, 9(4):767–808, January 1997. * [10] Lichtenstein A I and Katsnelson M I. Ab initio calculations of quasiparticle band structure in correlated systems: LDA++ approach. Phys. Rev.B, 57(12):6884–6895, March 1998. * [11] Savrasov S Y, Kotliar G, and Abrahams E. Correlated electrons in delta-plutonium within a dynamical mean-field picture. Nature, 410(6830):793–5, April 2001. * [12] Andersen O K and Saha-Dasgupta T. Muffin-tin orbitals of arbitrary order. Phys. Rev.B, 62(24):R16219–R16222, December 2000. * [13] Anisimov V I, Kondakov D E, Kozhevnikov A V, Nekrasov I A, Pchelkina Z V, Allen J W, Mo S K, Kim H D, Metcalf P, Suga S, Sekiyama A, Keller G, Leonov I, Ren X, and Vollhardt D. Full orbital calculation scheme for materials with strongly correlated electrons. Phys. Rev.B, 71(12):125119, March 2005. * [14] Souza I, Marzari N, and Vanderbilt D. Maximally localized Wannier functions for entangled energy bands. Phys. Rev.B, 65(3):035109, December 2001. * [15] Marzari N and Vanderbilt D. Maximally localized generalized Wannier functions for composite energy bands. Phys. Rev.B, 56(20):12847–12865, November 1997. * [16] Amadon B, Lechermann F, Georges A, Jollet F, Wehling T O, and Lichtenstein A I. Plane-wave based electronic structure calculations for correlated materials using dynamical mean-field theory and projected local orbitals. Phys. Rev.B, 77(20):205112, May 2008. * [17] Trimarchi G, Leonov I, Binggeli N, Korotin Dm, and Anisimov V I. LDA+DMFT implemented with the pseudopotential plane-wave approach. J. Phys.: Condens. Matter, 20(13):135227, April 2008. * [18] Korotin DM, Kozhevnikov A V, Skornyakov S L, Leonov I, Binggeli N, Anisimov V I, and Trimarchi G. Construction and solution of a Wannier-functions based Hamiltonian in the pseudopotential plane-wave framework for strongly correlated materials. Eur. Phys. J. B, 98:91–98, 2008. * [19] Vanderbilt D. Soft self-consistent pseudopotentials in a generalized eigenvalue formalism. Phys. Rev.B, 41(11):7892–7895, April 1990. * [20] Hamann D R, Schlüter M, and Chiang C. Norm-Conserving Pseudopotentials. Phys. Rev. Lett., 43(20):1494–1497, November 1979. * [21] Held K. Electronic structure calculations using dynamical mean field theory. Advances in Physics, 56(6):829–926, November 2007. * [22] Savrasov S Y, Haule K, and Kotliar G. Many-Body Electronic Structure of Americium Metal. Phys. Rev. Lett., 96(3):036404, January 2006. * [23] Amadon B, Biermann S, Georges A, and Aryasetiawan F. The $\alpha$-$\gamma$ Transition of Cerium Is Entropy Driven. Phys. Rev. Lett., 96(6):066402, February 2006. * [24] Tian M F, Deng X Y, Fang Z, and Dai X. The intermediate pressure phases of cerium: a LDA+Gutzwiller method study. arXiv, page 1104.0156, April 2011. * [25] Haule K, Oudovenko V, Savrasov S Y, and Kotliar G. The $\alpha\rightarrow\gamma$ transition in ce: A theoretical view from optical spectroscopy. Phys. Rev. Lett., 94(3):036401, Jan 2005. * [26] Amadon B. Fully self-consistent LDA+DMFT Calculation: A plane wave and projector augmented wave implementation and application to Cerium, Ce2O3 and Pu2O3. arXiv, page 1101.0539, January 2011. * [27] Jeong I K, Darling T W, Graf M J, Proffen Th, Heffner R H, Lee Y, Vogt T, and Jorgensen J D. Role of the lattice in the $\gamma\rightarrow\alpha$ phase transition of ce: A high-pressure neutron and x-ray diffraction study. Phys. Rev. Lett., 92(10):105702, Mar 2004. * [28] Decremps F, Antonangeli D, Amadon B, and Schmerber G. Role of the lattice in the two-step evolution of $\gamma$-cerium under pressure. Phys. Rev.B, 80(13):132103, October 2009. * [29] Huang L and Chen C A. The lattice dynamics of $\alpha$\- and $\gamma$-Ce: a first-principles approach. J. Phys.: Condens. Matter, 19(47):476206, November 2007. * [30] Amadon B, Jollet F, and Torrent M. $\gamma$ and $\beta$ cerium: LDA+U calculations of ground-state parameters. Phys. Rev.B, 77(15):155104, April 2008. * [31] Wuilloud E, Moser H R, Schneider W D, and Baer Y. Electronic structure of $\gamma$\- and $\alpha$-ce. Phys. Rev.B, 28(12):7354–7357, Dec 1983. * [32] Wieliczka D, Weaver J H, Lynch D W, and Olson C G. Photoemission studies of the $\gamma$-$\alpha$ phase transition in ce: Changes in 4$f$ character. Phys. Rev.B, 26(12):7056–7059, Dec 1982. * [33] Moore K T and van der Laan G. Nature of the 5f states in actinide metals. Rev. Mod. Phys., 81(1):235–298, February 2009. * [34] Naegele J R, Manes L, Spirlet J C, and Müller W. Localization of 5f Electrons in Americium: A Photoemission Study. Phys. Rev. Lett., 52(20):1834–1837, May 1984. * [35] Heathman S, Haire R G, Le Bihan T, Lindbaum A, Litfin K, Méresse Y, and Libotte H. Pressure Induces Major Changes in the Nature of Americium’s 5f Electrons. Phys. Rev. Lett., 85(14):2961–2964, October 2000. * [36] Carnall W T and Wybourne B G. Electronic Energy Levels of the Lighter Actinides: U3+, Np3+, Pu3+, Am3+, and Cm3+. J. Chem. Phys., 40(11):3428, 1964. * [37] Lindbaum A, Heathman S, Litfin K, Méresse Y, Haire R G, Le Bihan T, and Libotte H. High-pressure studies of americium metal: Insights into its position in the actinide series. Phys. Rev.B, 63(21):214101, April 2001. * [38] Kutepov A and Kutepova S. First-principles study of electronic and magnetic structure of alpha-Pu, delta-Pu, americium, and curium. J. Magn. Magn. Mater., 272-276:E329–E330, May 2004. * [39] Ren X, Leonov I, Keller G, Kollar M, Nekrasov I, and Vollhardt D. LDA+DMFT computation of the electronic spectrum of NiO. Phys. Rev.B, 74(19):195114, November 2006. * [40] Kuneš J, Anisimov V I, Lukoyanov A V, and Vollhardt D. Local correlations and hole doping in NiO: A dynamical mean-field study. Phys. Rev.B, 75(16):4–7, April 2007. * [41] Sawatzky G and Allen J. Magnitude and Origin of the Band Gap in NiO. Phys. Rev. Lett., 53(24):2339–2342, December 1984.	arxiv-papers	2011-11-09T10:02:43	2024-09-04T02:49:24.149333	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Jian-Zhou Zhao, Jia-Ning Zhuang, Xiao-Yu Deng, Yan Bi, Ling-Cang Cai,\n Zhong Fang, Xi Dai", "submitter": "Feng Lu", "url": "https://arxiv.org/abs/1111.2157" }
1111.2183	See pages 1-3 of StartDoc.pdf See pages 1-6 of icrc0726.pdf See pages 1-4 of icrc0684.pdf See pages 1-4 of icrc0935.pdf See pages 1-4 of icrc2011_0666_v2.pdf See pages 1-4 of icrc0936.pdf See pages 1-4 of icrc0706_v2.pdf See pages 1-4 of icrc0580.pdf See pages 1-4 of icrc1119_Kubo.pdf See pages 1-4 of icrc1091_final.pdf See pages 1-4 of icrc0941_v2.pdf See pages 1-4 of icrc0692_submitted.pdf See pages 1-4 of 1065_v3.pdf See pages 1-4 of icrc1021_v2.pdf See pages 1-4 of icrc0688.pdf See pages 1-4 of icrc0668_v5.pdf See pages 1-4 of icrc0408.pdf See pages 1-4 of Orr-Krennrich-2011-icrc1156.pdf See pages 1-4 of icrc1086_v2.pdf See pages 1-4 of icrc0883_v2.pdf	arxiv-papers	2011-11-09T11:56:41	2024-09-04T02:49:24.158214	{ "license": "Public Domain", "authors": "The CTA Consortium", "submitter": "Bruno Kh\\'elifi", "url": "https://arxiv.org/abs/1111.2183" }
1111.2272	# Numerical and variational solutions of the dipolar Gross-Pitaevskii equation in reduced dimensions P. Muruganandam Instituto de Física Teórica, UNESP - Universidade Estadual Paulista, 01.140-070 São Paulo, São Paulo, Brazil School of Physics, Bharathidasan University, Palkalaiperur Campus, Tiruchirappalli 620024, Tamilnadu, India S. K. Adhikari Instituto de Física Teórica, UNESP - Universidade Estadual Paulista, 01.140-070 São Paulo, São Paulo, Brazil ###### Abstract We suggest a simple Gaussian Lagrangian variational scheme for the reduced time-dependent quasi-one- and quasi-two-dimensional Gross-Pitaevskii (GP) equations of a dipolar Bose-Einstein condensate (BEC) in cigar and disk configurations, respectively. The variational approximation for stationary states and breathing oscillation dynamics in reduced dimensions agrees well with the numerical solution of the GP equation even for moderately large short-range and dipolar nonlinearities. The Lagrangian variational scheme also provides much physical insight about soliton formation in dipolar BEC. ###### pacs: 03.75.Hh,03.75.Kk ## I Introduction The time-dependent mean-field Gross-Pitaevskii (GP) equation can accurately describe many static and dynamic properties of a harmonically trapped Bose- Einstein condensate (BEC) rev ; rev-2 ; lpl-1 ; lpl-2 ; lpl-3 ; lpl-4 ; ref-a01 ; ref-a02 ; ref-a03 ; ref-a04 ; ref-b01 ; ref-b02 ; ref-b03 ; ref-b04 ; ref-b05 ; ref-b06 ; balaz2011 . However, the numerical solution of the three-dimensional (3D) GP equation could often be a difficult task due to a large nonlinear term CPC ; CPC-1 . Fortunately, in many experimental situations the 3D axially symmetric harmonic trap has extreme symmetry so that the BEC has either a cigar or a disk shape geom . In these cases the essential statics and dynamics of a BEC take place in reduced dimensions. By integrating out the unimportant dimensional variable(s), reduced GP equations have been derived in lower dimensions luca ; other ; other-1 , which give a faithful description of the BEC in disk and cigar shapes. For disk and cigar shapes the reduced GP equation is written in two (2D) and one dimensions (1D), respectively. The numerical solution of such 2D, or 1D equation, although simpler than that of the original 3D equation, remain complex due to the nonlinear nature of the GP equation. Hence, for small values of the nonlinearity parameter, a Gaussian variational approximation is much useful for the solution of these equations var . The alkali metal atoms used in early BEC experiments have negligible dipole moment. However, most bosonic atoms and molecules have large dipole moments and a 52Cr lahaye ; pfau ; pfau-1 ; pfau-4 , and 164Dy dy ; dy-2 BEC with a larger long-range dipolar interaction superposed on the short-range atomic interaction, has been realized. Other atoms, like 166Er otherdi ; otherdi-1 , and molecules, such as 7Li-133Cs becmol , with much larger dipole moment are being considered for BEC experiments. A 3D GP equation for a dipolar BEC with a nonlocal nonlinear interaction has been suggested lahaye and successfully used to describe many properties of these condensates dipsol ; jb ; YY ; Yi2000 ; Dutta2007-1 ; Dutta2007-2 ; Dutta2007-3 ; Dutta2007-5 ; Dutta2007-6 . The applicability of the nonlocal GP equation to the case of dipolar BEC has been a subject of intensive study lahaye . After a detailed analysis, You and Yi YY ; Yi2000 concluded that the GP equation is valid for the dipolar BEC. Further support on the validity of this equation came from the study of Bortolotti et al. BB ; BB-1 . They compared the solution of the dipolar GP equation with the results of diffusive Monte Carlo calculations and found good agreement between the two. However, the 3D GP equation for a dipolar BEC with the nonlocal dipolar interaction has a complex structure and its numerical solution, involving the Fourier transformation of the dipolar nonlinear term to momentum space jb ; YY ; Yi2000 , is even more challenging than that of the GP equation of a non-dipolar BEC. Here we reconsider the dimensional reduction SS ; deu of the GP equation to 1D form for cigar-shaped dipolar BEC and obtain the precise 1D potential with a dipolar contact-interaction term. Previous derivations SS ; deu of the 1D reduced equation for dipolar BEC did not include the proper contact- interaction term, lacking which the 1D model will not provide a correct description of the full 3D system. We also consider the reduced 2D GP equation fisch ; PS for a disk-shaped dipolar BEC. Though these reduced GP equations for dipolar BEC are computationally less expensive than their 3D counterparts, the numerical solution procedure remains complicated due to repeated forward and backward Fourier transformations of the non-local dipolar term. As an alternative, here we suggest time-dependent Gaussian Lagrangian variational approximation of the 1D and 2D reduced equations. A direct attempt to derive the variational Lagrangian density of the reduced equations is not straightforward due to nonlocal integrals with error functions. We present an indirect evaluation of the Lagrangian density avoiding the above complex procedure. Thus, the present variational approximation involves algebraical quantities without requiring any Fourier transformation to momentum space. In case of dipolar BEC of Cr and Dy atoms we consider the numerical solution of the 3D and the reduced 1D and 2D GP equations for cigar and disk shapes to demonstrate the appropriateness of the solution of the reduced equations. The variational approximation of the reduced equations provided results for density, root-mean-square (rms) size, chemical potential, and breathing oscillation dynamics in good agreement with the numerical solution of the reduced and full 3D GP equations. ## II Analytical formulation ### II.1 3D GP Equation We study a dipolar BEC of $N$ atoms, each of mass $m$, using the dimensionless GP equation pfau ; pfau-1 ; pfau-4 $\displaystyle i\frac{\partial\phi({\bf r},t)}{\partial t}$ $\displaystyle\,=\biggr{[}-\frac{1}{2}\nabla^{2}+V({\bf r})+4\pi aN\|\phi({\bf r},t)\|^{2}$ $\displaystyle\,+N\int U_{dd}({\bf r-r^{\prime}})\|\phi({\bf r^{\prime}},t)\|^{2}d^{3}{r^{\prime}}\biggr{]}\phi({\bf r},t),$ (1) with dipolar interaction $U_{dd}({\bf R})=3a_{dd}(1-3\cos^{2}\theta)$ $/R^{3},$ $\quad{\bf R=r-r^{\prime}}.$ Here $V({\bf r})$ is the confining axially symmetric harmonic potential, $\phi({\bf r},t)$ the wave function at time $t$ with normalization $\int\|\phi({\bf r},t)\|^{2}d{\bf r}=1$, $a$ the atomic scattering length, $\theta$ the angle between $\bf R$ and the polarization direction $z$. The constant $a_{dd}=\mu_{0}\bar{\mu}^{2}m/(12\pi\hbar^{2})$ is a length characterizing the strength of dipolar interaction and its experimental value for 52Cr is $15a_{0}$ pfau ; pfau-1 ; pfau-4 , with $a_{0}$ the Bohr radius, $\bar{\mu}$ the (magnetic) dipole moment of a single atom, and $\mu_{0}$ the permeability of free space. In equation (II.1) length is measured in units of characteristic harmonic oscillator length $l\equiv\sqrt{\hbar/m\omega}$, angular frequency of trap in units of $\omega$, time $t$ in units of $\omega^{-1}$, and energy in units of $\hbar\omega$. The axial and radial angular frequencies of the trap are $\Omega_{z}\omega$ and $\Omega_{\rho}\omega$, respectively. The dimensionless 3D harmonic trap is $V({\bf r})=\frac{1}{2}\Omega_{\rho}^{2}\rho^{2}+\frac{1}{2}\Omega_{z}^{2}z^{2},$ (2) where ${\bf r}\equiv(\vec{\rho},z)$, with $\vec{\rho}$ the radial coordinate and $z$ the axial coordinate. The Lagrangian density of equation (II.1) is given by $\displaystyle{\mathcal{L}}$ $\displaystyle\,=\frac{i}{2}(\phi\phi^{\star}_{t}-\phi^{\star}\phi_{t})+\frac{\|\nabla\phi\|^{2}}{2}+V({\bf r})\|\phi\|^{2}$ $\displaystyle\,+2\pi aN\|\phi\|^{4}+\frac{N}{2}\|\phi\|^{2}\int U_{dd}({\mathbf{r}}-{\mathbf{r}^{\prime}})\|\phi({\mathbf{r}^{\prime}})\|^{2}d^{3}{r}^{\prime}.$ (3) We use the Gaussian ansatz var ; jb ; YY ; Yi2000 $\phi({\bf r},t)=\frac{\pi^{-3/4}}{w_{\rho}\sqrt{w_{z}}}\exp\left(-\frac{\rho^{2}}{2w_{\rho}^{2}}-\frac{z^{2}}{2w_{z}^{2}}+i\alpha\rho^{2}+i\beta z^{2}\right)$ (4) for a variational calculation, where $w_{\rho}$ and $w_{z}$ are time-dependent radial and axial widths, and $\alpha$ and $\beta$ time-dependent phases. The effective Lagrangian $L\equiv\int{\mathcal{L}}d^{3}{r}$ (per particle) becomes $\displaystyle L$ $\displaystyle=$ $\displaystyle\,\left(w_{\rho}^{2}\dot{\alpha}+\frac{w_{z}^{2}\dot{\beta}}{2}\right)+\frac{\Omega_{\rho}^{2}w_{\rho}^{2}}{2}+\frac{\Omega_{z}^{2}w_{z}^{2}}{4}+\frac{1}{2{w_{\rho}^{2}}}+\frac{1}{4w_{z}^{2}}$ (5) $\displaystyle\,+2w_{\rho}^{2}\alpha^{2}+w_{z}^{2}\beta^{2}+\frac{N}{(\sqrt{2\pi}w_{\rho}^{2}w_{z})}\left[{a}-{a_{dd}}f(\kappa)\right],$ with $\displaystyle f(\kappa)=\frac{1+2\kappa^{2}-3\kappa^{2}d(\kappa)}{(1-\kappa^{2})},$ (6) $\displaystyle d(\kappa)=\frac{\mbox{atanh}\sqrt{1-\kappa^{2}}}{\sqrt{1-\kappa^{2}}},\;\;\kappa=\frac{w_{\rho}}{w_{z}}.$ (7) The Euler-Lagrange equations for variational parameters $w_{\rho},w_{z},\alpha$ and $\beta$ yield the following equations for widths $w_{\rho}$ and $w_{z}$ $\displaystyle\ddot{w}_{\rho}+\Omega_{\rho}^{2}w_{\rho}=\frac{1}{w_{\rho}^{3}}+\frac{N}{\sqrt{2\pi}}\frac{\left[2{a}-a_{dd}{g(\kappa)}\right]}{w_{\rho}^{3}w_{z}},$ (8) $\displaystyle\ddot{w}_{z}+\Omega_{z}^{2}w_{z}=\frac{1}{w_{z}^{3}}+\frac{2N}{\sqrt{2\pi}}\frac{\left[{a}-a_{dd}c(\kappa)\right]}{w_{\rho}^{2}w_{z}^{2}},$ (9) with $\displaystyle g(\kappa)=\frac{2-7\kappa^{2}-4\kappa^{4}+9\kappa^{4}d(\kappa)}{(1-\kappa^{2})^{2}},$ (10) $\displaystyle c(\kappa)=\frac{1+10\kappa^{2}-2\kappa^{4}-9\kappa^{2}d(\kappa)}{(1-\kappa^{2})^{2}}.$ (11) The chemical potential $\mu$ for a stationary state is $\displaystyle\mu=$ $\displaystyle\,\frac{1}{2w_{\rho}^{2}}+\frac{1}{4w_{z}^{2}}+\frac{2N[a-a_{dd}f(\kappa)]}{\sqrt{2\pi}w_{z}w_{\rho}^{2}}+\frac{\Omega_{\rho}^{2}w_{\rho}^{2}}{2}+\frac{\Omega_{z}^{2}w_{z}^{2}}{4}.$ (12) ### II.2 1D reduction For a cigar-shaped dipolar BEC with a strong radial trap $(\Omega_{\rho}>\Omega_{z})$ one can write the following effective 1D equation (details given in Appendix) $\displaystyle i\frac{\partial\phi_{1D}(z,t)}{\partial t}=\biggr{[}-\frac{\partial_{z}^{2}}{2}+\frac{\Omega_{z}^{2}z^{2}}{2}+\frac{2aN}{d_{\rho}^{2}}\|\phi_{1D}\|^{2}+\frac{2a_{dd}N}{d_{\rho}^{2}}$ $\displaystyle\times\int_{-\infty}^{\infty}\frac{dk_{z}}{2\pi}e^{ik_{z}z}\tilde{n}(k_{z})s_{1D}\left(\frac{k_{z}d_{\rho}}{\sqrt{2}}\right)\biggr{]}\phi_{1D}(z,t),$ (13) where $s_{1D}$ is defined by equation (35) and $d_{\rho}\equiv 1/\sqrt{\Omega_{\rho}}$ is the radial harmonic oscillator length. To solve equation (II.2), we use the Gaussian variational ansatz $\phi_{1D}(z)=\frac{\pi^{-1/4}}{\sqrt{w_{z}}}\exp\left[-\frac{z^{2}}{2w_{z}^{2}}+i\beta z^{2}\right].$ (14) From equation (28) we see that the variational 1D ansatz (14) corresponds to the following 3D wave function $\phi({\bf r},t)=\frac{\pi^{-3/4}}{d_{\rho}\sqrt{w_{z}}}\exp\left(-\frac{\rho^{2}}{2d_{\rho}^{2}}-\frac{z^{2}}{2w_{z}^{2}}+i\beta z^{2}\right).$ (15) The present variational wave function (15) is a special case of the 3D variational wave function (4) with $w_{\rho}=d_{\rho}$ and $\alpha=0$. Hence, the 1D variational Lagrangian can be written from the 3D Lagrangian (5), (using $w_{\rho}=d_{\rho}$ and $\alpha=0$,) as $\displaystyle L_{1D}$ $\displaystyle=\frac{w_{z}^{2}\dot{\beta}}{2}+\frac{1}{4w_{z}^{2}}+w_{z}^{2}\beta^{2}+\frac{\Omega_{z}^{2}w_{z}^{2}}{4}$ $\displaystyle+\frac{N}{\sqrt{2\pi}d_{\rho}^{2}w_{z}}\left[{a}-{a_{dd}}f(\kappa_{0})\right];\quad\kappa_{0}=\frac{d_{\rho}}{w_{z}},$ (16) where we have removed the constant terms. This inductive derivation of the 1D Lagrangian (16) avoids the construction of Lagrangian density involving error functions in the 1D potential (36) and subsequent integration to obtain the Lagrangian. The Euler-Lagrange equation for the variational parameter $w_{z}$ of Lagrangian (16) is $\displaystyle\ddot{w}_{z}+\Omega_{z}^{2}w_{z}=\frac{1}{w_{z}^{3}}+\frac{2N[a-a_{dd}c(\kappa_{0})]}{\sqrt{2\pi}w_{z}^{2}d_{\rho}^{2}}.$ (17) The variational chemical potential is given by $\displaystyle\mu=\frac{1}{4w_{z}^{2}}+\frac{2N[a-a_{dd}f(\kappa_{0})]}{\sqrt{2\pi}w_{z}d_{\rho}^{2}}+\frac{\Omega_{z}^{2}w_{z}^{2}}{4}.$ (18) Not only are the above variational results simple and yield a good approximation to the 1D GP equation, much physical insight about the system can be obtained from the variational Lagrangian (16). In a quasi-1D system, the axial width is much larger than the transverse oscillator length: $w_{z}\gg d_{\rho}$. Consequently, $\kappa_{0}\to 0$ and $f(\kappa_{0})\to 1$. From equation (16), we see that the interaction term becomes in this limit $N(a-a_{dd})/(\sqrt{2\pi}d_{\rho}^{2}w_{z})$. In equation (II.2), the dipolar term involves a nonlocal integral. However, the variational approximation suggests that the effect of the dipolar interaction integral is to reduce the contact interaction term in equation (II.2) replacing the scattering length $a$ by $(a-a_{dd})$. Immediately, one can conclude that the system effectively becomes attractive for $a_{dd}>a$. So one can have the formation of bright soliton even for positive (repulsive) scattering length $a$, provided that $a_{dd}>a$. ### II.3 2D reduction In the disk-shape, with a strong axial trap ($\Omega_{z}>\Omega_{\rho}$), the dipolar BEC is assumed to be in the ground state $\phi(z)=\exp(-z^{2}/2d_{z}^{2})/{(\pi d_{z}^{2})}^{1/4}$ of the axial trap and the wave function $\phi({\bf r})$ can be written as fisch ; PS $\phi({\bf r})=\frac{1}{{(\pi d_{z}^{2})}^{1/4}}\exp\left(-\frac{z^{2}}{2d_{z}^{2}}\right)\phi_{2D}(x,y),$ (19) where $\phi_{2D}(x,y)$ is the 2D wave function and $d_{z}=\sqrt{1/\Omega_{z}}$. Using ansatz (19) in equation (II.1), the $z$ dependence can be integrated out to obtain the following effective 2D equation fisch ; PS $\displaystyle\,i\frac{\partial\phi_{2D}(\vec{\rho},t)}{\partial t}=\biggr{[}-\frac{\nabla_{\rho}^{2}}{2}+\frac{\Omega_{\rho}^{2}\rho^{2}}{2}+\frac{4\pi aN}{\sqrt{2\pi}d_{z}}\|\phi_{2D}\|^{2}+\frac{4\pi a_{dd}N}{\sqrt{2\pi}d_{z}}$ $\displaystyle\,\quad\times\int\frac{d^{2}k_{\rho}}{(2\pi)^{2}}\exp({i{\bf k}_{\rho}\cdot{\vec{\rho}}}){\tilde{n}}({\bf k_{\rho}})h_{2D}(\frac{k_{\rho}d_{z}}{\sqrt{2}})\biggr{]}\phi_{2D}(\vec{\rho},t),$ (20) $\displaystyle\,\tilde{n}({\bf k}_{\rho})=\int\exp\left(i{\bf k}_{\rho}\cdot\vec{\rho}\right)\|\phi_{2D}(\vec{\rho})\|^{2}d\vec{\rho},$ (21) where $h_{2D}(\xi)=2-3\sqrt{\pi}\xi e^{\xi^{2}}{\mbox{erfc}}(\xi)$ PS , ${\bf k}_{\rho}\equiv(k_{x},k_{y})$, and the dipolar term is written in Fourier space. To solve equation (II.3), we use the Gaussian ansatz $\phi_{2D}(\rho)=\frac{1}{w_{\rho}\sqrt{\pi}}\exp\left(-\frac{\rho^{2}}{2w_{\rho}^{2}}+i\alpha\rho^{2}\right).$ (22) From equation (19) we see that the 2D wave function (22) corresponds to the following 3D wave function $\phi({\bf r},t)=\frac{\pi^{-3/4}}{w_{\rho}\sqrt{d_{z}}}\exp\left(-\frac{\rho^{2}}{2w_{\rho}^{2}}-\frac{z^{2}}{2d_{z}^{2}}+i\alpha\rho^{2}\right).$ (23) The present variational wave function (23) is a special case of the 3D variational wave function (4) with $w_{z}=d_{z}$ and $\beta=0$. Hence, the 2D variational Lagrangian can be written from the 3D Lagrangian (5) as $\displaystyle L_{2D}$ $\displaystyle\,={w_{\rho}^{2}\dot{\alpha}}+\frac{w_{\rho}^{2}\Omega_{\rho}^{2}}{2}+\frac{1}{2w_{\rho}^{2}}+2w_{\rho}^{2}\alpha^{2}$ $\displaystyle\,+\frac{N}{\sqrt{2\pi}w_{\rho}^{2}d_{z}}\left[{a}-{a_{dd}}f(\bar{\kappa})\right];\quad\bar{\kappa}=\frac{w_{\rho}}{d_{z}},$ (24) where we have removed the constant terms. The Euler-Lagrange variational equation for width $w_{\rho}$ becomes $\displaystyle\ddot{w}_{\rho}+{w_{\rho}\Omega_{\rho}^{2}}=\frac{1}{w_{\rho}^{3}}+\frac{N}{\sqrt{2\pi}}\frac{\left[2{a}-a_{dd}{g(\bar{\kappa})}\right]}{w_{\rho}^{3}d_{z}}.$ (25) The chemical potential $\mu$ for a stationary state is $\displaystyle\mu$ $\displaystyle=$ $\displaystyle\frac{1}{2w_{\rho}^{2}}+\frac{2N[a-a_{dd}f(\bar{\kappa})]}{\sqrt{2\pi}d_{z}w_{\rho}^{2}}+\frac{w_{\rho}^{2}\Omega_{\rho}^{2}}{2}.$ (26) In a quasi-2D system, the radial width is much larger than the axial oscillator length: $w_{\rho}\gg d_{z}$. Consequently, $\bar{\kappa}\to\infty$ and $f(\bar{\kappa})\to-2$. From equation (II.3), we see that the interaction term becomes in this limit $N(a+2a_{dd})/(\sqrt{2\pi}w_{\rho}^{2}d_{z})$. The variational approximation suggests that the effect of the dipolar interaction in equation (II.3) is to increase the contact interaction term replacing $a$ by $(a+2a_{dd})$. Hence, for positive $a$, there cannot be any bright soliton in 2D, which was found from a solution of the 2D GP equation (II.3) and Bogoliubov theory tick . However, effectively the sign of the dipolar term in the GP equation can be changed by rotating the external field that orients the dipoles much faster than any other relevant time scale in the system nath . In this fashion Nath et al. nath suggest changing the dipole interaction term by a factor of $-1/2$, which changes the effective scattering length in the Lagrange variational approximation to $(a-a_{dd})$, (as discussed in the 1D case above,) leading to the formation of bright 2D solitons for $a_{dd}>a$. These solitons were obtained by Nath et al. from a solution of the 2D GP equation (II.3). ## III Numerical results We solve the 1D, 2D, and 3D GP equations employing imaginary- and real-time propagation with Crank-Nicolson method CPC ; CPC-1 . The dipolar interaction is evaluated by fast Fourier transform jb ; YY . We present results for 52Cr and 164Dy atoms. The 52Cr has a moderate dipole moment with $a_{dd}=15a_{0}$ pfau ; pfau-1 ; pfau-4 , while the 164Dy atom has a large dipole moment with $a_{dd}=130a_{0}$ dy ; dy-2 . In both cases we present results for dipolar BEC of up to 10,000 atoms for $0<a<10$ nm and choose the frequency $\omega$ such that the oscillator length $l=1\mu$m. First, we present the results for density profiles obtained from a solution of the reduced 1D and 2D equation and compare with the full 3D results. It is known that the densities obtained from the reduced equations agree well with the full 3D density, as the nonlinearity tends to zero and/or the trap asymmetry is extreme luca . Hence in this study we consider a moderately small trap asymmetry and a relatively large nonlinearity of experimental interest. In the cigar (1D) case we consider 52Cr atoms with $a=6$ nm, and in the disk (2D) case we consider 164Dy atoms with $a=6$ nm. Figure 1: Linear density of a cigar-shaped 52Cr dipolar BEC of 1,000 atoms of $a=6$ nm, with trap parameters $\Omega_{z}=1$ and (a) $\Omega_{\rho}=4,$ and (b) $\Omega_{\rho}=9$ from a numerical (N) solution of the 3D equation (II.1) and 1D equation (II.2), and its variational (V) result. Radial density of a disk-shaped 164Dy dipolar BEC of 1,000 atoms of $a=6$ nm, with trap parameters $\Omega_{\rho}=1$ and (c) $\Omega_{z}=4,$ and (d) $\Omega_{z}=9$ from a numerical solution of the 3D equation (II.1) and 2D equation (II.3), and its variational result. In Figs. 1 (a) and (b), we plot results for linear density of a cigar-shaped 52Cr dipolar BEC of 1,000 atoms as calculated from the numerical solution of the 3D equation (II.1) and the 1D equation (II.2) and its variational result (17) for $\Omega_{z}=1$ and $\Omega_{\rho}=4$ and 9. We find, as the trap asymmetry increases by changing $\Omega_{\rho}$ from 4 to 9, the agreement between 3D and 1D models improves. In Figs. 1 (c) and (d), we plot results for radial density of a disk-shaped 164Dy dipolar BEC of 1,000 atoms as calculated from the numerical solution of the 3D equation (II.1) and the 2D equation (II.3) and its variational approximation (25) for $\Omega_{\rho}=1$ and $\Omega_{z}=4$ and 9. We find that, with the increase of the trap asymmetry from $\Omega_{z}=4$ to 9, the agreement between the 3D and 2D models enhances. In all cases the variational results of the reduced 1D and 2D equations are in good agreement with those of the full 3D model. After having established the appropriateness of the reduced 1D and 2D equations in the cigar and disk shapes, it is realized that although the numerical solution of these reduced GP equations are simpler than that of the full 3D GP equation, they are still complicated due to the presence of the nonlocal dipolar interaction. The variational approximation of these equations presented here is relatively simple and could be used for approximate solution of these equations. Now we test the variational results of the reduced 1D and 2D equations by comparing with the numericalsolution of these equations. Figure 2: The numerical (N) and variational (V) rms size $\langle\rho\rangle$ versus scattering length $a$ of a disk-shaped dipolar BEC of 10,000 (a) 52Cr and (b) 164Dy atoms for trap parameters $\Omega_{\rho}=1$ and $\Omega_{z}=4$ and 9 from a solution of the reduced 2D GP equation (II.3). The corresponding chemical potential $\mu$ in these cases for (c) 52Cr and (d) 164Dy atoms. In Figs. 2 we present the results for rms size $\langle\rho\rangle$ and chemical potential $\mu$ of a disk-shaped 52Cr and 164Dy dipolar BEC of 10,000 atoms with the trap parameters $\Omega_{\rho}=1$ and $\Omega_{z}=4$ and 9 for $0<a<10$ nm as calculated from numerical and variational approaches of the reduced 2D equation (II.3). In Figs. 3 we exhibit the results for rms size $\langle z\rangle$ and chemical potential $\mu$ of a cigar-shaped 52Cr and 164Dy dipolar BEC of 10,000 atoms with the trap parameters $\Omega_{z}=1$ and $\Omega_{\rho}=4$ and 9 for $0<a<20$ nm as calculated from numerical and variational approaches of the reduced 1D equation (II.2). Figure 3: The numerical (N) and variational (V) rms length $\langle z\rangle$ versus scattering length $a$ of a cigar-shaped dipolar BEC of 10,000 (a) 52Cr and (b) 164Dy atoms for trap parameters $\Omega_{z}=1$ and $\Omega_{\rho}=4$ and 9 from a solution of the reduced 1D GP equation (II.2). The corresponding chemical potential $\mu$ in these cases for (c) 52Cr and (d) 164Dy atoms. The dipolar interaction changes from strongly attractive in the extreme cigar shape ($\Omega_{\rho}\gg\Omega_{z}$) to strongly repulsive in the extreme disk shape ($\Omega_{\rho}\ll\Omega_{z}$) and its effect is minimum (nearly zero) for $\Omega_{\rho}$ slightly less than $\Omega_{z}$. In Fig. 2 the dipolar interaction is slightly attractive for $\Omega_{z}=4$ and $\Omega_{\rho}=1$. Hence in the absence of any short-range interaction ($a=0$), the system will collapse and no stable solution of the GP equation can be obtained. For Cr atoms the dipolar interaction is weak, and for $a\geq 1$ nm, the short-range repulsion for 10,000 atoms surpluses the dipolar attraction and a stable state can be obtained for $\Omega_{z}=4$. For Dy atoms the dipolar interaction is stronger, and a stable state can be obtained only for $a\geq 2$ nm for $\Omega_{z}=4$. For $\Omega_{z}=9$, the dipolar interaction for both Cr and Dy atoms are repulsive and a stable state is obtained in this case for $a>0$. In Fig. 3 the dipolar interaction is attractive for both $\Omega_{\rho}=4$ and 9. Hence the dipolar BEC can be stable only for scattering length $a$ greater than a critical value. This is why the curves in this figure start above this critical value. This critical value is larger for Dy atoms and $\Omega_{\rho}=9$ compared to that of Cr atoms and $\Omega_{\rho}=4$ as can be found in Fig. 3. As there is no real collapse in 1D models with cubic nonlinearity; for confirming the collapse correctly one must solve the full 3D GP equation. Figure 4: The rms sizes $\langle z\rangle$ of a cigar-shaped Cr dipolar BEC of 1,000 atoms versus time $t$ for (a) $\Omega_{\rho}=4$ and (b) 9 from a numerical (N) and variational (V) results of the reduced 1D equation. The rms sizes $\langle\rho\rangle$ of a disk-shaped Dy dipolar BEC of 1,000 atoms versus time $t$ for (c) $\Omega_{z}=4$ and (d) 9. The oscillation was initiated by jumping the scattering length $a$ from 6 nm to $6.15$ nm for Cr ($6.3$ nm for Dy) at $t=0$ from a solution of the reduced 2D equation. Next we study, by numerical and variational solutions of the reduced 1D and 2D equations, the dynamics of breathing oscillation of the four dipolar BEC of cigar- and disk-shaped Cr and Dy atoms shown in Figs. 1 started by a small change of the scattering length. This can be implemented experimentally by a Feshbach resonance fesh . In Fig. 4 this dynamics is shown for a cigar-shaped Cr dipolar BEC of 1,000 atoms for (a) $\Omega_{\rho}=4$ and (b) $\Omega_{\rho}=9$ from a solution of the reduced 1D equation, and for a disk- shaped Dy dipolar BEC of 1,000 atoms for (c) $\Omega_{z}=4$ and (d) $\Omega_{z}=9$ from a solution of the reduced 2D equation. In these figures we also show the results from a numerical solution of the 3D Eq. (II.1). The agreement between the numerical and variational results is good in all cases. We also calculated the angular frequencies of these oscillations. In case of Cr in Figs. 4 (a) and (b), the axial frequencies are 1.75 (variational, 1D), 1.76 (numerical, 1D) and 1.63 (numerical, 3D), and in case of Dy in Figs. 4 (c) and (d), the radial frequencies are 1.93 (variational, 2D), 1.89 (numerical, 2D) and 1.76 (numerical, 3D). For quasi-linear systems, these angular frequencies are expected to be 2 stringari . The deviation from this value is due to the large nonlinearity of the dipolar BECs considered here. ## IV Conclusion The usual GP equation provides a good description of statics and dynamics of a normal nondipolar BEC. For a dipolar BEC the numerical solution of the GP equation is a difficult task due to the nonlocal dipolar interaction. For a cigar- and disk-shaped dipolar BEC, the reduced 1D and 2D equations provide an alternative to the full 3D equation. Nevertheless, the solution of these reduced equations is also challenging involving Fourier and inverse Fourier transformations. As an alternative, we suggest a time-dependent variational scheme for these reduced equations, not requiring any Fourier transformation. The variational approximation of these reduced equations provides results for stationary cigar- and disk-shaped dipolar BEC as well as for breathing oscillation of the same in good agreement with the numerical solution of the respective GP equations. This is illustrated for large Cr and Dy dipolar BECs of 10,000 atoms and large atomic scattering lengths $a$ up to 20 nm. We also study the breathing oscillation of a bright soliton of 1,000 Cr atoms using the numerical solution of the 3D equation as well as the numerical and variational approaches to the 1D equation. A typical dipolar BEC considered here corresponds to a large short-range cubic nonlinearity of about $4\pi aN\approx 1250$ for $a=10$ nm and $N=10,000$ and a large dipolar nonlinearity of $4\pi a_{dd}\approx 865$ for Dy atoms for $a_{dd}=130a_{0}$ and $N=10,000$. The variational approximations considered here provided good results for such large nonlinearities and should be useful for analyzing the statics and dynamics of realistic dipolar BECs under appropriate experimental conditions. ###### Acknowledgements. We thank FAPESP (Brazil), CNPq (Brazil), DST (India), and CSIR (India) for partial support. ## Appendix A 1D reduction For a cigar-shaped dipolar BEC with a strong radial trap ($\Omega_{\rho}>\Omega_{z}$), we assume that in the radial direction the dipolar BEC is confined in the ground state $\displaystyle\phi({\bf\rho})=\exp(-(\rho^{2}/2d_{\rho}^{2})/(d_{\rho}\sqrt{\pi})$ (27) of the transverse trap and the wave function $\phi({\bf r})=\phi_{1D}(z)$ $\times\phi({\bf\rho})$ can be written as SS ; deu $\displaystyle\phi({\bf r})=\frac{1}{\sqrt{\pi d_{\rho}^{2}}}\exp\left[-\frac{\rho^{2}}{2d_{\rho}^{2}}\right]\phi_{1D}(z);\quad\Omega_{\rho}d_{\rho}^{2}=1,$ (28) where $d_{\rho}$ is the radial harmonic oscillator length. The contribution of the dipole potential to energy is $\displaystyle H_{dd}$ $\displaystyle=$ $\displaystyle\frac{N}{2}\int d^{3}r\int d^{3}r^{\prime}n({\bf r})U_{dd}({\bf r-r^{\prime}})n({\bf r^{\prime}}),$ (29) $\displaystyle=$ $\displaystyle\frac{1}{2}\frac{N}{(2\pi)^{3}}\int d^{3}k\tilde{n}({\bf k})\tilde{U}_{dd}({\bf k})\tilde{n}(-{\bf k}),$ (30) where $n({\bf r})\equiv\|\phi({\bf r})\|^{2}$ is the density and in Eq. (30) we used a convolution of the respective variables to Fourier space and where tilde denotes Fourier transformations: jb ; YY ; fisch ; PS $\displaystyle\tilde{U}_{dd}({\bf k})$ $\displaystyle=\frac{4\pi}{3}3a_{dd}\biggr{[}\frac{3k_{z}^{2}}{k^{2}}-1\biggr{]},$ (31) $\displaystyle\tilde{n}({\bf k})$ $\displaystyle=\exp\left[-\frac{k_{\rho}^{2}d_{\rho}^{2}}{4}\right]\tilde{n}_{1D}(k_{z}).$ (32) The $k_{x},k_{y}$ integrals in (30) can now be done and $\displaystyle H_{dd}$ $\displaystyle=$ $\displaystyle\frac{4\pi N}{3}\frac{3a_{dd}}{2}\frac{1}{2\pi}\int_{-\infty}^{\infty}dk_{z}\tilde{n}_{1D}(k_{z})\tilde{n}_{1D}(-k_{z})\frac{1}{(2\pi)^{2}}$ (33) $\displaystyle\times$ $\displaystyle\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}dk_{x}dk_{y}\left[\frac{3k_{z}^{2}}{k_{\rho}^{2}+k_{z}^{2}}-1\right]\exp\left[-\frac{k_{\rho}^{2}d_{\rho}^{2}}{2}\right],$ $\displaystyle\equiv$ $\displaystyle\frac{N}{2}\frac{1}{2\pi}\int_{-\infty}^{\infty}dk_{z}\tilde{n}_{1D}(k_{z})\tilde{n}_{1D}(-k_{z})V_{1D}(k_{z}),$ where the 1D potential in Fourier space is $\displaystyle V_{1D}(k_{z})=$ $\displaystyle\,{2a_{dd}}\int_{0}^{\infty}dk_{\rho}k_{\rho}\left[\frac{3k_{z}^{2}}{k_{\rho}^{2}+k_{z}^{2}}-1\right]\exp\left[-\frac{k_{\rho}^{2}d_{\rho}^{2}}{2}\right],$ $\displaystyle\equiv$ $\displaystyle\,\frac{2a_{dd}}{d_{\rho}^{2}}s_{1D}(\frac{k_{z}d_{\rho}}{\sqrt{2}}),$ (34) $\displaystyle s_{1D}(\zeta)=\int_{0}^{\infty}du\left[\frac{3\zeta^{2}}{u+\zeta^{2}}-1\right]e^{-u}.$ (35) The 1D potential in configuration space is $\displaystyle U_{dd}^{1D}(Z)=\frac{1}{2\pi}\int_{-\infty}^{\infty}dk_{z}e^{ik_{z}z}V_{1D}(k_{z})$ $\displaystyle=\frac{6a_{dd}}{(\sqrt{2}d_{\rho})^{3}}\left[\frac{4}{3}\delta(\sqrt{t})+2\sqrt{t}-\sqrt{\pi}(1+2t)e^{t}{\mbox{erfc}}(\sqrt{t})\right],$ (36) where $t=[Z/(\sqrt{2}d_{\rho})]^{2},Z=\|z-z^{\prime}\|$. Similar, but not identical, 1D reduced potential was derived in SS ; deu , where the $\delta$-function term was absent. To derive the effective 1D equation for the cigar-shaped dipolar BEC, we substitute the ansatz (28) in Eq. (II.1), multiply by the ground-state wave function $\phi(\rho)$ and integrate in $\rho$ to get the 1D equation $\displaystyle i\frac{\partial\phi_{1D}(z,t)}{\partial t}=\biggr{[}-\frac{\partial_{z}^{2}}{2}+\frac{\Omega_{z}^{2}z^{2}}{2}+\frac{2aN}{d_{\rho}^{2}}\|\phi_{1D}\|^{2}$ $\displaystyle+\frac{2a_{dd}N}{d_{\rho}^{2}}\int_{-\infty}^{\infty}\frac{dk_{z}}{2\pi}e^{ik_{z}z}\tilde{n}(k_{z})s_{1D}(\frac{k_{z}d_{\rho}}{\sqrt{2}})\biggr{]}\phi_{1D}(z,t),$ (37) $\displaystyle\equiv\biggr{[}-\frac{\partial_{z}^{2}}{2}+\frac{\Omega_{z}^{2}z^{2}}{2}+\frac{2aN\|\phi_{1D}\|^{2}}{d_{\rho}^{2}}$ $\displaystyle+N\int_{-\infty}^{\infty}U_{dd}^{1D}(Z)\|\phi_{1D}({z^{\prime}},t)\|^{2}d{z^{\prime}}\biggr{]}\phi_{1D}(z,t).$ (38) ## References * (1) F. Dalfovo, S. Giorgin, L. P. Pitaevskii, and S. Stringari, Rev. Mod. Phys. 71, 463 (1999). * (2) P. W. Courteille, V. S. Bagnato, and V. I. Yukalov, Laser Phys. 11, 659 (2001). * (3) Y. Cheng and S. K. Adhikari, Laser Phys. Lett. 7, 824 (2010). * (4) V. I. Yukalov, Laser Phys. Lett. 8, 485 (2011). * (5) V. I. Yukalov and V. S. Bagnato, Laser Phys. Lett. 6, 399 (2009). * (6) V. I. Yukalov and E. P. Yukalova, Laser Phys. Lett. 6, 235 (2009). * (7) J. A. Seman, E.A.L. Henn, R. F. Shiozaki, G. Roati, F. J. Poveda-Cuevas, K. M. F. Magalhães, V.I. Yukalov, M. Tsubota, M. Kobayashi, K. Kasamatsu, and V. S. Bagnato, Laser Phys. Lett. 8, 691 (2011). * (8) H. W. Xiong and B. Wu, Laser Phys. Lett. 8, 398 (2011). * (9) V. I. Yukalov, Laser Phys. Lett. 7, 467 (2010). * (10) L. Salasnich, F. Ancilotto, and F. Toigo, Laser Phys. Lett. 7, 78 (2010). * (11) V. I. Yukalov and E. P. Yukalova, Laser Phys. 21, 1448 (2011). * (12) L. Tomio, M. T. Yamashita, T. Frederico, and F. Bringas, Laser Phys. 21, 1464 (2011). * (13) T. D. Graß, F. E. A. dos Santos, and A. Pelster, Laser Phys. 21, 1459 (2011). * (14) L. Chen, S. W. Bi, and B. Z. Lu, Laser Phys. 21, 1202 (2011). * (15) Z. -B. Lu, J. -S. Chen, and J. -R. Li, Laser Phys. 20, 910 (2010). * (16) B. S. Marangoni, C. R. Menegatti, and L. G. Marcassa, Laser Phys. 20, 557 (2010). * (17) I. Vidanović, A. Balaž, H. Al-Jibbouri, and A. Pelster, Phys. Rev. A 84, 013618 (2011). * (18) P. Muruganandam and S. K. Adhikari, Comput. Phys. Commun. 180, 1888 (2009). * (19) S. K. Adhikari and P. Muruganandam, J. Phys. B 35, 2831 (2002). * (20) A. Görlitz, J. M. Vogels, A. E. Leanhardt, C. Raman, T. L. Gustavson, J. R. Abo-Shaeer, A. P. Chikkatur, S. Gupta, S. Inouye, T. Rosenband, and W. Ketterle, Phys. Rev. Lett. 87, 130402 (2001). * (21) L. Salasnich, A. Parola, and L. Reatto, Phys. Rev. A 65, 043614 (2002). * (22) C. A. G. Buitrago and S. K. Adhikari, J. Phys. B 42, 215306 (2009). * (23) A. Münoz Mateo and V. Delgado, Phys. Rev. A 77, 013617 (2008). * (24) V. M. Pérez-García, H. Michinel, J. I. Cirac, M. Lewenstein, and P. Zoller, Phys. Rev. A 56, 1424 (1997). * (25) T Lahaye, C Menotti, L Santos, M Lewenstein and T Pfau, Rep. Prog. Phys. 72, 126401 (2009). * (26) T. Koch, T. Lahaye, J. Metz, B. Fröhlich, A. Griesmaier and T. Pfau Nature Phys. 4, 218 (2008). * (27) T. Lahaye, T. Koch, B. Fröhlich, M. Fattori, J. Metz, A. Griesmaier, S. Giovanazzi and T. Pfau, Nature 448, 672 (2007). * (28) A. Griesmaier, J. Stuhler, T. Koch, M. Fattori, T. Pfau, and S. Giovanazzi, Phys. Rev. Lett. 97, 250402 (2006). * (29) S. H. Youn, M. W. Lu, U. Ray, and B. L. Lev, Phys. Rev. A 82, 043425 (2010). * (30) M. Lu, N. Q. Burdick, Seo Ho Youn, and B. L. Lev, Phys. Rev. Lett. 107, 190401 (2011). * (31) C. I. Hancox, S. C. Doret, M. T. Hummon, L. J. Luo, and J. M. Doyle, Nature 431, 281 (2004). * (32) J. J. McClelland and J. L. Hanssen, Phys. Rev. Lett. 96, 143005 (2006). * (33) J. Deiglmayr, A. Grochola, M. Repp, K. Mörtlbauer, C. Glück, J. Lange, O. Dulieu†, R. Wester, and M. Weidemüller, Phys. Rev. Lett. 101, 133004 (2008). * (34) I. Tikhonenkov, B. A. Malomed, and A. Vardi, Phys. Rev. Lett. 100, 090406 (2008). * (35) K. Góral and L. Santos, Phys. Rev. A 66, 023613 (2002). * (36) S. Yi and L. You, Phys. Rev. A 63, 053607 (2001). * (37) S. Yi and L. You, Phys. Rev. A 61, 041604 (2000). * (38) P. Muruganandam, R. Kishor Kumar, and S. K. Adhikari, J. Phys. B 43, 205305 (2010). * (39) P. Muruganandam and S. K. Adhikari, J. Phys. B 44, 121001 (2011). * (40) N. G. Parker, C. Ticknor, A. M. Martin, and D. H. J. O’Dell, Phys. Rev. A 79, 013617 (2009). * (41) R. M. Wilson and J. L. Bohn, Phys. Rev. A 83, 023623 (2011). * (42) M. Asad-uz-Zaman and D. Blume, Phys. Rev. A 83, 033616 (2011). * (43) D. C. E. Bortolotti, S. Ronen, J. L. Bohn, and D. Blume, Phys. Rev. Lett. 97, 160402 (2006). * (44) S. Ronen, D. C. E. Bortolotti, D. Blume, and J. L. Bohn, Phys. Rev. A 74, 033611 (2006). * (45) S. Sinha and L. Santos, Phys. Rev. Lett. 99, 140406 (2007). * (46) F. Deuretzbacher, J. C. Cremon, and S. M. Reimann, Phys. Rev. A 81, 063616 (2010). * (47) U. R. Fischer, Phys. Rev. A 73, 031602 (2006). * (48) P. Pedri and L. Santos, Phys. Rev. Lett. 95, 200404 (2005). * (49) C. Ticknor, R. M. Wilson, and J. L. Bohn, Phys. Rev. Lett. 106, 065301 (2011). * (50) R. Nath, P. Pedri, and L. Santos, Phys. Rev. Lett. 102, 050401 (2009). * (51) S. Inouye, M. R. Andrews, J. Stenger, H.-J. Miesner, D. M. Stamper-Kurn and W. Ketterle, Nature 392, 151 (1998). * (52) S. Stringari, Phys. Rev. Lett. 77, 2360 (1996).	arxiv-papers	2011-11-09T16:56:46	2024-09-04T02:49:24.163958	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "P. Muruganandam and S. K. Adhikari", "submitter": "Paulsamy Muruganandam", "url": "https://arxiv.org/abs/1111.2272" }
1111.2357	EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN) CERN-PH-EP-2011-172 LHCb-PAPER-2011-018 Measurement of $b$ hadron production fractions in 7 TeV $pp$ collisions The LHCb Collaboration 111Authors are listed on the following pages. Measurements of $b$ hadron production ratios in proton-proton collisions at a centre-of-mass energy of 7 TeV with an integrated luminosity of 3 pb-1 are presented. We study the ratios of strange $B$ meson to light $B$ meson production $f_{s}/(f_{u}+f_{d})$ and $\mathchar 28931\relax_{b}^{0}$ baryon to light $B$ meson production $f_{\Lambda_{b}}/(f_{u}+f_{d})$ as a function of the charmed hadron-muon pair transverse momentum $p_{\rm T}$ and the $b$ hadron pseudorapidity $\eta$, for $p_{\rm T}$ between 0 and 14 GeV and $\eta$ between 2 and 5. We find that $f_{s}/(f_{u}+f_{d})$ is consistent with being independent of $p_{\rm T}$ and $\eta$, and we determine $f_{s}/(f_{u}+f_{d})$ = 0.134$\pm$0.004${}^{+0.011}_{-0.010}$, where the first error is statistical and the second systematic. The corresponding ratio $f_{\Lambda_{b}}/(f_{u}+f_{d})$ is found to be dependent upon the transverse momentum of the charmed hadron-muon pair, $f_{\Lambda_{b}}/(f_{u}+f_{d})=(0.404\pm 0.017{\rm(stat)}\pm 0.027{\rm(syst)}\pm 0.105{\rm(Br)})\times[1-(0.031\pm 0.004{\rm(stat)}\pm 0.003{\rm(syst)})\times p_{\rm T}{\rm(GeV)}]$, where Br reflects an absolute scale uncertainty due to the poorly known branching fraction ${\cal B}(\mathchar 28931\relax_{c}^{+}\rightarrow pK^{-}\pi^{+})$. We extract the ratio of strange $B$ meson to light neutral $B$ meson production $f_{s}/f_{d}$ by averaging the result reported here with two previous measurements derived from the relative abundances of $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D^{+}_{s}\pi^{-}$ to $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow D^{+}K^{-}$ and $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow D^{+}\pi^{-}$. We obtain $f_{s}/f_{d}=0.267^{+0.021}_{-0.020}$. R. Aaij23, C. Abellan Beteta35,n, B. Adeva36, M. Adinolfi42, C. Adrover6, A. Affolder48, Z. Ajaltouni5, J. Albrecht37, F. Alessio37, M. Alexander47, G. Alkhazov29, P. Alvarez Cartelle36, A.A. Alves Jr22, S. Amato2, Y. Amhis38, J. Anderson39, R.B. Appleby50, O. Aquines Gutierrez10, F. Archilli18,37, L. Arrabito53, A. Artamonov 34, M. Artuso52,37, E. Aslanides6, G. Auriemma22,m, S. Bachmann11, J.J. Back44, D.S. Bailey50, V. Balagura30,37, W. Baldini16, R.J. Barlow50, C. Barschel37, S. Barsuk7, W. Barter43, A. Bates47, C. Bauer10, Th. Bauer23, A. Bay38, I. Bediaga1, K. Belous34, I. Belyaev30,37, E. Ben- Haim8, M. Benayoun8, G. Bencivenni18, S. Benson46, J. Benton42, R. Bernet39, M.-O. Bettler17, M. van Beuzekom23, A. Bien11, S. Bifani12, A. Bizzeti17,h, P.M. Bjørnstad50, T. Blake37, F. Blanc38, C. Blanks49, J. Blouw11, S. Blusk52, A. Bobrov33, V. Bocci22, A. Bondar33, N. Bondar29, W. Bonivento15, S. Borghi47, A. Borgia52, T.J.V. Bowcock48, C. Bozzi16, T. Brambach9, J. van den Brand24, J. Bressieux38, D. Brett50, S. Brisbane51, M. Britsch10, T. Britton52, N.H. Brook42, H. Brown48, A. Büchler-Germann39, I. Burducea28, A. Bursche39, J. Buytaert37, S. Cadeddu15, J.M. Caicedo Carvajal37, O. Callot7, M. Calvi20,j, M. Calvo Gomez35,n, A. Camboni35, P. Campana18,37, A. Carbone14, G. Carboni21,k, R. Cardinale19,i,37, A. Cardini15, L. Carson36, K. Carvalho Akiba2, G. Casse48, M. Cattaneo37, M. Charles51, Ph. Charpentier37, N. Chiapolini39, K. Ciba37, X. Cid Vidal36, G. Ciezarek49, P.E.L. Clarke46,37, M. Clemencic37, H.V. Cliff43, J. Closier37, C. Coca28, V. Coco23, J. Cogan6, P. Collins37, A. Comerma-Montells35, F. Constantin28, G. Conti38, A. Contu51, A. Cook42, M. Coombes42, G. Corti37, G.A. Cowan38, R. Currie46, B. D’Almagne7, C. D’Ambrosio37, P. David8, I. De Bonis4, S. De Capua21,k, M. De Cian39, F. De Lorenzi12, J.M. De Miranda1, L. De Paula2, P. De Simone18, D. Decamp4, M. Deckenhoff9, H. Degaudenzi38,37, M. Deissenroth11, L. Del Buono8, C. Deplano15, D. Derkach14,37, O. Deschamps5, F. Dettori24, J. Dickens43, H. Dijkstra37, P. Diniz Batista1, F. Domingo Bonal35,n, S. Donleavy48, F. Dordei11, A. Dosil Suárez36, D. Dossett44, A. Dovbnya40, F. Dupertuis38, R. Dzhelyadin34, S. Easo45, U. Egede49, V. Egorychev30, S. Eidelman33, D. van Eijk23, F. Eisele11, S. Eisenhardt46, R. Ekelhof9, L. Eklund47, Ch. Elsasser39, D. Esperante Pereira36, L. Estève43, A. Falabella16,e, E. Fanchini20,j, C. Färber11, G. Fardell46, C. Farinelli23, S. Farry12, V. Fave38, V. Fernandez Albor36, M. Ferro-Luzzi37, S. Filippov32, C. Fitzpatrick46, M. Fontana10, F. Fontanelli19,i, R. Forty37, M. Frank37, C. Frei37, M. Frosini17,f,37, S. Furcas20, A. Gallas Torreira36, D. Galli14,c, M. Gandelman2, P. Gandini51, Y. Gao3, J-C. Garnier37, J. Garofoli52, J. Garra Tico43, L. Garrido35, D. Gascon35, C. Gaspar37, N. Gauvin38, M. Gersabeck37, T. Gershon44,37, Ph. Ghez4, V. Gibson43, V.V. Gligorov37, C. Göbel54, D. Golubkov30, A. Golutvin49,30,37, A. Gomes2, H. Gordon51, M. Grabalosa Gándara35, R. Graciani Diaz35, L.A. Granado Cardoso37, E. Graugés35, G. Graziani17, A. Grecu28, E. Greening51, S. Gregson43, B. Gui52, E. Gushchin32, Yu. Guz34, T. Gys37, G. Haefeli38, C. Haen37, S.C. Haines43, T. Hampson42, S. Hansmann-Menzemer11, R. Harji49, N. Harnew51, J. Harrison50, P.F. Harrison44, J. He7, V. Heijne23, K. Hennessy48, P. Henrard5, J.A. Hernando Morata36, E. van Herwijnen37, E. Hicks48, K. Holubyev11, P. Hopchev4, W. Hulsbergen23, P. Hunt51, T. Huse48, R.S. Huston12, D. Hutchcroft48, D. Hynds47, V. Iakovenko41, P. Ilten12, J. Imong42, R. Jacobsson37, A. Jaeger11, M. Jahjah Hussein5, E. Jans23, F. Jansen23, P. Jaton38, B. Jean-Marie7, F. Jing3, M. John51, D. Johnson51, C.R. Jones43, B. Jost37, M. Kaballo9, S. Kandybei40, M. Karacson37, T.M. Karbach9, J. Keaveney12, U. Kerzel37, T. Ketel24, A. Keune38, B. Khanji6, Y.M. Kim46, M. Knecht38, P. Koppenburg23, A. Kozlinskiy23, L. Kravchuk32, K. Kreplin11, M. Kreps44, G. Krocker11, P. Krokovny11, F. Kruse9, K. Kruzelecki37, M. Kucharczyk20,25,37, S. Kukulak25, R. Kumar14,37, T. Kvaratskheliya30,37, V.N. La Thi38, D. Lacarrere37, G. Lafferty50, A. Lai15, D. Lambert46, R.W. Lambert37, E. Lanciotti37, G. Lanfranchi18, C. Langenbruch11, T. Latham44, R. Le Gac6, J. van Leerdam23, J.-P. Lees4, R. Lefèvre5, A. Leflat31,37, J. Lefrançois7, O. Leroy6, T. Lesiak25, L. Li3, L. Li Gioi5, M. Lieng9, M. Liles48, R. Lindner37, C. Linn11, B. Liu3, G. Liu37, J.H. Lopes2, E. Lopez Asamar35, N. Lopez-March38, J. Luisier38, F. Machefert7, I.V. Machikhiliyan4,30, F. Maciuc10, O. Maev29,37, J. Magnin1, S. Malde51, R.M.D. Mamunur37, G. Manca15,d, G. Mancinelli6, N. Mangiafave43, U. Marconi14, R. Märki38, J. Marks11, G. Martellotti22, A. Martens7, L. Martin51, A. Martín Sánchez7, D. Martinez Santos37, A. Massafferri1, Z. Mathe12, C. Matteuzzi20, M. Matveev29, E. Maurice6, B. Maynard52, A. Mazurov16,32,37, G. McGregor50, R. McNulty12, C. Mclean14, M. Meissner11, M. Merk23, J. Merkel9, R. Messi21,k, S. Miglioranzi37, D.A. Milanes13,37, M.-N. Minard4, S. Monteil5, D. Moran12, P. Morawski25, R. Mountain52, I. Mous23, F. Muheim46, K. Müller39, R. Muresan28,38, B. Muryn26, M. Musy35, J. Mylroie-Smith48, P. Naik42, T. Nakada38, R. Nandakumar45, I. Nasteva1, M. Nedos9, M. Needham46, N. Neufeld37, C. Nguyen-Mau38,o, M. Nicol7, V. Niess5, N. Nikitin31, A. Nomerotski51, A. Novoselov34, A. Oblakowska-Mucha26, V. Obraztsov34, S. Oggero23, S. Ogilvy47, O. Okhrimenko41, R. Oldeman15,d, M. Orlandea28, J.M. Otalora Goicochea2, P. Owen49, K. Pal52, J. Palacios39, A. Palano13,b, M. Palutan18, J. Panman37, A. Papanestis45, M. Pappagallo13,b, C. Parkes47,37, C.J. Parkinson49, G. Passaleva17, G.D. Patel48, M. Patel49, S.K. Paterson49, G.N. Patrick45, C. Patrignani19,i, C. Pavel-Nicorescu28, A. Pazos Alvarez36, A. Pellegrino23, G. Penso22,l, M. Pepe Altarelli37, S. Perazzini14,c, D.L. Perego20,j, E. Perez Trigo36, A. Pérez-Calero Yzquierdo35, P. Perret5, M. Perrin-Terrin6, G. Pessina20, A. Petrella16,37, A. Petrolini19,i, E. Picatoste Olloqui35, B. Pie Valls35, B. Pietrzyk4, T. Pilar44, D. Pinci22, R. Plackett47, S. Playfer46, M. Plo Casasus36, G. Polok25, A. Poluektov44,33, E. Polycarpo2, D. Popov10, B. Popovici28, C. Potterat35, A. Powell51, T. du Pree23, J. Prisciandaro38, V. Pugatch41, A. Puig Navarro35, W. Qian52, J.H. Rademacker42, B. Rakotomiaramanana38, M.S. Rangel2, I. Raniuk40, G. Raven24, S. Redford51, M.M. Reid44, A.C. dos Reis1, S. Ricciardi45, K. Rinnert48, D.A. Roa Romero5, P. Robbe7, E. Rodrigues47, F. Rodrigues2, P. Rodriguez Perez36, G.J. Rogers43, S. Roiser37, V. Romanovsky34, M. Rosello35,n, J. Rouvinet38, T. Ruf37, H. Ruiz35, G. Sabatino21,k, J.J. Saborido Silva36, N. Sagidova29, P. Sail47, B. Saitta15,d, C. Salzmann39, M. Sannino19,i, R. Santacesaria22, C. Santamarina Rios36, R. Santinelli37, E. Santovetti21,k, M. Sapunov6, A. Sarti18,l, C. Satriano22,m, A. Satta21, M. Savrie16,e, D. Savrina30, P. Schaack49, M. Schiller24, S. Schleich9, M. Schmelling10, B. Schmidt37, O. Schneider38, A. Schopper37, M.-H. Schune7, R. Schwemmer37, B. Sciascia18, A. Sciubba18,l, M. Seco36, A. Semennikov30, K. Senderowska26, I. Sepp49, N. Serra39, J. Serrano6, P. Seyfert11, B. Shao3, M. Shapkin34, I. Shapoval40,37, P. Shatalov30, Y. Shcheglov29, T. Shears48, L. Shekhtman33, O. Shevchenko40, V. Shevchenko30, A. Shires49, R. Silva Coutinho54, H.P. Skottowe43, T. Skwarnicki52, A.C. Smith37, N.A. Smith48, E. Smith51,45, K. Sobczak5, F.J.P. Soler47, A. Solomin42, F. Soomro49, B. Souza De Paula2, B. Spaan9, A. Sparkes46, P. Spradlin47, F. Stagni37, S. Stahl11, O. Steinkamp39, S. Stoica28, S. Stone52,37, B. Storaci23, M. Straticiuc28, U. Straumann39, N. Styles46, V.K. Subbiah37, S. Swientek9, M. Szczekowski27, P. Szczypka38, T. Szumlak26, S. T’Jampens4, E. Teodorescu28, F. Teubert37, E. Thomas37, J. van Tilburg11, V. Tisserand4, M. Tobin39, S. Topp-Joergensen51, N. Torr51, E. Tournefier4,49, M.T. Tran38, A. Tsaregorodtsev6, N. Tuning23, M. Ubeda Garcia37, A. Ukleja27, P. Urquijo52, U. Uwer11, V. Vagnoni14, G. Valenti14, R. Vazquez Gomez35, P. Vazquez Regueiro36, S. Vecchi16, J.J. Velthuis42, M. Veltri17,g, K. Vervink37, B. Viaud7, I. Videau7, X. Vilasis-Cardona35,n, J. Visniakov36, A. Vollhardt39, D. Voong42, A. Vorobyev29, H. Voss10, S. Wandernoth11, J. Wang52, D.R. Ward43, A.D. Webber50, D. Websdale49, M. Whitehead44, D. Wiedner11, L. Wiggers23, G. Wilkinson51, M.P. Williams44,45, M. Williams49, F.F. Wilson45, J. Wishahi9, M. Witek25, W. Witzeling37, S.A. Wotton43, K. Wyllie37, Y. Xie46, F. Xing51, Z. Xing52, Z. Yang3, R. Young46, O. Yushchenko34, M. Zavertyaev10,a, F. Zhang3, L. Zhang52, W.C. Zhang12, Y. Zhang3, A. Zhelezov11, L. Zhong3, E. Zverev31, A. Zvyagin 37. 1Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, Brazil 2Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil 3Center for High Energy Physics, Tsinghua University, Beijing, China 4LAPP, Université de Savoie, CNRS/IN2P3, Annecy-Le-Vieux, France 5Clermont Université, Université Blaise Pascal, CNRS/IN2P3, LPC, Clermont- Ferrand, France 6CPPM, Aix-Marseille Université, CNRS/IN2P3, Marseille, France 7LAL, Université Paris-Sud, CNRS/IN2P3, Orsay, France 8LPNHE, Université Pierre et Marie Curie, Université Paris Diderot, CNRS/IN2P3, Paris, France 9Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany 10Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany 11Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg, Germany 12School of Physics, University College Dublin, Dublin, Ireland 13Sezione INFN di Bari, Bari, Italy 14Sezione INFN di Bologna, Bologna, Italy 15Sezione INFN di Cagliari, Cagliari, Italy 16Sezione INFN di Ferrara, Ferrara, Italy 17Sezione INFN di Firenze, Firenze, Italy 18Laboratori Nazionali dell’INFN di Frascati, Frascati, Italy 19Sezione INFN di Genova, Genova, Italy 20Sezione INFN di Milano Bicocca, Milano, Italy 21Sezione INFN di Roma Tor Vergata, Roma, Italy 22Sezione INFN di Roma La Sapienza, Roma, Italy 23Nikhef National Institute for Subatomic Physics, Amsterdam, Netherlands 24Nikhef National Institute for Subatomic Physics and Vrije Universiteit, Amsterdam, Netherlands 25Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of Sciences, Cracow, Poland 26Faculty of Physics & Applied Computer Science, Cracow, Poland 27Soltan Institute for Nuclear Studies, Warsaw, Poland 28Horia Hulubei National Institute of Physics and Nuclear Engineering, Bucharest-Magurele, Romania 29Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia 30Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia 31Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow, Russia 32Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN), Moscow, Russia 33Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State University, Novosibirsk, Russia 34Institute for High Energy Physics (IHEP), Protvino, Russia 35Universitat de Barcelona, Barcelona, Spain 36Universidad de Santiago de Compostela, Santiago de Compostela, Spain 37European Organization for Nuclear Research (CERN), Geneva, Switzerland 38Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland 39Physik-Institut, Universität Zürich, Zürich, Switzerland 40NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine 41Institute for Nuclear Research of the National Academy of Sciences (KINR), Kyiv, Ukraine 42H.H. Wills Physics Laboratory, University of Bristol, Bristol, United Kingdom 43Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom 44Department of Physics, University of Warwick, Coventry, United Kingdom 45STFC Rutherford Appleton Laboratory, Didcot, United Kingdom 46School of Physics and Astronomy, University of Edinburgh, Edinburgh, United Kingdom 47School of Physics and Astronomy, University of Glasgow, Glasgow, United Kingdom 48Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom 49Imperial College London, London, United Kingdom 50School of Physics and Astronomy, University of Manchester, Manchester, United Kingdom 51Department of Physics, University of Oxford, Oxford, United Kingdom 52Syracuse University, Syracuse, NY, United States 53CC-IN2P3, CNRS/IN2P3, Lyon-Villeurbanne, France, associated member 54Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de Janeiro, Brazil, associated to 2 aP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS), Moscow, Russia bUniversità di Bari, Bari, Italy cUniversità di Bologna, Bologna, Italy dUniversità di Cagliari, Cagliari, Italy eUniversità di Ferrara, Ferrara, Italy fUniversità di Firenze, Firenze, Italy gUniversità di Urbino, Urbino, Italy hUniversità di Modena e Reggio Emilia, Modena, Italy iUniversità di Genova, Genova, Italy jUniversità di Milano Bicocca, Milano, Italy kUniversità di Roma Tor Vergata, Roma, Italy lUniversità di Roma La Sapienza, Roma, Italy mUniversità della Basilicata, Potenza, Italy nLIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain oHanoi University of Science, Hanoi, Viet Nam ## 1 Introduction The fragmentation process, in which a primary $b$ quark forms either a $b\bar{q}$ meson or a $bq_{1}q_{2}$ baryon, cannot be reliably predicted because it is driven by strong dynamics in the non-perturbative regime. Thus fragmentation functions for the various hadron species must be determined experimentally. The LHCb experiment at the LHC explores a unique kinematic region: it detects $b$ hadrons produced in a cone centered around the beam axis covering a region of pseudorapidity $\eta$, defined in terms of the polar angle $\theta$ with respect to the beam direction as $-\ln(\tan{\theta/2})$, ranging approximately between 2 and 5. Knowledge of the fragmentation functions allows us to relate theoretical predictions of the $b\bar{b}$ quark production cross-section, derived from perturbative QCD, to the observed hadrons. In addition, since many absolute branching fractions of $B^{-}$ and $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ decays have been well measured at $e^{+}e^{-}$ colliders [1], it suffices to measure the ratio of $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ production to either $B^{-}$ or $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ production to perform precise absolute $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ branching fraction measurements. In this paper we describe measurements of two ratios of fragmentation functions: $f_{s}/(f_{u}+f_{d})$ and $f_{\Lambda_{b}}/(f_{u}+f_{d})$, where $f_{q}\equiv{\cal B}(b\rightarrow B_{q})$ and $f_{\Lambda_{b}}\equiv{\cal B}(b\rightarrow\Lambda_{b})$. The inclusion of charged conjugate modes is implied throughout the paper, and we measure the average production ratios. Previous measurements of these fractions have been made at LEP [2] and at CDF [3]. More recently, LHCb measured the ratio $f_{s}/f_{d}$ using the decay modes $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow D^{+}\pi^{-}$, $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow D^{+}K^{-}$, and $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D^{+}_{s}\pi^{-}$ [4] and theoretical input from QCD factorization [5, 6]. Here we measure this ratio using semileptonic decays without any significant model dependence. A commonly adopted assumption is that the fractions of these different species should be the same in high energy $b$ jets originating from $Z^{0}$ decays and high $p_{\rm T}$ $b$ jets originating from $p\bar{p}$ collisions at the Tevatron or $pp$ collisions at LHC, based on the notion that hadronization is a non-perturbative process occurring at the scale of $\Lambda_{\rm QCD}$. Nonetheless, the results from different experiments are discrepant in the case of the $b$ baryon fraction [2]. The measurements reported in this paper are performed using the LHCb detector [7], a forward spectrometer designed to study production and decays of hadrons containing $b$ or $c$ quarks. LHCb includes a vertex detector (VELO), providing precise locations of primary $pp$ interaction vertices, and of detached vertices of long lived hadrons. The momenta of charged particles are determined using information from the VELO together with the rest of the tracking system, composed of a large area silicon tracker located before a 4 Tm dipole magnet, and a combination of silicon strip and straw drift chamber detectors located after the magnet. Two Ring Imaging Cherenkov (RICH) detectors are used for charged hadron identification. Photon detection and electron identification are implemented through an electromagnetic calorimeter followed by a hadron calorimeter. A system of alternating layers of iron and chambers provides muon identification. The two calorimeters and the muon system provide the energy and momentum information to implement a first level (L0) hardware trigger. An additional trigger level is software based, and its algorithms are tuned to the experiment’s operating condition. In this analysis we use a data sample of 3 pb-1 collected from 7 TeV centre- of-mass energy $pp$ collisions at the LHC during 2010. The trigger selects events where a single muon is detected without biasing the impact parameter distribution of the decay products of the $b$ hadron, nor any kinematic variable relevant to semileptonic decays. These features reduce the systematic uncertainty in the efficiency. Our goal is to measure two specific production ratios: that of $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ relative to the sum of $B^{-}$ and $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$, and that of $\mathchar 28931\relax_{b}^{0}$, relative to the sum of $B^{-}$ and $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$. The sum of the $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$, $B^{-}$, $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ and $\mathchar 28931\relax_{b}^{0}$ fractions does not equal one, as there is other $b$ production, namely a very small rate for $B_{c}^{-}$ mesons, bottomonia, and other $b$ baryons that do not decay strongly into $\mathchar 28931\relax_{b}^{0}$, such as the $\Xi_{b}$. We measure relative fractions by studying the final states $D^{0}\mu^{-}\overline{\nu}X$, $D^{+}\mu^{-}\overline{\nu}X$, $D^{+}_{s}\mu^{-}\overline{\nu}X$, $\mathchar 28931\relax_{c}^{+}\mu^{-}\overline{\nu}X$, $D^{0}K^{+}\mu^{-}\overline{\nu}X$, and $D^{0}p\mu^{-}\overline{\nu}X$. We do not attempt to separate $f_{u}$ and $f_{d}$, but we measure the sum of $D^{0}$ and $D^{+}$ channels and correct for cross-feeds from $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ and $\mathchar 28931\relax_{b}^{0}$ decays. We assume near equality of the semileptonic decay width of all $b$ hadrons, as discussed below. Charmed hadrons are reconstructed through the modes listed in Table 1, together with their branching fractions. We use all $D^{+}_{s}\rightarrow K^{-}K^{+}\pi^{+}$ decays rather than a combination of the resonant $\phi\pi^{+}$ and $\overline{K}^{0}K^{+}$ contributions, because these $D^{+}_{s}$ decays cannot be cleanly isolated due to interference effects of different amplitudes. Table 1: Charmed hadron decay modes and branching fractions. Particle \| Final state \| Branching fraction (%) ---\|---\|--- $D^{0}$ \| $K^{-}\pi^{+}$ \| 3.89$\pm$0.05 [1] $D^{+}$ \| $K^{-}\pi^{+}\pi^{+}$ \| 9.14$\pm$0.20 [19] $D_{s}^{+}$ \| $K^{-}K^{+}\pi^{+}$ \| 5.50$\pm$0.27 [20] $\Lambda_{c}^{+}$ \| $pK^{-}\pi^{+}$ \| 5.0$\pm$1.3 [1] Each of these different charmed hadron plus muon final states can be populated by a combination of initial $b$ hadron states. $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ mesons decay semileptonically into a mixture of $D^{0}$ and $D^{+}$ mesons, while $B^{-}$ mesons decay predominantly into $D^{0}$ mesons with a smaller admixture of $D^{+}$ mesons. Both include a tiny component of $D^{+}_{s}K$ meson pairs. $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ mesons decay predominantly into $D_{s}^{+}$ mesons, but can also decay into $D^{0}K^{+}$ and $D^{+}K_{S}^{0}$ mesons; this is expected if the $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ decays into a $D_{s}^{}$ state that is heavy enough to decay into a $DK$ pair. In this paper we measure this contribution using $D^{0}K^{+}X\mu^{-}\overline{\nu}$ events. Finally, $\mathchar 28931\relax_{b}^{0}$ baryons decay mostly into $\mathchar 28931\relax_{c}^{+}$ final states. We determine other contributions using $D^{0}pX\mu^{-}\overline{\nu}$ events. We ignore the contributions of $b\rightarrow u$ decays that comprise approximately 1% of semileptonic $b$ hadron decays [8], and constitute a roughly equal portion of each $b$ species in any case. The corrected yields for $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ or $B^{-}$ decaying into $D^{0}\mu^{-}\overline{\nu}X$ or $D^{+}\mu^{-}\overline{\nu}X$, $n_{\rm corr}$, can be expressed in terms of the measured yields, $n$, as $\displaystyle n_{\rm corr}(B\rightarrow D^{0}\mu)$ $\displaystyle=$ $\displaystyle\frac{1}{{\cal{B}}(D^{0}\rightarrow K^{-}\pi^{+})\epsilon(B\rightarrow D^{0})}\times$ $\displaystyle\left[n(D^{0}\mu)-n(D^{0}K^{+}\mu)\frac{\epsilon(\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D^{0})}{\epsilon(\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D^{0}K^{+})}-n(D^{0}p\mu)\frac{\epsilon(\mathchar 28931\relax_{b}^{0}\rightarrow D^{0})}{\epsilon(\mathchar 28931\relax_{b}^{0}\rightarrow D^{0}p)}\right],$ where we use the shorthand $n(D\mu)\equiv n(DX\mu^{-}\overline{\nu})$. An analogous abbreviation $\epsilon$ is used for the total trigger and detection efficiencies. For example, the ratio $\epsilon(\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D^{0})/\epsilon(\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D^{0}K^{+})$ gives the relative efficiency to reconstruct a charged $K$ in semi-muonic $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ decays producing a $D^{0}$ meson. Similarly $\displaystyle n_{\rm corr}(B\rightarrow D^{+}\mu)$ $\displaystyle=$ $\displaystyle\frac{1}{\epsilon(B\rightarrow D^{+})}\left[\frac{n(D^{+}\mu^{-})}{{\cal{B}}(D^{+}\rightarrow K^{-}\pi^{+}\pi^{+})}-\right.$ (2) $\displaystyle\left.\frac{n(D^{0}K^{+}\mu^{-})}{{\cal{B}}(D^{0}\rightarrow K^{-}\pi^{+})}\frac{\epsilon(\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D^{+})}{\epsilon(\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D^{0}K^{+})}\right.$ $\displaystyle\left.-\frac{n(D^{0}p\mu^{-})}{{\cal{B}}(D^{0}\rightarrow K^{-}\pi^{+})}\frac{\epsilon(\mathchar 28931\relax_{b}\rightarrow D^{+})}{\epsilon(\mathchar 28931\relax_{b}\rightarrow D^{0}p)}\right].$ Both the $D^{0}X\mu^{-}\overline{\nu}$ and the $D^{+}X\mu^{-}\overline{\nu}$ final states contain small components of cross-feed from $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ decays to $D^{0}K^{+}X\mu^{-}\overline{\nu}$ and to $D^{+}K^{0}X\mu^{-}\overline{\nu}$. These components are accounted for by the two decays $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D_{s1}^{+}X\mu^{-}\overline{\nu}$ and $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D_{s2}^{+}X\mu^{-}\overline{\nu}$ as reported in a recent LHCb publication [9]. The third terms in Eqs. 1 and 2 are due to a similar small cross-feed from $\mathchar 28931\relax_{b}^{0}$ decays. The number of $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ resulting in $D^{+}_{s}X\mu^{-}\overline{\nu}$ in the final state is given by $\displaystyle n_{\rm corr}(\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D^{+}_{s}\mu)$ $\displaystyle=$ $\displaystyle\frac{1}{\epsilon(\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D^{+}_{s})}\left[\frac{n(D^{+}_{s}\mu)}{{\cal B}(D^{+}_{s}\rightarrow K^{+}K^{-}\pi^{+})}-\right.$ $\displaystyle\left.N(\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}+B^{-}){\cal B}(B\rightarrow D^{+}_{s}K\mu)\epsilon(\bar{B}\rightarrow D^{+}_{s}K\mu)\right],$ where the last term subtracts yields of $D^{+}_{s}KX\mu^{-}\overline{\nu}$ final states originating from $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ or $B^{-}$ semileptonic decays, and $N(\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}+B^{-})$ indicates the total number of $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ and $B^{-}$ produced. We derive this correction using the branching fraction ${\cal B}(B\rightarrow D_{s}^{()+}K\mu\nu)=(6.1\pm 1.2)\times 10^{-4}$ [10] measured by the BaBar experiment. In addition, $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ decays semileptonically into $DKX\mu^{-}\overline{\nu}$, and thus we need to add to Eq. 1 $n_{\rm corr}(\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow DK\mu)=2\frac{n(D^{0}K^{+}\mu)}{{\cal B}(D^{0}\rightarrow K^{-}\pi^{+})\epsilon(\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D^{0}K^{+}\mu)},$ (4) where, using isospin symmetry, the factor of 2 accounts for $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{0}X\mu^{-}\overline{\nu}$ semileptonic decays. The equation for the ratio $f_{s}/(f_{u}+f_{d})$ is $\frac{f_{s}}{f_{u}+f_{d}}=\frac{n_{\rm corr}(\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D\mu)}{n_{\rm corr}(B\rightarrow D^{0}\mu)+n_{\rm corr}(B\rightarrow D^{+}\mu)}\frac{\tau_{B^{-}}+\tau_{\kern 1.25995pt\overline{\kern-1.25995ptB}{}^{0}}}{2\tau_{\kern 1.25995pt\overline{\kern-1.25995ptB}{}^{0}_{s}}}.$ (5) where $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D\mu$ represents $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ semileptonic decays to a final charmed hadron, given by the sum of the contributions shown in Eqs. 1 and 4, and the symbols $\tau_{B_{i}}$ indicate the $B_{i}$ hadron lifetimes, that are all well measured [1]. We use the average $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ lifetime, 1.472$\pm$0.025 ps [1]. This equation assumes equality of the semileptonic widths of all the $b$ meson species. This is a reliable assumption, as corrections in HQET arise only to order 1/$m_{b}^{2}$ and the SU(3) breaking correction is quite small, of the order of 1% [11, 12, 13]. The $\mathchar 28931\relax_{b}^{0}$ corrected yield is derived in an analogous manner. We determine $n_{\rm corr}(\mathchar 28931\relax_{b}^{0}\rightarrow D\mu)=\frac{n(\Lambda_{c}^{+}\mu^{-})}{{\cal{B}}(\Lambda_{c}^{+}\rightarrow pK^{-}\pi^{+})\epsilon(\mathchar 28931\relax_{b}^{0}\rightarrow\mathchar 28931\relax_{c}^{+})}+2\frac{n(D^{0}p\mu^{-})}{{\cal{B}}(D^{0}\rightarrow K^{-}\pi^{+})\epsilon(\mathchar 28931\relax_{b}^{0}\rightarrow D^{0}p)},$ (6) where $D$ represents a generic charmed hadron, and extract the $\Lambda_{b}^{0}$ fraction using $\frac{f_{\Lambda_{b}}}{f_{u}+f_{d}}=\frac{n_{\rm corr}(\Lambda_{b}^{0}\rightarrow D\mu)}{n_{\rm corr}(B\rightarrow D^{0}\mu)+n_{\rm corr}(B\rightarrow D^{+}\mu)}\frac{\tau_{B^{-}}+\tau_{\kern 1.25995pt\overline{\kern-1.25995ptB}{}^{0}}}{2\tau_{\Lambda_{b}^{0}}}(1-\xi).$ (7) Again, we assume near equality of the semileptonic widths of different $b$ hadrons, but we apply a small adjustment $\xi=4\pm$2%, to account for the chromomagnetic correction, affecting $b$-flavoured mesons but not $b$ baryons [11, 12, 13]. The uncertainty is evaluated with very conservative assumptions for all the parameters of the heavy quark expansion. ## 2 Analysis method To isolate a sample of $b$ flavoured hadrons with low backgrounds, we match charmed hadron candidates with tracks identified as muons. Right-sign (RS) combinations have the sign of the charge of the muon being the same as the charge of the kaon in $D^{0}$, $D^{+}$, or $\mathchar 28931\relax_{c}^{+}$ decays, or the opposite charge of the pion in $D^{+}_{s}$ decays, while wrong- sign (WS) combinations comprise combinations with opposite charge correlations. WS events are useful to estimate certain backgrounds. This analysis follows our previous investigation of $b\rightarrow D^{0}X\mu^{-}\overline{\nu}$ [14]. We consider events where a well-identified muon with momentum greater than 3 GeV and transverse momentum greater than 1.2 GeV is found. Charmed hadron candidates are formed from hadrons with momenta greater than 2 GeV and transverse momenta greater than 0.3 GeV, and we require that the average transverse momentum of the hadrons forming the candidate be greater than 0.7 GeV. Kaons, pions, and protons are identified using the RICH system. The impact parameter (IP), defined as the minimum distance of approach of the track with respect to the primary vertex, is used to select tracks coming from charm decays. We require that the $\chi^{2}$, formed by using the hypothesis that each track’s IP is equal to 0, is greater than 9. Moreover, the selected tracks must be consistent with coming from a common vertex: the $\chi^{2}$ per number of degrees of freedom of the vertex fit must be smaller than 6. In order to ensure that the charm vertex is distinct from the primary $pp$ interaction vertex, we require that the $\chi^{2}$, based on the hypothesis that the decay flight distance from the primary vertex is zero, is greater than 100. Charmed hadrons and muons are combined to form a partially reconstructed $b$ hadron by requiring that they come from a common vertex, and that the cosine of the angle between the momentum of the charmed hadron and muon pair and the line from the $D\mu$ vertex to the primary vertex be greater than 0.999. As the charmed hadron is a decay product of the $b$ hadron, we require that the difference in $z$ component of the decay vertex of the charmed hadron candidate and that of the beauty candidate be greater than 0. We explicitly require that the $\eta$ of the $b$ hadron candidate be between 2 and 5. We measure $\eta$ using the line defined by connecting the primary event vertex and the vertex formed by the $D$ and the $\mu$. Finally, the invariant mass of the charmed hadron and muon system must be between 3 and 5 GeV for $D^{0}\mu^{-}$ and $D^{+}\mu^{-}$ candidates, between 3.1 and 5.1 GeV for $D^{+}_{s}\mu^{-}$ candidates, and between 3.3 and 5.3 GeV for $\mathchar 28931\relax_{c}^{+}\mu^{-}$ candidates. We perform our analysis in a grid of 3 $\eta$ and 5 $p_{\rm T}$ bins, covering the range $2<\eta<5$ and $p_{\rm T}\leq 14$ GeV. The $b$ hadron signal is separated from various sources of background by studying the two dimensional distribution of charmed hadron candidate invariant mass and ln(IP/mm). This approach allows us to determine the background coming from false combinations under the charmed hadron signal mass peak directly. The study of the ln(IP/mm) distribution allows the separation of prompt charm decay candidates from charmed hadron daughters of $b$ hadrons [14]. We refer to these samples as Prompt and Dfb respectively. ### 2.1 Signal extraction We describe the method used to extract the charmed hadron-$\mu$ signal by using the $D^{0}X\mu^{-}\overline{\nu}$ final state as an example; the same procedure is applied to the final states $D^{+}X\mu^{-}\overline{\nu}$, $D^{+}_{s}X\mu^{-}\overline{\nu}$, and $\mathchar 28931\relax_{c}^{+}X\mu^{-}\overline{\nu}$. We perform unbinned extended maximum likelihood fits to the two-dimensional distributions in $K^{-}\pi^{+}$ invariant mass over a region extending $\pm$80 MeV from the $D^{0}$ mass peak, and ln(IP/mm). The parameters of the IP distribution of the Prompt sample are found by examining directly produced charm [14] whereas a shape derived from simulation is used for the Dfb component. Figure 1: The logarithm of the IP distributions for (a) RS and (c) WS $D^{0}$ candidate combinations with a muon. The dotted curves show the false $D^{0}$ background, the small red-solid curves the Prompt yields, the dashed curves the Dfb signal, and the larger green-solid curves the total yields. The invariant $K^{-}\pi^{+}$ mass spectra for (b) RS combinations and (d) WS combinations are also shown. An example fit for $D^{0}\mu^{-}\overline{\nu}X$, using the whole $p_{\rm T}$ and $\eta$ range, is shown in Fig. 1. The fitted yields for RS are 27666$\pm$187 Dfb, 695$\pm$43 Prompt, and 1492$\pm$30 false $D^{0}$ combinations, inferred from the fitted yields in the sideband mass regions, spanning the intervals between 35 and 75 MeV from the signal peak on both sides. For WS we find 362$\pm$39 Dfb, 187$\pm$18 Prompt, and 1134$\pm$19 false $D^{0}$ combinations. The RS yield includes a background of around 0.5% from incorrectly identified $\mu$ candidates. As this paper focuses on ratios of yields, we do not subtract this component. Figure 2 shows the corresponding fits for the $D^{+}X\mu^{-}\overline{\nu}$ final state. The fitted yields consist of 9257$\pm$110 Dfb events, 362$\pm$34 Prompt, and 1150$\pm$22 false $D^{+}$ combinations. For WS we find 77$\pm$22 Dfb, 139$\pm$14 Prompt and 307$\pm$10 false $D^{+}$ combinations. Figure 2: The logarithm of the IP distributions for (a) RS and (c) WS $D^{+}$ candidate combinations with a muon. The grey-dotted curves show the false $D^{+}$ background, the small red-solid curves the Prompt yields, the blue- dashed curves the Dfb signal, and the larger green-solid curves the total yields. The invariant $K^{-}\pi^{+}\pi^{+}$ mass spectra for (b) RS combinations and (d) WS combinations are also shown. The analysis for the $D_{s}^{+}X\mu^{-}\overline{\nu}$ mode follows in the same manner. Here, however, we are concerned about the reflection from $\Lambda_{c}^{+}\rightarrow pK^{-}\pi^{+}$ where the proton is taken to be a kaon, since we do not impose an explicit proton veto. Using such a veto would lose 30% of the signal and also introduce a systematic error. We choose to model separately this particular background. We add a probability density function (PDF) determined from simulation to model this, and the level is allowed to float within the estimated error on the size of the background. The small peak near 2010 MeV in Fig. 3(b) is due to $D^{+}\rightarrow\pi^{+}D^{0},D^{0}\rightarrow K^{+}K^{-}$. We explicitly include this term in the fit, assuming the shape to be the same as for the $D^{+}_{s}$ signal, and we obtain 4$\pm$1 events in the RS signal region and no events in the WS signal region. The measured yields in the RS sample are 2192$\pm$64 Dfb, 63$\pm$16 Prompt, 985$\pm$145 false $D^{+}_{s}$ background, and 387$\pm$132 $\Lambda_{c}^{+}$ reflection background. The corresponding yields in the WS sample are 13$\pm$19, 20$\pm$7, 499$\pm$16, and 3$\pm$3 respectively. Figure 3 shows the fit results. Figure 3: The logarithm of the IP distributions for (a) RS and (c) WS $D^{+}_{s}$ candidate combinations with a muon. The grey-dotted curves show the false $D^{+}_{s}$ background, the small red-solid curves the Prompt yields, the blue-dashed curves the Dfb signal, the purple dash-dotted curves represent the background originating from $\mathchar 28931\relax_{c}^{+}$ reflection, and the larger green-solid curves the total yields. The invariant $K^{-}K^{+}\pi^{+}$ mass spectra for RS combinations (b) and WS combinations (d) are also shown. Figure 4: The logarithm of the IP distributions for (a) RS and (c) WS $\Lambda_{c}^{+}$ candidate combinations with a muon. The grey- dotted curves show the false $\Lambda_{c}^{+}$ background, the small red-solid curves the Prompt yields, the blue-dashed curves the Dfb signal, and the larger green-solid curves the total yields. The invariant $pK^{-}\pi^{+}$ mass spectra for RS combinations (b) and WS combinations (d) are also shown. The last final state considered is $\mathchar 28931\relax_{c}^{+}X\mu^{-}\overline{\nu}$. Figure 4 shows the data and fit components to the ln(IP/mm) and $pK^{-}\pi^{+}$ invariant mass combinations for events with $2<\eta<5$. This fit gives 3028$\pm$112 RS Dfb events, 43$\pm$17 RS Prompt events, 589$\pm$27 RS false $\mathchar 28931\relax_{c}^{+}$ combinations, 9$\pm$16 WS Dfb events, 0.5$\pm$4 WS Prompt events, and 177$\pm$10 WS false $\mathchar 28931\relax_{c}^{+}$ combinations. Figure 5: (a) Invariant mass of $D^{0}p$ candidates that vertex with each other and together with a RS muon (black closed points) and for a $\overline{p}$ (red open points) instead of a $p$; (b) fit to $D^{0}$ invariant mass for RS events with the invariant mass of $D^{0}p$ candidate in the signal mass difference window; (c) fit to $D^{0}$ invariant mass for WS events with the invariant mass of $D^{0}p$ candidate in the signal mass difference window. The $\mathchar 28931\relax_{b}^{0}$ may also decay into $D^{0}pX\mu^{-}\overline{\nu}$. We search for these decays by requiring the presence of a track well identified as a proton and detached from any primary vertex. The resulting $D^{0}p$ invariant mass distribution is shown in Fig. 5. We also show the combinations that cannot arise from $\Lambda_{b}^{0}$ decay, namely those with $D^{0}\overline{p}$ combinations. There is a clear excess of RS over WS combinations especially near threshold. Fits to the $K^{-}\pi^{+}$ invariant mass in the $[m(K^{-}\pi^{+}p)-m(K^{-}\pi^{+})+m(D^{0})_{\rm PDG}]$ region shown in Fig. 5(a) give 154$\pm$13 RS events and 55$\pm$8 WS events. In this case, we use the WS yield for background subtraction, scaled by the RS/WS background ratio determined with a MC simulation including $(B^{-}+\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow D^{0}X\mu^{-}\overline{\nu})$ and generic $b\overline{b}$ events. This ratio is found to be 1.4$\pm$0.2. Thus, the net signal is 76$\pm$17$\pm$11, where the last error reflects the uncertainty in the ratio between RS and WS background. ### 2.2 Background studies Apart from false $D$ combinations, separated from the signal by the two- dimensional fit described above, there are also physical background sources that affect the RS Dfb samples, and originate from $b\overline{b}$ events, which are studied with a MC simulation. In the meson case, the background mainly comes from $b\rightarrow DDX$ with one of the $D$ mesons decaying semi- muonically, and from combinations of tracks from the $pp\rightarrow b\bar{b}X$ events, where one $b$ hadron decays into a $D$ meson and the other $b$ hadron decays semi-muonically. The background fractions are (1.9$\pm$0.3)% for $D^{0}X\mu^{-}\overline{\nu}$, (2.5$\pm$0.6)% for $D^{+}X\mu^{-}\overline{\nu}$, and (5.1$\pm$1.7)% for $D_{s}^{+}X\mu^{-}\overline{\nu}$. The main background component for $\mathchar 28931\relax_{b}^{0}$ semileptonic decays is $\mathchar 28931\relax_{b}^{0}$ decaying into $D^{-}_{s}\mathchar 28931\relax_{c}^{+}$, and the $D^{-}_{s}$ decaying semi-muonically. Overall, we find a very small background rate of (1.0$\pm$0.2)%, where the error reflects only the statistical uncertainty in the simulation. We correct the candidate $b$ hadron yields in the signal region with the predicted background fractions. A conservative 3% systematic uncertainty in the background subtraction is assigned to reflect modelling uncertainties. Figure 6: Projections of the two-dimensional fit to the $q^{2}$ and $m(D^{+}_{s}\mu)$ distributions of semileptonic decays including a $D^{+}_{s}$ meson. The $D_{s}^{}/D_{s}$ ratio has been fixed to the measured $D^{}/D$ ratio in light $B$ decays (2.42$\pm$0.10), and the background contribution is obtained using the sidebands in the $K^{+}K^{-}\pi^{+}$ mass spectrum. The different components are stacked: the background is represented by a black dot-dashed line, $D^{+}_{s}$ by a red dashed line, $D_{s}^{+}$ by a blue dash-double dotted line and $D_{s}^{+}$ by a green dash-dotted line. ### 2.3 Monte Carlo simulation and efficiency determination In order to estimate the detection efficiency, we need some knowledge of the different final states which contribute to the Cabibbo favoured semileptonic width, as some of the selection criteria affect final states with distinct masses and quantum numbers differently. Although much is known about the $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ and $B^{-}$ semileptonic decays, information on the corresponding $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ and $\mathchar 28931\relax_{b}^{0}$ semileptonic decays is rather sparse. In particular, the hadronic composition of the final states in $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ decays is poorly known [9], and only a study from CDF provides some constraints on the branching ratios of final states dominant in the corresponding $\mathchar 28931\relax_{b}^{0}$ decays [15]. In the case of the $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D^{+}_{s}$ semileptonic decays, we assume that the final states are $D^{+}_{s}$, $D^{+}_{s}$, $D_{s0}^{}(2317)^{+}$, $D_{s1}(2460)^{+}$, and $D_{s1}(2536)^{+}$. States above $DK$ threshold decay predominantly into $D^{()}K$ final states. We model the decays to the final states $D^{+}_{s}\mu^{-}\overline{\nu}$ and $D^{+}_{s}\mu^{-}\overline{\nu}$ with HQET form factors using normalization coefficients derived from studies of the corresponding $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ and $B^{-}$ semileptonic decays [1], while we use the ISGW2 form factor model [16] to describe final states including higher mass resonances. In order to determine the ratio between the different hadron species in the final state, we use the measured kinematic distributions of the quasi- exclusive process $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D^{+}_{s}\mu^{-}\overline{\nu}X$. To reconstruct the squared invariant mass of the $\mu^{-}\overline{\nu}$ pair ($q^{2}$), we exploit the measured direction of the $b$ hadron momentum, which, together with energy and momentum conservation, assuming no missing particles other than the neutrino, allow the reconstruction of the $\nu$ 4-vector, up to a two-fold ambiguity, due to its unknown orientation with respect to the $B$ flight path in its rest frame. We choose the solution corresponding to the lowest $b$ hadron momentum. This method works well when there are no missing particles, or when the missing particles are soft, as in the case when the charmed system is a $D^{}$ meson. We then perform a two-dimensional fit to the $q^{2}$ versus $m(\mu D^{+}_{s})$ distribution. Figure 6 shows stacked histograms of the $D^{+}_{s}$, $D^{+}_{s}$, and $D_{s}^{+}$ components. In the fit we constrain the ratio ${\cal B}(\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D^{+}_{s}\mu^{-}\overline{\nu})/{\cal B}(\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D^{+}_{s}\mu^{-}\overline{\nu})$ to be equal to the average $D^{}\mu^{-}\overline{\nu}/D\mu^{-}\overline{\nu}$ ratio in semileptonic $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ and $B^{-}$ decays (2.42$\pm$0.10) [1]. This constraint reduces the uncertainty of one $D^{}$ fraction. We have also performed fits removing this assumption, and the variation between the different components is used to assess the modelling systematic uncertainty. A similar procedure is applied to the $\mathchar 28931\relax_{c}^{+}\mu^{-}$ sample and the results are shown in Fig. 7. In this case we consider three final states, $\mathchar 28931\relax_{c}^{+}\mu^{-}\overline{\nu}$, ${\it\Lambda_{c}}(2595)^{+}\mu^{-}\overline{\nu}$, and ${\it\Lambda_{c}}(2625)^{+}\mu^{-}\overline{\nu}$, with form factors from the model of Ref. [17]. We constrain the two highest mass hadrons to be produced in the ratio predicted by this theory. Figure 7: Projections of the two-dimensional fit to the $q^{2}$ and $m(\mathchar 28931\relax_{c}^{+}\mu^{-})$ distributions of semileptonic decays including a $\mathchar 28931\relax_{c}^{+}$ baryon. The different components are stacked: the dotted line represents the combinatoric background, the bigger dashed line (red) represents the $\mathchar 28931\relax_{c}^{+}\mu^{-}\overline{\nu}$ component, the smaller dashed line (blue) the ${\it\Lambda_{c}}(2595)^{+}$, and the solid line represents the ${\it\Lambda_{c}}(2625)^{+}$ component. The ${\it\Lambda_{c}}(2595)^{+}/{\it\Lambda_{c}}(2625)^{+}$ ratio is fixed to its predicted value, as described in the text. Figure 8: Measured proton identification efficiency as a function of the $\mathchar 28931\relax_{c}^{+}\mu^{-}$ $p_{\rm T}$ for $2<\eta<3$, $3<\eta<4$, $4<\eta<5$ respectively, and for the selection criteria used in the $\mathchar 28931\relax_{c}^{+}\rightarrow pK^{-}\pi^{+}$ reconstruction. The measured pion, kaon and proton identification efficiencies are determined using $K_{\rm S}^{0}$, $D^{+}$, and $\Lambda^{0}$ calibration samples where $p$, $K$, and $\pi$ are selected without utilizing the particle identification criteria. The efficiency is obtained by fitting simultaneously the invariant mass distributions of events either passing or failing the identification requirements. Values are obtained in bins of the particle $\eta$ and $p_{\rm T}$, and these efficiency matrices are applied to the MC simulation. Alternatively, the particle identification efficiency can be determined by using the measured efficiencies and combining them with weights proportional to the fraction of particle types with a given $\eta$ and $p_{\rm T}$ for each $\mu$ charmed hadron pair $\eta$ and $p_{\rm T}$ bin. The overall efficiencies obtained with these two methods are consistent. An example of the resulting particle identification efficiency as a function of the $\eta$ and $p_{\rm T}$ of the $\mathchar 28931\relax_{c}^{+}\mu^{-}$ pair is shown in Fig. 8. As the functional forms of the fragmentation ratios in terms of $p_{\rm T}$ and $\eta$ are not known, we determine the efficiencies for the final states studied as a function of $p_{\rm T}$ and $\eta$ within the LHCb acceptance. Figure 9 shows the results. ## 3 Evaluation of the ratios ${f_{s}/(f_{u}+f_{d})}$ and ${f_{\Lambda_{b}}/(f_{u}+f_{d})}$ Perturbative QCD calculations lead us to expect the ratios $f_{s}/(f_{u}+f_{d})$ and $f_{\Lambda_{b}}/(f_{u}+f_{d})$ to be independent of $\eta$, while a possible dependence upon the $b$ hadron transverse momentum $p_{\rm T}$ is not ruled out, especially for ratios involving baryon species [18]. Thus we determine these fractions in different $p_{\rm T}$ and $\eta$ bins. For simplicity, we use the transverse momentum of the charmed hadron-$\mu$ pair as the $p_{\rm T}$ variable, and do not try to unfold the $b$ hadron transverse momentum. In order to determine the corrected yields entering the ratio $f_{s}/(f_{u}+f_{d})$, we determine yields in a matrix of three $\eta$ and five $p_{\rm T}$ bins and divide them by the corresponding efficiencies. We then use Eq. 5, with the measured lifetime ratio $(\tau_{B^{-}}+\tau_{\kern 1.25995pt\overline{\kern-1.25995ptB}{}^{0}})/2\tau_{\kern 1.25995pt\overline{\kern-1.25995ptB}{}^{0}_{s}}=1.07\pm 0.02$ [1] to derive the ratio $f_{s}/(f_{u}+f_{d})$ in two $\eta$ bins. The measured ratio is constant over the whole $\eta$-$p_{\rm T}$ domain. Figure 10 shows the $f_{s}/(f_{u}+f_{d})$ fractions in bins of $p_{\rm T}$ in two $\eta$ intervals. Figure 9: Efficiencies for $D^{0}\mu^{-}\overline{\nu}X$, $D^{+}\mu^{-}\overline{\nu}X$, $D^{+}_{s}\mu^{-}\overline{\nu}X$, $\mathchar 28931\relax_{c}^{+}\mu^{-}\overline{\nu}X$ as a function of $\eta$ and $p_{\rm T}$. $\begin{array}[]{cc}\includegraphics[width=433.62pt]{fig10}\end{array}$ Figure 10: Ratio between $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ and light $B$ meson production fractions as a function of the transverse momentum of the $D^{+}_{s}\mu^{-}$ pair in two bins of $\eta$. The errors shown are statistical only. Table 2: Systematic uncertainties on the relative $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ production fraction. Source \| Error (%) ---\|--- Bin-dependent errors \| 1.0 ${\cal B}(D^{0}\rightarrow K^{-}\pi^{+})$ \| 1.2 ${\cal B}(D^{+}\rightarrow K^{-}\pi^{+}\pi^{+})$ \| 1.5 ${\cal B}(D^{+}_{s}\rightarrow K^{-}K^{+}\pi^{+})$ \| 4.9 $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ semileptonic decay modelling \| 3.0 Backgrounds \| 2.0 Tracking efficiency \| 2.0 Lifetime ratio \| 1.8 PID efficiency \| 1.5 $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D^{0}K^{+}X\mu^{-}\overline{\nu}$ \| ${}^{+4.1}_{-1.1}$ ${\cal B}((B^{-},\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0})\rightarrow D_{s}^{+}KX\mu^{-}\overline{\nu})$ \| 2.0 Total \| ${}^{+8.6}_{-7.7}$ Figure 11: $f_{+}/f_{0}$ as a function of $p_{\rm T}$ for $\eta$=(2,3) (a) and $\eta$=(3,5) (b). The horizontal line shows the average value. The error shown combines statistical and systematic uncertainties accounting for the detection efficiency and the particle identification efficiency. By fitting a single constant to all the data, we obtain $f_{s}/(f_{u}+f_{d})=0.134\pm 0.004^{+0.011}_{-0.010}$ in the interval $2<\eta<5$, where the first error is statistical and the second is systematic. The latter includes several different sources listed in Table 2. The dominant systematic uncertainty is caused by the experimental uncertainty on ${\cal{B}}(D_{s}^{+}\rightarrow K^{+}K^{-}\pi^{+})$ of 4.9%. Adding in the contributions of the $D^{0}$ and $D^{+}$ branching fractions we have a systematic error of 5.5% due to the charmed hadron branching fractions. The $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ semileptonic modelling error is derived by changing the ratio between different hadron species in the final state obtained by removing the SU(3) symmetry constrain, and changing the shapes of the less well known $D^{*}$ states. The tracking efficiency errors mostly cancel in the ratio since we are dealing only with combinations of three or four tracks. The lifetime ratio error reflects the present experimental accuracy [1]. We correct both for the bin-dependent PID efficiency obtained with the procedure detailed before, accounting for the statistical error of the calibration sample, and the overall PID efficiency uncertainty, due to the sensitivity to the event multiplicity. The latter is derived by taking the kaon identification efficiency obtained with the method described before, without correcting for the different track multiplicities in the calibration and signal samples. This is compared with the results of the same procedure performed correcting for the ratio of multiplicities in the two samples. The error due to $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D^{0}K^{+}X\mu^{-}\overline{\nu}$ is obtained by changing the RS/WS background ratio predicted by the simulation within errors, and evaluating the corresponding change in $f_{s}/(f_{u}+f_{d})$. Finally, the error due to $(B^{-},\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0})\rightarrow D_{s}^{+}KX\mu^{-}\overline{\nu}$ reflects the uncertainty in the measured branching fraction. Isospin symmetry implies the equality of $f_{d}$ and $f_{u}$, which allows us to compare $f_{+}/f_{0}\equiv n_{\rm corr}(D^{+}\mu)/n_{\rm corr}(D^{0}\mu)$ with its expected value. It is not possible to decouple the two ratios for an independent determination of $f_{u}/f_{d}$. Using all the known semileptonic branching fractions [1], we estimate the expected relative fraction of the $D^{+}$ and $D^{0}$ modes from $B^{+/0}$ decays to be $f_{+}/f_{0}=0.375\pm 0.023,$ where the error includes a 6% theoretical uncertainty associated to the extrapolation of present experimental data needed to account for the inclusive $b\rightarrow c\mu^{-}\overline{\nu}$ semileptonic rate. Our corrected yields correspond to $f_{+}/f_{0}=0.373\pm 0.006$ (stat) $\pm$ 0.007 (eff) $\pm$ 0.014, for a total uncertainty of 4.5%. The last error accounts for uncertainties in $B$ background modelling, in the $D^{0}K^{+}\mu^{-}\overline{\nu}$ yield, the $D^{0}p\mu^{-}\overline{\nu}$ yield, the $D^{0}$ and $D^{+}$ branching fractions, and tracking efficiency. The other systematic errors mostly cancel in the ratio. Our measurement of $f_{+}/f_{0}$ is not seen to be dependent upon $p_{\rm T}$ or $\eta$, as shown in Fig. 11, and is in agreement with expectation. Figure 12: Fragmentation ratio $f_{\mathchar 28931\relax_{b}}/(f_{u}+f_{d})$ dependence upon $p_{\rm T}(\mathchar 28931\relax_{c}^{+}\mu^{-})$. The errors shown are statistical only. We follow the same procedure to derive the fraction $f_{\mathchar 28931\relax_{b}}/(f_{u}+f_{d})$, using Eq. 7 and the ratio $(\tau_{B^{-}}+\tau_{\kern 1.25995pt\overline{\kern-1.25995ptB}{}^{0}})/(2\tau_{\mathchar 28931\relax_{b}^{0}})=1.14\pm 0.03$ [1]. In this case, we observe a $p_{\rm T}$ dependence in the two $\eta$ intervals. Figure 12 shows the data fitted to a straight line $\frac{f_{\mathchar 28931\relax_{b}}}{f_{u}+f_{d}}=a[1+b\times p_{\rm T}({\rm GeV})].$ (8) Table 3 summarizes the fit results. A corresponding fit to a constant shows that a $p_{\rm T}$ independent $f_{\mathchar 28931\relax_{b}}/(f_{u}+f_{d})$ is excluded at the level of four standard deviations. The systematic errors reported in Table 3 include only the bin-dependent terms discussed above. Table 4 summarizes all the sources of absolute scale systematic uncertainties, that include several components. Their definitions mirror closely the corresponding uncertainties for the $f_{s}/(f_{u}+f_{d})$ determination, and are assessed with the same procedures. The term $\mathchar 28931\relax_{b}\rightarrow D^{0}pX\mu^{-}\overline{\nu}$ accounts for the uncertainty in the raw $D^{0}pX\mu^{-}\overline{\nu}$ yield, and is evaluated by changing the RS/WS background ratio (1.4$\pm$0.2) within the quoted uncertainty. In addition, an uncertainty of 2% is associated with the derivation of the semileptonic branching fraction ratios from the corresponding lifetimes, labelled $\Gamma_{\rm sl}$ in Table 4. The uncertainty is derived assigning conservative errors to the parameters affecting the chromomagnetic operator that influences the $B$ meson total decay widths, but not the $\mathchar 28931\relax_{b}^{0}$. By far the largest term is the poorly known ${\cal B}(\mathchar 28931\relax_{c}^{+}\rightarrow pK^{-}\pi^{+}$); thus it is quoted separately. Table 3: Coefficients of the linear fit describing the $p_{\rm T}(\mathchar 28931\relax_{c}^{+}\mu^{-})$ dependence of $f_{\mathchar 28931\relax_{b}}/(f_{u}+f_{d})$. The systematic uncertainties included are only those associated with the bin-dependent MC and particle identification errors. $\eta$ range \| $a$ \| $b$ ---\|---\|--- 2-3 \| 0.434$\pm$0.040$\pm$0.025 \| -0.036$\pm$0.008$\pm$0.004 3-5 \| 0.397$\pm$0.020$\pm$0.009 \| -0.028$\pm$0.006$\pm$0.003 2-5 \| 0.404$\pm$0.017$\pm$0.009 \| -0.031$\pm$0.004$\pm$0.003 Table 4: Systematic uncertainties on the absolute scale of $f_{\Lambda_{b}}/(f_{u}+f_{d})$. Source \| Error (%) ---\|--- Bin dependent errors \| 2.2 ${\cal B}(\mathchar 28931\relax_{b}^{0}\rightarrow D^{0}pX\mu^{-}\overline{\nu})$ \| 2.0 Monte Carlo modelling \| 1.0 Backgrounds \| 3.0 Tracking efficiency \| 2.0 $\Gamma_{\rm sl}$ \| 2.0 Lifetime ratio \| 2.6 PID efficiency \| 2.5 Subtotal \| 6.3 ${\cal B}(\mathchar 28931\relax_{c}^{+}\rightarrow pK^{-}\pi^{+})$ \| 26.0 Total \| $26.8$ In view of the observed dependence upon $p_{\rm T}$, we present our results as $\left[\frac{f_{\mathchar 28931\relax_{b}}}{f_{u}+f_{d}}\right](p_{\rm T})=(0.404\pm 0.017\pm 0.027\pm 0.105)\times[1-(0.031\pm 0.004\pm 0.003)\times p_{\rm T}{\rm(GeV)}],$ (9) where the scale factor uncertainties are statistical, systematic, and the error on ${\cal B}(\mathchar 28931\relax_{c}\rightarrow pK^{-}\pi^{+})$ respectively. The correlation coefficient between the scale factor and the slope parameter in the fit with the full error matrix is $-0.63$. Previous measurements of this fraction have been made at LEP and the Tevatron [3]. LEP obtains 0.110$\pm$0.019 [2]. This fraction has been calculated by combining direct rate measurements with time-integrated mixing probability averaged over an unbiased sample of semi-leptonic $b$ hadron decays. CDF measures $f_{\mathchar 28931\relax_{b}}/(f_{u}+f_{d})=0.281\pm 0.012^{+0.011+0.128}_{-0.056-0.086}$, where the last error reflects the uncertainty in ${\cal B}(\mathchar 28931\relax_{c}^{+}\rightarrow pK^{-}\pi^{+})$. It has been suggested [3] that the difference between the Tevatron and LEP results is explained by the different kinematics of the two experiments. The average $p_{\rm T}$ of the $\mathchar 28931\relax_{c}^{+}\mu^{-}$ system is 10 GeV for CDF, while the $b$-jets, at LEP, have $p\approx 40$ GeV. LHCb probes an even lower $b$ $p_{\rm T}$ range, while retaining some sensitivity in the CDF kinematic region. These data are consistent with CDF in the kinematic region covered by both experiments, and indicate that the baryon fraction is higher in the lower $p_{\rm T}$ region. ## 4 Combined result for the production fraction $f_{s}/f_{d}$ from LHCb From the study of $b$ hadron semileptonic decays reported above, and assuming isospin symmetry, namely $f_{u}=f_{d}$, we obtain $\left(\frac{f_{s}}{f_{d}}\right)_{\rm sl}=0.268\pm 0.008({\rm stat})^{+0.022}_{-0.020}({\rm syst}),$ where the first error is statistical and the second is systematic. Measurements of this quantity have also been made by LHCb by using hadronic $B$ meson decays [4]. The ratio determined using the relative abundances of $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D^{+}_{s}\pi^{-}$ to $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow D^{+}K^{-}$ is $\left(\frac{f_{s}}{f_{d}}\right)_{\rm h1}=0.250\pm 0.024(\rm stat)\pm 0.017({\rm syst})\pm 0.017({\rm theor}),$ while that from the relative abundances of $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D^{+}_{s}\pi^{-}$ to $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow D^{+}\pi^{-}$ [4] is $\left(\frac{f_{s}}{f_{d}}\right)_{\rm h2}=0.256\pm 0.014({\rm stat})\pm 0.019({\rm syst})\pm 0.026({\rm theor}).$ The first uncertainty is statistical, the second systematic and the third theoretical. The theoretical uncertainties in both cases include non- factorizable SU(3)-breaking effects and form factor ratio uncertainties. The second ratio is affected by an additional source, accounting for the $W$-exchange diagram in the $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow D^{+}\pi^{-}$ decay. In order to average these results, we consider the correlations between different sources of systematic uncertainties, as shown in Table 5. We then utilise a generator of pseudo-experiments, where each independent source of uncertainty is generated as a random variable with Gaussian distribution, except for the component $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D^{0}K^{+}\mu^{-}\overline{\nu}_{\mu}X$, which is modeled with a bifurcated Gaussian with standard deviations equal to the positive and negative errors shown in Table 5. This approach to the averaging procedure is motivated by the goal of proper treatment of asymmetric errors [21]. We assume that the theoretical errors have a Gaussian distribution. Table 5: Summary of the systematic and theoretical uncertainties in the three LHCb measurements of $f_{s}/f_{d}$. Source \| Error (%) \| ---\|---\|--- \| $(f_{s}/f_{d})_{\rm sl}$ \| $(f_{s}/f_{d})_{\rm h1}$ \| $(f_{s}/f_{d})_{\rm h2}$ \| Bin dependent error \| 1.0 \| - \| - \| Uncorrelated Semileptonic decay modelling \| 3.0 \| - \| - \| Uncorrelated Backgrounds \| 2.0 \| - \| - \| Uncorrelated Fit model \| - \| 2.8 \| 2.8 \| Uncorrelated Trigger simulation \| - \| 2.0 \| 2.0 \| Uncorrelated Tracking efficiency \| 2.0 \| - \| - \| Uncorrelated ${\cal B}(\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D^{0}K^{+}X\mu^{-}\overline{\nu})$ \| ${}^{+4.1}_{-1.1}$ \| - \| - \| Uncorrelated ${\cal B}(\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}/B^{-}\rightarrow D^{+}_{s}KX\mu^{-}\overline{\nu})$ \| 2.0 \| - \| - \| Uncorrelated Particle identification calibration \| 1.5 \| 1.0 \| 2.5 \| Correlated $B$ lifetimes \| 1.5 \| 1.5 \| 1.5 \| Correlated ${\cal B}(D^{+}_{s}\rightarrow K^{+}K^{-}\pi^{+})$ \| 4.9 \| 4.9 \| 4.9 \| Correlated ${\cal B}(D^{+}\rightarrow K^{-}\pi^{+}\pi^{-})$ \| 1.5 \| 1.5 \| 1.5 \| Correlated SU(3) and form factors \| - \| 6.1 \| 6.1 \| Correlated $W$-exchange \| - \| - \| 7.8 \| Uncorrelated We define the average fraction as $f_{s}/f_{d}=\alpha_{1}(f_{s}/f_{d})_{\rm sl}+\alpha_{2}(f_{s}/f_{d})_{\rm h1}+\alpha_{3}(f_{s}/f_{d})_{\rm h2},$ (10) where $\alpha_{1}+\alpha_{2}+\alpha_{3}=1.$ (11) The RMS value of $f_{s}/f_{d}$ is then evaluated as a function of $\alpha_{1}$ and $\alpha_{2}$. We derive the most probable value $f_{s}/f_{d}$ by determining the coefficients $\alpha_{i}$ at which the RMS is minimum, and the total errors by computing the boundaries defining the 68% CL, scanning from top to bottom along the axes $\alpha_{1}$ and $\alpha_{2}$ in the range comprised between 0 and 1. The optimal weights determined with this procedure are $\alpha_{1}=0.73$, and $\alpha_{2}=0.14$, corresponding to the most probable value $f_{s}/f_{d}=0.267^{+0.021}_{-0.020}.$ The most probable value differs slightly from a simple weighted average of the three measurements because of the asymmetry of the error distribution in the semileptonic determination. By switching off different components we can assess the contribution of each source of uncertainty. Table 6 summarizes the results. Table 6: Uncertainties in the combined value of $f_{s}/f_{d}$. Source \| Error (%) ---\|--- Statistical \| 2.8 Experimental systematic (symmetric) \| 3.3 ${\cal B}(\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D^{0}K^{+}X\mu^{-}\overline{\nu})$ \| ${}^{+3.0}_{-0.8}$ ${\cal B}(D^{+}\rightarrow K^{-}\pi^{+}\pi^{-})$ \| 2.2 ${\cal B}(D^{+}_{s}\rightarrow K^{+}K^{-}\pi^{+})$ \| 4.9 $B$ lifetimes \| 1.5 ${\cal B}(\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}/B^{-}\rightarrow D^{+}_{s}KX\mu^{-}\overline{\nu})$ \| 1.5 Theory \| 1.9 ## 5 Conclusions We measure the ratio of the $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ production fraction to the sum of those for $B^{-}$ and $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ mesons $f_{s}/(f_{u}+f_{d})$ = 0.134$\pm$0.004${}^{+0.011}_{-0.010}$, and find it consistent with being independent of $\eta$ and $p_{\rm T}$. Our results are more precise than, and in agreement with, previous measurements in different kinematic regions. We combine the LHCb measurements of the ratio of $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ to $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ production fractions obtained using $b$ hadron semileptonic decays, and two different ratios of branching fraction of exclusive hadronic decays to derive $f_{s}/f_{d}=0.267^{+0.021}_{-0.020}$. The ratio of the $\Lambda_{b}^{0}$ baryon production fraction to the sum of those for $B^{-}$ and $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ mesons varies with the $p_{\rm T}$ of the charmed hadron muon pair. Assuming a linear dependence up to $p_{\rm T}=14$ GeV, we obtain $\frac{f_{\mathchar 28931\relax_{b}}}{f_{u}+f_{d}}=(0.404\pm 0.017\pm 0.027\pm 0.105)\times[1-(0.031\pm 0.004\pm 0.003)\times p_{\rm T}{\rm(GeV)}],$ (12) where the errors on the absolute scale are statistical, systematic and error on ${\cal B}(\mathchar 28931\relax_{c}^{+}\rightarrow pK^{-}\pi^{+})$ respectively. No $\eta$ dependence is found. ## Acknowledgements We express our gratitude to our colleagues in the CERN accelerator departments for the excellent performance of the LHC. We thank the technical and administrative staff at CERN and at the LHCb institutes, and acknowledge support from the National Agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil); CERN; NSFC (China); CNRS/IN2P3 (France); BMBF, DFG, HGF and MPG (Germany); SFI (Ireland); INFN (Italy); FOM and NWO (the Netherlands); SCSR (Poland); ANCS (Romania); MinES of Russia and Rosatom (Russia); MICINN, XuntaGal and GENCAT (Spain); SNSF and SER (Switzerland); NAS Ukraine (Ukraine); STFC (United Kingdom); NSF (USA). We also acknowledge the support received from the ERC under FP7 and the Region Auvergne. ## References [1] K. Nakamura et al. [Particle Data Group], J. Phys. G 37, 075021 (2010). * [2] D. Asner et al. [Heavy Flavor Averaging Group], [arXiv:1010.1589 [hep-ex]] (2010); online updates are available at http://www.slac.stanford.edu/xorg/hfag/osc/PDG_2010 (2010). * [3] T. Aaltonen et al. [CDF collaboration], Phys. Rev. D 77, 072003 (2008) [arXiv:0801.4375v1 [hep-ex]]. * [4] R. Aaij et al. [LHCb collaboration], LHCb-PAPER-2011-006, [arXiv:1106.4435 [hep-ex]] (2011). * [5] R. Fleischer, N. Serra, and N. Tuning, Phys. Rev. D 82, 034038 (2010) [arXiv:1004.3982 [hep-ph]]. * [6] R. Fleischer, N. Serra, and N. Tuning, Phys. Rev. D 83, 014017 (2011) [arXiv:1012.2784 [hep-ph]]. * [7] A. Augusto Alves Jr. et al. [LHCb collaboration], JINST 3, S08005 (2008). * [8] M. Artuso, E. Barberio, and S. Stone, PMC Phys. A 3, 3 (2009) [arXiv:0902.3743v4 [hep-ph]]. * [9] R. Aaij et al. [LHCb collaboration], Phys. Lett. B 698, 14 (2011) [arXiv:1102.0348v3 [hep-ex]] * [10] P. del Amo Sanchez et al. [BABAR collaboration], Phys. Rev. Lett. 107, 041804 (2011) [arXiv:1012.4158 [hep-ex]]. * [11] A. V. Manohar and M. B. Wise, Phys. Rev. D 49, 1310 (1994) [arXiv:hep-ph/9308246v2]. * [12] I. I. Y. Bigi, M. A. Shifman, N. G. Uraltsev, and A. I. Vainshtein, Phys. Rev. Lett. 71, 496 (1993) [arXiv:hep-ph/9304225v1]. * [13] I.I Bigi, T. Mannel, and N. Uraltsev, [arXiv:1105.4574v1 [hep-ph]] (2011). * [14] R. Aaij et al. [LHCb collaboration], Phys. Lett. B 694, 209 (2010) [arXiv:1009.2731v2 [hep-ex]]. * [15] T Aaltonen et al. [CDF collaboration], Phys. Rev. D 79, 032001 (2009) [arXiv:0810.3213v3 [hep-ex]]. * [16] D. Scora and N. Isgur, Phys. Rev. D 52, 2783 (1995) [arXiv:hep-ph/9503486v1]. * [17] M. Pervin, W. Roberts, and S. Capstick, Phys. Rev. C 72, 035201 (2005) [arXiv:nucl-th/0503030]. * [18] M. Mangano, talk given at “Charm and bottom quark production at the LHC” CERN, Dec 3, 2010. * [19] S. Dobbs et al. [CLEO collaboration], Phys. Rev. D 76, 112001 (2007) [arXiv:0709.3783 [hep-ex]]. * [20] J. Alexander et al. [CLEO collaboration], Phys. Rev. Lett. 100, 161804 (2008) [arXiv:0801.0680v2 [hep-ex]]. * [21] R. Barlow, [arXiv:physics/0406120v1] (2004).	arxiv-papers	2011-11-09T21:57:41	2024-09-04T02:49:24.173970	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "The LHCb Collaboration: R. Aaij, C. Abellan Beteta, B. Adeva, M.\n Adinolfi, C. Adrover, A. Affolder, Z. Ajaltouni, J. Albrecht, F. Alessio, M.\n Alexander, G. Alkhazov, P. Alvarez Cartelle, A. A. Alves Jr, S. Amato, Y.\n Amhis, J. Anderson, R. B. Appleby, O. Aquines Gutierrez, F. Archilli, L.\n Arrabito, A. Artamonov, M. Artuso, E. Aslanides, G. Auriemma, S. Bachmann, J.\n J. Back, D. S. Bailey, V. Balagura, W. Baldini, R. J. Barlow, C. Barschel, S.\n Barsuk, W. Barter, A. Bates, C. Bauer, Th. Bauer, A. Bay, I. Bediaga, S.\n Belogurov, K. Belous, I. Belyaev, E. Ben-Haim, M. Benayoun, G. Bencivenni, S.\n Benson, J. Benton, R. Bernet, M.-O. Bettler, M. van Beuzekom, A. Bien, S.\n Bifani, A. Bizzeti, P.M. Bj{\\o}rnstad, T. Blake, F. Blanc, C. Blanks, J.\n Blouw, S. Blusk, A. Bobrov, V. Bocci, A. Bondar, N. Bondar, W. Bonivento, S.\n Borghi, A. Borgia, T. J. V. Bowcock, C. Bozzi, T. Brambach, J. van den Brand,\n J. Bressieux, D. Brett, S. Brisbane, M. Britsch, T. Britton, N. H. Brook, H.\n Brown, A. B\\\"uchler-Germann, I. Burducea, A. Bursche, J. Buytaert, S.\n Cadeddu, J. M. Caicedo Carvajal, O. Callot, M. Calvi, M. Calvo Gomez, A.\n Camboni, P. Campana, A. Carbone, G. Carboni, R. Cardinale, A. Cardini, L.\n Carson, K. Carvalho Akiba, G. Casse, M. Cattaneo, M. Charles, Ph.\n Charpentier, N. Chiapolini, K. Ciba, X. Cid Vidal, G. Ciezarek, P. E. L.\n Clarke, M. Clemencic, H. V. Cliff, J. Closier, C. Coca, V. Coco, J. Cogan, P.\n Collins, A. Comerma-Montells, F. Constantin, G. Conti, A. Contu, A. Cook, M.\n Coombes, G. Corti, G. A. Cowan, R. Currie, B. D'Almagne, C. D'Ambrosio, P.\n David, I. De Bonis, S. De Capua, M. De Cian, F. De Lorenzi, J. M. De Miranda,\n L. De Paula, P. De Simone, D. Decamp, M. Deckenhoff, H. Degaudenzi, M.\n Deissenroth, L. Del Buono, C. Deplano, D. Derkach, O. Deschamps, F. Dettori,\n J. Dickens, H. Dijkstra, P. Diniz Batista, F. Domingo Bonal, S. Donleavy, F.\n Dordei, A. Dosil Su\\'arez, D. Dossett, A. Dovbnya, F. Dupertuis, R.\n Dzhelyadin, A. Dziurda, S. Easo, U. Egede, V. Egorychev, S. Eidelman, D. van\n Eijk, F. Eisele, S. Eisenhardt, R. Ekelhof, L. Eklund, Ch. Elsasser, D.\n Elsby, D. Esperante Pereira, L. Est\\'eve, A. Falabella, E. Fanchini, C.\n F\\\"arber, G. Fardell, C. Farinelli, S. Farry, V. Fave, V. Fernandez Albor, M.\n Ferro-Luzzi, S. Filippov, C. Fitzpatrick, M. Fontana, F. Fontanelli, R.\n Forty, M. Frank, C. Frei, M. Frosini, S. Furcas, A. Gallas Torreira, D.\n Galli, M. Gandelman, P. Gandini, Y. Gao, J-C. Garnier, J. Garofoli, J. Garra\n Tico, L. Garrido, D. Gascon, C. Gaspar, N. Gauvin, M. Gersabeck, T. Gershon,\n Ph. Ghez, V. Gibson, V. V. Gligorov, C. G\\\"obel, D. Golubkov, A. Golutvin, A.\n Gomes, H. Gordon, M. Grabalosa G\\'andara, R. Graciani Diaz, L. A. Granado\n Cardoso, E. Graug\\'es, G. Graziani, A. Grecu, E. Greening, S. Gregson, B.\n Gui, E. Gushchin, Yu. Guz, T. Gys, G. Haefeli, C. Haen, S. C. Haines, T.\n Hampson, S. Hansmann-Menzemer, R. Harji, N. Harnew, J. Harrison, P. F.\n Harrison, J. He, V. Heijne, K. Hennessy, P. Henrard, J. A. Hernando Morata,\n E. van Herwijnen, E. Hicks, K. Holubyev, P. Hopchev, W. Hulsbergen, P. Hunt,\n T. Huse, R. S. Huston, D. Hutchcroft, D. Hynds, V. Iakovenko, P. Ilten, J.\n Imong, R. Jacobsson, A. Jaeger, M. Jahjah Hussein, E. Jans, F. Jansen, P.\n Jaton, B. Jean-Marie, F. Jing, M. John, D. Johnson, C. R. Jones, B. Jost, M.\n Kaballo, S. Kandybei, M. Karacson, T. M. Karbach, J. Keaveney, I. R. Kenyon,\n U. Kerzel, T. Ketel, A. Keune, B. Khanji, Y. M. Kim, M. Knecht, P.\n Koppenburg, A. Kozlinskiy, L. Kravchuk, K. Kreplin, M. Kreps, G. Krocker, P.\n Krokovny, F. Kruse, K. Kruzelecki, M. Kucharczyk, R. Kumar, T.\n Kvaratskheliya, V. N. La Thi, D. Lacarrere, G. Lafferty, A. Lai, D. Lambert,\n R. W. Lambert, E. Lanciotti, G. Lanfranchi, C. Langenbruch, T. Latham, R. Le\n Gac, J. van Leerdam, J.-P. Lees, R. Lef\\'evre, A. Leflat, J. Lefran\\c{c}ois,\n O. Leroy, T. Lesiak, L. Li, L. Li Gioi, M. Lieng, M. Liles, R. Lindner, C.\n Linn, B. Liu, G. Liu, J. H. Lopes, E. Lopez Asamar, N. Lopez-March, J.\n Luisier, F. Machefert, I. V. Machikhiliyan, F. Maciuc, O. Maev, J. Magnin, S.\n Malde, R. M. D. Mamunur, G. Manca, G. Mancinelli, N. Mangiafave, U. Marconi,\n R. M\\\"arki, J. Marks, G. Martellotti, A. Martens, L. Martin, A. Mart\\'in\n S\\'anchez, D. Martinez Santos, A. Massafferri, Z. Mathe, C. Matteuzzi, M.\n Matveev, E. Maurice, B. Maynard, A. Mazurov, G. McGregor, R. McNulty, C.\n Mclean, M. Meissner, M. Merk, J. Merkel, R. Messi, S. Miglioranzi, D. A.\n Milanes, M.-N. Minard, S. Monteil, D. Moran, P. Morawski, R. Mountain, I.\n Mous, F. Muheim, K. M\\\"uller, R. Muresan, B. Muryn, M. Musy, J.\n Mylroie-Smith, P. Naik, T. Nakada, R. Nandakumar, I. Nasteva, M. Nedos, M.\n Needham, N. Neufeld, C. Nguyen-Mau, M. Nicol, S. Nies, V. Niess, N. Nikitin,\n A. Nomerotski, A. Novoselov, A. Oblakowska-Mucha, V. Obraztsov, S. Oggero, S.\n Ogilvy, O. Okhrimenko, R. Oldeman, M. Orlandea, J. M. Otalora Goicochea, P.\n Owen, K. Pal, J. Palacios, A. Palano, M. Palutan, J. Panman, A. Papanestis,\n M. Pappagallo, C. Parkes, C. J. Parkinson, G. Passaleva, G. D. Patel, M.\n Patel, S. K. Paterson, G. N. Patrick, C. Patrignani, C. Pavel-Nicorescu, A.\n Pazos Alvarez, A. Pellegrino, G. Penso, M. Pepe Altarelli, S. Perazzini, D.\n L. Perego, E. Perez Trigo, A. P\\'erez-Calero Yzquierdo, P. Perret, M.\n Perrin-Terrin, G. Pessina, A. Petrella, A. Petrolini, A. Phan, E. Picatoste\n Olloqui, B. Pie Valls, B. Pietrzyk, T. Pilar, D. Pinci, R. Plackett, S.\n Playfer, M. Plo Casasus, G. Polok, A. Poluektov, E. Polycarpo, D. Popov, B.\n Popovici, C. Potterat, A. Powell, T. du Pree, J. Prisciandaro, V. Pugatch, A.\n Puig Navarro, W. Qian, J. H. Rademacker, B. Rakotomiaramanana, M. S. Rangel,\n I. Raniuk, G. Raven, S. Redford, M. M. Reid, A. C. dos Reis, S. Ricciardi, K.\n Rinnert, D. A. Roa Romero, P. Robbe, E. Rodrigues, F. Rodrigues, P. Rodriguez\n Perez, G. J. Rogers, S. Roiser, V. Romanovsky, M. Rosello, J. Rouvinet, T.\n Ruf, H. Ruiz, G. Sabatino, J. J. Saborido Silva, N. Sagidova, P. Sail, B.\n Saitta, C. Salzmann, M. Sannino, R. Santacesaria, C. Santamarina Rios, R.\n Santinelli, E. Santovetti, M. Sapunov, A. Sarti, C. Satriano, A. Satta, M.\n Savrie, D. Savrina, P. Schaack, M. Schiller, S. Schleich, M. Schmelling, B.\n Schmidt, O. Schneider, A. Schopper, M.-H. Schune, R. Schwemmer, B. Sciascia,\n A. Sciubba, M. Seco, A. Semennikov, K. Senderowska, I. Sepp, N. Serra, J.\n Serrano, P. Seyfert, B. Shao, M. Shapkin, I. Shapoval, P. Shatalov, Y.\n Shcheglov, T. Shears, L. Shekhtman, O. Shevchenko, V. Shevchenko, A. Shires,\n R. Silva Coutinho, H. P. Skottowe, T. Skwarnicki, A. C. Smith, N. A. Smith,\n E. Smith, K. Sobczak, F. J. P. Soler, A. Solomin, F. Soomro, B. Souza De\n Paula, B. Spaan, A. Sparkes, P. Spradlin, F. Stagni, S. Stahl, O. Steinkamp,\n S. Stoica, S. Stone, B. Storaci, M. Straticiuc, U. Straumann, N. Styles, V.\n K. Subbiah, S. Swientek, M. Szczekowski, P. Szczypka, T. Szumlak, S.\n T'Jampens, E. Teodorescu, F. Teubert, C. Thomas, E. Thomas, J. van Tilburg,\n V. Tisserand, M. Tobin, S. Topp-Joergensen, N. Torr, E. Tournefier, M. T.\n Tran, A. Tsaregorodtsev, N. Tuning, M. Ubeda Garcia, A. Ukleja, P. Urquijo,\n U. Uwer, V. Vagnoni, G. Valenti, R. Vazquez Gomez, P. Vazquez Regueiro, S.\n Vecchi, J. J. Velthuis, M. Veltri, K. Vervink, B. Viaud, I. Videau, X.\n Vilasis-Cardona, J. Visniakov, A. Vollhardt, D. Voong, A. Vorobyev, H. Voss,\n K. Wacker, S. Wandernoth, J. Wang, D. R. Ward, N. K. Watson, A. D. Webber, D.\n Websdale, M. Whitehead, D. Wiedner, L. Wiggers, G. Wilkinson, M. P. Williams,\n M. Williams, F. F. Wilson, J. Wishahi, M. Witek, W. Witzeling, S. A. Wotton,\n K. Wyllie, Y. Xie, F. Xing, Z. Xing, Z. Yang, R. Young, O. Yushchenko, M.\n Zavertyaev, F. Zhang, L. Zhang, W. C. Zhang, Y. Zhang, A. Zhelezov, L. Zhong,\n E. Zverev, A. Zvyagin", "submitter": "Marina Artuso", "url": "https://arxiv.org/abs/1111.2357" }
1111.2380	# Age and structure parameters of a remote M31 globular cluster B514 based on HST, 2MASS, GALEX and BATC observations Jun Ma11affiliation: National Astronomical Observatories, Chinese Academy of Sciences, Beijing 100012, P.R. China; majun@nao.cas.cn 22affiliation: Key Laboratory of Optical Astronomy, National Astronomical Observatories, Chinese Academy of Sciences, Beijing, 100012, China , Song Wang11affiliation: National Astronomical Observatories, Chinese Academy of Sciences, Beijing 100012, P.R. China; majun@nao.cas.cn 33affiliation: Graduate University of Chinese Academy of Sciences, A19 Yuquan Road, Shijingshan District, Beijing 100049, China , Zhenyu Wu11affiliation: National Astronomical Observatories, Chinese Academy of Sciences, Beijing 100012, P.R. China; majun@nao.cas.cn , Zhou Fan11affiliation: National Astronomical Observatories, Chinese Academy of Sciences, Beijing 100012, P.R. China; majun@nao.cas.cn , Tianmen Zhang11affiliation: National Astronomical Observatories, Chinese Academy of Sciences, Beijing 100012, P.R. China; majun@nao.cas.cn , Jianghua Wu11affiliation: National Astronomical Observatories, Chinese Academy of Sciences, Beijing 100012, P.R. China; majun@nao.cas.cn , Xu Zhou11affiliation: National Astronomical Observatories, Chinese Academy of Sciences, Beijing 100012, P.R. China; majun@nao.cas.cn , Zhaoji Jiang11affiliation: National Astronomical Observatories, Chinese Academy of Sciences, Beijing 100012, P.R. China; majun@nao.cas.cn and Jiansheng Chen11affiliation: National Astronomical Observatories, Chinese Academy of Sciences, Beijing 100012, P.R. China; majun@nao.cas.cn ###### Abstract B514 is a remote M31 globular cluster which locating at a projected distance of $R_{p}\simeq 55$ kpc. Deep observations with the Advanced Camera for Surveys (ACS) on the Hubble Space Telescope (HST) are used to provide the accurate integrated light and star counts of B514. By coupling analysis of the distribution of the integrated light with star counts, we are able to reliably follow the profile of the cluster out to $\sim 40\arcsec$. Based on the combined profile, we study in detail its surface brightness distribution in F606W and F814W filters, and determine its structural parameters by fitting a single-mass isotropic King model. The results showed that, the surface brightness distribution departs from the best-fit King model for $r>10^{\prime\prime}$. B514 is quite flatted in the inner region, and has a larger half-light radius than majority of normal globular clusters of the same luminosity. It is interesting that, in the $M_{V}$ versus $\log R_{h}$ plane, B514 lies nearly on the threshold for ordinary globular clusters as defined by Mackey & van den Bergh. In addition, B514 was observed as part of the Beijing- Arizona-Taiwan-Connecticut (BATC) Multicolor Sky Survey, using 13 intermediate-band filters covering a wavelength range of 3000–8500 Å. Based on aperture photometry, we obtain its SEDs as defined by the 13 BATC filters. We determine the cluster’s age and mass by comparing its SEDs (from 2267 to 20000Å, comprising photometric data in the near-ultraviolet of GALEX, 5 SDSS bands, 13 BATC intermediate-band, and 2MASS near-infrared $JHK_{\rm s}$ filters) with theoretical stellar population synthesis models, resulting in age of $11.5\pm 3.5$ Gyr. This age confirms the previous suggestion that B514 is an old GC in M31. B514 has a mass of $0.96-1.08\times 10^{6}\rm M_{\odot}$, and is a medium-mass globular cluster in M31. ###### Subject headings: galaxies: evolution – galaxies: individual (M31) – star clusters: B514 ††slugcomment: AJ, in press ## 1\. Introduction In hierarchical cosmological models, galaxies are built up through the continual accretion and merger of smaller ones. The signature of these system assembly processes is expected to be seen in the outskirts of a galactic halo. Globular clusters (GCs), as luminous compact objects that are found out to distant radii in the haloes of massive galaxies, can serve as excellent tracers of substructures in the outer region of their parent galaxy. So, detailed studies on GCs in the outer haloes of the local galaxies are very important. M31, with a distance modulus of 24.47 (Holland, 1998; Stanek & Garnavich, 1998; McConnachie et al., 2005), is an ideal nearby galaxy for studying GCs, since it is so near, and contains more GCs than all other Local Group galaxies combined (Battistini et al., 1987; Racine, 1991; Harris, 1991; Fusi Pecci et al., 1993). The study of GCs in M31 was initiated by Hubble (1932), who discovered 140 GC candidates with $m_{pg}\leq 18$ mag. And then, a number of catalogs of GC candidates were published. For example, the Bologna Group (Battistini et al., 1980, 1987, 1993) did independent searches of GC candidates and compiled them with their own Bologna number. Bologna catalog contains a total of 827 objects, and all the objects were classified into five classes by authors’ degree of confidence. 353 of these candidates were considered as class A or class B by high level of confidence, and the others fell into class C, D, or E. V magnitude and $B-V$ color for most candidates were given in the Bologna catalog. There are recent works dealing with the searched and the catalogs of M31 GCs (e.g., Mochejska et al., 1998; Barmby & Huchra, 2001; Galleti et al., 2004, 2006, 2007; Huxor et al., 2005, 2008, 2011; Kim et al., 2007; Mackey et al., 2006, 2007, 2010; Martin et al., 2006; Caldwell et al., 2009, 2011; Peacock et al., 2010). The continued importance of the study of GCs in this galaxy has been reviewed by Barmby et al. (2000). From the spatial structure and internal stellar kinematics of GCs, we can get information on both their formation conditions and dynamical evolution within the tidal fields of their host galaxies. For example, the structural parameters of GCs indicate on what timescales the cluster is bound to dissolve. However, the integrated properties of GCs, such as age and metallicity, are believed to reflect conditions in the early stages of galaxy formation (Brodie & Strader 2006). The most direct method to determine a cluster’s age is by employing main- sequence photometry, since the absolute magnitude of the main-sequence turnoff is predominantly affected by age (see Puzia et al., 2002, and references therein). However, until recently (cf. Perina et al., 2009), this method was only applied to the star clusters in the Milky Way and its satellites (e.g., Rich et al., 2001), although Brown et al. (2004) estimated the age of an M31 GC using extremely deep images observed with the HST’s Advanced Camera for Surveys (ACS). Generally, the ages of extragalactic star clusters are determined by comparing their observed spectral-energy distributions (SEDs) and/or spectroscopy with the predictions of SSP models (Williams & Hodge, 2001a, b; de Grijs et al., 2003b, c, a; Bik et al., 2003; Jiang et al., 2003; Beasley et al., 2004; Puzia et al., 2005; Fan et al., 2006; Ma et al., 2006a, 2007a, 2009a, 2009b, 2011; Caldwell et al., 2009, 2011; Wang et al., 2010; Perina et al., 2011). M31 GC B514 (B for ‘Bologna’, see Battistini 1987), which was detected by Galleti et al. (2005) based on the XSC sources of the All Sky Data Release of the Two Micron All Sky Survey (2MASS) within a $\sim 9^{\circ}\times 9^{\circ}$ area centered on M31, is the outermost cluster known in M31 at that time, which locating at a projected distance of $R_{p}\simeq 55$ kpc. Now, many new members of M31 halo GC system, which extending to very large radii, have been discovered (e.g., Huxor et al., 2005, 2008, 2011; Mackey et al., 2006, 2007, 2010; Martin et al., 2006). Galleti et al. (2006) presented deep color-magnitude diagram for B514 in F606W and F814W photometry obtained with the ACS/HST, which reveals a steep red giant branch (RGB) and a horizontal branch (HB) extending blueward of the instability strip showing that B514 is a classical old metal-poor GC. Federici et al. (2007) studied the density profile of B514 based on the HST/ACS observations as Galleti et al. (2006) had, and they found that the light and the star-count profiles show a departure from the best-fitted empirical models of King (1962) for $r\geq 8^{\prime\prime}$ – as a surface brightness excess at large radii, and the star-count profile shows a clear break in the correspondence of the estimated tidal radius; they also found that B514 has a significantly larger half-light radius than ordinary GCs of the same luminosity. Clementini et al. (2009) identified a rich harvest of RR Lyrae stars in B514, based on HST Wide Field Planetary Camera 2 (WFPC2) and HST/ACS time-series observations. Since B514 is located in the halo of M31, i.e., far away from the galaxy’s disk, it is (for all practical purposes) only affected by the Galactic foreground extinction. About the foreground Galactic reddening in the direction of M31, it was discussed by many authors (e.g., van den Bergh, 1969; McClure & Racine, 1969; Frogel et al., 1980; Fusi Pecci et al., 2005), and nearly similar values were determined such as $E(B-V)=0.08$ by van den Bergh (1969), 0.11 by McClure & Racine (1969) and Hodge (1992), 0.08 by Frogel et al. (1980). We argue that the reddening value of B514 should not smaller than the foreground Galactic reddening in the direction of M31. In this paper, we adopt the reddening value of $E(B-V)=0.10$ from Galleti et al. (2006). The reddening law from Cardelli et al. (1989) is employed in this paper. In addition, throughout this paper we adopt a distance to M31 of $783\pm 25$ kpc ($1{\arcsec}$ subtends 3.8 pc), corresponding to a distance modulus of $(m-M)_{0}=24.47\pm 0.07$ mag (McConnachie et al., 2005). In this paper, we describe the details of the observations and our approach to the data reduction with the HST/ACS and the BATC system in §2 and §3. We will study the surface distribution of B514 with the King models of King (1966) in details, which are developed by Michie (1963) and King (1966) based on the assumption that the GCs are formed by single-mass, isotropic, lowered isothermal spheres (hereafter ‘King models’). We determine the age and mass of B514 by comparing observational SEDs with population synthesis models in §4. We provide a summary in §5. ## 2\. Observation and photometric data with HST/ACS The images of B514 used in this paper were observed with the ACS/Wide Field Camera (WFC) in the F606W and F814W filters on 2005 July 19 (program ID GO 10394, PI: N. Tanvir), covering the period 2005, July 19–20 in F606W (total $t_{\rm exp}=1776$ s) and F814W (total $t_{\rm exp}=2505$ s). Upon retrieval from the STScI archive, all images were processed by the standard ACS calibration pipeline, in which bias and dark subtractions, flatfield division, and the masking of known bad pixels are included. Subsequently, photometric header keywords are populated. In the final stage of the pipeline, the MultiDrizzle software is used to correct the geometric distortion presented in the images. Finally, any cosmic rays are rejected while individual images in each band are combined into a final single image. We checked the images, and did not find saturated cluster stars. Figure 1 show the images observed with the ACS/WFC in the F606W and F814W. The ACS/WFC spatial resolution is $0.05\arcsec$ pixel-1. ### 2.1. Ellipticity, position angle, and surface brightness profile Surface photometry of the cluster is obtained from the drizzled images, using the iraf task ellipse. Its center position was fixed at a value derived by object locator of ellipse task, however an initial center position was determined by centroiding. Elliptical isophotes were fitted to the data, with no sigma clipping. We ran two passes of ellipse task, the first pass was run in the usual way, with ellipticity and position angle allowed to vary with the isophote semimajor axis. In the second pass, surface brightness profiles on fixed, zero-ellipticity isophotes were measured, since we choose to fit circular models for the intrinsic cluster structure and the point spread function (PSF) as Barmby et al. (2007) did (see §2.3 for details). The background value was derived as the mean of a region of $100\times 100$ pixels in “empty” areas far away from the cluster. We performed the photometric calibration using the results of Sirianni et al. (2005): 26.398 in F606W and 25.501 in F814W zero-points. Magnitudes are derived in the ACS/WFC vegamag system. Tables 1 and 2 list the ellipticity, $\epsilon=1-b/a$, and the position angle (P.A.) as a function of the semi-major axis length, $a$, from the center of annulus in the F606W and F814W filters, respectively. These observables have also been plotted in Figure 2; the errors were generated by the iraf task ellipse, in which the ellipticity errors are obtained from the internal errors in the harmonic fit, after removal of the first and second fitted harmonics. From Table 1 and Figure 2, we can see that, the values of ellipticity and position angle cannot be obtained beyond $0.9744\arcsec$ in the F606W filter because of low signal-to-noise ratio. In addition, Figure 2 shows that the ellipticity varies significantly with position along the semimajor axis radius smaller than $\sim 0.1752\arcsec$. Beyond $\sim 0.1752\arcsec$, the ellipticity does not vary significantly as a function of the cluster s semimajor axis. The P.A. does not vary as a function of the cluster s semimajor axis within $\sim 0.1752\arcsec$ because of high signal-to-noise ratio; however, beyond this position, it varies significantly with great errors because of low signal-to-noise ratio. Tables 3 and 4 list the surface brightness profile, $\mu$, of B514, and its integrated magnitude, $m$, as a function of radius in the F606W and F814W filters, respectively. The errors in the surface brightness were generated by the iraf task ellipse, in which they are obtained directly from the root mean square scatter of the intensity data along the zero-ellipticity isophotes. In addition, the surface photometries at radii where the ellipticity and position angle cannot be measured, are obtained based on the last ellipticity and position angle as the iraf task ellipse is designed. In order to derive the surface brightness profile of B514 in its outer region, we use the profile from star counts. We used the DOLPHOT111http://americano.dolphinsim.com/dolphot/ photometry software (Dolphin 2000a), specifically the ACS module, to photometer our images. DOLPHOT performs PSF fitting using PSFs especially tailored to the ACS camera. Photometry was done simultaneously on all the flat-fielded images from the STScI archive (both filters), relative to a deep reference image–we used the drizzled combination of the F606W image. DOLPHOT accounts for the hot-pixel and cosmic-ray masking information attached to each flat-fielded image, fits the sky locally around each detected source, and automatically applies the correction for the charge-transfer efficiency (CTE, Dolphin 2000b). It then transforms instrumental magnitude to the vegamag system (Dolphin 2000b). A variety of quality information is listed with each detected object, including the object type (stellar, extended, etc.), $\chi^{2}$ of the PSF fit, sharpness and roundness of the object, and a “crowding” parameter which describes how much brighter an object would have been had neighboring objects not been fitted simultaneously. We used the quality information provided by DOLPHOT to clean the resulting detection lists, selecting only stellar detections, with valid photometry on all input images, global sharpness parameter between $-0.3$ and 0.3 in each filter, and crowding parameter less than 0.25 in each filter. We joined the two profiles into one based on the method of Federici et al. (2007). This involves matching the intensity scales of the two profiles by fitting both profiles to smooth curves in the region $r=9-16^{\prime\prime}$. The star count profile is listed in Table 5. The errors for the star counts take account of Poisson statistical uncertainties. The joined profile covered the full $0<r\leq 40^{\prime\prime}$ range, as shown in Figure 3. Figure 1.— The images of GC B514 observed in the F606W and F814W filters of ACS/HST. The image size is $20\arcsec\times 20\arcsec$ for each panel. Figure 2.— Ellipticity and P.A. as a function of the semimajor axis in the F606W and F814W filters of ACS/HST. ### 2.2. Point spread function At a distance of 783 kpc, the ACS/WFC has a scale of $\rm{0.05^{\prime\prime}=0.19~{}pc~{}pixel^{-1}}$, and thus M31 clusters are clearly resolved with it. Their observed core structures, however, are still affected by the PSF. We chose not to deconvolve the data, instead fitting structural models after convolving them with a simple analytic description of the PSF as Barmby et al. (2007) and McLaughlin et al. (2008) did (see Barmby et al., 2007; McLaughlin et al., 2008; Ma et al., 2011, for details). In addition, since this PSF formula is radially symmetric and the models of King (1966) we fit are intrinsically spherical, the convolved models to be fitted to the data are also circularly symmetric. ### 2.3. Models and fits #### 2.3.1 Structural models After elliptical galaxies, GCs are the best understood and most thoroughly modelled class of stellar systems. For example, a large majority of the $\sim 150$ Galactic GCs have been fitted by the simple models of single-mass, isotropic, lowered isothermal spheres developed by Michie (1963) and King (1966) (i.e. King models), yielding comprehensive catalogs of cluster structural parameters and physical properties (see McLaughlin & van der Marel, 2005, and references therein). For extragalactic GCs, HST imaging data have been used to fit King models to a large number of GCs in M31 (e.g., Barmby et al., 2002, 2007, 2009, and references therein), in M33 (Larsen et al., 2002), and in NGC 5128 (e.g., Harris et al., 2002; McLaughlin et al., 2008, and references therein). In addition, there are other models used to fit the surface profile of GCs, including Wilson (1975), Elson et al. (1987), and Sérsic (1968). In this paper, we fit King models to the density profile of B514 observed with ACS/WFC. #### 2.3.2 Fits Our fitting procedure involves computing in full large numbers of King structural models, spanning a wide range of fixed values of the appropriate shape parameter $W_{0}$ (see McLaughlin & van der Marel, 2005, in detail). And then the models are convolved with the ACS/WFC PSF for the F606W and F814W filters (see Barmby et al., 2007, for details): ${\widetilde{I}_{\rm mod}^{}(R\|r_{0})=\int\\!\\!\\!\int_{-\infty}^{\infty}\widetilde{I}_{\rm mod}(R^{\prime}/r_{0})\times\widetilde{I}_{\rm PSF}\left[(x-x^{\prime}),(y-y^{\prime})\right]}$ $\ dx^{\prime}\,dy^{\prime}\ ,$ (1) where $\widetilde{I}_{\rm mod}\equiv I_{\rm mod}/I_{0}$ (see McLaughlin et al., 2008, in detail). We changed the luminosity density to surface brightness $\widetilde{\mu}_{\rm mod}^{}=-2.5\,\log\,[\widetilde{I}_{\rm mod}^{}]$ before fitting them to the observed surface brightness profile of B514, $\mu=\mu_{0}-2.5\,\log\,[I(R/r_{0})/I_{0}]$, finding the radial scale $r_{0}$ and central surface brightness $\mu_{0}$ which minimizing $\chi^{2}$ for every given value of $W_{0}$. The $(W_{0},r_{0},\mu_{0})$ combination that yields the global minimum $\chi_{\rm min}^{2}$ over the grid used defines the best- fit model of that type: $\chi^{2}=\sum_{i}{\frac{\left[\mu_{\rm obs}(R_{i})-\widetilde{\mu}_{\rm mod}^{}(R_{i}\|r_{0})\right]^{2}}{\sigma_{i}^{2}}},$ (2) in which $\sigma_{i}$ is the error in the surface brightness. Estimates of the one-sigma uncertainties on these basic fit parameters follow from their extreme values over the subgrid of fits with $\chi^{2}/\nu\leq\chi_{\rm min}^{2}/\nu+1$, here $\nu$ is the number of free parameters. Figure 3 shows our best King fits to B514. In Figure 3, open squares are ellipse data points included in the least-squares model fitting, and the asterisks are points not used to constrain the fit; black circles are star-counts points included in the $\chi^{2}$ model fitting, and red circles are star-counts points not used to constrain the fits. These observed data points shown by asterisks are included in the radius of $R<2~{}\rm{pixels}=0\farcs 1$, and the isophotal intensity is dependent on its neighbors. As Barmby et al. (2007) pointed out that, the ellipse output contains brightnesses for 15 radii inside 2 pixel, but they are all measured from the same 13 central pixels and are not statistically independent. So, to avoid excessive weighting of the central regions of B514 in the fits, we only used intensities at radii $R_{\rm min}$, $R_{\rm min}+(0.5,1.0,2.0)~{}{\rm pixels}$, or $R>2.5$ as Barmby et al. (2007) used. Table 6 summarizes the results obtained in this paper. From Figure 3, we can note that the surface brightness distribution departs from the best-fit King model for $r>10^{\prime\prime}$, which can be interpreted as the presence of a population of extratidal stars around the cluster. In fact, Federici et al. (2007) have reported this population of extratidal stars (see their Fig. 5 and their discussions). Figure 3.— Surface brightness profile of B514 measured in the F606W and F814 filters. Dashed curves (blue) trace the PSF intensity profiles and solid (red) curves are the PSF-convolved best-fit models. Open squares are ellips data points and black circles are star-counts profiles included in the $\chi^{2}$ model fitting, and the asterisks are ellips data points and red circles are star-counts profiles not used to constrain the fits (see the text in detail). Figure 4.— $M_{V}$ vs. $R_{h}$ for interesting stellar systems. The plotted line is the threshold for ordinary cluster in this plane as defined by Mackey & van den Bergh (2005), $\log R_{h}({\rm pc})=0.25M_{V}({\rm mag})+2.95$. Red circles are Galactic globular clusters from the on-line data base of Harris (1996) (2010 update); green crosses are M31 globular clusters from Barmby et al. (2007); green filled triangles are M31 young massive clusters from Barmby et al. (2009); green filled circles are M31 extended clusters from Mackey et al. (2006); green open triangles are outer M31 GCs from Mackey et al. (2007); black filled circles are M33 outer halo clusters from Cockcroft et al. (2011); the black square is B514 derived here. ### 2.4. Distribution of B514 in the $M_{V}$ vs. $\log R_{h}$ plane The distribution of stellar systems in the $M_{V}$ vs. $\log R_{h}$ plane can provide interesting information on the evolutionary history of these objects (e.g., van den Bergh & Mackey, 2004; Mackey & van den Bergh, 2005). In this plane, the half-light radius is an important parameter, which can be used to trace the initial size of a cluster, since it changes little in evolution process (see Spitzer & Thuan, 1972; Henon, 1973; Lightman & Shapiro, 1978; Murphy et al., 1990, for details). Recently, van den Bergh & Mackey (2004) and Mackey & van den Bergh (2005) showed that in a plot of luminosity versus half-light radius, the overwhelming majority of normal Galactic GCs lie below (or to the right) of the line: $\log R_{h}({\rm pc})=0.25M_{V}({\rm mag})+2.95.$ (3) Exceptions to this rule are massive clusters, such as M54 and $\omega$ Centauri in the Milky Way, and G1 in M31, which are widely believed to be the remnant cores of now defunct dwarf galaxies (Zinnecker et al., 1988; Freeman, 1993; Meylan et al., 2001). Because the well-known giant GC NGC 2419 (van den Bergh & Mackey, 2004) in the Galaxy and 037-B327 (Ma et al., 2006b) in M31 also lie above this line, it has been speculated that these two objects might also be the remnant cores of dwarf galaxies (but see de Grijs et al., 2005, for doubts regarding NGC 2419). With the value of $R_{h}$ (i.e. $r_{h}$) in the F606W filter obtained in this paper, we plot the relationship of $M_{V}$ versus $\log R_{h}$ in Figure 4, in which $M_{V}=-9.02$ which being derived based on $m_{V}=15.76$ from Huxor et al. (2008). It is interesting that, on this plot B514 is seen to lie nearly on the line defined by equation (3). Considering the uncertainties of $R_{h}$ and $M_{V}$, a certain conclusion may not be presented here. However, we argued that, B514 is a medium-mass GC in M31 (see §4.4 for details), and is not as massive as G1 and 037–B327 (see Ma et al., 2006a, b, 2007b, 2009b, for details). Furthermore, and for completeness, in Figure 4 we have also included GCs in the Milky Way, M31 and M33. Galactic GCs are from the on-line data base of Harris (1996) (2010 update). This new revision of the McMaster catalog of Galactic GCs is the first update since 2003 and the biggest single revision since the original version of the catalog published in 1996. The starting points for the present list of structural parameters are the major compilations of McLaughlin & van der Marel (2005) and Trager et al. (1995). McLaughlin & van der Marel (2005) used the same raw data as Trager et al. (1995), and derived structural parameter values from King (1966) dynamical profile models. M31 GCs are from recent compilations of data by Barmby et al. (2007, 2009), Mackey et al. (2006, 2007). M33 GCs are from Cockcroft et al. (2011). Barmby et al. (2007) derived structural parameters for 34 GCs in M31 based on ACS/HST observations, and the derived structural parameters are combined with corrected versions of those measured in an earlier survey in order to construct a comprehensive catalog of structural and dynamical parameters for 93 M31 GCs. Barmby et al. (2009) measured structural parameters for 23 bright young clusters in M31 based on the HST/WFPC2 observations, and suggested that on average they are larger and more concentrated than typical old clusters. Mackey et al. (2006) determined structural parameters for 4 extended, luminous globular clusters in the outskirts of M31 based on ACS/HST observations. These objects were discovered by Huxor et al. (2005) and Martin et al. (2006). Mackey et al. (2007) derived structural parameters for 10 classical GCs in the far outer regions of M31 based on ACS/WFC observations. Cockcroft et al. (2011) searched for outer halo star clusters in M33 as part of the Pan-Andromeda Archaeological Survey using the images taken with the Canada-France-Hawaii Telescope (CFHT)/MegaCam, and found one new unambiguous star cluster in addition to the five previously known in the M33 outer halo, and determined structural parameters for these 6 outer halo clusters. From Figure 4, we can see that, for the data of Galactic GCs which updated in 2010, in addition to the clusters already noted by Mackey & van den Bergh (2005), i.e. M 54 (NGC 6715), ${\omega{\rm~{}Cen}}$ (NGC 5139), NGC 2419, there are three other bright GCs (NGC 104, NGC 5272, and NGC 5024) lying above the “ordinary globular clusters” threshold. For M31 star clusters, in addition to G1, there are 26 other bright clusters lying above the line defined by equation (3). All of these objects are classified as GCs (Barmby et al., 2007; Mackey et al., 2007) or bright young clusters (Barmby et al., 2009). For 4 extended, luminous GCs in the outskirts of M31, they all lie above the line defined by equation (3). Based on F606W and F814W images of B514 obtained with the ACS/HST (program ID GO 10565, PI: S. Galleti), Federici et al. (2007) also studied in detail its surface brightness distribution in F606W and F814W filters, and determine its structural parameters by fitting a King (1962) model to a surface brightness profile. Comparing the results of Federici et al. (2007) with Table 6 of this paper, we find that our model fits produce smaller tidal radii, which resulting in smaller half-light, or effective, radii of a model. In addition, Federici et al. (2007) adopted $M_{V}=-9.1$ being brighter than $M_{V}=-9.02$ adopted here. So, in Federici et al. (2007), B514 lied above and brightward of the line defined by equation (3). ## 3\. Archival images of the BATC Multicolor Sky Survey, 2MASS and GALEX and photometric data of SDSS In this section, we will determine the magnitudes of B514 based on the archival images of the BATC Multicolor Sky Survey, 2MASS and GALEX using a standard aperture photometry approach, i.e., the phot routine in daophot (Stetson, 1987). In addition, we will introduce the photometric data of B514 from the Sloan Digital Sky Survey (SDSS) obtained by Peacock et al. (2010) ### 3.1. Intermediate-band photometry of B514 Observations of B514 were also obtained with the BATC 60/90cm Schmidt telescope located at the Xinglong station of the National Astronomical Observatory of China (NAOC). This telescope is equipped with 15 intermediate- band filters covering the optical wavelength range from 3000 to 10000 Å(see Fan et al., 2009, for details). Figure 5 shows a finding chart of B514 in the BATC $b$ band (centered at 5795 Å). Figure 5.— Image of B514 in the BATC $b$ band, obtained with the NAOC 60/90cm Schmidt telescope. B514 is circled using an aperture with a radius of $13^{\prime\prime}$. The field of view of the image is $4.3^{\prime}\times 4.3^{\prime}$. The BATC survey team obtained 47 images of B514 in 13 BATC filters between 2005 March 1 and 2006 December 9. Table 7 contains the observation log, including the BATC filter names, the central wavelength and bandwidth of each filter, the number of images observed through each filter, and the total observing time per filter. Multiple images through the same filter were combined to improve image quality (i.e., increase the signal-to-noise ratio and remove spurious signal). Calibration of the magnitude zero level in the BATC photometric system is similar to that of the spectrophotometric AB magnitude system. For flux calibration, the Oke-Gunn (Oke & Gunn, 1983) primary flux standard stars HD 19445, HD 84937, BD +26∘2606, and BD +17∘4708 were observed during photometric nights (Yan et al., 2000). Column (6) of Table 7 gives the zero-point errors in magnitude for the standard stars through each filter. The formal errors obtained for these stars in the 13 BATC filters used are $\lesssim 0.02$ mag, which implies that we can define photometrically the BATC system to an accuracy of better than 0.02 mag. We determined the intermediate-band magnitudes of B514 on the combined images. The (radial) photometric asymptotic growth curves, in all BATC bands, flatten out at a radius of $\sim 13^{\prime\prime}$. Inspection ensured that this aperture is adequate for photometry, i.e., B514 does not show any obvious signal beyond this radius. Therefore, we use an aperture with $r\approx 13^{\prime\prime}$ for integrated photometry. Since B514 is located in the M31 halo, contamination from background fluctuations can be neglected. We adopted annuli for background subtraction spanning between 14 to $20^{\prime\prime}$. The calibrated photometry of B514 in 13 filters is summarized in column (7) of Table 7, in conjunction with the $1\sigma$ magnitude uncertainties, which include uncertainties from the calibration errors in magnitude from daophot. ### 3.2. Near-infrared 2MASS photometry of B514 B514 was detected by Galleti et al. (2005) based on the XSC sources of the All Sky Data Release of 2MASS within a $\sim 9^{\circ}\times 9^{\circ}$ area centered on M31. In order to obtain accurate photometry for B514 in $JHK_{s}$, we download the images in $JHK_{s}$ filters including B514. The image in each filter is combined using 6 frames of 1.3 seconds, so the total exposure time of image in each filter is 7.8 seconds. The mosaic pixel scale of the final atlas image is resampled to $1^{\prime\prime}$ (see Skrutskie et al., 2006, for details). The relevant zero-points for photometry are 20.9210, 20.7089, and 20.0783 in $J$, $H$, and $K_{s}$ magnitudes, respectively, which are presented in photometric header keywords. We use an aperture with $r=13^{\prime\prime}$ for integrated photometry, and annuli for background subtraction spanning between $14^{\prime\prime}$ to $19^{\prime\prime}$. The calibrated photometry of B514 in $J$, $H$, and $K_{s}$ filters is summarized in Table 8, in conjunction with the $1\sigma$ magnitude uncertainties obtained from daophot. ### 3.3. GALEX Ultraviolet photometry of B514 While the principle science goal of the Galaxy Evolution Explorer (GALEX, Martin et al. (2005); Morrissey et al. (2007)) has been the study of star formation in the local and intermediate-redshift universe, nearby galaxies such as M31 have also been surveyed, taking advantage of the wide ($1.2^{\circ}$) field of view of GALEX. The B514 images were obtained as part of the guest program carried out by GALEX in two UV bands: far-ultraviolet (FUV) ($\mbox{$\lambda_{\rm eff}$}=1539$ Å, FWHM $\approx 270$ Å), and near- ultraviolet (NUV) ($\mbox{$\lambda_{\rm eff}$}=2316$ Å, FWHM $\approx 615$ Å) with resolution $4.2^{\prime\prime}$ (FUV) and $5.3^{\prime\prime}$ (NUV) (Morrissey et al., 2007). The exposure times are 1616 seconds in FUV and 1704 seconds in NUV. The images are sampled with $1.5^{\prime\prime}$ pixels. The data was downloaded from the MAST archive. The relevant zero-points for photometry are 20.08 and 18.82 in NUV and FUV magnitudes, respectively (Morrissey et al., 2007). We use an aperture with $r=12^{\prime\prime}$ for integrated photometry, and annuli for background subtraction spanning between $13.5^{\prime\prime}$ to $19.5^{\prime\prime}$. The calibrated photometry of B514 in NUV and FUV filters is summarized in Table 8. From Table 8, we can see that the $1\sigma$ magnitude uncertainties are great, especially the magnitude uncertainty in FUV is very great (2.3 magnitude), i.e. the signal-to-noise ratios of these images are low, especially the signal-to-noise ratio of the image in FUV is very low. Since the magnitude uncertainty in FUV is very great, we will not use it when fitting to derive the age of B514 in §4. ### 3.4. Photometric data of B514 from SDSS Peacock et al. (2010) presented an updated catalog of M31 GCs based on images from the Wide Field Camera (WFCAM) on the United Kingdom Infrared Telescope and from the SDSS, in which $ugriz$ and $K$-band photometry are determined. In this catalog, B514 is named H6 from Huxor et al. (2008), and $ugriz$ photometry is presented. ## 4\. Stellar population of B514 ### 4.1. Metallicity of B514 Cluster SEDs are determined by the combination of their ages and metallicities, which is often referred to as the age-metallicity degeneracy. Therefore, the age of a cluster can only be constrained accurately if the metallicity is known with confidence, from independent determinations. There exist four metallicity determinations for B514: namely, $\rm{[Fe/H]}=-1.8\pm 0.3$ (spectroscopic from Galleti et al. 2005), $-1.8\pm 0.15$ (from the CMD; Galleti et al. 2006), $-2.14\pm 0.15$ (from the CMD; Mackey et al. 2007), and $-2.06\pm 0.16$ (spectroscopic from Galleti et al. 2009), which are consistent. In order to adopt a reasonable value of metallicity for B514, the mean value of these four independent determinations, i.e. $\rm{[Fe/H]=-1.95}$, is adopted in this paper. ### 4.2. Stellar populations and synthetic photometry To determine the age and mass of B514, we compared its SEDs with theoretical stellar population synthesis models. The SEDs consist of photometric data in NUV of GALEX, 13 BATC intermediate-band and 2MASS near-infrared $JHK_{s}$ filters obtained in this paper, and of the photometric data in 5 SDSS filters obtained by Peacock et al. (2010). We will not include the photometric datum in the FUV band when constraining the age of B514 because of its large photometric error (2.3 magnitude), i.e. the photometric datum is not accurate. B514 is a very metal poor GC (see discussions above). So, we use the SSP models of Bruzual & Charlot (2003) (hereafter BC03), which have been upgraded from the earlier Bruzual & Charlot (1993, 1996) versions, and now provide the evolution of the spectra and photometric properties for a wide range of stellar metallicities. For example, BC03 SSP models based on the Padova-1994 evolutionary tracks include six initial metallicities, $Z=0.0001,0.0004,0.004,0.008,0.02\,(Z_{\odot})$, and 0.05, corresponding to ${\rm[Fe/H]}=-2.25$, $-1.65$, $-0.64$, $-0.33$, $+0.09$, and $+0.56$. BC03 provides 26 SSP models (both of high and low spectral resolution) using the Padova-1994 evolutionary tracks, half of which were computed based on the Salpeter (1955) IMF with lower and upper-mass cut-offs of $m_{\rm L}=0.1~{}M_{\odot}$ and $m_{\rm U}=100~{}M_{\odot}$, respectively. The other thirteen were computed using the Chabrier (2003) IMF with the same mass cut- offs. In addition, BC03 provide 26 SSP models using the Padova-2000 evolutionary tracks which including six partially different initial metallicities, $Z=0.0004$, 0.001, 0.004, 0.008, 0.019 $(Z_{\odot})$, and 0.03, i.e., ${\rm[Fe/H]}=-1.65,-1.25,-0.64,-0.33,+0.07$, and $+0.29$. In this paper, we adopt the high-resolution SSP models using the Padova-1994 evolutionary tracks to determine the most appropriate age for B514 since its metallicity is $\rm[Fe/H]=-1.95$, and a Salpeter (1955) IMF is used. These SSP models contain 221 spectra describing the spectral evolution of SSPs from $1.0\times 10^{5}$ yr to 20 Gyr. The evolving spectra include the contribution of the stellar component at wavelengths from 91Å to $160~{}\mu$m. Since our observational data are integrated luminosities through a given set of filters, we convolved the theoretical SSP SEDs of BC03 with the GALEX NUV, SDSS $ugriz$, BATC $a-n$ and 2MASS $JHK_{\rm s}$ filter response curves to obtain synthetic optical and NIR photometry for comparison (see Ma et al., 2009a, b; Wang et al., 2010; Ma et al., 2011, for details). ### 4.3. Fit results We use a $\chi^{2}$ minimization approach to examine which SSP models are most compatible with the observed SEDs, following $\chi^{2}=\sum_{i=1}^{22}{\frac{[m_{\lambda_{i}}^{\rm intr}-m_{\lambda_{i}}^{\rm mod}(t)]^{2}}{\sigma_{i}^{2}}},$ (4) where $m_{\lambda_{i}}^{\rm mod}(t)$ is the integrated magnitude in the $i{\rm th}$ filter of a theoretical SSP at age $t$, $m_{\lambda_{i}}^{\rm intr}$ represents the intrinsic integrated magnitude in the same filter, and $\sigma_{i}$ is the magnitude uncertainty, defined as $\sigma_{i}^{2}=\sigma_{{\rm obs},i}^{2}+\sigma_{{\rm mod},i}^{2}+\sigma_{{\rm md},i}^{2}.$ (5) Here, $\sigma_{{\rm obs},i}$ is the observational uncertainty from Tables 7 and 8 of this paper, and Table 1 of Peacock et al. (2010), $\sigma_{{\rm mod},i}$ is the uncertainty associated with the model itself, and $\sigma_{{\rm md},i}$ is associated with the uncertainty with the distance modulus adopted here. Charlot et al. (1996) estimated the uncertainty associated with the term $\sigma_{{\rm mod},i}$ by comparing the colors obtained from different stellar evolutionary tracks and spectral libraries. Following Ma et al. (2009a), Ma et al. (2009b), Wang et al. (2010) and Ma et al. (2011), we adopt $\sigma_{{\rm mod},i}=0.05$ mag. For $\sigma_{{\rm md},i}$, we adopt 0.07 from McConnachie et al. (2005). Before fitting, we obtained the the theoretical SEDs for the metallicity $\rm[Fe/H]=-1.95$ model by interpolation of between ${\rm[Fe/H]}=-2.25$ and $-1.65$ models. Since the observed magnitudes in the 2MASS photometric systems are given in the Vega system, we transformed them to the AB system for our fits. The photometric offsets in the 2MASS filters between the Vega and AB systems were obtained based on equations (7) and (8) in the manual provided by Bruzual & Charlot (2003) (bc03.ps). The best-reduced $\chi^{2}_{\rm min}/\nu=0.8$ is achieved with an age of $11.5\pm 3.5$ Gyr (1 $\sigma$ uncertainties), $\nu=21$ is the number of free parameters, i.e., the number of observational data points minus the number of parameters used in the theoretical model. In Figure 6, we show the intrinsic SEDs of B514, the integrated SEDs of the best-fitting model, and the spectra of the best-fitting model. From Figure 6, we can see that the BC03 SSP models cannot fit the photometric data point in $H$ band as well as the other 21 data points, i.e. the observed magnitude is brighter than the model one in the $H$ band. However, the photometric data point in the $H$ band from Galleti et al. (2005) can be fitted by BC03 SSP models as well as the other 21 data points, and the fitting result (the age of B514) is in agreement with one obtained above ($11.5\pm 3.5$ Gyr) within the uncertainty. much better fitted by BC03 SSP models than the data value derived in this paper, and the fitting results are consistent within the uncertainty. ### 4.4. Mass of B514 We next determined the mass of B514. The BC03 SSP models are normalized to a total mass of $1M_{\odot}$ in stars at age $t=0$. The absolute magnitudes (in the Vega system) in $V$, SDSS $ugriz$ and 2MASS $JHK_{\rm s}$ filters are included in the BC03 SSP models. The difference between the intrinsic absolute magnitudes and those given by the model provides a direct measurement of the cluster mass. To reduce mass uncertainties resulting from photometric uncertainties based on only magnitudes in one filter (in general the $V$ band is used), we estimated the mass of B514 using magnitudes in the $V$, $ugriz$ and $JHK_{\rm s}$ bands. The resulting mass determinations for B514 are listed in Table 9 with their $1\sigma$ uncertainties. From Table 9, we can see that the mass of B514 obtained based on the magnitudes in different filters is consistent except for one in the $H$ band. In fact, the observed magnitude is brighter than the model one in $H$ band (see discussion in §4.3). So, the mass of B514 derived based on the magnitude in the $H$ band is more massive than its true one. The mass of B514 derived based on the magnitude in the $H$ band from Galleti et al. (2005) is in agreement with ones derived based on the magnitudes in the other 8 bands. Table 9, we know that the obtained mass of B514 is between $0.96-1.08\times 10^{6}\rm M_{\odot}$ not including one in the $H$ band. Comparing with 037-B327 [$\mathcal{M}_{\rm 037-B327}\sim 8.5\times 10^{6}\rm M_{\odot}$ (Barmby et al., 2002) or $\mathcal{M}_{\rm 037-B327}\sim 3.0\pm 0.5\times 10^{7}\rm M_{\odot}$ (Ma et al., 2006a)] and G1 [$\mathcal{M}_{\rm G1}\sim(7-17)\times 10^{6}\rm M_{\odot}$ (Meylan et al., 2001) or $\mathcal{M}_{\rm G1}\sim(5.8-10.6)\times 10^{6}\rm M_{\odot}$ (Ma et al., 2009b)] in M31 and $\omega$ Cen [$\mathcal{M}_{\omega{\rm~{}Cen}}\sim(2.9-5.1)\times 10^{6}$M⊙ (Meylan, 2002)] in the Milky Way, the most massive clusters in the Local Group, B514 is only a medium-mass globular cluster. Figure 6.— Best-fitting, integrated theoretical BC03 SEDs compared to the intrinsic SED of B514. The photometric measurements are shown as symbols with error bars. Open circles represent the calculated magnitudes of the model SED for each filter. ## 5\. Summary In this paper, we determined the structural parameters of one remote globular cluster B514 known in M31 based on F606W and F814W images obtained with the ACS/HST. By performing a fit to the surface brightness distribution of a single-mass isotropic King model, we derive its parameters: the best-fitting scale radii $r_{0}=0.36^{+0.09}_{-0.05}~{}\rm{arcsec}~{}(=1.35^{+0.35}_{-0.19}~{}\rm{pc})$ and $0.36^{+0.09}_{-0.06}~{}\rm{arcsec}~{}(=1.35^{+0.32}_{-0.22}~{}\rm{pc})$, tidal radii $r_{t}=16.08^{+2.11}_{-1.35}~{}\rm{arcsec}~{}(=61.11^{+8.00}_{-5.14}~{}\rm{pc})$ and $16.79^{+1.74}_{-1.48}~{}\rm{arcsec}~{}(=63.78^{+6.62}_{-5.64}~{}\rm{pc})$, and concentration indexes $c=\log(r_{t}/r_{0})=1.66^{+0.05}_{-0.04}$ and $1.68^{+0.04}_{-0.04}$ in F606W and F814W, respectively; the central surface brightnesses are $16.25^{+0.57}_{-0.56}$ mag arcsec-2 and $15.64^{+0.80}_{-0.64}$ mag arcsec-2 in F606W and F814W, respectively; the half-light, or effective, radius of a model that contains half the total luminosity in projection, at $r_{h}=1.31^{+0.14}_{-0.08}~{}\rm{arcsec}~{}(=5.00^{+0.55}_{-0.32}~{}\rm{pc})$ and $1.36^{+0.13}_{-0.09}~{}\rm{arcsec}~{}(=5.17^{+0.48}_{-0.34}~{}\rm{pc})$ in F606W and F814W, respectively. The results show that, the surface brightness distribution departs from the best-fit King model for $r>10^{\prime\prime}$. In addition, B514 was observed as part of the BATC Multicolor Sky Survey, using 13 intermediate-band filters covering a wavelength range of 3000–80,000 Å. Based on aperture photometry, we obtain its SEDs as defined by the 13 BATC filters. We determine the cluster’s age by comparing its SEDs (from 2267 to 20,000Å, comprising photometric data in the NUV of GALEX, 13 BATC intermediate-band filters, and 5 SDSS filters, and 2MASS near-infrared $JHK_{\rm s}$ data) with theoretical stellar population synthesis models, resulting in an age of $11.5\pm 3.5$ Gyr. This age confirms previous suggestions that B514 is an old GC in M31. B514 has a mass of $0.96-1.08\times 10^{6}\rm M_{\odot}$, and is a medium-mass globular cluster in M31. We would like to thank the anonymous referee for providing rapid and thoughtful report that helped improve the original manuscript greatly. This work is partly based on observations made with the NASA/ESA Hubble Space Telescope, obtained at the Space Telescope Science Institute, which is operated by AURA, Inc., under NASA contract NAS 5-26555. These observations are associated with proposal 10394. This work was supported by the Chinese National Natural Science Foundation grands No. 10873016, 10633020, 10803007, 11003021, and 11073032, and by National Basic Research Program of China (973 Program), No. 2007CB815403. ## References * Barmby et al. (2002) Barmby, P., Holland, S., & Huchra, J. 2002, AJ, 123, 1937 * Barmby et al. (2000) Barmby, P., Huchra, J., Brodie, J., Forbes, D., Schroder, L., & Grillmair, C. 2000, AJ, 119, 727 * Barmby & Huchra (2001) Barmby, P., & Huchra, J. P. 2001, AJ, 122, 2458 * Barmby et al. (2007) Barmby, P., McLaughlin, D. E., Harris, W. E., Harris, G. L. H., & Forbes, D. A. 2007, AJ, 133, 2764 * Barmby et al. (2009) Barmby, P., et al. 2009, AJ, 138, 1667 * Battistini et al. (1987) Battistini, P., Bònoli, F., Braccesi, A., Federici, L., Fusi Pecci F., Marano, B., & Börngen, F. 1987, A&AS, 67, 447 * Battistini et al. (1980) Battistini, P., Bònoli, F., Braccesi, A., Fusi Pecci F., Malagnini, M. L., & Marano, B. 1980, A&AS, 42, 357 * Battistini et al. (1993) Battistini, P., Bònoli, F., Casavecchia, M., Ciotti, L., Federici, L., & Fusi Pecci F. 1993, A&A, 272, 77 * Beasley et al. (2004) Beasley, M. A., et al. 2004, AJ, 128, 1623 * Bik et al. (2003) Bik, A., Lamers, H. J. G. L. M., Bastian, N., Panagia, N., & Romaniello, M. 2003, A&A, 397, 473 * Brown et al. (2004) Brown, T. M., et al. 2004, ApJ, 613, L125 * Bruzual & Charlot (1993) Bruzual, A. G., & Charlot, S. 1993, ApJ, 405, 538 * Bruzual & Charlot (1996) Bruzual, A. G., & Charlot, S. 1996, unpublished * Bruzual & Charlot (2003) Bruzual, A. G., & Charlot, S. 2003, MNRAS, 344, 1000 * Caldwell et al. (2009) Caldwell, N., Harding, P., Morrison, H., Rose, J. A., Schiavon, R., & Kriessler, J. 2009, AJ, 137, 94 * Caldwell et al. (2011) Caldwell, N., Schiavon, R., Morrison, H., Rose, J. A., & Harding, P. 2011, AJ, 141, 61 * Cardelli et al. (1989) Cardelli, J. A., Clayton, G. C., & Mathis, J. S. 1989, ApJ, 345, 245 * Chabrier (2003) Chabrier, G. 2003, PASP, 115, 763 * Charlot et al. (1996) Charlot, S., Worthey, G., & Bressan, A. 1996, ApJ, 457, 625 * Clementini et al. (2009) Clementini, G., et al. 2009, ApJ, 704L, 103 * Cockcroft et al. (2011) Cockcroft, R., et al. 2011, ApJ, 730, 112 * Dolphin (2000a) Dolphin, A. E. 2000a, PASP, 112, 1383 * Dolphin (2000b) Dolphin, A. E. 2000b, PASP, 112, 1397 * de Grijs et al. (2003a) de Grijs, R., Anders, P., Lynds, R., Bastian, N., Lamers, H. J. G. L. M., & O’Neill, E. J. Jr. 2003a, MNRAS, 343, 1285 * de Grijs et al. (2003b) de Grijs, R., Bastian, N., & Lamers, H. J. G. L. M. 2003b, MNRAS, 340, 197 * de Grijs et al. (2003c) de Grijs, R., Fritze-v. Alvensleben, U., Anders, P., Gallagher, J. S., Bastian, N., Taylor, V. A., & Windhorst, R. A. 2003c, MNRAS, 342, 259 * de Grijs et al. (2005) de Grijs, R., Wilkinson, M. I., & Tadhunter, C. N. 2005, MNRAS, 361, 311 * Elson et al. (1987) Elson, R. A. W., Fall, S. M., & Freeman, K. C. 1987, ApJ, 323, 54 * Fan et al. (2006) Fan, Z., Ma, J., de Grijs, R., Yang Y., & Zhou X. 2006, MNRAS, 371, 1648 * Fan et al. (2009) Fan, Z., Ma, J., & Zhou X. 2009, RAA, 9, 993 * Federici et al. (2007) Federici, L., Bellazzini, M., Galleti, S., Fusi Pecci, F., Buzzoni, A., & Parmeggiani, G. 2007, A&A, 473, 429 * Freeman (1993) Freeman, K. C. 1993, in IAU Symp. 153, Galactic Bulges, eds. H. Dejonghe and H. J. Habing (Dordrecht: Kluwer), p. 263 * Frogel et al. (1980) Frogel, J. A., Persson, S. E., & Cohen, J. G. 1980, ApJ, 240, 785 * Fusi Pecci et al. (2005) Fusi Pecci, F., Bellazini, M., De Simone, E., & Federici, L. 2005, AJ, 130, 544 * Fusi Pecci et al. (1993) Fusi Pecci, F., Cacciari, C., Federici, L., & Pasquali, A. 1993, in: The Globular Cluster-Galaxy Connection, eds. G. H. Smith and J. P. Brodie, Vol. 48, p. 410 * Galleti et al. (2009) Galleti, S., Bellazzini, M., Buzzoni, A., Federici, L., & Fusi Pecci, F. 2009, A&A, 508, 1285 * Galleti et al. (2007) Galleti, S., Bellazzini, M., Federici, L., Buzzoni, A., & Fusi Pecci, F. 2007, A&A, 471, 127 * Galleti et al. (2005) Galleti, S., Bellazzini, M., Federici, L., & Fusi Pecci, F. 2005, A&A, 436, 535 * Galleti et al. (2006) Galleti, S., Federici, L., Bellazzini, M., Buzzoni, A., & Fusi Pecci, F. 2006, ApJ, 650, L107 * Galleti et al. (2004) Galleti, S., Federici, L., Bellazzini, M., Fusi Pecci, F., & Macrina, S. 2004, A&A, 426, 917 * Harris (1991) Harris, W. E. 1991, ARA&A, 29, 543 * Harris (1996) Harris, W. E. 1996, AJ, 112, 1487 * Harris et al. (2002) Harris, W. E., Harris, G. L. H., Holland, S. T., & McLaughlin, D. E. 2002, AJ, 124, 1435 * Henon (1973) Henon, M. 1973, A&A, 24, 229 * Hodge (1992) Hodge, P. W. 1992, The Andromeda Galaxy (Dordrecht: Kluwer) * Holland (1998) Holland, S. 1998, AJ, 115, 1916 * Hubble (1932) Hubble, E. P. 1932, ApJ, 76, 44 * Huxor et al. (2011) Huxor, A. P., et al. 2011, MNRAS, 414, 770 * Huxor et al. (2008) Huxor, A. P., Tanvir, N. R., Ferguson, A. M. N., Irwin, M. J., Ibata, R. A., Bridges, T., & Lewis, G. F. 2008, MNRAS, 385, 1989 * Huxor et al. (2005) Huxor, A. P., Tanvir, N. R., Irwin, M. J., Ibata, R., Collett, J. L., Ferguson, A. M. N., Bridges, T., & Lewis, G. F. 2005, MNRAS, 360, 1007 * Jiang et al. (2003) Jiang, L., Ma, J., Zhou, X., Chen, J., Wu, H., & Jiang, Z. 2003, AJ, 125, 727 * Kim et al. (2007) Kim, S., et al. 2007, AJ, 134, 706 * King (1962) King, I. 1962, AJ, 67, 471 * King (1966) King, I. 1966, AJ, 71, 64 * Larsen et al. (2002) Larsen, S. S., Brodie, J. P., Sarajedini, A., & Huchra, J. P. 2002, AJ, 124, 2615 * Lightman & Shapiro (1978) Lightman, A. P., & Shapiro, S. L. 1978, Rev. Mod. Phys., 50, 437 * Ma (2011) Ma, J. 2011, RAA, 11, 524 * Ma et al. (2006a) Ma, J., de Grijs, R., Yang, Y., Zhou, X., Chen, J., Jiang, Z., Wu, Z., & Wu, J. 2006a, MNRAS, 368, 1443 * Ma et al. (2006b) Ma, J., et al. 2006b, ApJ, 636, L93 * Ma et al. (2007a) Ma, J., et al. 2007a, ApJ, 659, 359 * Ma et al. (2007b) Ma, J., et al. 2007b, MNRAS, 376, 1621 * Ma et al. (2009a) Ma, J., et al. 2009a, AJ, 137, 4884 * Ma et al. (2009b) Ma, J., et al. 2009b, RAA, 9, 641 * Ma et al. (2011) Ma, J., et al. 2011, AJ, 141, 86 * Mackey & van den Bergh (2005) Mackey, A., & van den Bergh, S. 2005, MNRAS, 360, 631 * Mackey et al. (2006) Mackey, A. D., et al. 2006, ApJ, 653, L105 * Mackey et al. (2007) Mackey, A. D., et al. 2007, ApJ, 655, L85 * Mackey et al. (2010) Mackey, A. D., et al. 2010, MNRAS, 401, 533 * Martin et al. (2005) Martin, D. C., et al. 2006, ApJ, 619, L1 * Martin et al. (2006) Martin, N. F., et al. 2006, MNRAS, 371, 1983 * McClure & Racine (1969) McClure, R. D., & Racine, R. 1969, AJ, 74, 1000 * McConnachie et al. (2005) McConnachie, A. W., Irwin, M. J., Ferguson, A. M. N., Ibata, R. A., Lewis, G. F., & Tanvir, N. 2005, MNRAS, 356, 979 * McLaughlin et al. (2008) McLaughlin, D. E., Barmby, P., Harris, W. E., Forbes, D. A., & Harris, G. L. H. 2008, MNRAS, 384, 563 * McLaughlin & van der Marel (2005) McLaughlin, D. E., & van der Marel, R. P. 2005, ApJS, 161, 304 * Meylan (2002) Meylan, G. 2002, in: Extragalactic Star Clusters, Geisler, D., Grevel, E. K., & Minniti, D., eds., (ASP: San Francisco), IAUS, 207, 555 * Meylan et al. (2001) Meylan, G., Sarajedini, A., Jablonka, P., Djorgovski, S., Bridges, T., & Rich, R. 2001, AJ, 122, 830 * Michie (1963) Michie, R. W. 1963, MNRAS, 125, 127 * Mochejska et al. (1998) Mochejska, B. J., Kaluzny, J., Krockenberger, M., Sasselov, D. D., & Stanek, K. Z. 1998, AcA, 48, 455 * Morrissey et al. (2007) Morrissey, P., et al. 2007, ApJS, 173, 682 * Murphy et al. (1990) Murphy, B. W., Cohn, H. N., & Hut, P. 1990, MNRAS, 245, 335 * Oke & Gunn (1983) Oke, J. B., & Gunn, J. E. 1983, ApJ, 266, 713 * Perina et al. (2009) Perina, S., et al. 2009, A&A, 494, 933 * Perina et al. (2011) Perina, S., Galleti, S., Fusi Pecci, F., Bellazzini, M., Federici, L., & Buzzoni, A. 2011, A&A, 531, 155 * Peacock et al. (2010) Peacock, M., et al. 2010, MNRAS, 402, 803 * Puzia et al. (2005) Puzia, T. H., Perrett, K. M., & Bridges, T. J. 2005, A&A, 434, 909 * Puzia et al. (2002) Puzia, T. H., Zepf, S. E., Kissler-Patig, M., Hilker, M., Minniti, D., & Goudfrooij, P. 2002, A&A, 391, 453 * Racine (1991) Racine, R. 1991, AJ, 101, 865 * Rich et al. (2001) Rich, R. M., Shara, M. M., & Zurek, D. 2001, AJ, 122, 842 * Salpeter (1955) Salpeter, E. E. 1955, ApJ, 121, 161 * Sérsic (1968) Sérsic, J.-L. 1968, Atlas de Galaxias Australes (Cordoba: Observatoria Astronomico) * Sirianni et al. (2005) Sirianni, M., et al. 2005, PASP, 117, 1049 * Skrutskie et al. (2006) Skrutskie, M. F., et al. 2006, AJ, 131, 1163 * Spitzer & Thuan (1972) Spitzer, L., & Thuan, T. X. 1972, ApJ, 175, 31 * Stanek & Garnavich (1998) Stanek, K. Z., & Garnavich, P. M. 1998, ApJ, 503, 131 * Stetson (1987) Stetson, P. B. 1987, PASP, 99, 191 * Trager et al. (1995) Trager, S. C., King, I. R., & Djorgovski, S. 1995, AJ, 109, 218 * van den Bergh (1969) van den Bergh, S. 1969, ApJS, 19, 145 * van den Bergh & Mackey (2004) van den Bergh, S., & Mackey, A. D. 2004, MNRAS, 354, 713 * Wang et al. (2010) Wang, S., Fan, Z., Ma, J., de Grijs, R., & Zhou, X. 2010, AJ, 139, 1438 * Williams & Hodge (2001a) Williams, B. F., & Hodge, P. W. 2001a, ApJ, 548, 190 * Williams & Hodge (2001b) Williams, B. F., & Hodge, P. W. 2001b, ApJ, 559, 851 * Wilson (1975) Wilson, C. P. 1975, AJ, 80, 175 * Yan et al. (2000) Yan, H. J., et al. 2000, PASP, 112, 691 * Zinnecker et al. (1988) Zinnecker, H., Keable, C. J., Dunlop, J. S., Cannon, R. D., & Griffiths, W. K. 1988, in IAU Symp. 126: The Harlow-Shapley Symposium on Globular Cluster Systems in Galaxies, 603 Table 1B514: Ellipticity, $\epsilon$, and position angle (P.A.) as a function of the semimajor axis, $a$, in the F606W filter of HST ACS-WFC $a$ \| $\epsilon$ \| P.A. \| $a$ \| $\epsilon$ \| P.A. ---\|---\|---\|---\|---\|--- (arcsec) \| \| (deg) \| (arcsec) \| \| (deg) 0.0260 \| $0.360\pm 0.082$ \| $159.5\pm 8.3$ \| 0.1752 \| $0.598\pm 0.046$ \| $147.9\pm 3.4$ 0.0287 \| $0.380\pm 0.087$ \| $158.6\pm 8.3$ \| 0.1928 \| $0.150\pm 0.150$ \| $150.4\pm 32.0$ 0.0315 \| $0.402\pm 0.092$ \| $157.7\pm 8.5$ \| 0.2120 \| $0.150\pm 0.183$ \| $107.3\pm 38.4$ 0.0347 \| $0.426\pm 0.098$ \| $156.9\pm 8.6$ \| 0.2333 \| $0.171\pm 0.111$ \| $84.8\pm 20.6$ 0.0381 \| $0.449\pm 0.104$ \| $155.6\pm 8.8$ \| 0.2566 \| $0.203\pm 0.066$ \| $76.4\pm 10.5$ 0.0420 \| $0.474\pm 0.112$ \| $154.3\pm 9.2$ \| 0.2822 \| $0.199\pm 0.049$ \| $79.3\pm 8.0$ 0.0461 \| $0.503\pm 0.123$ \| $153.1\pm 9.7$ \| 0.3105 \| $0.205\pm 0.062$ \| $94.7\pm 9.8$ 0.0508 \| $0.535\pm 0.134$ \| $152.3\pm 10.2$ \| 0.3415 \| $0.210\pm 0.059$ \| $102.4\pm 9.1$ 0.0558 \| $0.570\pm 0.133$ \| $151.6\pm 9.7$ \| 0.3757 \| $0.197\pm 0.059$ \| $112.5\pm 9.5$ 0.0614 \| $0.596\pm 0.137$ \| $150.6\pm 9.7$ \| 0.4132 \| $0.171\pm 0.066$ \| $129.9\pm 12.1$ 0.0676 \| $0.622\pm 0.146$ \| $149.6\pm 10.1$ \| 0.4545 \| $0.090\pm 0.094$ \| $143.8\pm 31.9$ 0.0743 \| $0.649\pm 0.159$ \| $148.8\pm 10.8$ \| 0.5000 \| $0.090\pm 0.071$ \| $20.8\pm 23.6$ 0.0818 \| $0.678\pm 0.134$ \| $147.8\pm 8.8$ \| 0.5500 \| $0.123\pm 0.041$ \| $95.8\pm 10.1$ 0.0899 \| $0.694\pm 0.114$ \| $147.4\pm 7.4$ \| 0.6050 \| $0.102\pm 0.024$ \| $117.4\pm 7.2$ 0.0989 \| $0.686\pm 0.092$ \| $146.9\pm 6.0$ \| 0.6655 \| $0.119\pm 0.024$ \| $98.6\pm 6.2$ 0.1088 \| $0.650\pm 0.075$ \| $145.4\pm 5.0$ \| 0.7321 \| $0.158\pm 0.032$ \| $93.4\pm 6.2$ 0.1197 \| $0.636\pm 0.059$ \| $146.1\pm 4.0$ \| 0.8053 \| $0.067\pm 0.049$ \| $54.6\pm 21.3$ 0.1317 \| $0.619\pm 0.050$ \| $146.6\pm 3.6$ \| 0.8858 \| $0.065\pm 0.034$ \| $4.8\pm 15.5$ 0.1448 \| $0.636\pm 0.037$ \| $146.6\pm 2.7$ \| 0.9744 \| $0.067\pm 0.096$ \| $132.5\pm 42.6$ 0.1593 \| $0.610\pm 0.064$ \| $147.1\pm 4.5$ \| \| \| Table 2B514: Ellipticity, $\epsilon$, and position angle (P.A.) as a function of the semimajor axis, $a$, in the F814W filter of HST ACS-WFC $a$ \| $\epsilon$ \| P.A. \| $a$ \| $\epsilon$ \| P.A. ---\|---\|---\|---\|---\|--- (arcsec) \| \| (deg) \| (arcsec) \| \| (deg) 0.0260 \| $0.296\pm 0.081$ \| $162.5\pm 9.5$ \| 0.6655 \| $0.139\pm 0.026$ \| $103.1\pm 5.6$ 0.0287 \| $0.308\pm 0.084$ \| $161.4\pm 9.5$ \| 0.7321 \| $0.175\pm 0.028$ \| $94.5\pm 5.1$ 0.0315 \| $0.323\pm 0.087$ \| $160.4\pm 9.5$ \| 0.8053 \| $0.116\pm 0.043$ \| $35.2\pm 11.4$ 0.0347 \| $0.342\pm 0.091$ \| $159.5\pm 9.6$ \| 0.8858 \| $0.099\pm 0.044$ \| $11.7\pm 13.4$ 0.0381 \| $0.363\pm 0.097$ \| $158.5\pm 9.7$ \| 0.9744 \| $0.099\pm 0.075$ \| $137.4\pm 22.8$ 0.0420 \| $0.386\pm 0.104$ \| $157.6\pm 9.9$ \| 1.0718 \| $0.074\pm 0.117$ \| $146.5\pm 46.9$ 0.0461 \| $0.412\pm 0.112$ \| $156.6\pm 10.1$ \| 1.1790 \| $0.246\pm 0.144$ \| $111.3\pm 19.3$ 0.0508 \| $0.437\pm 0.117$ \| $155.1\pm 10.2$ \| 1.2969 \| $0.277\pm 0.066$ \| $129.1\pm 8.1$ 0.0558 \| $0.462\pm 0.128$ \| $153.6\pm 10.7$ \| 1.4266 \| $0.134\pm 0.058$ \| $93.4\pm 13.2$ 0.0614 \| $0.488\pm 0.150$ \| $152.5\pm 12.0$ \| 1.5692 \| $0.134\pm 0.062$ \| $41.3\pm 14.2$ 0.0676 \| $0.529\pm 0.146$ \| $152.0\pm 11.1$ \| 1.7261 \| $0.162\pm 0.082$ \| $70.6\pm 16.1$ 0.0743 \| $0.571\pm 0.138$ \| $151.5\pm 10.0$ \| 1.8987 \| $0.086\pm 0.124$ \| $144.4\pm 43.2$ 0.0818 \| $0.606\pm 0.127$ \| $151.0\pm 8.9$ \| 2.0886 \| $0.155\pm 0.073$ \| $0.1\pm 14.8$ 0.0899 \| $0.633\pm 0.129$ \| $150.9\pm 8.8$ \| 2.2975 \| $0.384\pm 0.069$ \| $121.7\pm 6.8$ 0.0989 \| $0.663\pm 0.114$ \| $151.1\pm 7.6$ \| 2.5272 \| $0.131\pm 0.103$ \| $146.7\pm 24.2$ 0.1088 \| $0.697\pm 0.087$ \| $151.0\pm 5.6$ \| 2.7800 \| $0.131\pm 0.074$ \| $161.3\pm 17.5$ 0.1197 \| $0.703\pm 0.077$ \| $148.9\pm 5.0$ \| 3.0580 \| $0.196\pm 0.114$ \| $108.0\pm 18.6$ 0.1317 \| $0.704\pm 0.157$ \| $148.3\pm 10.1$ \| 3.3638 \| $0.130\pm 0.072$ \| $117.6\pm 16.9$ 0.1448 \| $0.702\pm 0.064$ \| $149.1\pm 4.4$ \| 3.7001 \| $0.209\pm 0.075$ \| $124.5\pm 11.4$ 0.1593 \| $0.759\pm 0.055$ \| $149.1\pm 3.6$ \| 4.0701 \| $0.199\pm 0.041$ \| $97.7\pm 6.5$ 0.1752 \| $0.772\pm 0.033$ \| $147.9\pm 2.2$ \| 4.4772 \| $0.067\pm 0.051$ \| $110.7\pm 22.6$ 0.1928 \| $0.726\pm 0.106$ \| $147.9\pm 6.9$ \| 4.9249 \| $0.078\pm 0.048$ \| $94.6\pm 18.3$ 0.2120 \| $0.380\pm 0.149$ \| $150.9\pm 14.4$ \| 5.4174 \| $0.128\pm 0.061$ \| $105.3\pm 14.5$ 0.2333 \| $0.100\pm 0.159$ \| $121.0\pm 48.4$ \| 5.9591 \| $0.011\pm 0.043$ \| $115.7\pm 118.0$ 0.2566 \| $0.125\pm 0.125$ \| $113.3\pm 30.5$ \| 6.5550 \| $0.007\pm 0.059$ \| $65.8\pm 244.4$ 0.2822 \| $0.232\pm 0.125$ \| $116.3\pm 17.7$ \| 7.2105 \| $0.078\pm 0.046$ \| $57.5\pm 17.6$ 0.3105 \| $0.299\pm 0.094$ \| $118.4\pm 10.9$ \| 7.9316 \| $0.096\pm 0.041$ \| $89.4\pm 12.8$ 0.3415 \| $0.265\pm 0.056$ \| $118.4\pm 7.2$ \| 8.7247 \| $0.218\pm 0.042$ \| $89.4\pm 6.1$ 0.3757 \| $0.229\pm 0.064$ \| $119.4\pm 9.1$ \| 9.5972 \| $0.160\pm 0.039$ \| $94.7\pm 7.6$ 0.4132 \| $0.226\pm 0.072$ \| $120.4\pm 10.5$ \| 10.5569 \| $0.333\pm 0.029$ \| $87.5\pm 3.0$ 0.4545 \| $0.146\pm 0.096$ \| $139.6\pm 20.5$ \| 11.6126 \| $0.307\pm 0.023$ \| $87.5\pm 2.6$ 0.5000 \| $0.184\pm 0.246$ \| $110.0\pm 42.6$ \| 12.7738 \| $0.171\pm 0.072$ \| $90.6\pm 13.3$ 0.5500 \| $0.325\pm 0.071$ \| $102.3\pm 7.5$ \| 14.0512 \| $0.234\pm 0.058$ \| $82.6\pm 8.1$ 0.6050 \| $0.127\pm 0.030$ \| $108.9\pm 7.3$ \| 15.4564 \| $0.234\pm 0.064$ \| $82.6\pm 7.4$ Table 3B514: Surface brightness, $\mu$, and integrated magnitude, $m$, as a function of the radius in the F606W filter of HST ACS-WFC $R$ \| $\mu$ \| $m$ \| $R$ \| $\mu$ \| $m$ ---\|---\|---\|---\|---\|--- (arcsec) \| ($\rm{mag/arcsec^{2}}$) \| (mag) \| (arcsec) \| ($\rm{mag/arcsec^{2}}$) \| (mag) 0.0260 \| $16.438\pm 0.034$ \| 22.868 \| 0.6655 \| $18.091\pm 0.055$ \| 17.008 0.0287 \| $16.445\pm 0.038$ \| 22.868 \| 0.7321 \| $18.212\pm 0.071$ \| 16.900 0.0315 \| $16.452\pm 0.043$ \| 22.868 \| 0.8053 \| $18.374\pm 0.080$ \| 16.804 0.0347 \| $16.461\pm 0.048$ \| 22.868 \| 0.8858 \| $18.588\pm 0.079$ \| 16.705 0.0381 \| $16.470\pm 0.054$ \| 22.868 \| 0.9744 \| $18.799\pm 0.086$ \| 16.615 0.0420 \| $16.480\pm 0.062$ \| 22.868 \| 1.0718 \| $18.808\pm 0.134$ \| 16.531 0.0461 \| $16.491\pm 0.070$ \| 22.868 \| 1.1790 \| $18.953\pm 0.289$ \| 16.434 0.0508 \| $16.503\pm 0.080$ \| 21.221 \| 1.2969 \| $19.283\pm 0.175$ \| 16.362 0.0558 \| $16.517\pm 0.090$ \| 21.221 \| 1.4266 \| $19.545\pm 0.170$ \| 16.283 0.0614 \| $16.534\pm 0.102$ \| 21.221 \| 1.5692 \| $19.839\pm 0.141$ \| 16.212 0.0676 \| $16.553\pm 0.114$ \| 21.221 \| 1.7261 \| $19.890\pm 0.243$ \| 16.149 0.0743 \| $16.574\pm 0.126$ \| 20.628 \| 1.8987 \| $20.258\pm 0.141$ \| 16.087 0.0818 \| $16.598\pm 0.135$ \| 20.628 \| 2.0886 \| $20.308\pm 0.232$ \| 16.024 0.0899 \| $16.623\pm 0.146$ \| 20.628 \| 2.2975 \| $20.770\pm 0.147$ \| 15.968 0.0989 \| $16.650\pm 0.159$ \| 20.628 \| 2.5272 \| $20.905\pm 0.220$ \| 15.917 0.1088 \| $16.680\pm 0.156$ \| 20.281 \| 2.7800 \| $21.196\pm 0.162$ \| 15.866 0.1197 \| $16.708\pm 0.148$ \| 19.809 \| 3.0580 \| $21.565\pm 0.252$ \| 15.819 0.1317 \| $16.713\pm 0.143$ \| 19.809 \| 3.3638 \| $21.967\pm 0.094$ \| 15.785 0.1448 \| $16.725\pm 0.132$ \| 19.638 \| 3.7001 \| $22.090\pm 0.274$ \| 15.751 0.1593 \| $16.764\pm 0.117$ \| 19.254 \| 4.0701 \| $22.568\pm 0.117$ \| 15.720 0.1752 \| $16.788\pm 0.102$ \| 19.254 \| 4.4772 \| $22.904\pm 0.102$ \| 15.695 0.1928 \| $16.810\pm 0.090$ \| 19.060 \| 4.9249 \| $23.113\pm 0.115$ \| 15.673 0.2120 \| $16.858\pm 0.075$ \| 18.837 \| 5.4174 \| $23.359\pm 0.147$ \| 15.649 0.2333 \| $16.900\pm 0.054$ \| 18.652 \| 5.9591 \| $23.743\pm 0.096$ \| 15.622 0.2566 \| $16.959\pm 0.049$ \| 18.418 \| 6.5550 \| $24.078\pm 0.101$ \| 15.604 0.2822 \| $17.028\pm 0.057$ \| 18.341 \| 7.2105 \| $24.331\pm 0.099$ \| 15.580 0.3105 \| $17.109\pm 0.074$ \| 18.150 \| 7.9316 \| $24.692\pm 0.082$ \| 15.560 0.3415 \| $17.199\pm 0.076$ \| 17.991 \| 8.7247 \| $25.101\pm 0.116$ \| 15.545 0.3757 \| $17.285\pm 0.066$ \| 17.835 \| 9.5972 \| $25.328\pm 0.118$ \| 15.534 0.4132 \| $17.366\pm 0.060$ \| 17.662 \| 10.5569 \| $25.526\pm 0.127$ \| 15.518 0.4545 \| $17.439\pm 0.064$ \| 17.532 \| 11.6126 \| $25.840\pm 0.164$ \| 15.496 0.5000 \| $17.490\pm 0.074$ \| 17.382 \| 12.7738 \| $26.212\pm 0.170$ \| 15.483 0.5500 \| $17.630\pm 0.072$ \| 17.249 \| 14.0512 \| $26.527\pm 0.176$ \| 15.469 0.6050 \| $17.881\pm 0.058$ \| 17.108 \| 15.4564 \| $26.951\pm 0.233$ \| 15.462 Table 4B514: Surface brightness, $\mu$, and integrated magnitude, $m$, as a function of the radius in the F814W filter of HST ACS-WFC $R$ \| $\mu$ \| $m$ \| $R$ \| $\mu$ \| $m$ ---\|---\|---\|---\|---\|--- (arcsec) \| ($\rm{mag/arcsec^{2}}$) \| (mag) \| (arcsec) \| ($\rm{mag/arcsec^{2}}$) \| (mag) 0.0260 \| $15.810\pm 0.050$ \| 22.224 \| 0.6655 \| $17.332\pm 0.068$ \| 16.278 0.0287 \| $15.819\pm 0.055$ \| 22.224 \| 0.7321 \| $17.451\pm 0.077$ \| 16.166 0.0315 \| $15.828\pm 0.061$ \| 22.224 \| 0.8053 \| $17.601\pm 0.105$ \| 16.068 0.0347 \| $15.839\pm 0.068$ \| 22.224 \| 0.8858 \| $17.837\pm 0.096$ \| 15.966 0.0381 \| $15.850\pm 0.076$ \| 22.224 \| 0.9744 \| $18.048\pm 0.098$ \| 15.875 0.0420 \| $15.862\pm 0.085$ \| 22.224 \| 1.0718 \| $18.007\pm 0.179$ \| 15.791 0.0461 \| $15.874\pm 0.095$ \| 22.224 \| 1.1790 \| $18.194\pm 0.245$ \| 15.686 0.0508 \| $15.888\pm 0.107$ \| 20.601 \| 1.2969 \| $18.447\pm 0.336$ \| 15.613 0.0558 \| $15.903\pm 0.118$ \| 20.601 \| 1.4266 \| $18.755\pm 0.190$ \| 15.533 0.0614 \| $15.917\pm 0.126$ \| 20.601 \| 1.5692 \| $19.082\pm 0.151$ \| 15.460 0.0676 \| $15.931\pm 0.135$ \| 20.601 \| 1.7261 \| $19.078\pm 0.335$ \| 15.394 0.0743 \| $15.947\pm 0.144$ \| 20.013 \| 1.8987 \| $19.444\pm 0.186$ \| 15.328 0.0818 \| $15.965\pm 0.152$ \| 20.013 \| 2.0886 \| $19.540\pm 0.232$ \| 15.261 0.0899 \| $15.982\pm 0.160$ \| 20.013 \| 2.2975 \| $19.965\pm 0.199$ \| 15.203 0.0989 \| $15.999\pm 0.169$ \| 20.013 \| 2.5272 \| $20.101\pm 0.259$ \| 15.150 0.1088 \| $16.015\pm 0.169$ \| 19.656 \| 2.7800 \| $20.402\pm 0.164$ \| 15.096 0.1197 \| $16.034\pm 0.159$ \| 19.170 \| 3.0580 \| $20.718\pm 0.310$ \| 15.045 0.1317 \| $16.024\pm 0.157$ \| 19.170 \| 3.3638 \| $21.153\pm 0.126$ \| 15.012 0.1448 \| $16.022\pm 0.151$ \| 18.989 \| 3.7001 \| $21.253\pm 0.322$ \| 14.975 0.1593 \| $16.055\pm 0.137$ \| 18.589 \| 4.0701 \| $21.778\pm 0.116$ \| 14.944 0.1752 \| $16.072\pm 0.128$ \| 18.589 \| 4.4772 \| $22.117\pm 0.088$ \| 14.920 0.1928 \| $16.078\pm 0.118$ \| 18.384 \| 4.9249 \| $22.297\pm 0.119$ \| 14.898 0.2120 \| $16.120\pm 0.098$ \| 18.153 \| 5.4174 \| $22.484\pm 0.174$ \| 14.874 0.2333 \| $16.171\pm 0.070$ \| 17.959 \| 5.9591 \| $22.916\pm 0.113$ \| 14.845 0.2566 \| $16.225\pm 0.059$ \| 17.715 \| 6.5550 \| $23.213\pm 0.123$ \| 14.826 0.2822 \| $16.287\pm 0.070$ \| 17.639 \| 7.2105 \| $23.470\pm 0.125$ \| 14.800 0.3105 \| $16.367\pm 0.092$ \| 17.441 \| 7.9316 \| $23.880\pm 0.088$ \| 14.778 0.3415 \| $16.454\pm 0.098$ \| 17.275 \| 8.7247 \| $24.263\pm 0.121$ \| 14.763 0.3757 \| $16.558\pm 0.089$ \| 17.116 \| 9.5972 \| $24.457\pm 0.132$ \| 14.752 0.4132 \| $16.646\pm 0.067$ \| 16.944 \| 10.5569 \| $24.607\pm 0.158$ \| 14.735 0.4545 \| $16.708\pm 0.070$ \| 16.814 \| 11.6126 \| $24.935\pm 0.155$ \| 14.710 0.5000 \| $16.740\pm 0.082$ \| 16.659 \| 12.7738 \| $25.294\pm 0.197$ \| 14.697 0.5500 \| $16.878\pm 0.098$ \| 16.524 \| 14.0512 \| $25.649\pm 0.231$ \| 14.681 0.6050 \| $17.137\pm 0.079$ \| 16.380 \| 15.4564 \| $26.034\pm 0.379$ \| 14.674 Table 5B514: Surface brightness profiles $\mu$ from star counts $R$ \| $\mu_{\rm{F606W}}$ \| $R$ \| $\mu_{\rm{F814W}}$ ---\|---\|---\|--- (arcsec) \| ($\rm{mag/arcsec^{2}}$) \| (arcsec) \| ($\rm{mag/arcsec^{2}}$) \| \| 7.9625 \| $23.831\pm 0.066$ 9.1875 \| $24.961\pm 0.072$ \| 9.1875 \| $24.181\pm 0.072$ 10.4125 \| $25.297\pm 0.079$ \| 10.4125 \| $24.517\pm 0.079$ 11.6375 \| $25.622\pm 0.087$ \| 11.6375 \| $24.842\pm 0.087$ 12.8625 \| $25.905\pm 0.095$ \| 12.8625 \| $25.125\pm 0.095$ 14.0875 \| $26.361\pm 0.111$ \| 14.0875 \| $25.581\pm 0.111$ 15.3125 \| $26.694\pm 0.125$ \| 15.3125 \| $25.914\pm 0.125$ 16.5375 \| $26.998\pm 0.138$ \| 16.5375 \| $26.218\pm 0.138$ 17.7625 \| $27.310\pm 0.154$ \| 17.7625 \| $26.530\pm 0.154$ 18.9875 \| $27.360\pm 0.152$ \| 18.9875 \| $26.580\pm 0.152$ 20.2125 \| $27.517\pm 0.158$ \| 20.2125 \| $26.737\pm 0.158$ 21.4375 \| $27.703\pm 0.168$ \| 21.4375 \| $26.923\pm 0.168$ 22.6625 \| $27.816\pm 0.172$ \| 22.6625 \| $27.036\pm 0.172$ 23.8875 \| $28.050\pm 0.186$ \| 23.8875 \| $27.270\pm 0.186$ 25.1125 \| $28.240\pm 0.198$ \| 25.1125 \| $27.460\pm 0.198$ 26.3375 \| $28.292\pm 0.198$ \| 26.3375 \| $27.512\pm 0.198$ 27.5625 \| $28.630\pm 0.226$ \| 27.5625 \| $27.850\pm 0.226$ 28.7875 \| $28.725\pm 0.231$ \| 28.7875 \| $27.945\pm 0.231$ 30.0125 \| $28.722\pm 0.226$ \| 30.0125 \| $27.942\pm 0.226$ 31.2375 \| $28.864\pm 0.237$ \| 31.2375 \| $28.084\pm 0.237$ 32.4625 \| $29.201\pm 0.271$ \| 32.4625 \| $28.421\pm 0.271$ 33.6875 \| $29.242\pm 0.271$ \| 33.6875 \| $28.462\pm 0.271$ 34.9125 \| $29.350\pm 0.280$ \| 34.9125 \| $28.570\pm 0.280$ 36.1375 \| $29.543\pm 0.301$ \| 36.1375 \| $28.763\pm 0.301$ 37.3625 \| $29.761\pm 0.327$ \| 37.3625 \| $28.981\pm 0.327$ 38.5875 \| $30.014\pm 0.362$ \| 38.5875 \| $29.234\pm 0.362$ 39.8125 \| $30.321\pm 0.410$ \| 39.8125 \| $29.541\pm 0.410$ 41.0375 \| $30.208\pm 0.384$ \| 41.0375 \| $29.428\pm 0.384$ Table 6 Structural parameters of B514 Parameters \| F606W \| F814W ---\|---\|--- $r_{0}$ \| $0.36^{+0.09}_{-0.05}~{}\rm{arcsec}~{}(=1.35^{+0.35}_{-0.19}~{}\rm{pc})$ \| $0.36^{+0.09}_{-0.06}~{}\rm{arcsec}~{}(=1.35^{+0.32}_{-0.22}~{}\rm{pc})$ $r_{t}$ \| $16.08^{+2.11}_{-1.35}~{}\rm{arcsec}~{}(=61.11^{+8.00}_{-5.14}~{}\rm{pc})$ \| $16.79^{+1.74}_{-1.48}~{}\rm{arcsec}~{}(=63.78^{+6.62}_{-5.64}~{}\rm{pc})$ $c=\log(r_{t}/r_{0})$ \| $1.66^{+0.05}_{-0.04}$ \| $1.68^{+0.04}_{-0.04}$ $r_{h}$ \| $1.31^{+0.14}_{-0.08}~{}\rm{arcsec}~{}(=5.00^{+0.55}_{-0.32}~{}\rm{pc})$ \| $1.36^{+0.13}_{-0.09}~{}\rm{arcsec}~{}(=5.17^{+0.48}_{-0.34}~{}\rm{pc})$ $\mu_{0}$ (${\rm mag~{}arcsec^{-2}}$) \| $16.25^{+0.57}_{-0.56}$ \| $15.64^{+0.80}_{-0.64}$ Table 7BATC photometry of B514 Filter \| Central wavelength \| Bandwidth \| Number of images \| Exposure time \| rms \| Magnitude ---\|---\|---\|---\|---\|---\|--- \| (Å) \| (Å) \| \| (hours) \| (mag) \| $a$ \| 3360 \| 222 \| 6 \| 2:00 \| 0.010 \| $17.59\pm 0.05$ $b$ \| 3890 \| 187 \| 6 \| 2:00 \| 0.010 \| $16.82\pm 0.02$ $c$ \| 4210 \| 185 \| 4 \| 1:00 \| 0.002 \| $16.52\pm 0.01$ $d$ \| 4550 \| 222 \| 4 \| 1:20 \| 0.015 \| $16.25\pm 0.02$ $e$ \| 4920 \| 225 \| 3 \| 1:00 \| 0.007 \| $16.05\pm 0.01$ $f$ \| 5270 \| 211 \| 3 \| 1:00 \| 0.014 \| $15.85\pm 0.02$ $g$ \| 5795 \| 176 \| 3 \| 1:00 \| 0.010 \| $15.64\pm 0.01$ $h$ \| 6075 \| 190 \| 3 \| 0:50 \| 0.005 \| $15.56\pm 0.01$ $i$ \| 6660 \| 312 \| 3 \| 0:50 \| 0.004 \| $15.44\pm 0.01$ $j$ \| 7050 \| 121 \| 3 \| 1:00 \| 0.006 \| $15.33\pm 0.01$ $k$ \| 7490 \| 125 \| 3 \| 1:00 \| 0.011 \| $15.25\pm 0.01$ $m$ \| 8020 \| 179 \| 3 \| 1:00 \| 0.003 \| $15.19\pm 0.01$ $n$ \| 8480 \| 152 \| 3 \| 1:00 \| 0.005 \| $15.12\pm 0.01$ Table 82MASS and $GALEX$ photometry of B514 Filter \| Magnitude ---\|--- $J$ \| $14.23\pm 0.07$ $H$ \| $13.32\pm 0.06$ $K_{s}$ \| $13.63\pm 0.10$ NUV \| $19.57\pm 0.86$ FUV \| $20.44\pm 2.30$ Table 9Mass estimates (and uncertainties) of B514 based on the BC03 models $V$ \| $u$ \| $g$ \| $r$ \| $i$ \| $z$ \| $J$ \| $H$ \| $K_{\rm s}$ ---\|---\|---\|---\|---\|---\|---\|---\|--- \| Mass $(10^{6}~{}M_{\odot})$ \| $1.08\pm 0.03$ \| $0.99\pm 0.03$ \| $1.06\pm 0.03$ \| $1.02\pm 0.03$ \| $1.0\pm 0.03$ \| $0.98\pm 0.03$ \| $0.98\pm 0.06$ \| $1.41\pm 0.08$ \| $0.96\pm 0.09$	arxiv-papers	2011-11-10T01:33:41	2024-09-04T02:49:24.186116	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Jun Ma (1,2), Song Wang (1,3), Zhenyu Wu (1), Zhou Fan (1), Tianmeng\n Zhang (1) and Jianghua Wu (1) ((1) National Astronomical Observatories,\n Chinese Academy of Sciences, (2) Key Laboratory of Optical Astronomy,\n National Astronomical Observatories, Chinese Academy of Sciences, Beijing,\n China, (3) Graduate University of Chinese Academy of Sciences, Shijingshan\n District, Beijing, China)", "submitter": "Jun Ma", "url": "https://arxiv.org/abs/1111.2380" }
1111.2620	# Search for the rare decays $B^{0}_{(s)}\to\mu^{+}\mu^{-}$ at LHCb CPPM, Aix-Marseille Université, CNRS/IN2P3, Marseille, France E-mail On behalf of the LHCb collaboration ###### Abstract: A search for the $B_{s}\to\mu^{+}\mu^{-}$ and $B_{d}\to\mu^{+}\mu^{-}$ decays is presented using $\sim 300$ pb-1 of $pp$ collisions at $\sqrt{s}$ = 7 TeV collected by the LHCb experiment at the Large Hadron Collider at CERN. The measured upper limit for the branching ratio of the $B_{s}\to\mu^{+}\mu^{-}$ decay is $\cal B$($B_{s}\to\mu^{+}\mu^{-}$)$<1.3\ (1.6)\times 10^{-8}$ at 90 % (95 %) confidence level (CL), while in the case of the $B_{d}\to\mu^{+}\mu^{-}$ decay the measured upper limit is $\cal B$($B_{d}\to\mu^{+}\mu^{-}$)$<4.2\ (5.1)\times 10^{-9}$ at 90 % (95 %) CL. A combination with the 2010 dataset results in $\cal B$($B_{s}\to\mu^{+}\mu^{-}$)$<1.2\ (1.5)\times 10^{-8}$ at 90 % (95 %) CL. ## 1 Introduction Measurements at low energies may provide interesting indirect constraints on the masses of particles that are too heavy to be produced directly. This is particularly true for Flavour Changing Neutral Currents (FCNC) processes which are highly suppressed in the Standard Model (SM) and can only occur through higher order diagrams. The SM prediction for the branching ratios ($\cal B$) of the FCNC decays $B_{s}\to\mu^{+}\mu^{-}$ and $B_{d}\to\mu^{+}\mu^{-}$ have been computed to be $(3.2\pm 0.2)\times 10^{-9}$ and $(0.10\pm 0.01)\times 10^{-9}$ respectively [1]. However New Physics (NP) contributions can significantly enhance these values. The best published limits from the Tevatron at $95\%$ CL are obtained using 6.1 fb-1 by the D0 collaboration [2], and using 2 fb-1 by the CDF collaboration [3]. The CDF collaboration has also presented a preliminary result [4] with 6.9 fb-1 in which an excess of $B_{s}\to\mu^{+}\mu^{-}$candidates is reported, compatible with a $\cal B$($B_{s}\to\mu^{+}\mu^{-}$)= $(1.8^{+1.1}_{-0.9})\times 10^{-8}$. The LHCb collaboration has previously obtained the limits $\cal B$($B_{s}\to\mu^{+}\mu^{-}$)$<5.4\times 10^{-8}$ and $\cal B$($B_{d}\to\mu^{+}\mu^{-}$)$<1.5\times 10^{-8}$ at 95$\%$ CL based on 37 pb-1 of luminosity collected in the 2010 run [5]. We present here a measurement based on 300 pb-1 of integrated luminosity collected between March and June 2011. ## 2 Analysis strategy The general structure of the analysis is similar to the one described in Ref. [5] and is detailed in Ref. [6]. The selection procedure treats signal and control/normalization channels in the same way in order to minimize the systematic uncertainties. Assuming the SM branching ratio and the $b\bar{b}$ cross-section, measured within the LHCb acceptance, of $\sigma_{b\overline{b}}=75\pm 14\,\mu$b [7], approximately $3.4~{}(0.32)$ $B^{0}_{s}\to\mu^{+}\mu^{-}$ ($B^{0}\to\mu^{+}\mu^{-}$) events are expected to be reconstructed and selected in the analysed sample. After the selection, each event is given a probability to be signal or background in a two-dimensional space defined by two independent likelihoods: the invariant mass and the output of a Boosted Decision Tree (BDT) from the TMVA package [8]. The combination of variables entering the BDT is optimized using Monte Carlo (MC) simulation. The following variables have been used: the $B$ lifetime, impact parameter and transverse momentum of the $B$, the minimum impact parameter significance of the muons, the distance of closest approach between the two muons, the degree of isolation of the two muons with respect to any other track in the event, the cosine of the polarization angle, the $B$ isolation and the minimum $p_{T}$ of the muons. The BDT distribution is then transformed in order to be flat for the signal and peaked at 0 for the background. The calibration of the invariant mass and the BDT likelihoods are obtained from data using control samples. The signal BDT shape is obtained from $B^{0}_{(s)}\to h^{+}h^{-}$ events free from trigger biases while the background shape is obtained using sideband $B^{0}_{(s)}\to\mu^{+}\mu^{-}$ candidates. The resulting distributions are shown in Fig. 1. Figure 1: BDT calibration for signal and background. The parameters describing the invariant mass line shape of the signal are extracted from data using control samples. The average mass values are obtained from $B^{0}\to K^{+}\pi^{-}$ and $B^{0}_{s}\to K^{+}K^{-}$ exclusive samples. The $B^{0}_{s}$ and $B^{0}$ mass resolutions are estimated by interpolating the ones obtained with the dimuon resonances ($J/\psi,\psi(2S)$ and $\Upsilon(1S,2S,3S)$) and cross-checked via a fit to the invariant mass distribution of the $B^{0}_{(s)}\to h^{+}h^{-}$ inclusive decays and of the $B^{0}\to K^{+}\pi^{-}$ exclusive decay. The interpolation yields $\sigma(B)=24.6\pm 0.2_{\rm stat}\pm 1.0_{\rm syst}$. The number of expected signal events is obtained by normalizing to channels of known branching ratios, $B^{+}\\!\to J/\psi K^{+}$, $B^{0}_{s}\\!\to J/\psi\phi$, and $B^{0}\\!\to K^{+}\pi^{-}$, that are selected in a way as similar as possible to the signal. The probability for a background event to have a given BDT and invariant mass value is obtained by a fit of the mass distribution of events in the mass sidebands, in bins of BDT. Different fit functions and mass ranges are used to compute the systematics uncertainties. The two-dimensional space formed by the invariant mass and BDT is binned, and for each bin we compute how many events are observed in data, how many signal events are expected for a given $\cal B$ hypothesis and luminosity, and how many background events are expected for a given luminosity. The compatibility of the observed distribution of events in all bins with the expected one for a given $\cal B$ hypothesis is computed using the CLs method [9], which allows to exclude a given hypothesis at a given confidence level. In order to avoid unconscious biases, the data in the mass region defined by $M_{B^{0}}-60\,{\rm MeV}/c^{2}$ and $M_{B^{0}_{s}}+60\,{\rm MeV}/c^{2}$ have been blinded until the completion of the analysis. ## 3 Results Figure 2: Distribution of selected dimuon events in the invariant mass vs BDT plane. The orange short-dashed (green long-dashed) lines indicate the $\pm 60\,{\rm MeV}/c^{2}$ search window around the $B^{0}_{s}$ ($B^{0}$). The distribution of events in the invariant mass versus BDT plane is reported Fig. 2. The expected limit at 90 (95) % CL for the $B_{s}\to\mu^{+}\mu^{-}$ is $0.8~{}(1.0)\times 10^{-8}$ in the case of background only hypothesis. When adding signal events according to the SM branching fraction, these limits become $1.2~{}(1.5)\times 10^{-8}$. The observed values for the $B_{s}\to\mu^{+}\mu^{-}$channel is $1.3~{}(1.6)\times 10^{-8}$ with a CLb value of 0.80. The observed events are in good agreement with the background expectations and the presence of $B_{s}\to\mu^{+}\mu^{-}$events according to SM predictions. For the $B_{d}\to\mu^{+}\mu^{-}$, the expected limit at 90 (95) % CL is $2.4~{}(3.1)\times 10^{-9}$ in the case of background only hypothesis. The observed values is $4.2~{}(5.2)\times 10^{-9}$ with a CLb value of 0.79. The comparison of the observed distribution of events with the expected background distribution results in a p-value (1-$\textrm{CL}_{\textrm{b}}$) of 20 % (21 %) for the $B_{s}\to\mu^{+}\mu^{-}$ ($B_{d}\to\mu^{+}\mu^{-}$) decays. In the case of $B_{d}\to\mu^{+}\mu^{-}$, the slightly low p-value is due to an excess of the observed events in the most sensitive BDT bin with respect to the background expectations. A larger data sample will allow to clarify the situation. In the case of $B_{s}\to\mu^{+}\mu^{-}$, when a signal is included at the level expected in the Standard Model, the p-value increases to 50 %. Finally, the $B_{s}\to\mu^{+}\mu^{-}$ limit is combined with the one published from the 2010 data to obtain $\cal B$($B_{s}\to\mu^{+}\mu^{-}$) ¡ $1.2~{}(1.5)\times 10^{-8}$ at 90 % (95 %) CL. This 90 % CL upper limit is still 3.8 times above the standard model prediction. ## References * [1] A.J. Buras, G. Isidori and P. Paradisi, “EDMs vs CPV in $B_{s,d}$ mixing in two Higgs doublet models with MFV”, arXiv:1007.5291 [hep-ph] (2010). * [2] V. Abazov et al. [D0 Collaboration], “Search for the rare decay $B^{0}_{s}\to\mu^{+}\mu^{-}$”, Phys. Lett. B 693 (2010) 539. * [3] T. Aaltonen et al. [CDF Collaboration], “Search for $B_{s}\to\mu^{+}\mu^{-}$and $B_{d}\to\mu^{+}\mu^{-}$Decays with 2$\mbox{\,fb}^{-1}$ of $p\bar{p}$ Collisions” Phys. Rev. Lett. 100 (2008) 101802. * [4] T. Aaltonen et al. [CDF Collaboration], “Search for $B_{s}\to\mu^{+}\mu^{-}$and $B_{d}\to\mu^{+}\mu^{-}$Decays with CDF-II” , arXiv:1107.2304v1 [hep-ex] (2011). * [5] R. Aaji et al. [LHCb Collaboration], “Search for the rare decays $B^{0}_{s}\to\mu^{+}\mu^{-}$ and $B^{0}\to\mu^{+}\mu^{-}$”, Phys. Lett. B 699 (2011) 330. * [6] R. Aaji et al. [LHCb Collaboration], LHCb-CONF-2011-037. * [7] R. Aaij et al. [LHCb Collaboration], “Measurement of $\sigma$(pp $\rightarrow b\bar{b}$ X) at $\sqrt{s}$=7 TeV in the forward region”, Phys. Lett. B 694 (2010) 209. * [8] TMVA, Toolkit for Multi Variate analysis with ROOT, http://tmva.sourceforge.net. * [9] A. Read, “Presentation of Search Results: The $\textrm{CL}_{\textrm{s}}$ Technique”, J. Phys. G 28 (2002) 2693.	arxiv-papers	2011-11-10T21:38:38	2024-09-04T02:49:24.201240	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Justine Serrano", "submitter": "Justine Serrano", "url": "https://arxiv.org/abs/1111.2620" }
1111.2724	Temperature anisotropy and differential streaming of solar wind ions – Correlations with transverse fluctuations Sofiane Bourouaine1, Eckart Marsch1 and Fritz M. Neubauer2 1 Max-Planck-Institut für Sonnensystemforschung, 37191 Katlenburg-Lindau, Germany 2 Institut für Geophysik und Meteorologie, Universität zu Köln, Albertus-Magnus-Platz, Köln, 50923, Germany Abstract. We study correlations of the temperature ratio (which is an indicator for perpendicular ion heating) and the differential flow of the alpha particles with the power of transverse fluctuations that have wave numbers between $0.01$ and $0.1$ (normalized to $k_{p}=1/l_{p}$, where $l_{p}$ is the proton inertial length). We found that both the normalized differential ion speed, $V_{\alpha p}/V_{\mathrm{A}}$ (where $V_{\mathrm{A}}$ is the Alfvén speed) and the proton temperature anisotropy, $T_{\perp p}/T_{\parallel p}$, increase when the relative wave power is growing. Furthermore, if the normalized differential ion speed stays below 0.5, the alpha-particle temperature anisotropy, $T_{\perp\alpha}/T_{\parallel\alpha}$, correlates positively with the relative power of the transverse fluctuations. However, if $V_{\alpha p}/V_{\mathrm{A}}$ is higher than 0.6, then the alpha-particle temperature anisotropy tends to become lower and attain even values below unity despite the presence of transverse fluctuations of relatively high amplitudes. Our findings appear to be consistent with the expectations from kinetic theory for the resonant interaction of the ions with Alfvén/ion-cyclotron waves and the resulting wave dissipation. ## 1 Introduction The in-situ measurements of protons and heavier ions in fast solar wind revealed distinct non-thermal kinetic features, such as the core temperature anisotropy and beam of the protons, or the preferential heating and acceleration of alpha particles and other minor species (see, e.g., the review of Marsch (2006) on this subject). Similarly, solar remote-sensing observations of coronal holes, which are known as the sources of the fast solar wind provided evidence via significant broadenings of ultraviolet emission lines, for strong perpendicular heating of oxygen and heavy ions in the solar corona (see, e.g., the review of Cranmer (2009)). These observed kinetic features have usually been interpreted as signatures of heating by ion-cyclotron waves (Isenberg et al. 2001; Marsch and Tu 2001; Hollweg and Isenberg 2002; Matteini et al. 2007). Their origin remains unclear, though, and is still a matter of debate. Some authors suggested that a nonlinear parallel cascade (via parametric decay) of low-frequency Alfvén waves may ultimately generate ion-cyclotron waves, since their numerical simulations showed that protons and heavy ions can be heated perpendicularly by wave absorption through cyclotron resonance (Araneda et al. 2008, 2009). Moreover, it has been argued theoretically that in low-beta plasma condition, the ion temperature anisotropy and the preferential (stochastic) heating of heavy species could even be caused by Alfvénic fluctuations with frequencies well below the local ion-cyclotron frequency (Wu and Yoon 2007; Bourouaine et al. 2008; Chandran 2010). On the other hand, it has also been argued that the energy of Alfvénic turbulence does not cascade effectively to high-frequency parallel cyclotron waves, but rather is transferred into low-frequency and highly oblique kinetic Alfvén waves. Then dissipation would still take place at the proton gyroscale, but via Landau damping acting mostly on electrons (Howes et al. 2008). In this letter, while benefiting from the high-resolution magnetic field data and the detailed proton and alpha-particle velocity distribution functions (VDFs) obtained (Marsch et al. 1982) simultaneously by the Helios 2 wave and plasma experiments, we analysed typical solar wind data from a heliocentric distance of about 0.7 AU. While studying them in detail we found that the prominent kinetic features of the alpha particles and protons are closely correlated with the power of the transverse high-frequency waves. The results of our study place new empirical constraints on solar wind models and further limit the theoretical assumptions involved. We will first present a statistical analysis of the relevant plasma and field parameters, and then discuss them in the light of kinetic plasma theory. ## 2 Observations Here we analyse plasma and magnetic field data provided by the Helios 2 spacecraft in 1976. For the purpose of a solid statistical study, we selected a week of continuous measurements, i.e., the days with DOY (day of year) numbers 67 to 73, on which the spacecraft was at a solar distance of about 0.7 AU. As is obvious from Figure 1, the selected set covers data from measurements made in slow and fast solar wind. The slightly inclined dashed line in that figure separates the data obtained in fast solar wind from those in slow wind. As we would expect, the domain with a relatively high number of collisions mainly corresponds to slow solar wind (with $A_{c}>0.2$), and that with a low number of collisions corresponds to fast solar wind which is weakly collisional (with $A_{c}<0.1$). Here collisionality is quantified by the so- called collision age (see e.g., Salem et al. 2003), $A_{c}=l/(V_{\mathrm{sw}}\tau_{\alpha p})$, where, $l$ is the distance from the sun and $\tau_{\alpha p}$ is the collision-exchange time between helium ions and protons.Moreover, for this data set the proton parallel plasma beta, $\beta_{\parallel p}$, is comparatively small and mainly varies between 0.1 and 0.7. Because the solar wind speed is clearly higher than the Alfvén speed, we can safely assume the so-called Taylor hypothesis to be valid. Therefore, the spacecraft frequency of the magnetic field fluctuations (obtained from fast Fourier transform) is simply given by, $2\pi f_{\mathrm{sc}}\simeq kV_{\mathrm{sw}}$, where $V_{\mathrm{sw}}$ is the solar wind speed, and $k$ is the wave number of the magnetic field fluctuations. Here, we deal with the average wave power density, $\delta B^{2}$, which is obtained in the spacecraft frame by integration over a frequency range that corresponds to the wave-number range (0.01–0.1)$k_{p}$ (where $k_{\rho}=1/l_{p}$, $l_{p}$ is the proton inertial length). Therefore, we are dealing with turbulent fluctuations that still belong to the inertial range but are near the dissipation range of the turbulence. We have chosen this frequency domain to avoid the inclusion of higher-frequency fluctuations that might stem from local wave excitation (e.g., owing to a large temperature anisotropy or high normalized ion differential speed) below but near the proton gyro-kinetic scale. Consistently with our choice of this narrow frequency range, we define the mean field (required as the reference value for the superposed fluctuations) by the average of the full magnetic-field vector over the short time period of about one hundred times $l_{p}/V_{\mathrm{sw}}$. Figure 1: Distributions of the number of data (presented in colour with the coding indicated by the right-hand bar), which are plotted as a function of the collision age, $A_{c}$, and the solar wind speed, $V_{sw}$. The inclined dashed line separates the data from fast and slow solar wind. Figure 2: Distributions (presented in colour with coding indicated by the right-hand bars) of the number of our data (a) and the compressibility, $\delta B_{\parallel}^{2}/(\delta B_{\perp}^{2}+\delta B_{\parallel}^{2})$ (b), plotted in the parameter plane of the normalized wave power versus the proton temperature anisotropy, $T_{\perp p}/T_{\parallel p}$. (c) Weighted mean value of the proton temperature anisotropy for $A_{c}<0.1$ (black), and $A_{c}>0.2$ (green) displayed versus normalized wave power. The short vertical bars indicate the small uncertainties of the weighted mean values. In Figure 2 a the distribution of the normalized wave power, $\delta B^{2}/B^{2}_{0}$, is plotted versus the proton temperature anisotropy, $T_{\perp p}/T_{\parallel p}$. It turns out that the latter parameter ranges between 0.4 and 2 and correlates positively with the normalized wave power. According to Figure 2 b, the relative compressive component (coded in colour) of the fluctuations is fairly small and not higher than $5\%$ of the overall wave amplitude, which means that the fluctuations are mainly transverse and essentially incompressible. It also appears that there is no striking correlation between magnetic compressibility and normalized wave power. Unfortunately, we do not have the simultaneous small-scale velocity- fluctuations owing to the lack of an adequate time resolution of the plasma experiment, and therefore we cannot scrutinize the nature of the magnetic fluctuations or confirm by polarization study whether they are Alfvénic or not. But presumably they are Alfvén waves, because these waves with periods longer than 40 s are observed widely in fast solar wind (Tu and Marsch 1995) streams at all distances between 0.3 and 1 AU. In Figure 2c we plot the weighted mean value (expectation value) of the proton temperature anisotropy as a function of the normalized wave power for different solar wind regimes. In fast solar wind (i.e., where $A_{c}<0.1$, black curve), the proton temperature anisotropy is below unity whenever the value of the normalized wave power is below 0.01. However, above 0.01 the temperature anisotropy steeply increases until the normalized wave power reaches a value of about 0.1, where the temperature anisotropy tends to saturate at a value of about 1.5. In contrast, when collisions are relatively strong (i.e., where $A_{c}>0.2$, green curve), the proton temperature anisotropy also increases with the wave amplitude but reaches lower values than those when collisions are weak. The temperature anisotropy ranges between 0.6 and about 1.1, while the normalized wave power varies between 0.01 and 0.1. Our study shows that the proton temperature anisotropy correlates positively with the normalized wave power.Moreover, a positive correlation between the wave power and the proton temperature anisotropy was found earlier by Bourouaine et al. (2010), but for another set of Helios data restricted to fast solar wind and at various solar distances. However, the Helios measurements selected for the present study were made at a fixed heliocentric distance which is about 0.7 AU. Therefore, the new correlation established here between the proton temperature anisotropy and normalized wave power is certainly of local nature and not affected by radial evolution. According to Figure 2c, it seems that collisions play a major role in reducing the effects waves have on the local heating of the protons. Therefore, protons can become heated (if wave energy is available) more easily in regions with relatively weak collisions than in regions where collisions are relatively strong. Figure 3 a shows the data distribution of the normalized wave power versus the normalized ion differential speed. It hardly exceeds unity, a result which is consistent with the prediction of kinetic theory for a linear plasma instability. The theory says that, whenever the ion differential speed exceeds the Alfvén speed, the plasma should become unstable and excite magnetosonic waves (see, e.g., Li and Habbal (2000)). As Bourouaine et al. (2011) found previously, the present study also indicates a positive correlation between the normalized ion differential speed and the normalized wave power (see the white curve, which represents the weighted mean values of the normalized wave power shown in Figure 3 a. ) Figure 3: Distributions (presented in colour with coding indicated by the right-hand bars) of the number of our data (a), the collision age, $A_{c}$ (b) and the ratio between the temperature anisotropies of alpha particles and protons, $(T_{\perp\alpha}T_{\parallel p})/(T_{\parallel\alpha}T_{\perp p})$, (c). All parameter are plotted in the plane of the normalized wave-power, $\delta B^{2}/B_{0}^{2}$, versus normalized ion differential speed, $V_{\alpha p}/V_{\mathrm{A}}$; (d) Weighted mean values of the collisional age, $A_{c}$, and the ratio between the temperature anisotropies of alpha particles and protons, $(T_{\perp\alpha}T_{\parallel p})/(T_{\parallel\alpha}T_{\perp p})$, displayed versus normalized ion differential speed, $V_{\alpha p}/V_{\mathrm{A}}$. The white dots in panel (a) represent the weighted mean values. The vertical dotted line in panel (d) separates the regimes of fast and slow solar wind. It is commonly believed (see, e.g., the review of Marsch (2006)) that low collisional friction would permit a relatively high differential speed to occur between the two main ion species in the solar wind. This notion is confirmed by the results of our Figure 3 b, which shows that higher values of the normalized ion differential speed correspond to lower collision ages. In slow solar wind, when the value of the collision age is higher than 0.2, the corresponding normalized differential speed is low (i.e., $V_{\alpha p}/V_{A}<0.3$), but it is higher than 0.3 for comparatively low values of the collision age (as is indicated later in Figure 3 d). In Figure 3c the coloured pixels represent the ratio of the alpha-particle-to- proton temperature anisotropy, $(T_{\perp\alpha}T_{\parallel p})/(T_{\perp p}T_{\parallel\alpha})$, plotted as a function of the relative ion differential speed and the normalized wave power. Interestingly, this figure clearly shows that when $V_{\alpha p}/V_{A}\leq 0.4$, the temperature anisotropy of the alpha particles, $T_{\perp\alpha}/T_{\parallel\alpha}$, is higher than the anisotropy of the protons, $T_{\perp p}/T_{\parallel p}$. However, the ratio of the ion temperature anisotropies tends to decrease to lower values of about 0.6 when $V_{\alpha p}/V_{\mathrm{A}}>0.6$, as is quantitatively shown in Figure 3 d. The curve in Figure 3d, which represents the weighted mean value of the ratio of the ion temperature anisotropies (black symbols), clearly indicates that alpha particles are preferentially heated (perpendicularly to the mean magnetic field) with respect to the protons whenever $V_{\alpha p}/V_{\mathrm{A}}\leq 0.4$, and this is true even for a relatively low wave energy (indicated by the white curve in Figure 3a) and at high collision rates (green symbols). One would expect that the plasma tends to thermal equilibrium, in coincidence with the lowest values of the normalized ion differential flow speed, if a relatively high collision rate. Then the temperature ratios of the ion species should also be near unity. However, observationally it seems that preferential perpendicular heating of the alpha particles with respect to the protons can persist even in regions where collision rates are high (with $V_{\alpha p}/V_{A}\leq 0.4$). This is possible because a wave-related local ion heating mechanism may be acting on time scales much lower than the long cumulative collision time. Such a fast wave-heating mechanism can drive the plasma far away from thermal equilibrium, and therefore may cause a significant ion temperature anisotropy, because on the other side collisions are not fast enough to enforce thermal equilibrium. We found in our previous paper (Bourouaine et al. (2011)) that the helium ion abundance for this selected data set varies mainly between 0.02 and 0.04. The helium abundance does not show a clear dependence on the normalized differential ion speed. However, we showed that there is an anti-correlation between the alpha-to-proton temperature ratio and the helium abundance at a fixed $V_{\alpha p}/V_{\mathrm{A}}$. Figure 4: Left ordinate: The mean values of the temperature anisotropy of the protons, $T_{\perp p}/T_{\parallel p}$ (red dots), and of the alpha particles, $T_{\perp\alpha}/T_{\parallel\alpha}$ (black diamonds). Right ordinate: Mean normalized wave power (blue squares). The bars indicate the uncertainties of the mean values. All quantities are plotted in bins versus the relative ion differential speed, $V_{\alpha p}/V_{\mathrm{A}}$. The vertical dotted line separates fast from slow solar wind regimes. Figure 4 is a plot of the mean values of the alpha-particle temperature anisotropy, $T_{\perp\alpha}/T_{\parallel\alpha}$, the proton temperature anisotropy, $T_{\perp p}/T_{\parallel p}$, and the mean relative wave power versus $V_{\alpha p}/V_{\mathrm{A}}$. The mean values are obtained by averaging the data within bins of a width $\Delta(V_{\alpha p}/V_{\mathrm{A}})=0.1$, and the vertical error bars indicate the related uncertainties of the mean values. As mentioned above, it turns out that the proton temperature ratio is strictly correlated with the normalized wave power, which indicates perpendicular heating of the protons whenever the power of the transverse fluctuations is enhanced. An interesting behaviour of the temperature ratio of the alpha particles can be inferred from Figure 4. This ratio increases with increasing normalized wave power as long as the normalized ion differential speed stays below about 0.5. Beyond this value of $V_{\alpha p}/V_{\mathrm{A}}$, the alpha-particle temperature ratio becomes roughly constant, until $V_{\alpha p}/V_{\mathrm{A}}$ exceeds a value of about 0.7, but then it decreases towards a value below unity when $V_{\alpha p}/V_{\mathrm{A}}$ reaches one. In the slow solar wind region, where the collisions are expected to be relatively high, the proton temperature anisotropy is ranging between 0.8 and 1, however, the alpha temperature anisotropy is not in the same range but higher, and varies between 0.9 and 1.2. Most likely strong collisionality at low $V_{\alpha p}/V_{\mathrm{A}}$ tends to isotropize the alphas, whereas the weakening of the resonance at higher alpha/proton speeds leads to a decreasing of the alpha anisotropy, as in Gary et al. (2005). The monotonic increase of the proton temperature anisotropy with the normalized differential ion speed appears to be consistent with the findings of Kasper et al. (2008) and with the ACE observations of Gary et al. (2005)). However, there is clear difference between the trend of the alpha temperature anisotropy in Figure 4 and the results found by Kasper et al. (2008). In their paper, the alpha temperature anisotropy reaches a minimum value at $V_{\alpha p}\sim 0.5V_{\mathrm{A}}$, whereas our Figure 3 shows that the alpha temperature anisotropy reaches its maximum value when $V_{\alpha p}\sim 0.5V_{\mathrm{A}}$. Moreover, the results of Gary et al. (2005) show that the average alpha temperature anisotropy is monotonically decreasing with increasing alpha/proton relative speed. Unlike what Kasper et al. (2008) and Gary et al. (2005) found previously, our Figure 4 shows that the perpendicular heating is reduced when $V_{\alpha p}/V_{A}$ is near zero. This reduced ion heating corresponds to an observed concurrent decrease in the wave power of the transverse waves. This results would be expected when the heating ultimately rests with the energy in those waves. If $V_{\alpha p}/V_{A}$ is near zero, we expect that the alpha-particle temperature ratio increases resulting in a strong perpendicular ion heating for sufficient wave power. However, although the wave power is empirically found to be weak when the normalized ion differential speed is low, the alpha particles are still heated perpendicularly much more than the protons. For both ion species the interaction with transverse waves is expected to work against the radial trend caused by the solar wind expansion in a magnetic mirror, which tends to build up a much larger parallel than perpendicular temperature, and accordingly a strong fire-hose-type anisotropy. ## 3 Discussion and conclusions If we assume that the transverse fluctuations provide the energy input for the observed preferential perpendicular heating of the alpha particles, then their temperature-ratio profile as given in Figure 4 could be a signature of cyclotron-wave heating of the alpha particles. It has been claimed that the alpha particles can only be heated through ion- cyclotron wave dissipation if the differential speed between alpha-particles and protons is approximately less than $0.5V_{A}$ (see e.g., Gary et al. 2005). Therefore, alpha-particles can be heated in the perpendicular direction as long as they stay in resonance with the ion-cyclotron waves. Moreover, the wave energy is an important input parameter that controls the heating of alpha-particles. We expect that high wave energy can cause strong perpendicular heating of those alpha particles that are in resonance with ion- cyclotron waves. Figure 4 shows that the perpendicular heating is reduced when $V_{\alpha p}/V_{\mathrm{A}}$ is near zero. This reduced ion heating corresponds to a decrease in the observed wave power of the transverse fluctuations, a result that is expected because the potential for heating ultimately rests with the energy contained in those waves. If $V_{\alpha p}/V_{\mathrm{A}}$ is near zero, we expect that the alpha-particle temperature ratio increases, resulting in strong perpendicular ion heating for sufficient wave power. However, although the wave power is found empirically to be weak when the normalized ion differential speed is small, the alpha particles can still be perpendicularly heated much more than the protons. Yet, for both ion species the interaction with transverse fluctuations works against the trend caused by the solar wind expansion in a magnetic-field mirror configuration, which tends to build up a high parallel temperature anisotropy. Furthermore, Coulomb collisions tend to thermalize the solar wind plasma and to reduce the differential speed between alpha particles and protons. But collisions are effective in removing the non-thermal ion features merely in the comparatively cold and dense slow solar wind, in which the collision age is found to be high and the average wave power observed to be weak. Another possible scenario that may occur as well is that the long-wavelength and high-amplitude fluctuations of the inertial-range turbulence may stochastically heat (Chandran 2010) the ions in the perpendicular direction with respect to the background magnetic field (or non-resonantly drive a slowly varying $T_{\perp}/T_{\parallel}$ on both ion species). The presence of this anisotropy can give rise to Alfvén-cyclotron instabilities, which lead to the growth of relatively high-frequency modes (with frequencies $\omega\sim\Omega_{p}$, where $\Omega_{p}$ is the proton cyclotron frequency). In other words, the ion temperature anisotropies could first be caused by low- frequency fluctuations in the inertial range, and then this thermal energy may be exchanged between ions and waves at the proton kinetic scale. ## References * Araneda et al. (2009) Araneda, J. A., Y. Maneva, and E. Marsch (2009), Preferential Heating and Acceleration of $\alpha$ Particles by Alfvén-Cyclotron Waves, Phys. Rev. Lett. 102, 175001 * Araneda et al. (2008) Araneda, J. A., E. Marsch, and A. F.-Viñas (2008), Proton Core Heating and Beam Formation via Parametrically Unstable Alfvén-Cyclotron Waves, Phys. Rev. Lett. 100, 125003 * Bourouaine et al. (2011) Bourouaine, S., E. Marsch, and F. M. Neubauer (2011), On the relative speed and temperature ratio of solar wind alpha particles and protons: Collisions versus wave effects, ApJ. 728, L3 * Bourouaine et al. (2008) Bourouaine, S., E. Marsch, and C. Vocks (2008), On the Efficiency of Nonresonant Ion Heating by Coronal Alfvén Waves, ApJ. 684, L119 * Bourouaine et al. (2010) Bourouaine, S., E. Marsch, and F. M. Neubauer (2010), Correlations between the proton temperature anisotropy and transverse high-frequency waves in the solar wind, Geophys. Res. Lett., 37, L14104 * Chandran (2010) Chandran, Benjamin D. G. (2010), Alfvén Wave Turbulence and Perpendicular Ion Temperatures in Coronal Holes, ApJ, 720, 548 * Cranmer (2009) Cranmer, S. R., (2009) Coronal Holes, Living Rev. Solar Phys, 6, 3 * Gary et al. (2005) Gary, P. S, C. W. Smith, and R. M. Skoug (2005), Signatures of Alfvén-cyclotron wave-ion scattering: Advanced Composition Explorer (ACE) solar wind observations, J. Geophys. Res. 110, A07 108 * Howes et al. (2008) Howes, G. G. W. Dorland, S. C. Cowley, G. W. Hammett, E. Quataert, A. A. Schekochihin, and T. Tatsuno (2008), Kinetic Simulations of Magnetized Turbulence in Astrophysical Plasmas, Phys. Rev. Lett., 100, 065004 * Matteini et al. (2007) Matteini, L., S. Landi, P. Hellinger, F. Pantellini, M. Maksimovic, M. Velli, B. E. Goldstein, and E. Marsch (2007), Evolution of the solar wind proton temperature anisotropy from 0.3 to 2.5 AU, Geophys. Res. Lett., 34, L20105 * Hollweg and Isenberg (2002) Hollweg, Joseph V. and Isenberg, Philip A (2002), Generation of the fast solar wind: A review with emphasis on the resonant cyclotron interaction, J. Geophys. Res. 107 (A7), 1147 * Isenberg et al. (2001) Isenberg, Philip A., Lee, Martin A. and Hollweg, Joseph V (2001), The kinetic shell model of coronal heating and acceleration by ion cyclotron waves 1. Outward propagating waves, J. Geophys. Res. 106, 5649 * Kasper et al. (2008) Kasper, J. C., A. J. Lazarus, and S. P. Gary (2008), Hot Solar-Wind Helium: Direct Evidence for Local Heating by Alfvén-Cyclotron Dissipation, Phys. Rev. Lett. 101, 261103 * Li and Habbal (2000) Li. X and S. R. Habbal (2000), Proton/alpha magnetosonic instability in the fast solar wind, Geophys. Res. 105, 7483 * Marsch et al. (1982) Marsch, E., R. Schwenn, H. Rosenbauer, K. H. Mühlhäuser, W. Pilipp, and F. M. Neubauer (1982b), Solar wind protons - Three-dimensional velocity distributions and derived plasma parameters measured between 0.3 and 1 AU, J. Geophys. Res. 87, 52 * Marsch and Tu (2001) Marsch, E., and C.-Y. Tu (2001), Heating and acceleration of coronal ions interacting with plasma waves through cyclotron and Landau resonance, J. Geophys. Res. 106, 227 * Marsch (2006) Marsch, E. (2006), Kinetic Physics of the Solar Corona and Solar Wind, Living Rev. Solar Phys. 3, 1 * Salem et al. (2003) Salem, C., D. Hubert, C. Lacombe, S. D. Bale, A. Mangeney, D. E. Larson , and R. P. Lin (2003), Electron Properties and Coulomb Collisions in the Solar Wind at 1 AU: Wind Observations, ApJ. 585, 1147 * Tu and Marsch (1995) Tu, C.-Y. and E. Marsch (1995), MHD structures, waves and turbulence in the solar wind: Observations and theories, Space Science Rev. 73, 1 * Wu and Yoon (2007) Wu, C. S., and P. H. Yoon (2007), Proton Heating via Nonresonant Scattering Off Intrinsic Alfvénic Turbulence, Phys. Rev. Lett. 99, 075001	arxiv-papers	2011-11-11T12:40:14	2024-09-04T02:49:24.208178	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "S. Bourouaine, E. Marsch and F. M. Neubauer", "submitter": "Sofiane Bourouaine", "url": "https://arxiv.org/abs/1111.2724" }
1111.2881	# Semileptonic decays of the Higgs boson at the Tevatron MisterX Joseph D. Lykken , Adam O. Martin Theoretical Physics Department Fermi National Accelerator Laboratory, Batavia, IL 60510, USA Jan Winter PH-TH, CERN, CH-1211 Geneva 23, Switzerland lykken@fnal.govaomartin@fnal.govjwinter@cern.ch ###### Abstract We examine the prospects for extending the Tevatron reach for a Standard Model Higgs boson by including the semileptonic Higgs boson decays $h\to WW\to\ell\nu_{\ell}\,jj$ for $M_{h}\gtrsim 2\,M_{W}$, and $h\to~{}Wjj\to\ell\nu_{\ell}\,jj$ for $M_{h}\lesssim 2\,M_{W}$, where $j$ is a hadronic jet. We employ a realistic simulation of the signal and backgrounds using the SHERPA Monte Carlo event generator. We find kinematic selections that enhance the signal over the dominant $W$+jets background. The resulting sensitivity could be an important addition to ongoing searches, especially in the mass range $120\lesssim M_{h}\lesssim 150\ \mathrm{GeV}$. The techniques described can be extended to Higgs boson searches at the Large Hadron Collider. FERMILAB-PUB-11-499-T CERN-PH-TH-2011-237 ###### Contents 1. 1 Introduction 2. 2 Strategy and key observables 3. 3 Inclusive cross sections and event generation 1. 3.1 Standard Model Higgs boson production and decay 2. 3.2 Relevant background processes 1. 3.2.1 $W$ boson plus jets background 3. 3.3 Monte Carlo simulation of signal and backgrounds using SHERPA 1. 3.3.1 Generation of signal events 2. 3.3.2 Generation of background events for $W$ boson plus jets production 3. 3.3.3 Generation of background events for electroweak and top-pair production 4. 4 Signal versus background studies based on Monte Carlo simulations using SHERPA 1. 4.1 Baseline selection 2. 4.2 Higgs boson reconstruction based on invariant masses 1. 4.2.1 Reconstruction below the on-shell diboson mass threshold 2. 4.2.2 Effect of the subdominant backgrounds 3. 4.3 More realistic Higgs boson reconstruction methods 1. 4.3.1 More realistic reconstruction below the on-shell diboson mass threshold 4. 4.4 Optimized selection – analyses refinements and (further) significance improvements 5. 5 Conclusions, caveats, and prospects 6. A Appendix: Monte Carlo event generation 1. A.1 Leading-order cross sections 2. A.2 NLO calculations versus CKKW ME+PS merging 7. B Appendix: Analysis side studies and additional material 1. B.1 Ideal Higgs boson reconstruction analyses 2. B.2 More realistic Higgs boson reconstruction analyses 3. B.3 Directions for additional improvements ## 1 Introduction The Standard Model (SM) predicts a neutral Higgs boson particle whose couplings to other particles are proportional to the particle masses, and that couples to photons and gluons via one-loop-generated effective couplings. While the Higgs boson mass is not predicted, the relation between the Higgs boson mass and its width is fixed from the predicted couplings. Virtual Higgs boson contributions to electroweak precision observables have been computed, and precision data favor $M_{h}<169\ \mathrm{GeV}$ at 95% confidence level [1]. Searches at the CERN Large Hadron Collider have produced 95% confidence level exclusion of a SM Higgs boson for a broad mass range above 145 GeV [2, 3]. Because of small signal cross sections and large backgrounds, the search for the Higgs boson in experiments at the Fermilab Tevatron is very challenging, even with the large final datasets approaching $10\ \mathrm{fb}^{-1}$ per experiment. Nevertheless both the CDF and DØ experiments have achieved steady improvements in their sensitivities in multiple channels to a SM Higgs boson, and their individual results already exclude a SM Higgs boson in the mass range $156.7$–$173.8\ \mathrm{GeV}$ and $162$–$170\ \mathrm{GeV}$, respectively, at 95% confidence level [4, 5, 6]. This exclusion relies mainly on sensitivity to the dilepton final-state decay chain analyzed in Refs. [7, 8, 9] : $h\to W^{+}W^{-}\\!\to\ell^{+}\nu_{\ell}\;\ell^{-}\bar{\nu}_{\ell}$ where $\ell^{\pm}=e^{\pm}$ or $\mu^{\pm}$. Here we examine the prospects for extending the Tevatron reach by including a search for the semileptonic Higgs boson decays $h\to WW\to\ell\nu_{\ell}\;jj$ for $M_{h}\gtrsim 2\,M_{W}$, and $h\to Wjj\to\ell\nu_{\ell}\;jj$ for $M_{h}\lesssim 2\,M_{W}$, where $j$ is a hadronic jet. This process was first considered as a potential Higgs boson discovery channel for the SSC [10, 11, 12, 13], emphasizing the case of a very heavy Higgs boson, where the $h\to ZZ\to 4\,\ell$ “golden mode” becomes limited by its small branching fraction and the broad Higgs boson width. Similar to the golden mode, the semileptonic $h\to WW$ modes are (almost) fully reconstructible: assuming that the leptonic $W$ is close to on-shell, the mass constraint gives an estimate of the unmeasured longitudinal momentum of the neutrino, up to a two-fold ambiguity [13]. For $M_{h}\gtrsim 140\ \mathrm{GeV}$ the overall decay rate is 6 times larger than any other SM Higgs boson decay mode with a triggerable lepton. Including these semileptonic channels thus offers the distinct possibility of significantly extending the Tevatron reach over a rather broad mass range. This channel suffers from large backgrounds from SM processes with a leptonically decaying $W$ boson. These include diboson production, top quark production, and direct inclusive $W$+2-jet production. There is also a purely QCD background that is difficult to estimate absent a dedicated analysis with data. The dominant background is inclusive $W$+2-jets; from this background alone we have estimated a signal to background ratio ($S/B$) of $3\times 10^{-4}$, after nominal preselections. Though worrisome, this is not smaller than the analogous $S/B\simeq 4\times 10^{-5}$ for the $e^{+}e^{-}$ and $\mu^{+}\mu^{-}$ modes after preselection in the successful Tevatron analyses of $h\to W^{+}W^{-}\\!\to\ell^{+}\nu_{\ell}\;\ell^{-}\bar{\nu}_{\ell}$ [14, 15, 16, 17, 18, 19]. A drastic reduction in both the $W$+2-jet and diboson backgrounds to semileptonic Higgs boson decay can be achieved by forward jet tagging, i.e. by restricting to Higgs boson production from vector boson fusion (VBF) [10, 20]; it is estimated that the additional requirement of forward jet tagging then gives a factor of $\sim$ 100 reduction in these backgrounds. However the reduction in the Higgs signal, versus inclusive Higgs boson production, is also severe: a factor of $\sim$ 10 at the Tevatron [21]. Looking at the similar trade-off for the dilepton $h\to W^{+}W^{-}\\!\to\ell^{+}\nu_{\ell}\;\ell^{-}\bar{\nu}_{\ell}$ channel, a Tevatron study [22] concluded that the overall sensitivity does not improve by restricting to VBF Higgs boson versus inclusive Higgs boson production. We do not know of any comparable analysis for the semileptonic channel. For inclusive Higgs boson production at the Tevatron, the semileptonic channels were first studied by Han and Zhang [7, 8]. In a parton-level study with some jet smearing they found that, after basic acceptance cuts together with a veto on extra energetic jets designed to suppress the $t\bar{t}$ background, the remaining background is completely dominated by $W$+2-jets. Han and Zhang then made additional kinematic selections that enhance the signal to background ratio $S/B$. For $M_{h}=140\ (160)\ \mathrm{GeV}$ they thus obtained a significance estimate of $S/\sqrt{B}=1.0\ (3.3)$ for $30\ \mathrm{fb}^{-1}$ of integrated Tevatron luminosity. The fully differential Higgs boson decay width for this process was exhibited by Dobrescu and Lykken [23], who analyzed the basic kinematics and angular distributions that characterize the Higgs signal. We improve on these studies by including realistic parton showering (since parton-level jet smearing is inadequate), an NLO-rate improved treatment of the Higgs boson decays (including off-shell effects), and a resummed NNLO estimate of the $gg\to h$ production cross section. The first two improvements are incorporated by the use of SHERPA [24, 25], a general purpose showering Monte Carlo program, for simulation of both the signal and the inclusive $W$+2-jets background. The NNLO signal cross section is modeled by a $K$-factor. Our purpose is to study these semileptonic Higgs boson decay channels in a systematic way, but not to mimic a fully-optimized experimental analysis. The DØ experiment has already reported on a semileptonic Higgs boson search using $5.4\ \mathrm{fb}^{-1}$ of Tevatron data [26, 27]; this analysis uses multivariate decision trees to enhance the significance of the result. Here we will limit ourselves to simple cuts, in order to make the features of the analysis and the underlying physics more explicit. We study the Higgs signal in the mass range $110$–$220\ \mathrm{GeV}$ to reasonably cover the below, near and above threshold regions for Higgs boson decay to two on-shell $W$ bosons. In Section 2 we outline the strategy and define several useful observables. In Section 3 we discuss inclusive Higgs boson production from the dominant gluon–gluon fusion mechanism, and implement a $K$-factor correction to the SHERPA result. In Section 4 we introduce basic preselections and develop cuts implemented in SHERPA to enhance $S/\sqrt{B}$; this section also contains our main results. We conclude in Section 5 with caveats about the limitations of our analysis and suggestions for further improvements. Cross-checks and additional material are presented in the appendices. ## 2 Strategy and key observables We are interested in the Higgs boson decay $h\to W^{}W^{}\to e\nu_{e}\;jj\ ,$ (1) and the similar decay with a muon in the final state. In general we take both $W$ bosons off shell. We will write $e\nu_{e}\,jj$ as $e\nu_{e}\,jj^{\prime}$ where $j$ is the jet with higher transverse momentum ($p_{T}$), and noting that physical observables will be symmetric under $j\leftrightarrow j^{\prime}$. We use a baseline selection adopted from the DØ analysis to define reconstructed jets and leptons and impose realistic acceptance cuts. We will assume that events with more than one reconstructed lepton are vetoed, but we want to allow the possibility of extra jets in order to increase signal efficiency. When more than two jets are present there is a combinatorial problem; we will define a Higgs boson candidate selection algorithm that assigns which two jets to use in the Higgs boson reconstruction; these jets may or may not correspond to the two leading jets in the event. It is important to note that this algorithm is chosen so as to optimize the signal sensitivity after the full selection, which is not equivalent to maximizing the number of correctly reconstructed signal events. When the leptonically decaying $W$ boson is (close to) on-shell, these decays are fully reconstructible up to a two-fold ambiguity in the neutrino momentum without making any assumption about the Higgs boson mass. Here we are assuming that the transverse momentum of the neutrino is well estimated given a measurement of the missing transverse energy (MET), as has been demonstrated by both Tevatron experiments in the determination of the $W$ boson mass. The semileptonic channel’s advantage of being, in principle, completely reconstructible offers a great way to separate signal from backgrounds. However, when the leptonically decaying $W$ boson is far off shell, a straightforward full reconstruction is not possible. There are then three generic possibilities for how to proceed: > * • > > Use only transverse observables. > > * • > > Perform an approximate event-by-event reconstruction using an estimate of > the off-shell $W$ boson mass. > > * • > > Perform an approximate event-by-event reconstruction using a (hypothesized) > Higgs boson mass constraint. > > Since it is not clear a priori which of these approaches maximizes the Higgs boson sensitivity, we will pursue all three and compare the results. Given an event-by-event approximate combinatorial full reconstruction of the putative decaying Higgs boson, one can approximately reproduce the kinematics in the Higgs boson rest frame. The true Higgs boson rest frame is given by a longitudinal boost from the lab frame together with a transverse boost defined by the transverse momentum $p_{T,h}$ of the Higgs boson. An explicit representation for the four-momenta in the Higgs boson rest frame is given by: $\displaystyle p_{e}$ $\displaystyle=$ $\displaystyle\frac{1}{2}\;m_{e\nu_{e}}\biggl{(}\gamma_{e\nu_{e}}(1+\beta_{e\nu_{e}}{\rm cos}\,\theta_{\ell}),\;{\rm sin}\,\theta_{\ell}\,{\rm cos}\,\varphi_{\ell},\;{\rm sin}\,\theta_{\ell}\,{\rm sin}\,\varphi_{\ell},\;\gamma_{e\nu_{e}}(\beta_{e\nu_{e}}+{\rm cos}\,\theta_{\ell})\biggr{)}\;,$ $\displaystyle p_{\nu_{e}}$ $\displaystyle=$ $\displaystyle\frac{1}{2}\;m_{e\nu_{e}}\biggl{(}\gamma_{e\nu_{e}}(1-\beta_{e\nu_{e}}{\rm cos}\,\theta_{\ell}),\;-{\rm sin}\,\theta_{\ell}\,{\rm cos}\,\varphi_{\ell},\;-{\rm sin}\,\theta_{\ell}\,{\rm sin}\,\varphi_{\ell},\;-\gamma_{e\nu_{e}}(\beta_{e\nu_{e}}-{\rm cos}\,\theta_{\ell})\biggr{)}\;,$ $\displaystyle p_{j}$ $\displaystyle=$ $\displaystyle\frac{1}{2}\;m_{jj^{\prime}}\biggl{(}\gamma_{jj^{\prime}}(1+\beta_{jj^{\prime}}{\rm cos}\,\theta_{j}),\;{\rm sin}\,\theta_{j},\;0,\;-\gamma_{jj^{\prime}}(\beta_{jj^{\prime}}+{\rm cos}\,\theta_{j})\biggr{)}\;,$ $\displaystyle p_{j^{\prime}}$ $\displaystyle=$ $\displaystyle\frac{1}{2}\;m_{jj^{\prime}}\biggl{(}\gamma_{jj^{\prime}}(1-\beta_{jj^{\prime}}{\rm cos}\,\theta_{j}),\;-{\rm sin}\,\theta_{j},\;0,\;-\gamma_{jj^{\prime}}(\beta_{jj^{\prime}}-{\rm cos}\,\theta_{j})\biggr{)}\;,$ (2) where we have chosen the dijet plane to coincide to the $x$–$z$ plane, and have chosen the positive $z$-axis to be the direction of the leptonically decaying $W$ boson. The boost factors of the two $W$ bosons relative to the Higgs boson rest frame are given by $\displaystyle\gamma_{jj^{\prime}}$ $\displaystyle=$ $\displaystyle\frac{M_{h}}{2\,m_{jj^{\prime}}}\left(1+\frac{m_{jj^{\prime}}^{2}-m_{e\nu_{e}}^{2}}{M_{h}^{2}}\right)\;,$ $\displaystyle\gamma_{e\nu_{e}}$ $\displaystyle=$ $\displaystyle\frac{M_{h}}{2\,m_{e\nu_{e}}}\left(1-\frac{m_{jj^{\prime}}^{2}-m_{e\nu_{e}}^{2}}{M_{h}^{2}}\right)\;,$ (3) and we note the identities $\displaystyle M_{h}$ $\displaystyle=$ $\displaystyle m_{jj^{\prime}}\,\gamma_{jj^{\prime}}+m_{e\nu_{e}}\,\gamma_{e\nu_{e}}\;,$ $\displaystyle m_{jj^{\prime}}\,\beta_{jj^{\prime}}\,\gamma_{jj^{\prime}}$ $\displaystyle=$ $\displaystyle m_{e\nu_{e}}\,\beta_{e\nu_{e}}\,\gamma_{e\nu_{e}}\;.$ (4) Note that $\theta_{j}$ is the angle between jet $j$ and the direction of the hadronic $W$ boson, as seen in the $W$ rest frame, while $\theta_{e}$ is the angle between the charged lepton and the direction of the leptonic $W$ boson as seen in the $W$ rest frame. The azimuthal angle $\varphi_{e}$ is the angle between the dilepton and dijet planes. Defining $\displaystyle r_{jj^{\prime}}$ $\displaystyle=$ $\displaystyle\beta_{jj^{\prime}}^{2}\,\gamma_{jj^{\prime}}^{2}\,{\rm sin}^{2}\theta_{j}\ ,$ (5) we can calculate the angle $\theta_{jj^{\prime}}$ between the two jets as seen in the Higgs boson rest frame: $\displaystyle{\rm cos}\,\theta_{jj^{\prime}}$ $\displaystyle=$ $\displaystyle\frac{r_{jj^{\prime}}-1}{r_{jj^{\prime}}+1}\ .$ (6) Signal events have a minimum opening angle between the jets as seen in the Higgs boson rest frame: $\displaystyle-1\;\leq\;{\rm cos}\,\theta_{jj^{\prime}}\;\leq\;2\,\beta_{jj^{\prime}}^{2}-1\ .$ (7) Figure 1: Examples of observables discriminating between Higgs signal and (the dominant $W$+jets) backgrounds. Distributions are shown from the upper left to the lower right, for the reconstructed masses of the $\\{e,\nu_{e},j,j^{\prime}\\}$ final states, their transverse momenta, the boost of the dijet subsystem with respect to the parent system and the scalar sum of the $p_{T}$ of the two jets $j$ and $j^{\prime}$. All results are obtained after the combinatorial Higgs boson candidate selection based on an ideal mass reconstruction of the potential resonance. The selection is facilitated by symmetric mass window constraints for both the 4-object and dijet mass: $\|m_{e\nu_{e}jj^{\prime}}-M_{h}\|<\Delta$ and $\|m_{jj^{\prime}}-M_{W}\|<\delta$, respectively. More detailed explanations will follow in Section 4. Note that the background differential cross sections are scaled down by the respective factors indicated in the legends. In the approximate reconstructions that we will employ in our analysis, the Higgs boson mass $M_{h}$ is approximated by a 4-object invariant mass $m_{e\nu_{e}jj^{\prime}}$. The transverse momentum of the Higgs boson is approximated by the 4-object transverse momentum $p_{T,e\nu_{e}jj^{\prime}}$. The dijet boost $\gamma_{jj^{\prime}}$ defined in the Higgs boson rest frame is approximated by $\gamma_{jj^{\prime}\|e\nu_{e}}$, which is the dijet boost defined in the 4-object rest frame, in which we can compute this boost factor via $\gamma_{jj^{\prime}\|e\nu_{e}}=E_{jj^{\prime}}/m_{jj^{\prime}}$. This is equivalent to using Eqs. (2) where one inserts for each invariant mass its reconstructed counterpart. All three of these observables discriminate between Higgs signal and backgrounds. As seen in Figure 1 (top left), $m_{e\nu_{e}jj^{\prime}}$ for signal events peaks strongly near the true Higgs boson mass, with a width determined primarily by parton shower effects. Thus a simple mass window selection significantly enhances the signal, and since we are interested in Higgs boson exclusion there is no “look-elsewhere” effect associated with imposing mass windows [28]. Note that the backgrounds are not necessarily flat in the mass windows: as seen in Figure 1 the dominant $W$+jets background is rather flat in the high mass window, but is steeply rising in the lower mass window because of the underlying kinematics. In Figure 1 (top right) one sees that the 4-object transverse momentum $p_{T,e\nu_{e}jj^{\prime}}$ has a harder spectrum for Higgs signal events than for the $W$+jets backgrounds, independent of the Higgs boson mass. The reconstructed dijet boost $\gamma_{jj^{\prime}\|e\nu_{e}}$ has qualitatively different behaviour depending on the underlying Higgs boson mass. When the Higgs boson mass is close to $2\,M_{W}$, the distribution of the dijet boost for signal events is strongly peaked near one compared to the distribution for $W$+jets, as seen in Figure 1 (lower left). For larger Higgs boson masses the signal distribution of the dijet boost is instead rather strongly peaked around the value $M_{h}/(2\,M_{W})$, as expected from Eq. (2). Other physical observables of interest for signal versus background discrimination are defined directly in the lab frame. This includes the 4-object pseudo-rapidity $\eta_{e\nu_{e}jj^{\prime}}$, the pseudo-rapidity difference of the two jets, $\Delta\eta_{j,j^{\prime}}$, and the scalar sum of the two selected jet transverse momenta $H_{T,jj^{\prime}}$. As seen in Figure 1 (lower right), the distribution of $H_{T,jj^{\prime}}$ for signal events is harder than for the $W$+jets background, independent of the Higgs boson mass. From the distributions shown in Figure 12 one sees that the dijet pseudo- rapidity difference has a maximum at zero for signal events (which tend to be confined to the central region), but peaks at a larger value for the $W$+jets background. Similarly from Figure 13 one notes that the 4-object pseudo- rapidity distribution is more central for signal than for the $W$+jets background. We will also employ various transverse masses, both as signal discriminators and as inputs to the algorithms for the approximate reconstruction of the Higgs boson candidates. One class of $m_{T}$ observables is solely constructed out of transverse degrees of freedom, $\vec{p}_{T,i}=(p_{x,i},p_{y,i})$; we define these $m_{T}$ observables as $m_{T,ij..}\;=\;\sqrt{(\lvert\vec{p}_{T,i}\rvert+\lvert\vec{p}_{T,j}\rvert+\ldots)^{2}-(\vec{p}_{T,i}+\vec{p}_{T,j}+\ldots)^{2}}\;\leq\;m_{ij..}\ ,$ (8) where, for the purposes of this study, the labels $ij..$ refer to two, three, or four final-state physics objects (charged lepton, MET, and the two selected jets). We also investigate how our results change when we adopt a slightly different definition that includes the full information from the invariant mass of the visible subset: $\left(m^{(k..)}_{T,ik..l..}\right)^{2}\;=\;m^{2}_{il..}\,+\;2\,\bigl{(}E_{T,il..}\,p_{T,k..}-\vec{p}_{T,il..}\cdot\vec{p}_{T,k..}\bigr{)}\;\leq\;m_{ik..l..}\ ,$ (9) where $E^{2}_{T,il..}=p^{2}_{T,il..}+m^{2}_{il..}$. Here, we have separated the event into a “visible” ($il..$) and “invisible” ($k..$) part. The transverse masses are all approximately bounded from above by a kinematic edge; this gives us another handle when fully reconstructing the event. Schematically, we have $m_{T,i[k..]l..}\;\leq\;m^{(k..)}_{T,ik..l..}\;\leq\;m_{ik..l..}$ (10) where $m_{T,i[k..]l..}$ just indicates that in the presence of multiple invisible objects the $[k..]$ subsystem enters as a whole when computing Eq. (8). In the 2-particle case, the two transverse-mass definitions coincide provided the single objects are massless. Given the above arsenal of kinematic discriminators and approximate reconstruction techniques, our basic strategy will be to find the most promising combinations of selections as a function of the Higgs boson mass. Since we are only performing a cut and count analysis, and are lacking a realistic detector description, there is no point in attempting a complete optimization. Instead we will concentrate on providing a comprehensive look at the physics that distinguishes signal from background. ## 3 Inclusive cross sections and event generation We use the multi-purpose Monte Carlo event generator SHERPA [24, 25] to pursue our analysis of semileptonic Higgs boson decays in Higgs boson production via gluon fusion. This way we can easily include all (hard and soft) initial-state radiation and final-state radiation (ISR and FSR) effects and arrive at a fairly realistic description of final states as used for detector simulations. Furthermore, sophisticated cuts can be implemented in a straightforward manner owing to the convenient analysis features that come with the SHERPA package. We also want to make use of SHERPA’s capabilities in providing an enhanced modeling of multi-jet final states with respect to a treatment by parton showers only. Apart from a handful of key processes, the SHERPA Monte Carlo program evaluates cross sections at leading order/tree level utilizing its integrated automated matrix-element generators AMEGIC++ [29] and/or COMIX [30]. However, in a number of studies, SHERPA has been shown to generate predictions that are in sufficient, often good agreement with the shapes of kinematic distributions obtained from measurements as well as higher-order calculations; we will give more details in the respective subsections that follow up. Hence, we wish to identify appropriate constant $K$-factors between the most accurate theory results and the leading-order predictions. We in turn want to apply these $K$-factors to correct SHERPA’s predictions for the inclusion of exact higher-order rate effects. Therefore, we study signal and background fixed-order and resummed cross sections at Tevatron Run II energies for the processes $P\bar{P}\to\ell\nu_{\ell}\;pp$ (11) leading to final states consisting of an isolated lepton, missing transverse energy and at least two jets. The label $\ell$ denotes electrons, $e$, and muons, $\mu$; the parton label $p$ contains light/massless-quark flavours and gluons. Note that final-state gluons may only occur in background hard processes or through the inclusion of I/FSR effects. For the respective $K$-factors, it is not clear a priori at which level of cuts they are defined most accurately. The most convenient definition should be given in terms of the total inclusive cross sections, while one may define more exclusive $K$-factors, if the higher-order tools allow for the specification of the desired cuts. We do not expect a strong dependence on the exact $K$-factor definition provided the shapes are comparable. These issues are examined in more detail below with the goal of determining reasonable signal and background $K$-factors that can be used to rescale the respective leading- order cross sections $\sigma^{(0)}$ of the SHERPA predictions, which we take to pursue our signal versus background studies. ### 3.1 Standard Model Higgs boson production and decay The signal processes for the final states of interest are summarized by $P\bar{P}\to h\to W^{(\ast)}W^{(\ast)}\to\ell\nu_{\ell}\;pp$ (12) where the Higgs particles are produced through gluon–gluon fusion and decay into $W$ boson pairs that split further into the desired semileptonic final states. There are other Higgs boson production mechanisms that can contribute. In particular, the associated production $P\bar{P}\to Vh$ with the additional vector boson $V$ decaying hadronically and the production via vector-boson fusion (VBF) ought to be mentioned in this context. For all production channels, up-to-date theory predictions for the total inclusive cross sections of these events are needed to arrive at reliable acceptance estimates for various Higgs boson masses. Refs. [4, 31, 32] give the most recent overview of the theory calculations and results that are used as input for the ongoing Tevatron (and LHC) SM Higgs boson searches. For the Higgs boson masses we are interested in, it is appropriate to separate the Higgs boson production from its subsequent decays and multiply the production rates by the respective branching fractions, which we obtain from Hdecay [33, 34, 35, 36] for $h\to W^{\ast}W^{\ast}$ and the Particle Data Group (PDG) listings [37] for the subsequent decays of the $W$ bosons. For our main production channel, the Standard Model Higgs boson production via gluon fusion, we want to use the most precise theoretical predictions that have become available over the last few years, for a great review, we refer to Ref. [32]. Using an effective theory approach, this production channel is known at NNLO including electroweak and mixed QCD–electroweak contributions [38, 39, 40, 41]. For a wide range of Higgs boson masses, these NNLO cross sections have been shown to reproduce the latest results obtained from soft- gluon resummation up to NNLL accuracy, cf. Refs. [42, 43, 44]. To mimic the resummation effects, the optimal scale choice at NNLO is found to be $\mbox{$\mu_{\mathrm{F}}$}=\mbox{$\mu_{\mathrm{R}}$}=M_{h}/2$, while for the NNLL calculation one employs common scales of $\mu=M_{h}$. Both higher-order calculations take the most recent parametrization of PDFs at next-to-next-to- leading order into account where the corresponding PDF sets have been provided by the MSTW group in 2008 [45]. The Tevatron Higgs boson searches use these higher-order $gg\to h$ cross section predictions to report their combined CDF and DØ upper limits on Standard Model Higgs boson production in the $W^{+}W^{-}$ decay mode [14, 15, 16, 4, 46, 17, 18, 19]. Hence, it makes sense to input the same theory cross sections in our studies to guarantee a reasonable level of compatibility between our work and the experimental searches. However, one should keep in mind that different viewpoints exist concerning the determination of the best $gg\to h$ cross section numbers. For example, in Ref. [47] the authors argue that the 10–15% enhancement seen in the inclusive rates is unlikely to survive the cuts applied in the Tevatron analyses and, therefore, should not be included in the calculation of the limits [48, 49]. On the contrary, the renormalization-group improved resummed NNLO cross sections discussed by Ahrens, Becher, Neubert and Yang in Refs. [50, 51, 52] would yield a further 5–6% increase of the NNLL $gg\to h$ rates. This is because in their approach Ahrens et al. do not only resum threshold logarithms from soft-gluon emission but also $\pi^{2}$-enhanced terms, which arise in the analytic continuation of the gluon form factor to timelike momentum transfer. $M_{h}$ \| $\Gamma_{h}$ \| $\sigma^{\mathrm{NNLL}}_{ggh}$ \| $B^{\mbox{\tiny Ref.~{}\cite[cite]{[\@@bibref{}{:2011cb}{}{}]}}}_{W^{\ast}W^{\ast}}$ \| $\ \sigma^{\mathrm{NNLO}}_{ggh}$ \| $B_{W^{\ast}W^{\ast}}$ \| $\ \;\sigma^{(0)}_{S,\mathrm{all}}$ \| $\sigma^{(0)}_{S,\mathrm{NNLO}}$ \| $\ \sigma^{(0)}_{S}$ \| $K_{S}$ \| $\ \sigma^{(0)}_{S,\mathrm{66}}$ \| $K^{\mathrm{66}}_{S}$ ---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|--- $\mathrm{[GeV]}$ \| $\mathrm{[GeV]}$ \| $\mathrm{[fb]}$ \| \| $\mathrm{[fb]}$ \| \| $\mathrm{[fb]}$ \| $\mathrm{[fb]}$ \| $\mathrm{[fb]}$ \| \| $\mathrm{[fb]}$ \| $110$ \| $0.002939$ \| $1385.0$ \| $0.0482$ \| $1428$ \| $0.04633$ \| $11.81$ \| $9.660$ \| $3.350$ \| 2.88 \| $2.550$ \| 3.79 $120$ \| $0.003595$ \| $1072.3$ \| $0.143$ \| $1102$ \| $0.1380$ \| $27.11$ \| $22.21$ \| $7.717$ \| 2.88 \| $5.990$ \| 3.71 $130$ \| $0.004986$ \| $842.9$ \| $0.305$ \| $863$ \| $0.2976$ \| $45.83$ \| $37.50$ \| $13.06$ \| 2.87 \| $10.15$ \| 3.69 $140$ \| $0.008222$ \| $670.6$ \| $0.504$ \| $685$ \| $0.4959$ \| $60.53$ \| $49.60$ \| $17.29$ \| 2.87 \| $13.46$ \| 3.68 $150$ \| $0.01726$ \| $539.1$ \| $0.699$ \| $550$ \| $0.6927$ \| $67.88$ \| $55.63$ \| $19.33$ \| 2.88 \| $15.08$ \| 3.69 $165$ \| $0.2429$ \| $383.7$ \| $0.960$ \| $389$ \| $0.9595$ \| $66.72$ \| $54.50$ \| $19.35$ \| 2.82 \| $15.13$ \| 3.60 $170$ \| $0.3759$ \| $344.0$ \| $0.965$ \| $347$ \| $0.9642$ \| $60.02$ \| $48.85$ \| $17.77$ \| 2.75 \| $13.91$ \| 3.51 $180$ \| $0.6290$ \| $279.2$ \| $0.932$ \| $283$ \| $0.9327$ \| $47.33$ \| $38.54$ \| $14.19$ \| 2.72 \| $11.15$ \| 3.46 $190$ \| $1.036$ \| $228.0$ \| $0.786$ \| $229$ \| $0.7871$ \| $32.50$ \| $26.32$ \| $9.862$ \| 2.67 \| $7.775$ \| 3.39 $200$ \| $1.426$ \| $189.1$ \| $0.741$ \| $190$ \| $0.7426$ \| $25.40$ \| $20.60$ \| $7.827$ \| 2.63 \| $6.191$ \| 3.33 $210$ \| $1.841$ \| \| \| $159$ \| $0.7250$ \| \| $16.83$ \| $6.473$ \| 2.60 \| $5.131$ \| 3.28 $220$ \| $2.301$ \| \| \| $134$ \| $0.7160$ \| \| $14.01$ \| $5.420$ \| 2.58 \| $4.317$ \| 3.25 Table 1: Signal cross sections $\sigma^{(0)}_{S}$ at NNLO and LO plus the resulting signal $K$-factors for several different SM Higgs boson masses. The Higgs boson widths are found from Hdecay calculations. Columns 5 and 6 show our input NNLO cross sections taken from recent Fehip calculations [53, 54, 55, 38] and input branching fractions for $h\to W^{\ast}W^{\ast}$ obtained from Hdecay [34, 35], respectively. They are comparable to the values given in the 3rd and 4th columns that are used by the Tevatron experimentalists in their ongoing Higgs boson searches [4]. The signal cross sections labelled “all” account for contributions stemming not only from gluon–gluon fusion but also from Higgs strahlung and VBF Higgs boson production processes. All calculations use MSTW2008 parton distributions except those to extract the LO values denoted by “$\mathrm{66}$”, which result from using CTEQ6.6 PDFs. For various Higgs boson masses, Table 1 summarizes the signal cross sections $\sigma^{(0)}_{S}$ that are of relevance for our studies. Separated from the other entries, the left and right parts of the table show parameters, which we use as input to our analysis. We have used the NNLO $gg\to h$ inclusive cross sections, $\sigma^{\mathrm{NNLO}}_{ggh}$, given in the 5th column and multiplied them with the branching ratios listed in the column to the right of it, which we have computed with Hdecay version 3.51.111To obtain the values in Table 1, the NNLO MSTW fit result for the strong coupling, $\alpha_{\mathrm{s}}(M_{Z})=0.11707$, has been employed, the running of quark masses at NNLO has been enabled and the top and bottom quark masses have been set to $m_{t}=173.1$ and $m_{b}=4.8\ \mathrm{GeV}$, respectively. The Higgs boson widths shown in the 2nd column are also taken from the Hdecay calculation; they slightly differ from the older values given in [21]. To arrive at the higher-order prediction of our signal cross sections depicted in the 8th column, $\sigma^{(0)}_{S,\mathrm{NNLO}}$, we furthermore have accounted for the $W$ decays described by the branching ratios for one light lepton species, $B(W\to\ell\nu_{\ell})=0.108$, and for jets, $B(W\to pp)=0.676$, and a combinatorial factor of 2 reflecting that either of the $W$ bosons may decay leptonically. The NNLO cross sections used here [55] are updated values with respect to the ones published in [38]. The difference can be traced back to the addition of the electroweak real-radiation corrections as encoded in Ref. [39] and the change to top masses of $m_{t}=173.1\ \mathrm{GeV}$. In the 3rd and 4th columns we respectively also give the $gg\to h$ cross sections and $h\to W^{\ast}W^{\ast}$ branching fractions as used by the Tevatron experimentalists in their ongoing searches [4]. These values are in good agreement with the respective numbers used in our study. The other signal cross sections listed in the rightmost part of Table 1 are the LO rates obtained from SHERPA, where the one labelled $\sigma^{(0)}_{S,\mathrm{66}}$ refers to the use of CTEQ6.6 PDFs [56]. We will discuss these LO results and the corresponding $K$-factors in Section 3.3.1. In the 7th column we show an upper estimate for the signal cross sections $\sigma^{(0)}_{S,\mathrm{all}}$ if one were to include the contributions from the $Wh$, $Zh$ and VBF production channels. Over the considered Higgs boson mass range the extra processes would enhance the signal rates resulting from gluon–gluon fusion by about 22–23%. We have determined these estimates by adding to the $gg\to h$ rates the theory predictions for $\sigma_{Wh}$, $\sigma_{Zh}$ and $\sigma_{\mathrm{VBF}}$ as presented in the July 2010 CDF and DØ Higgs boson searches combination paper [46] (for updated values, cf. [4]) including all necessary branching fractions to arrive at the $pp\;\ell\nu_{\ell}\,pp$ final states.222For all production channels, we consider the Higgs boson decay as specified in (12), since our selection and reconstruction procedures are tailored to this decay mode, see Section 4. In our analysis we deal with $\ell$, MET and multiple jets (from the decays as well as I/FSR), thus, the presence of additional jets stemming from other hard decays or VBF does not alter our analysis procedures considerably, in other words the Higgs boson reconstruction and selection procedures are designed in a robust way with respect to additional jet activity. The dominant Higgs strahlung contributions to the semileptonic final states arise from the hadronic decays of the associated vector bosons, where we have used the PDG values $B(W\to pp)=0.676$ and $B(Z\to pp)=0.6991$. All other combinations are suppressed by about an order of magnitude. Also, we do not consider more than one lepton, i.e. we implicitly assume an exclusive one-lepton cut. We also neglect the $Vh$ cases where the $Z$ boson decays invisibly and the associated $W$ boson splits up leptonically while $h\to pppp$. These modes will fail our $h$ boson reconstruction. The additional 20% increase resulting from these considerations should be born in mind when acceptances and significances are evaluated in a signal versus background study. For the purpose of the analysis we are pursuing here, we however want to be on the conservative side and solely concentrate on the gluon–gluon fusion events. ### 3.2 Relevant background processes Processes that give rise to major background contributions are $V$+jets and multi-jet production, where the latter comes into play because of jets faking isolated leptons and/or MET. The muon channel suffers less from jet fakes reducing the multi-jet background by a factor of 5 with respect to the electron channel. The $Z$+jets background can also contribute in cases where one lepton goes missing or a jet mimics a lepton while the $Z$ decays invisibly. Minor contributions stem from $VV$ and $t\bar{t}$ production. A first DØ search in the semileptonic Higgs boson decay channel shows how these backgrounds compare to each other after basic selection cuts, see Ref. [26]. The largest fraction of 83% occurs from $V$+jets, followed by multi-jets, $t\bar{t}$ and $VV$ contributing with 12%, 3% and 2% to the overall background. The $V$+jets contribution is totally dominated by $W$+jets production; the $Z(\to\ell^{+}\ell^{-})$+jets background, where one of the leptons is missed, is small and makes up less than 1% of the total background. To gain a better understanding of the backgrounds, we will have a closer look at the major contributor $W$+jets. The multi-jet background cannot be simulated straightforwardly, since it requires detailed knowledge of the experiments and measured fake rates etc. Regarding the minor background contributors, we will study the $t\bar{t}$ as well as the $WW$ – or, more exactly, electroweak – background. Even though they enter at a rather low level after the basic selection compared to $W$+jets, it is necessary to cross-check what number of events remain after more selective cuts have been applied, as will be discussed in Section 4. Another background contribution that has been discussed is gluon-initiated vector boson pair production [57, 58, 59]. This (quark-loop-induced) process occurs at ${\mathcal{O}}(\alpha^{4}_{\mathrm{ew}}\alpha^{2}_{\mathrm{s}})$, the same order as the signal. This background formally arises at NNLO, but under realistic experimental cuts this production channel has been shown to significantly increase e.g. the $WW\to 2\,\ell\;2\,\nu_{\ell}$ background at the LHC. At the Tevatron the gluon densities are small, so the impact of $gg\to WW$ is expected to be negligible. This expectation was confirmed in Ref. [60] (a 4‰ effect with respect to the NLO cross section for this decay channel). A more important effect also recently pointed out by Campbell et al. in Ref. [60] is the interference between $gg\to WW$ and $gg\to h\to WW$, which can result in ${\mathcal{O}}(0.1)$ corrections to the Higgs boson signal cross section. However, interference effects are considerably reduced by requiring the transverse mass of the leptons plus MET system to be smaller than $M_{h}$. This type of transverse cut is frequently used in our analyses, so we can safely neglect interference effects in our study. #### 3.2.1 $W$ boson plus jets background For our first study of $W$+jets production, we explore the dependence of inclusive $W$+jet cross sections on the number of jets and the variation of the common scale $\mu$ used to specify the factorization and renormalization scales, $\mu_{\mathrm{F}}$ and $\mu_{\mathrm{R}}$, respectively. This information will help us identify an optimal definition of the $W$+jets $K$-factor, which we take to improve the rates of the SHERPA predictions. We calculate inclusive $W$+$\,n\leq 2$-jet cross sections with MCFM version 5.8 to obtain results that are accurate at NLO in the strong-coupling constant [61, 62, 63]. We also run MCFM at LO to determine explicit NLO-to-LO theoretical $K$-factors.333We only consider $W^{+}$ bosons decaying into $e^{+}\nu_{e}$ pairs; the charge conjugated process will just double the cross section owing to the $P\bar{P}$ initial states at the Tevatron. We employ the LO and NLO MSTW2008 PDFs [45] with $\alpha_{\mathrm{s}}(M_{Z})=0.13939$ and $\alpha_{\mathrm{s}}(M_{Z})=0.12018$, respectively, and impose cuts according to the parameters given in Section 4.1. Note that we do not account for the so-called triangle cut relating the transverse mass of the $W$ boson and the missing energy. Other parameters, such as the electroweak input values of the Standard Model, have been taken according to the MCFM default settings. \| Inclusive $W^{+}$+$\,n$-jet cross sections in pb. ---\|--- $n$ \| \| $\mu=M_{\perp,W}/2$ \| $=M_{\perp,W}$ \| $=2\,M_{\perp,W}$ \| % \| \| $\mu=\hat{H}_{T}/2$ \| $=\hat{H}_{T}$ \| $=2\,\hat{H}_{T}$ \| % $0$ \| LO \| $457$ \| $465$ \| $469$ \| ${}^{-1.7}_{+0.9}$ \| \| $453$ \| $463$ \| $468$ \| ${}^{-2.2}_{+1.1}$ \| NLO \| $625$ \| $619$ \| $616$ \| ${}^{+1.0}_{-0.5}$ \| \| $606$ \| $602$ \| $602$ \| ${}^{+0.7}_{-0.0}$ \| $K$ \| 1.37 \| 1.33 \| 1.31 \| \| \| 1.34 \| 1.30 \| 1.29 \| $1$ \| LO \| $66.2$ \| $55.6$ \| $47.3$ \| ${}^{+19.1}_{-14.9}$ \| \| $62.5$ \| $52.7$ \| $45.1$ \| ${}^{+18.6}_{-14.4}$ \| NLO \| $79.8$ \| $74.6$ \| $69.3$ \| ${}^{+7.0}_{-7.1}$ \| \| $74.2$ \| $70.2$ \| $65.7$ \| ${}^{+5.7}_{-6.4}$ \| $K$ \| 1.21 \| 1.34 \| 1.47 \| \| \| 1.19 \| 1.33 \| 1.46 \| \| $R_{\mathrm{LO}}^{(1,0)}$ \| $0.145$ \| $0.120$ \| $0.101$ \| \| \| $0.138$ \| $0.114$ \| $0.096$ \| \| $R_{\mathrm{NLO}}^{(1,0)}$ \| $0.128$ \| $0.121$ \| $0.113$ \| \| \| $0.122$ \| $0.117$ \| $0.109$ \| $2$ \| LO \| $14.4$ \| $10.1$ \| $7.40$ \| ${}^{+42.6}_{-26.7}$ \| \| $10.9$ \| $7.89$ \| $5.89$ \| ${}^{+38.1}_{-25.3}$ \| NLO \| $12.8$ \| $11.7$ \| $10.4$ \| ${}^{+9.4}_{-11.1}$ \| \| $12.0$ \| $10.1$ \| $8.95$ \| ${}^{+18.8}_{-11.4}$ \| $K$ \| 0.89 \| 1.16 \| 1.41 \| \| \| 1.10 \| 1.28 \| 1.52 \| \| $R_{\mathrm{LO}}^{(2,1)}$ \| $0.218$ \| $0.182$ \| $0.156$ \| \| \| $0.174$ \| $0.150$ \| $0.131$ \| \| $R_{\mathrm{NLO}}^{(2,1)}$ \| $0.160$ \| $0.157$ \| $0.150$ \| \| \| $0.162$ \| $0.144$ \| $0.136$ \| Table 2: Inclusive $W^{+}$+$\,n$-jet cross sections $\sigma_{n}$ in pb at LO and NLO in QCD for different scale choices and jet multiplicities using MSTW2008 PDFs. The variations with respect to the nominal choices, $M_{\perp,W}$ with $M^{2}_{\perp,W}=M^{2}_{W}+p^{2}_{T,W}$ and $\hat{H}_{T}$, are given in the columns labelled by “%”. Numerical integration uncertainties are not displayed, since they are at least one order of magnitude below the accuracy indicated here. NLO-to-LO $K$-factors and $n$-to-($n-1$)-jet cross section ratios are also shown for all possible instances. We display our MCFM results in Table 2 for different inclusive jet bins $n$ and scale choices $\mu$. As expected, for each $n$-jet multiplicity, the NLO cross sections are more stable under scale variations with the largest deviations occurring for the more complex $W^{+}$+2-jet processes. This is also reflected by the various NLO-to-LO $K$-factors, which vary from about 0.9 to 1.5 for $n=2$ while they are rather constant for $n=0$ ranging from about 1.3 to 1.4 only. For illustrative purposes, we also list the LO and NLO inclusive jet-rate ratios $R^{(n,n-1)}=\sigma_{n}/\sigma_{n-1}$ starting with $n=1$. The $W^{+}$+$\,n\leq 1$-jet cross sections do not deviate substantially for the two nominal scales chosen, $\mu\sim M_{\perp,W}$ where $M^{2}_{\perp,W}=M^{2}_{W}+p^{2}_{T,W}$ and $\mu\sim\hat{H}_{T}$, which are determined dynamically for each event. Note that $\hat{H}_{T}$ is the scalar sum of the transverse momenta of all particles (partons) in the event, i.e. no jet clustering has taken place. The $\mu\sim M_{\perp,W}$ scales lead to slightly larger rates when compared to those obtained for $\mu\sim\hat{H}_{T}$. This can be traced back to the occurrence of $\mu$-values that are on average larger in the latter case, since $\langle\hat{H}_{T}\rangle\gtrsim\langle M_{\perp,W}\rangle$ for $n\geq 1$. For the same reason, the cross section differences become more manifest for $n=2$. The presence of the second jet gives an extra $p_{T}$ contribution to $\hat{H}_{T}$ per event whereas $M_{\perp,W}$ is less affected. This further enhances the deviation of the $\hat{H}_{T}$ and $M_{\perp,W}$ averages. Given the numbers of Table 2 we can conclude that our knowledge of the $W$+2-jet background is accurate on the level of $\lesssim$ 20%. A $K$-factor of about $1.5$ should be viewed as the upper limit for correcting LO results; in Section 3.3.2 we will however compare the SHERPA background rates more closely with the results of Table 2 and determine a $K$-factor accordingly. ### 3.3 Monte Carlo simulation of signal and backgrounds using SHERPA For reasons outlined at the beginning of Section 3, we use SHERPA version 1.1.3 [24, 25] to generate the $\ell\nu_{\ell}$+jets signal and background events that are needed to understand the potential of a Standard Model Higgs boson analysis in the lepton + MET + jets channel.444Version 1.1.3 was the last of the previous SHERPA generation; for all our purposes, it models the necessary physics equally well compared to the upgraded versions of the current (1.3.x) generation. Cross-comparisons have confirmed this result. We will employ the results of the previous two subsections to settle the inclusive $K$-factors needed to re-scale SHERPA’s LO predictions and include higher-order rate effects. The signal and background simulations share a number of common parameters and options that have been set as follows: we simulate all events at the parton- shower level, i.e. we include initial- and final-state QCD radiation, but do not account for hadronization effects and corrections owing to the underlying event, since their impact is considerably smaller with respect to additional QCD radiation arising from the hard processes. The intrinsic transverse motion of quarks and gluons inside the colliding hadrons is however modeled by an intrinsic Gaussian $k_{T}$-smearing of $\mu(k_{T})=0.2$ and $\sigma(k_{T})=0.8\ \mathrm{GeV}$. The electroweak parameters are explicitly given: $M_{W}=80.419$, $\Gamma_{W}=2.06$, $M_{Z}=91.188$, $\Gamma_{Z}=2.49\ \mathrm{GeV}$; the Higgs boson masses and widths are mutable, taken according to Table 1; the couplings are specified by $\alpha_{\mathrm{ew}}(0)=1/137.036$, $\sin^{2}_{\mathrm{W}}=0.2222$ and the Higgs field vacuum expectation value and its quartic coupling are given as $246\ \mathrm{GeV}$ and $0.47591$, respectively. The CKM matrix is simply parametrized by the identity matrix. The bottom and top quark masses are set to $m_{b}=4.8$ and $m_{t}=173.1\ \mathrm{GeV}$, respectively, and all other quark masses are zero. To avoid any bias owing to the utilization of different PDFs and in order to develop a consistent picture, signal and background events are generated using the same parton distributions. Our first choice of PDFs is the LO MSTW set MSTW2008lo90cl [45], because its NNLO version has been the preferred PDF set used for the recent calculations of the gluon–gluon fusion Higgs boson production cross sections. The strong coupling is determined by one-loop running with $\alpha_{\mathrm{s}}(M_{Z})=0.13939$, which is the advertised fit value of the LO MSTW2008 set. To gain some understanding of PDF effects, we compare our MSTW2008 results against predictions generated with a different PDF set. To fully establish the comparison on the same level as for the MSTW2008 PDFs, signal and background rates have to be predicted from theory using the alternative PDF libraries. We cannot follow this approach here, instead we start out from the same normalization that has been used for the SHERPA predictions calculated with MSTW2008 PDFs. After the application of our cuts we then focus on the differences induced by the alternative PDF set. As our second choice we employ the CTEQ6.6 PDF libraries [56] where the strong coupling is set by $\alpha_{\mathrm{s}}(M_{Z})=0.118$ and the running of the coupling is again computed at one loop. Notice that SHERPA invokes a 6-flavour running for all strong-coupling evaluations. #### 3.3.1 Generation of signal events We simulate signal events with electrons or positrons in the final state according to $P\bar{P}\to h\to e\nu_{e}\;pp\to e\nu_{e}+\mathrm{jets}\ .$ (13) The hard process composed as $gg\to h\to e\nu_{e}\,pp$ is calculated at LO. The incoming gluons and the quarks arising from the decay undergo further parton showering, which automatically is taken care of by the SHERPA simulation. One ends up with the $e\nu_{e}+\mathrm{jets}$ final states generated at shower level. The hard-process tree-level matrix elements and subsequent parton showers needed for the simulation are provided by the SHERPA modules AMEGIC++ and APACIC++, respectively. For our purpose, it is sufficient to treat the muon final states in exactly the same manner as the electron final states, i.e. the muon decay channel is included by multiplication with the lepton factor $f_{\ell}=2$ at the appropriate places. The Higgs boson production occurs through gluon–gluon fusion via intermediate heavy-quark loops. In SHERPA this is modeled at LO by an effective $gg\to h$ coupling where the top quarks have been integrated out. The EHC (Effective Higgs Couplings) implementation of SHERPA includes all interactions up to 5-point vertices that result from the effective-theory Lagrangian. These effective vertices can simply be added to the Standard Model. We do not work in the infinite top-mass limit, because we also want to consider Higgs bosons heavier than the top quark, the approximation however is well applicable only as long as $m_{t}>M_{h}$. The Higgs boson decays are described by $1\to 4$ processes, i.e. we directly consider $h\to e\nu_{e}\,pp$. We thereby make use of SHERPA’s feature to decompose processes on the amplitude level into the production and decays of unstable intermediate particles while the colour and spin correlations are fully preserved between the production and decay amplitudes [25]. This way one can focus on certain resonant contributions instead of calculating the full set of diagrams contributing to a given final state, which in our case would lead to the inclusion of contributions from the backgrounds. The intermediate propagators are allowed to be off-shell, such that finite-width effects are naturally incorporated into the simulation. This comes in handy especially for Higgs boson masses below the $WW$ mass threshold as the $1\to 4$ decays moreover guarantee the inclusion of off-shell $W$-boson effects. A consistent LO treatment would require the use of total Higgs boson widths as computed at LO. We instead put in the values from the Hdecay calculations [34, 35] as listed in Table 1. This modifies the Higgs boson propagators and one arrives at a more accurate description of the finite-width effects of the Higgs boson decays. The effect on the total rate, $\sigma^{(0)}_{S}\;=\;\frac{\Gamma(h\to e\nu_{e}\,pp)}{\Gamma_{h}}\;\sigma^{\mathrm{LO}}_{ggh}\ ,$ (14) is nullified, since we eventually correct for the NNLO rates $\sigma^{(0)}_{S,\mathrm{NNLO}}$ worked out in Section 3.1. In Ref. [35] a comparative study has been presented for Higgs boson production via gluon fusion at the LHC. Amongst a variety of predictions including those given by HNNLO [64, 65], the SHERPA versions 1.1.3 and 1.2.1 have been validated to produce very reasonable results for the shapes of distributions like the rapidity and transverse momentum of the Higgs boson, pseudo- rapidities and transverse momenta of associated jets and jet–jet $\Delta R$ separations. We hence rely on a well validated approach that works not only for pure parton showering in addition to the Higgs boson production and decays, but also beyond in the context of merging higher-order tree-level matrix elements with parton showers. Nevertheless, we have carried out a number of cross-checks to convince ourselves of the correctness of the SHERPA calculations; for the details, we refer the reader to Appendix A.1. Finally we turn to the discussion of the $K$-factors. Recalling our findings of Section 3.1, we want to re-scale SHERPA’s leading-order signal cross sections $\sigma^{(0)}_{S}$ to the fixed-order NNLO predictions given by Fehip for Higgs boson production in $gg\to h$ fusion via intermediate heavy-quark loops [53, 54, 38, 55]. To be consistent, the renormalization and factorization scales of the LO hard-process evaluations are chosen as for the higher-order calculations, which employ $\mu=\mbox{$\mu_{\mathrm{R}}$}=\mbox{$\mu_{\mathrm{F}}$}=M_{h}/2$. The resulting cross sections ultimately define our signal $K$-factors: $K_{S}\;=\;\frac{\sigma^{(0)}_{S,\mathrm{NNLO}}}{\sigma^{(0)}_{S}}\ .$ (15) We have determined two sets of $K$-factors for our two choices of PDFs where the $K$-factors and LO cross sections labelled by “$\mathrm{66}$” refer to the case of utilizing the CTEQ6.6 libraries when calculating the LO cross sections. Our results have already been summarized in Section 3.1, they are presented in the right part of Table 1. The $K$-factors are remarkably stable varying slowly from 2.8 to 2.6 over the entire Higgs boson mass range when relying on MSTW2008 PDFs. In the CTEQ6.6 case, where we have employed $\mu=\mbox{$\mu_{\mathrm{R}}$}=\mbox{$\mu_{\mathrm{F}}$}=\sqrt{\hat{s}}/2\approx M_{h}/2$, they are larger due to the smaller LO rates but their magnitude still remains $\lesssim$ 3.6.555The LO rates calculated with the CTEQ PDF libraries are diminished for two reasons mainly, the value of $\alpha_{\mathrm{s}}$ at $M_{Z}$ is considerably lower and the altered scale choice entails a further reduction of the cross sections. In addition to the default scale choice of $\mu=M_{h}/2$ that we used for the MSTW runs, we have explored other options by essentially varying this default setting for $\mu$ by factors of 2. We obtained results for $\mu=M_{h}/4$, $\mu=\sqrt{\hat{s}}/2\approx M_{h}/2$ and $\mu=\sqrt{\hat{s}}\approx M_{h}$ with the effect that the LO rates were varied by +20% to -15% but – as expected – no shape changes were induced. #### 3.3.2 Generation of background events for $W$ boson plus jets production We restrict ourselves to the Monte Carlo simulation of the $e^{\pm}$ channels. Their final states are generated through $P\bar{P}\to e\nu_{e}+0,1,2\,p\to e\nu_{e}+\mathrm{jets}$ (16) using an inclusive $W$+2-jets sample obtained from the Catani–Krauss–Kuhn–Webber (CKKW) merging of the corresponding tree-level matrix elements with the parton showers (ME+PS) [66, 67]. In these $W$+2-jets calculations the electroweak order is tied to $\alpha^{2}_{\mathrm{ew}}$. Unlike the NLO calculation we do include matrix elements where the extra partons may occur as $b$ quarks; effectively, they are however treated as massless quarks in the evaluation of the matrix elements and generation of the radiation pattern. The events are corrected for the $b$-quark mass after the parton showering. This approach generates slightly harder $p_{T}$ spectra but as part of being more conservative in estimating this background it is totally reasonable. Similarly, we simply assume no effect of a $b$-jet veto in removing $W$+jets events. The parameters of the matrix-element parton-shower merging are the jet separation scale $Q_{\mathrm{jet}}$ and the $D$-parameter, which is used to fix the minimal separation of the parton jets. These parameters are respectively set to $Q_{\mathrm{jet}}=20\ \mathrm{GeV}$ and $D=0.4$ in correspondence to the jet $p_{T}$ threshold and cone definitions of our analysis, see Section 4.1. $Q_{\mathrm{jet}}$ denotes the scale at which – according to the internal $k_{T}$-jet measure incorporating the $D$-parameter – the multi-jet phase space is divided into the two domains of $Q>Q_{\mathrm{jet}}$ where the jets are produced through exact tree-level matrix elements and $Q_{\mathrm{jet}}>Q>Q_{\textrm{cut-off}}\sim 1\ \mathrm{GeV}$ where the parton-shower intra-jet evolution takes place. We generate predictions from samples that merge matrix elements with up to $n^{\mathrm{max}}_{p}=2$ partons. Although we could increase this maximum number, at this point we do not want to include matrix elements with more than two partons in order to be consistent with our signal event generation where the jets beyond those arising from the $W$-boson decays are produced by parton showers only. If one wishes to further improve on the description of additional hard jets, both background and signal simulations should be extended on the same footing. $n$ \| \| $\mu=M_{\perp,W}/2$ \| $=M_{\perp,W}$ \| $=2\,M_{\perp,W}$ \| \| $\mu=\hat{H}_{T}/2$ \| $=\hat{H}_{T}$ \| $=2\,\hat{H}_{T}$ \| \| $\sigma_{\mathrm{{CKKW}}}/\mathrm{pb}$ ---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|--- LO \| $0$ \| 0.92 \| 0.94 \| 0.95 \| \| 0.91 \| 0.93 \| 0.94 \| \| $496$ \| $2$ \| 1.45 \| 1.02 \| 0.75 \| \| 1.10 \| 0.80 \| 0.59 \| \| $9.90$ NLO \| $0$ \| 1.26 \| 1.25 \| 1.24 \| \| 1.22 \| 1.21 \| 1.21 \| \| \| $2$ \| 1.29 \| 1.18 \| 1.05 \| \| 1.21 \| 1.02 \| 0.90 \| \| Table 3: Ratios at LO and QCD NLO taken between rates of MCFM and SHERPA CKKW (rightmost column) for inclusive $W^{+}$+$\,n$-jet production at different choices of scales in MCFM using MSTW2008 PDFs in all cases. The MCFM cross sections are listed in Table 2. The $V$+jets predictions of SHERPA have been extensively studied and validated over the last few years. Studies exist for comparisons against other Monte Carlo tools [68, 69, 70, 71, 72], NLO calculations [68, 69, 73] and Tevatron Run I and II data [68, 74, 75, 25, 76, 77, 78, 79]. They have helped improve SHERPA gradually and provided evidence that SHERPA gives a good description of the shapes of the $V$+jet final-state distributions missing a global scaling factor only, which can be extracted from the data [25] or higher-order calculations [73].In Appendix A.2 we briefly highlight to what extent the CKKW ME+PS merging includes important features of NLO computations. We use the results of Table 2 to identify a reasonable $K$-factor for our simulated $W$+jets backgrounds. Relying on MSTW2008 PDFs, the SHERPA numbers for the inclusive $W^{+}$ and $W^{+}$+2-jet cross sections are 496 pb and 9.90 pb, respectively. The 0-jet SHERPA rate thereby is about 7% larger than the corresponding LO rates given by MCFM. The differences occur because on the one hand MCFM by default invokes a non-diagonal CKM matrix and a somewhat larger $W$-boson width 666Switching to an unity CKM matrix and using SHERPA’s input parameters, one finds 486 pb at $\mu=M_{W}$., on the other hand SHERPA’s merged-sample generation relies on a very different scale-setting procedure compared to the leading fixed-order calculations. These differences have no effect on the kinematic distributions – and are fully absorbed by the $K$-factor, i.e. CKM effects may eventually enter through the correction of SHERPA’s rate. Table 3 summarizes the ratios between the MCFM predictions of Table 2 and SHERPA’s CKKW cross sections mentioned above. This overview neatly points to the two options that give the most stable ratios; they are found at NLO for $\mu=M_{\perp,W}/2$ and $\mu=\hat{H}_{T}/2$ where the latter scale choice has been reported to be well suitable for even higher jet multiplicities [80, 73, 81]. Based on these observations, we can hence conclude that it is fair to apply a $K$-factor of $K_{B}\;=\;1.25$ (17) to the $W$+jets backgrounds employed in our study. The number found here compares well to global $K$-factors as reported throughout the literature. As outlined at the beginning of Section 3.3, we want to normalize the backgrounds obtained with CTEQ6.6 to those computed with MSTW2008 PDFs. In the CTEQ case the SHERPA CKKW cross sections amount to 544 pb and 8.13 pb for the inclusive $W^{+}$ and $W^{+}$+2-jet final states, respectively. Since the latter selection of $W$+2-jet events is more exclusive, we re-scale the CTEQ backgrounds according to $K^{\mathrm{66}}_{B}\times 8.13\ \mathrm{pb}=K_{B}\times 9.90\ \mathrm{pb}$ and arrive at $K^{\mathrm{66}}_{B}\;=\;1.52\ .$ (18) #### 3.3.3 Generation of background events for electroweak and top-pair production The $WW$ background enters at ${\mathcal{O}}(\alpha^{4}_{\mathrm{ew}})$ of the electroweak coupling constant $\alpha_{\mathrm{ew}}$, i.e. it is suppressed by more than two orders of magnitude with respect to the $W$+2-jets contribution occurring at ${\mathcal{O}}(\alpha^{2}_{\mathrm{ew}}\alpha^{2}_{\mathrm{s}})$. Still, without running the simulation we cannot say for sure whether the continuum $WW$ production remains an 1% effect after application of the analysis cuts and – if necessary – what handles exist to distinguish it from the signal. Because of the large resemblance between the topologies of the Higgs boson decay and the dominant $WW$ production channels, we anticipate some of the cuts to be equally efficient for both signal and minor background. This makes it hard to estimate a priori the extent to which the Higgs boson signal will be diluted by the electroweak production type of processes. For the same reasons, the $t\bar{t}$ production final states can be expected to enhance the signal dilution on a similar level. Certainly, whether we end up with an 1% or 10% effect, this time it is sufficient to apply $K$-factors taken from the literature. For the simulation of the diboson production background, we take the complete set of electroweak diagrams occurring at ${\mathcal{O}}(\alpha^{4}_{\mathrm{ew}})$ into account including interference effects. This way we comprise physics effects beyond the plain $WW$ production with subsequent decays of the gauge bosons.777Relying on the full set of electroweak processes is more conservative: the rate increases by about 20%; the effect on the shapes is rather small in general, although we observe slightly harder tails in $p_{T}$ distributions. As before we only generate the processes regarding the first lepton family: $P\bar{P}\to e\nu_{e}\;pp\to e\nu_{e}+\mathrm{jets}$ (19) where additional jets are produced by the parton shower. Similar setups have been validated for SHERPA in [82] and more recently in [83, 84]. Here, we employ a dynamic choice, $\mu=\sqrt{\hat{s}}\sim 2\,M_{V}$, to calculate the scales of the LO processes. Parton-level jets are generated as in Section 3.3.2 using the same jet-finder algorithm and the same parameters ($Q_{\mathrm{jet}}=20\ \mathrm{GeV}$ and $D=0.4$). Processes with bottom quarks are included; just as in the $W$+2-jets case, they are treated as massless. The $t\bar{t}$ background events are generated according to $P\bar{P}\to t\bar{t}\to b\bar{b}\;e\nu_{e}\;pp\to e\nu_{e}+\mathrm{jets}$ (20) again utilizing the parton shower to describe any additional jet activity beyond that generated by the top quark decays. We only consider the semileptonic channel. The fully hadronic channel has to be considered together with the QCD background, and the fully leptonic channel will suffer from smaller branching fractions, the single isolated-lepton requirement and any dijet mass window that we impose around the $W$ mass. The LO processes are calculated at the scale $\mu=m_{t}$, the mass of the $b$ quarks is fully taken into account and the partonic phase-space generation is subject to the same jet-finding constraints as used for the compilation of the electroweak background. In addition we place mild generation cuts on the $b$ quarks: $p_{T,b}>10\ \mathrm{GeV}$ and $\Delta R_{b,p}>0.3$. We also examined the impact of $Z$+jets production on our analyses, and found that this contribution makes up less than 1% of the total background. Since $Z$+jets has kinematics similar to $W$+jets, we will not study it further. In SHERPA the minor backgrounds are computed at LO. As in all other cases, we correct the total inclusive cross section for NLO effects by multiplying with global $K$-factors, which for both electroweak and $t\bar{t}$ production are larger than 1. Tevatron diboson searches like [85, 86] measure cross sections in good agreement with the prediction given by Campbell and Ellis ($16.1\pm 0.9\ \mathrm{pb}$ for $WW$+$WZ$). From their work [87] (Table III) we infer an NLO-to-LO $K$-factor ranging from $1.30$ to $1.35$. For our analysis, we will then use the conservative estimate 888NLO corrections to $VV$ production can become large, for a recent example, see [88] where $K$-factors as large as $1.77$ have been reported; taken this value, we would certainly overestimate the electroweak contribution, since the CDF-type cuts employed in [88] are more exclusive. As for the shapes, we found them reliably described in a cross-check against an electroweak $VV$+1-jet merged sample, including matrix-element contributions at $\mathcal{O}(\alpha^{4}_{\mathrm{ew}}\alpha_{\mathrm{s}})$. $K_{B,\mathrm{ew}}\;=\;1.35\ .$ (21) For the inclusive $t\bar{t}$ production, we can safely estimate a conservative $K$-factor of $1.30$ by comparing the cross section results given for the Tevatron in Ref. [89]. Adopting a $b$-tagging efficiency of the order of 50% would give us a 75% chance of vetoing $t\bar{t}$ events with at least one $b$-quark jet, i.e. we were able to remove about 3/4 of the $t\bar{t}$ background; again, we will be more conservative here and assume that about 40% of the $t\bar{t}$ events will pass; hence, for our purpose, we finally assign $K^{b\textrm{-}\mathrm{veto}}_{B,t\bar{t}}\;=\;0.52\ .$ (22) ## 4 Signal versus background studies based on Monte Carlo simulations using SHERPA We report the successive improvements of the $S/\sqrt{B}$ significances when applying a series of cuts that preserve most of the signal and reduce the inclusive $W$+2-jets background significantly. ### 4.1 Baseline selection We follow the event-selection procedure as used by the DØ collaboration [26]: hadronic jets $j$ are identified by a seeded midpoint cone algorithm using the $E$-scheme for recombining the momenta [90]. The cone size is taken as $R=0.5$ and selection cuts of $p^{\mathrm{jet}}_{T}>20\ \mathrm{GeV}$ and $\lvert\eta^{\mathrm{jet}}\rvert<2.5$ are imposed. Additionally, we require a lepton–jet isolation of $\Delta R^{\mathrm{lep}\textrm{--}\mathrm{jet}}>0.4$. For the leptonic sector, we apply transverse-momentum and pseudo-rapidity cuts of $p^{\mathrm{lep}}_{T}>15\ \mathrm{GeV}$ and $\lvert\eta^{\mathrm{lep}}\rvert<1.1$, respectively, supplemented by a missing-energy cut via $\not{p}_{T}>15\ \mathrm{GeV}$. In addition, we also account for $M_{T,W}+\not{E}_{T}/2>40\ \mathrm{GeV}$, which is known as triangle cut.999The cut is applied to the leptonic $W$ boson where $M_{T,W}=\sqrt{(\lvert\vec{p}_{T,\ell}\rvert+\lvert\not{\vec{p}}_{T}\rvert)^{2}-(\vec{p}_{T,\ell}+\not{\vec{p}}_{T})^{2}}\equiv m_{T,\ell\nu_{\ell}}$, cf. Eq. (8). For Higgs boson masses above the $WW$ threshold, the rate reduction and shape changes induced by this cut are marginal. ### 4.2 Higgs boson reconstruction based on invariant masses After the application of the basic cuts, we identify the best-fit $\\{e,\nu_{e},j,j^{\prime}\\}$ set from all possible candidates allowed by combinatorics. The algorithm we use to identify the best-fit object is referred to as the Higgs boson candidate selection. Several different selection algorithms are possible, however for now, we will use an invariant mass (or invm) selection: the four particles (reconstructed in a more or less ideal way) whose combined mass $m_{e\nu_{e}jj^{\prime}}$ is closest to a “test” Higgs boson mass $M_{h}$ are chosen. Of course, in the context of the analysis, the Higgs boson mass enters as a hypothesis and, thus, is treated as a parameter. Regardless of the selection algorithm, we refer to $j$ and $j^{\prime}$ as the two selected jets, which are not necessarily the hardest jets in the event. After selection, we impose a requirement on the absolute difference between $m_{e\nu_{e}jj^{\prime}}$ and the hypothesized Higgs boson mass; events are kept only if they reconstruct a mass that lies within the window $M_{h}-\Delta<m_{e\nu_{e}jj^{\prime}}<M_{h}+\Delta$. This completes our combinatorial Higgs boson reconstruction, which we label as “comb. $h$-reco” in our tables. On top of this selection, we may include an additional dijet mass constraint of $M_{W}-\delta<m_{jj^{\prime}}<M_{W}+\delta$ (marked by $\lvert\tilde{m}_{jj^{\prime}}\\!-80\rvert\\!<\\!\tilde{\delta}$ in the tables). The selection procedure will certainly shape – to some extent – the remaining background to look like the signal, however the primary effect we are interested in concerns the reduction of the background rate while we want to preserve as many signal events as possible. One may ask whether the reconstruction of the Higgs particle candidate can be achieved more easily by selecting the set containing the respective hardest particles, in particular, by choosing the two hardest jets, $j_{1}$ and $j_{2}$, to reconstruct the hadronically decaying $W$ boson. We will refer to this approach as the naive Higgs boson reconstruction, denoted as “naive $h$-reco” later on (as before we use $\lvert\tilde{m}_{j_{1}j_{2}}\\!-80\rvert\\!<\\!\tilde{\delta}$ in the tables to indicate that a dijet mass constraint has been imposed in addition). There are no combinatorial issues in the naive scheme. However, as we show in our tables, it yields poorer significances than the selection based on combinatorics. We calculate the number of $S$ signal and $B$ background events for different Higgs boson masses assuming a total integrated luminosity of ${\mathcal{L}}=10\ \mathrm{fb}^{-1}$. This seems to be a good ${\mathcal{L}}$ estimate for what each of the two Tevatron experiments, CDF and DØ, were able to collect before the eventual Run II shutdown in September 2011. We compute the numbers according to $\displaystyle S$ $\displaystyle=$ $\displaystyle K_{S}\,\varepsilon_{S}\,\sigma^{(0)}_{S}\;\times\;2\,f_{\ell}\,{\mathcal{L}}\;=\;K_{S}\,\sigma_{S}\;\times\;2\,f_{\ell}\,{\mathcal{L}}\ ,$ $\displaystyle B$ $\displaystyle=$ $\displaystyle K_{B}\,\varepsilon_{B}\,\sigma^{(0)}_{B}\;\times\;2\,f_{\ell}\,{\mathcal{L}}\;=\;K_{B}\,\sigma_{B}\;\times\;2\,f_{\ell}\,{\mathcal{L}}$ (23) where $\varepsilon$ and $K$ respectively denote the total cut efficiencies and the $K$-factors, which we have worked out in Section 3.1, cf. Table 1, and Section 3.3, cf. Eqs. (17), (18), (21) and (22). The total efficiencies are a product of single-step efficiencies, i.e. $\varepsilon=\prod_{i}\varepsilon_{i}$. The factor $f_{\ell}=2$ accounts for including the decay channels that involve muons and their associate neutrinos. Notice that the Higgs boson mass enters in our simulation in two, potentially different ways. In practice, the Higgs boson mass that we used to generate the signal need not be the same as the Higgs boson mass we use to formulate the analysis. We refer to the former as the injected mass $M^{\mathrm{inj}}_{h}$ in the text, while we have already introduced the terminology of the latter as the “test” or “hypothesis” Higgs boson mass $M_{h}$. However, for simplicity we take the generation level Higgs boson mass and the analysis level Higgs boson mass to be equal, $M^{\mathrm{inj}}_{h}=M_{h}$. A discussion on how different generation versus analysis masses would change our results can be found in the Appendix B.1. cuts & \| $2\,\Delta/$ \| $\sigma_{S}/\mathrm{fb}$ \| $\sigma_{B}/\mathrm{fb}$ \| $S/B$ \| $\sigma_{S}/\mathrm{fb}$ \| $\sigma_{B}/\mathrm{fb}$ \| $S/B$ \| $\sigma_{S}/\mathrm{fb}$ \| $\sigma_{B}/\mathrm{fb}$ \| $S/B$ ---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|--- selections \| $\mathrm{GeV}$ \| $\varepsilon_{S}$ \| $\varepsilon_{B}$ \| $S/\sqrt{B}$ \| $\varepsilon_{S}$ \| $\varepsilon_{B}$ \| $S/\sqrt{B}$ \| $\varepsilon_{S}$ \| $\varepsilon_{B}$ \| $S/\sqrt{B}$ $M_{h}/\mathrm{GeV}\quad[\delta/\mathrm{GeV}]$ \| $165\quad[20]$ \| $170\quad[20]$ \| $180\quad[20]$ $\sigma^{(0)}$ \| \| $19.35$ \| $220\mbox{$\cdot\;\\!\\!$}10^{4}$ \| 20$\mbox{$\cdot\;\\!\\!$}10^{\textrm{-}6}$ \| $17.77$ \| $220\mbox{$\cdot\;\\!\\!$}10^{4}$ \| 18$\mbox{$\cdot\;\\!\\!$}10^{\textrm{-}6}$ \| $14.19$ \| $220\mbox{$\cdot\;\\!\\!$}10^{4}$ \| 14$\mbox{$\cdot\;\\!\\!$}10^{\textrm{-}6}$ \| \| 1.0 \| 1.0 \| 0.21 \| 1.0 \| 1.0 \| 0.19 \| 1.0 \| 1.0 \| 0.15 lepton & \| \| $10.66$ \| $984\mbox{$\cdot\;\\!\\!$}10^{3}$ \| 24$\mbox{$\cdot\;\\!\\!$}10^{\textrm{-}6}$ \| $9.869$ \| $984\mbox{$\cdot\;\\!\\!$}10^{3}$ \| 22$\mbox{$\cdot\;\\!\\!$}10^{\textrm{-}6}$ \| $7.946$ \| $984\mbox{$\cdot\;\\!\\!$}10^{3}$ \| 18$\mbox{$\cdot\;\\!\\!$}10^{\textrm{-}6}$ MET cuts \| \| 0.551 \| 0.45 \| 0.17 \| 0.555 \| 0.45 \| 0.15 \| 0.560 \| 0.45 \| 0.12 as above & \| \| $8.572$ \| $191\mbox{$\cdot\;\\!\\!$}10^{2}$ \| 0.0010 \| $7.967$ \| $191\mbox{$\cdot\;\\!\\!$}10^{2}$ \| 92$\mbox{$\cdot\;\\!\\!$}10^{\textrm{-}5}$ \| $6.471$ \| $191\mbox{$\cdot\;\\!\\!$}10^{2}$ \| 74$\mbox{$\cdot\;\\!\\!$}10^{\textrm{-}5}$ $\geq 2$ jets \| \| 0.443 \| 0.0087 \| 0.99 \| 0.448 \| 0.0087 \| 0.90 \| 0.456 \| 0.0087 \| 0.72 as above & \| \| $5.195$ \| $6997$ \| 0.0017 \| $4.735$ \| $6997$ \| 0.0015 \| $3.691$ \| $6997$ \| 0.0011 $\lvert\tilde{m}_{j_{1}j_{2}}\\!-80\rvert\\!<\\!\tilde{\delta}$ \| \| 0.269 \| 0.0032 \| 0.99 \| 0.266 \| 0.0032 \| 0.88 \| 0.260 \| 0.0032 \| 0.68 naive $h$-reco \| $50$ \| $5.422$ \| $6492$ \| 0.0019 \| $4.983$ \| $6492$ \| 0.0017 \| $3.911$ \| $6749$ \| 0.0013 \| \| 0.280 \| 0.0030 \| 1.07 \| 0.280 \| 0.0030 \| 0.96 \| 0.276 \| 0.0031 \| 0.73 naive $h$-reco \| $30$ \| $3.948$ \| $4108$ \| 0.0022 \| $3.897$ \| $4108$ \| 0.0021 \| $3.039$ \| $4199$ \| 0.0016 \| \| 0.204 \| 0.0019 \| 0.98 \| 0.219 \| 0.0019 \| 0.95 \| 0.214 \| 0.0019 \| 0.72 naive $h$-reco \| $48$ \| $4.657$ \| $2965$ \| 0.0035 \| $4.214$ \| $3210$ \| 0.0029 \| $3.232$ \| $3539$ \| 0.0020 $\lvert\tilde{m}_{j_{1}j_{2}}\\!-80\rvert\\!<\\!\tilde{\delta}$ \| \| 0.241 \| 0.0013 \| 1.36 \| 0.237 \| 0.0015 \| 1.16 \| 0.228 \| 0.0016 \| 0.84 naive $h$-reco \| $20$ \| $3.080$ \| $1374$ \| 0.0051 \| $2.876$ \| $1512$ \| 0.0042 \| $2.219$ \| $1676$ \| 0.0029 $\lvert\tilde{m}_{j_{1}j_{2}}\\!-80\rvert\\!<\\!\tilde{\delta}$ \| \| 0.159 \| 62$\mbox{$\cdot\;\\!\\!$}10^{\textrm{-}5}$ \| 1.33 \| 0.162 \| 69$\mbox{$\cdot\;\\!\\!$}10^{\textrm{-}5}$ \| 1.15 \| 0.156 \| 76$\mbox{$\cdot\;\\!\\!$}10^{\textrm{-}5}$ \| 0.83 comb. $h$-reco \| $50$ \| $7.105$ \| $6816$ \| 0.0024 \| $6.557$ \| $7117$ \| 0.0020 \| $5.241$ \| $7396$ \| 0.0015 \| \| 0.367 \| 0.0031 \| 1.37 \| 0.369 \| 0.0032 \| 1.21 \| 0.369 \| 0.0034 \| 0.94 comb. $h$-reco \| $20$ \| $4.827$ \| $3094$ \| 0.0035 \| $4.577$ \| $3191$ \| 0.0032 \| $3.657$ \| $3255$ \| 0.0024 \| \| 0.249 \| 0.0014 \| 1.38 \| 0.258 \| 0.0015 \| 1.26 \| 0.258 \| 0.0015 \| 0.99 comb. $h$-reco \| $50$ \| $6.346$ \| $3336$ \| 0.0043 \| $5.884$ \| $3697$ \| 0.0035 \| $4.679$ \| $4098$ \| 0.0025 $\lvert\tilde{m}_{jj^{\prime}}\\!-80\rvert\\!<\\!\tilde{\delta}$ \| \| 0.328 \| 0.0015 \| 1.75 \| 0.331 \| 0.0017 \| 1.51 \| 0.330 \| 0.0019 \| 1.12 comb. $h$-reco \| $30$ \| $5.586$ \| $2217$ \| 0.0057 \| $5.159$ \| $2488$ \| 0.0046 \| $4.083$ \| $2756$ \| 0.0032 $\lvert\tilde{m}_{jj^{\prime}}\\!-80\rvert\\!<\\!\tilde{\delta}$ \| \| 0.289 \| 0.0010 \| 1.89 \| 0.290 \| 0.0011 \| 1.61 \| 0.288 \| 0.0013 \| 1.20 comb. $h$-reco \| $20$ \| $4.616$ \| $1525$ \| 0.0068 \| $4.280$ \| $1731$ \| 0.0054 \| $3.404$ \| $1933$ \| 0.0038 $\lvert\tilde{m}_{jj^{\prime}}\\!-80\rvert\\!<\\!\tilde{\delta}$ \| \| 0.239 \| 69$\mbox{$\cdot\;\\!\\!$}10^{\textrm{-}5}$ \| 1.89 \| 0.241 \| 79$\mbox{$\cdot\;\\!\\!$}10^{\textrm{-}5}$ \| 1.60 \| 0.240 \| 88$\mbox{$\cdot\;\\!\\!$}10^{\textrm{-}5}$ \| 1.19 comb. $h$-reco \| $16$ \| $4.075$ \| $1235$ \| 0.0074 \| $3.784$ \| $1396$ \| 0.0060 \| $3.017$ \| $1575$ \| 0.0042 $\lvert\tilde{m}_{jj^{\prime}}\\!-80\rvert\\!<\\!\tilde{\delta}$ \| \| 0.211 \| 56$\mbox{$\cdot\;\\!\\!$}10^{\textrm{-}5}$ \| 1.85 \| 0.213 \| 63$\mbox{$\cdot\;\\!\\!$}10^{\textrm{-}5}$ \| 1.58 \| 0.213 \| 72$\mbox{$\cdot\;\\!\\!$}10^{\textrm{-}5}$ \| 1.17 comb. $h$-reco \| $10$ \| $3.025$ \| $787.9$ \| 0.0087 \| $2.624$ \| $905.0$ \| 0.0064 \| $2.103$ \| $1006$ \| 0.0046 $\lvert\tilde{m}_{jj^{\prime}}\\!-80\rvert\\!<\\!\tilde{\delta}$ \| \| 0.156 \| 36$\mbox{$\cdot\;\\!\\!$}10^{\textrm{-}5}$ \| 1.72 \| 0.148 \| 41$\mbox{$\cdot\;\\!\\!$}10^{\textrm{-}5}$ \| 1.36 \| 0.148 \| 46$\mbox{$\cdot\;\\!\\!$}10^{\textrm{-}5}$ \| 1.02 Table 4: Impact of the different levels of cuts on the $e\nu_{e}$+jets final states for the $gg\to h\to WW$ production and decay signal and the $W$+jets background as obtained from SHERPA. Cross sections $\sigma_{S}$, $\sigma_{B}$, acceptances $\varepsilon_{S}$, $\varepsilon_{B}$ and $S/B$, $S/\sqrt{B}$ ratios are shown for Higgs boson masses of $M_{h}=165\ \mathrm{GeV}$, $M_{h}=170\ \mathrm{GeV}$ and $M_{h}=180\ \mathrm{GeV}$. Note that $\tilde{m}_{ij}=m_{ij}/\mathrm{GeV}$ and $\tilde{\delta}=\delta/\mathrm{GeV}$. Significances were calculated using Eqs. (4.2) assuming ${\mathcal{L}}=10\ \mathrm{fb}^{-1}$ of integrated luminosity, counting both electrons and muons and combining Tevatron experiments. We can now go ahead and calculate the $S/B$ ratios and $S/\sqrt{B}$ significances. For various Higgs boson mass hypotheses, Table 4 and Tables 7–10 of Appendix B.1 list signal and $W$+jet-background cross sections, acceptances, $S/B$ ratios and significances at different levels of cuts for the selection procedures discussed in this subsection. The SHERPA simulation runs obtained with the MSTW2008 LO PDFs have been used to extract the results of all tables except those of Table 9 presented in Appendix B.1 which are based on a set of runs taken with the CTEQ6.6 PDFs. In Appendix B.1 we will then also briefly discuss the differences that can be seen between the predictions for the two PDF sets. We now turn to the discussion of the tables. Their setup is as follows: the rows represent different stages in the cut-flow, Higgs boson reconstruction strategies, and mass window cuts, while the third through fifth columns contain the outcomes for different Higgs boson masses. The second column indicates the mass window cut (in $\mathrm{GeV}$, referred to as $\Delta$ in the text), which has been applied to all reconstructed Higgs boson candidates. Similarly, the number in square brackets next to each Higgs boson mass is the dijet mass window cut (referred to as $\delta$, also in $\mathrm{GeV}$). At every analysis level, six numbers are displayed for each Higgs boson mass. The top row displays the LO signal cross section (in $\mathrm{fb}$), the LO $W$+jets cross section (in $\mathrm{fb}$) and $S/B$ at $10\ \mathrm{fb}^{-1}$ of integrated luminosity, calculated including $K$-factors and factors of 2 following Eqs. (4.2). The bottom three numbers in each table entry are the signal and background efficiencies and $S/\sqrt{B}$. Of these entries, $S/\sqrt{B}$ is displayed in bold. For the first set of tables, Table 4 and 7 (see Appendix B.1), we concentrate on Higgs boson masses greater than $\approx 162\ \mathrm{GeV}$ – above the $WW$ threshold. Higgs boson masses below the $WW$ threshold have additional challenges, which we explore in a later subsection. The rows are divided into three groups. In the first group, rows 1–4, the baseline selection cuts, as described in Section 4.1, are applied.101010Note that the “lepton & MET cuts” level also includes a lepton–jet separation of $\Delta R^{\mathrm{lep}\textrm{--}\mathrm{jet}}>0.4$ in the presence of jets. In the second group, rows 5–8, events are selected using the “naive” criteria, then retained if their reconstructed sum falls within various Higgs boson and dijet mass windows. Finally, in the last set of rows, 9–15, we select events with the “comb. $h$-reco” algorithm, then apply several different mass windows. The effect of the mass window cuts, with either the “naive” or “comb. $h$-reco" selection scheme, are fairly intuitive; mass windows always help because they emphasize the peaks in the signal in comparison to a featureless $W$+jets background. Tighter mass windows are usually, but not always, better. Clearly, among the three groups the combinatorial selections give the best significances, followed by the naive ones, which already improve over the baseline selection cuts. Comparing rows with identical cuts but different selections (“naive” versus “comb. $h$-reco”), such as rows 5 and 9, or 7 and 11, the combinatorial Higgs boson reconstruction is better across all Higgs boson masses by roughly 30%. The difference can be traced to events where one of the hardest jets comes from I/FSR rather than from one of the jets of the $W$ decay. Had we truncated our treatment of the background at the matrix-element level (or even at matrix-element level plus some Gaussian smearing, as in Ref. [7], additional jet activity arising from I/FSR would be absent and the “comb. $h$-reco” scheme would give the same result as the “naive $h$-reco” scheme. Incorporating these relevant I/FSR jets using a complete, matrix-element plus parton-shower treatment of the background, we notice that the “naive” scheme is no longer the best option. The ME+PS merging thereby allows us a fully inclusive description of $W$+2-jet events on almost equal footing with the related NLO calculation, however with the advantage of accounting for multiple parton emissions at leading-logarithmic accuracy. These effects are pivotal to obtain reliable results for the combinatorial selections. Showering effects are not just limited to the background. In particular, the width of the Higgs boson candidates reconstructed from showered events is much broader than the reconstructed width derived from parton level. In fact, after showering, the reconstructed Higgs boson peak is typically so broad that the tightest mass windows used in the tables ($\Delta=5\ \mbox{and}\ 8\ \mathrm{GeV}$) cut out some of the signal and yield worse significances than broader windows. For example, the combinatorial selections supplemented by a dijet mass window yield FWHM of about $10\ \mathrm{GeV}$ at the shower level, while the FWHM at the parton level are reduced down to $2\ \mathrm{GeV}$ – that basically is the width of one bin. If we relied on the matrix-element level results, we would obtain far too promising $S/\sqrt{B}$. Focusing on the $M_{h}=180\ \mathrm{GeV}$ test point and the “comb. $h$-reco” with a dijet mass window, we would find the significances increasing from $1.5$ for $\Delta=25\ \mathrm{GeV}$, $2.2$ for $\Delta=10\ \mathrm{GeV}$ to $3.0$ for $\Delta=5\ \mathrm{GeV}$. These numbers should be contrasted with those in Table 4, namely $1.12$, $1.19$ and $1.02$, respectively. Figure 2: $S/\sqrt{B}$ significances for Higgs boson masses varying from $M_{h}=110$ to $220\ \mathrm{GeV}$ after different levels of cuts. The numbers are taken from Tables 4 and 7–10, which reflect in more detail the outcome of the analysis based on the invm selection procedure for $e\nu_{e}$+jets final states originating from the $gg\to h\to WW$ signal and the $W$+jets background. Results are shown for Higgs boson masses below and above the $WW$ mass threshold; the threshold region has been left out though. All significances were calculated according to Eqs. (4.2) under the assumption of an integrated luminosity of ${\mathcal{L}}=10\ \mathrm{fb}^{-1}$ and including electron and muon channels, i.e. $f_{\ell}=2$. To conclude this discussion, it is illustrative to show a plot of the significances versus Higgs boson masses for various selections as presented in the tables (Table 4 and Tables 7–10 in Appendix B.1), all of which is summarized in Figure 2. The significance, at least after this level of analysis, reaches a maximum of $\sim 2.0$. The highest significance occurs, as expected, near the $WW$ threshold. For heavier Higgs bosons, as the $WW$ decay mode becomes subdominant to the $ZZ$ mode, the significance drops slowly, reaching $\sim 1.0$ at $M_{h}\sim 185\ \mathrm{GeV}$. By gradually enhancing the selections the gain in significance remains approximately equal over the whole region of large $M_{h}$; this is indicated by the parallel shifts of the respective significance curves. Hence, the differences seen in the significances per $M_{h}$ test point are mainly driven by the behaviour of the total inclusive cross section for the signal. Looking back at Tables 4 and 7, we in fact realize that the acceptances $\varepsilon_{S}$ and $\varepsilon_{B}$ are rather similar at any selection step (for each row), only mildly varying across the different Higgs boson test mass points. For Higgs boson masses below threshold (Table 8 in Appendix B.1), as we will discuss in the next sections, the drop-off is more severe. Not only does the branching fraction to $WW^{}$ fall rapidly, but the signal becomes more background-like once the two $W$ bosons from the Higgs boson decay cannot both be produced on-shell. The significances shown in Figure 2 reflect our best estimates, however, we have also performed several checks on the stability of these significances under slight variations in the analysis. These checks not only include – as mentioned earlier – varying parton distribution functions, but also varying jet definitions, etc. and are summarized in Appendix B.1. #### 4.2.1 Reconstruction below the on-shell diboson mass threshold In the above-threshold case there is good hope that the idealized approach of considering the neutrino as a fully measurable particle will not lead to results, which are sizably different from those obtained by a realistic treatment of neutrinos. This is based on the fact that in most cases the leptonic $W$ will be on its mass shell. The approximation $m_{e\nu_{e}}\approx M_{W}$ can in principle be used to determine the neutrino’s longitudinal momentum – up to a twofold ambiguity – by employing the lepton and missing transverse energy measurements. Below the $WW$ mass threshold one of the $W$ bosons will be off-shell, so that the simple ansatz in calculating $p_{\parallel,\nu_{e}}$ will be rather inaccurate. Hence, it a priori is not clear whether an event selection based on invariant-mass windows will give an overall picture that can be maintained in more realistic scenarios. Nevertheless here we briefly establish what kind of significances may be achievable assuming we had knowledge about the off-shellness (the actual mass) of the leptonic $W$ boson. This will give us a benchmark, which we may use to assess more realistic reconstruction approaches. When we apply the same analysis as above the $WW$ threshold, we find significances as summarized in Table 8 of Appendix B.1. They are visualized in Figure 2. The numbers demonstrate that we quickly lose sensitivity below the $WW$ threshold, in particular for test points $M_{h}\lesssim 130\ \mathrm{GeV}$. This happens for three reasons (which apply to the signal only): one factor is the decline of the total inclusive signal cross section $\sigma^{(0)}_{S,\mathrm{NNLO}}$ towards lower $M_{h}$, which actually is comparable to that seen for large $M_{h}$. As shown in Table 1, this effect is not as drastic as one would assume from the drop in the $h\to W^{}W^{}$ branching ratios; it is partly compensated by the rising gluon–gluon fusion rate for low $M_{h}$. In contrast to the above-threshold case, there are yet two more factors coming into the equation. Firstly, the basic selection cuts affect the signal more severely;111111For $M_{h}=110\ \mathrm{GeV}$, only about 7% of the events survive, while 45–49% of the signal is kept above threshold (cf. the respective 1st rows in Table 8 and 3rd rows in Tables 4 and 7). secondly, the low $M_{h}$ signals that pass the baseline selection are often penalized because of substantial off-shell effects. In particular, the Higgs boson propagator can be pushed far off-shell and the Higgs boson reconstruction will fall outside the mass window, such that the event will be discarded. The tendency for lighter Higgs bosons to go off-shell increases, since the basic cuts make it extremely unlikely for the leptonic and hadronic $W$ masses to drop below $\sim 30\ \mathrm{GeV}$. Figure 3: Mass spectra $m_{e\nu_{e}jj^{\prime}}$ after the combinatorial reconstruction of Higgs boson candidates for very wide Higgs boson mass windows. Results are shown for $M_{h}=130\ \mathrm{GeV}$ and $M_{h}=180\ \mathrm{GeV}$ and $e\nu_{e}$+jets final states originating from the $gg\to h\to WW$ signal (peaked distributions) and the $W$+jets background (flat distributions). Figure 3 shows the $m_{e\nu_{e}jj^{\prime}}$ spectra including shower effects for signals and backgrounds at $M_{h}=130$ and $180\ \mathrm{GeV}$ after the combinatorial Higgs boson reconstruction has been applied using wide Higgs boson mass windows ($\Delta\equiv M_{h}$). The parton showering washes out the peaks, therefore reduces and broadens them. Both signal distributions develop a softer tail above $M_{h}$ as a result of the jet combinatorics. For $M_{h}=130\ \mathrm{GeV}$, the tail plateaus due to the off-shell effects mentioned earlier. Figure 3 also illustrates why the value of the significance jumps up significantly (as shown in Figure 2) when we tighten the Higgs boson mass window from $\Delta=25$ to $10\ \mathrm{GeV}$ for $M_{h}=130\ \mathrm{GeV}$. This effect arises because we place our window cuts in a steeply rising $W$+jets background. When we studied which choice of mass window gives us the best results in terms of separating signal from background, it came as somewhat of a surprise that we did not have to alter the additional dijet mass constraint of $M_{W}-\delta<m_{jj^{\prime}}<M_{W}+\delta$, $\delta=20\ \mathrm{GeV}$. Our studies indicate that it is helpful to have the hadronically decaying $W$ boson to be close to its on-shell mass $M_{W}$. The $W$ boson decaying leptonically is then forced to go off-shell ($m_{e\nu_{e}}<M_{W}$), a kinematic configuration at odds with most $W$+jets events.Cutting on $m_{jj^{\prime}}$ therefore helps suppress the dominant background and, moreover, should also be convenient to demote the production of multi-jets efficiently. For the tighter Higgs boson mass windows, our results show that a simple one- sided lower cut on $m_{jj^{\prime}}$, i.e. $m_{jj^{\prime}}>M_{W}-\delta$ is slightly more efficient than using any type of dijet mass window. The one- sided cut improves the significances as given in Table 8 by 1–2%. The removal of the upper bound on $m_{jj^{\prime}}$ has however negligible effects on selections using broad Higgs boson mass windows. As a consequence of keeping an $m_{jj^{\prime}}$ constraint the leptonically decaying $W$ will almost always be off-shell, such that the reconstruction of the longitudinal component of the neutrino’s four-momentum cannot succeed without a good guess of the mass of the $e\nu_{e}$ pair. We will address this issue in Section 4.3. #### 4.2.2 Effect of the subdominant backgrounds In this section, we examine to what extent the significances of the ideal Higgs boson reconstruction will be diluted by contributions from the electroweak and top-pair production of the $e\nu_{e}$+jets final states. To this end we apply the analysis as established so far, without any modification. Figure 4: Cut efficiencies $\varepsilon_{B}$ and $\varepsilon_{S}$ for $e\nu_{e}$+jets final states and different invm naive and combinatorial Higgs boson reconstructions taking the mass points $M_{h}=130$, $170$ and $210\ \mathrm{GeV}$. The selections are labelled by the row numbers as assigned in Tables 4 and 7–10; row 3 marks the baseline selection (used as benchmark). The left pane exhibits the efficiencies found for the minor backgrounds – electroweak and top–antitop pair production (dashed and solid lines, respectively) – whereas the right pane displays the $\varepsilon_{B}$ for the $W$+jets background (dashed lines) as well as the $\varepsilon_{S}$ of the $gg\to h\to WW$ signal (solid lines). The two plots to the right compared with each other nicely visualize why the combinatorial outperforms the naive selection: the signal cut efficiencies get increased, while, for $W$+jets, the cuts remain about as effective as for the naive approach. Also notice the drop of the $M_{h}=130\ \mathrm{GeV}$ signal curves – they show the penalty in employing the same basic cuts as above the $WW$ threshold. Figure 5: Single- background and total significances as a function of $M_{h}$ for three different Higgs boson candidate invm selections as denoted on top of each panel. The $e\nu_{e}$+jets final states are generated from the $gg\to h\to WW$ signal, $W$+jets, electroweak and $t\bar{t}$ production backgrounds. All $S/\sqrt{B_{i}}$ were calculated according to Eqs. (4.2) assuming an integrated luminosity of ${\mathcal{L}}=10\ \mathrm{fb}^{-1}$ and including electron and muon channels, i.e. $f_{\ell}=2$. The total-background significances were obtained with Eq. (24). The lower plots hold the ratios of $S/\sqrt{B_{\mathrm{tot}}}$ over $S/\sqrt{B}$ for $W$+jets only. Note that the large $t\bar{t}$ significances obtained by the naive selection (left panel) result from the failure of the relatively harder leading-jet pairs to satisfy the mass window constraints. The first thing to notice is the total inclusive LO cross sections for these minor backgrounds are ${\mathcal{O}}(1)\ \mathrm{pb}$ – substantially smaller than the $W$ production contribution. After the application of the basic cuts, the inclusive $e\nu_{e}$+2-jet cross sections drop to about $0.5\ \mathrm{pb}$, a factor of 40 below the major background. Including all the various $K$-factors, see Table 1 and Eqs. (17), (21), (22), we find that the total significance, $\frac{S}{\sqrt{B_{\mathrm{tot}}}}\;=\;\frac{1}{\sqrt{\,\sum_{i}\left(\frac{S}{\sqrt{B_{i}}}\right)^{-2}}\,}\;=\;\left(\left(\frac{S}{\sqrt{B}}\right)^{-2}+\left(\frac{S}{\sqrt{B_{\mathrm{ew}}}}\right)^{-2}+\left(\frac{S}{\sqrt{B^{b\textrm{-}\mathrm{veto}}_{t\bar{t}}}}\right)^{-2}\right)^{-\frac{1}{2}}\ ,$ (24) at the basic selection level is only 2% smaller compared to the significance $S/\sqrt{B}$ using only $W$+jets.121212The cross sections stated are LO-like cross sections as obtained with SHERPA: before (after) the basic cuts, we find $1.21$ and $0.886\ \mathrm{pb}$ ($540$ and $508\ \mathrm{fb}$) for the electroweak and $t\bar{t}$ backgrounds, respectively; the resulting single- background significances turn out to be almost 6 and 10 times larger than the $W$+jets $S/\sqrt{B}$. Switching from the baseline selection to a combinatorial selection and finite dijet mass window, the single minor-background significances improve by up to 50% above (100% below) the $WW$ threshold. So, for beyond-baseline $h$ reconstructions, the significance corrections owing to the inclusion of the minor backgrounds will be of the same order as before. This is documented in both Figures 4 and 5. In the former we compare the cut efficiencies between all backgrounds and the signal (shown together with the $W$+jets background in the plots to the right in Figure 4) for various naive and combinatorial Higgs boson candidate selections. Firstly, no background cut efficiency ever exceeds any of the $\varepsilon_{S}$. In all cases the background curves decrease more strongly when tightening the selection. Secondly, the pattern we observe for the minor-background cut efficiency curves resembles by and large those of the major background.131313In particular, the minor-background cut efficiencies show more pronounced drops, if one enhances the baseline to a Higgs boson candidate selection, introduces the dijet mass window $\delta$ or tightens the $\Delta$ Higgs boson mass range. Notice that the naive Higgs boson reconstruction very efficiently beats down the $t\bar{t}$ background. This is because the two leading jets turn out harder compared to all other cases. The presence of a sufficient number of subleading jets however makes the selection based on jet combinatorics pick a pair of soft jets, and, on the contrary almost too effective for Higgs boson masses above the $WW$ threshold. For these reasons, the single-background significances plotted versus $M_{h}$ follow the trend found for the $W$+jets contribution, but remain well above the $W$+jets significances.141414Both the electroweak and $t\bar{t}$ production significances show the same strong enhancement around $M_{h}=130\ \mathrm{GeV}$ as a result of the effect discussed around Figure 3 which is due to the use of a tight Higgs boson mass window. As for $W$+jets, the minor backgrounds fall rapidly for decreasing $M_{h}$. All of which is exemplified in Figure 5 using three Higgs boson candidate selections, which impose a dijet mass window (corresponding to the rows 8, 11 and 13 in Tables 4, 7 and 10), namely the naive method with $\Delta=10\ \mathrm{GeV}$ (left panel) and the combinatorial method with the same and broader window of $\Delta=25\ \mathrm{GeV}$ (middle panel). The total significances $S/\sqrt{B_{\mathrm{tot}}}$ resulting from combining the three single backgrounds are also shown. In fact, they only decrease by 1–5% as demonstrated by the ratio plots in the lower part of Figure 5. The high $M_{h}$ region is found to receive the larger, ${\mathcal{O}}(\mbox{5\%})$ corrections once the electroweak and $t\bar{t}$ contributions are included in the overall background. As noted early in Section 3.2, experimenters have estimated this fraction of events with 5% implying a 2.5% drop in significance. It is reassuring to be able to confirm this expectation with our results. Slightly contrary to the expectation, we identify the electroweak as the leading minor background in all selections. Based on these results it is easy to conclude; at this stage of our analysis we do not have to worry about contributions from minor backgrounds. Although additional handles exist to further reduce these backgrounds or supplement the (here conservatively chosen) $b$-jet veto, it is of far more importance to find ways to diminish the $W$+jets background. We postpone this discussion until Section 4.4. ### 4.3 More realistic Higgs boson reconstruction methods Up to this point we have ignored one big problem, namely the neutrino problem. In our selection based on the reconstruction of invariant masses – which we dubbed invm approach – we currently treat neutrinos as if we were able to measure them like leptons. This is, of course, unrealistic and before we can talk about further significance improvements, we have to investigate in which way our analysis may fall short when switching to more experimentally motivated Higgs boson candidate selections. Under experimental conditions, missing energy is taken from the $\vec{p}_{T}$ imbalance in the event. However, in our analysis we then make a small simplification and identify the missing energy with the neutrino’s transverse momentum as given by the Monte Carlo simulation. There are multiple choices for how to proceed.Recall that whatever method we pick acts as a selection criterion; we decide which two jets to keep in the event based on these variables, therefore we want to design variables, which are best at correctly picking out the jets from a Higgs boson decay. One way to proceed is to give up complete reconstruction and to work solely with transverse quantities; this is clean and unambiguous, but we throw out information. The second approach is to attempt to guess the longitudinal neutrino momentum by requiring that some or all of the final-state objects reconstruct an object we expect, such as a $W$ or Higgs boson. Full reconstruction then gives access to a larger set of observables, therefore keeps more handles and information, but it is also more ambiguous. $M_{h}/M_{W}$ \| \| pzmw \| \| pzmh \| \| mt \| \| mtp ---\|---\|---\|---\|---\|---\|---\|---\|--- $<2$ \| \| $m^{(\nu_{e})}_{T,e\nu_{e}jj^{\prime}}$ \| \| $m^{(\nu_{e})}_{T,e\nu_{e}jj^{\prime}}$ \| \| $m^{(\nu_{e})}_{T,e\nu_{e}jj^{\prime}}$ \| \| $m^{(\nu_{e})}_{T,e\nu_{e}jj^{\prime}}$ $>2$ \| \| $m_{T,e\nu_{e}jj^{\prime}}$ \| \| $m^{(\nu_{e})}_{T,e\nu_{e}jj^{\prime}}$ \| \| $m^{(\nu_{e})}_{T,e\nu_{e}jj^{\prime}}$ \| \| $m_{T,e\nu_{e}jj^{\prime}}$ Table 5: The preferred choice of definition for the 4-particle transverse mass shown for each of the more realistic Higgs boson candidate selections. The $m_{T}$ definitions are given in Eqs. (8) and (9). To remove some of the combinatorial headache, we use $e\nu_{e}jj^{\prime}$ and $jj^{\prime}$ (transverse) mass windows as before; moreover, we can impose criteria on subsets of the event. For example, if the (3-particle) mass of the visible system $m_{ejj^{\prime}}$ is greater than the test Higgs boson mass, that particular choice of jets is unphysical and we can move on to the next choice. A second constraint we often impose is that the 4-particle transverse mass does not exceed the upper bound on the Higgs boson mass window: $m_{T,e\nu_{e}jj^{\prime}}\leq M_{h}+\Delta$. As to the definition of $m_{T}$, we generally use the definitions stated in Section 2, see Eqs. (8) and (9). For our selections, we found that the distinction of the two $m_{T}$ definitions in fact only matters when we calculate the 4-particle transverse masses. Accordingly, each selection comes in two versions either using $m_{T,e\nu_{e}jj^{\prime}}$ or $m^{(\nu_{e})}_{T,e\nu_{e}jj^{\prime}}$. In Table 5 we summarize for each type which version is more appropriate to use and in what context. Whenever we refer to a specific selection in due course, we understand it according to the findings listed in Table 5. With these criteria in hand, the different selection methods are specified as follows: • mt: we want to test a method where the final state of the Higgs boson decay will be identified purely with the help of transverse masses rather than invariant masses. To this end we calculate $m_{T,e\nu_{e}jj^{\prime}}$ according to Eqs. (8) or (9) and prefer the final state giving us the 4-particle transverse mass closest to $M_{h}$. Owing to $m_{T,e\nu_{e}jj^{\prime}}\leq m_{e\nu_{e}jj^{\prime}}$ the test mass window is placed on the lower side only, $M_{h}-2\,\Delta<m_{T,e\nu_{e}jj^{\prime}}<M_{h}$, with double the size as compared to the other selections. The advantage, but also disadvantage of the method is there is no reconstruction. Avoiding reconstruction eliminates uncertainties owing to constraining masses plus resolving ambiguities, but means we have no access to longitudinal and invariant-mass observables involving the neutrino. For the next two selections, we aim to approximately determine the longitudinal momentum of the neutrino, $p_{\parallel,\nu_{e}}$, using knowledge about which value for $m_{e\nu_{e}}$ should likely be reconstructed by the combined system,$p_{e}+p_{\nu_{e}}$.151515Provided the MET cut was passed, we assume here that all MET in the event has been produced by a single neutrino. When we write $\displaystyle m^{2}_{\ast,i\nu_{e}k..}\;\approx\;m^{2}_{i\nu_{e}k..}\;=\;$ (25) $\displaystyle m^{2}_{ik..}\,+\,2\left(\sqrt{m^{2}_{ik..}+\,\vec{p}^{2}_{ik..}}\sqrt{\vec{p}^{2}_{T,\nu_{e}}+p^{2}_{\parallel,\nu_{e}}}-\vec{p}_{T,ik..}\cdot\vec{p}_{T,\nu_{e}}-p_{\parallel,ik..}\,p_{\parallel,\nu_{e}}\right)\ ,$ using the “(in)visible” subsystem notation, we note that such problems can be solved up to a twofold ambiguity. The difference among the two selections lies in how particles are grouped in Eq. (25) and how the twofold ambiguity is resolved.161616If the solutions are complex-valued, we only assign the real part to describe $p_{\parallel,\nu_{e}}$ with no ambiguity left to resolve. * • pzmw: in this selection we use the $W$ mass constraint to solve for the neutrino momentum: $m^{2}_{,e\nu_{e}}=M^{2}_{W}$. The ambiguity is then resolved by picking the neutrino $p_{z}$ solution, which brings the reconstructed mass $m_{e\nu_{e}jj^{\prime}}$ more closely to the Higgs boson test mass $M_{h}$. For true signal events, it then is more likely to find the reconstructed $m_{e\nu_{e}jj^{\prime}}$ matching the Higgs boson mass. The tricky part is to pick the best choice for the $m^{2}_{,e\nu_{e}}$ constraint – meaning is $M^{2}_{W}$ always optimal given that the $W$ boson may be off-shell? For pzmw, we do the following: first, we inspect the transverse mass, $m_{T,e\nu_{e}}$, of the $e\nu_{e}$ subsystem in each event. If $m_{T,e\nu_{e}}\geq M_{W}$ we choose $m_{,e\nu_{e}}=m_{T,e\nu_{e}}$, otherwise we pick $m_{,e\nu_{e}}=M_{W}$ as long as $M_{h}>2\,M_{W}$ or $0.9<m_{T,e\nu_{e}}/M_{W}<1.0$. That is, above and around the $WW$ threshold, we take $m_{e\nu_{e}}$ towards $M_{W}$. If below threshold and $m_{T,e\nu_{e}}/M_{W}<0.9$, the mass estimate is chosen taking various subsystem invariant and transverse masses into account but enforcing $m_{,e\nu_{e}}$ to lie between $m_{T,e\nu_{e}}$ and $M_{W}$. For example, if $m_{jj^{\prime}}>2\,m_{T,jj^{\prime}}$ we set $m_{,e\nu_{e}}=m_{T,e\nu_{e}jj^{\prime}}-m_{jj^{\prime}}$ while otherwise $m_{,e\nu_{e}}=m_{T,e\nu_{e}jj^{\prime}}-m_{T,jj^{\prime}}$ unless $m_{ejj^{\prime}}>m_{T,e\nu_{e}jj^{\prime}}$ where we say $m_{,e\nu_{e}}=m_{T,e\nu_{e}}$. * • pzmh: we again infer the neutrino’s longitudinal momentum from mass constraints. Although technically similar to pzmw – with the “visible” subsystem entering Eq. (25) now being $\\{e,j,j^{\prime}\\}$ – we here turn the idea around and already require $m_{e\nu_{e}jj^{\prime}}\approx M_{h}$ in order to solve for $p_{\parallel,\nu_{e}}$. That is to say we enforce the combined system, $p_{ejj^{\prime}}+p_{\nu_{e}}$, to mimic a Higgs boson signal mass while leaving us with reasonable leptonic $W$ masses $m_{e\nu_{e}}$ at the same time. When reconstructing the signal these observables are likely correlated, while for the background they are uncorrelated apart from kinematic constraints. The details of the method are: we specify the target mass via $m_{,e\nu_{e}jj^{\prime}}=M_{h}$ unless we find $m_{T,e\nu_{e}jj^{\prime}}/M_{h}\geq 0.94$, i.e. the 4-particle transverse mass turns out too large already so that $m^{2}_{,e\nu_{e}jj^{\prime}}=m^{2}_{T,e\nu_{e}jj^{\prime}}/0.95$ is the more appropriate choice. We approximate the leptonic $W$ boson mass by $m_{,e\nu_{e}}=M_{h}-m_{jj^{\prime}}$ freezing it at $m_{,e\nu_{e}}=M_{W}$ if this difference exceeds $M_{W}$. We however require $\min(m_{,e\nu_{e}})=m_{T,e\nu_{e}}$. Taking this estimate, we can form the absolute difference $\delta m_{e\nu_{e}}=\|m_{e\nu_{e}}-m_{,e\nu_{e}}\|$ using the reconstructed mass $m_{e\nu_{e}}$ for each possible neutrino solution. In the presence of two solutions we define, as a measure of the longitudinal activity, $b_{ij}=m_{\perp,ij}\exp\|y_{ij}\|=\max\\{E_{ij}\pm p_{\parallel,ij}\\}$ with $m^{2}_{\perp,ij}=m^{2}_{ij}+p^{2}_{T,ij}$ and pick the solution that generates the smaller $b_{e\nu_{e}}$, i.e. the $e\nu_{e}$ subsystem less likely going forward. We do so unless the other solution’s $\delta m_{e\nu^{\prime}_{e}}$ drops below $\delta m_{e\nu_{e}}$ and $(b_{jj^{\prime}}+b_{e\nu^{\prime}_{e}})/(m_{jj^{\prime}}+m_{e\nu^{\prime}_{e}})<(b_{jj^{\prime}}+b_{e\nu_{e}})/(m_{jj^{\prime}}+m_{e\nu_{e}})+\delta x$ is satisfied; this is when we pick conversely ($\delta x=0.5$ if $M_{h}<2\,M_{W}$ otherwise $\delta x=1.0$). Finally, we ensure that the $\\{e,\nu_{e},j,j\\}$ set minimizing $\delta m_{e\nu_{e}}$ will be preferred by the overall selection among all sets reconstructing the same 4-particle mass. Note that we do not reject the selected ensemble if the $\delta m_{e\nu_{e}}$ deviation becomes too large; we leave this potential to be exploited by supplemental cuts, which we discuss in Section 4.4. Figure 6: Single $W$+jets background significances as a function of $M_{h}$ for 4 different realistic Higgs boson candidate selections using jet combinatorics. The selection types are denoted on top of each panel. Results are shown for 4 different mass window parameter settings each, overlaying the sort of optimal case defined by the invm combinatorial $h$ reconstruction for $\Delta=10$, $\delta=20\ \mathrm{GeV}$, which also serves as the main reference. The $e\nu_{e}$+jets final states are generated from the signal, $gg\to h\to WW$, and the $W$+jets background. All $S/\sqrt{B}$ were calculated according to Eqs. (4.2) assuming an integrated luminosity of ${\mathcal{L}}=10\ \mathrm{fb}^{-1}$ and including electron and muon channels, i.e. $f_{\ell}=2$. The lower plots hold the ratios of realistic over ideal $S/\sqrt{B}$. The thick lines are drawn with respect to the main reference, while the thin lines are taken from comparing to the invm combinatorial selection relying on the same window parameters. Figure 7: $S/B$ ratios as a function of $M_{h}$ for 4 types of realistic combinatorial Higgs boson candidate selections, each with their window parameters chosen as to reach maximal significance. The $e\nu_{e}$+jets final states are generated from the $gg\to h\to WW$ signal and $W$+jets background. The $S/B$ were calculated from the $\sigma_{S}$ and $\sigma_{B}$ as obtained after the selection, the signal $K$-factors of Table 1 and $K_{B}$ as given in Eq. (17). We explored in detail how each of these selection criteria compare to the ideal case. Figure 6 shows the significances per $M_{h}$ test point that we can achieve running the different combinatorial selections for various, reasonable window parameters. In the upper panels we display them directly on top of the (quasi optimal) ideal case, i.e. the invm combinatorial $h$ candidate selection. The plots on the right and in the center respectively exhibit the results of the mt and pzmh methods for the whole $M_{h}$ test range. The pzmw method yields similar, yet slightly worse results below the $WW$ mass threshold compared to pzmh. Therefore, we split the leftmost pane into two subplots: on the right, one finds the pzmw results for masses above threshold; on the left we then already reveal the outcome of the mtp selection, whose discussion we postpone until the next subsection. The bottom-row plots of Figure 6 depict the respective survival fractions: the significance ratios for each selection and different window parameters always taken with respect to the quasi optimal case (cf. the thick lines with symbols). To indicate the net effect of the more realistic approaches, we also show in each case the significance loss relative to the invm selection using exactly the same mass windows as the realistic one (cf. the thin lines with no symbols). We observe serious losses, larger than 50%, if one uses selections with tighter window parameters and/or runs for $M_{h}$ points further away from $2\,M_{W}$. The reconstruction methods (pzwm, pzmh) do not work better than 90% of the quasi optimal case. This only is improved by the mt approach making use of broad Higgs boson mass windows where one can reach up to about 100%. However, lowering $\Delta$ here quickly results in losing sensitivity. Related to the quasi optimal invm selection, the various results point us to work with medium-sized Higgs boson mass windows always imposing the dijet mass cuts. Tighter $\Delta$ constraints may help improve the outcome of the reconstruction types, but are of disadvantage in the measurement. In all selections the below-threshold region is especially problematic. One might settle for 65–80% of the ideal significances, but if we want to get a better handle on the low $h$ boson masses, in particular include the $M_{h}=130\ \mathrm{GeV}$ mass point, we have to push further – which we do so in Section 4.3.1. Focusing on the above-threshold region, we see that pzmw slightly outperforms pzmh over the whole range; only for the near-threshold region up to $M_{h}=180\ \mathrm{GeV}$ this is topped by the mt selection for medium-sized $\Delta$ windows. This is somewhat surprising, but seems plausible, if one considers that signal events are central in rapidity and manifest themselves in larger transverse activity on average. This has to be opposed to the $W$+2-jets background whose events tend to populate phase space more along the beam direction, which generates $y_{e\nu_{e}jj^{\prime}}$ distributions peaking about half an unit away from zero rapidity. Nevertheless the differences between the 3 methods are not conclusive per se; to some extent the selections will shape distributions differently and it is easy to imagine the picture changing if additional cuts are imposed. But, one has to bear in mind, there is a second, very important criterion, the ratio $S/B$, which one wants to maximize. For their optimal window parameters, we show in Figure 7 the $S/B$ curves of the more realistic and ideal selections as functions of $M_{h}$. Even more surprisingly than before, the mt outperforms the neutrino reconstruction methods and, moreover, the ideal $S/B$ are also beaten unless $M_{h}<2\,M_{W}$. Based on this observation one may prefer the methods where the selection utilizes transverse masses, with the only drawback of having no $p_{\parallel,\nu_{e}}$ estimate available. #### 4.3.1 More realistic reconstruction below the on-shell diboson mass threshold We have just seen that the significances achievable in more realistic scenarios drop off considerably below the $WW$ mass threshold, amplifying the loss already present in the ideal case. Therefore, it is of great importance to learn how the reconstruction methods described above can be applied more efficiently in the below-threshold region. We noticed that the mt selection picks up background events, which often fall outside (mostly above) the Higgs boson mass window. As a result, a somewhat different class of background events survives the mt selection procedure compared to utilizing the invm, ideal, approach. This is no surprise since we have already argued that the Higgs boson decays yield an enhanced transverse production with regard to the $W$+2-jet background. Using $M_{h}=130\ \mathrm{GeV}$, Figure 8 exemplifies this by means of the $m_{e\nu_{e}jj^{\prime}}$ and $m_{T,e\nu_{e}jj^{\prime}}/m_{e\nu_{e}jj^{\prime}}$ ratio distributions. We clearly see the large impact on the $W$+jets results being a consequence of enforcing a transverse- rather than invariant-mass window. Imposing the constraint $80\leq m_{T,e\nu_{e}jj^{\prime}}/\mathrm{GeV}\leq 130$ on the background is fairly equivalent to choosing events with larger longitudinal components. This drives the associated invariant masses to higher values whereas the $m_{T}/m$ ratios are shifted to lower ones. Figure 8: Differential 4-particle mass spectra $m_{e\nu_{e}jj^{\prime}}$ (see left panel) and $m_{T,e^{-}\bar{\nu}_{e}jj^{\prime}}/m_{e^{-}\bar{\nu}_{e}jj^{\prime}}$ ratio distributions after combinatorial selection of Higgs boson candidates using a $50\ \mathrm{GeV}$ symmetric mass window centered at a mass of $M_{h}=130\ \mathrm{GeV}$. The mt selection characteristics is compared to the ideal case of choosing candidates according to the invm criteria. Eq. (8) was used to compute the mt criteria. All results are shown for $e\nu_{e}$+jets final states originating from the $gg\to h\to WW$ signal (solid lines) and the $W$+jets background (dashed lines). To give a more quantitative example, we consider the case $M_{h}=130\ \mathrm{GeV}$ for broad (tight) Higgs boson mass windows and $\delta=20\ \mathrm{GeV}$. If we select events using mt, then discard all those events with an invariant mass $m_{e\nu_{e}jj^{\prime}}$ outside $M_{h}\pm 25\ (8)\ \mathrm{GeV}$, we find a quite impressive gain of 82% (360%). Similarly, if we use the invm selection and apply further cuts removing events with $m_{T,e\nu_{e}jj^{\prime}}$ values greater than $M_{h}$ or less than $M_{h}-2\,\Delta=M_{h}-50\ (16)\ \mathrm{GeV}$, we observe that $S/\sqrt{B}$ improves by 12% (drops by 20%). As the invm selection has a better starting $S/\sqrt{B}$ than mt, the final significances are similar in all cases, however the essential point is we can improve the significance by combining the transverse- and invariant-mass selections. To exploit this potential in a more realistic scenario, we use (in a first phase) the mt selection to pick out the 4-particle system, then (in a second phase) reconstruct the neutrino momentum following the pzmw procedure. Compared to the ideal case, this reconstruction works rather inefficiently in identifying background events that yield invariant masses $m_{e\nu_{e}jj^{\prime}}>M_{h}+\Delta$. As it is optimized to the features of the Higgs boson decay signal, pzmw generates $m_{,e\nu_{e}}$ estimates by assuming $m_{T,e\nu_{e}}/m_{e\nu_{e}}$ ratios close to 1. For $W$+jets, these choices often turn out to be sufficiently smaller than the actual masses of the leptonically decaying $W$ boson. The $W$+jets background usually contains an on-shell $W$, a lower $m_{,e\nu_{e}}$ is not ideal and tends to bring down the deconstructed, associated 4-particle masses $m_{e\nu_{e}jj^{\prime}}$. As a result a good fraction of background events possessing true invariant masses exceeding the upper bound on $m_{e\nu_{e}jj^{\prime}}$ (see left plot of Figure 8) are shuffled back into the Higgs boson mass window. Hence, we make adjustments to the pzmw reconstruction used in this symbiosis of selections so that it performs better below threshold. The basic idea is to maximally exploit the differences of signal and background in the leptonic and hadronic $W$ mass distributions. If, after initial mt selection, we constrain the transverse masses of the $e\nu_{e}$ and dijet subsystems by placing cuts that favor $m_{T,e\nu_{e}}\approx M_{W}/2$ and $m_{T,jj^{\prime}}\approx M_{W}$, we enforce off-shell $W\to e\nu_{e}$ decays while keeping those of $W\to jj^{\prime}$ on-shell. This is beneficial to the signal and suppresses, at the same time, the $W$+jets background. Here, we only add requirements on the hadronic subsystem by imposing a minimum value for $m_{T,jj}$. We leave the $m_{T,e\nu_{e}}$ potential to be exploited by the $M_{h}$-dependent cut optimization (which we discuss in Section 4.4), since demanding an upper bound on $m_{T,e\nu_{e}}$ would not only enhance mt but all realistic selections. With this major adjustment, we can now design a promising mt+pzmw selection, which we call mtp: * • mtp: the very first part of computing $m_{T,e\nu_{e}jj^{\prime}}$ is identical to the mt selection. Subsequently, we require $m_{T,jj^{\prime}}$ to be greater than $m^{\mathrm{min}}_{T,jj^{\prime}}=3.51\,M_{h}-0.015\,M^{2}_{h}-135.4\ \mathrm{GeV}$, which we have parametrized in terms of $M_{h}$ for convenience. This concludes the phase of testing the transverse criteria. Accepted $\\{e,\nu_{e},j,j^{\prime}\\}$ candidates are subject to the neutrino reconstruction, whose implementation deviates from pzmw to some extent: we choose $m_{,e\nu_{e}}=0.55\,m_{T,e\nu_{e}}+0.45\,Q$ freezing it below $m_{T,e\nu_{e}}$; here we employ $Q=M_{h}-m_{jj^{\prime}}$ but keep it constant above $M_{W}+\delta x$ ($\delta x=4\ \mathrm{GeV}$). As in all other reconstruction methods we then solve for the longitudinal momentum of the neutrino, cf. Eq. (25), and reject the particular $\\{e,\nu_{e},j,j^{\prime}\\}$ choice if the solutions are degenerate or generate a mean $m_{e\nu_{e}jj^{\prime}}$ mass deviating from $M_{h}$ by more than $\Delta^{\prime}=\max\\{\Delta,20\ \mathrm{GeV}\\}$.171717Here, we do not adopt tight $\Delta$ choices owing to the uncertainties intrinsic to the reconstruction of the $p_{\parallel,\nu_{e}}$ component. In all other cases, we pick the solution giving the smaller $\|m_{e\nu_{e}jj^{\prime}}-M_{h}\|$ and accept it if the reconstructed $h$ mass falls inside the $\Delta^{\prime}$ window around $M_{h}$. The mtp results we can reach in terms of significance and sensitivity are known already from Figures 6 and 7. The increase in $S/\sqrt{B}$ (leftmost plot in Figure 6) and $S/B$ (Figure 7) is highly visible for all of the $h$ test masses below threshold. For the mtp analyses with broadest $M_{h}$ windows, the significance effect is huge compared to the respective ideal selections using the same $\Delta$ parameter. Likewise the signal over background ratio turns out similarly or even better than in the quasi optimal case. Not only does mtp profit from the better selection performance, but we end up with a fully reconstructed neutrino vector, giving us access to a larger set of observables. However, because of the hard cut on $m_{T,jj^{\prime}}$, we notice that the mtp-selected backgrounds are strongly sculpted, washing out a number of shape differences. The effect of the $m_{T,jj^{\prime}}$ cut moreover deteriorates as soon as $M_{h}>2\,M_{W}$. We obtain significances similar to the other methods and since they sculpt the background less, there is no real advantage to applying mtp above threshold, hence, we restrict its use to the low-mass $h$ region. On that note, it remains to be studied whether the $S/\sqrt{B}$, $S/B$ improvements came at the expense of further handles for the $M_{h}$-dependent optimization. As a possible consequence the mtp selection actually might be superseded by an – at this level – inferior selection once enhanced by appropriately designed cuts. ### 4.4 Optimized selection – analyses refinements and (further) significance improvements Having established the more realistic overall picture, we here discuss steps to achieve better signal over background discrimination. After baseline and combinatorial selections, we are interested in cuts that help further increase the significances obtained so far, i.e. $\varepsilon_{B,i}<\varepsilon^{2}_{S,i}$. Our aim is to identify observables for each $M_{h}$ test point that are sufficiently uncorrelated such that simultaneous selections yield a total significance gain in the range: $\max\left\\{\frac{\varepsilon_{S,1}}{\sqrt{\varepsilon_{B,1}}}\,,\,\frac{\varepsilon_{S,2}}{\sqrt{\varepsilon_{B,2}}}\right\\}\;<\;\frac{\varepsilon_{S}}{\sqrt{\varepsilon_{B}}}\;\leq\;\frac{\varepsilon_{S,1}\,\varepsilon_{S,2}}{\sqrt{\varepsilon_{B,1}\,\varepsilon_{B,2}}}\ .$ (26) Above relation is written out for the example of two extra handles, but easily extensible to multi-cut scenarios. As we have shown, the subdominant backgrounds have negligible effects at this level; we therefore concentrate on reducing the $W$+jets background, although the other backgrounds are still included in computing the total significance. In order to be conservative about mass resolutions for hadronic final states at the Tevatron, we will fix the mass window parameters as $\Delta=\delta=20\ \mathrm{GeV}$. As we have done in earlier sections, we divide the optimized selection into three broad Higgs boson mass ranges: below threshold, near threshold, and above threshold. As we vary the Higgs boson mass, we probe different kinematic configurations for the background. For the lowest Higgs boson mass, many background distributions ($H_{T}$, $p_{T}$, etc.) are steeply rising in the region of interest, cut off from below by baseline kinematics. For intermediate Higgs boson masses, the backgrounds tend to be flatter or peaked, while the higher Higgs boson mass region overlaps with backgrounds that are sharply falling. This basic shape behind (many) background distributions drives which cuts are optimal for a given Higgs boson mass. We present a number of distributions in Appendix B.2 to back up the many findings presented in this subsection. The optimized analysis employs only observables constructed out of the momenta of the selected 4-particle system $\\{e,\nu_{e},j,j^{\prime}\\}$. More inclusive observables may be useful: for example, the scalar sum of the two selected-jet $p_{T}$ versus the two hardest-jet $p_{T}$, $H_{T,jj^{\prime}}\leftrightarrow H_{T,12}$. However such observables may also be subject to larger uncertainties from, e.g. the modeling of hard initial state radiation. To reduce these uncertainties one could extend the ME+PS program here, e.g. to inclusive $W$+3-jets, however this is beyond the scope of this study. In the optimized analysis the longitudinal observables, $\Delta\eta_{j,j^{\prime}}$ or $\eta_{e\nu_{e}jj^{\prime}}$, offer only moderate gains in significance (typically, we obtain gains on the order of 3–10% with larger gains occurring for heavy $M_{h}$), but they are insensitive to the exact value of the Higgs boson mass and are therefore more broadly applicable. The total 4-particle pseudo-rapidity $\eta_{e\nu_{e}jj^{\prime}}$ can only be used in the pzmw, pzmh or mtp selections, since reconstruction of the neutrino is necessary. By the same logic, because the pseudo-rapidity difference between the selected jets is independent of the neutrino, cuts on $\Delta\eta_{j,j^{\prime}}$ can be used with all selections (though they are not efficient for mtp). We find that the requirements $\|\Delta\eta_{j,j^{\prime}}\|\lesssim 1.5$ and $\|\eta_{e\nu_{e}jj^{\prime}}\|\lesssim 3.0$ work well for the entire range of Higgs boson masses we are interested in, so we include these cuts into our optimized (pzmh/w and mt) selection. In Appendix B.2 we present examples of $\|\Delta\eta_{j,j^{\prime}}\|$ distributions (Figure 12) and $\eta_{e\nu_{e}jj^{\prime}}$ spectra (Figure 13) after un-optimized combinatorial selections, documenting the usefulness of these cuts. The other useful observables we have found are all transverse, or in some cases based on invariant masses. Unlike the longitudinal variables, the optimal transverse variables and cut values depend strongly on the Higgs boson mass. We also find that transverse and longitudinal observables are largely uncorrelated in this study, so any gains in significance from selections in the transverse observables will add to the gains from $\|\Delta\eta_{j,j^{\prime}}\|$ and $\|\eta_{e\nu_{e}jj^{\prime}}\|$. We refer to Figures 14 and 15 of Appendix B.2 conveniently illustrating this decorrelation for the case $\|\Delta\eta_{j,j^{\prime}}\|$ versus $m_{e\nu_{e}jj^{\prime}}$. $M_{h}$ \| comb. $h$-reco \| leading = major cut \| gain \| subleading cut \| gain \| \| minor cut \| gain ---\|---\|---\|---\|---\|---\|---\|---\|--- $\mathrm{[GeV]}$ \| selection \| $\mathrm{[range\ in\ GeV]}$ \| $\mathrm{[\%]}$ \| $\mathrm{[range\ in\ GeV]}$ \| $\mathrm{[\%]}$ \| \| $\mathrm{[range\ in\ GeV]}$ \| $\mathrm{[\%]}$ \| mtp \| $m_{jj^{\prime}}\ \ [75,\infty]$ \| 17 \| $H_{T,jj^{\prime}}\ \ [76,\infty]$ \| 9 \| \| $p_{T,j}\ \ [38,\infty]$ \| 4 $120$ \| mt \| $m_{T,jj^{\prime}}\ \ [72,\infty]$ \| 70 \| $H_{T,jj^{\prime}}\ \ [72,\infty]$ \| 52 \| \| $H_{T,e\nu_{e}jj^{\prime}}\ \ [108,\infty]$ \| 6 \| pzmw \| $m_{T,e\nu_{e}}\ \ [0,40]$ \| 73 \| $m^{(\nu_{e})}_{T,e\nu_{e}jj^{\prime}}\ \ [0,120]$ \| 38 \| \| $H_{T,jj^{\prime}}\ \ [64,\infty]$ \| 10 \| pzmh \| $m^{(\nu_{e})}_{T,e\nu_{e}jj^{\prime}}\ \ [0,120]$ \| 63 \| $m_{e\nu_{e}jj^{\prime}}\ \ [0,124]$ \| 59 \| \| $m_{\perp,jj^{\prime}}\ \ [76,\infty]$ \| 48 \| mtp \| $p_{T,e\nu_{e}jj^{\prime}}\ \ [9,\infty]$ \| 6 \| $m_{T,e\nu_{e}}\ \ [0,48]$ \| 4 \| \| $m_{\perp,jj^{\prime}}\ \ [76,\infty]$ \| 4 $130$ \| mt \| $H_{T,jj^{\prime}}\ \ [72,\infty]$ \| 38 \| $m_{T,jj^{\prime}}\ \ [68,\infty]$ \| 31 \| \| $m_{jj^{\prime}}\ \ [73,\infty]$ \| 4 \| pzmw \| $m_{T,e\nu_{e}}\ \ [0,48]$ \| 53 \| $m^{(\nu_{e})}_{T,e\nu_{e}jj^{\prime}}\ \ [0,130]$ \| 42 \| \| $m_{\perp,jj^{\prime}}\ \ [72,\infty]$ \| 11 \| pzmh \| $m^{(\nu_{e})}_{T,e\nu_{e}jj^{\prime}}\ \ [0,130]$ \| 48 \| $m_{e\nu_{e}jj^{\prime}}\ \ [0,134]$ \| 46 \| \| $H_{T,jj^{\prime}}\ \ [68,\infty]$ \| 25 \| mtp \| $p_{T,e\nu_{e}jj^{\prime}}\ \ [12,\infty]$ \| 8 \| $H_{T,jj^{\prime}}\ \ [68,\infty]$ \| 3 \| \| $m_{T,e\nu_{e}}\ \ [0,60]$ \| 3 $140$ \| mt \| $H_{T,jj^{\prime}}\ \ [68,\infty]$ \| 24 \| $p_{T,e\nu_{e}jj^{\prime}}\ \ [16,\infty]$ \| 15 \| \| $p_{T,e\nu_{e}jj^{\prime}}\ \ [15,\infty]$ \| 6 \| pzmw \| $m_{T,e\nu_{e}}\ \ [0,56]$ \| 30 \| $m^{(\nu_{e})}_{T,e\nu_{e}jj^{\prime}}\ \ [0,140]$ \| 30 \| \| $m_{\perp,jj^{\prime}}\ \ [70,\infty]$ \| 6 \| pzmh \| $m^{(\nu_{e})}_{T,e\nu_{e}jj^{\prime}}\ \ [0,140]$ \| 29 \| $m_{e\nu_{e}jj^{\prime}}\ \ [0,144]$ \| 28 \| \| $H_{T,jj^{\prime}}\ \ [68,\infty]$ \| 14 \| mtp \| $p_{T,e\nu_{e}jj^{\prime}}\ \ [17,\infty]$ \| 14 \| $H_{T,e\nu_{e}jj^{\prime}}\ \ [116,\infty]$ \| 3 \| \| $H_{T,e\nu_{e}jj^{\prime}}\ \ [116,\infty]$ \| $150$ \| mt \| $p_{T,e\nu_{e}jj^{\prime}}\ \ [20,\infty]$ \| 18 \| $H_{T,jj^{\prime}}\ \ [60,\infty]$ \| 10 \| \| $H_{T,jj^{\prime}}\ \ [56,\infty]$ \| * \| pzmw \| $m^{(\nu_{e})}_{T,e\nu_{e}jj^{\prime}}\ \ [0,150]$ \| 20 \| $p_{T,e\nu_{e}jj^{\prime}}\ \ [18,\infty]$ \| 18 \| \| $p_{T,e\nu_{e}jj^{\prime}}\ \ [18,\infty]$ \| 9 \| pzmh \| $m^{(\nu_{e})}_{T,e\nu_{e}jj^{\prime}}\ \ [0,150]$ \| 19 \| $m_{e\nu_{e}jj^{\prime}}\ \ [0,154]$ \| 19 \| \| $p_{T,e\nu_{e}jj^{\prime}}\ \ [18,\infty]$ \| 9 \| mt \| $p_{T,e\nu_{e}jj^{\prime}}\ \ [18,\infty]$ \| 18 \| $H_{T,e\nu_{e}jj^{\prime}}\ \ [136,\infty]$ \| 12 \| \| $\Delta\phi_{e,\nu_{e}}\geq 1.9$ \| 3 $165$ \| pzmw \| $p_{T,e\nu_{e}jj^{\prime}}\ \ [18,\infty]$ \| 18 \| $m_{e\nu_{e}jj^{\prime}}\ \ [0,170]$ \| 17 \| \| $\gamma_{jj^{\prime}\|e\nu_{e}}\leq 1.06$ \| 20 \| pzmh \| $p_{T,e\nu_{e}jj^{\prime}}\ \ [18,\infty]$ \| 18 \| $m_{e\nu_{e}jj^{\prime}}\ \ [0,170]$ \| 13 \| \| $\gamma_{jj^{\prime}\|e\nu_{e}}\leq 1.09$ \| 15 \| mt \| $p_{T,e\nu_{e}jj^{\prime}}\ \ [21,\infty]$ \| 20 \| $H_{T,e\nu_{e}jj^{\prime}}\ \ [140,\infty]$ \| 16 \| \| $\Delta\phi_{e,\nu_{e}}\geq 1.7$ \| * $170$ \| pzmw \| $p_{T,e\nu_{e}jj^{\prime}}\ \ [19,\infty]$ \| 20 \| $m_{e\nu_{e}jj^{\prime}}\ \ [0,176]$ \| 13 \| \| $\gamma_{jj^{\prime}\|e\nu_{e}}\leq 1.12$ \| 11 \| pzmh \| $p_{T,e\nu_{e}jj^{\prime}}\ \ [20,\infty]$ \| 20 \| $m_{e\nu_{e}jj^{\prime}}\ \ [0,176]$ \| 9 \| \| $\gamma_{jj^{\prime}\|e\nu_{e}}\leq 1.16$ \| 9 \| mt \| $p_{T,e\nu_{e}jj^{\prime}}\ \ [22,\infty]$ \| 24 \| $H_{T,e\nu_{e}jj^{\prime}}\ \ [148,\infty]$ \| 22 \| \| $H_{T,e\nu_{e}jj^{\prime}}\ \ [140,\infty]$ \| * $180$ \| pzmw \| $p_{T,e\nu_{e}jj^{\prime}}\ \ [21,\infty]$ \| 23 \| $H_{T,jj^{\prime}}\ \ [64,\infty]$ \| 11 \| \| $1.06\leq\gamma_{jj^{\prime}\|e\nu_{e}}\leq 1.22$ \| 5 \| pzmh \| $p_{T,e\nu_{e}jj^{\prime}}\ \ [22,\infty]$ \| 24 \| $H_{T,e\nu_{e}jj^{\prime}}\ \ [140,\infty]$ \| 10 \| \| $m^{(\nu_{e})}_{T,e\nu_{e}jj^{\prime}}\ \ [136,182]$ \| 5 \| mt \| $p_{T,e\nu_{e}jj^{\prime}}\ \ [24,\infty]$ \| 28 \| $H_{T,e\nu_{e}jj^{\prime}}\ \ [156,\infty]$ \| 27 \| \| $H_{T,e\nu_{e}jj^{\prime}}\ \ [148,\infty]$ \| * $190$ \| pzmw \| $p_{T,e\nu_{e}jj^{\prime}}\ \ [23,\infty]$ \| 24 \| $H_{T,e\nu_{e}jj^{\prime}}\ \ [148,\infty]$ \| 15 \| \| $1.12\leq\gamma_{jj^{\prime}\|e\nu_{e}}\leq 1.30$ \| 5 \| pzmh \| $p_{T,e\nu_{e}jj^{\prime}}\ \ [24,\infty]$ \| 29 \| $H_{T,e\nu_{e}jj^{\prime}}\ \ [144,\infty]$ \| 17 \| \| $m^{(\nu_{e})}_{T,e\nu_{e}jj^{\prime}}\ \ [142,194]$ \| 3 \| mt \| $H_{T,e\nu_{e}jj^{\prime}}\ \ [164,\infty]$ \| 31 \| $p_{T,e\nu_{e}jj^{\prime}}\ \ [24,\infty]$ \| 28 \| \| $p_{T,e\nu_{e}jj^{\prime}}\ \ [15,\infty]$ \| 9 $200$ \| pzmw \| $p_{T,e\nu_{e}jj^{\prime}}\ \ [24,\infty]$ \| 25 \| $H_{T,e\nu_{e}jj^{\prime}}\ \ [156,\infty]$ \| 20 \| \| $1.18\leq\gamma_{jj^{\prime}\|e\nu_{e}}\leq 1.40$ \| 6 \| pzmh \| $p_{T,e\nu_{e}jj^{\prime}}\ \ [27,\infty]$ \| 32 \| $H_{T,e\nu_{e}jj^{\prime}}\ \ [156,\infty]$ \| 25 \| \| $H_{T,e\nu_{e}jj^{\prime}}\ \ [144,\infty]$ \| 4 \| mt \| $H_{T,e\nu_{e}jj^{\prime}}\ \ [172,\infty]$ \| 36 \| $p_{T,e\nu_{e}jj^{\prime}}\ \ [25,\infty]$ \| 27 \| \| $p_{T,e\nu_{e}jj^{\prime}}\ \ [15,\infty]$ \| 8 $210$ \| pzmw \| $H_{T,e\nu_{e}jj^{\prime}}\ \ [160,\infty]$ \| 24 \| $p_{T,e\nu_{e}jj^{\prime}}\ \ [24,\infty]$ \| 23 \| \| $1.25\leq\gamma_{jj^{\prime}\|e\nu_{e}}\leq 1.45$ \| 14 \| pzmh \| $H_{T,e\nu_{e}jj^{\prime}}\ \ [162,\infty]$ \| 36 \| $p_{T,e\nu_{e}jj^{\prime}}\ \ [30,\infty]$ \| 36 \| \| $1.25\leq\gamma_{jj^{\prime}\|e\nu_{e}}\leq 1.54$ \| 7 \| mt \| $H_{T,e\nu_{e}jj^{\prime}}\ \ [180,\infty]$ \| 39 \| $m_{T,e\nu_{e}jj^{\prime}}\ \ [174,\infty]$ \| 26 \| \| $p_{T,e\nu_{e}jj^{\prime}}\ \ [12,\infty]$ \| 8 $220$ \| pzmw \| $H_{T,e\nu_{e}jj^{\prime}}\ \ [168,\infty]$ \| 29 \| $p_{T,e\nu_{e}jj^{\prime}}\ \ [24,\infty]$ \| 22 \| \| $1.31\leq\gamma_{jj^{\prime}\|e\nu_{e}}\leq 1.53$ \| 15 \| pzmh \| $H_{T,e\nu_{e}jj^{\prime}}\ \ [172,\infty]$ \| 49 \| $p_{T,e\nu_{e}}\ \ [56,\infty]$ \| 43 \| \| $1.30\leq\gamma_{jj^{\prime}\|e\nu_{e}}\leq 1.58$ \| 8 Table 6: Leading (optimal/major) and subleading cuts for each Higgs boson mass and selection criteria (all selections are combinatorial with window parameters $\Delta=\delta=20\ \mathrm{GeV}$). Gain in $S/\sqrt{B}$, in percent, is shown after each cut. Having used the major discriminators including pseudo-rapidity cuts (see text), next in the cut hierarchy are the minor cuts shown in the two rightmost columns. The significance gains associated with them are understood in addition to the major cut improvements. Cuts marked with an asterisk have less than 2% improvement. ##### Below-threshold region: For pzmw and pzmh, the reconstruction selections, the transverse mass of the 4-particle system, $m^{(\nu_{e})}_{T,e\nu_{e}jj^{\prime}}$, and (for pzmw at low $M_{h}$) the transverse mass of the leptonic system, $m_{T,e\nu_{e}}$, are the best observables. This result is really just a reiteration of the fact that simple reconstruction selections work poorly for below-threshold Higgs bosons. In our effort to make pzmh/w more flexible and apply them to below- threshold scenarios, we have allowed the possibility for background configurations that are inconsistent with a single parent resonance – such as 4-particle systems with $m^{(\nu_{e})}_{T,e\nu_{e}jj^{\prime}}>M_{h}$. Removing this inconsistent region results in gains of ${\mathcal{O}}(\mbox{40\%})$. For mt, the transverse mass of the dijet system or the scalar $p_{T}$ sum formed with the selected jets, $H_{T,jj^{\prime}}$, are the most optimal, with improvements of ${\mathcal{O}}(\mbox{50\%})$. The former cut takes advantage of the fact that the signal jets originate in a (real or virtual) $W$ boson, while the background jets come primarily from ISR – information that is not exploited in the initial mt selection. Somewhat smaller gains come from cutting on the total selected system’s transverse momentum, $p_{T,e\nu_{e}jj^{\prime}}$. For mtp, there is little optimization to be done. Much of the physics that is behind the optimal cuts in the pzmh/w or mt cases has already been incorporated into the selection process. Some improvement is possible by cutting out the region with very low 4-particle transverse momentum, $p_{T,e\nu_{e}jj^{\prime}}$. We illustrate our findings by showing and commenting on a small collection of spectra resulting from the baseline combinatorial selections; for more details, we refer the reader to the discussion around Figure 16 of Appendix B.2. ##### Near-threshold region: For Higgs boson masses close to the $WW$ threshold, the 4-particle $p_{T}$ is the most powerful additional handle. In Higgs boson production, as in other colour-singlet resonance production, the $p_{T,e\nu_{e}jj^{\prime}}$ distribution is cut off at low values by soft-gluon resummation, and falls off at high values because of parton kinematics. The result is a peaked distribution. The hard scale of the process, dictated by the Higgs boson mass, sets the initial ISR scale, thereby influencing where $p_{T,e\nu_{e}jj^{\prime}}$ peaks. Since the Higgs boson is heavier than the $W$ boson, $p_{T,e\nu_{e}jj^{\prime}}$ always peaks at higher values for the signal compared to $W$ production. Even though the dominant background for our study is $W$+2-jets, rather than $W$+0-jets, the argument still holds. The peak in $p_{T,e\nu_{e}jj^{\prime}}$ for $W$+2-jets is still governed by $M_{W}$ and continues to peak at lower values than the Higgs boson signal. Selection criteria change the tails of the background $p_{T,e\nu_{e}jj^{\prime}}$ distribution, but do not affect the location of the peak. Cutting out the low-$p_{T,e\nu_{e}jj^{\prime}}$ region, improvements on the order of 20% are possible. Distributions such as $H_{T,jj^{\prime}}$, the scalar $p_{T}$ sum of the two selected jets, or $H_{T,e\nu_{e}jj^{\prime}}$, the $H_{T}$ of the 4-particle system, also have potential discriminating power. Signal versus background distributions in the relevant variables after baseline combinatorial selections are shown in Figure 17, see Appendix B.2. ##### Above-threshold region: For higher Higgs boson masses, the total amount of (transverse) energy in the $W$+$jj^{\prime}$ system becomes the most powerful discriminator between the signal and the background. Specifically, once $M_{h}\gtrsim 200\ \mathrm{GeV}$, the $H_{T}$ of the selected 4-particle system peaks at significantly higher values than the background, regardless of the selection technique. By cutting away the low-$H_{T}$ region, we find gains of order 25% are possible. The 4-object $p_{T}$ remains a very useful observable, as does the 4-particle transverse mass. Examples of signal versus background distributions are shown in Figure 18, see Appendix B.2. ##### Results: The optimal or “leading” or “major” cuts for the different Higgs boson mass categories and selection methods are summarized in Table 6. To give some idea how useful the single best discriminator is compared to other observables, we also show the percent increase in significance for the second best or “subleading” single discriminator. Separately we have also determined which combinations of the leading discriminator (supplemented by the respective pseudo-rapidity cuts discussed above) with a second observable give the largest (additional) increase in significance. The second “minor” cuts of these optimal two-variable selections are summarized in the last two columns of Table 6. Note that in most cases these minor cuts do not involve the same observables as the subleading cut; this is because the subleading discriminator is typically strongly correlated with the leading discriminator, and thus does not add much to the combined significance. Some ideas beyond the application of minor cuts exist; we comment in Appendix B.3 on a few possible routes one can take to enhance the optimized analyses presented here. When looking for minor cuts, we found in addition to variables we have already discussed, such as $\gamma_{jj^{\prime}\|e\nu_{e}}$ (in Section 2), $H_{T,(e\nu_{e})jj^{\prime}}$ and the 4-object $p_{T}$, a few other observables, namely $p_{T,j}$, $\Delta\phi_{e,\nu_{e}}$ and $m_{\perp,jj^{\prime}}$ to be beneficial. The first two, $p_{T,j}$ and $\Delta\phi_{e,\nu_{e}}$ are common, so we do not repeat their definitions here. The last minor cut observable, $m_{\perp,jj^{\prime}}$ is defined through $m^{2}_{\perp,jj^{\prime}}=m^{2}_{jj^{\prime}}+p^{2}_{T,jj^{\prime}}$ exhibiting yet another way of defining a transverse mass. The additional gains from the minor cuts are typically small, except close to the $WW$ threshold and for the largest Higgs boson masses considered here. In particular for $M_{h}>2\,M_{W}$, the boost of the $jj^{\prime}$ system in the reconstructed 4-object rest frame stands out as a helpful discriminator of secondary order; for more details, we refer again to Appendix B.2 and the discussion around Figure 19. Also, the dijet-system based handles, $m_{\perp,jj^{\prime}}$ and $H_{T,jj^{\prime}}$ yield fairly substantial extra gains, but only at low $M_{h}$ if we rely on the pzmh method. Once jets are picked stemming from the backgrounds, the strict $M_{h}$ mass reconstruction of the selected 4-object, as encoded in pzmh and amplified by the major cut given through $m^{(\nu_{e})}_{T,e\nu_{e}jj^{\prime}}$, gives rise to the selection of less energetic $j$, $j^{\prime}$ jets with preferred pair masses of $m_{jj^{\prime}}\sim M_{W}-\delta$. The observables $m_{\perp,jj^{\prime}}$ and $H_{T,jj^{\prime}}$ exploit this fact conveniently, hence facilitate such secondary improvements, as shown in Figure 19 for the example of $M_{h}=130\ \mathrm{GeV}$. Figure 9: Total significances as a function of $M_{h}$ after including the subdominant backgrounds and all major plus minor cuts (as specified in the text and Table 6). The results (lines with squares) are shown for the four types of more realistic Higgs boson candidate selections, which have been advertised in this work. They are denoted on top of each panel. In all cases the combinatorial approach using window parameters $\Delta=\delta=20\ \mathrm{GeV}$ has been applied for selecting the candidate set of particles. As before in Figure 6, the ideal case (i.e. the invm combinatorial reconstruction of the Higgs boson candidates using $2\,\Delta=\delta=20\ \mathrm{GeV}$) is taken as the main reference to compare the different results. For each combinatorial selection, the outcome (lines with circles) with no cuts applied (but using a slightly smaller mass window, $\Delta=15\ \mathrm{GeV}$) is also displayed to emphasize the effect of the cut optimization. Note that the effect of the minor backgrounds has been neglected in computing each of these reference curves. The $e\nu_{e}$+jets final states are generated from the signal, $gg\to h\to WW$, the $W$+jets, electroweak and $t\bar{t}$ backgrounds. All $S/\sqrt{B_{i}}$ were calculated using Eqs. (4.2) and combined according to Eq. (24) assuming an integrated luminosity of ${\mathcal{L}}=10\ \mathrm{fb}^{-1}$ and including electron and muon channels, i.e. $f_{\ell}=2$. The lower plots depict for each selection the ratios of optimized over un-optimized $S/\sqrt{B}$ (lines with squares) and optimized over ideal-case reference $S/\sqrt{B}$ (lines with circles). The thin blue lines visualize in how far the final $S/\sqrt{B_{\mathrm{(tot)}}}$ results suffer from the presence of the minor backgrounds. Figure 10: The $S/B$ ratios associated with the total significances presented in Figure 9. The ratios are shown as a function of $M_{h}$ for the final selections: the combinatorial pzmw (triangles), pzmh (circles), mt (squares) and mtp (diamonds) selections using the window parameters $\Delta=\delta=20\ \mathrm{GeV}$ each supplemented by their specific optimization cuts, see Table 6. The reference curve (dashed line with diamonds) is the same as used in Figure 7 representing the invm combinatorial reconstruction with $2\,\Delta=\delta=20\ \mathrm{GeV}$, no extra cuts applied and neglecting the impact of the minor backgrounds. The $e\nu_{e}$+jets final states were generated from $gg\to h\to WW$ signal events, the $W$+jets, electroweak and $t\bar{t}$ backgrounds. The $S/B_{i}$ were calculated using the $\sigma_{S}$ and $\sigma_{B}$ as obtained after the final selections, the signal $K$-factors of Table 1 and the $K_{B}$ as given in Eqs. (17), (21) and (22). Single ratios were combined according to $S/B_{\mathrm{(tot)}}=1/\sum_{i}(S/B_{i})^{-1}$. Finally, the baseline combinatorial and optimized combinatorial significances are displayed as a function of the Higgs boson mass in Figure 9. The ratio plots associated with each selection in the lower part of the figure visualize the significance increase achieved by the optimization. Independent of the selection, they also indicate a $\mathcal{O}(\mbox{10\%})$ drop of significances caused by the subdominant backgrounds. Focusing on the $S/\sqrt{B}$ ratios taken with respect to the ideal case (orange lines with circles), these ratio plots emphasize that the optimized mt(p) (transverse) and pzmw selections work best below and above the $WW$ mass threshold, respectively. The related $S/B$ ratios presented in Figure 10 confirm these findings. They turn out to be rather small, as a consequence of maximizing the significance and trying to preserve most of the signal; both of which does not allow for imposing too restrictive cuts. Advantageously, the actual number of signal events, $S$, present in this $h\to WW$ channel is not small. Except for the Higgs signal at $M_{h}=120\ \mathrm{GeV}$ (with $\mathcal{O}(\mbox{4})$ expected events), the optimized analyses usually leave us with hundreds of signal events (50–300), if we assume an integrated luminosity of ${\mathcal{L}}=10\ \mathrm{fb}^{-1}$. Even the $\sim$ 25 signal events for $M_{h}=130\ \mathrm{GeV}$ are sufficient, particularly as $S/B$ increases to $\sim 0.04$. Clearly, as seen from Figure 9, the optimized significances for the four different Higgs boson reconstruction methods are very similar. The best significance is for Higgs boson masses close to the $WW$ threshold, suggesting that it would be possible to achieve 95% confidence level exclusion in a stand-alone analysis. For a Higgs boson mass in the range $130\lesssim M_{h}\lesssim 150\ \mathrm{GeV}$, the optimized significance is between $0.7$ and $1.4$. Given the additional improvements expected from a full multivariate analysis, this indicates that the semileptonic Higgs boson decay channel can make a significant contribution to Higgs boson exclusion in this mass range. ## 5 Conclusions, caveats, and prospects We have presented a systematic study of the prospects for extending the Tevatron exclusion reach for a Standard Model Higgs boson by including the final states arising from semileptonic Higgs boson decays. We have used a realistic simulation of the Higgs signal and the relevant Standard Model background processes to exhibit the kinematic differences between the signal and background. We have used three qualitatively different approaches to extracting the event kinematics, one based on transverse observables and the others based on approximate even-by-event full reconstruction. We have shown that all three approaches give similar results when one optimizes selections based on several discriminating observables. The details of the optimization depend on the Higgs boson mass, and in particular on whether it is below, near, or above the threshold for decay to two on-shell $W$ bosons. The optimized significances that we have achieved are not sufficient for stand-alone Higgs boson exclusion except in the most favorable case where the Higgs boson mass is close to threshold. However the sensitivities shown here are certainly promising as ingredients to a combined multi-channel analysis. One important caveat is that the signal to background ratios for this type of analysis are fairly small, on the order of a percent, as illustrated in Figure 10. This means that Higgs boson exclusion is sensitive to relatively small systematic errors in the modeling of the backgrounds, notably the dominant background from $W$+jets. However we have shown here that an experimental analysis has multiple over-constrained handles on the kinematic features of the data, providing extra cross-checks. In addition, the Higgs boson candidate selection employed here to identify the two jets from the Higgs boson decay is by design rather stable against effects from extra hard radiation; this reduces the uncertainty in the background modeling. The techniques described here are applicable to Higgs boson searches at the CERN Large Hadron Collider. At 7 TeV center-of-mass collision energy, the Higgs boson production cross section increases by a factor of $\sim$ 30, while the $W$+jets background increases by a factor of $\sim$ 20\. With no hard upper limit of the amount of data available, one can use more restrictive selections and improve both the signal to background ratio and the overall significance. A recent analysis by the ATLAS collaboration [91] employed a 4-object invariant mass reconstruction and a dijet mass window similar to the baseline naive analysis used here, but applied to the heavy Higgs boson mass region $M_{h}>240\ \mathrm{GeV}$. A CMS study [92] looked at the enhanced signal to background ratio provided by focusing on VBF production (and thus requiring two extra forward jets). Because of the large branching fraction, the $h\to WW$ semileptonic mode might prove to be observable at the LHC over a larger mass range than the $h\to WW$ dilepton mode; thus it could play a critical role in establishing the nature of electroweak symmetry breaking, in the event that the Higgs boson mass is $\sim$ 120 GeV. ## Acknowledgments We would like to thank Bogdan Dobrescu for many lively and stimulating discussions. We would also like to thank Thomas Becher, Li Lin Yang, Tanju Gleisberg, Frank Petriello, John Campbell and Ciaran Williams for their help in accomplishing necessary cross-checks. We are grateful to Frank Krauss and Frank Siegert, as well as all other members of the SHERPA collaboration for their continuous support. We gratefully acknowledge useful discussions with Bob Hirosky and Lidija Zivkovic, as well as Lance Dixon, Gavin Salam, Walter Giele, Rakhi Mahbubani, Patrick Fox and Agostino Patella. Fermilab is operated by Fermi Research Alliance, LLC, under contract DE- AC02-07CH11359 with the United States Department of Energy. ## Appendix A Appendix: Monte Carlo event generation ### A.1 Leading-order cross sections Here we briefly report on the tests we have done to convince ourselves of the correctness of the SHERPA leading-order cross section calculations. We found satisfactory or better agreement in all our cross-checks, which we briefly summarize here: * • For the case of $h\to e^{-}\bar{\nu}_{e}\,pp$ decays, we have compared SHERPA’s branching ratios with those obtained by multiplying the Hdecay results for $B_{W^{\ast}W^{\ast}}$ given in Table 1 times the PDG literature numbers for $B(W\to e\nu_{e})\times B(W\to pp)=0.1075\times 0.676$. Using AMEGIC++’s mode of calculating partial widths of $1\to N$ processes, we determined $B_{e^{-}\bar{\nu}_{e}pp}=\Gamma(h\to e^{-}\bar{\nu}_{e}\,pp)/\Gamma_{h}$ for the various Higgs boson masses and respective widths of Table 1. The differences seen are at most on the few- percent level. * • With the explicit knowledge of the SHERPA branching fractions we were able to extract Higgs boson production rates at LO from the SHERPA signal cross section calculations according to $\sigma^{\mathrm{LO}}_{ggh}\;=\;\frac{\Gamma_{h}}{\Gamma(h\to e^{-}\bar{\nu}_{e}\,pp)}\;\;\sigma^{(0)}_{e^{-}\bar{\nu}_{e}pp}\;=\;\frac{\sigma^{(0)}_{e^{-}\bar{\nu}_{e}pp}}{B_{e^{-}\bar{\nu}_{e}pp}}\ .$ (27) The numbers that we obtained from this procedure compare well to numbers of other LO calculations, for example the LO rates as evaluated in MCFM or provided by Becher and Yang for verification purposes [93]. * • SHERPA LO rates were computed for both finite top masses and in the infinite top-mass limit. The ratio of the former over the latter cross section given as $\frac{\sigma^{(0)}_{S}}{\sigma^{(0)}_{S,m_{t}\to\infty}}\;=\;\left\|\,I\left(\frac{m^{2}_{t}}{M^{2}_{h}}\right)\right\|^{2}$ (28) singles out the dependence on the top mass versus Higgs boson mass ratio, which is encoded by the function $\displaystyle I(x)\;=\;6x+3x\,(4x-1)$ $\displaystyle\Biggl{\\{}$ $\displaystyle\frac{\Theta(1-4x)}{2}\left[\ln\left(\frac{1+\sqrt{1-4x}}{1-\sqrt{1-4x}}\right)-i\pi\right]^{2}-$ (29) $\displaystyle 2\,\Theta(4x-1)\arcsin^{2}\left(\frac{1}{2\sqrt{x}}\right)\;\,\Biggl{\\}}\ .$ Note that $\|I(x)\|^{2}$ attains unity as $x\to\infty$, while it vanishes for $x\to 0$. Comparing the numerical cross section ratios with the analytical values for $\|I(x)\|^{2}$, we found excellent agreement over the entire Higgs boson mass range considered in this study. ### A.2 NLO calculations versus CKKW ME+PS merging When compared to NLO calculations, it is evident that SHERPA’s CKKW merging approach does not account for the virtual corrections to $V$+jets in their entirety.181818For a brief summary of the basics of the CKKW merging, see Ref. [70]. For the current generation of SHERPA Monte Carlo programs, the ME+PS facilities have been extended to allow for truncated showering, which is a major refinement over the CKKW approach, see Refs. [76, 94, 95]. The only contributions enter through Sudakov form-factor terms at leading-logarithmic accuracy used in the parton shower and to reweight the tree-level matrix elements. The real-emission corrections however are included on a fairly comparable level with respect to full NLO calculations.191919More recent versions of SHERPA, from version 1.2.3 on, have been enhanced by the means to generate, for a number of important processes, events at the hadron level with a rate correct at next-to-leading order in $\alpha_{\mathrm{s}}$ [83, 84, 96]. To make this work, SHERPA relies on interfacing external one-loop amplitude generators like MCFM [61, 97], BlackHat [98, 99] or, more generally, via the Binoth Les Houches Accord [100]. Because of the complexity of the procedure, such improvements are not yet available for arbitrary processes, in particular the multi-jet final states we are interested in. Unless one decides for a fixed-scale choice at NLO, both approaches determine the strong-coupling scales dynamically, i.e. on an event-by-event basis, taking the kinematic configuration of the event into account. For all these reasons, it then occurs that the CKKW shapes of distributions emerge in many cases quite similarly to those evaluated at NLO, making an application of global $K$-factors feasible.202020For example, in [25] a global $K$-factor of magnitude $1.33$ with respect to the total inclusive cross section as measured by CDF [101] was applied to achieve a good agreement between the data and the SHERPA predictions for inclusive jet multiplicity and transverse momentum distributions. The treatment to fix the strong couplings is different when multiple scales are present. While at NLO the scales are set uniformly such that all $\alpha_{\mathrm{s}}$ factors obtain the same value, in the CKKW method they are set locally by the procedure itself, cf. [68, 102, 70] for example. This is known as $\alpha_{\mathrm{s}}$ reweighting, constituting the second component of the matrix-element reweighting of the CKKW method. The assignment of the scales proceeds hierarchically based on the splitting history, which is identified by the $k_{T}$-jet cluster algorithm when applied to the initial matrix-element configuration considering physical parton combinations only. The nodal $k_{T}$ values found by the clustering can be interpreted as the relative transverse momenta of the identified splittings. They are then used as the scales for the strong-coupling constants replacing the predefined choice of the initial matrix-element generation. It would be interesting to see if a hierarchical scale setting can further stabilize NLO results, but no such $\alpha_{\mathrm{s}}$ reweighting has been completely worked out yet for NLO calculations. ## Appendix B Appendix: Analysis side studies and additional material cuts & \| $2\,\Delta/$ \| $\sigma_{S}/\mathrm{fb}$ \| $\sigma_{B}/\mathrm{fb}$ \| $S/B$ \| $\sigma_{S}/\mathrm{fb}$ \| $\sigma_{B}/\mathrm{fb}$ \| $S/B$ \| $\sigma_{S}/\mathrm{fb}$ \| $\sigma_{B}/\mathrm{fb}$ \| $S/B$ ---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|--- selections \| $\mathrm{GeV}$ \| $\varepsilon_{S}$ \| $\varepsilon_{B}$ \| $S/\sqrt{B}$ \| $\varepsilon_{S}$ \| $\varepsilon_{B}$ \| $S/\sqrt{B}$ \| $\varepsilon_{S}$ \| $\varepsilon_{B}$ \| $S/\sqrt{B}$ $M_{h}/\mathrm{GeV}\quad[\delta/\mathrm{GeV}]$ \| $190\quad[20]$ \| $200\quad[20]$ \| $210\quad[20]$ $\sigma^{(0)}$ \| \| $9.862$ \| $220\mbox{$\cdot\;\\!\\!$}10^{4}$ \| 96$\mbox{$\cdot\;\\!\\!$}10^{\textrm{-}7}$ \| $7.827$ \| $220\mbox{$\cdot\;\\!\\!$}10^{4}$ \| 75$\mbox{$\cdot\;\\!\\!$}10^{\textrm{-}7}$ \| $6.473$ \| $220\mbox{$\cdot\;\\!\\!$}10^{4}$ \| 61$\mbox{$\cdot\;\\!\\!$}10^{\textrm{-}7}$ \| \| 1.0 \| 1.0 \| 0.10 \| 1.0 \| 1.0 \| 0.08 \| 1.0 \| 1.0 \| 0.06 lepton & \| \| $5.561$ \| $984\mbox{$\cdot\;\\!\\!$}10^{3}$ \| 12$\mbox{$\cdot\;\\!\\!$}10^{\textrm{-}6}$ \| $4.433$ \| $984\mbox{$\cdot\;\\!\\!$}10^{3}$ \| 95$\mbox{$\cdot\;\\!\\!$}10^{\textrm{-}7}$ \| $3.689$ \| $984\mbox{$\cdot\;\\!\\!$}10^{3}$ \| 78$\mbox{$\cdot\;\\!\\!$}10^{\textrm{-}7}$ MET cuts \| \| 0.564 \| 0.45 \| 0.08 \| 0.566 \| 0.45 \| 0.07 \| 0.570 \| 0.45 \| 0.05 as above & \| \| $4.586$ \| $191\mbox{$\cdot\;\\!\\!$}10^{2}$ \| 51$\mbox{$\cdot\;\\!\\!$}10^{\textrm{-}5}$ \| $3.709$ \| $191\mbox{$\cdot\;\\!\\!$}10^{2}$ \| 41$\mbox{$\cdot\;\\!\\!$}10^{\textrm{-}5}$ \| $3.128$ \| $191\mbox{$\cdot\;\\!\\!$}10^{2}$ \| 34$\mbox{$\cdot\;\\!\\!$}10^{\textrm{-}5}$ $\geq 2$ jets \| \| 0.465 \| 0.0087 \| 0.50 \| 0.474 \| 0.0087 \| 0.40 \| 0.483 \| 0.0087 \| 0.33 as above & \| \| $2.533$ \| $6997$ \| 77$\mbox{$\cdot\;\\!\\!$}10^{\textrm{-}5}$ \| $2.007$ \| $6997$ \| 60$\mbox{$\cdot\;\\!\\!$}10^{\textrm{-}5}$ \| $1.671$ \| $6997$ \| 50$\mbox{$\cdot\;\\!\\!$}10^{\textrm{-}5}$ $\lvert\tilde{m}_{j_{1}j_{2}}\\!-80\rvert\\!<\\!\tilde{\delta}$ \| \| 0.257 \| 0.0032 \| 0.46 \| 0.256 \| 0.0032 \| 0.36 \| 0.258 \| 0.0032 \| 0.29 naive $h$-reco \| $50$ \| $2.699$ \| $6644$ \| 87$\mbox{$\cdot\;\\!\\!$}10^{\textrm{-}5}$ \| $2.146$ \| $6303$ \| 72$\mbox{$\cdot\;\\!\\!$}10^{\textrm{-}5}$ \| $1.789$ \| $5837$ \| 64$\mbox{$\cdot\;\\!\\!$}10^{\textrm{-}5}$ \| \| 0.274 \| 0.0030 \| 0.50 \| 0.274 \| 0.0029 \| 0.40 \| 0.276 \| 0.0027 \| 0.34 naive $h$-reco \| $30$ \| $2.082$ \| $4062$ \| 0.0011 \| $1.649$ \| $3805$ \| 91$\mbox{$\cdot\;\\!\\!$}10^{\textrm{-}5}$ \| $1.374$ \| $3521$ \| 81$\mbox{$\cdot\;\\!\\!$}10^{\textrm{-}5}$ \| \| 0.211 \| 0.0018 \| 0.49 \| 0.211 \| 0.0017 \| 0.40 \| 0.212 \| 0.0016 \| 0.34 naive $h$-reco \| $48$ \| $2.177$ \| $3483$ \| 0.0013 \| $1.699$ \| $3151$ \| 0.0011 \| $1.397$ \| $2649$ \| 0.0011 $\lvert\tilde{m}_{j_{1}j_{2}}\\!-80\rvert\\!<\\!\tilde{\delta}$ \| \| 0.221 \| 0.0016 \| 0.56 \| 0.217 \| 0.0014 \| 0.45 \| 0.216 \| 0.0012 \| 0.40 naive $h$-reco \| $20$ \| $1.488$ \| $1565$ \| 0.0020 \| $1.159$ \| $1312$ \| 0.0019 \| $0.9518$ \| $1080$ \| 0.0018 $\lvert\tilde{m}_{j_{1}j_{2}}\\!-80\rvert\\!<\\!\tilde{\delta}$ \| \| 0.151 \| 71$\mbox{$\cdot\;\\!\\!$}10^{\textrm{-}5}$ \| 0.57 \| 0.148 \| 60$\mbox{$\cdot\;\\!\\!$}10^{\textrm{-}5}$ \| 0.48 \| 0.147 \| 49$\mbox{$\cdot\;\\!\\!$}10^{\textrm{-}5}$ \| 0.43 comb. $h$-reco \| $50$ \| $3.666$ \| $7296$ \| 0.0011 \| $2.937$ \| $6946$ \| 89$\mbox{$\cdot\;\\!\\!$}10^{\textrm{-}5}$ \| $2.464$ \| $6465$ \| 79$\mbox{$\cdot\;\\!\\!$}10^{\textrm{-}5}$ \| \| 0.372 \| 0.0033 \| 0.65 \| 0.375 \| 0.0032 \| 0.52 \| 0.381 \| 0.0029 \| 0.45 comb. $h$-reco \| $20$ \| $2.545$ \| $3145$ \| 0.0017 \| $2.031$ \| $2964$ \| 0.0014 \| $1.699$ \| $2756$ \| 0.0013 \| \| 0.258 \| 0.0014 \| 0.69 \| 0.260 \| 0.0013 \| 0.56 \| 0.262 \| 0.0013 \| 0.48 comb. $h$-reco \| $50$ \| $3.243$ \| $4088$ \| 0.0017 \| $2.573$ \| $3755$ \| 0.0014 \| $2.138$ \| $3211$ \| 0.0014 $\lvert\tilde{m}_{jj^{\prime}}\\!-80\rvert\\!<\\!\tilde{\delta}$ \| \| 0.329 \| 0.0019 \| 0.77 \| 0.329 \| 0.0017 \| 0.62 \| 0.330 \| 0.0015 \| 0.55 comb. $h$-reco \| $30$ \| $2.806$ \| $2662$ \| 0.0023 \| $2.205$ \| $2314$ \| 0.0020 \| $1.823$ \| $1925$ \| 0.0020 $\lvert\tilde{m}_{jj^{\prime}}\\!-80\rvert\\!<\\!\tilde{\delta}$ \| \| 0.284 \| 0.0012 \| 0.82 \| 0.282 \| 0.0011 \| 0.68 \| 0.282 \| 88$\mbox{$\cdot\;\\!\\!$}10^{\textrm{-}5}$ \| 0.61 comb. $h$-reco \| $20$ \| $2.333$ \| $1830$ \| 0.0027 \| $1.829$ \| $1555$ \| 0.0025 \| $1.507$ \| $1293$ \| 0.0024 $\lvert\tilde{m}_{jj^{\prime}}\\!-80\rvert\\!<\\!\tilde{\delta}$ \| \| 0.237 \| 83$\mbox{$\cdot\;\\!\\!$}10^{\textrm{-}5}$ \| 0.82 \| 0.234 \| 71$\mbox{$\cdot\;\\!\\!$}10^{\textrm{-}5}$ \| 0.69 \| 0.233 \| 59$\mbox{$\cdot\;\\!\\!$}10^{\textrm{-}5}$ \| 0.62 comb. $h$-reco \| $16$ \| $2.066$ \| $1484$ \| 0.0030 \| $1.617$ \| $1255$ \| 0.0027 \| $1.331$ \| $1029$ \| 0.0027 $\lvert\tilde{m}_{jj^{\prime}}\\!-80\rvert\\!<\\!\tilde{\delta}$ \| \| 0.209 \| 67$\mbox{$\cdot\;\\!\\!$}10^{\textrm{-}5}$ \| 0.81 \| 0.207 \| 57$\mbox{$\cdot\;\\!\\!$}10^{\textrm{-}5}$ \| 0.68 \| 0.206 \| 47$\mbox{$\cdot\;\\!\\!$}10^{\textrm{-}5}$ \| 0.61 comb. $h$-reco \| $10$ \| $1.436$ \| $921.6$ \| 0.0033 \| $1.121$ \| $773.9$ \| 0.0030 \| $0.9215$ \| $632.3$ \| 0.0030 $\lvert\tilde{m}_{jj^{\prime}}\\!-80\rvert\\!<\\!\tilde{\delta}$ \| \| 0.146 \| 42$\mbox{$\cdot\;\\!\\!$}10^{\textrm{-}5}$ \| 0.71 \| 0.143 \| 35$\mbox{$\cdot\;\\!\\!$}10^{\textrm{-}5}$ \| 0.60 \| 0.142 \| 29$\mbox{$\cdot\;\\!\\!$}10^{\textrm{-}5}$ \| 0.54 Table 7: Impact of the different levels of cuts on the $e\nu_{e}$+jets final states for the $gg\to h\to WW$ production and decay signal and the $W$+jets background as obtained from SHERPA. Cross sections $\sigma_{S}$, $\sigma_{B}$, acceptances $\varepsilon_{S}$, $\varepsilon_{B}$ and $S/B$, $S/\sqrt{B}$ ratios are shown for Higgs boson masses of $M_{h}=190$, $200$ and $210\ \mathrm{GeV}$. Note that $\tilde{m}_{ij}=m_{ij}/\mathrm{GeV}$ and $\tilde{\delta}=\delta/\mathrm{GeV}$. Significances were calculated using Eqs. (4.2) assuming ${\mathcal{L}}=10\ \mathrm{fb}^{-1}$ of integrated luminosity, counting both electrons and muons and combining Tevatron experiments. cuts & \| $2\,\Delta/$ \| $\sigma_{S}/\mathrm{fb}$ \| $\sigma_{B}/\mathrm{fb}$ \| $S/B$ \| $\sigma_{S}/\mathrm{fb}$ \| $\sigma_{B}/\mathrm{fb}$ \| $S/B$ \| $\sigma_{S}/\mathrm{fb}$ \| $\sigma_{B}/\mathrm{fb}$ \| $S/B$ ---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|--- selections \| $\mathrm{GeV}$ \| $\varepsilon_{S}$ \| $\varepsilon_{B}$ \| $S/\sqrt{B}$ \| $\varepsilon_{S}$ \| $\varepsilon_{B}$ \| $S/\sqrt{B}$ \| $\varepsilon_{S}$ \| $\varepsilon_{B}$ \| $S/\sqrt{B}$ $M_{h}/\mathrm{GeV}\quad[\delta/\mathrm{GeV}]$ \| $110\quad[20]$ \| $120\quad[20]$ \| $130\quad[20]$ lepton & MET \| \| $0.2254$ \| $19080$ \| 27$\mbox{$\cdot\;\\!\\!$}10^{\textrm{-}6}$ \| $0.8017$ \| $19080$ \| 97$\mbox{$\cdot\;\\!\\!$}10^{\textrm{-}6}$ \| $2.260$ \| $19080$ \| 27$\mbox{$\cdot\;\\!\\!$}10^{\textrm{-}5}$ cuts & $\geq 2$ jets \| \| 0.0673 \| 0.0087 \| 0.027 \| 0.104 \| 0.0087 \| 0.095 \| 0.173 \| 0.0087 \| 0.27 naive $h$-reco \| $50$ \| $0.02723$ \| $321.2$ \| 20$\mbox{$\cdot\;\\!\\!$}10^{\textrm{-}5}$ \| $0.2132$ \| $969.2$ \| 51$\mbox{$\cdot\;\\!\\!$}10^{\textrm{-}5}$ \| $0.9580$ \| $1963$ \| 0.0011 \| \| 0.00813 \| 15$\mbox{$\cdot\;\\!\\!$}10^{\textrm{-}5}$ \| 0.025 \| 0.0276 \| 44$\mbox{$\cdot\;\\!\\!$}10^{\textrm{-}5}$ \| 0.11 \| 0.0733 \| 89$\mbox{$\cdot\;\\!\\!$}10^{\textrm{-}5}$ \| 0.35 naive $h$-reco \| $48$ \| $0.00873$ \| $34.23$ \| 59$\mbox{$\cdot\;\\!\\!$}10^{\textrm{-}5}$ \| $0.1160$ \| $90.62$ \| 0.0029 \| $0.5804$ \| $363.7$ \| 0.0037 $\lvert\tilde{m}_{j_{1}j_{2}}\\!-80\rvert\\!<\\!\tilde{\delta}$ \| \| 0.00261 \| 16$\mbox{$\cdot\;\\!\\!$}10^{\textrm{-}6}$ \| 0.024 \| 0.0150 \| 41$\mbox{$\cdot\;\\!\\!$}10^{\textrm{-}6}$ \| 0.20 \| 0.0444 \| 17$\mbox{$\cdot\;\\!\\!$}10^{\textrm{-}5}$ \| 0.49 comb. $h$-reco \| $50$ \| $0.04236$ \| $392.8$ \| 25$\mbox{$\cdot\;\\!\\!$}10^{\textrm{-}5}$ \| $0.3000$ \| $1131$ \| 61$\mbox{$\cdot\;\\!\\!$}10^{\textrm{-}5}$ \| $1.253$ \| $2239$ \| 0.0013 \| \| 0.0126 \| 18$\mbox{$\cdot\;\\!\\!$}10^{\textrm{-}5}$ \| 0.035 \| 0.0389 \| 51$\mbox{$\cdot\;\\!\\!$}10^{\textrm{-}5}$ \| 0.15 \| 0.0959 \| 0.0010 \| 0.43 comb. $h$-reco \| $50$ \| $0.01252$ \| $41.87$ \| 69$\mbox{$\cdot\;\\!\\!$}10^{\textrm{-}5}$ \| $0.1515$ \| $124.0$ \| 0.0028 \| $0.7673$ \| $461.3$ \| 0.0038 $\lvert\tilde{m}_{jj^{\prime}}\\!-80\rvert\\!<\\!\tilde{\delta}$ \| \| 0.00374 \| 19$\mbox{$\cdot\;\\!\\!$}10^{\textrm{-}6}$ \| 0.032 \| 0.0196 \| 56$\mbox{$\cdot\;\\!\\!$}10^{\textrm{-}6}$ \| 0.22 \| 0.0587 \| 21$\mbox{$\cdot\;\\!\\!$}10^{\textrm{-}5}$ \| 0.58 comb. $h$-reco \| $20$ \| $0.00607$ \| $8.805$ \| 0.0016 \| $0.1017$ \| $23.97$ \| 0.0098 \| $0.4995$ \| $53.81$ \| 0.021 $\lvert\tilde{m}_{jj^{\prime}}\\!-80\rvert\\!<\\!\tilde{\delta}$ \| \| 0.00181 \| 40$\mbox{$\cdot\;\\!\\!$}10^{\textrm{-}7}$ \| 0.033 \| 0.0132 \| 11$\mbox{$\cdot\;\\!\\!$}10^{\textrm{-}6}$ \| 0.34 \| 0.0382 \| 24$\mbox{$\cdot\;\\!\\!$}10^{\textrm{-}6}$ \| 1.11 $M_{h}/\mathrm{GeV}\quad[\delta/\mathrm{GeV}]$ \| \| $140\quad[20]$ \| $150\quad[20]$ lepton & MET \| \| \| \| \| $4.316$ \| $19080$ \| 52$\mbox{$\cdot\;\\!\\!$}10^{\textrm{-}5}$ \| $6.343$ \| $19080$ \| 77$\mbox{$\cdot\;\\!\\!$}10^{\textrm{-}5}$ cuts & $\geq 2$ jets \| \| \| \| \| 0.250 \| 0.0087 \| 0.51 \| 0.328 \| 0.0087 \| 0.75 naive $h$-reco \| $50$ \| \| \| \| $2.213$ \| $3230$ \| 0.0016 \| $3.642$ \| $4593$ \| 0.0018 \| \| \| \| \| 0.128 \| 0.0015 \| 0.63 \| 0.188 \| 0.0021 \| 0.88 naive $h$-reco \| $48$ \| \| \| \| $1.374$ \| $924.4$ \| 0.0034 \| $2.509$ \| $1662$ \| 0.0035 $\lvert\tilde{m}_{j_{1}j_{2}}\\!-80\rvert\\!<\\!\tilde{\delta}$ \| \| \| \| \| 0.0795 \| 42$\mbox{$\cdot\;\\!\\!$}10^{\textrm{-}5}$ \| 0.73 \| 0.130 \| 76$\mbox{$\cdot\;\\!\\!$}10^{\textrm{-}5}$ \| 1.00 comb. $h$-reco \| $50$ \| \| \| \| $2.902$ \| $3637$ \| 0.0018 \| $4.778$ \| $5103$ \| 0.0022 \| \| \| \| \| 0.168 \| 0.0017 \| 0.78 \| 0.247 \| 0.0023 \| 1.09 comb. $h$-reco \| $50$ \| \| \| \| $1.877$ \| $1121$ \| 0.0038 \| $3.487$ \| $1968$ \| 0.0041 $\lvert\tilde{m}_{jj^{\prime}}\\!-80\rvert\\!<\\!\tilde{\delta}$ \| \| \| \| \| 0.109 \| 51$\mbox{$\cdot\;\\!\\!$}10^{\textrm{-}5}$ \| 0.91 \| 0.180 \| 89$\mbox{$\cdot\;\\!\\!$}10^{\textrm{-}5}$ \| 1.28 comb. $h$-reco \| $20$ \| \| \| \| $1.213$ \| $234.1$ \| 0.012 \| $2.462$ \| $704.8$ \| 0.0080 $\lvert\tilde{m}_{jj^{\prime}}\\!-80\rvert\\!<\\!\tilde{\delta}$ \| \| \| \| \| 0.0701 \| 11$\mbox{$\cdot\;\\!\\!$}10^{\textrm{-}5}$ \| 1.29 \| 0.127 \| 32$\mbox{$\cdot\;\\!\\!$}10^{\textrm{-}5}$ \| 1.51 Table 8: Impact of the different levels of cuts on the $e\nu_{e}$+jets final states for the $gg\to h\to WW$ production and decay signal and the $W$+jets background as obtained from SHERPA. Cross sections $\sigma_{S}$, $\sigma_{B}$, acceptances $\varepsilon_{S}$, $\varepsilon_{B}$ and $S/B$, $S/\sqrt{B}$ ratios are shown for Higgs boson masses below the on-shell diboson mass threshold from $M_{h}=110\ \mathrm{GeV}$ to $M_{h}=150\ \mathrm{GeV}$. Note that $\tilde{m}_{ij}=m_{ij}/\mathrm{GeV}$ and $\tilde{\delta}=\delta/\mathrm{GeV}$. The significances were calculated according to Eqs. (4.2) assuming ${\mathcal{L}}=10\ \mathrm{fb}^{-1}$ of integrated luminosity, counting both electrons and muons and combining Tevatron experiments. The layout of the table is the same as in Tables 4 and 7, however a smaller number of Higgs boson candidate selections is shown. ### B.1 Ideal Higgs boson reconstruction analyses We first complete the presentation of our Section 4.2 main results by showing Tables 7 and 8 where we display the numbers associated with the high and low Higgs boson mass region, respectively. As in Table 4 we list the signal and $W$+jet background cross sections, selection efficiencies, $S/B$ ratios and significances at different analysis levels. All cuts, the selection procedures, the layout of the tables and the interpretation of the results given in the tables have been discussed in detail in Section 4.2. Note that the rightmost column of Table 10 carries the outcomes for the test mass point $M_{h}=220\ \mathrm{GeV}$. One remark shall be added regarding the magnitude of off-shell effects. The loss one faces due to off-shell Higgs bosons can be read off Table 8 by comparing the acceptances after baseline (1st rows) and combinatorial Higgs boson selection (4th rows). While above the $WW$ mass threshold the loss on the signal (background) is mild (significant) ranging from $1.2$–$1.3$ ($2.6$–$3.2$), it steadily increases for decreasing $M_{h}$, approaching $1.8$ and $5.3$ at $M_{h}=130\ \mathrm{GeV}$ and $M_{h}=110\ \mathrm{GeV}$, respectively. The background loss factors turn huge (up to 48) because of the steeply falling $m_{e\nu_{e}jj^{\prime}}$ spectrum (cf. Figure 3), but this cannot overcome the smallness of $S/\sqrt{B}$ due to the signal reduction. We now present the results of our side studies, which we decided to put in an appendix in order to not distract the flow of the main body. Figure 11: $S/\sqrt{B}$ significances as a function of injected Higgs boson masses varying from $M^{\mathrm{inj}}_{h}=165\ \mathrm{GeV}$ to $200\ \mathrm{GeV}$ for different mass window parameters $\Delta$ and $\delta$. Results of two combinatorial analyses based on the invm reconstruction are shown: one using the default setting $M_{h}=M^{\mathrm{inj}}_{h}$ (dashed lines), the other where the hypothesized Higgs boson mass is fixed at $M_{h}=180\ \mathrm{GeV}$ (solid lines). For the different injected Higgs boson masses, the $e\nu_{e}$+jets final states are generated from the $gg\to h\to WW$ signal and the $W$+jets production background. All significances were calculated according to Eqs. (4.2) taking only the dominant background into account and under the assumption of an integrated luminosity of ${\mathcal{L}}=10\ \mathrm{fb}^{-1}$, including electron and muon channels, i.e. $f_{\ell}=2$. cuts & \| $2\,\Delta/$ \| $\sigma_{S}/\mathrm{fb}$ \| $\sigma_{B}/\mathrm{fb}$ \| $S/B$ \| $\sigma_{S}/\mathrm{fb}$ \| $\sigma_{B}/\mathrm{fb}$ \| $S/B$ \| $\sigma_{S}/\mathrm{fb}$ \| $\sigma_{B}/\mathrm{fb}$ \| $S/B$ ---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|--- selections \| $\mathrm{GeV}$ \| $\varepsilon_{S}$ \| $\varepsilon_{B}$ \| $S/\sqrt{B}$ \| $\varepsilon_{S}$ \| $\varepsilon_{B}$ \| $S/\sqrt{B}$ \| $\varepsilon_{S}$ \| $\varepsilon_{B}$ \| $S/\sqrt{B}$ $M_{h}/\mathrm{GeV}\quad[\delta/\mathrm{GeV}]$ \| $165\quad[20]$ \| $170\quad[20]$ \| $180\quad[20]$ naive $h$-reco \| $50$ \| $4.495$ \| $5317$ \| 0.0020 \| $4.121$ \| $5317$ \| 0.0018 \| $3.232$ \| $5527$ \| 0.0013 \| \| 0.297 \| 0.0022 \| 1.14 \| 0.296 \| 0.0022 \| 1.02 \| 0.290 \| 0.0023 \| 0.77 naive $h$-reco \| $48$ \| $3.965$ \| $2371$ \| 0.0040 \| $3.585$ \| $2571$ \| 0.0032 \| $2.759$ \| $2849$ \| 0.0022 $\lvert\tilde{m}_{j_{1}j_{2}}\\!-80\rvert\\!<\\!\tilde{\delta}$ \| \| 0.262 \| 0.0010 \| 1.50 \| 0.258 \| 0.0011 \| 1.27 \| 0.248 \| 0.0012 \| 0.92 comb. $h$-reco \| $50$ \| $5.603$ \| $5511$ \| 0.0024 \| $5.149$ \| $5758$ \| 0.0021 \| $4.105$ \| $5982$ \| 0.0016 \| \| 0.370 \| 0.0023 \| 1.39 \| 0.370 \| 0.0024 \| 1.22 \| 0.368 \| 0.0025 \| 0.94 comb. $h$-reco \| $50$ \| $5.081$ \| $2632$ \| 0.0046 \| $4.690$ \| $2923$ \| 0.0037 \| $3.722$ \| $3251$ \| 0.0026 $\lvert\tilde{m}_{jj^{\prime}}\\!-80\rvert\\!<\\!\tilde{\delta}$ \| \| 0.336 \| 0.0011 \| 1.83 \| 0.337 \| 0.0012 \| 1.56 \| 0.334 \| 0.0014 \| 1.16 comb. $h$-reco \| $20$ \| $3.890$ \| $1206$ \| 0.0076 \| $3.592$ \| $1357$ \| 0.0061 \| $2.846$ \| $1518$ \| 0.0043 $\lvert\tilde{m}_{jj^{\prime}}\\!-80\rvert\\!<\\!\tilde{\delta}$ \| \| 0.257 \| 51$\mbox{$\cdot\;\\!\\!$}10^{\textrm{-}5}$ \| 2.07 \| 0.258 \| 57$\mbox{$\cdot\;\\!\\!$}10^{\textrm{-}5}$ \| 1.76 \| 0.255 \| 64$\mbox{$\cdot\;\\!\\!$}10^{\textrm{-}5}$ \| 1.30 $M_{h}/\mathrm{GeV}\quad[\delta/\mathrm{GeV}]$ \| $190\quad[20]$ \| $200\quad[20]$ \| $210\quad[20]$ naive $h$-reco \| $50$ \| $2.231$ \| $5435$ \| 92$\mbox{$\cdot\;\\!\\!$}10^{\textrm{-}5}$ \| $1.779$ \| $5158$ \| 76$\mbox{$\cdot\;\\!\\!$}10^{\textrm{-}5}$ \| $1.480$ \| $4787$ \| 67$\mbox{$\cdot\;\\!\\!$}10^{\textrm{-}5}$ \| \| 0.287 \| 0.0023 \| 0.53 \| 0.287 \| 0.0022 \| 0.42 \| 0.288 \| 0.0020 \| 0.36 naive $h$-reco \| $48$ \| $1.867$ \| $2821$ \| 0.0015 \| $1.465$ \| $2560$ \| 0.0013 \| $1.203$ \| $2161$ \| 0.0012 $\lvert\tilde{m}_{j_{1}j_{2}}\\!-80\rvert\\!<\\!\tilde{\delta}$ \| \| 0.240 \| 0.0012 \| 0.61 \| 0.237 \| 0.0011 \| 0.49 \| 0.234 \| 91$\mbox{$\cdot\;\\!\\!$}10^{\textrm{-}5}$ \| 0.44 comb. $h$-reco \| $50$ \| $2.865$ \| $5892$ \| 0.0011 \| $2.302$ \| $5607$ \| 90$\mbox{$\cdot\;\\!\\!$}10^{\textrm{-}5}$ \| $1.925$ \| $5221$ \| 80$\mbox{$\cdot\;\\!\\!$}10^{\textrm{-}5}$ \| \| 0.368 \| 0.0025 \| 0.65 \| 0.372 \| 0.0024 \| 0.53 \| 0.375 \| 0.0022 \| 0.45 comb. $h$-reco \| $50$ \| $2.575$ \| $3243$ \| 0.0018 \| $2.052$ \| $2988$ \| 0.0015 \| $1.702$ \| $2557$ \| 0.0014 $\lvert\tilde{m}_{jj^{\prime}}\\!-80\rvert\\!<\\!\tilde{\delta}$ \| \| 0.331 \| 0.0014 \| 0.79 \| 0.331 \| 0.0013 \| 0.64 \| 0.332 \| 0.0011 \| 0.57 comb. $h$-reco \| $20$ \| $1.949$ \| $1445$ \| 0.0030 \| $1.536$ \| $1242$ \| 0.0027 \| $1.265$ \| $1029$ \| 0.0027 $\lvert\tilde{m}_{jj^{\prime}}\\!-80\rvert\\!<\\!\tilde{\delta}$ \| \| 0.251 \| 61$\mbox{$\cdot\;\\!\\!$}10^{\textrm{-}5}$ \| 0.89 \| 0.248 \| 53$\mbox{$\cdot\;\\!\\!$}10^{\textrm{-}5}$ \| 0.74 \| 0.247 \| 44$\mbox{$\cdot\;\\!\\!$}10^{\textrm{-}5}$ \| 0.66 Table 9: Impact of the different levels of cuts on the $e\nu_{e}$+jets final states for the $gg\to h\to WW$ production and decay signal and the $W$+jets background as obtained from SHERPA when using the CTEQ6.6 PDF libraries. Cross sections $\sigma_{S}$, $\sigma_{B}$, acceptances $\varepsilon_{S}$, $\varepsilon_{B}$ and $S/B$, $S/\sqrt{B}$ ratios are shown for Higgs boson masses from $M_{h}=165\ \mathrm{GeV}$ to $M_{h}=210\ \mathrm{GeV}$. Note that $\tilde{m}_{ij}=m_{ij}/\mathrm{GeV}$ and $\tilde{\delta}=\delta/\mathrm{GeV}$. All significances were calculated according to Eqs. (4.2) assuming ${\mathcal{L}}=10\ \mathrm{fb}^{-1}$ of integrated luminosity, counting both electron and muon channels and combining Tevatron experiments. The Higgs boson masses used in the analyses are, of course, hypothetical. However, we only considered the obvious scenario where the test mass $M_{h}$ has been set equal to the Higgs boson mass $M^{\mathrm{inj}}_{h}$ injected while generating the signal predictions. It is clear that one scans over a range of masses when pursuing an analysis, and here we do it in steps of 10 GeV; still the actual Higgs boson mass could deviate as much as 5 GeV from the assumed mass. Hence, we want to briefly study how strongly the invm Higgs boson candidate selections and their related significances depend on the match between the test and injected Higgs boson mass. To this end we generated other than default signal predictions for Higgs boson masses of $M^{\mathrm{inj}}_{h}=165,175,185,195\ \mathrm{GeV}$ and input them into the analyses using $M_{h}=180\ \mathrm{GeV}$. Figure 11 shows the outcome of this side study where, for both types of analyses, we collected results for several mass window settings. We learn two things from plotting the significance as a function of the Higgs boson generation mass. First, the significances that we attain if we keep the selection parameters ($M_{h}$, $\Delta$, $\delta$) constant are fairly robust over a broader range of generation masses. Yet the maximum $S/\sqrt{B}$ occur for $M_{h}=M^{\mathrm{inj}}_{h}$. Secondly we learn, asymmetric window placements such that $M_{h}<M^{\mathrm{inj}}_{h}$ are beneficial to achieve significance gains. By focusing on a single generation mass, e.g. $M^{\mathrm{inj}}_{h}=190\ \mathrm{GeV}$, we see that the response in significance can easily get as large as 40%. The strong sensitivity can be understood by comparing the signal and background shapes as visualized in Figure 3 or the upper left plot of Figure 1. cuts & \| $2\,\Delta/$ \| $\sigma_{S}/\mathrm{fb}$ \| $\sigma_{B}/\mathrm{fb}$ \| $S/B$ \| $\sigma_{S}/\mathrm{fb}$ \| $\sigma_{B}/\mathrm{fb}$ \| $S/B$ \| $\sigma_{S}/\mathrm{fb}$ \| $\sigma_{B}/\mathrm{fb}$ \| $S/B$ ---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|--- selections \| $\mathrm{GeV}$ \| $\varepsilon_{S}$ \| $\varepsilon_{B}$ \| $S/\sqrt{B}$ \| $\varepsilon_{S}$ \| $\varepsilon_{B}$ \| $S/\sqrt{B}$ \| $\varepsilon_{S}$ \| $\varepsilon_{B}$ \| $S/\sqrt{B}$ $M_{h}/\mathrm{GeV}\quad[\delta/\mathrm{GeV}]$ \| $180\quad[15]$ \| $180\quad[20]\,[p^{\mathrm{jet}}_{T}/\mathrm{GeV}\\!>\\!30]$ \| $220\quad[20]$ $\sigma^{(0)}$ \| \| $14.19$ \| $220\mbox{$\cdot\;\\!\\!$}10^{4}$ \| 14$\mbox{$\cdot\;\\!\\!$}10^{\textrm{-}6}$ \| $14.19$ \| $220\mbox{$\cdot\;\\!\\!$}10^{4}$ \| 14$\mbox{$\cdot\;\\!\\!$}10^{\textrm{-}6}$ \| $5.420$ \| $220\mbox{$\cdot\;\\!\\!$}10^{4}$ \| 51$\mbox{$\cdot\;\\!\\!$}10^{\textrm{-}7}$ \| \| 1.0 \| 1.0 \| 0.15 \| 1.0 \| 1.0 \| 0.15 \| 1.0 \| 1.0 \| 0.05 lepton & \| \| $7.946$ \| $984\mbox{$\cdot\;\\!\\!$}10^{3}$ \| 18$\mbox{$\cdot\;\\!\\!$}10^{\textrm{-}6}$ \| $8.127$ \| $987\mbox{$\cdot\;\\!\\!$}10^{3}$ \| 18$\mbox{$\cdot\;\\!\\!$}10^{\textrm{-}6}$ \| $3.101$ \| $984\mbox{$\cdot\;\\!\\!$}10^{3}$ \| 65$\mbox{$\cdot\;\\!\\!$}10^{\textrm{-}7}$ MET cuts \| \| 0.560 \| 0.45 \| 0.12 \| 0.573 \| 0.45 \| 0.13 \| 0.572 \| 0.45 \| 0.05 as above & \| \| $6.471$ \| $191\mbox{$\cdot\;\\!\\!$}10^{2}$ \| 74$\mbox{$\cdot\;\\!\\!$}10^{\textrm{-}5}$ \| $3.907$ \| $6715$ \| 0.0013 \| $2.661$ \| $191\mbox{$\cdot\;\\!\\!$}10^{2}$ \| 29$\mbox{$\cdot\;\\!\\!$}10^{\textrm{-}5}$ $\geq 2$ jets \| \| 0.456 \| 0.0087 \| 0.72 \| 0.275 \| 0.0031 \| 0.73 \| 0.491 \| 0.0087 \| 0.28 as above & \| \| $3.169$ \| $5272$ \| 0.0013 \| $1.932$ \| $2194$ \| 0.0019 \| $1.413$ \| $6997$ \| 42$\mbox{$\cdot\;\\!\\!$}10^{\textrm{-}5}$ $\lvert\tilde{m}_{j_{1}j_{2}}\\!-80\rvert\\!<\\!\tilde{\delta}$ \| \| 0.223 \| 0.0024 \| 0.67 \| 0.136 \| 0.0010 \| 0.63 \| 0.261 \| 0.0032 \| 0.25 naive $h$-reco \| $50$ \| $3.911$ \| $6749$ \| 0.0013 \| $1.948$ \| $1429$ \| 0.0030 \| $1.510$ \| $5342$ \| 58$\mbox{$\cdot\;\\!\\!$}10^{\textrm{-}5}$ \| \| 0.276 \| 0.0031 \| 0.73 \| 0.137 \| 65$\mbox{$\cdot\;\\!\\!$}10^{\textrm{-}5}$ \| 0.79 \| 0.279 \| 0.0024 \| 0.30 naive $h$-reco \| $30$ \| $3.039$ \| $4199$ \| 0.0016 \| $1.551$ \| $907.0$ \| 0.0037 \| $1.161$ \| $3196$ \| 75$\mbox{$\cdot\;\\!\\!$}10^{\textrm{-}5}$ \| \| 0.214 \| 0.0019 \| 0.72 \| 0.109 \| 41$\mbox{$\cdot\;\\!\\!$}10^{\textrm{-}5}$ \| 0.79 \| 0.214 \| 0.0015 \| 0.30 naive $h$-reco \| $48$ \| $2.857$ \| $2778$ \| 0.0022 \| $1.631$ \| $925.5$ \| 0.0038 \| $1.169$ \| $2167$ \| 0.0011 $\lvert\tilde{m}_{j_{1}j_{2}}\\!-80\rvert\\!<\\!\tilde{\delta}$ \| \| 0.201 \| 0.0013 \| 0.83 \| 0.115 \| 42$\mbox{$\cdot\;\\!\\!$}10^{\textrm{-}5}$ \| 0.82 \| 0.216 \| 99$\mbox{$\cdot\;\\!\\!$}10^{\textrm{-}5}$ \| 0.37 naive $h$-reco \| $20$ \| $2.118$ \| $1351$ \| 0.0034 \| $1.170$ \| $454.3$ \| 0.0056 \| $0.7991$ \| $874.9$ \| 0.0019 $\lvert\tilde{m}_{j_{1}j_{2}}\\!-80\rvert\\!<\\!\tilde{\delta}$ \| \| 0.149 \| 61$\mbox{$\cdot\;\\!\\!$}10^{\textrm{-}5}$ \| 0.89 \| 0.0825 \| 21$\mbox{$\cdot\;\\!\\!$}10^{\textrm{-}5}$ \| 0.84 \| 0.147 \| 40$\mbox{$\cdot\;\\!\\!$}10^{\textrm{-}5}$ \| 0.39 comb. $h$-reco \| $50$ \| $5.241$ \| $7396$ \| 0.0015 \| $2.484$ \| $1534$ \| 0.0035 \| $2.088$ \| $5942$ \| 73$\mbox{$\cdot\;\\!\\!$}10^{\textrm{-}5}$ \| \| 0.369 \| 0.0034 \| 0.94 \| 0.175 \| 70$\mbox{$\cdot\;\\!\\!$}10^{\textrm{-}5}$ \| 0.98 \| 0.385 \| 0.0027 \| 0.40 comb. $h$-reco \| $20$ \| $3.657$ \| $3255$ \| 0.0024 \| $1.718$ \| $671.7$ \| 0.0056 \| $1.440$ \| $2508$ \| 0.0012 \| \| 0.258 \| 0.0015 \| 0.99 \| 0.121 \| 31$\mbox{$\cdot\;\\!\\!$}10^{\textrm{-}5}$ \| 1.02 \| 0.266 \| 0.0011 \| 0.42 comb. $h$-reco \| $50$ \| $4.248$ \| $3241$ \| 0.0029 \| $2.185$ \| $1024$ \| 0.0046 \| $1.798$ \| $2662$ \| 0.0014 $\lvert\tilde{m}_{jj^{\prime}}\\!-80\rvert\\!<\\!\tilde{\delta}$ \| \| 0.299 \| 0.0015 \| 1.15 \| 0.154 \| 47$\mbox{$\cdot\;\\!\\!$}10^{\textrm{-}5}$ \| 1.05 \| 0.332 \| 0.0012 \| 0.51 comb. $h$-reco \| $30$ \| $3.845$ \| $2220$ \| 0.0038 \| $1.916$ \| $699.8$ \| 0.0060 \| $1.528$ \| $1581$ \| 0.0020 $\lvert\tilde{m}_{jj^{\prime}}\\!-80\rvert\\!<\\!\tilde{\delta}$ \| \| 0.271 \| 0.0010 \| 1.26 \| 0.135 \| 32$\mbox{$\cdot\;\\!\\!$}10^{\textrm{-}5}$ \| 1.11 \| 0.282 \| 72$\mbox{$\cdot\;\\!\\!$}10^{\textrm{-}5}$ \| 0.56 comb. $h$-reco \| $20$ \| $3.260$ \| $1564$ \| 0.0045 \| $1.617$ \| $492.9$ \| 0.0071 \| $1.264$ \| $1058$ \| 0.0025 $\lvert\tilde{m}_{jj^{\prime}}\\!-80\rvert\\!<\\!\tilde{\delta}$ \| \| 0.230 \| 71$\mbox{$\cdot\;\\!\\!$}10^{\textrm{-}5}$ \| 1.27 \| 0.114 \| 22$\mbox{$\cdot\;\\!\\!$}10^{\textrm{-}5}$ \| 1.12 \| 0.233 \| 48$\mbox{$\cdot\;\\!\\!$}10^{\textrm{-}5}$ \| 0.57 comb. $h$-reco \| $16$ \| $2.895$ \| $1273$ \| 0.0049 \| $1.444$ \| $399.0$ \| 0.0079 \| $1.115$ \| $846.6$ \| 0.0027 $\lvert\tilde{m}_{jj^{\prime}}\\!-80\rvert\\!<\\!\tilde{\delta}$ \| \| 0.204 \| 58$\mbox{$\cdot\;\\!\\!$}10^{\textrm{-}5}$ \| 1.25 \| 0.102 \| 18$\mbox{$\cdot\;\\!\\!$}10^{\textrm{-}5}$ \| 1.11 \| 0.206 \| 38$\mbox{$\cdot\;\\!\\!$}10^{\textrm{-}5}$ \| 0.56 comb. $h$-reco \| $10$ \| $2.010$ \| $805.0$ \| 0.0054 \| $1.038$ \| $258.5$ \| 0.0087 \| $0.7722$ \| $525.2$ \| 0.0030 $\lvert\tilde{m}_{jj^{\prime}}\\!-80\rvert\\!<\\!\tilde{\delta}$ \| \| 0.142 \| 37$\mbox{$\cdot\;\\!\\!$}10^{\textrm{-}5}$ \| 1.09 \| 0.0731 \| 12$\mbox{$\cdot\;\\!\\!$}10^{\textrm{-}5}$ \| 0.99 \| 0.142 \| 24$\mbox{$\cdot\;\\!\\!$}10^{\textrm{-}5}$ \| 0.49 Table 10: Impact of the different levels of cuts on the $e\nu_{e}$+jets final states for the $gg\to h\to WW$ production and decay signal and the $W$+jets background as obtained from SHERPA. Cross sections $\sigma_{S}$, $\sigma_{B}$, acceptances $\varepsilon_{S}$, $\varepsilon_{B}$ and $S/B$, $S/\sqrt{B}$ ratios are given for Higgs boson masses of $M_{h}=180\ \mathrm{GeV}$ and $M_{h}=220\ \mathrm{GeV}$. For the former mass point, the central column shows the values for a jet $p_{T}$ threshold increased by $10\ \mathrm{GeV}$, while the left column has the values for a smaller dijet mass window of $\delta=15\ \mathrm{GeV}$. A further decrease of the dijet mass window to $\delta=10\ \mathrm{GeV}$ yields $S/B=0.0034$ as well as $S/\sqrt{B}=1.14$ and $S/B=0.0058$ as well as $S/\sqrt{B}=1.36$ for the combinatorial Higgs boson reconstruction and mass windows of $\tilde{\Delta}=50$ and $\tilde{\Delta}=20$, respectively. Note that $\tilde{m}_{ij}=m_{ij}/\mathrm{GeV}$; all other mass variables denoted by a tilde are understood in the same way. All significances were calculated according to Eqs. (4.2) assuming ${\mathcal{L}}=10\ \mathrm{fb}^{-1}$ of integrated luminosity, counting both electron and muon channels and combining Tevatron experiments. In the remainder of this appendix, we outline the impact of PDF variations and parameter variations other than $M_{h}$, $\Delta$ and $\delta$ on our analyses. The SHERPA calculations resulting from using the CTEQ6.6 PDF libraries give a similar pattern with significances that are about 1–10% larger. This can be read off Table 9 and seen in Figure 2. By normalizing the Monte Carlo predictions for both PDF choices, MSTW2008 and CTEQ6.6, to the same respective theory cross sections, it is altogether reassuring to see that the cut and event selection procedures only induce deviations of the order of 10% or below. For the combinatorial Higgs boson candidate selection, one finds almost the same $S/\sqrt{B}$ ratios, provided a wide Higgs boson mass window is used. The significances of the CTEQ6.6 calculations outperform those obtained with MSTW2008, once the mass windows for the Higgs bosons ($\Delta$) and dijets ($\delta$) are tightened. We find the differences being more pronounced for Higgs boson masses just above the $WW$ threshold. Speaking of shape differences triggered by the use of different PDFs, we note that the MSTW2008 PDF set accounts for a larger transverse activity, i.e. rapidity distributions turn out steeper while the $p_{T}$ spectra develop 10–25% harder tails compared to the CTEQ6.6 predictions. Also, for MSTW2008, mass peaks are more washed out, again resulting in differences of the order of 25%. Lastly, apart from the rightmost column, Table 10 gives more details for the choice $M_{h}=M^{\mathrm{inj}}_{h}=180\ \mathrm{GeV}$ including the cases where either the dijet mass window has been tightened from $\delta=20\ \mathrm{GeV}$ to $\delta=15\ \mathrm{GeV}$ or the threshold of the jet transverse momenta has been enhanced to $p^{\mathrm{jet}}_{T}>30\ \mathrm{GeV}$. ### B.2 More realistic Higgs boson reconstruction analyses We collect here in this appendix additional material to substantiate our findings presented in Section 4.4. Figures 12–18 display a number of distributions resulting from the baseline plus combinatorial analyses. In each plot we show four curves, two predictions each for the signal and the $W$+jets background as obtained after the ideal (invm) and one of the more realistic combinatorial selections. While we vary the Higgs boson masses and the choice of the realistic $h$ reconstruction method, we keep the window parameters of the combinatorial selections fixed at $\tilde{\Delta}=\Delta/\mathrm{GeV}=25$ and $\tilde{\delta}=\delta/\mathrm{GeV}=20$ over the whole set of spectra shown in these figures. In Figure 19 we look at distribution taken at the intermediate optimization level, including the effects of the major cuts but before the application of the respective minor cuts as listed in Table 6. In these plots we compare the Monte Carlo predictions for the Higgs boson signals of different $M_{h}$ with all, dominant and subdominant background predictions. All these predictions follow from using realistic selection procedures where Higgs boson and dijet mass windows of $\tilde{\Delta}=\tilde{\delta}=20$ have been employed. Each one-dimensional distribution is supplemented by an one-minus-ratio subplot. The first prediction in the respective legend is always taken as the reference, which we have arranged to be a $W$+jets prediction for all plots shown here. The ratio subplots nicely visualize why we place the cuts as given in Table 6. Note that in all plots we only compare the shapes, i.e. all distributions are normalized to unit area. Figure 12: Pseudo-rapidity difference between the two selected jets. Predictions for $gg\to h\to e\nu_{e}$+jets production obtained after invm (dashed) and more realistic (solid) combinatorial selections (top: pzmh, center: pzmw, bottom: mt) are compared with each other and to the corresponding predictions for the $W(\to e\nu_{e})$+jets background. Left panes show results for $M_{h}=150\ \mathrm{GeV}$; in all other cases $M_{h}=190\ \mathrm{GeV}$ was used. Figure 13: Pseudo-rapidity of the selected $\\{e,\nu_{e},j,j^{\prime}\\}$ set, the Higgs boson candidate. The predictions for $gg\to h\to e\nu_{e}$+jets production (coloured lines) obtained after the combinatorial invm (dashed) and more realistic $h$ reconstruction (solid) selections (upper: pzmh, lower: pzmw) are compared with each other and to the corresponding predictions of the $W(\to e\nu_{e})$+jets background (black lines). Left panes show results for $M_{h}=150\ \mathrm{GeV}$, while in the right panes the outcomes for $M_{h}=190\ \mathrm{GeV}$ are shown. We start by showing the $\|\Delta\eta_{j,j^{\prime}}\|$ distributions in Figure 12. The differences in the results of the ideal and more realistic selections are immaterial; there are essentially no differences in the above-threshold cases. Furthermore, the shapes are very stable under $M_{h}$ variations. We see that placing a cut around $\|\Delta\eta_{j,j^{\prime}}\|=1.5$ keeps most of the signal, while it removes a large fraction of the $W$+jets events. We note it is only the $W$+jets background featuring a peak location away from zero, all other backgrounds (not shown here) behave similarly to the signal. When working with the reconstruction methods, pzmh/w, we have access to another longitudinal variable: we can include a cut on the $h$ candidate’s pseudo-rapidity to supplement the constraints from the major and $\|\Delta\eta_{j,j^{\prime}}\|$ cuts. Figure 13 displays various $\eta_{e\nu_{e}jj^{\prime}}$ distributions. Again, all predicted shapes are rather independent of the choice of the test mass $M_{h}$. The $W$+jets background (as well as the electroweak background which is not shown here) tends to preferably populate the forward rapidities while the signal (and the $t\bar{t}$ contribution also not shown here) shows up more central. This leads us to require $\|\eta_{e\nu_{e}jj^{\prime}}\|\lesssim 3.0$ as pointed out in Section 4.4 to achieve additional significance gains. Deviations between the ideal and more realistic reconstructions become visible; they now are $\mathcal{O}(\mbox{25\%})$, this is clearly because one needs information about the neutrino to form this observable. We observe that the $W$+jets background receives the larger corrections compared to the signal. Figure 14: Two-dimensional distributions showing the selected-jet pseudo- rapidity difference, $\|\Delta\eta_{j,j^{\prime}}\|$, plotted versus the reconstructed mass $m_{e\nu_{e}jj^{\prime}}$ of the selected $\\{e,\nu_{e},j,j^{\prime}\\}$ combinations. The predictions for $gg\to h\to e\nu_{e}$+jets production obtained after pzmw reconstruction of Higgs boson candidates are compared with each other and to the corresponding predictions given by the $W(\to e\nu_{e})$+jets background. The upper (lower) plots represent the signal (background) predictions, while the left (right) panes show results for $M_{h}=140\ (180)\ \mathrm{GeV}$. Figure 15: Two-dimensional distributions showing the selected-jet pseudo- rapidity difference, $\|\Delta\eta_{j,j^{\prime}}\|$, plotted versus the reconstructed mass $m_{e\nu_{e}jj^{\prime}}$ of the selected $\\{e,\nu_{e},j,j^{\prime}\\}$ combinations. The predictions for $gg\to h\to e\nu_{e}$+jets production obtained after combinatorial selection according to the mt procedure are compared with each other and to the corresponding predictions given by the $W(\to e\nu_{e})$+jets background. The upper (lower) plots represent the signal (background) predictions, while the left (right) panes show results for $M_{h}=140\ (180)\ \mathrm{GeV}$. Note that for the purpose of illustration, the $m_{e\nu_{e}jj^{\prime}}$ quantities are reconstructed as in the ideal case. We add one more comment regarding longitudinal quantities. In this study, as stated in Section 4.4, the pseudo-rapidity variables, which we discussed above, occur largely uncorrelated with the transverse observables as well as invariant masses. Schematically, we illustrate this on the basis of two- dimensional $\|\Delta\eta_{j,j^{\prime}}\|$ versus $m_{e\nu_{e}jj^{\prime}}$ distributions for both the Higgs boson signal and the $W$+jets background. In Figure 14 we show these distributions as resulting from the combinatorial pzmw reconstruction for two different Higgs boson masses, below ($M_{h}=140\ \mathrm{GeV}$) and above ($M_{h}=180\ \mathrm{GeV}$) the diboson mass threshold. Similarly, Figure 15 exhibits the results obtained with the mt selection where the $m_{e\nu_{e}jj^{\prime}}$ quantities were reconstructed as in the ideal case. When confronted with the respective $W$+jets backgrounds, we notice that the predictions originating from the production of Higgs bosons cover rather different parameter regions in the $m_{e\nu_{e}jj^{\prime}}$ – $\|\Delta\eta_{j,j^{\prime}}\|$ plane. This happens independent of the value of $M_{h}$ and the chosen combinatorial selection. Based on these observations, we expect the total $S/\sqrt{B}$ increase to almost completely factorize into a product of single $S/\sqrt{B}$ improvement factors. Figure 16: Examples of leading and subleading cut observables below the $2\,M_{W}$ threshold, for $M_{h}=130\ \mathrm{GeV}$. Predictions are shown for the $gg\to h$ and $W$ production of $e\nu_{e}$+jets final states using the invm (dashed) and more realistic (solid) combinatorial selections. Top to bottom, left panes: $m_{T,e^{-}\bar{\nu}_{e}}$ (pzmw leading), $m^{(\nu_{e})}_{T,e^{+}\nu_{e}jj^{\prime}}$ and $m_{e^{+}\nu_{e}jj^{\prime}}$ (pzmh leading and subleading); right panes: $H_{T,jj^{\prime}}$, $m_{T,jj^{\prime}}$ and $m_{jj^{\prime}}$ (mt leading to subsubleading). Figure 17: Examples of (sub)leading cut observables for Higgs boson masses $M_{h}\sim 2\,M_{W}$. Predictions are shown for the $gg\to h$ and $W$ production of $e\nu_{e}$+jets final states using the invm (dashed) and more realistic (solid) combinatorial selections. Top to bottom, left panes: $p_{T,e^{+}\nu_{e}jj^{\prime}}$ and $m_{e^{-}\bar{\nu}_{e}jj^{\prime}}$ (pzmh leading and subleading), $H_{T,e^{-}\bar{\nu}_{e}jj^{\prime}}$ (mt subleading); right panes: $m^{(\nu_{e})}_{T,e^{-}\bar{\nu}_{e}jj^{\prime}}$, $m_{e^{-}\bar{\nu}_{e}jj^{\prime}}$ and $H_{T,jj^{\prime}}$ (pzmw leading to subsubleading). In Figure 16 we depict examples of potential discriminators below the diboson mass threshold. From the upper left to the lower right we present, for the choice of $M_{h}=130\ \mathrm{GeV}$, the transverse mass $m_{T,e^{-}\bar{\nu}_{e}}$ of the leptonically decaying $W^{-}$, the scalar $p_{T}$ sum $H_{T,jj^{\prime}}$ of the selected jets, the transverse masses of the selected 4-particle set, $m^{(\nu_{e})}_{T,e^{+}\nu_{e}jj^{\prime}}$, and of the selected jets, $m_{T,jj^{\prime}}$, as well as their corresponding invariant masses, $m_{e^{+}\nu_{e}jj^{\prime}}$ and $m_{jj^{\prime}}$. The first three plots in that order are the leading cut variables for pzmw, mt and pzmh followed by the subleading cut variables for mt and pzmh, cf. Table 6. In the lower left plot we display the subsubleading cut variable for the mt method, $m_{jj^{\prime}}$, which would yield a significance gain of 19% if we were to demand $m_{jj^{\prime}}\geq 72\ \mathrm{GeV}$. Unlike the transverse observables depicted in the upper and center panes of Figure 16, the invariant-mass distributions are affected by the modifications of the ideal selection owing to a realistic neutrino treatment. The selected 4-object invariant mass (lower left) exemplifies to what degree shapes can get distorted by the pzmh approach. The shoulder above $M_{h}=130\ \mathrm{GeV}$ emerges because complex solutions cannot be completely avoided in the reconstruction of the neutrino momenta; the lower tail arises from the $m_{T,e\nu_{e}jj^{\prime}}$ constraints on the target mass $m_{,e\nu_{e}jj^{\prime}}$, see Section 4.3. The 2-particle transverse masses shown in the upper left and center right of Figure 16 together with the $m_{jj^{\prime}}$ spectra document why we exploit the signal’s preference for on-shell hadronic and off-shell leptonic decays of the $W$ bosons. All backgrounds considered in this study disfavor this correlation. The 4-particle transverse mass (center left) exhibits – as expected – a nice kinematic edge for the signal at $m^{(\nu_{e})}_{T,e^{+}\nu_{e}jj^{\prime}}=M_{h}$, while all backgrounds are continuous in this variable reaching their broad maxima above the applied mass window. The features of the $H_{T,jj^{\prime}}$ handle (upper right) have been already described in Section 2. For this variable, the electroweak background turns out signal-like whereas the $t\bar{t}$ background generates (by far) the hardest tails. Figure 18: Leading and subleading cut observables for $M_{h}$ choices well above the $2\,M_{W}$ threshold. Predictions are shown for the $gg\to h$ and $W$ production of $e\nu_{e}$+jets final states using the invm (dashed) and more realistic (solid) combinatorial selections. Upper panes: $H_{T,e^{-}\bar{\nu}_{e}jj^{\prime}}$ (leading, pzmh (left) and mt); center panes: $p_{T,e^{-}\bar{\nu}_{e}jj^{\prime}}$ (subleading, pzmh (left) and pzmw); lower panes (subleading), left: $p_{T,e^{-}\bar{\nu}_{e}}$ (pzmh), right: $m_{T,e^{+}\nu_{e}jj^{\prime}}$ (mt). We now discuss some of the near-threshold discriminators where most of the examples are given for the test point $M_{h}=170\ \mathrm{GeV}$. Figure 17 shows in its top row the two leading cut variables that we found for this mass region. In the upper right we have the transverse mass $m^{(\nu_{e})}_{T,e^{-}\bar{\nu}_{e}jj^{\prime}}$ of the selected $\\{e,\nu_{e},j,j^{\prime}\\}$ objects as resulting after the pzmw reconstruction with $M_{h}=150\ \mathrm{GeV}$. We discover properties very similar to those discussed for the case of low Higgs boson masses, cf. Figure 16. In the upper left we have depicted the first-rank discriminator for medium Higgs boson masses – the selected 4-object transverse momentum distribution. It was introduced early on in Section 2. Here, we present the $p_{T,e^{+}\nu_{e}jj^{\prime}}$ distribution as obtained after the combinatorial pzmh selection. We remark that the curves of the other two realistic approaches, pzmw and mt, (both not shown here) deviate even less from the respective curves of the ideal selection. Also not shown in Figure 17 but worthwhile to mention, the electroweak background would yield spectra similar to those of the $W$+jets background whereas the top-pair production would turn up significantly harder than the signal’s $p_{T}$ spectra. The other example plots of Figure 17 are chosen from the set of second-leading cut variables. The center panes display the invariant mass distributions $m_{e^{-}\bar{\nu}_{e}jj^{\prime}}$ resulting from the pzmh/w selections. As opposed to the – by construction – sculpted shapes of the pzmh method, we observe that the pzmw selection reproduces the invariant mass shapes of the invm ideal reconstruction to a large extent. The lower panes demonstrate the potential possessed by scalar transverse momentum sums that we exemplify by means of the selected-set $H_{T,e^{-}\bar{\nu}_{e}jj^{\prime}}$ observable as given by the mt selection (lower left) and the selected-jet $H_{T,jj^{\prime}}$ observable resulting from the pzmw reconstruction (lower right). Remarkably, based on the 4-object $H_{T}$, we can achieve an even clearer separation between the signal and the $W$+jets background once we select $h$ candidates according to the mt method. Figure 18 summarizes the types of discriminating observables as identified in Table 6 for the region of large Higgs boson masses; we use $M_{h}=210$ and $220\ \mathrm{GeV}$ in the example plots. The upper panels represent $H_{T,e^{-}\bar{\nu}_{e}jj^{\prime}}$ distributions obtained after pzmh (left) and mt combinatorial selections. Between these two cases we detect only marginal differences, and similarly between the predictions of the ideal and pzmw reconstructions (not shown here). At large Higgs boson masses, the signal develops the peak at considerably larger $H_{T}$ values compared to the $W$+jets background. This characterizes the selected-set $H_{T}$ as the strongest handle we have above the $WW$ threshold. Moreover, the pzmh and mt realistic selections further enhance the separation between the two peak regions. For the subdominant backgrounds (not shown here), we noticed a strong similarity between the $H_{T}$ spectra arising from the electroweak production and the $W$+jets background. The $t\bar{t}$ background however yields the hardest spectra both in terms of the peak position as well as the tail of the $H_{T}$ distributions. The center panes of Figure 18 show two examples of selected $h$ candidate $p_{T}$ distributions. These variables do not constitute the best discriminators anymore, but still quite often rank second best in separating signal from $W$+jets production in the domain of large $M_{h}$. As the pzmw reconstruction works extremely well for heavy Higgs boson decays into on-shell $W$ bosons, it gives $p_{T,e^{-}\bar{\nu}_{e}jj^{\prime}}$ shapes almost identical with the ideal selection. For the $M_{h}=220\ \mathrm{GeV}$ point, we present two different subleading cut variables in the lower pane of Figure 18. To the left, one finds the $p_{T,e^{-}\bar{\nu}_{e}}$ distribution of the reconstructed $W^{-}$ resulting after selecting Higgs boson candidates according to pzmh. The purely transverse mass $m_{T,e^{+}\nu_{e}jj^{\prime}}$ of the mt-selected 4-object set, cf. Eq. (8), is depicted on the lower right. For these variables, we recognize similar features as for the $H_{T,e\nu_{e}jj^{\prime}}$ spectra concerning the minor backgrounds and the comparison with the invm Higgs boson candidate reconstruction. Figure 19: Examples of minor cut observables after application of major cuts and rapidity constraints including spectra resulting from subdominant backgrounds. Predictions using the more realistic combinatorial $h$ candidate selections are shown for $e\nu_{e}$+jets final states arising from $gg\to h$ (solid red), $W$+jets (solid black), $t\bar{t}$ (dashed blue) and electroweak (dashed green) production at Tevatron Run II. See text for more details. In Figure 19 we give a brief overview of discriminators that lead to a further increase in significance after implementing all major and rapidity constraints discussed in Section 4.4. The boost factor $\gamma_{jj^{\prime}\|e\nu_{e}}$, as introduced in Section 2, proves very helpful in separating signal from backgrounds over a large range of above-threshold Higgs boson masses. Since $\gamma_{jj^{\prime}\|e\nu_{e}}$ develops a peak for the signal only, it is advantageous to isolate the boost factor peak region in order to exploit that a not too broad scalar resonance has been produced over a multitude of continuous backgrounds. This is exemplified in the upper plots of Figure 19 where we present two boost factor $\gamma_{jj^{\prime}\|e^{-}\bar{\nu}_{e}}$ example distributions for $M_{h}=165\ \mathrm{GeV}$ (left) and $M_{h}=220\ \mathrm{GeV}$ when selecting via the pzmw method. For the pzmh method in particular, we identified two transverse variables that, if constrained from below, are very yielding in the low Higgs boson mass region, even at this more involved level of the analysis. We exhibit these variables, $H_{T,jj^{\prime}}$ on the left and $m_{\perp,jj^{\prime}}$ on the right, in the center panes of Figure 19 choosing $M_{h}=130\ \mathrm{GeV}$. As in similar cases the significance gain originates from exploiting the different peak locations associated with the signal, at higher values, and the dominant background, preferring the low values. For $M_{h}$ very close above threshold, we can use the azimuthal angle between the lepton and MET or the two selected jets (as suggested by Han and Zhang in Ref. [7, 8]) to further suppress the backgrounds. A low-$p_{T}$ Higgs boson produced at $WW$ threshold gives rise to two longitudinally moving $W$ bosons, which in turn each decay into two objects oriented almost back-to-back in the transverse plane. This is demonstrated by the example plot in the lower left of Figure 19 where signal and background distributions are shown for the $\Delta\phi_{e^{-},\bar{\nu}_{e}}$ observable when employing the mt selection for $M_{h}=165\ \mathrm{GeV}$. Finally, we display in the lower right of Figure 19 an example of a global- event observable, namely $H_{T}$ as calculated from the entire event, not vetoed by the selection and major cuts. It illustrates the hierarchy of scales intrinsic to the heavy Higgs boson signal ($M_{h}=220\ \mathrm{GeV}$) and different background processes; it also visualizes the leftover potential when considering the global $H_{T}$ for the implementation of additional cuts, see Appendix B.3. ### B.3 Directions for additional improvements The final sets of events surviving our optimized combinatorial selections of Higgs boson candidates are perfect for use as input to a full multivariate analysis. This is primarily because the analysis types presented here were geared towards significance maximization, so that a sufficiently large number of events can be preserved. • $S/B$ improvements: since we are rather safe from a statistics point of view, there is in many cases potential to improve $S/B$ by simply requiring more restrictive constraints accepting (mild) significance losses at the same time. The simplest way is to cut harder in the tails of the major observables; for instance for pzmw at $M_{h}=220\ \mathrm{GeV}$, using $H_{T,e\nu_{e}jj^{\prime}}\geq 188\ \mathrm{GeV}$ (instead of the bound given in Table 6) results in a 40% gain in $S/B$ while the significance only drops by 10%. Of course, a change in variable sometimes is more beneficial for maximizing $S/B$ and keeping a reasonable significance. Consider for example mt at $M_{h}=140\ \mathrm{GeV}$; hardening the $H_{T,jj^{\prime}}$ constraint by cutting out the region below $92\ \mathrm{GeV}$ maximizes $S/B$ by doubling it, but reduces the significance by a factor of $3.4$. In contrast, using $m_{T,jj^{\prime}}\geq 72\ \mathrm{GeV}$ gives a factor $1.6$ increase in $S/B$, yet only a factor $1.2$ decrease in significance. Certainly one can opt for the (other) extreme and totally maximize $S/B$, e.g. for the pzmw case just mentioned, we find a huge $S/B$ gain of 600% by demanding $p_{T,j^{\prime}}\geq 64\ \mathrm{GeV}$ but we actually just traded a reasonable significance associated with half a percent $S/B$ for a $\sim$ 4% $S/B$ of very low significance diminished by a factor of $6.5$. This behaviour is typical owing to the limited amount of data taken at the Tevatron. * • Overall $H_{T}$ cut: the philosophy of this study is to only constrain variables involving the candidate set of particles. Allowing cut observables sensitive to the whole event structure can lead to additional significance improvements but the related uncertainties are larger since such observables are more prone to hard radiative corrections that need to be described appropriately, often beyond parton-shower modeling. At the level of identifying minor cuts, as given in Table 6, we found that an overall $H_{T}$ cut on the selected events yields in most cases significance gains of the order of 20–40% near and above threshold. This is a conservative estimate considering the larger uncertainties on such cuts. An example is shown in the lower right of Figure 19 where we see why one benefits by constraining $H_{T}$ from below. In addition one could suppress the $t\bar{t}$ background by introducing an upper $H_{T}$ bound, or equally, exploit the fact that the leading jets in $t\bar{t}$ production yield a substantially harder $H_{T,2}=H_{T,j_{1}j_{2}}$ spectrum. * • Asymmetric mass windows for reconstruction methods: as touched during the discussion of Figure 11, the use of asymmetric test mass windows, i.e. $(M_{h}-\Delta_{\mathrm{low}},M_{h}+\Delta_{\mathrm{up}})$, can improve the realistic selections that either approximate or set the selected 4-object mass, $m_{e\nu_{e}jj^{\prime}}$. For all $M_{h}$ values, it is advantageous to choose $\Delta_{\mathrm{low}}$ larger by a few GeV than $\Delta_{\mathrm{up}}$ where $2\,\Delta=\Delta_{\mathrm{low}}+\Delta_{\mathrm{up}}$. The results presented in Figure 1 (upper left) and Figure 3 clarify why this is a good idea: first, the Higgs boson resonance is deformed by radiative losses and resolution effects (there is no perfect jet finding let alone (sub)event reconstruction) amounting to a larger portion of cross section below the peak; second, the backgrounds are either sufficiently flat or – below the $WW$ threshold – steeply falling towards smaller masses amplifying the asymmetry effect further. We checked the asymmetric window scenario for the pzmw method, where this led to significance gains up to 10%. For the dijet mass windows, the effects turned out too weak. ## References * [1] M. Baak et. al., Updated Status of the Global Electroweak Fit and Constraints on New Physics, arXiv:1107.0975. * [2] ATLAS Collaboration, Combination of the Searches for the Higgs Boson in $\sim$ 1 fb-1 of Data Taken with the ATLAS Detector at 7 TeV Center-of-Mass Energy, Tech. Rep. ATLAS-CONF-2011-112, CERN, Geneva, August, 2011. * [3] CMS Collaboration, Combination of Higgs Searches, Tech. Rep. CMS-PAS-HIG-11-022, CERN, Geneva, 2011. * [4] For CDF and DØ, the TEVNPH Working Group Collaboration, Combined CDF and DØ upper limits on Standard Model Higgs boson production with up to 8.6 fb-1 of data, arXiv:1107.5518. FERMILAB-CONF-11-354-E, CDF Note 10606 and DØ Note 6226. * [5] On behalf of the CDF Collaboration, A. Buzatu, Standard Model Higgs boson search combination at CDF, in talk presented at the International Europhysics Conference on High Energy Physics, Grenoble, 21–27 July 2011. * [6] On behalf of the DØ Collaboration, S. Greder, Combined upper limits on SM Higgs, in talk presented at the International Europhysics Conference on High Energy Physics, Grenoble, 21–27 July 2011. * [7] T. Han and R.-J. Zhang, Extending the Higgs boson reach at upgraded Tevatron, Phys. Rev. Lett. 82 (1999) 25–28, [hep-ph/9807424]. * [8] T. Han, A. S. Turcot, and R.-J. Zhang, Exploiting $h\to W^{}W^{}$ decays at the upgraded Fermilab Tevatron, Phys. Rev. D59 (1999) 093001, [hep-ph/9812275]. * [9] Higgs Working Group Collaboration, M. S. Carena et. al., Report of the Tevatron Higgs working group, hep-ph/0010338. * [10] W. Stirling, R. Kleiss, and S. Ellis, $W^{+}W^{-}$ Pair Production in High-Energy Hadronic Collisions: Signal Versus Background, Phys. Lett. B163 (1985) 261. * [11] J. Gunion, Z. Kunszt, and M. Soldate, A Background to Higgs Detection, Phys. Lett. B163 (1985) 389. * [12] J. Gunion, P. Kalyniak, M. Soldate, and P. Galison, Searching for the intermediate mass Higgs boson, Phys. Rev. D34 (1986) 101. * [13] J. Gunion and M. Soldate, Overcoming a Critical Background to Higgs Detection, Phys. Rev. D34 (1986) 826. * [14] For CDF and DØ, the TEVNPH Working Group Collaboration, D. Benjamin, Combined CDF and DØ upper limits on $gg\to H\to W^{+}W^{-}$ and constraints on the Higgs boson mass in fourth-generation fermion models with up to 8.2 fb-1 of data, arXiv:1108.3331. * [15] CDF Collaboration, T. Aaltonen et. al., Search for $H\to WW^{}$ Production at CDF using 8.2 fb-1 of data, CDF Note 10599 (2011). [16] DØ Collaboration, V. M. Abazov et. al., Search for Higgs boson production in dilepton plus missing transverse energy final states with 8.1 fb-1 of $p\bar{p}$ collisions at $\sqrt{s}$ = 1.96 TeV, DØ Note 6219-CONF (2011). * [17] CDF and DØ Collaboration, T. Aaltonen et. al., Combination of Tevatron searches for the Standard Model Higgs boson in the $W^{+}W^{-}$ decay mode, Phys. Rev. Lett. 104 (2010) 061802, [arXiv:1001.4162]. * [18] CDF Collaboration, T. Aaltonen et. al., Inclusive search for Standard Model Higgs boson production in the $WW$ decay channel using the CDF II detector, Phys. Rev. Lett. 104 (2010) 061803, [arXiv:1001.4468]. * [19] DØ Collaboration, V. M. Abazov et. al., Search for Higgs boson production in dilepton and missing energy final states with 5.4 fb-1 of $p\bar{p}$ collisions at $\sqrt{s}$ = 1.96 TeV, Phys. Rev. Lett. 104 (2010) 061804, [arXiv:1001.4481]. * [20] K. Iordanidis and D. Zeppenfeld, Searching for a heavy Higgs boson via the $H\to\ $lepton neutrino jet jet mode at the CERN LHC, Phys. Rev. D57 (1998) 3072–3083, [hep-ph/9709506]. * [21] A. Djouadi, The Anatomy of electro-weak symmetry breaking. I: The Higgs boson in the standard model, Phys. Rept. 457 (2008) 1–216, [hep-ph/0503172]. * [22] B. Mellado, W. Quayle, and S. L. Wu, Feasibility of Searches for a Higgs Boson using $H\to WW\to\ell\ell$ \+ MET and High PT Jets at the Tevatron, Phys. Rev. D76 (2007) 093007, [arXiv:0708.2507]. * [23] B. A. Dobrescu and J. D. Lykken, Semileptonic decays of the standard Higgs boson, JHEP 1004 (2010) 083, [arXiv:0912.3543]. * [24] T. Gleisberg et. al., SHERPA 1.alpha, a proof-of-concept version, JHEP 02 (2004) 056, [hep-ph/0311263]. * [25] T. Gleisberg et. al., Event generation with SHERPA 1.1, JHEP 02 (2009) 007, [arXiv:0811.4622]. * [26] DØ Collaboration, V. M. Abazov et. al., A search for the Standard Model Higgs boson in $H\to WW\to\ $leptons + jets in 5.4 fb-1 of $p\bar{p}$ collisions at $\sqrt{s}$ = 1.96 TeV, DØ Note 6095-CONF (July, 2010). * [27] DØ Collaboration, V. M. Abazov et. al., Search for the Standard Model Higgs Boson in the $H\to WW\to\ $lepton + neutrino + $q^{\prime}\bar{q}$ Decay Channel, Phys. Rev. Lett. 106 (2011) 171802, [arXiv:1101.6079]. * [28] CMS Collaboration , according to CMS Statistics Committee, September, 2011. * [29] F. Krauss, R. Kuhn, and G. Soff, AMEGIC++ 1.0: A Matrix element generator in C++, JHEP 02 (2002) 044, [hep-ph/0109036]. * [30] T. Gleisberg and S. Höche, Comix, a new matrix element generator, JHEP 12 (2008) 039, [arXiv:0808.3674]. * [31] For CDF and DØ, the TEVNPH Working Group Collaboration, Combined CDF and DØ upper limits on Standard Model Higgs boson production with up to 8.2 fb-1 of data, arXiv:1103.3233. FERMILAB-CONF-11-044-E, CDF Note 10441 and DØ Note 6184. * [32] LHC Higgs Cross Section Working Group Collaboration, S. Dittmaier et. al., Handbook of LHC Higgs Cross Sections: 1. Inclusive Observables, arXiv:1101.0593. * [33] M. Spira, QCD effects in Higgs physics, Fortsch. Phys. 46 (1998) 203–284, [hep-ph/9705337]. * [34] A. Djouadi, J. Kalinowski, and M. Spira, HDECAY: A program for Higgs boson decays in the Standard Model and its supersymmetric extension, Comput. Phys. Commun. 108 (1998) 56–74, [hep-ph/9704448]. * [35] Les Houches 2009 Tools and Monte Carlo Working Group Collaboration, J. M. Butterworth et. al., The Tools and Monte Carlo working group: Summary Report, arXiv:1003.1643. * [36] A. Denner, S. Heinemeyer, I. Puljak, D. Rebuzzi, and M. Spira, Standard Model Higgs-Boson Branching Ratios with Uncertainties, Eur. Phys. J. C71 (2011) 1753, [arXiv:1107.5909]. * [37] Particle Data Group Collaboration, C. Amsler et. al., Review of particle physics, Phys. Lett. B667 (2008) 1. * [38] C. Anastasiou, R. Boughezal, and F. Petriello, Mixed QCD-electroweak corrections to Higgs boson production in gluon fusion, JHEP 04 (2009) 003, [arXiv:0811.3458]. * [39] W.-Y. Keung and F. J. Petriello, Electroweak and finite quark-mass effects on the Higgs boson transverse momentum distribution, Phys. Rev. D80 (2009) 013007, [arXiv:0905.2775]. * [40] S. Actis, G. Passarino, C. Sturm, and S. Uccirati, NLO Electroweak Corrections to Higgs Boson Production at Hadron Colliders, Phys. Lett. B670 (2008) 12–17, [arXiv:0809.1301]. * [41] S. Actis, G. Passarino, C. Sturm, and S. Uccirati, NNLO Computational Techniques: The Cases $H\to\gamma\gamma$ and $H\to gg$, Nucl. Phys. B811 (2009) 182–273, [arXiv:0809.3667]. * [42] D. de Florian and M. Grazzini, Higgs production through gluon fusion: updated cross sections at the Tevatron and the LHC, Phys. Lett. B674 (2009) 291–294, [arXiv:0901.2427]. * [43] S. Catani, D. de Florian, M. Grazzini, and P. Nason, Soft gluon resummation for Higgs boson production at hadron colliders, JHEP 0307 (2003) 028, [hep-ph/0306211]. * [44] S. Moch and A. Vogt, Higher-order soft corrections to lepton pair and Higgs boson production, Phys. Lett. B631 (2005) 48–57, [hep-ph/0508265]. * [45] A. D. Martin, W. J. Stirling, R. S. Thorne, and G. Watt, Parton distributions for the LHC, Eur. Phys. J. C63 (2009) 189–285, [arXiv:0901.0002]. * [46] For CDF and DØ, the TEVNPH Working Group Collaboration, Combined CDF and DØ upper limits on Standard Model Higgs boson production with up to 6.7 fb-1 of data, arXiv:1007.4587. FERMILAB-CONF-10-257-E, CDF Note 10241 and DØ Note 6096. * [47] J. Baglio and A. Djouadi, Predictions for Higgs production at the Tevatron and the associated uncertainties, JHEP 1010 (2010) 064, [arXiv:1003.4266]. [Addendum arXiv:1009.1363]. * [48] J. Baglio, A. Djouadi, S. Ferrag, and R. Godbole, The Tevatron Higgs exclusion limits and theoretical uncertainties: A Critical appraisal, Phys. Lett. B699 (2011) 368–371, [arXiv:1101.1832]. * [49] J. Baglio, A. Djouadi, and R. Godbole, Clarifications on the impact of theoretical uncertainties on the Tevatron Higgs exclusion limits, arXiv:1107.0281. * [50] V. Ahrens, T. Becher, M. Neubert, and L. L. Yang, Origin of the Large Perturbative Corrections to Higgs Production at Hadron Colliders, Phys. Rev. D79 (2009) 033013, [arXiv:0808.3008]. * [51] V. Ahrens, T. Becher, M. Neubert, and L. L. Yang, Renormalization-Group Improved Prediction for Higgs Production at Hadron Colliders, Eur. Phys. J. C62 (2009) 333–353, [arXiv:0809.4283]. * [52] V. Ahrens, T. Becher, M. Neubert, and L. L. Yang, Updated Predictions for Higgs Production at the Tevatron and the LHC, Phys. Lett. B698 (2011) 271–274, [arXiv:1008.3162]. * [53] C. Anastasiou, K. Melnikov, and F. Petriello, Higgs boson production at hadron colliders: Differential cross sections through next-to-next-to-leading order, Phys. Rev. Lett. 93 (2004) 262002, [hep-ph/0409088]. * [54] C. Anastasiou, K. Melnikov, and F. Petriello, Fully differential Higgs boson production and the di-photon signal through next-to-next-to-leading order, Nucl. Phys. B724 (2005) 197–246, [hep-ph/0501130]. * [55] F. J. Petriello , private communication, July, 2010. * [56] P. M. Nadolsky et. al., Implications of CTEQ global analysis for collider observables, Phys. Rev. D78 (2008) 013004, [arXiv:0802.0007]. * [57] L. J. Dixon and M. Siu, Resonance continuum interference in the diphoton Higgs signal at the LHC, Phys. Rev. Lett. 90 (2003) 252001, [hep-ph/0302233]. * [58] T. Binoth, M. Ciccolini, N. Kauer, and M. Kramer, Gluon-induced WW background to Higgs boson searches at the LHC, JHEP 0503 (2005) 065, [hep-ph/0503094]. * [59] T. Binoth, M. Ciccolini, N. Kauer, and M. Kramer, Gluon-induced W-boson pair production at the LHC, JHEP 0612 (2006) 046, [hep-ph/0611170]. * [60] J. M. Campbell, R. Ellis, and C. Williams, Gluon-gluon contributions to W+ W- production and Higgs interference effects, JHEP 1110 (2011) 005, [arXiv:1107.5569]. * [61] J. M. Campbell and R. K. Ellis, MCFM – Monte Carlo for FeMtobarn processes. http://mcfm.fnal.gov. * [62] J. M. Campbell and R. K. Ellis, Next-to-leading order corrections to $W$ \+ 2-jet and $Z$ \+ 2-jet production at hadron colliders, Phys. Rev. D65 (2002) 113007, [hep-ph/0202176]. * [63] J. M. Campbell, R. K. Ellis, and D. L. Rainwater, Next-to-leading order QCD predictions for $W$ \+ 2-jet and $Z$ \+ 2jet production at the CERN LHC, Phys. Rev. D68 (2003) 094021, [hep-ph/0308195]. * [64] S. Catani and M. Grazzini, An NNLO subtraction formalism in hadron collisions and its application to Higgs boson production at the LHC, Phys. Rev. Lett. 98 (2007) 222002, [hep-ph/0703012]. * [65] M. Grazzini, NNLO predictions for the Higgs boson signal in the $H\to WW\to\ell\nu\ell\nu$ and $H\to ZZ\to\ $4$\ell$ decay channels, JHEP 02 (2008) 043, [arXiv:0801.3232]. * [66] S. Catani, F. Krauss, R. Kuhn, and B. R. Webber, QCD Matrix Elements + Parton Showers, JHEP 11 (2001) 063, [hep-ph/0109231]. * [67] F. Krauss, Matrix elements and parton showers in hadronic interactions, JHEP 08 (2002) 015, [hep-ph/0205283]. * [68] F. Krauss, A. Schälicke, S. Schumann, and G. Soff, Simulating $W/Z$ \+ jets production at the Tevatron, Phys. Rev. D70 (2004) 114009, [hep-ph/0409106]. * [69] F. Krauss, A. Schälicke, S. Schumann, and G. Soff, Simulating $W/Z$ \+ jets production at the CERN LHC, Phys. Rev. D72 (2005) 054017, [hep-ph/0503280]. * [70] J. Alwall et. al., Comparative study of various algorithms for the merging of parton showers and matrix elements in hadronic collisions, Eur. Phys. J. C53 (2008) 473–500, [arXiv:0706.2569]. * [71] J. Winter, QCD jet evolution at high and low scales. PhD thesis, Technische Universität Dresden, 2008. * [72] P. Lenzi and J. M. Butterworth, A study on Matrix Element corrections in inclusive $Z/\gamma^{\ast}$ production at LHC as implemented in PYTHIA, HERWIG, ALPGEN and SHERPA, arXiv:0903.3918. * [73] Les Houches 2009 SM and NLO Multileg Working Group Collaboration, J. R. Andersen et. al., The SM and NLO multileg working group: Summary report, arXiv:1003.1241. * [74] DØ Collaboration, V. M. Abazov et. al., $Z$+jet production in the DØ experiment: A comparison between data and the PYTHIA and SHERPA Monte Carlos, DØ Note 5066-CONF (March, 2006). * [75] DØ Collaboration, V. M. Abazov et. al., Measurement of differential $Z/\gamma^{}$ \+ jet + X cross sections in $p\bar{p}$ collisions at $\sqrt{s}$ = 1.96 TeV, Phys. Lett. B669 (2008) 278–286, [arXiv:0808.1296]. [76] S. Höche, F. Krauss, S. Schumann, and F. Siegert, QCD matrix elements and truncated showers, JHEP 05 (2009) 053, [arXiv:0903.1219]. * [77] DØ Collaboration, V. M. Abazov et. al., Measurements of differential cross sections of $Z/\gamma^{\ast}$ \+ jets + X events in proton anti-proton collisions at $\sqrt{s}$ = 1.96 TeV, Phys. Lett. B678 (2009) 45–54, [arXiv:0903.1748]. * [78] DØ Collaboration, V. M. Abazov et. al., Measurement of $Z/\gamma^{\ast}$ \+ jet + X angular distributions in $p\bar{p}$ collisions at $\sqrt{s}$ = 1.96 TeV, Phys. Lett. B682 (2010) 370–380, [arXiv:0907.4286]. * [79] DØ Collaboration, V. M. Abazov et. al., Measurement of the normalized $Z/\gamma^{\ast}\to\mu^{+}\mu^{-}$ transverse momentum distribution in $p\bar{p}$ collisions at $\sqrt{s}$ = 1.96 TeV, arXiv:1006.0618. * [80] C. F. Berger et. al., Next-to-Leading Order QCD Predictions for $W$ \+ 3-Jet Distributions at Hadron Colliders, Phys. Rev. D80 (2009) 074036, [arXiv:0907.1984]. * [81] C. F. Berger et. al., Next-to-Leading Order QCD Predictions for $Z,\gamma^{}$ \+ 3-Jet Distributions at the Tevatron, arXiv:1004.1659. [82] T. Gleisberg, F. Krauss, A. Schälicke, S. Schumann, and J. Winter, Studying $W^{+}W^{-}$ production at the Fermilab Tevatron with SHERPA, Phys. Rev. D72 (2005) 034028, [hep-ph/0504032]. * [83] S. Höche, F. Krauss, M. Schönherr, and F. Siegert, Automating the POWHEG method in Sherpa, JHEP 1104 (2011) 024, [arXiv:1008.5399]. * [84] S. Höche, F. Krauss, M. Schönherr, and F. Siegert, NLO matrix elements and truncated showers, JHEP 1108 (2011) 123, [arXiv:1009.1127]. * [85] CDF Collaboration, T. Aaltonen et. al., Measurement of the $WW$ \+ $WZ$ Production Cross Section Using the Lepton + Jets Final State at CDF II, Phys. Rev. Lett. 104 (2010) 101801, [arXiv:0911.4449]. * [86] DØ Collaboration, V. Abazov et. al., Measurement of trilinear gauge boson couplings from $WW+WZ\to\ell\nu jj$ events in p anti-p collisions at $\sqrt{s}$ = 1.96 TeV, Phys. Rev. D80 (2009) 053012, [arXiv:0907.4398]. * [87] J. M. Campbell and R. K. Ellis, An update on vector boson pair production at hadron colliders, Phys. Rev. D60 (1999) 113006, [hep-ph/9905386]. * [88] J. M. Campbell, A. Martin, and C. Williams, NLO predictions for a lepton, missing transverse momentum and dijets at the Tevatron, Phys. Rev. D84 (2011) 036005, [arXiv:1105.4594]. * [89] M. Cacciari, S. Frixione, M. L. Mangano, P. Nason, and G. Ridolfi, Updated predictions for the total production cross sections of top and of heavier quark pairs at the Tevatron and at the LHC, JHEP 0809 (2008) 127, [arXiv:0804.2800]. * [90] G. C. Blazey et. al., Run II jet physics, hep-ex/0005012. * [91] For the ATLAS Collaboration, M. S. Neubauer, Search for the Standard Model Higgs Boson in the Lepton + Missing Transverse Energy + Jets Final State in ATLAS, arXiv:1110.2265. * [92] CMS Collaboration, G. Bayatian et. al., CMS technical design report, volume II: Physics performance, J. Phys. G G34 (2007) 995–1579. * [93] T. Becher and L. L. Yang , private communication, March, 2010. * [94] S. Höche, S. Schumann, and F. Siegert, Hard photon production and matrix-element parton-shower merging, Phys. Rev. D81 (2010) 034026, [arXiv:0912.3501]. * [95] T. Carli, T. Gehrmann, and S. Höche, Hadronic final states in deep-inelastic scattering with Sherpa, Eur. Phys. J. C67 (2010) 73–97, [arXiv:0912.3715]. * [96] S. Höche, F. Krauss, M. Schönherr, and F. Siegert, A critical appraisal of NLO+PS matching methods, arXiv:1111.1220. * [97] J. M. Campbell and R. Ellis, MCFM for the Tevatron and the LHC, Nucl. Phys. Proc. Suppl. 205-206 (2010) 10–15, [arXiv:1007.3492]. * [98] C. Berger, Z. Bern, L. Dixon, F. Febres Cordero, D. Forde, et. al., An Automated Implementation of On-Shell Methods for One-Loop Amplitudes, Phys. Rev. D78 (2008) 036003, [arXiv:0803.4180]. * [99] C. Berger, Z. Bern, L. J. Dixon, F. Febres Cordero, D. Forde, et. al., Vector Boson + Jets with BlackHat and Sherpa, Nucl. Phys. Proc. Suppl. 205-206 (2010) 92–97, [arXiv:1005.3728]. * [100] T. Binoth, F. Boudjema, G. Dissertori, A. Lazopoulos, A. Denner, et. al., A Proposal for a standard interface between Monte Carlo tools and one-loop programs, Comput. Phys. Commun. 181 (2010) 1612–1622, [arXiv:1001.1307]. Dedicated to the memory of, and in tribute to, Thomas Binoth, who led the effort to develop this proposal for Les Houches 2009. * [101] CDF Collaboration, T. Aaltonen et. al., Measurement of the cross section for $W$ boson production in association with jets in $p\bar{p}$ collisions at $\sqrt{s}$ = 1.96 TeV, Phys. Rev. D77 (2008) 011108, [arXiv:0711.4044]. * [102] A. Schälicke and F. Krauss, Implementing the ME+PS merging algorithm, JHEP 07 (2005) 018, [hep-ph/0503281].	arxiv-papers	2011-11-11T23:00:41	2024-09-04T02:49:24.221206	{ "license": "Public Domain", "authors": "Joseph D. Lykken, Adam O. Martin, Jan-Christopher Winter", "submitter": "Jan Winter", "url": "https://arxiv.org/abs/1111.2881" }
1111.2925	# Low Mach number limit for the full compressible magnetohydrodynamic equations with general initial data Song Jiang LCP, Institute of Applied Physics and Computational Mathematics, P.O. Box 8009, Beijing 100088, P.R. China jiang@iapcm.ac.cn , Qiangchang Ju Institute of Applied Physics and Computational Mathematics, P.O. Box 8009-28, Beijing 100088, P.R. China qiangchang_ju@yahoo.com , Fucai Li∗ Department of Mathematics, Nanjing University, Nanjing 210093, P.R. China fli@nju.edu.cn and Zhouping Xin The Institute of Mathematical Sciences, The Chinese University of Hong Kong, Shatin, NT, Hong Kong zpxin@ims.cuhk.edu.hk ###### Abstract. The low Mach number limit for the full compressible magnetohydrodynamic equations with general initial data is rigorously justified in the whole space $\mathbb{R}^{3}$. The uniform in Mach number estimates of the solutions in a Sobolev space are obtained on a time interval independent of the Mach number. The limit is proved by using the established uniform estimates and a theorem due to Métiver and Schochet [Arch. Ration. Mech. Anal. 158 (2001), 61-90] for the Euler equations that gives the local energy decay of the acoustic wave equations. ###### Key words and phrases: Full compressible magnetohydrodynamic equations, local smooth solution, low Mach number limit, general initial data ###### 2000 Mathematics Subject Classification: 76W05, 35B40 ∗Corresponding author ## 1\. Introduction In this paper we study the low Mach number limit of local smooth solutions to the following full compressible magnetohydrodynamic (MHD) equations with general initial data in the whole space $\mathbb{R}^{3}$ (see [20, 27, 37, 43]): $\displaystyle\partial_{t}\rho+{\rm div}(\rho{\mathbf{u}})=0,$ (1.1) $\displaystyle\partial_{t}(\rho{\mathbf{u}})+{\rm div}\left(\rho{\mathbf{u}}\otimes{\mathbf{u}}\right)+{\nabla P}=\frac{1}{4\pi}({\rm curl\,}\mathbf{H})\times\mathbf{H}+{\rm div}\Psi({\mathbf{u}}),$ (1.2) $\displaystyle\partial_{t}\mathbf{H}-{\rm curl\,}({\mathbf{u}}\times\mathbf{H})=-{\rm curl\,}(\nu\,{\rm curl\,}\mathbf{H}),\quad{\rm div}\mathbf{H}=0,$ (1.3) $\displaystyle\partial_{t}{\mathcal{E}}+{\rm div}\left({\mathbf{u}}({\mathcal{E}}^{\prime}+P)\right)=\frac{1}{4\pi}{\rm div}(({\mathbf{u}}\times\mathbf{H})\times\mathbf{H})$ $\displaystyle\qquad\qquad\qquad\qquad\qquad\,\,+{\rm div}\Big{(}\frac{\nu}{4\pi}\mathbf{H}\times({\rm curl\,}\mathbf{H})+{\mathbf{u}}\Psi({\mathbf{u}})+\kappa\nabla\theta\Big{)}.$ (1.4) Here the unknowns $\rho$, ${\mathbf{u}}=(u_{1},u_{2},u_{3})\in{\mathbb{R}}^{3}$ , $\mathbf{H}=(H_{1},H_{2},H_{3})\in{\mathbb{R}}^{3}$, and $\theta$ denote the density, velocity, magnetic field, and temperature, respectively; $\Psi({\mathbf{u}})$ is the viscous stress tensor given by $\Psi({\mathbf{u}})=2\mu\mathbb{D}({\mathbf{u}})+\lambda{\rm div}{\mathbf{u}}\;\mathbf{I}_{3}$ with $\mathbb{D}({\mathbf{u}})=(\nabla{\mathbf{u}}+\nabla{\mathbf{u}}^{\top})/2$, $\mathbf{I}_{3}$ the $3\times 3$ identity matrix, and $\nabla{\mathbf{u}}^{\top}$ the transpose of the matrix $\nabla{\mathbf{u}}$; ${\mathcal{E}}$ is the total energy given by ${\mathcal{E}}={\mathcal{E}}^{\prime}+\|\mathbf{H}\|^{2}/({8\pi})$ and ${\mathcal{E}}^{\prime}=\rho\left(e+\|{\mathbf{u}}\|^{2}/2\right)$ with $e$ being the internal energy, $\rho\|{\mathbf{u}}\|^{2}/2$ the kinetic energy, and $\|\mathbf{H}\|^{2}/({8\pi})$ the magnetic energy. The viscosity coefficients $\lambda$ and $\mu$ of the flow satisfy $\mu>0$ and $2\mu+3\lambda>0$. The parameter $\nu>0$ is the magnetic diffusion coefficient of the magnetic field and $\kappa>0$ the heat conductivity. For simplicity, we assume that $\mu,\lambda,\nu$ and $\kappa$ are constants. The equations of state $P=P(\rho,\theta)$ and $e=e(\rho,\theta)$ relate the pressure $P$ and the internal energy $e$ to the density $\rho$ and the temperature $\theta$ of the flow. Multiplying (1.2) by ${\mathbf{u}}$ and (1.3) by $\mathbf{H}/({4\pi})$ and summing over, one finds that $\displaystyle\frac{d}{dt}\Big{(}\frac{1}{2}\rho\|{\mathbf{u}}\|^{2}+\frac{1}{8\pi}\|\mathbf{H}\|^{2}\Big{)}+\frac{1}{2}{\rm div}\left(\rho\|{\mathbf{u}}\|^{2}{\mathbf{u}}\right)+\nabla P\cdot{\mathbf{u}}$ $\displaystyle\quad={\rm div}\Psi\cdot{\mathbf{u}}+\frac{1}{4\pi}({\rm curl\,}\mathbf{H})\times\mathbf{H}\cdot{\mathbf{u}}+\frac{1}{4\pi}{\rm curl\,}({\mathbf{u}}\times\mathbf{H})\cdot\mathbf{H}$ $\displaystyle\,\,\qquad\qquad-\frac{\nu}{4\pi}{\rm curl\,}({\rm curl\,}\mathbf{H})\cdot\mathbf{H}.$ (1.5) Due to the identities $\displaystyle{\rm div}(\mathbf{H}\times({\rm curl\,}\mathbf{H}))=\|{\rm curl\,}\mathbf{H}\|^{2}-{\rm curl\,}({\rm curl\,}\mathbf{H})\cdot\mathbf{H},$ $\displaystyle{\rm div}(({\mathbf{u}}\times\mathbf{H})\times\mathbf{H})=({\rm curl\,}\mathbf{H})\times\mathbf{H}\cdot{\mathbf{u}}+{\rm curl\,}({\mathbf{u}}\times\mathbf{H})\cdot\mathbf{H},$ (1.6) one can subtract (1) from (1.4) to rewrite the energy equation (1.4) in terms of the internal energy as $\partial_{t}(\rho e)+{\rm div}(\rho{\mathbf{u}}e)+({\rm div}{\mathbf{u}})P=\frac{\nu}{4\pi}\|{\rm curl\,}\mathbf{H}\|^{2}+\Psi({\mathbf{u}}):\nabla{\mathbf{u}}+\kappa\Delta\theta,$ (1.7) where $\Psi({\mathbf{u}}):\nabla{\mathbf{u}}$ denotes the scalar product of two matrices: $\Psi({\mathbf{u}}):\nabla{\mathbf{u}}=\sum^{3}_{i,j=1}\frac{\mu}{2}\left(\frac{\partial u^{i}}{\partial x_{j}}+\frac{\partial u^{j}}{\partial x_{i}}\right)^{2}+\lambda\|{\rm div}{\mathbf{u}}\|^{2}=2\mu\|\mathbb{D}({\mathbf{u}})\|^{2}+\lambda\|\mbox{tr}\mathbb{D}({\mathbf{u}})\|^{2}.$ To establish the low Mach number limit for the system (1.1)–(1.3) and (1.7), in this paper we shall focus on the ionized fluids obeying the following perfect gas relations $\displaystyle P=\mathfrak{R}\rho\theta,\quad e=c_{V}\theta,$ (1.8) where the parameters $\mathfrak{R}>0$ and $c_{V}\\!>\\!0$ are the gas constant and the heat capacity at constant volume, respectively, which will be assumed to be one for simplicity of the presentation. We also ignore the coefficient $1/(4\pi)$ in the magnetic field. Let $\epsilon$ be the Mach number, which is a dimensionless number. Consider the system (1.1)–(1.3), (1.7) in the physical regime: $\displaystyle P\sim P_{0}+O(\epsilon),\quad{\mathbf{u}}\sim O(\epsilon),\quad\mathbf{H}\sim O(\epsilon),\quad\nabla\theta\sim O(1),$ where $P_{0}>0$ is a certain given constant which is normalized to be $P_{0}=1$. Thus we consider the case when the pressure $P$ is a small perturbation of the given state $1$, while the temperature $\theta$ has a finite variation. As in [2], we introduce the following transformation to ensure positivity of $P$ and $\theta$ $\displaystyle\ P(x,t)=e^{\epsilon p^{\epsilon}(x,\epsilon t)},\quad\theta(x,t)=e^{\theta^{\epsilon}(x,\epsilon t)},$ (1.9) where a longer time scale $t=\tau/\epsilon$ (still denote $\tau$ by $t$ later for simplicity) is introduced in order to seize the evolution of the fluctuations. Note that (1.8) and (1.9) imply that $\rho(x,t)=e^{\epsilon p^{\epsilon}(x,\epsilon t)-\theta^{\epsilon}(x,\epsilon t)}$ since $\mathfrak{R}\equiv c_{V}\equiv 1$. Set $\displaystyle{\mathbf{H}}(x,t)=\epsilon\mathbf{H}^{\epsilon}(x,\epsilon t),\quad{{\mathbf{u}}}(x,t)=\epsilon{\mathbf{u}}^{\epsilon}(x,\epsilon t),$ (1.10) and $\displaystyle\mu=\epsilon\mu^{\epsilon},\quad\lambda=\epsilon\lambda^{\epsilon},\quad\nu=\epsilon\nu^{\epsilon},\quad\kappa=\epsilon\kappa^{\epsilon}.$ Under these changes of variables and coefficients, the system, (1.1)–(1.3), (1.7) with (1.8), takes the following equivalent form: $\displaystyle\partial_{t}p^{\epsilon}+({\mathbf{u}}^{\epsilon}\cdot\nabla)p^{\epsilon}+\frac{1}{\epsilon}{\rm div}(2{\mathbf{u}}^{\epsilon}-\kappa^{\epsilon}e^{-\epsilon p^{\epsilon}+\theta^{\epsilon}}\nabla\theta^{\epsilon})$ $\displaystyle\quad\quad\qquad=\epsilon e^{-\epsilon p^{\epsilon}}[\nu^{\epsilon}\|{\rm curl\,}\mathbf{H}^{\epsilon}\|^{2}+\Psi({\mathbf{u}}^{\epsilon}):\nabla{\mathbf{u}}^{\epsilon}]+\kappa^{\epsilon}e^{-\epsilon p^{\epsilon}+\theta^{\epsilon}}\nabla p^{\epsilon}\cdot\nabla\theta^{\epsilon},$ (1.11) $\displaystyle e^{-\theta^{\epsilon}}[\partial_{t}{\mathbf{u}}^{\epsilon}+({\mathbf{u}}^{\epsilon}\cdot\nabla){\mathbf{u}}^{\epsilon}]+\frac{\nabla p^{\epsilon}}{\epsilon}=e^{-\epsilon p^{\epsilon}}[({\rm curl\,}\mathbf{H}^{\epsilon})\times\mathbf{H}^{\epsilon}+{\rm div}\Psi^{\epsilon}({\mathbf{u}}^{\epsilon})],$ (1.12) $\displaystyle\partial_{t}\mathbf{H}^{\epsilon}-{\rm curl\,}({\mathbf{u}}^{\epsilon}\times\mathbf{H}^{\epsilon})-\nu^{\epsilon}\Delta\mathbf{H}^{\epsilon}=0,\quad{\rm div}\mathbf{H}^{\epsilon}=0,$ (1.13) $\displaystyle\partial_{t}\theta^{\epsilon}+({\mathbf{u}}^{\epsilon}\cdot\nabla)\theta^{\epsilon}+{\rm div}{\mathbf{u}}^{\epsilon}$ $\displaystyle\quad\quad\qquad=\epsilon^{2}e^{-\epsilon p^{\epsilon}}[\nu^{\epsilon}\|{\rm curl\,}\mathbf{H}^{\epsilon}\|^{2}+\Psi^{\epsilon}({\mathbf{u}}^{\epsilon}):\nabla{\mathbf{u}}^{\epsilon}]+\kappa^{\epsilon}e^{-\epsilon p^{\epsilon}}{\rm div}(e^{\theta^{\epsilon}}\nabla\theta^{\epsilon}),$ (1.14) where $\Psi^{\epsilon}({\mathbf{u}}^{\epsilon})=2\mu^{\epsilon}\mathbb{D}({\mathbf{u}}^{\epsilon})+\lambda^{\epsilon}{\rm div}{\mathbf{u}}^{\epsilon}\,\mathbf{I}_{3}$, and the identity ${\rm curl\,}({\rm curl\,}\mathbf{H}^{\epsilon})=\nabla{\rm div}\mathbf{H}^{\epsilon}-\Delta\mathbf{H}^{\epsilon}$ and the constraint that ${\rm div}\mathbf{H}^{\epsilon}=0$ have been used. We shall study the limit as $\epsilon\to 0$ of solutions to the system (1.11)–(1.14). Formally, as $\epsilon$ goes to zero, if the sequence $(p^{\epsilon},{\mathbf{u}}^{\epsilon},\mathbf{H}^{\epsilon},\theta^{\epsilon})$ converges strongly to a limit $(1,{\mathbf{w}},{\mathbf{B}},\vartheta)$ in some sense, and $(\mu^{\epsilon},\lambda^{\epsilon},\nu^{\epsilon},\kappa^{\epsilon})$ converges to a constant vector $(\bar{\mu},\bar{\lambda},\bar{\nu},\bar{\kappa})$, then taking the limit to (1.11)–(1.14), we have $\displaystyle{\rm div}(2{\mathbf{w}}-\bar{\kappa}\,e^{\vartheta}\nabla\vartheta)=0,$ (1.15) $\displaystyle e^{-\vartheta}[\partial_{t}{\mathbf{w}}+({\mathbf{w}}\cdot\nabla){\mathbf{w}}]+\nabla\pi=({\rm curl\,}{\mathbf{B}})\times{\mathbf{B}}+{\rm div}\Phi({\mathbf{w}}),$ (1.16) $\displaystyle\partial_{t}{\mathbf{B}}-{\rm curl\,}({\mathbf{w}}\times{\mathbf{B}})-\bar{\nu}\Delta{\mathbf{B}}=0,\quad{\rm div}{\mathbf{B}}=0,$ (1.17) $\displaystyle\partial_{t}\vartheta+({\mathbf{w}}\cdot\nabla)\vartheta+{\rm div}{\mathbf{w}}=\bar{\kappa}\,{\rm div}(e^{\vartheta}\nabla\vartheta),$ (1.18) with some function $\pi$, where $\Phi({\mathbf{w}})$ is defined by $\Phi({\mathbf{w}})=2\bar{\mu}\mathbb{D}({\mathbf{w}})+\bar{\lambda}{\rm div}{\mathbf{w}}\,\mathbf{I}_{3}.$ The purpose of this paper is to establish the above limit process rigorously. For this purpose, we supplement the system (1.11)–(1.14) with the following initial conditions $\displaystyle(p^{\epsilon},{\mathbf{u}}^{\epsilon},\mathbf{H}^{\epsilon},\theta^{\epsilon})\|_{t=0}=(p^{\epsilon}_{\rm in}(x),{\mathbf{u}}^{\epsilon}_{\rm in}(x),\mathbf{H}^{\epsilon}_{\rm in}(x),\theta^{\epsilon}_{\rm in}(x)),\quad x\in\mathbb{R}^{3}.$ (1.19) For simplicity of presentation, we shall assume that $\mu^{\epsilon}\equiv\bar{\mu}>0$, $\nu^{\epsilon}\equiv\bar{\nu}>0$, $\kappa^{\epsilon}\equiv\bar{\kappa}>0$, and $\lambda^{\epsilon}\equiv\bar{\lambda}$. The general case $\mu^{\epsilon}\rightarrow\bar{\mu}>0$, $\nu^{\epsilon}\rightarrow\bar{\nu}>0$, $\kappa^{\epsilon}\rightarrow\bar{\kappa}>0$ and $\lambda^{\epsilon}\rightarrow\bar{\lambda}$ simultaneously as $\epsilon\rightarrow 0$ can be treated by slightly modifying the arguments presented here. As in [2], we will use the notation $\\|v\\|_{H_{\eta}^{\sigma}}:=\\|v\\|_{H^{\sigma-1}}+\eta\\|v\\|_{H^{\sigma}}$ for any $\sigma\in\mathbb{R}$ and $\eta\geq 0$. For each $\epsilon>0$, $t\geq 0$ and $s\geq 0$, we will also use the following norm: $\displaystyle\\|(p^{\epsilon},{\mathbf{u}}^{\epsilon},\mathbf{H}^{\epsilon},\theta^{\epsilon}-\bar{\theta})(t)\\|_{s,\epsilon}$ $\displaystyle\qquad:=\sup_{\tau\in[0,t]}\big{\\{}\\|(p^{\epsilon},{\mathbf{u}}^{\epsilon},\mathbf{H}^{\epsilon})(\tau)\\|_{H^{s}}+\\|(\epsilon p^{\epsilon},\epsilon{\mathbf{u}}^{\epsilon},\epsilon\mathbf{H}^{\epsilon},\theta^{\epsilon}-\bar{\theta})(\tau)\\|_{H_{\epsilon}^{s+2}}\big{\\}}$ $\displaystyle\qquad\ \ \quad+\Big{\\{}\int^{t}_{0}[\\|\nabla(p^{\epsilon},{\mathbf{u}}^{\epsilon},\mathbf{H}^{\epsilon})\\|^{2}_{H^{s}}+\\|\nabla(\epsilon{\mathbf{u}}^{\epsilon},\epsilon H^{\epsilon},\theta^{\epsilon})\\|^{2}_{H_{\epsilon}^{s+2}}](\tau)d\tau\Big{\\}}^{1/2}.$ Then, the main result of this paper reads as follows. ###### Theorem 1.1. Let $s\geq 4$. Assume that the initial data $(p^{\epsilon}_{\rm in},{\mathbf{u}}^{\epsilon}_{\rm in},\mathbf{H}^{\epsilon}_{\rm in},\theta^{\epsilon}_{\rm in})$ satisfy $\displaystyle\\|(p^{\epsilon}_{\rm in},{\mathbf{u}}^{\epsilon}_{\rm in},\mathbf{H}^{\epsilon}_{\rm in})\\|_{H^{s}}+\\|(\epsilon p^{\epsilon}_{\rm in},\epsilon{\mathbf{u}}^{\epsilon}_{\rm in},\epsilon\mathbf{H}^{\epsilon}_{\rm in},\theta^{\epsilon}_{\rm in}-\bar{\theta})\\|_{H_{\epsilon}^{s+2}}\leq L_{0}$ (1.20) for all $\epsilon\in(0,1]$ and two given positive constants $\bar{\theta}$ and $L_{0}$. Then there exist positive constants $T_{0}$ and $\epsilon_{0}<1$, depending only on $L_{0}$ and $\bar{\theta}$, such that the Cauchy problem (1.11)–(1.14), (1.19) has a unique solution $(p^{\epsilon},{\mathbf{u}}^{\epsilon},\mathbf{H}^{\epsilon},\theta^{\epsilon})$ satisfying $\displaystyle\\|(p^{\epsilon},{\mathbf{u}}^{\epsilon},\mathbf{H}^{\epsilon},\theta^{\epsilon}-\bar{\theta})(t)\\|_{s,\epsilon}\leq L,\qquad\forall\,\,t\in[0,T_{0}],\ \forall\,\epsilon\in(0,\epsilon_{0}],$ (1.21) where $L$ depends only on $L_{0}$, $\bar{\theta}$ and $T_{0}$. Moreover, assume further that the initial data satisfy the following conditions $\displaystyle\|\theta^{\epsilon}_{0}(x)-\bar{\theta}\|\leq{N}_{0}\|x\|^{-1-\zeta},\quad\|\nabla\theta^{\epsilon}_{0}(x)\|\leq N_{0}\|x\|^{-2-\zeta},\quad\forall\,\epsilon\in(0,1],$ (1.22) $\displaystyle\\!\\!\\!\\!\big{(}p^{\epsilon}_{\rm in},{\rm curl\,}(e^{-\theta^{\epsilon}_{\rm in}}{\mathbf{u}}^{\epsilon}_{\rm in}),\mathbf{H}^{\epsilon}_{\rm in},\theta^{\epsilon}_{\rm in}-\bar{\theta}\big{)}\rightarrow(0,{\mathbf{w}}_{0},{\mathbf{B}}_{0},\vartheta_{0}-\bar{\theta})\;\mbox{ in }H^{s}(\mathbb{R}^{3})$ (1.23) as $\epsilon\rightarrow 0$, where $N_{0}$ and $\zeta$ are fixed positive constants. Then the solution sequence $(p^{\epsilon},{\mathbf{u}}^{\epsilon},\mathbf{H}^{\epsilon},$ $\theta^{\epsilon}-\bar{\theta})$ converges weakly in $L^{\infty}(0,T_{0};H^{s}({\mathbb{R}}^{3}))$ and strongly in $L^{2}(0,T_{0};$ $H^{s_{2}}_{\mathrm{loc}}({\mathbb{R}}^{3}))$ for all $0\leq s_{2}<s$ to the limit $(0,{\mathbf{w}},{\mathbf{B}},\vartheta-\bar{\theta})$, where $({\mathbf{w}},{\mathbf{B}},\vartheta)$ satisfies the system (1.15)–(1.18) with initial data $({\mathbf{w}},{\mathbf{B}},\vartheta)\|_{t=0}=({\mathbf{w}}_{0},{\mathbf{B}}_{0},\vartheta_{0})$. We now give some comments on the proof of Theorem 1.1. The key point in the proof is to establish the uniform estimates in Sobolev norms for the acoustic components of solutions, which are propagated by the wave equations whose coefficients are functions of the temperature. Our main strategy is to bound the norm of $(\nabla p^{\epsilon},{\rm div}{\mathbf{u}}^{\epsilon})$ in terms of the norm of $(\epsilon\partial_{t})(p^{\epsilon},{\mathbf{u}}^{\epsilon},\mathbf{H}^{\epsilon})$ and $(\epsilon p^{\epsilon},\epsilon{\mathbf{u}}^{\epsilon},\epsilon\mathbf{H}^{\epsilon},\theta^{\epsilon})$ through the density and the momentum equations. This approach is motivated by the previous works on the compressible Navier-Stokes equations due to Alazard in [2], and Levermore, Sun and Trivisa [38]. It should be pointed out that the analysis for (1.11)–(1.14) is complicated and difficult due to the strong coupling of the hydrodynamic motion and the magnetic fields. Moreover, it is observed that the terms $({\rm curl\,}{\mathbf{H}^{\epsilon}})\times{\mathbf{H}^{\epsilon}}$ in the momentum equations, ${\rm curl\,}({{\mathbf{u}}^{\epsilon}}\times{\mathbf{H}^{\epsilon}})$ in the magnetic field equation, and $\|\nabla\times{\mathbf{H}^{\epsilon}}\|^{2}$ in the temperature equation change basically the structure of the system. More efforts should be paid on the estimates involving these terms, in particular, on the estimate of higher order spatial derivatives. We shall exploit the special structure of the system to obtain the tamed estimate on higher order derivatives, so that we can close our estimates on the uniform boundedness of the solutions. Once the uniform boundedness of the solutions has been established, one can the convergence result in Theorem 1.1 by applying the compactness arguments and the dispersive estimates on the acoustic wave equations in the whole space developed in [40]. ###### Remark 1.1. The positivity of the coefficients $\mu$, $\nu$ and $\kappa$ plays an fundamental role in the proof of Theorem 1.1. The arguments given in this paper can not be applied to the case when one of them disappears. Recently, Jiang, Ju and Li [31] have studied the incompressible limit of the compressible non-isentropic ideal MHD equations with general initial data in the whole space $\mathbb{R}^{d}$ ($d=2,3$) when the initial initial data belong to $H^{s}(\mathbb{R}^{d})$ with $s$ being an even integer. We emphasize that the restriction on the Sobolev index $s$ to be even plays a crucial role in the proof since in this case the nonstandard highest order derivative operators applied to the momentum equations are not intertwined with the pressure equation, and thus we can apply the same operators to the magnetic field equations to close the estimates on ${\mathbf{u}}$ and $\mathbf{H}$. On the other hand, the proof presented in [31] fully exploits the structure of the ideal MHD equations and can not be directly extended to the full compressible MHD equations studied in the current paper, where the heat conductivity is positive. We point out that the low Mach number limit is an interesting topic in fluid dynamics and applied mathematics. Now we briefly review some related results on the Euler, Navier-Stokes and MHD equations. In [48], Schochet obtained the convergence of the non-isentropic compressible Euler equations to the incompressible non-isentropic Euler equations in a bounded domain for local smooth solutions and well-prepared initial data. As mentioned above, in [40] Métivier and Schochet proved rigorously the incompressible limit of the compressible non-isentropic Euler equations in the whole space with general initial data, see also [1, 2, 38] for further extensions. In [41] Métivier and Schochet showed the incompressible limit of the one-dimensional non-isentropic Euler equations in a periodic domain with general data. For compressible heat- conducting flows, Hagstrom and Lorenz established in [18] the low Mach number limit under the assumption that the variation of the density and temperature is small. In the case of without heat conductivity, Kim and Lee [33] investigated the incompressible limit to the non-isentropic Navier-Stokes equations in a periodic domain with well-prepared data, while Jiang and Ou [32] investigated the incompressible limit in three-dimensional bounded domains, also for well-prepared data. The justification of the low Mach number limit for the non-isentropic Euler or Navier-Stokes equations with general initial data in bounded domains or multi-dimensional periodic domains is still open. We refer the interested reader to [6] on formal computations for viscous polytropic gases, and to [41, 5] for the study on the acoustic waves of the non-isentropic Euler equations in periodic domains. Compared with the non- isentropic case, the description of the propagation of oscillations in the isentropic case is simpler and there are many articles on this topic (isentropic flows) in the literature, see, for example, Ukai [50], Asano [3], Desjardins and Grenier [11] in the whole space case; Isozaki [25, 26] in the case of exterior domains; Iguchi [24] in the half space case; Schochet [47] and Gallagher [16] in the case of periodic domains; and Lions and Masmoudi [44], and Desjardins, et al. [12] in the case of bounded domains. For the compressible isentropic MHD equations, the justification of the low Mach number limit has been established in several aspects. In [34] Klainerman and Majda studied the low Mach number limit to the compressible isentropic MHD equations in the spatially periodic case with well-prepared initial data. Recently, the low Mach number limit to the compressible isentropic viscous (including both viscosity and magnetic diffusivity) MHD equations with general data was studied in [23, 28, 29]. In [23] Hu and Wang obtained the convergence of weak solutions to the compressible viscous MHD equations in bounded domains, periodic domains and the whole space. In [28] Jiang, Ju and Li employed the modulated energy method to verify the limit of weak solutions of the compressible MHD equations in the torus to the strong solution of the incompressible viscous or partially viscous MHD equations (zero shear viscosity but with magnetic diffusion), while in [29] the convergence of weak solutions of the viscous compressible MHD equations to the strong solution of the ideal incompressible MHD equations in the whole space was established by using the dispersion property of the wave equation, as both shear viscosity and magnetic diffusion coefficients go to zero. For the full compressible MHD equations, the incompressible limit in the framework of the so-called variational solutions was established in [35, 36, 42]. Recently, the low Mach number limit for the full compressible MHD equations with small entropy or temperature variation was justified rigourously in [30]. Besides the references mentioned above, the interested reader can refer to the monograph [14] and the survey papers [9, 45, 49] for more related results on the low Mach number limit to fluid models. We also mention that there are a lot of articles in the literatures on the other topics related to the compressible MHD equations due to theirs physical importance, complexity, rich phenomena, and mathematical challenges, see, for example, [4, 7, 8, 20, 39, 10, 13, 15, 19, 43, 21, 22, 52] and the references cited therein. This paper is arranged as follows. In Section 2, we describe some notations, recall basic facts and present commutators estimates. In Section 3 we first establish a priori estimates on $(\mathbf{H}^{\epsilon},\theta^{\epsilon})$, $(\epsilon p^{\epsilon},\epsilon{\mathbf{u}}^{\epsilon},\epsilon\mathbf{H}^{\epsilon},\theta^{\epsilon})$ and on $(p^{\epsilon},{\mathbf{u}}^{\epsilon})$. Then, with the help of these estimates we establish the uniform boundeness of the solutions and prove the existence part of Theorem 1.1. Finally, in Section 4 we study the local energy decay for the acoustic wave equations and prove the convergence part of Theorem 1.1. ## 2\. Preliminary In this section, we give some notations and recall basic facts which will be frequently used throughout the paper. We also present some commutators estimates introduced in [38] and state the results on local solutions to the Cauchy problem (1.11)–(1.14), (1.19). We denote by $\langle\cdot,\cdot\rangle$ the standard inner product in $L^{2}({\mathbb{R}}^{3})$ with norm $\langle f,f\rangle=\\|f\\|^{2}_{L^{2}}$ and by $H^{k}$ the standard Sobolev space $W^{k,2}$ with norm $\\|\cdot\\|_{H^{k}}$. The notation $\\|(A_{1},\dots,A_{k})\\|_{L^{2}}$ means the summation of $\\|A_{i}\\|_{L^{2}},i=1,\cdots,k$, and it also applies to other norms. For a multi-index $\alpha=(\alpha_{1},\alpha_{2},\alpha_{3})$, we denote $\partial^{\alpha}=\partial^{\alpha_{1}}_{x_{1}}\partial^{\alpha_{2}}_{x_{2}}\partial^{\alpha_{3}}_{x_{3}}$ and $\|\alpha\|=\|\alpha_{1}\|+\|\alpha_{2}\|+\|\alpha_{3}\|$. We will omit the spatial domain ${\mathbb{R}}^{3}$ in integrals for convenience. We use $l_{i}>0$ ($i\in\mathbb{N}$) to denote given constants. We also use the symbol $K$ or $C_{0}$ to denote generic positive constants, and $C(\cdot)$ to denote a smooth function which may vary from line to line. For a scalar function $f$, vector functions $\mathbf{a}$ and $\mathbf{b}$, we have the following basic vector identities: $\displaystyle{\rm div}(f\mathbf{a})$ $\displaystyle=f{\rm div}\mathbf{a}+\nabla f\cdot\mathbf{a},$ (2.1) $\displaystyle{\rm curl\,}(f\mathbf{a})$ $\displaystyle=f\cdot{\rm curl\,}\mathbf{a}-\nabla f\times\mathbf{a},$ (2.2) $\displaystyle{\rm div}(\mathbf{a}\times\mathbf{b})$ $\displaystyle=\mathbf{b}\cdot{\rm curl\,}\mathbf{a}-\mathbf{a}\cdot{\rm curl\,}\mathbf{b},$ (2.3) $\displaystyle{\rm curl\,}(\mathbf{a}\times\mathbf{b})$ $\displaystyle=(\mathbf{b}\cdot\nabla)\mathbf{a}-(\mathbf{a}\cdot\nabla)\mathbf{b}+\mathbf{a}({\rm div}\mathbf{b})-\mathbf{b}({\rm div}\mathbf{a}),$ (2.4) $\displaystyle\nabla(\mathbf{a}\cdot\mathbf{b})$ $\displaystyle=(\mathbf{a}\cdot\nabla)\mathbf{b}+(\mathbf{b}\cdot\nabla)\mathbf{a}+\mathbf{a}\times({\rm curl\,}\mathbf{b})+\mathbf{b}\times({\rm curl\,}\mathbf{a}).$ (2.5) Below we recall some results on commutators estimates. ###### Lemma 2.1. Let $\alpha=(\alpha_{1},\alpha_{2},\alpha_{3})$ be a multi-index such that $\|\alpha\|=k$. Then, for any $\sigma\geq 0$, there exists a positive constant $C_{0}$, such that for all $f,g\in H^{k+\sigma}({\mathbb{R}}^{3})$, $\displaystyle\\|[f,\partial^{\alpha}]g\\|_{H^{\sigma}}\leq$ $\displaystyle C_{0}(\\|f\\|_{W^{1,\infty}}\\|g\\|_{H^{\sigma+k-1}}+\\|f\\|_{H^{\sigma+k}}\\|g\\|_{L^{\infty}}).$ (2.6) ###### Lemma 2.2. Let $s>5/2$. Then there exists a positive constant $C_{0}$, such that for all $\epsilon\in(0,1]$, $T>0$ and multi-index $\beta=(\beta_{1},\beta_{2},\beta_{3})$ satisfying $0\leq\|\beta\|\leq s-1$, and any $f,g\in C^{\infty}([0,T],H^{s}({\mathbb{R}}^{3}))$, it holds that $\displaystyle\\|[f,\partial^{\beta}(\epsilon\partial_{t})]g\\|_{L^{2}}\leq$ $\displaystyle\epsilon C_{0}(\\|f\\|_{H^{s-1}}\\|\partial_{t}g\\|_{H^{s-2}}+\\|\partial_{t}f\\|_{H^{s-1}}\\|g\\|_{H^{s-1}}).$ (2.7) Since the system (1.1)–(1.3), (1.7), (1.8) is hyperbolic-parabolic, so the classical result of Vol’pert and Hudiaev [51] implies that ###### Proposition 2.3. Let $s\geq 4$. Assume that the initial data $(\rho_{0},{\mathbf{u}}_{0},\mathbf{H}_{0},\theta_{0})$ satisfy $\displaystyle\\|(\rho_{0}-\underline{\rho},{\mathbf{u}}_{0},\mathbf{H}_{0},\theta_{0}-\underline{\theta})\\|_{H^{s}}\leq C_{0}$ for some positive constants $\underline{\rho}$, $\underline{\theta}$ and $C_{0}$. Then there exists a $\tilde{T}>0$, such that the system (1.1)–(1.3), (1.7), and (1.8) with these initial data has a unique classical solution $(\rho,{\mathbf{u}},\mathbf{H},\theta)$ enjoying $\rho-\underline{\rho}\in C([0,\tilde{T}],H^{s}({\mathbb{R}}^{3}))$, $({\mathbf{u}},\mathbf{H},\theta-\bar{\theta})\in C([0,\tilde{T}],\linebreak H^{s}({\mathbb{R}}^{3}))\cap L^{2}(0,\tilde{T};H^{s+1}({\mathbb{R}}^{3}))$, and $\displaystyle\sup_{0\leq t\leq\tilde{T}}\\|(\rho-\underline{\rho},{\mathbf{u}},\mathbf{H},\theta-\bar{\theta})\\|^{2}_{H^{s}}$ $\displaystyle+\int^{\tilde{T}}_{0}\Big{\\{}\mu\\|\mathbb{D}({\mathbf{u}})\\|^{2}_{H^{s}}+\lambda\\|{\rm div}{\mathbf{u}}\\|^{2}_{H^{s}}$ $\displaystyle\qquad\quad+\nu\\|\nabla\mathbf{H}\\|^{2}_{H^{s}}+\kappa\\|\nabla\theta\\|^{2}_{H^{s}}\Big{\\}}(\tau)d\tau\leq 4C_{0}^{2}.$ It follows from Proposition 2.3, and the transforms (1.9) and (1.10) that there exists a $T_{\epsilon}>0$, depending on $\epsilon$ and $L_{0}$, such that for each fixed $\epsilon$ and any initial data (1.19) satisfying (1.20), the Cauchy problem (1.11)–(1.14), (1.19) has a unique solution $(p^{\epsilon},{\mathbf{u}}^{\epsilon},\mathbf{H}^{\epsilon},\theta^{\epsilon}-\bar{\theta})$ satisfying $(p^{\epsilon},{\mathbf{u}}^{\epsilon},\mathbf{H}^{\epsilon},\theta^{\epsilon}-\bar{\theta})\in C([0,T_{\epsilon}),H^{s}({\mathbb{R}}^{3}))$ and $({\mathbf{u}}^{\epsilon},\mathbf{H}^{\epsilon},\theta^{\epsilon}-\bar{\theta})\in L^{2}(0,T_{\epsilon};H^{s+1}({\mathbb{R}}^{3}))$. Moreover, let $T_{\epsilon}^{}$ be the maximal time of existence of such a smooth solution, then if $T_{\epsilon}^{}$ is finite, one has $\displaystyle{\underset{t\rightarrow T_{\epsilon}^{}}{\lim\sup}}\,\left\\{\\|(p^{\epsilon},{\mathbf{u}}^{\epsilon},\mathbf{H}^{\epsilon},\theta^{\epsilon})(t)\\|_{W^{1,\infty}}+\\|({\mathbf{u}}^{\epsilon},\mathbf{H}^{\epsilon},\theta^{\epsilon})(t)\\|_{W^{2,\infty}}\right\\}=\infty.$ Therefore, we shall see by the same argument as in [40] that the existence part of Theorem 1.1 is a consequence of the above assertion and the following key a priori estimates which will be shown in the next section. ###### Proposition 2.4. For any given $s\geq 4$ and fixed $\epsilon>0$, let $(p^{\epsilon},{\mathbf{u}}^{\epsilon},\mathbf{H}^{\epsilon},\theta^{\epsilon})$ be the classical solution to the Cauchy problem (1.11)–(1.14) and (1.19). Denote $\displaystyle\mathcal{O}(T):=$ $\displaystyle\\|(p^{\epsilon},{\mathbf{u}}^{\epsilon},\mathbf{H}^{\epsilon},\theta^{\epsilon}-\bar{\theta})(T)\\|_{s,\epsilon},$ $\displaystyle\mathcal{O}_{0}:=$ $\displaystyle\\|(p^{\epsilon}_{\rm in},{\mathbf{u}}^{\epsilon}_{\rm in},\mathbf{H}^{\epsilon}_{\rm in})\\|_{H^{s}}+\\|(\epsilon p^{\epsilon}_{\rm in},\epsilon{\mathbf{u}}^{\epsilon}_{\rm in},\epsilon\mathbf{H}^{\epsilon}_{\rm in},\theta^{\epsilon}_{\rm in}-\bar{\theta})\\|_{H_{\epsilon}^{s+2}}.$ Then there exist positive constants $\hat{T}_{0}$ and $\epsilon_{0}<1$, and an increasing positive function $C(\cdot)$, such that for all $T\in[0,\hat{T}_{0}]$ and $\epsilon\in(0,\epsilon_{0}]$, $\displaystyle\mathcal{O}(T)\leq C(\mathcal{O}_{0})\exp\\{(\sqrt{T}+\epsilon)C(\mathcal{O}(T))\\}.$ ## 3\. Uniform estimates In this section we shall establish the uniform bounds of the solutions to the Cauchy problem (1.11)–(1.14) and (1.19) stated in Proposition 2.4 by modifying the approaches developed in [40, 2, 38] and making careful use of the special structure of the system (1.11)–(1.14). In the rest of this section, we will drop the superscripts $\epsilon$ of the variables in the Cauchy problem and denote $\Psi({\mathbf{u}})=2\bar{\mu}\mathbb{D}({\mathbf{u}})+\bar{\lambda}{\rm div}{\mathbf{u}}\;\mathbf{I}_{3}.$ Recall that it has been assumed that $\mu^{\epsilon}\equiv\bar{\mu}>0$, $\nu^{\epsilon}\equiv\bar{\nu}>0$, $\kappa^{\epsilon}\equiv\bar{\kappa}>0$, and $\lambda^{\epsilon}\equiv\bar{\lambda}$ independent of $\epsilon$. ### 3.1. $H^{s}$-estimates on $(\mathbf{H},\theta)$ and $(\epsilon p,\epsilon{\mathbf{u}})$ To prove Proposition 2.4, we first give some estimates derived directly from the system (1.11)–(1.14). Denoting $\displaystyle\mathcal{Q}:$ $\displaystyle=\\|(p,{\mathbf{u}},\mathbf{H},\theta-\bar{\theta})\\|_{H^{s}}+\\|(\epsilon p,\epsilon{\mathbf{u}},\epsilon\mathbf{H},\theta-\bar{\theta})\\|_{H_{\epsilon}^{s+2}},$ $\displaystyle\mathcal{S}:$ $\displaystyle=\\|(\nabla{\mathbf{u}},\nabla p,\nabla\mathbf{H})\\|_{H^{s}}+\\|\nabla(\epsilon{\mathbf{u}},\epsilon\mathbf{H},\theta)\\|_{H_{\epsilon}^{s+2}},$ one has ###### Lemma 3.1. Let $s\geq 4$ and $(p,{\mathbf{u}},\mathbf{H},\theta)$ be a solution to the problem (1.11)–(1.14), (1.19) on $[0,T_{1}]$. There exists an increasing function $C(\cdot)$ such that, for any $\epsilon\in(0,1]$ and $t\in[0,T]$, $T=\min\\{T_{1},1\\}$, it holds that $\displaystyle\sup_{\tau\in[0,t]}\\{\\|(\mathbf{H},\theta)(\tau)\\|_{H^{s}}+\\|\epsilon\mathbf{H}(\tau)\\|_{H_{\epsilon}^{s+1}}\\}$ $\displaystyle\quad+\Big{\\{}\int^{t}_{0}\big{(}\\|\nabla(\mathbf{H},\theta)(\tau)\\|_{H^{s}}^{2}+\\|\nabla(\epsilon\mathbf{H})(\tau)\\|^{2}_{H_{\epsilon}^{s+1}}\Big{)}d\tau\Big{\\}}^{1/2}$ $\displaystyle\qquad\leq C(\mathcal{O}_{0})\exp\\{\sqrt{T}C(\mathcal{O}(T))\\}.$ ###### Proof. For any multi-index $\alpha=(\alpha_{1},\alpha_{2},\alpha_{3})$ satisfying $\|\alpha\|\leq s$, let $\mathbf{H}_{\alpha}=\partial^{\alpha}\mathbf{H}$. Then $\displaystyle\partial_{t}\mathbf{H}_{\alpha}+({\mathbf{u}}\cdot\nabla)\mathbf{H}_{\alpha}-\bar{\nu}\Delta\mathbf{H}_{\alpha}=-[\partial^{\alpha},{\mathbf{u}}]\cdot\nabla\mathbf{H}-\partial^{\alpha}(\mathbf{H}{\rm div}{\mathbf{u}})+\partial^{\alpha}((\mathbf{H}\cdot\nabla){\mathbf{u}}).$ Taking inner product of the above equations with $\mathbf{H}_{\alpha}$ and integrating by parts, we have $\displaystyle\frac{1}{2}\frac{d}{dt}\\|\mathbf{H}_{\alpha}\\|_{L^{2}}^{2}+\bar{\nu}\\|\nabla\mathbf{H}_{\alpha}\\|_{L^{2}}^{2}=$ $\displaystyle-\langle({\mathbf{u}}\cdot\nabla)\mathbf{H}_{\alpha},\mathbf{H}_{\alpha}\rangle-\langle[\partial^{\alpha},{\mathbf{u}}]\cdot\nabla\mathbf{H},\mathbf{H}_{\alpha}\rangle$ $\displaystyle-\langle\partial^{\alpha}(\mathbf{H}{\rm div}{\mathbf{u}}),\mathbf{H}_{\alpha}\rangle+\langle\partial^{\alpha}((\mathbf{H}\cdot\nabla){\mathbf{u}}),\mathbf{H}_{\alpha}\rangle.$ By integration by parts we obtain $\displaystyle-\langle({\mathbf{u}}\cdot\nabla)\mathbf{H}_{\alpha},\mathbf{H}_{\alpha}\rangle=\frac{1}{2}\int{\rm div}{\mathbf{u}}\|\mathbf{H}_{\alpha}\|^{2}dx\leq C(\mathcal{Q})\\|\mathbf{H}_{\alpha}\\|^{2}_{L^{2}}.$ It follows from the commutator inequality (2.6) that $\\|[\partial^{\alpha},{\mathbf{u}}]\cdot\nabla\mathbf{H}\\|_{L^{2}}\leq C_{0}(\\|{\mathbf{u}}\\|_{W^{1,\infty}}\\|\nabla\mathbf{H}\\|_{H^{s-1}}+\\|{\mathbf{u}}\\|_{H^{s}}\\|\nabla\mathbf{H}\\|_{L^{\infty}})\leq C(\mathcal{Q}).$ By Sobolev’s inequality, one gets $-\langle\partial^{\alpha}(\mathbf{H}{\rm div}{\mathbf{u}}),\mathbf{H}_{\alpha}\rangle+\langle\partial^{\alpha}(\mathbf{H}\cdot\nabla{\mathbf{u}}),\mathbf{H}_{\alpha}\rangle\leq C_{0}\\|\mathbf{H}\\|_{H^{s}}^{2}\\|{\mathbf{u}}\\|_{H^{s+1}}\leq C(\mathcal{Q})\mathcal{S}.$ Thus, we conclude that $\displaystyle\ \ \ \sup_{\tau\in[0,t]}\\|\mathbf{H}(\tau)\\|_{H^{s}}+\bar{\nu}\Big{\\{}\int^{t}_{0}\\|\nabla\mathbf{H}\\|^{2}_{H^{s}}d\tau\Big{\\}}^{1/2}$ $\displaystyle\leq C(\mathcal{O}_{0})+C(\mathcal{O}(t)t+C(\mathcal{O}(t))\int_{0}^{t}\mathcal{S}(\tau)d\tau$ $\displaystyle\leq C(\mathcal{O}_{0})+C(\mathcal{O}(t)t+C(\mathcal{O}(t))\sqrt{t}$ $\displaystyle\leq C(\mathcal{O}_{0})+C(\mathcal{O}(T))\sqrt{T}$ $\displaystyle\leq C(\mathcal{O}_{0})\exp\\{\sqrt{T}C(\mathcal{O}(T))\\}.$ Now denote $\hat{\mathbf{H}}=\epsilon\mathbf{H}$ and $\hat{\mathbf{H}}_{\alpha}=\partial^{\alpha}(\epsilon\mathbf{H})$ for $\|\alpha\|=s+1$. Then, $\hat{\mathbf{H}}_{\alpha}$ satisfies $\displaystyle\partial_{t}\hat{\mathbf{H}}_{\alpha}+({\mathbf{u}}\cdot\nabla)\hat{\mathbf{H}}_{\alpha}-\bar{\nu}\Delta\hat{\mathbf{H}}_{\alpha}=-\epsilon[\partial^{\alpha},{\mathbf{u}}]\cdot\nabla\mathbf{H}-\epsilon\partial^{\alpha}(\mathbf{H}{\rm div}{\mathbf{u}})+\epsilon\partial^{\alpha}((\mathbf{H}\cdot\nabla){\mathbf{u}}).$ (3.1) The commutator inequality (2.6) implies that $\\|-\epsilon[\partial^{\alpha},{\mathbf{u}}]\cdot\nabla\mathbf{H}\\|_{L^{2}}\leq C_{0}(\\|{\mathbf{u}}\\|_{W^{1,\infty}}\\|\nabla\hat{\mathbf{H}}\\|_{H^{s}}+\\|\epsilon{\mathbf{u}}\\|_{H^{s+1}}\\|\nabla\mathbf{H}\\|_{L^{\infty}})\leq C(\mathcal{Q}),$ while an integration by parts and Sobolev’s inequality lead to $\displaystyle-\langle\epsilon\partial^{\alpha}(\mathbf{H}{\rm div}{\mathbf{u}}),\hat{\mathbf{H}}_{\alpha}\rangle+\langle\epsilon\partial^{\alpha}((\mathbf{H}\cdot\nabla){\mathbf{u}}),\hat{\mathbf{H}}_{\alpha}\rangle$ $\displaystyle\leq\frac{\bar{\nu}}{2}\\|\nabla\hat{\mathbf{H}}_{\alpha}\\|_{L^{2}}^{2}+C_{0}\\|\mathbf{H}\\|_{H^{s}}^{2}\\|\epsilon{\mathbf{u}}\\|_{H^{s+1}}^{2}$ $\displaystyle\leq\frac{\bar{\nu}}{2}\\|\nabla\hat{\mathbf{H}}_{\alpha}\\|_{L^{2}}^{2}+C(\mathcal{Q}).$ Hence, we obtain $\displaystyle\sup_{\tau\in[0,t]}\\|\epsilon\mathbf{H}(\tau)\\|_{H^{s+1}}+\Big{\\{}\int^{t}_{0}\bar{\nu}\\|\nabla(\epsilon\mathbf{H})(\tau)\\|^{2}_{H^{s+1}}d\tau\Big{\\}}^{1/2}$ $\displaystyle\leq C(\mathcal{O}_{0})\exp\\{\sqrt{T}C(\mathcal{O}(T))\\}.$ Similarly, we can obtain $\displaystyle\sup_{\tau\in[0,t]}\epsilon^{2}\\|\mathbf{H}(\tau)\\|_{H^{s+2}}+\epsilon^{2}\Big{\\{}\int^{t}_{0}\bar{\nu}\\|\nabla\mathbf{H}(\tau)\\|^{2}_{H^{s+2}}d\tau\Big{\\}}^{1/2}$ $\displaystyle\leq C(\mathcal{O}_{0})\exp\\{\sqrt{T}C(\mathcal{O}(T))\\}.$ Next, we estimate $\theta$. Using Sobolev’s inequality, one finds that $\displaystyle\\|\partial^{s}(\epsilon^{2}e^{-\epsilon p}[\bar{\nu}\|{\rm curl\,}\mathbf{H}\|^{2}+\Psi({\mathbf{u}}):\nabla{\mathbf{u}}])\\|_{L^{2}}$ $\displaystyle\qquad\qquad\qquad\leq C_{0}\\|\epsilon p\\|_{H^{s}}(\\|\epsilon\nabla\mathbf{H}\\|^{2}_{H^{s}}+\\|\epsilon\nabla{\mathbf{u}}\\|^{2}_{H^{s}})\leq C(\mathcal{Q}).$ Employing arguments similar to those used for $\mathbf{H}$, we can obtain $\displaystyle\sup_{\tau\in[0,t]}\\|\theta(\tau)\\|_{H^{s}}+\Big{\\{}\int^{t}_{0}\bar{\kappa}\\|\nabla\theta(\tau)\\|^{2}_{H^{s}}d\tau\Big{\\}}^{1/2}$ $\displaystyle\leq C(\mathcal{O}_{0})\exp\\{\sqrt{T}C(\mathcal{O}(T))\\}.$ Thus, the lemma is proved. ∎ ###### Lemma 3.2. Let $s\geq 4$ and $(p,{\mathbf{u}},\mathbf{H},\theta)$ be a solution to the problem (1.11)–(1.14), (1.19) on $[0,T_{1}]$. Then there exists an increasing function $C(\cdot)$ such that, for any $\epsilon\in(0,1]$ and $t\in[0,T],T=\min\\{T_{1},1\\}$, it holds that $\displaystyle\sup_{\tau\in[0,t]}\\|(\epsilon p,\epsilon{\mathbf{u}})(\tau)\\|_{H^{s}}+\Big{\\{}\int^{t}_{0}\bar{\mu}\\|\nabla(\epsilon{\mathbf{u}})(\tau)\\|^{2}_{H^{s}}d\tau\Big{\\}}^{1/2}\leq C(\mathcal{O}_{0})\exp\\{\sqrt{T}C(\mathcal{O}(T))\\}.$ ###### Proof. Let $\check{p}=\epsilon p$, and $\check{p}_{\alpha}=\partial^{\alpha}(\epsilon p)$ for any multi-index $\alpha=(\alpha_{1},\alpha_{2},\alpha_{3})$ satisfying $\|\alpha\|\leq s$. Then $\displaystyle\partial_{t}\check{p}_{\alpha}+({\mathbf{u}}\cdot\nabla)\check{p}_{\alpha}=$ $\displaystyle-[\partial^{\alpha},{\mathbf{u}}]\cdot(\nabla\check{p})-\partial^{\alpha}[{\rm div}(2{\mathbf{u}}-\kappa a(\epsilon p)b(\theta)\nabla\theta)]$ $\displaystyle+\partial^{\alpha}\\{a(\epsilon p)[\nu\|{\rm curl\,}(\epsilon\mathbf{H})\|^{2}+\Psi(\epsilon{\mathbf{u}}):\nabla(\epsilon{\mathbf{u}})]\\}$ $\displaystyle+\kappa\partial^{\alpha}\\{a(\epsilon p)b(\theta)\nabla(\epsilon p)\cdot\nabla\theta\\}$ $\displaystyle:=$ $\displaystyle h_{1}+h_{2}+h_{3}+h_{4},$ (3.2) where, for simplicity of presentation, we have set $a(\epsilon p):=e^{-\epsilon p},\quad b(\theta):=e^{\theta}.$ It is easy to see that the energy estimate for (3.2) gives $\displaystyle\frac{1}{2}\frac{d}{dt}\\|\check{p}_{\alpha}\\|_{L^{2}}^{2}=-\langle({\mathbf{u}}\cdot\nabla)\check{p}_{\alpha},\check{p}_{\alpha}\rangle+\langle h_{1}+h_{2}+h_{3}+h_{4},\check{p}_{\alpha}\rangle,$ (3.3) where we have to estimate each term on the right-hand side of (3.3). First, an integration by parts yields $\displaystyle-\langle({\mathbf{u}}\cdot\nabla)\check{p}_{\alpha},\check{p}_{\alpha}\rangle=\frac{1}{2}\int{\rm div}{\mathbf{u}}\|\check{p}_{\alpha}\|^{2}dx\leq C(\mathcal{Q})\\|\check{p}_{\alpha}\\|^{2}_{L^{2}},$ while the commutator inequality leads to $\displaystyle\\|h_{1}\\|\leq C_{0}(\\|{\mathbf{u}}\\|_{W^{1,\infty}}\\|\nabla\check{p}\\|_{H^{s-1}}+\\|{\mathbf{u}}\\|_{H^{s}}\\|\nabla\check{p}\\|_{L^{\infty}})\leq C(\mathcal{Q}).$ Consequently, $\displaystyle\langle h_{1},\check{p}_{\alpha}\rangle\leq\\|\check{p}_{\alpha}\\|_{L^{2}}\\|h_{1}\\|_{L^{2}}\leq C(\mathcal{Q}).$ From Sobolev’s inequality one gets $\\|h_{2}\\|\leq C_{0}\\|{\mathbf{u}}\\|_{H^{s+1}}+\\|\theta\\|_{H^{s}}\\|\epsilon p\\|_{H^{s}}\\|\theta\\|_{H^{s+2}}\leq C(\mathcal{S})(1+C(\mathcal{Q})),$ whence, $\displaystyle\langle h_{2},\check{p}_{\alpha}\rangle\leq$ $\displaystyle C(\mathcal{Q})C(\mathcal{S}).$ Similarly, one can prove that $\displaystyle\langle(h_{3}+h_{4}),\check{p}_{\alpha}\rangle\leq$ $\displaystyle C(\mathcal{Q}).$ Hence, we conclude that $\displaystyle\sup_{\tau\in[0,t]}\\|\epsilon p(\tau)\\|_{H^{s}}\leq C(\mathcal{O}_{0})\exp\\{\sqrt{T}C(\mathcal{O}(T))\\}.$ In a similar way, we can estimate ${\mathbf{u}}$. Thus the proof of the lemma is completed. ∎ Next, we control the term $\\|({\mathbf{u}},p)\\|_{H^{s}}$. The idea is to bound the norm of $({\rm div}{\mathbf{u}},$ $\nabla p)$ in terms of the suitable norm of $(\epsilon{\mathbf{u}},\epsilon p,\epsilon\mathbf{H},\theta)$ and $\epsilon(\partial_{t}{\mathbf{u}},\partial_{t}p)$ by making use of the structure of the system. To this end, we first estimate $\\|(\epsilon{\mathbf{u}},\epsilon p,\theta)\\|_{H^{s+1}}$. ### 3.2. $H^{s+1}$-estimates on $(\epsilon{\mathbf{u}},\epsilon p,\epsilon\mathbf{H},\theta)$ Following [2], we set $\displaystyle(\hat{p},\hat{\mathbf{u}},\hat{\mathbf{H}},\hat{\theta}):=(\epsilon p-\theta,\epsilon{\mathbf{u}},\epsilon\mathbf{H},\theta-\bar{\theta}).$ A straightforward calculation results in that $(\hat{p},\hat{\mathbf{u}},\hat{\mathbf{H}},\hat{\theta})$ solves the following system: $\displaystyle\partial_{t}\hat{p}+({\mathbf{u}}\cdot\nabla)\hat{p}+\frac{1}{\epsilon}{\rm div}\hat{\mathbf{u}}=0,$ (3.4) $\displaystyle b(-\theta)[\partial_{t}\hat{\mathbf{u}}+({\mathbf{u}}\cdot\nabla)\hat{\mathbf{u}}]+\frac{1}{\epsilon}(\nabla\hat{p}+\nabla\hat{\theta})$ $\displaystyle\qquad\qquad\qquad\qquad\qquad=a(\epsilon p)[({\rm curl\,}\mathbf{H})\times\hat{\mathbf{H}}+{\rm div}\Psi(\hat{\mathbf{u}})],$ (3.5) $\displaystyle\partial_{t}\hat{\mathbf{H}}+{\mathbf{u}}\cdot\nabla\hat{\mathbf{H}}+\mathbf{H}{\rm div}\hat{\mathbf{u}}-\mathbf{H}\cdot\nabla\hat{\mathbf{u}}-\bar{\nu}\Delta\hat{\mathbf{H}}=0,\quad{\rm div}\hat{\mathbf{H}}=0,$ (3.6) $\displaystyle\partial_{t}\hat{\theta}+({\mathbf{u}}\cdot\nabla)\hat{\theta}+\frac{1}{\epsilon}{\rm div}\hat{\mathbf{u}}=\epsilon a(\epsilon p)[\bar{\nu}\,{\rm curl\,}\mathbf{H}:{\rm curl\,}\hat{\mathbf{H}}+\epsilon a(\epsilon p)\Psi({\mathbf{u}}):\nabla\hat{\mathbf{u}}]$ $\displaystyle\qquad\qquad\qquad\qquad\qquad\ \quad+\bar{\kappa}a(\epsilon p){\rm div}(b(\theta)\nabla\hat{\theta}).$ (3.7) We have ###### Lemma 3.3. Let $s\geq 4$ and $(p,{\mathbf{u}},\mathbf{H},\theta)$ be a solution to the problem (1.11)–(1.14), (1.19) on $[0,T_{1}]$. Then there exist a constant $l_{1}>0$ and an increasing function $C(\cdot)$ such that, for any $\epsilon\in(0,1]$ and $t\in[0,T]$, $T=\min\\{T_{1},1\\}$, it holds that $\displaystyle\sup_{\tau\in[0,t]}\\|(\epsilon q,\epsilon{\mathbf{u}},\theta-\bar{\theta})(\tau)\\|_{H^{s+1}}+l_{1}\Big{\\{}\int^{t}_{0}\\|\nabla(\epsilon{\mathbf{u}},\theta)\\|^{2}_{H^{s+1}}(\tau)d\tau\Big{\\}}^{1/2}$ $\displaystyle\leq C(\mathcal{O}_{0})\exp\\{\sqrt{T}C(\mathcal{O}(T))\\}.$ (3.8) ###### Proof. Let $\alpha=(\alpha_{1},\alpha_{2},\alpha_{3})$ be a multi-index such that $\|\alpha\|=s+1$. Set $\displaystyle(\hat{p}_{\alpha},\hat{{\mathbf{u}}}_{\alpha},\hat{H}_{\alpha},\hat{\theta}_{\alpha}):=\left(\partial^{\alpha}(\epsilon p-\theta),\partial^{\alpha}(\epsilon{\mathbf{u}}),\partial^{\alpha}(\epsilon\mathbf{H}),\partial^{\alpha}(\theta-\bar{\theta})\right).$ Then, $\hat{\mathbf{H}}_{\alpha}$ satisfies (3.1) and $(\hat{p}_{\alpha},\hat{{\mathbf{u}}}_{\alpha},\hat{\theta}_{\alpha})$ solves $\displaystyle\partial_{t}\hat{p}_{\alpha}+({\mathbf{u}}\cdot\nabla)\hat{p}_{\alpha}+\frac{1}{\epsilon}{\rm div}\hat{{\mathbf{u}}}_{\alpha}=g_{1},$ (3.9) $\displaystyle b(-\theta)[\partial_{t}\hat{{\mathbf{u}}}_{\alpha}+({\mathbf{u}}\cdot\nabla)\hat{{\mathbf{u}}}_{\alpha}]+\frac{1}{\epsilon}(\nabla\hat{p}_{\alpha}+\nabla\hat{\theta}_{\alpha})$ $\displaystyle\quad\quad\qquad\quad=a(\epsilon p)({\rm curl\,}\mathbf{H})\times\hat{\mathbf{H}}_{\alpha}+a(\epsilon p){\rm div}\Psi(\hat{{\mathbf{u}}}_{\alpha})+g_{2},$ (3.10) $\displaystyle\partial_{t}\hat{\theta}_{\alpha}+({\mathbf{u}}\cdot\nabla)\hat{\theta}_{\alpha}+\frac{1}{\epsilon}{\rm div}\hat{{\mathbf{u}}}_{\alpha}=\epsilon a(\epsilon p)[\bar{\nu}\,{\rm curl\,}\mathbf{H}:{\rm curl\,}\hat{\mathbf{H}}_{\alpha}+\Psi({\mathbf{u}}):\nabla\hat{\mathbf{u}}_{\alpha}]$ $\displaystyle\quad\quad\qquad\quad+\bar{\kappa}a(\epsilon p){\rm div}(b(\theta)\nabla\hat{\theta}_{\alpha})+g_{3},$ (3.11) with initial data $\displaystyle(\hat{p}_{\alpha},\hat{{\mathbf{u}}}_{\alpha},\hat{\mathbf{H}}_{\alpha},\hat{\theta}_{\alpha})\|_{t=0}:=\big{(}$ $\displaystyle\partial^{\alpha}(\epsilon p_{\rm in}(x)-\theta_{\rm in}(x)),\partial^{\alpha}(\epsilon{\mathbf{u}}_{\rm in}(x)),$ $\displaystyle\partial^{\alpha}\mathbf{H}_{\rm in}(x)),\partial^{\alpha}(\theta_{\rm in}(x)-\bar{\theta})\big{)},$ (3.12) where $\displaystyle g_{1}:=$ $\displaystyle-[\partial^{\alpha},{\mathbf{u}}]\cdot\nabla(\epsilon p-\theta),$ $\displaystyle g_{2}:=$ $\displaystyle-[\partial^{\alpha},b(-\theta)]\partial_{t}(\epsilon{\mathbf{u}})-[\partial^{\alpha},b(-\theta){\mathbf{u}}]\cdot\nabla(\epsilon{\mathbf{u}})$ $\displaystyle+[\partial^{\alpha},a(\epsilon p){\rm curl\,}(\epsilon\mathbf{H})]\times\mathbf{H}+[\partial^{\alpha},a(\epsilon p)]{\rm div}\Psi(\epsilon{\mathbf{u}}),$ $\displaystyle g_{3}:=$ $\displaystyle-[\partial^{\alpha},{\mathbf{u}}]\cdot\nabla\theta+\bar{\nu}\,[\partial^{\alpha},a(\epsilon p){\rm curl\,}(\epsilon\mathbf{H})]:{\rm curl\,}(\epsilon\mathbf{H})$ $\displaystyle+\epsilon[\partial^{\alpha},a(\epsilon p)\Psi({\mathbf{u}})]:\nabla(\epsilon{\mathbf{u}})+\bar{\kappa}\partial^{\alpha}\big{(}a(\epsilon p){\rm div}(b(\theta)\nabla\theta)\big{)}$ $\displaystyle-\bar{\kappa}a(\epsilon p){\rm div}(b(\theta)\nabla\hat{\theta}_{\alpha}).$ It follows from Proposition 2.3 and the positivity of $a(\cdot)$ and $b(\cdot)$ that $a(\cdot)$ and $b(\cdot)$ are bounded away from $0$ uniformly with respect to $\epsilon$, i.e. $\displaystyle a(\epsilon p)\geq\underline{a}>0,\;\;\;b(-\theta)\geq\underline{b}>0.$ (3.13) The standard $L^{2}$-energy estimates for (3.9), (3.10) and (3.11) yield that $\displaystyle\frac{1}{2}\frac{d}{dt}\big{(}\\|\hat{p}_{\alpha}\\|_{L^{2}}^{2}+\langle b(-\theta)\hat{{\mathbf{u}}}_{\alpha},\hat{{\mathbf{u}}}_{\alpha}\rangle+\\|\hat{\theta}_{\alpha}\\|_{L^{2}}^{2}\big{)}$ $\displaystyle\leq\frac{1}{2}\langle b_{t}(\theta)\hat{\mathbf{u}}_{\alpha},\hat{\mathbf{u}}_{\alpha}\rangle-\langle({\mathbf{u}}\cdot\nabla)\hat{p}_{\alpha},\hat{p}_{\alpha}\rangle-\langle b(-\theta)({\mathbf{u}}\cdot\nabla)\hat{{\mathbf{u}}}_{\alpha},\hat{{\mathbf{u}}}_{\alpha}\rangle-\langle({\mathbf{u}}\cdot\nabla)\hat{\theta}_{\alpha},\hat{\theta}_{\alpha}\rangle$ $\displaystyle\quad+\langle a(\epsilon p)({\rm curl\,}\mathbf{H})\times\hat{\mathbf{H}}_{\alpha},\hat{\mathbf{u}}_{\alpha}\rangle+\langle a(\epsilon p){\rm div}\Psi(\hat{{\mathbf{u}}}_{\alpha}),\hat{\mathbf{u}}_{\alpha}\rangle$ $\displaystyle\qquad\langle\epsilon a(\epsilon p)[\bar{\nu}\,{\rm curl\,}\mathbf{H}:{\rm curl\,}\hat{\mathbf{H}}_{\alpha}+\Psi({\mathbf{u}}):\nabla\hat{\mathbf{u}}_{\alpha}],\theta_{\alpha}\rangle+\langle\bar{\kappa}a(\epsilon p){\rm div}(b(\theta)\nabla\theta_{\alpha}),\hat{\theta}_{\alpha}\rangle$ $\displaystyle\quad+\langle g_{1},\hat{p}_{\alpha}\rangle+\langle g_{2},\hat{u}_{\alpha}\rangle+\langle g_{3},\hat{\theta}_{\alpha}\rangle.$ (3.14) It follows from equation (1.14), and the definition of $\mathcal{Q}$ and $\mathcal{S}$ that $\displaystyle\\|b_{t}(\theta)\\|_{L^{\infty}}\leq\\|b(\theta)\\|_{H^{s}}\\|\theta_{t}\\|_{H^{s}}\leq C(\mathcal{Q})(1+\mathcal{S}).$ Therefore, $\displaystyle\frac{1}{2}\langle b_{t}(\theta)\hat{\mathbf{u}}_{\alpha},\hat{\mathbf{u}}_{\alpha}\rangle\leq C(\mathcal{Q})(1+\mathcal{S}).$ On the other hand, it is easy to see that $\displaystyle-\langle({\mathbf{u}}\cdot\nabla)\hat{p}_{\alpha},\hat{p}_{\alpha}\rangle-\langle b(-\theta)({\mathbf{u}}\cdot\nabla)\hat{{\mathbf{u}}}_{\alpha},\hat{{\mathbf{u}}}_{\alpha}\rangle-\langle({\mathbf{u}}\cdot\nabla)\hat{\theta}_{\alpha},\hat{\theta}_{\alpha}\rangle\leq C(\mathcal{Q})$ and $\displaystyle\langle a(\epsilon p)({\rm curl\,}\mathbf{H})\times\hat{\mathbf{H}}_{\alpha},\hat{\mathbf{u}}_{\alpha}\rangle\leq C(\mathcal{Q}).$ By integration by parts we have $\displaystyle-\langle a(\epsilon p_{0}){\rm div}\Psi(\hat{\mathbf{u}}),\hat{\mathbf{u}}\rangle=$ $\displaystyle\int\mu a(\epsilon p_{0})[\|\nabla\hat{\mathbf{u}}_{\alpha}\|^{2}+(\mu+\lambda)\|{\rm div}\hat{\mathbf{u}}_{\alpha}\|^{2}]dx$ $\displaystyle+\mu\langle(\nabla a(\epsilon p)\cdot\nabla)\hat{\mathbf{u}}_{\alpha},\hat{\mathbf{u}}_{\alpha}\rangle$ $\displaystyle+(\mu+\lambda)\langle(\nabla a(\epsilon p){\rm div}\hat{\mathbf{u}}_{\alpha},\hat{\mathbf{u}}_{\alpha})dx$ $\displaystyle\equiv$ $\displaystyle\,d_{1}+d_{2}+d_{3}.$ (3.15) Thanks to the assumption that $\bar{\mu}>0$ and $2\bar{\mu}+3\bar{\lambda}>0$, there exists a positive constant $\xi_{1}$, such that $\displaystyle d_{1}$ $\displaystyle\geq{\underline{a}}\xi\int\|\nabla\hat{\mathbf{u}}_{\alpha}\|^{2}dx,$ (3.16) while Cauchy-Schwarz’s inequality implies $\displaystyle\|d_{2}\|+\|d_{3}\|\leq C(\mathcal{Q})\mathcal{S}.$ (3.17) Similarly, we can obtain $\displaystyle-\langle\bar{\kappa}a(\epsilon p){\rm div}(b(\theta)\nabla\hat{\theta}_{\alpha}),\hat{\theta}_{\alpha}\rangle\geq\bar{\kappa}{\underline{a}}\,{\underline{b}}\\|\nabla\hat{\theta}_{\alpha}\\|^{2}_{L^{2}}-C(\mathcal{Q})\mathcal{S}.$ (3.18) Easily, one has $\displaystyle\|\langle\epsilon a(\epsilon p)[\bar{\nu}\,{\rm curl\,}\mathbf{H}:{\rm curl\,}\hat{\mathbf{H}}_{\alpha}+\Psi({\mathbf{u}}):\nabla\hat{\mathbf{u}}_{\alpha}],\hat{\theta}_{\alpha}\rangle\|\leq C(\mathcal{Q})(1+\mathcal{S}).$ It remains to estimate $\langle g_{1},\hat{p}_{\alpha}\rangle$, $\langle g_{2},\hat{u}_{\alpha}\rangle$ and $\langle g_{3},\hat{\theta}_{\alpha}\rangle$ in (3.14). First, an application of Hölder’s inequality gives $\displaystyle\|\langle\hat{p}_{\alpha},g_{1}\rangle\|\leq C_{0}\\|\hat{p}_{\alpha}\\|_{L^{2}}\\|g_{1}\\|_{L^{2}},$ where $\\|g_{1}\\|_{L^{2}}$ can be bounded, by using (2.6), as follows $\displaystyle\\|g_{1}\\|_{L^{2}}=$ $\displaystyle\\|[\partial^{\alpha},{\mathbf{u}}]\cdot\nabla(\epsilon p-\theta)\\|_{L^{2}}$ $\displaystyle\leq$ $\displaystyle C_{0}(\\|{\mathbf{u}}\\|_{W^{1,\infty}}\\|\nabla(\epsilon p-\theta)\\|_{H^{s}}+\\|{\mathbf{u}}\\|_{H^{s+1}}\\|\nabla(\epsilon p-\theta)\\|_{L^{\infty}}).$ It follows from the definition of $\mathcal{Q}$ and Sobolev’s inequalities that $\displaystyle\\|\nabla(\epsilon p,\theta)\\|_{H^{s}}\leq\mathcal{Q},\;\;\;\;\;\\|\nabla(\epsilon p,\theta)\\|_{L^{\infty}}\leq\mathcal{Q}.$ Therefore, we obtain $\\|g_{1}\\|_{L^{2}}\leq C(\mathcal{Q})(1+\mathcal{S})$, and $\displaystyle\|\langle p_{\alpha},g_{1}\rangle\|\leq C(\mathcal{Q})(1+\mathcal{S}).$ (3.19) Next, we turn to the term $\|\langle{\mathbf{u}}_{\alpha},g_{2}\rangle\|$. Due to the equation (1.12), one has $\displaystyle-[\partial^{\alpha},b(-\theta)]\partial_{t}(\epsilon{\mathbf{u}})=$ $\displaystyle[\partial^{\alpha},b(-\theta)]\big{(}({\mathbf{u}}\cdot\nabla)(\epsilon{\mathbf{u}})\big{)}+[\partial^{\alpha},b(-\theta)]\big{(}b(\theta)\nabla p\big{)}$ $\displaystyle-[\partial^{\alpha},b(-\theta)]\big{(}b(\theta)a(\epsilon p)({\rm curl\,}\mathbf{H}\times(\epsilon\mathbf{H}))\big{)}$ $\displaystyle-[\partial^{\alpha},b(-\theta)]\big{(}b(\theta)a(\epsilon p){\rm div}\Psi(\epsilon{\mathbf{u}})\big{)}.$ (3.20) The inequality (2.6) implies that $\displaystyle\left\|\left\langle{\mathbf{u}}_{\alpha},[\partial^{\alpha},b(-\theta)]\big{(}({\mathbf{u}}\cdot\nabla)(\epsilon{\mathbf{u}})\big{)}\right\rangle\right\|$ $\displaystyle\leq C_{0}\\|{\mathbf{u}}_{\alpha}\\|_{L^{2}}\\|[\partial^{\alpha},b(-\theta)]\big{(}{\mathbf{u}}\cdot\nabla(\epsilon{\mathbf{u}})\big{)}\\|_{L^{2}}$ $\displaystyle\leq C(\mathcal{Q})\big{(}\\|b(-\theta)\\|_{W^{1,\infty}}\\|{\mathbf{u}}\cdot\nabla(\epsilon{\mathbf{u}})\\|_{H^{s}}+\\|b(-\theta)\\|_{H^{s+1}}\\|{\mathbf{u}}\cdot\nabla(\epsilon{\mathbf{u}})\\|_{L^{\infty}}$ $\displaystyle\leq C(\mathcal{Q}),$ and $\displaystyle\left\|\left\langle{\mathbf{u}}_{\alpha},[\partial^{\alpha},b(-\theta)]\big{(}b(\theta)\nabla p\big{)}\right\rangle\right\|$ $\displaystyle\leq C_{0}\\|{\mathbf{u}}_{\alpha}\\|_{L^{2}}\\|[\partial^{\alpha},b(-\theta)]\big{(}b(\theta)\nabla p\big{)}\\|_{L^{2}}$ $\displaystyle\leq C(\mathcal{Q})\big{(}\\|b(-\theta)\\|_{W^{1,\infty}}\\|b(\theta)\nabla p\\|_{H^{s}}+\\|b(-\theta)\\|_{H^{s+1}}\\|b(\theta)\nabla p\\|_{L^{\infty}}$ $\displaystyle\leq C(\mathcal{Q})(1+\mathcal{S}).$ The third term on the right-hand side of (3.2) can be treated in a similar manner, and we obtain $\displaystyle\left\|\left\langle{\mathbf{u}}_{\alpha},[\partial^{\alpha},b(-\theta)]\big{(}b(\theta)a(\epsilon p)({\rm curl\,}\mathbf{H}\times(\epsilon\mathbf{H}))\big{)}\right\rangle\right\|\leq C(\mathcal{Q})(1+\mathcal{S}).$ To bound the last term on the right-hand side of (3.2), we use (2.6) to deduce that $\displaystyle\left\langle{\mathbf{u}}_{\alpha},[\partial^{\alpha},b(-\theta)]\big{(}b(\theta)a(\epsilon p){\rm div}\Psi(\epsilon{\mathbf{u}})\big{)}\right\rangle$ $\displaystyle\leq C_{0}\\|{\mathbf{u}}_{\alpha}\\|_{L^{2}}\\|[\partial^{\alpha},b(-\theta)]\big{(}b(\theta)a(\epsilon p){\rm div}\Psi(\epsilon{\mathbf{u}})\big{)}\\|_{L^{2}}$ $\displaystyle\leq C(\mathcal{Q})(\\|b(-\theta)\\|_{W^{1,\infty}}\\|b(\theta)a(\epsilon p){\rm div}\Psi(\epsilon{\mathbf{u}})\\|_{H^{s}}$ $\displaystyle\quad+\\|b(-\theta)\\|_{H^{s+1}}\\|b(\theta)a(\epsilon p){\rm div}\Psi(\epsilon{\mathbf{u}})\\|_{L^{\infty}})$ $\displaystyle\leq C(\mathcal{Q})(1+\mathcal{S}).$ Hence, it holds that $\displaystyle\left\|\left\langle{\mathbf{u}}_{\alpha},g_{2}\right\rangle\right\|\leq C(\mathcal{Q})(1+\mathcal{S}).$ (3.21) Since $g_{3}$ is similar to $g_{1}$ in structure, we easily get $\displaystyle\big{\|}\big{\langle}\hat{\theta}_{\alpha},g_{3}\big{\rangle}\big{\|}\leq C(\mathcal{Q})(1+\mathcal{S}).$ (3.22) Therefore, it follows from (3.19), (3.21)–(3.22), the positivity of $b(-\theta)$, and the definition of $\mathcal{O}$, $\mathcal{O}_{0}$, $\mathcal{Q}$ and $\mathcal{S}$, that there exists a constant $l_{1}>0$, such that for $t\in[0,T]$ and $T=\min\\{T_{1},1\\}$, $\displaystyle\sup_{\tau\in[0,t]}\\|(\hat{p}_{\alpha},\hat{\mathbf{u}}_{\alpha},\hat{\theta}_{\alpha})(\tau)\\|^{2}_{H^{s+1}}+l_{1}\int^{t}_{0}\\|\nabla(\hat{\mathbf{u}}_{\alpha},\hat{\theta}_{\alpha})\\|^{2}_{H^{s+1}}(\tau)d\tau$ $\displaystyle\leq C(\mathcal{O}_{0})+C(\mathcal{O}(t))t+C(\mathcal{O}(t))\int^{t}_{0}\mathcal{S}(\tau)d\tau$ $\displaystyle\leq C(\mathcal{O}_{0})+C(\mathcal{O}(t))\sqrt{t}$ $\displaystyle\leq C(\mathcal{O}_{0})\exp\\{\sqrt{T}C(\mathcal{O}(T)\\}.$ Summing up the above estimates for all $\alpha$ with $0\leq\|\alpha\|\leq s+1$, we obtain the desired inequality (3.3). ∎ In a way similar to the proof of Lemma 3.3, we can show that ###### Lemma 3.4. Let $s\geq 4$ and $(p,{\mathbf{u}},\mathbf{H},\theta)$ be a solution to (1.11)–(1.14), (1.19) on $[0,T_{1}]$. Then there exist a constant $l_{2}>0$ and an increasing function $C(\cdot)$ such that, for any $\epsilon\in(0,1]$ and $t\in[0,T]$, $T=\min\\{T_{1},1\\}$, it holds that $\displaystyle\sup_{\tau\in[0,t]}\\|(\epsilon^{2}q,\epsilon^{2}{\mathbf{u}},\epsilon(\theta-\bar{\theta})(\tau)\\|_{H^{s+2}}+l_{2}\Big{\\{}\int^{t}_{0}\\|\nabla(\epsilon^{2}{\mathbf{u}},\epsilon\theta)\\|^{2}_{H^{s+2}}(\tau)d\tau\Big{\\}}^{1/2}$ $\displaystyle\leq C(\mathcal{O}_{0})\exp\\{(\sqrt{T})C(\mathcal{O}(T))\\}.$ Recalling Lemma 2.2 and the definition of $\mathcal{Q}$ and $\mathcal{S}$, we find that $\displaystyle\\|\partial_{t}(\epsilon p,\epsilon{\mathbf{u}},\epsilon\mathbf{H},\theta)\\|_{H^{s-1}}\leq C(\mathcal{Q}),$ (3.23) $\displaystyle\\|\partial_{t}(\epsilon p,\epsilon{\mathbf{u}},\epsilon\mathbf{H},\theta)\\|_{H^{s}}\leq C(\mathcal{Q})(1+\mathcal{S}),$ (3.24) $\displaystyle\epsilon\\|\partial_{t}(\epsilon p,\epsilon{\mathbf{u}},\epsilon\mathbf{H},\theta)\\|_{H^{s}}\leq C(\mathcal{Q}).$ (3.25) Moreover, it follows easily from Lemmas 3.1–3.4 and the equation (1.14) that for some constant $l_{3}>0$, $\displaystyle\sup_{\tau\in[0,t]}\\|\epsilon\partial_{t}\theta\\|_{H^{s}}^{2}+l_{3}\int_{0}^{t}\\|\nabla((\epsilon\partial_{t})\theta\\|_{H^{s}}^{2}(\tau)d\tau\leq C(\mathcal{O}_{0})\exp\\{\sqrt{T}C(\mathcal{O}(T)\\}.$ (3.26) ### 3.3. $H^{s-1}$-estimates on $({\rm div}{\mathbf{u}},\nabla p)$ To establish the estimates for $p$ and the acoustic part of ${\mathbf{u}}$, we first control the term $(\epsilon\partial_{t})(p,{\mathbf{u}})$. To this end, we start with a $L^{2}$-estimate for the linearized system. For a given state $(p_{0},{\mathbf{u}}_{0},\mathbf{H}_{0},\theta_{0})$, consider the following linearized system of (1.11)–(1.14): $\displaystyle\partial_{t}p+({\mathbf{u}}_{0}\cdot\nabla)p+\frac{1}{\epsilon}{\rm div}(2{\mathbf{u}}-\bar{\kappa}a(\epsilon p_{0})b(\theta_{0})\nabla\theta)=\epsilon a(\epsilon p_{0})[\bar{\nu}\,{\rm curl\,}\mathbf{H}_{0}:{\rm curl\,}\mathbf{H}]$ $\displaystyle\quad\quad\qquad\quad\quad\ \ +\epsilon a(\epsilon p_{0})\Psi({\mathbf{u}}_{0}):\nabla{\mathbf{u}}+\bar{\kappa}a(\epsilon p_{0})b(\theta_{0})\nabla p_{0}\cdot\nabla\theta+f_{1},$ (3.27) $\displaystyle b(-\theta_{0})[\partial_{t}{\mathbf{u}}+({\mathbf{u}}_{0}\cdot\nabla){\mathbf{u}}]+\frac{\nabla p}{\epsilon}=a(\epsilon p_{0})[({\rm curl\,}\mathbf{H}_{0})\times\mathbf{H}+{\rm div}\Psi({\mathbf{u}})]+f_{2},$ (3.28) $\displaystyle\partial_{t}\mathbf{H}-{\rm curl\,}({\mathbf{u}}_{0}\times\mathbf{H})-\bar{\nu}\Delta\mathbf{H}=f_{3},\quad{\rm div}\mathbf{H}=0,$ (3.29) $\displaystyle\partial_{t}\theta+({\mathbf{u}}_{0}\cdot\nabla)\theta+{\rm div}{\mathbf{u}}=\epsilon^{2}a(\epsilon p_{0})[\bar{\nu}\,{\rm curl\,}\mathbf{H}_{0}:{\rm curl\,}\mathbf{H}+\Psi({\mathbf{u}}_{0}):\nabla{\mathbf{u}}]$ $\displaystyle\quad\quad\qquad\quad\quad\ \ +\bar{\kappa}a(\epsilon p_{0}){\rm div}(b(\theta_{0})\nabla\theta)+f_{4},$ (3.30) where we have added the source terms $f_{i}$ ($1\leq i\leq 4$) on the right- hands sides of (3.27)–(3.30) for latter use, and used the following notations: $a(\epsilon p_{0}):=e^{-\epsilon p_{0}},\quad b(\theta_{0}):=e^{\theta_{0}}.$ The system (3.27)–(3.30) is supplemented with initial data $\displaystyle(p,{\mathbf{u}},\mathbf{H},\theta)\|_{t=0}=(p_{\rm in}(x),{\mathbf{u}}_{\rm in}(x),\mathbf{H}_{\rm in}(x),\theta_{\rm in}(x)),\quad x\in\mathbb{R}^{3}.$ (3.31) ###### Lemma 3.5. Let $(p,{\mathbf{u}},\mathbf{H},\theta)$ be a solution to the Cauchy problem (3.27)–(3.31) on $[0,\hat{T}]$. Then there exist a constant $l_{4}>0$ and an increasing function $C(\cdot)$ such that, for any $\epsilon\in(0,1]$ and $t\in[0,T]$, $T=\min\\{\hat{T},1\\}$, it holds that $\displaystyle\sup_{\tau\in[0,t]}\\|(p,{\mathbf{u}},\mathbf{H})(\tau)\\|^{2}_{L^{2}}+l_{4}\int^{t}_{0}\\|\nabla({\mathbf{u}},\mathbf{H})\\|_{L^{2}}^{2}(\tau)d\tau$ $\displaystyle\leq e^{TC(R_{0})}\\|(p,{\mathbf{u}},\mathbf{H})(0)\\|^{2}_{L^{2}}+C(R_{0})e^{TC(R_{0})}\sup_{\tau\in[0,T]}\\|\nabla\theta(\tau)\\|^{2}_{L^{2}}$ $\displaystyle\quad+C(R_{0})\int_{0}^{T}\\|\nabla(\epsilon{\mathbf{u}},\epsilon\mathbf{H})\\|_{L^{2}}^{2}d\tau+C(R_{0})\int^{T}_{0}\\|\nabla\theta\\|^{2}_{H^{1}}(\tau)d\tau$ $\displaystyle\quad+C(R_{0})\int^{T}_{0}\left\\{\\|f_{1}\\|^{2}_{L^{2}}+\\|f_{2}\\|^{2}_{L^{2}}+\\|f_{3}\\|^{2}_{L^{2}}+\\|\nabla f_{4}\\|_{L^{2}}^{2}\right\\}(\tau)d\tau,$ (3.32) where $\displaystyle R_{0}=\sup_{\tau\in[0,T]}\\{\\|\partial_{t}\theta_{0}(\tau)\\|_{L^{\infty}},\\|(p_{0},{\mathbf{u}}_{0},\mathbf{H}_{0},\theta_{0})(\tau)\\|_{W^{1,\infty}}\\}.$ (3.33) ###### Proof. Set $\displaystyle(\tilde{p},\tilde{\mathbf{u}},\tilde{\mathbf{H}},\tilde{\theta})=(p,2{\mathbf{u}}-\bar{\kappa}a(\epsilon p_{0})b(\theta_{0})\nabla\theta,\mathbf{H},\theta).$ Then $\tilde{p}$ and $\tilde{\mathbf{H}}$ satisfy $\displaystyle\partial_{t}\tilde{p}$ $\displaystyle+({\mathbf{u}}_{0}\cdot\nabla)\tilde{p}+\frac{1}{\epsilon}{\rm div}\tilde{\mathbf{u}}=\epsilon a(\epsilon p_{0})[\bar{\nu}\,{\rm curl\,}\mathbf{H}_{0}:{\rm curl\,}\tilde{\mathbf{H}}]+\frac{\epsilon}{2}a(\epsilon p_{0})\Psi({\mathbf{u}}_{0}):\nabla\tilde{\mathbf{u}}$ $\displaystyle+\frac{\epsilon}{2}a(\epsilon p_{0})\Psi({\mathbf{u}}_{0}):\nabla(\bar{\kappa}a(\epsilon p_{0})b(\theta_{0})\nabla\tilde{\theta})+\bar{\kappa}a(\epsilon p_{0})b(\theta_{0})\nabla p_{0}\cdot\nabla\tilde{\theta}+f_{1}$ (3.34) and $\displaystyle\partial_{t}\tilde{\mathbf{H}}-{\rm curl\,}({\mathbf{u}}_{0}\times\tilde{\mathbf{H}})-\bar{\nu}\Delta\tilde{\mathbf{H}}=f_{3},\quad{\rm div}\tilde{\mathbf{H}}=0,$ (3.35) respectively. One can derive the equation for $\tilde{\mathbf{u}}$ by applying the operator $\nabla$ to (3.30) to obtain $\displaystyle\partial_{t}\nabla\tilde{\theta}+({\mathbf{u}}_{0}\cdot\nabla)\nabla\tilde{\theta}+\frac{1}{2}\nabla{\rm div}\tilde{\mathbf{u}}+\frac{1}{2}\nabla{\rm div}(\bar{\kappa}a(\epsilon p_{0})b(\theta_{0})\nabla\tilde{\theta})$ $\displaystyle=\nabla\left\\{\epsilon^{2}a(\epsilon p_{0})[\bar{\nu}\,{\rm curl\,}\mathbf{H}_{0}:{\rm curl\,}\tilde{\mathbf{H}}]\right\\}+\frac{1}{2}\nabla\left\\{\epsilon^{2}a(\epsilon p_{0})\Psi({\mathbf{u}}_{0}):\nabla\tilde{\mathbf{u}}\right\\}$ $\displaystyle\quad+\frac{1}{2}\nabla\left\\{\epsilon^{2}a(\epsilon p_{0})\Psi({\mathbf{u}}_{0}):\nabla(\bar{\kappa}a(\epsilon p_{0})b(\theta_{0})\nabla\tilde{\theta})\right\\}$ $\displaystyle\quad+\nabla\left\\{\bar{\kappa}a(\epsilon p_{0}){\rm div}(b(\theta_{0})\nabla\tilde{\theta})\right\\}+[\nabla,{\mathbf{u}}_{0}]\cdot\nabla\tilde{\theta}+\nabla f_{4}.$ (3.36) If we multiply (3.3) with $\frac{1}{2}\bar{\kappa}a(\epsilon p_{0})$, we get $\displaystyle\frac{1}{2}b(-\theta_{0})\left\\{\partial_{t}(\bar{\kappa}a(\epsilon p_{0})b(\theta_{0})\nabla\tilde{\theta})+({\mathbf{u}}_{0}\cdot\nabla)[\bar{\kappa}a(\epsilon p_{0})b(\theta_{0})\nabla\tilde{\theta}]\right\\}$ $\displaystyle=\frac{\bar{\kappa}}{2}b(-\theta_{0})\partial_{t}\\{a(\epsilon p_{0})b(\theta_{0})\\}\nabla\tilde{\theta}+\frac{\bar{\kappa}}{2}b(-\theta_{0})\left\\{{\mathbf{u}}_{0}\cdot\nabla[a(\epsilon p_{0})b(\theta_{0})]\nabla\tilde{\theta}\right\\}$ $\displaystyle\quad-\frac{1}{2}\bar{\kappa}a(\epsilon p_{0})\nabla{\rm div}\tilde{\mathbf{u}}-\frac{1}{2}\bar{\kappa}a(\epsilon p_{0})\nabla{\rm div}(\bar{\kappa}a(\epsilon p_{0})b(\theta_{0})\nabla\tilde{\theta})$ $\displaystyle\quad+\frac{1}{2}\bar{\kappa}a(\epsilon p_{0})\nabla\left\\{\epsilon^{2}a(\epsilon p_{0})[\bar{\nu}\,{\rm curl\,}\mathbf{H}_{0}:{\rm curl\,}\tilde{\mathbf{H}}]\right\\}+\frac{1}{4}\bar{\kappa}a(\epsilon p_{0})\nabla\left\\{\epsilon^{2}a(\epsilon p_{0})\Psi({\mathbf{u}}_{0}):\nabla\tilde{\mathbf{u}}\right\\}$ $\displaystyle\quad+\frac{1}{4}\bar{\kappa}a(\epsilon p_{0})\nabla\left\\{\epsilon^{2}a(\epsilon p_{0})\Psi({\mathbf{u}}_{0}):\nabla(\bar{\kappa}a(\epsilon p_{0})b(\theta_{0})\nabla\tilde{\theta})\right\\}$ $\displaystyle\quad+\frac{1}{2}\bar{\kappa}a(\epsilon p_{0})\nabla\left\\{\bar{\kappa}a(\epsilon p_{0}){\rm div}(b(\theta_{0})\nabla\tilde{\theta})\right\\}$ $\displaystyle\quad+\frac{1}{2}\bar{\kappa}a(\epsilon p_{0})[\nabla,{\mathbf{u}}_{0}]\cdot\nabla\tilde{\theta}+\frac{1}{2}\bar{\kappa}a(\epsilon p_{0})\nabla f_{4}.$ (3.37) Subtracting (3.3) from (3.28) yields $\displaystyle\frac{1}{2}$ $\displaystyle b(-\theta_{0})[\partial_{t}\tilde{\mathbf{u}}+{\mathbf{u}}_{0}\cdot\nabla\tilde{\mathbf{u}}]+\frac{\nabla\tilde{p}}{\epsilon}$ $\displaystyle=$ $\displaystyle-\frac{\bar{\kappa}}{2}b(-\theta_{0})\partial_{t}\\{a(\epsilon p_{0})b(\theta_{0})\\}\nabla\tilde{\theta}-\frac{\bar{\kappa}}{2}b(-\theta_{0})\left\\{{\mathbf{u}}_{0}\cdot\nabla[a(\epsilon p_{0})b(\theta_{0})]\nabla\tilde{\theta}\right\\}$ $\displaystyle+\frac{1}{4}\bar{\kappa}a(\epsilon p_{0})\nabla{\rm div}\tilde{\mathbf{u}}+\frac{1}{4}\bar{\kappa}a(\epsilon p_{0})\nabla{\rm div}(\bar{\kappa}a(\epsilon p_{0})b(\theta_{0})\nabla\tilde{\theta})$ $\displaystyle-\frac{1}{2}\bar{\kappa}a(\epsilon p_{0})\nabla\left\\{\epsilon^{2}a(\epsilon p_{0})[\bar{\nu}\,{\rm curl\,}\mathbf{H}_{0}:{\rm curl\,}\tilde{\mathbf{H}}]\right\\}$ $\displaystyle-\frac{1}{4}\bar{\kappa}a(\epsilon p_{0})\nabla\left\\{\epsilon^{2}a(\epsilon p_{0})\Psi({\mathbf{u}}_{0}):\nabla\tilde{\mathbf{u}}\right\\}$ $\displaystyle-\frac{1}{4}\bar{\kappa}a(\epsilon p_{0})\nabla\left\\{\epsilon^{2}a(\epsilon p_{0})\Psi({\mathbf{u}}_{0}):\nabla(\bar{\kappa}a(\epsilon p_{0})b(\theta_{0})\nabla\tilde{\theta})\right\\}$ $\displaystyle-\frac{1}{2}\bar{\kappa}a(\epsilon p_{0})\nabla\left\\{\bar{\kappa}a(\epsilon p_{0}){\rm div}(b(\theta_{0})\nabla\tilde{\theta})\right\\}+a(\epsilon p_{0})[\bar{\nu}\,{\rm curl\,}\mathbf{H}_{0}:{\rm curl\,}\tilde{\mathbf{H}}]$ $\displaystyle+\frac{1}{2}a(\epsilon p_{0})[\Psi({\mathbf{u}}_{0}):\nabla\tilde{\mathbf{u}}]+\frac{1}{2}a(\epsilon p_{0})[\Psi({\mathbf{u}}_{0}):\nabla(\bar{\kappa}a(\epsilon p_{0})b(\theta_{0})\nabla\tilde{\theta})]$ $\displaystyle-\frac{1}{2}\bar{\kappa}a(\epsilon p_{0})[\nabla,{\mathbf{u}}_{0}]\cdot\nabla\tilde{\theta}+a(\epsilon p_{0})[({\rm curl\,}\mathbf{H}_{0})\times\tilde{\mathbf{H}}]$ $\displaystyle+\frac{1}{2}a(\epsilon p_{0}){\rm div}\Psi(\bar{\kappa}a(\epsilon p_{0})b(\theta_{0})\nabla\tilde{\theta})+\frac{1}{2}a(\epsilon p_{0}){\rm div}\Psi(\tilde{\mathbf{u}})-\frac{1}{2}\bar{\kappa}a(\epsilon p_{0})\nabla f_{4}+f_{2}$ $\displaystyle:=$ $\displaystyle\sum^{14}_{i=1}h_{i}+\frac{1}{2}a(\epsilon p_{0}){\rm div}\Psi(\tilde{\mathbf{u}})-\frac{1}{2}\bar{\kappa}a(\epsilon p_{0})\nabla f_{4}+f_{2}.$ (3.38) Multiplying (3.34) by $\tilde{p}$, (3.35) by $\tilde{\mathbf{H}}$, and (3.3) by $\tilde{\mathbf{u}}$ in $L^{2}(\mathbb{R}^{3})$ respectively, and summing up the resulting equations, we deduce that $\displaystyle\frac{d}{dt}$ $\displaystyle\Big{\\{}\frac{1}{2}\langle\tilde{p},\tilde{p}\rangle+\frac{1}{4}\langle b(-\theta_{0})\tilde{\mathbf{u}},\tilde{\mathbf{u}}\rangle+\frac{1}{2}\langle\tilde{\mathbf{H}},\mathbf{H}\rangle\Big{\\}}+\bar{\nu}\\|\nabla\tilde{\mathbf{H}}\\|^{2}_{L^{2}}$ $\displaystyle=$ $\displaystyle-\langle({\mathbf{u}}_{0}\cdot\nabla)\tilde{p},\tilde{p}\rangle+\frac{1}{4}\langle\partial_{t}b(-\theta_{0})\tilde{\mathbf{u}},\tilde{\mathbf{u}}\rangle-\frac{1}{2}\langle b(-\theta_{0})({\mathbf{u}}_{0}\cdot\nabla)\tilde{\mathbf{u}},\tilde{\mathbf{u}}\rangle$ $\displaystyle+\langle\epsilon a(\epsilon p_{0})[\bar{\nu}\,{\rm curl\,}\mathbf{H}_{0}:{\rm curl\,}\tilde{\mathbf{H}}],\tilde{p}\rangle+\frac{\epsilon}{2}\langle a(\epsilon p_{0})\Psi({\mathbf{u}}_{0}):\nabla\tilde{\mathbf{u}},\tilde{p}\rangle$ $\displaystyle+\frac{\epsilon}{2}\langle a^{2}(\epsilon p_{0})b(\theta_{0})\Psi({\mathbf{u}}_{0}):\nabla(\nabla\tilde{\theta}),\tilde{p}\rangle+\langle\bar{\kappa}a(\epsilon p_{0})b(\theta_{0})\nabla p_{0}\cdot\nabla\tilde{\theta},\tilde{p}\rangle$ $\displaystyle+\sum^{14}_{i=1}\left\langle h_{i},\tilde{\mathbf{u}}\right\rangle+\frac{1}{2}\langle a(\epsilon p_{0}){\rm div}\Psi(\tilde{\mathbf{u}}),\tilde{\mathbf{u}}\rangle$ $\displaystyle-\frac{1}{2}\left\langle\bar{\kappa}a(\epsilon p_{0})\nabla f_{4},\tilde{\mathbf{u}}\right\rangle+\left\langle f_{2},\tilde{\mathbf{u}}\right\rangle+\langle f_{3},\tilde{\mathbf{H}}\rangle+\langle f_{1},\tilde{p}\rangle,$ (3.39) where the singular terms have been canceled out. Now, the terms on the right-hand side of (3.3) can be estimated as follows. First, it follows from the regularity of $(p_{0},{\mathbf{u}}_{0},\mathbf{H}_{0},\theta_{0})$, a partial integration and Cauchy-Schwarz’s inequality that $\displaystyle\frac{1}{4}\left\|\langle\partial_{t}b(-\theta_{0})\tilde{\mathbf{u}},\tilde{\mathbf{u}}\rangle\right\|\leq\frac{1}{4}\\|\partial_{t}b(-\theta_{0})\\|_{L^{\infty}}\\|\tilde{\mathbf{u}}\\|^{2}_{L^{2}}\leq C(R_{0})\\|\tilde{\mathbf{u}}\\|^{2}_{L^{2}},$ $\displaystyle\|\langle({\mathbf{u}}_{0}\cdot\nabla)\tilde{p},\tilde{p}\rangle\|=\frac{1}{2}\left\|\int({\rm div}{\mathbf{u}}_{0})\|\tilde{p}\|^{2}dx\right\|\leq C(R_{0})\\|\tilde{p}\\|_{L^{2}}^{2},$ $\displaystyle\frac{1}{2}\|\langle b(-\theta_{0})({\mathbf{u}}_{0}\cdot\nabla)\tilde{\mathbf{u}},\tilde{\mathbf{u}}\rangle\|\leq C(R_{0})\\|\tilde{\mathbf{u}}\\|_{L^{2}}^{2},$ $\displaystyle\|\langle\epsilon a(\epsilon p_{0})[\bar{\nu}\,{\rm curl\,}\mathbf{H}_{0}:{\rm curl\,}\tilde{\mathbf{H}}],\tilde{p}\rangle\|\leq C(R_{0})(\\|\epsilon\nabla\tilde{\mathbf{H}}\\|^{2}_{L^{2}}+\\|\tilde{p}\\|^{2}_{L^{2}}),$ $\displaystyle\frac{\epsilon}{2}\|\langle a(\epsilon p_{0})\Psi({\mathbf{u}}_{0}):\nabla\tilde{\mathbf{u}},\tilde{p}\rangle\|\leq C(R_{0})(\\|\epsilon\nabla\tilde{\mathbf{u}}\\|^{2}_{L^{2}}+\\|\tilde{p}\\|^{2}_{L^{2}}),$ $\displaystyle\frac{\epsilon}{2}\|\langle a^{2}(\epsilon p_{0})b(\theta_{0})\Psi({\mathbf{u}}_{0}):\nabla(\nabla\tilde{\theta}),\tilde{p}\rangle\|\leq C(R_{0})\\|\tilde{p}\\|^{2}_{L^{2}}$ $\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad+G_{1}(\epsilon p_{0},\theta_{0})\sum_{\|\alpha\|=2}\\|\partial^{\alpha}(\epsilon\tilde{\theta})\\|^{2}_{L^{2}},$ $\displaystyle\|\langle\bar{\kappa}a(\epsilon p_{0})b(\theta_{0})\nabla p_{0}\cdot\nabla\tilde{\theta},\tilde{p}\rangle\|\leq C(R_{0})(\\|\nabla\tilde{\theta}\\|^{2}_{L^{2}}+\\|\tilde{p}\\|^{2}_{L^{2}}),$ where $G_{1}(\cdot,\cdot)$ is a smooth function. Similarly, one can bound the terms involving $h_{i}$ in (3.3) as follows. $\displaystyle\sum^{14}_{i=1}\|\langle h_{i},\tilde{\mathbf{u}}\rangle\|\leq$ $\displaystyle\frac{\bar{\nu}}{8}\\|\nabla\tilde{\mathbf{H}}\\|^{2}_{L_{2}}+\frac{\underline{a}\bar{\mu}}{8}\\|\nabla\tilde{\mathbf{u}}\\|^{2}_{L_{2}}+\frac{\underline{a}\bar{\nu}}{8}\\|{\rm div}\tilde{\mathbf{u}}\\|^{2}_{L_{2}}$ $\displaystyle+C(R_{0})\\|\tilde{\mathbf{u}}\\|^{2}_{L_{2}}+C(R_{0})\\|\nabla(\epsilon\tilde{\mathbf{u}},\tilde{\theta})\\|^{2}_{L_{2}}+G_{2}(\epsilon p_{0},\theta_{0})\\|\Delta\tilde{\theta}\\|^{2}_{L_{2}},$ where $G_{2}(\cdot,\cdot)$ is a smooth function. For the dissipative term $\frac{1}{2}\langle a(\epsilon p_{0}){\rm div}\Psi(\tilde{\mathbf{u}}),\tilde{\mathbf{u}}\rangle$, we can employ arguments similar to those used in the estimate of the slow motion in (3.2)–(3.17) to obtain that $\displaystyle-\frac{1}{2}\langle a(\epsilon p_{0}){\rm div}\Psi(\hat{\mathbf{u}}),\hat{\mathbf{u}}\rangle\geq\frac{\underline{a}\bar{\mu}}{4}(\\|\nabla\hat{\mathbf{u}}\\|^{2}_{L^{2}}+\\|{\rm div}\hat{\mathbf{u}}\\|^{2}_{L^{2}})-C(R_{0})\\|\hat{\mathbf{u}}\\|^{2}_{L^{2}}.$ Finally, putting all estimates above into (3.3) and applying Cauchy-Schwarz’s and Gronwall’s inequalities, we get (3.5). ∎ In the next lemma we utilize Lemma 3.5 to control $\big{(}(\epsilon\partial_{t})p,(\epsilon\partial_{t}){\mathbf{u}},(\epsilon\partial_{t})\mathbf{H}\big{)}$. ###### Lemma 3.6. Let $s\geq 4$ and $(p,{\mathbf{u}},\mathbf{H},\theta)$ be the solution to the Cauchy problem (1.11)–(1.14), (1.19) on $[0,T_{1}]$. Set $(p_{\beta},{\mathbf{u}}_{\beta},\mathbf{H}_{\beta},\theta_{\beta}):=\partial^{\beta}\big{(}(\epsilon\partial_{t})p,(\epsilon\partial_{t}){\mathbf{u}},(\epsilon\partial_{t})\mathbf{H},(\epsilon\partial_{t})\theta\big{)},$ where $1\leq\|\beta\|\leq s-1$. Then there exist a constant $l_{5}>0$ and an increasing function $C(\cdot)$ such that, for any $\epsilon\in(0,1]$ and $t\in[0,T]$, $T=\min\\{T_{1},1\\}$, it holds that $\displaystyle\sup_{\tau\in[0,t]}\\|(p_{\beta},{\mathbf{u}}_{\beta},\mathbf{H}_{\beta})(\tau)\\|^{2}_{L^{2}}+$ $\displaystyle l_{5}\int^{t}_{0}\\|\nabla({\mathbf{u}}_{\beta},\mathbf{H}_{\beta})\\|_{L^{2}}^{2}(\tau)d\tau$ $\displaystyle\leq C(\mathcal{O}_{0})\exp\\{(\sqrt{T})C(\mathcal{O}(T))\\}.$ (3.40) ###### Proof. An application of the operator $\partial^{\beta}(\epsilon\partial_{t})$ to the system (1.11)–(1.14) leads to $\displaystyle\partial_{t}p_{\beta}+({\mathbf{u}}\cdot\nabla)p_{\beta}+\frac{1}{\epsilon}{\rm div}(2{\mathbf{u}}_{\beta}-\bar{\kappa}a(\epsilon p)b(\theta)\nabla\theta_{\beta})=\epsilon a(\epsilon p)[\bar{\nu}\,{\rm curl\,}\mathbf{H}:{\rm curl\,}\mathbf{H}_{\beta}]$ $\displaystyle\quad\quad\qquad+\epsilon a(\epsilon p)\Psi({\mathbf{u}}):\nabla{\mathbf{u}}_{\beta}+\bar{\kappa}a(\epsilon p)b(\theta)\nabla p\cdot\nabla\theta_{\beta}+\tilde{g}_{1},$ (3.41) $\displaystyle b(-\theta)[\partial_{t}{\mathbf{u}}_{\beta}+({\mathbf{u}}\cdot\nabla){\mathbf{u}}_{\beta}]+\frac{\nabla p_{\beta}}{\epsilon}=a(\epsilon p)[({\rm curl\,}\mathbf{H})\times\mathbf{H}_{\beta}+{\rm div}\Psi({\mathbf{u}}_{\beta})]+\tilde{g}_{2},$ (3.42) $\displaystyle\partial_{t}\mathbf{H}_{\beta}-{\rm curl\,}({\mathbf{u}}\times\mathbf{H}_{\beta})-\bar{\nu}\Delta\mathbf{H}_{\beta}=\tilde{g}_{3},\quad{\rm div}\mathbf{H}_{\beta}=0,$ (3.43) $\displaystyle\partial_{t}\theta_{\beta}+({\mathbf{u}}\cdot\nabla)\theta_{\beta}+{\rm div}{\mathbf{u}}_{\beta}=\epsilon^{2}a(\epsilon p)[\bar{\nu}\,{\rm curl\,}\mathbf{H}:{\rm curl\,}\mathbf{H}_{\beta}]$ $\displaystyle\quad\quad\qquad+\epsilon^{2}a(\epsilon p)\Psi({\mathbf{u}}):\nabla{\mathbf{u}}_{\beta}+\bar{\kappa}a(\epsilon p){\rm div}(b(\theta)\nabla\theta_{\beta})+\tilde{g}_{4},$ (3.44) where $\displaystyle\tilde{g}_{1}:=$ $\displaystyle-[\partial^{\beta}(\epsilon\partial_{t}),{\mathbf{u}}]\cdot\nabla p+\frac{1}{\epsilon}[\partial^{\beta}(\epsilon\partial_{t}),(\bar{\kappa}a(\epsilon p)b(\theta))]\Delta\theta$ $\displaystyle+\frac{1}{\epsilon}[\partial^{\beta}(\epsilon\partial_{t}),\nabla(\bar{\kappa}a(\epsilon p)b(\theta))]\cdot\nabla\theta+\epsilon\bar{\nu}\,[\partial^{\beta}(\epsilon\partial_{t}),a(\epsilon p){\rm curl\,}\mathbf{H}]:{\rm curl\,}\mathbf{H}$ $\displaystyle+\epsilon[\partial^{\beta}(\epsilon\partial_{t}),a(\epsilon p)\Psi({\mathbf{u}})]:\nabla{\mathbf{u}}+[\partial^{\beta}(\epsilon\partial_{t}),\bar{\kappa}a(\epsilon p)b(\theta)\nabla p]\cdot\nabla\theta,$ $\displaystyle\tilde{g}_{2}:=$ $\displaystyle-[\partial^{\beta}(\epsilon\partial_{t}),b(-\theta)]\partial_{t}{\mathbf{u}}-[\partial^{\beta}(\epsilon\partial_{t}),b(-\theta){\mathbf{u}}]\cdot\nabla{\mathbf{u}},$ $\displaystyle+[\partial^{\beta}(\epsilon\partial_{t}),a(\epsilon p){\rm curl\,}\mathbf{H}]\times\mathbf{H}+[\partial^{\beta}(\epsilon\partial_{t}),a(\epsilon p)]{\rm div}\Psi({\mathbf{u}}),$ $\displaystyle\tilde{g}_{3}:=$ $\displaystyle\,\partial^{\beta}(\epsilon\partial_{t})\big{(}{\rm curl\,}({\mathbf{u}}\times\mathbf{H})\big{)}-{\rm curl\,}({\mathbf{u}}\times\mathbf{H}_{\beta}),$ $\displaystyle\tilde{g}_{4}:=$ $\displaystyle-[\partial^{\beta}(\epsilon\partial_{t}),{\mathbf{u}}]\cdot\nabla\theta+\epsilon^{2}\bar{\nu}\,[\partial^{\beta}(\epsilon\partial_{t}),a(\epsilon p){\rm curl\,}\mathbf{H}]:{\rm curl\,}\mathbf{H}$ $\displaystyle+\epsilon^{2}[\partial^{\beta}(\epsilon\partial_{t}),a(\epsilon p)\Psi({\mathbf{u}})]:\nabla{\mathbf{u}}$ $\displaystyle+\bar{\kappa}\partial^{\beta}(\epsilon\partial_{t})\big{(}a(\epsilon p){\rm div}(b(\theta)\nabla\theta_{\alpha})\big{)}-\bar{\kappa}a(\epsilon){\rm div}(b(\theta)\nabla\theta_{\alpha}).$ It follows from the linear estimate (3.5) that for some $l_{4}>0$, $\displaystyle\sup_{\tau\in[0,t]}\\|(p_{\beta},{\mathbf{u}}_{\beta},\mathbf{H}_{\beta})(\tau)\\|^{2}_{L^{2}}+l_{4}\int^{t}_{0}\\|\nabla({\mathbf{u}}_{\beta},\mathbf{H}_{\beta})\\|_{L^{2}}^{2}(\tau)d\tau$ $\displaystyle\leq e^{TC(R)}\\|(p_{\beta},\tilde{\mathbf{u}}_{\beta},\mathbf{H}_{\beta})(0)\\|^{2}_{L^{2}}+C(R)e^{TC(R)}\sup_{\tau\in[0,T]}\\|\nabla\theta_{\beta}(\tau)\\|^{2}_{L^{2}}$ $\displaystyle\quad+TC(R)\sup_{\tau\in[0,T]}\\|\nabla(\epsilon{\mathbf{u}}_{\beta},\epsilon\mathbf{H}_{\beta})(\tau)\\|^{2}_{L^{2}}+C(R)\int^{T}_{0}\\|\nabla\theta_{\beta}\\|^{2}_{H^{1}}(\tau)d\tau$ $\displaystyle\quad+C(R)\int^{T}_{0}\left\\{\\|\tilde{g}_{1}\\|^{2}_{L^{2}}+\\|\tilde{g}_{2}\\|^{2}_{L^{2}}+\\|\tilde{g}_{3}\\|^{2}_{L^{2}}+\\|\nabla\tilde{g}_{4}\\|_{L^{2}}^{2}\right\\}(\tau)d\tau,$ (3.45) where $R$ is defined as $R_{0}$ in (3.33) with $(p_{0},{\mathbf{u}}_{0},\mathbf{H}_{0},\theta_{0})$ replaced with $(p,{\mathbf{u}},\mathbf{H},\theta)$. It remains to control the terms $\\|\tilde{g}_{1}\\|_{L^{2}}^{2}$, $\\|\tilde{g}_{2}\\|_{L^{2}}^{2}$, $\\|\tilde{g}_{3}\\|_{L^{2}}^{2}$, and $\\|\nabla\tilde{g}_{4}\\|_{L^{2}}^{2}$. The first term of $\tilde{g}_{1}$ can be bounded as follows. $\displaystyle\left\\|[\partial^{\beta}(\epsilon\partial_{t}),{\mathbf{u}}]\cdot\nabla p\right\\|_{L^{2}}\leq$ $\displaystyle\epsilon C_{0}(\\|{\mathbf{u}}\\|_{H^{s-1}}\\|(\epsilon\partial_{t})\nabla p\\|_{H^{s-2}}+\\|(\epsilon\partial_{t}){\mathbf{u}}\\|_{H^{s-1}}\\|\nabla p\\|_{H^{s-1}})$ $\displaystyle\leq$ $\displaystyle C(\mathcal{Q}).$ Similarly, the second term of $\tilde{g}_{1}$ admits the following boundedness: $\displaystyle\frac{1}{\epsilon}\\|[\partial^{\beta}(\epsilon\partial_{t}),(\bar{\kappa}a(\epsilon p)b(\theta))]\Delta\theta\\|$ $\displaystyle\leq C_{0}\big{(}\\|a(\epsilon p)b(\theta)\\|_{H^{s-1}}\\|\partial_{t}\Delta\theta\\|_{H^{s-2}}+\\|\partial_{t}(a(\epsilon p)b(\theta))\\|_{H^{s-1}}\\|\Delta\theta\\|_{H^{s-1}}\big{)}$ $\displaystyle\leq C(\mathcal{Q})(1+\mathcal{S}).$ The other four terms in $\tilde{g}_{1}$ can be treated similarly and hence can be bounded from above by $C(\mathcal{Q})(1+\mathcal{S})$. For the first term of $\tilde{g}_{2}$, one has by the equation (1.12) that $\displaystyle[\partial^{\beta}(\epsilon\partial_{t}),b(-\theta)]\partial_{t}{\mathbf{u}}=$ $\displaystyle[\partial^{\beta}(\epsilon\partial_{t}),b(-\theta)]\\{({\mathbf{u}}\cdot\nabla){\mathbf{u}}\\}$ $\displaystyle+\frac{1}{\epsilon}[\partial^{\beta}(\epsilon\partial_{t}),b(-\theta)]\\{b^{-1}(-\theta_{0})\nabla p\\}$ $\displaystyle-[\partial^{\beta}(\epsilon\partial_{t}),b(-\theta)]\\{b^{-1}(-\theta)a(\epsilon p)[({\rm curl\,}\mathbf{H}_{0})\times\mathbf{H}]\\}$ $\displaystyle-[\partial^{\beta}(\epsilon\partial_{t}),b(-\theta)]\\{b^{-1}(-\theta)a(\epsilon p){\rm div}\Psi({\mathbf{u}})\\}.$ (3.46) Note that the terms on the right-hand side of (3.3) have similar structure as that of $\tilde{g}_{1}$. Thus, we see that $\displaystyle\\|[\partial^{\beta}(\epsilon\partial_{t}),b(-\theta)]\partial_{t}{\mathbf{u}}\\|_{L^{2}}\leq C(\mathcal{Q})(1+\mathcal{S}).$ Similarly, the other four terms of $\tilde{g}_{2}$ can be bounded from above by $C(\mathcal{Q})(1+\mathcal{S})$. Next, by the identity (2.4), one can rewrite $\tilde{g}_{3}$ as $\displaystyle\tilde{g}_{3}=$ $\displaystyle-[\partial^{\beta}(\epsilon\partial_{t}),{\rm div}{\mathbf{u}}]\mathbf{H}-[\partial^{\beta}(\epsilon\partial_{t}),{\mathbf{u}}]\cdot\nabla\mathbf{H}+\sum_{i=1}^{3}[\partial^{\beta}(\epsilon\partial_{t}),\nabla{\mathbf{u}}_{i}]\mathbf{H}.$ Following a process similar to that in the estimate of $\tilde{g}_{1}$, one gets $\displaystyle\\|\tilde{g}_{3}\\|_{L^{2}}\leq C(\mathcal{Q})(1+\mathcal{S}).$ And analogously, $\displaystyle\\|\nabla\tilde{g}_{4}\\|_{L^{2}}\leq C(\mathcal{Q})(1+\mathcal{S}).$ We proceed to control the other terms on the right-hand side of (3.3). It follows from (3.26) that $C(R)e^{TC(R)}\sup_{\tau\in[0,T]}\\|\nabla\theta_{\beta}(\tau)\\|^{2}_{L^{2}}\leq C(\mathcal{O}(T))\exp\\{\sqrt{T}C(\mathcal{O}(T)\\}$ and $\int^{T}_{0}\\|\Delta\theta_{\beta}\\|^{2}_{L_{2}}(\tau)d\tau\leq\int^{T}_{0}\\|(\epsilon\partial_{t})\theta\\|^{2}_{H^{s+1}}(\tau)d\tau\leq C(\mathcal{O}_{0})\exp\\{\sqrt{T}C(\mathcal{O}(T))\\}.$ Thanks to (3.25), one has $\displaystyle TC(R)\sup_{\tau\in[0,T]}\\|\nabla(\epsilon{\mathbf{u}}_{\beta},\epsilon\mathbf{H}_{\beta})(\tau)\\|^{2}_{L^{2}}$ $\displaystyle\leq TC(\mathcal{O}(T))\sup_{\tau\in[0,T]}\\|(\epsilon\partial_{t})(\epsilon{\mathbf{u}},\epsilon\mathbf{H})(\tau)\\|^{2}_{H^{s}}$ $\displaystyle\leq TC(\mathcal{O}(T)).$ Then, the desired inequality (3.6) follows from the above estimates and the inequality (3.3). ∎ Now we are in a position to estimate the Sobolev norm of $({\rm div}{\mathbf{u}},\nabla p)$ based on Lemma 3.6. ###### Lemma 3.7. Let $s\geq 4$ and $(p,{\mathbf{u}},\mathbf{H},\theta)$ be the solution to the Cauchy problem (1.11)–(1.14), (1.19) on $[0,T_{1}]$. Then there exist a constant $l_{6}>0$ and an increasing function $C(\cdot)$ such that, for any $\epsilon\in(0,1]$ and $t\in[0,T_{1}]$, $T=\min\\{T_{1},1\\}$, it holds that $\displaystyle\sup_{\tau\in[0,t]}\left\\{\\|p(\tau)\\|_{H^{s}}+\\|{\rm div}{\mathbf{u}}(\tau)\\|_{H^{s-1}}\right\\}$ $\displaystyle+l_{6}\int^{t}_{0}\left\\{\\|\nabla p\\|^{2}_{H^{s}}+\\|\nabla{\rm div}{\mathbf{u}}\\|^{2}_{H^{s-1}}\right\\}(\tau)d\tau$ $\displaystyle\leq C(\mathcal{O}_{0})\exp\big{\\{}(\sqrt{T}+\epsilon)C(\mathcal{O}(T))\big{\\}}.$ (3.47) ###### Proof. Rewrite the equations (1.11) and (1.12) as $\displaystyle{\rm div}{\mathbf{u}}=$ $\displaystyle-\frac{1}{2}(\epsilon\partial_{t})p-\frac{\epsilon}{2}({\mathbf{u}}\cdot\nabla)p+\frac{1}{2}{\rm div}(\bar{\kappa}a(\epsilon p)b(\theta)\nabla\theta)+\frac{\epsilon^{2}\bar{\nu}}{2}a(\epsilon p_{0})\|{\rm curl\,}\mathbf{H}\|^{2}$ $\displaystyle+\frac{\epsilon^{2}}{2}a(\epsilon p)\Psi({\mathbf{u}}):\nabla{\mathbf{u}}+\frac{\epsilon\bar{\kappa}}{2}a(\epsilon p)b(\theta)\nabla p\cdot\nabla\theta,$ (3.48) $\displaystyle{\nabla p}=$ $\displaystyle-b(-\theta)(\epsilon\partial_{t}){\mathbf{u}}-{\epsilon}b(-\theta)({\mathbf{u}}\cdot\nabla){\mathbf{u}}$ $\displaystyle+\epsilon a(\epsilon p)[({\rm curl\,}\mathbf{H})\times\mathbf{H}]+\epsilon a(\epsilon p){\rm div}\Psi({\mathbf{u}}).$ (3.49) Then, $\displaystyle\\|{\rm div}{\mathbf{u}}\\|_{H^{s-1}}\leq$ $\displaystyle C_{0}\\|(\epsilon\partial_{t})p\\|_{H^{s-1}}+C_{0}\epsilon\,\\|{\mathbf{u}}\\|_{H^{s-1}}\\|\nabla p\\|_{H^{s-1}}$ $\displaystyle+C_{0}\\|{\rm div}(\bar{\kappa}a(\epsilon p)b(\theta)\nabla\theta)\\|_{H^{s-1}}+C_{0}\\|a(\epsilon p_{0})\\|_{L^{\infty}}\\|\epsilon\,{\rm curl\,}\mathbf{H}\\|_{H^{s-1}}^{2}$ $\displaystyle+C_{0}\\|a(\epsilon p)\\|_{L^{\infty}}\\|\Psi(\epsilon{\mathbf{u}}):(\epsilon\nabla{\mathbf{u}})\\|_{H^{s-1}}$ $\displaystyle+C_{0}\\|a(\epsilon p)b(\theta)\\|_{L^{\infty}}\\|(\epsilon\nabla p)\\|_{H^{s-1}}\\|\nabla\theta\\|_{H^{s-1}}.$ (3.50) It follows from Lemmas 3.2–3.4 and 3.6, and the inequalities (3.23)–(3.26) that $\displaystyle\\|(\epsilon\partial_{t})p\\|_{H^{s-1}}\leq$ $\displaystyle C(\mathcal{O}_{0})\exp\\{(\sqrt{T}+\epsilon)C(\mathcal{O}(T))\\},$ $\displaystyle\epsilon\,\\|{\mathbf{u}}\\|_{H^{s-1}}\\|\nabla p\\|_{H^{s-1}}\leq$ $\displaystyle\epsilon C(\mathcal{O}),$ $\displaystyle\\|{\rm div}(\bar{\kappa}a(\epsilon p)b(\theta)\nabla\theta)\\|_{H^{s-1}}\leq$ $\displaystyle C_{0}\\|\Delta\theta\\|_{H^{s-1}}+C_{0}\\|\nabla\theta\\|_{H^{s-1}}$ $\displaystyle\leq$ $\displaystyle C(\mathcal{O}_{0})\exp\\{(\sqrt{T}+\epsilon)C(\mathcal{O}(T))\\},$ $\displaystyle\\|\epsilon\,{\rm curl\,}\mathbf{H}\\|_{H^{s-1}}\leq$ $\displaystyle C(\mathcal{O}_{0})\exp\\{(\sqrt{T}+\epsilon)C(\mathcal{O}(T))\\},$ $\displaystyle\\|\Psi(\epsilon{\mathbf{u}}):(\epsilon\nabla{\mathbf{u}})\\|_{H^{s-1}}\leq$ $\displaystyle C(\mathcal{O}_{0})\exp\\{(\sqrt{T}+\epsilon)C(\mathcal{O}(T))\\},$ $\displaystyle\\|(\epsilon\nabla p)\\|_{H^{s-1}}\\|\nabla\theta\\|_{H^{s-1}}\leq$ $\displaystyle C(\mathcal{O}_{0})\exp\\{(\sqrt{T}+\epsilon)C(\mathcal{O}(T))\\}.$ These bounds together with (3.50) imply that $\displaystyle\sup_{\tau\in[0,t]}\\|{\rm div}{\mathbf{u}}\\|(\tau)\\|_{H^{s-1}}\leq C(\mathcal{O}_{0})\exp\\{(\sqrt{T}+\epsilon)C(\mathcal{O}(T))\\}.$ Similar arguments applying to the equation (3.49) for $\nabla p$ yield $\displaystyle\sup_{\tau\in[0,t]}\\|p\\|_{H^{s}}+l_{6}\int^{t}_{0}\\|\nabla p\\|^{2}_{H^{s}}(\tau)d\tau\leq C(\mathcal{O}_{0})\exp\\{(\sqrt{T}+\epsilon)C(\mathcal{O}(T))\\}$ (3.51) for some positive constant $l_{6}>0$. To obtain the desired inequality (3.7), we shall establish the following estimate $\displaystyle\int^{T}_{0}\\|\nabla{\rm div}{\mathbf{u}}\\|^{2}_{H^{s-1}}(\tau)d\tau\leq C(\mathcal{O}_{0})\exp\\{(\sqrt{T}+\epsilon)C(\mathcal{O}(T))\\}.$ (3.52) In fact, for any multi-index $\alpha$ satisfying $1\leq\|\alpha\|\leq s$, one can apply the operator $\partial^{\alpha}$ to (3.48) and then take the inner product with $\partial^{\alpha}{\rm div}{\mathbf{u}}$ to obtain $\displaystyle\int^{T}_{0}\\|\partial^{\alpha}{\rm div}{\mathbf{u}}\\|^{2}_{L^{2}}(\tau)d\tau=$ $\displaystyle-\frac{1}{2}\int^{T}_{0}\langle\partial^{\alpha}(\epsilon\partial_{t})p,\partial^{\alpha}{\rm div}{\mathbf{u}}\rangle(\tau)d\tau$ $\displaystyle+\int^{T}_{0}\langle\Xi,\partial^{\alpha}{\rm div}{\mathbf{u}}\rangle(\tau)d\tau,$ (3.53) where $\displaystyle\Xi:=$ $\displaystyle-\frac{\epsilon}{2}({\mathbf{u}}\cdot\nabla)p+\frac{1}{2}{\rm div}(\bar{\kappa}a(\epsilon p)b(\theta)\nabla\theta)+\frac{\epsilon^{2}\bar{\nu}}{2}a(\epsilon p_{0})\|{\rm curl\,}\mathbf{H}\|^{2}$ $\displaystyle+\frac{\epsilon^{2}}{2}a(\epsilon p)\Psi({\mathbf{u}}):\nabla{\mathbf{u}}+\frac{\epsilon\bar{\kappa}}{2}a(\epsilon p)b(\theta)\nabla p\cdot\nabla\theta.$ It thus follows from (3.6) and similar arguments to those for (3.51), that for all $1\leq\|\alpha\|\leq s$, $\displaystyle\int^{T}_{0}\\|\partial^{\alpha}\Xi(\tau)\\|^{2}_{L^{2}}d\tau\leq C(\mathcal{O}_{0})\exp\\{(\sqrt{T}+\epsilon)C(\mathcal{O}(T))\\},$ whence, $\displaystyle\int^{T}_{0}\left\|\langle\Xi,\partial^{\alpha}{\rm div}{\mathbf{u}}\rangle\right\|(\tau)d\tau$ $\displaystyle\leq C(\mathcal{O}_{0})\exp\\{(\sqrt{T}+\epsilon)C(\mathcal{O}(T))\\}\left\\{\int^{T}_{0}\\|\partial^{\alpha}{\rm div}{\mathbf{u}}\\|^{2}_{L^{2}}(\tau)d\tau\right\\}^{1/2}$ $\displaystyle\leq C(\mathcal{O}_{0})\exp\\{(\sqrt{T}+\epsilon)C(\mathcal{O}(T))\\}+\frac{1}{4}\int^{T}_{0}\\|\partial^{\alpha}{\rm div}{\mathbf{u}}\\|^{2}_{L^{2}}(\tau)d\tau.$ For the first term on the right-hand side of (3.3), one gets by integration by parts that $\displaystyle-\frac{1}{2}\int^{T}_{0}\langle\partial^{\alpha}(\epsilon\partial_{t})p,\partial^{\alpha}{\rm div}{\mathbf{u}}\rangle(\tau)d\tau=$ $\displaystyle-\frac{1}{2}\langle\partial^{\alpha}p,\epsilon\partial^{\alpha}{\rm div}(\epsilon{\mathbf{u}})\rangle\Big{\|}^{T}_{0}$ $\displaystyle+\frac{1}{2}\int^{T}_{0}\langle\partial^{\alpha}\nabla p,\partial^{\alpha}(\epsilon\partial_{t}){\mathbf{u}}\rangle(\tau)d\tau.$ By virtue of the estimate (3.3) on $(\epsilon q,\epsilon{\mathbf{u}},\theta-\bar{\theta})$ and (3.51), we find that $\displaystyle\left\|\frac{1}{2}\langle\partial^{\alpha}p,\epsilon\partial^{\alpha}{\rm div}(\epsilon{\mathbf{u}})\rangle\Big{\|}^{T}_{0}\right\|\leq$ $\displaystyle\sup_{\tau\in[0,T]}\\{\\|p\\|_{H^{s}}\\|\epsilon{\mathbf{u}}\\|_{H^{s+1}}\\}$ $\displaystyle\leq$ $\displaystyle C(\mathcal{O}_{0})\exp\\{(\sqrt{T}+\epsilon)C(\mathcal{O}(T))\\},$ $\displaystyle\frac{1}{2}\bigg{\|}\int^{T}_{0}\langle\partial^{\alpha}\nabla p,\partial^{\alpha}(\epsilon\partial_{t}){\mathbf{u}}\rangle(\tau)d\tau\bigg{\|}\leq$ $\displaystyle\frac{1}{2}\bigg{\\{}\int^{T}_{0}\\|\partial^{\alpha}\nabla p\\|^{2}_{L^{2}}(\tau)d\tau\bigg{\\}}^{1/2}$ $\displaystyle\times\bigg{\\{}\int^{T}_{0}\\|\partial^{\alpha}(\epsilon\partial_{t}){\mathbf{u}}\\|^{2}_{L^{2}}(\tau)d\tau\bigg{\\}}^{1/2}$ $\displaystyle\leq$ $\displaystyle C(\mathcal{O}_{0})\exp\\{(\sqrt{T}+\epsilon)C(\mathcal{O}(T))\\}.$ These bounds, together with (3.3), yield the desired estimate (3.52). This completes the proof. ∎ ### 3.4. $H^{s-1}$-estimate on ${\rm curl\,}{\mathbf{u}}$ The another key point to obtain a uniform bound for ${\mathbf{u}}$ is the following estimate on ${\rm curl\,}{\mathbf{u}}$. ###### Lemma 3.8. Let $s>5/2$ be an integer and $(p,{\mathbf{u}},\mathbf{H},\theta)$ be the solution to the Cauchy problem (1.11)–(1.14), (1.19) on $[0,T_{1}]$. Then there exist a constant $l_{7}>0$ and an increasing function $C(\cdot)$ such that, for any $\epsilon\in(0,1]$ and $t\in[0,T_{1}]$, $T=\min\\{T_{1},1\\}$, it holds that $\displaystyle\sup_{\tau\in[0,t]}\left\\{\\|{\rm curl\,}(b({-\theta}){\mathbf{u}})(\tau)\\|^{2}_{H^{s-1}}+\\|{\rm curl\,}\mathbf{H}(\tau)\\|^{2}_{H^{s-1}}\right\\}$ $\displaystyle+l_{7}\int^{t}_{0}\left\\{\\|\nabla{\rm curl\,}(b({-\theta}){\mathbf{u}})\\|^{2}_{H^{s-1}}+\\|\nabla{\rm curl\,}\mathbf{H}(\tau)\\|^{2}_{H^{s-1}}\right\\}(\tau)d\tau$ $\displaystyle\leq$ $\displaystyle C(\mathcal{O}_{0})\exp\\{(\sqrt{T}+\epsilon)C(\mathcal{O}(T))\\}.$ (3.54) ###### Proof. Applying the operator _curl_ to the equations (1.12) and (1.13), using the identities (2.1) and (2.2), and the fact that ${\rm curl\,}\nabla=0$, one infers $\displaystyle\partial_{t}({\rm curl\,}(b(-\theta){\mathbf{u}}))+({\mathbf{u}}\cdot\nabla)({\rm curl\,}(b(-\theta){\mathbf{u}}))$ $\displaystyle\quad\ \ ={\rm curl\,}\\{a(\epsilon p)({\rm curl\,}\mathbf{H})\times\mathbf{H}\\}+\bar{\mu}{\rm div}\\{a(\epsilon p)b(\theta)\nabla({\rm curl\,}(b(-\theta){\mathbf{u}}))\\}+\Upsilon_{1},$ (3.55) $\displaystyle\partial_{t}({\rm curl\,}\mathbf{H})-{\rm curl\,}[{\rm curl\,}({\mathbf{u}}\times\mathbf{H})]-\bar{\nu}\Delta({\rm curl\,}\mathbf{H})=0,$ (3.56) where $\Upsilon_{1}$ is defined by $\displaystyle\Upsilon_{1}:=$ $\displaystyle\bar{\mu}{\rm div}\big{(}a(\epsilon p)(\nabla b(\theta))\otimes{\rm curl\,}(b(-\theta){\mathbf{u}})\big{)}-\bar{\mu}\nabla a(\epsilon p)\cdot\nabla(b(\theta){\rm curl\,}(b(-\theta){\mathbf{u}}))$ $\displaystyle-\bar{\mu}a(\epsilon p)\Delta((\nabla b(\theta))\times(b(-\theta){\mathbf{u}}))-\nabla a(\epsilon p)\times(\bar{\mu}\Delta{\mathbf{u}}+(\bar{\mu}+\bar{\lambda})\nabla{\rm div}{\mathbf{u}})$ $\displaystyle+{\rm curl\,}(b(-\theta){\mathbf{u}}\partial_{t}\theta)+[{\rm curl\,},{\mathbf{u}}]\cdot\nabla(b(-\theta){\mathbf{u}})+{\rm curl\,}(b(-\theta){\mathbf{u}}({\mathbf{u}}\cdot\nabla\theta)).$ For any multi-index $\alpha$ satisfying $0\leq\|\alpha\|\leq s-1$, we apply the operator $\partial^{\alpha}$ to (3.55) and (3.56) to obtain $\displaystyle\partial_{t}\partial^{\alpha}({\rm curl\,}(b(-\theta){\mathbf{u}}))+({\mathbf{u}}\cdot\nabla)[\partial^{\alpha}({\rm curl\,}(b(-\theta){\mathbf{u}}))]$ $\displaystyle\quad\quad\quad\ \ =\partial^{\alpha}{\rm curl\,}\\{a(\epsilon p)({\rm curl\,}\mathbf{H})\times\mathbf{H}\\}$ $\displaystyle\quad\quad\quad\quad\ \ +\bar{\mu}{\rm div}\\{a(\epsilon p)b(\theta)\nabla[\partial^{\alpha}({\rm curl\,}(b(-\theta){\mathbf{u}}))]\\}+\partial^{\alpha}\Upsilon_{1}+\Upsilon_{2},$ (3.57) $\displaystyle\partial_{t}\partial^{\alpha}({\rm curl\,}\mathbf{H})-\partial^{\alpha}{\rm curl\,}[{\rm curl\,}({\mathbf{u}}\times\mathbf{H})]-\bar{\nu}\Delta({\rm curl\,}\mathbf{H})=0,$ (3.58) where $\displaystyle\Upsilon_{2}:=$ $\displaystyle-[\partial^{\alpha},{\mathbf{u}}]\cdot\nabla[\partial^{\alpha}({\rm curl\,}(b(-\theta){\mathbf{u}}))]$ $\displaystyle-[\partial^{\alpha},{\rm div}(a(\epsilon p)b(\theta))]\nabla[\partial^{\alpha}({\rm curl\,}(b(-\theta){\mathbf{u}}))].$ Multiplying (3.57) by $\partial^{\alpha}({\rm curl\,}(b(-\theta){\mathbf{u}}))$ and (3.58) by $\partial^{\alpha}({\rm curl\,}\mathbf{H})$ respectively, summing up, and integrating over $\mathbb{R}^{3}$, we deduce that $\displaystyle\frac{1}{2}\frac{d}{dt}\\{\\|\partial^{\alpha}({\rm curl\,}(b(-\theta){\mathbf{u}}))\\|^{2}_{L^{2}}+\\|\partial^{\alpha}({\rm curl\,}\mathbf{H})\\|^{2}_{L^{2}}\\}+\bar{\nu}\\|\partial^{\alpha}({\rm curl\,}\mathbf{H})\\|^{2}_{L^{2}}$ $\displaystyle+\langle a(\epsilon p)b(\theta)\nabla[\partial^{\alpha}({\rm curl\,}(b(-\theta){\mathbf{u}}))],\nabla[\partial^{\alpha}({\rm curl\,}(b(-\theta){\mathbf{u}}))]\rangle$ $\displaystyle=$ $\displaystyle-\langle({\mathbf{u}}\cdot\nabla)[\partial^{\alpha}({\rm curl\,}(b(-\theta){\mathbf{u}}))],\partial^{\alpha}({\rm curl\,}(b(-\theta){\mathbf{u}}))\rangle$ $\displaystyle-\langle\partial^{\alpha}{\rm curl\,}\\{a(\epsilon p)({\rm curl\,}\mathbf{H})\times\mathbf{H}\\},\partial^{\alpha}({\rm curl\,}(b(-\theta){\mathbf{u}}))\rangle$ $\displaystyle+\langle\partial^{\alpha}{\rm curl\,}[{\rm curl\,}({\mathbf{u}}\times\mathbf{H})],\partial^{\alpha}({\rm curl\,}\mathbf{H})\rangle+\langle\partial^{\alpha}\Upsilon_{1}+\Upsilon_{2},\partial^{\alpha}({\rm curl\,}(b(-\theta){\mathbf{u}}))\rangle$ $\displaystyle:=$ $\displaystyle\mathcal{J}_{1}+\mathcal{J}_{2}+\mathcal{J}_{3}+\mathcal{J}_{4},$ (3.59) where $\mathcal{J}_{i}$ ($i=1,\cdots,4$) will be bounded as follows. An integration by parts leads to $\displaystyle\|\mathcal{J}_{1}\|\leq\\|{\rm div}{\mathbf{u}}\\|_{L^{\infty}}\\|\nabla[\partial^{\alpha}({\rm curl\,}(b(-\theta){\mathbf{u}}))]\\|^{2}_{L^{2}}.$ By virtue of (2.3), the Cauchy-Schwarz and Moser-type inequalities (see [34]), the term $\mathcal{J}_{2}$ can be bounded as follows. $\displaystyle\|\mathcal{J}_{2}\|\leq$ $\displaystyle\|\langle\partial^{\alpha}\\{a(\epsilon p)({\rm curl\,}\mathbf{H})\times\mathbf{H}\\},\partial^{\alpha}{\rm curl\,}({\rm curl\,}(b(-\theta){\mathbf{u}}))\rangle\|$ $\displaystyle\leq$ $\displaystyle\|\partial^{\alpha}\\{a(\epsilon p)({\rm curl\,}\mathbf{H})\times\mathbf{H}\\}\\|_{L^{2}}\\|\partial^{\alpha}\nabla({\rm curl\,}(b(-\theta){\mathbf{u}}))\\|_{L^{2}}$ $\displaystyle\leq$ $\displaystyle\eta_{2}\\|\partial^{\alpha}\nabla({\rm curl\,}(b(-\theta){\mathbf{u}}))\\|_{L^{2}}^{2}$ $\displaystyle+C_{\eta}\\{\\|{\rm curl\,}\mathbf{H}\\|^{2}_{L^{\infty}}\\|a(\epsilon p)\mathbf{H}\\|_{H^{s-1}}^{2}+\\|a(\epsilon p)\mathbf{H}\\|^{2}_{L^{\infty}}\\|{\rm curl\,}\mathbf{H}\\|^{2}_{H^{s-1}}\\},$ where $\eta_{2}>0$ is a sufficiently small constant independent of $\epsilon$. If we integrate by parts, make use of (2.3) and the fact that ${\rm curl\,}{\rm curl\,}\mathbf{a}=\nabla\,{\rm div}\,\mathbf{a}-\Delta\mathbf{a}$ and ${\rm div}\mathbf{H}=0$, we see that the term $\mathcal{J}_{3}$ can be rewritten as $\displaystyle\mathcal{J}_{3}=\left\langle\partial^{\alpha}{\rm curl\,}({\mathbf{u}}\times\mathbf{H}),\partial^{\alpha}\Delta\mathbf{H}\right\rangle,$ which, together with the Moser-type inequality, implies that $\displaystyle\|\mathcal{J}_{3}\|\leq C(\mathcal{S})+\eta_{3}\\|\mathbf{H}^{\epsilon}(\tau)\\|^{2}_{s+1},$ where $\eta_{3}>0$ is a sufficiently small constant independent of $\epsilon$. To handle $\mathcal{J}_{4}$, we note that the leading order terms in $\Upsilon_{1}$ are of third-order in $\theta$ and of second-order in ${\mathbf{u}}$, and the leading order terms in $\Upsilon_{2}$ are of order $s+1$ in ${\mathbf{u}}$ and of order $s+1$ in $(\epsilon p,\theta)$. Then it follows that $\displaystyle\|\mathcal{J}_{4}\|$ $\displaystyle\leq$ $\displaystyle C_{0}(\\|\partial^{\alpha}\Upsilon_{1}\\|_{L^{2}}+\\|\Upsilon_{2}\\|_{L^{2}})\\|\partial^{\alpha}({\rm curl\,}(b(-\theta){\mathbf{u}}))\\|_{L^{2}}$ $\displaystyle\leq$ $\displaystyle C(\mathcal{S})\\|\partial^{\alpha}({\rm curl\,}(b(-\theta){\mathbf{u}}))\\|_{L^{2}}.$ Putting the above estimates into the (3.4), choosing $\eta_{2}$ and $\eta_{3}$ sufficient small, summing over $\alpha$ for $0\leq\|\alpha\|\leq s-1$, and then integrating the result on $[0,t]$, we conclude $\displaystyle\sup_{\tau\in[0,t]}\left\\{\\|{\rm curl\,}(b({-\theta}){\mathbf{u}})(\tau)\\|^{2}_{H^{s-1}}+\\|{\rm curl\,}\mathbf{H}(\tau)\\|^{2}_{H^{s-1}}\right\\}$ $\displaystyle\quad+l_{7}\int^{t}_{0}\left\\{\\|\nabla{\rm curl\,}(b({-\theta}){\mathbf{u}})\\|^{2}_{H^{s-1}}+\\|\nabla{\rm curl\,}\mathbf{H}(\tau)\\|^{2}_{H^{s-1}}\right\\}(\tau)d\tau$ $\displaystyle\leq C_{0}\big{\\{}\\|{\rm curl\,}(b({-\theta}){\mathbf{u}})(0)\\|^{2}_{H^{s-1}}+\\|{\rm curl\,}\mathbf{H}(0)\\|^{2}_{H^{s-1}}\big{\\}}$ $\displaystyle\quad+C(\mathcal{O}_{0})\exp\\{(\sqrt{T}+\epsilon)C(\mathcal{O}(T))\\}$ $\displaystyle\leq C(\mathcal{O}_{0})\exp\\{(\sqrt{T}+\epsilon)C(\mathcal{O}(T))\\}.$ ∎ ###### Proof of Proposition 2.4. By virtue of the definition of the norm $\\|\cdot\\|_{s,\epsilon}$ and the fact that $\displaystyle\\|{\mathbf{v}}\\|_{H^{m+1}}\leq K\big{(}\\|{\rm div}{\mathbf{v}}\\|_{H^{m}}+\\|{\rm curl\,}{\mathbf{v}}\\|_{H^{m}}+\\|{\mathbf{v}}\\|_{H^{m}}\big{)},\quad\forall\,\,{\mathbf{v}}\in H^{m+1}(\mathbb{R}^{3}),$ Proposition 2.4 follows directly from Lemmas 3.1, 3.3, 3.7 and 3.8. ∎ Once Proposition 2.4 is established, the existence part of Theorem 1.1 can be proved by directly applying the same arguments as in [2, 40], and hence we omit the details here. ## 4\. Decay of the local energy and zero Mach number limit In this section, we shall prove the convergence part of Theorem 1.1 by modifying the arguments developed by Métivier and Schochet [40], see also some extensions in [1, 2, 38]. ###### Proof of the convergence part of Theorem 1.1. The uniform estimate (1.21) implies that $\displaystyle\sup_{\tau\in[0,T_{0}]}\\|(p^{\epsilon},{\mathbf{u}}^{\epsilon},\mathbf{H}^{\epsilon})(\tau)\\|_{H^{s}}+\sup_{\tau\in[0,T_{0}]}\\|\theta^{\epsilon}-\bar{\theta}\\|_{H^{s+1}}<+\infty.$ Thus, after extracting a subsequence, one has $\displaystyle(p^{\epsilon},{\mathbf{u}}^{\epsilon})\rightharpoonup(\bar{p},{\mathbf{w}})$ $\displaystyle\text{weakly-}\ast\ \text{in}$ $\displaystyle\quad\quad L^{\infty}(0,T_{0};H^{s}(\mathbb{R}^{3})),$ (4.1) $\displaystyle\mathbf{H}^{\epsilon}\rightharpoonup{\mathbf{B}}$ $\displaystyle\text{weakly-}\ast\ \text{in}$ $\displaystyle\qquad L^{\infty}(0,T_{0};H^{s}(\mathbb{R}^{3})),$ (4.2) $\displaystyle\theta^{\epsilon}-\bar{\theta}\rightharpoonup\vartheta-\bar{\theta}\\!\\!\\!\\!\\!\\!$ $\displaystyle\text{weakly-}\ast\ \text{in}$ $\displaystyle\qquad L^{\infty}(0,T_{0};H^{s+1}(\mathbb{R}^{3})).$ (4.3) It follows from the equations for $\mathbf{H}^{\epsilon}$ and $\theta^{\epsilon}$ that $\displaystyle\partial_{t}\mathbf{H}^{\epsilon},\,\partial_{t}\theta^{\epsilon}\in C([0,T_{0}],H^{s-2}(\mathbb{R}^{3})).$ (4.4) (4.2)–(4.4) implies, after further extracting a subsequence, that for all $s^{\prime}<s$, $\displaystyle\mathbf{H}^{\epsilon}\rightarrow{\mathbf{B}}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!$ strongly in $\displaystyle\quad C([0,T_{0}],H^{s^{\prime}}_{\mathrm{loc}}(\mathbb{R}^{3})),$ (4.5) $\displaystyle\theta^{\epsilon}-\bar{\theta}\rightarrow\vartheta-\bar{\theta}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!$ strongly in $\displaystyle\quad C([0,T_{0}],H^{s^{\prime}+1}_{\mathrm{loc}}(\mathbb{R}^{3})),$ (4.6) where the limit ${\mathbf{B}}\in C([0,T_{0}],H^{s^{\prime}}_{\mathrm{loc}}(\mathbb{R}^{3}))\cap L^{\infty}(0,T_{0};H^{s}_{\mathrm{loc}}(\mathbb{R}^{3}))$ and $(\vartheta-\bar{\theta})\in C([0,T_{0}],H^{s^{\prime}+1}_{\mathrm{loc}}(\mathbb{R}^{3}))\cap L^{\infty}(0,T_{0};H^{s+1}_{\mathrm{loc}}(\mathbb{R}^{3}))$. Similarly, from (3.8) we get $\displaystyle{\rm curl\,}\big{(}e^{-\theta^{\epsilon}}{\mathbf{u}}^{\epsilon}\big{)}\rightarrow{\rm curl\,}\big{(}e^{-\vartheta}{\mathbf{w}}\big{)}\quad\text{strongly in}\quad C([0,T_{0}],H^{s^{\prime}-1}_{\mathrm{loc}}(\mathbb{R}^{3}))$ (4.7) for all $s^{\prime}<s$. In order to obtain the limit system, one needs to show that the limits in (4.1) hold in the strong topology of $L^{2}(0,T_{0};H^{s^{\prime}}_{\mathrm{loc}}(\mathbb{R}^{3}))$ for all $s^{\prime}<s$. To this end, we first show that $\bar{p}=0$ and ${\rm div}(2{\mathbf{w}}-\bar{\kappa}e^{\vartheta}\nabla\vartheta)=0$. In fact, the equations (1.11) and (1.12) can be rewritten as $\displaystyle\epsilon\,\partial_{t}p^{\epsilon}+{\rm div}(2{\mathbf{u}}^{\epsilon}-\bar{\kappa}e^{-\epsilon p^{\epsilon}+\theta^{\epsilon}}\nabla\theta^{\epsilon})=\epsilon f^{\epsilon},$ (4.8) $\displaystyle\epsilon\,e^{-\theta^{\epsilon}}\partial_{t}{\mathbf{u}}^{\epsilon}+\nabla p^{\epsilon}=\epsilon\,\mathbf{g}^{\epsilon}.$ (4.9) By virtue of (1.21), $f^{\epsilon}$ and $\mathbf{g}^{\epsilon}$ are uniformly bounded in $C([0,T_{0}],H^{s-1}(\mathbb{R}^{3}))$. Passing to the weak limit in (4.8) and (4.9), respectively, we see that $\nabla\bar{p}=0$ and ${\rm div}(2{\mathbf{w}}-\bar{\kappa}e^{\vartheta}\nabla\vartheta)=0$. Since $\bar{p}\in L^{\infty}(0,T_{0};H^{s}(\mathbb{R}^{3}))$, we infer that $\bar{p}=0$. Notice that by virtue of (4.7), the strong compactness for the incompressible component of $e^{-\theta^{\epsilon}}{\mathbf{u}}^{\epsilon}$ holds. So, it is sufficient to prove the following proposition on the acoustic components in order to get the strong convergence of ${\mathbf{u}}^{\epsilon}$. ###### Proposition 4.1. Suppose that the assumptions in Theorem 1.1 hold. Then, $p^{\epsilon}$ converges to $0$ strongly in $L^{2}(0,T_{0};H^{s^{\prime}}_{\mathrm{loc}}(\mathbb{R}^{3}))$ and ${\rm div}(2{\mathbf{u}}^{\epsilon}-\bar{\kappa}e^{-\epsilon p^{\epsilon}+\theta^{\epsilon}}\nabla\theta^{\epsilon})$ converges to $0$ strongly in $L^{2}(0,T_{0};H^{s^{\prime}-1}_{\mathrm{loc}}(\mathbb{R}^{3}))$ for all $s^{\prime}<s$. The proof of Proposition 4.1 is based on the following dispersive estimates on the wave equation obtained by Métivier and Schochet [40] and reformulated in [2]. ###### Lemma 4.2. ([40, 2]) Let $T>0$ and $v^{\epsilon}$ be a bounded sequence in $C([0,T],H^{2}(\mathbb{R}^{3}))$, such that $\displaystyle\epsilon^{2}\partial_{t}(a^{\epsilon}\partial_{t}v^{\epsilon})-\nabla\cdot(b^{\epsilon}\nabla v^{\epsilon})=c^{\epsilon},$ where $c^{\epsilon}$ converges to $0$ strongly in $L^{2}(0,T;L^{2}(\mathbb{R}^{3}))$. Assume further that for some $s>3/2+1$, the coefficients $(a^{\epsilon},b^{\epsilon})$ are uniformly bounded in $C([0,T],H^{s}(\mathbb{R}^{3}))$ and converge in $C([0,T],H^{s}_{\mathrm{loc}}(\mathbb{R}^{3}))$ to a limit $(a,b)$ satisfying the decay estimates $\displaystyle\|a(x,t)-\hat{a}\|\leq C_{0}\|x\|^{-1-\zeta},\quad\|\nabla_{x}a(x,t)\|\leq C_{0}\|x\|^{-2-\zeta},$ $\displaystyle\|b(x,t)-\hat{b}\|\leq C_{0}\|x\|^{-1-\zeta},\quad\|\nabla_{x}b(x,t)\|\leq C_{0}\|x\|^{-2-\zeta},$ for some positive constants $\hat{a}$, $\hat{b}$, $C_{0}$ and $\zeta$. Then the sequence $v^{\epsilon}$ converges to $0$ strongly in $L^{2}(0,T;L^{2}_{\mathrm{loc}}(\mathbb{R}^{3}))$. ###### Proof of Proposition 4.1. We fist show that $p^{\epsilon}$ converges to $0$ strongly in $L^{2}(0,T_{0};\linebreak H^{s^{\prime}}_{\mathrm{loc}}(\mathbb{R}^{3}))$ for all $s^{\prime}<s$. Applying $\epsilon^{2}\partial_{t}$ to (1.11), we find that $\displaystyle\epsilon^{2}\partial_{t}\\{\partial_{t}p^{\epsilon}+({\mathbf{u}}^{\epsilon}\cdot\nabla)p^{\epsilon}\\}+{\epsilon}\partial_{t}\\{{\rm div}(2{\mathbf{u}}^{\epsilon}-\bar{\kappa}e^{-\epsilon p^{\epsilon}+\theta^{\epsilon}}\nabla\theta^{\epsilon})\\}$ $\displaystyle\quad\quad=\epsilon^{3}\partial_{t}\\{e^{-\epsilon p^{\epsilon}}[\bar{\nu}\|{\rm curl\,}\mathbf{H}^{\epsilon}\|^{2}+\Psi({\mathbf{u}}^{\epsilon}):\nabla{\mathbf{u}}^{\epsilon}]\\}+\epsilon^{2}\partial_{t}\\{\bar{\kappa}e^{-\epsilon p^{\epsilon}+\theta^{\epsilon}}\nabla p^{\epsilon}\cdot\nabla\theta^{\epsilon}\\}.$ (4.10) Dividing (1.12) by $e^{-\theta^{\epsilon}}$ and then applying the operator _div_ to the resulting equations, one gets $\displaystyle\epsilon\partial_{t}{\rm div}{\mathbf{u}}^{\epsilon}+{\rm div}\big{(}e^{\theta^{\epsilon}}{\nabla p^{\epsilon}}\big{)}=$ $\displaystyle-\epsilon{\rm div}\\{({\mathbf{u}}^{\epsilon}\cdot\nabla){\mathbf{u}}^{\epsilon}\\}$ $\displaystyle+\epsilon{\rm div}\big{\\{}e^{-\epsilon p^{\epsilon}+\theta^{\epsilon}}[({\rm curl\,}\mathbf{H})\times\mathbf{H}+{\rm div}\Psi^{\epsilon}({\mathbf{u}}^{\epsilon})]\big{\\}},$ (4.11) Subtracting (4) from (4), we have $\displaystyle\epsilon^{2}\partial_{t}\Big{(}\frac{1}{2}\partial_{t}p^{\epsilon}\Big{)}-{\rm div}\big{(}e^{\theta^{\epsilon}}{\nabla p^{\epsilon}}\big{)}=\epsilon F^{\epsilon}(p^{\epsilon},{\mathbf{u}}^{\epsilon},\mathbf{H}^{\epsilon},\theta^{\epsilon}),$ (4.12) where $F^{\epsilon}(p^{\epsilon},{\mathbf{u}}^{\epsilon},\mathbf{H}^{\epsilon},\theta^{\epsilon})$ is a smooth function in its variables with $F(0)=0$. By the uniform boundedness of $(p^{\epsilon},{\mathbf{u}}^{\epsilon},\mathbf{H}^{\epsilon},\theta^{\epsilon})$ one infers that $\displaystyle\epsilon F^{\epsilon}(p^{\epsilon},{\mathbf{u}}^{\epsilon},\mathbf{H}^{\epsilon},\theta^{\epsilon})\rightarrow 0\quad\text{strongly in}\quad L^{2}(0,T_{0};L^{2}(\mathbb{R}^{3})).$ By the the strong convergence of $\theta^{\epsilon}$, the initial conditions (1.22), and the arguments in Section 8.1 in [2], one can easily prove that the coefficients in (4.12) satisfy the conditions in Lemma 4.2. Therefore, we can apply Lemma 4.2 to obtain $\displaystyle p^{\epsilon}\rightarrow 0\quad\text{strongly in}\quad L^{2}(0,T_{0};L^{2}_{\mathrm{loc}}(\mathbb{R}^{3})).$ Since $p^{\epsilon}$ is bounded uniformly in $C([0,T_{0}],H^{s}(\mathbb{R}^{3}))$, an interpolation argument gives $\displaystyle p^{\epsilon}\rightarrow 0\quad\text{strongly in}\quad L^{2}(0,T_{0};H^{s^{\prime}}_{\mathrm{loc}}(\mathbb{R}^{3}))\ \ \text{for all}\ \ s^{\prime}<s.$ Similarly, we can obtain the strong convergence of ${\rm div}(2{\mathbf{u}}^{\epsilon}-\kappa^{\epsilon}e^{-\epsilon p^{\epsilon}+\theta^{\epsilon}}\nabla\theta^{\epsilon})$. This completes the proof. ∎ We continue our proof of Theorem 1.1. It follows from Proposition 4.1 and (4.6) that $\displaystyle{\rm div}\,{\mathbf{u}}^{\epsilon}\rightarrow{\rm div}\,{\mathbf{w}}\quad\text{strongly in}\quad L^{2}(0,T_{0};H^{s^{\prime}-1}_{\mathrm{loc}}(\mathbb{R}^{3})).$ Thus, using (4.7), one obtains $\displaystyle{\mathbf{u}}^{\epsilon}\rightarrow{\mathbf{w}}\quad\text{strongly in}\quad L^{2}(0,T_{0};H^{s^{\prime}}_{\mathrm{loc}}(\mathbb{R}^{3}))\qquad\mbox{for all }s^{\prime}<s.$ By (4.5), (4.6), and Proposition 4.1, we find that $\begin{array}[]{lcl}\nabla{\mathbf{u}}^{\epsilon}\rightarrow\nabla{\mathbf{w}}&\text{strongly in}&L^{2}(0,T_{0};H^{s^{\prime}-1}_{\mathrm{loc}}(\mathbb{R}^{3}));\\\ \nabla\mathbf{H}^{\epsilon}\rightarrow\nabla{\mathbf{B}}&\text{strongly in}&L^{2}(0,T_{0};H^{s^{\prime}-1}_{\mathrm{loc}}(\mathbb{R}^{3}));\\\ \nabla\theta^{\epsilon}\rightarrow\nabla\vartheta&\text{strongly in}&L^{2}(0,T_{0};H^{s^{\prime}-1}_{\mathrm{loc}}(\mathbb{R}^{3})).\end{array}$ Passing to the limits in the equations for $p^{\epsilon}$, $\mathbf{H}^{\epsilon}$, and $\theta^{\epsilon}$, one sees that the limit $(0,{\mathbf{w}},{\mathbf{B}},\vartheta)$ satisfies, in the sense of distributions, that $\displaystyle{\rm div}(2{\mathbf{w}}-\bar{\kappa}\,e^{\vartheta}\nabla\vartheta)=0,$ (4.13) $\displaystyle\partial_{t}{\mathbf{B}}-{\rm curl\,}({\mathbf{w}}\times{\mathbf{B}})-\bar{\nu}\Delta{\mathbf{B}}=0,\quad{\rm div}{\mathbf{B}}=0,$ (4.14) $\displaystyle\partial_{t}\vartheta+({\mathbf{w}}\cdot\nabla)\vartheta+{\rm div}{\mathbf{w}}=\bar{\kappa}\,{\rm div}(e^{\vartheta}\nabla\vartheta).$ (4.15) On the other hand, applying _curl_ to the momentum equations (1.12), using the equations (1.11) and (1.14) on $p^{\epsilon}$ and $\theta^{\epsilon}$, and then taking to the limit on the resulting equations, we deduce that $\displaystyle{\rm curl\,}\left\\{\partial_{t}\big{(}e^{-\vartheta}{\mathbf{w}})+{\rm div}\big{(}{\mathbf{w}}e^{-\vartheta}\otimes{\mathbf{w}}\big{)}-({\rm curl\,}{\mathbf{B}})\times{\mathbf{B}}-{\rm div}\Phi({\mathbf{w}})\right\\}=0$ holds in the sense of distributions. Therefore it follows from (4.13)–(4.15) that $\displaystyle e^{-\vartheta}[\partial_{t}{\mathbf{w}}+({\mathbf{w}}\cdot\nabla){\mathbf{w}}]+\nabla\pi=({\rm curl\,}{\mathbf{B}})\times{\mathbf{B}}+{\rm div}\Phi({\mathbf{w}}),$ (4.16) for some function $\pi$. Following the same arguments as those in the proof of Theorem 1.5 in [40], we conclude that $({\mathbf{w}},{\mathbf{B}},\vartheta)$ satisfies the initial condition $\displaystyle({\mathbf{w}},{\mathbf{B}},\vartheta)\|_{t=0}=({\mathbf{w}}_{0},{\mathbf{B}}_{0},\vartheta_{0}).$ (4.17) Moreover, the standard iterative method shows that the system (4.13)–(4.16) with initial data (4.17) has a unique solution $({\mathbf{w}}^{},{\mathbf{B}}^{},\vartheta^{}-\bar{\theta})\in C([0,T_{0}],H^{s}(\mathbb{R}^{3}))$. Thus, the uniqueness of solutions to the limit system (4.13)–(4.16) implies that the above convergence holds for the full sequence of $(p^{\epsilon},{\mathbf{u}}^{\epsilon},\mathbf{H}^{\epsilon},\theta^{\epsilon})$. Therefore the proof is completed. ∎ Acknowledgements: This work was partially done when Li was visiting the Institute of Mathematical Sciences, CUHK. He would like to thank the institute for hospitality. Jiang was supported by the National Basic Research Program under the Grant 2011CB309705 and NSFC (Grant No. 11229101). Ju was supported by NSFC (Grant No. 11171035). Li was supported by NSFC (Grant No. 11271184, 10971094), NCET-11-0227, PAPD, and the Fundamental Research Funds for the Central Universities. Xin was supported in part by the Zheng Ge Ru Foundation, Hong Kong RGC Earmarked Research Grants CUHK4042/08P and CUHK4041/11p. ## References * [1] T. Alazard, Incompressible limit of the nonisentropic Euler equations with solid wall boundary conditions, Adv. Differential Equations 10 (2005), 19-44. * [2] T. Alazard, Low Mach number limit of the full Navier-Stokes equations, Arch. Ration. Mech. Anal. 180 (2006), 1-73. * [3] K. Asano, On the incompressible limit of the compressible Euler equations, Japan J. Appl. Math. 4 (1987), 455-488. * [4] A. Blokhin, Yu. Trakhinin, Stability of strong discontinuities in fluids and MHD. In: Handbook of mathematical fluid dynamics, Vol. I, 545-652, North-Holland, Amsterdam, 2002. * [5] D. Bresch, B. Desjardins, E. Grenier, Oscillatory Limit with Changing Eigenvalues: A Formal Study. In: New Directions in Mathematical Fluid Mechanics, A.V. Fursikov, G.P. Galdi, V.V. Pukhnachev (eds.), 91-105, Birkhäuser Verlag, Basel, 2010. * [6] D. Bresch, B. Desjardins, E. Grenier, C.-K. Lin, Low Mach number limit of viscous polytropic flows: formal asymptotics in the periodic case, Stud. Appl. Math. 109 (2002), 125-149. * [7] G.Q. Chen and D.H. Wang, Global solutions of nonlinear magnetohydrodynamics with large initial data, J. Differential Equations, 182 (2002), no.1, 344-376. * [8] G.Q. Chen and D.H. Wang, Existence and continuous dependence of large solutions for the magnetohydrodynamic equations, Z. Angew. Math. Phys. 54 (2003), 608-632. * [9] R. Danchin, Low Mach number limit for viscous compressible flows, M2AN Math. Model. Numer. Anal. 39 (2005), 459-475. * [10] B. Ducomet and E. Feireisl, The equations of magnetohydrodynamics: on the interaction between matter and radiation in the evolution of gaseous stars, Comm. Math. Phys. 266 (2006), 595-629. * [11] B. Desjardins and E. Grenier, Low Mach number limit of viscous compressible flows in the whole space, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 455 (1999), 2271–2279. * [12] B. Desjardins, E. Grenier, P.-L. Lions, and N. Masmoudi, Incompressible limit for solutions of the isentropic Navier-Stokes equations with Dirichlet boundary conditions, J. Math. Pures Appl. 78 (1999), 461-471. * [13] J. Fan, S. Jiang and G. Nakamura, Vanishing shear viscosity limit in the magnetohydrodynamic equations, Comm. Math. Phys. 270 (2007), 691-708. * [14] E.Feireisl, A. Novotny, Singular Limits in Thermodynamics of Viscous Fluids, Birkhauser Basel, 2009. * [15] H. Freistühler and P. Szmolyan, Existence and bifurcation of viscous profiles for all intermediate magnetohydrodynamic shock waves, SIAM J. Math. Anal., 26 (1995), 112-128. * [16] I. Gallagher, A remark on smooth solutions of the weakly compressible periodic Navier-Stokes equations, J. Math. Kyoto Univ. 40 (2000), 525-540. * [17] S.K. Godunov, Symmetrization of magnetohydrodynamics equations. (In Russian) Chislennye Metody Mekhaniki Sploshnoi Sredy, Novosibirsk 3 (1972), 26-34. * [18] T. Hagstrom, J. Lorenz, On the stability of approximate solutions of hyperbolic-parabolic systems and the all-time existence of smooth, slightly compressible flows, Indiana Univ. Math. J., 51 (2002), 1339-1387. * [19] D. Hoff and E. Tsyganov, Uniqueness and continuous dependence of weak solutions in compressible magnetohydrodynamics, Z. Angew. Math. Phys. 56 (2005), 791-804. * [20] W.R. Hu, Cosmic magnetohydrodynamics, Science Press, Beijing, 1987 (in Chinese). * [21] X. Hu, D. Wang, Global existence and large-time behavior of solutions to the three-dimensional equations of compressible magnetohydrodynamic flows, Arch. Ration. Mech. Anal., 197 (2010), 203–238. * [22] X. P. Hu and D. H. Wang, Global solutions to the three-dimensional full compressible magnetohydrodynamic flows, Comm. Math. Phys. 283 (2008), 255-284. * [23] X.P. Hu and D.H. Wang, Low Mach number limit of viscous compressible magnetohydrodynamic flows, SIAM J. Math. Anal. 41 (2009), 1272-1294. * [24] T. Iguchi, The incompressible limit and the initial layer of the compressible Euler equation in $R_{n}^{+}$, Math. Methods Appl. Sci. 20 (1997), 945-958. * [25] H. Isozaki, Singular limits for the compressible Euler equation in an exterior domain, J. Reine Angew. Math. 381 (1987), 1-36. * [26] H. Isozaki, Singular limits for the compressible Euler equation in an exterior domain, I Bodies in an uniform flow, Osaka J. Math. 26 (1989), 399-410. * [27] A. Jeffrey and T. Taniuti, Non-Linear Wave Propagation. With applications to physics and magnetohydrodynamics. Academic Press, New York, 1964. * [28] S. Jiang, Q.C. Ju and F.C. Li, Incompressible limit of the compressible Magnetohydrodynamic equations with periodic boundary conditions, Comm. Math. Phys. 297 (2010), 371-400. * [29] S. Jiang, Q.C. Ju and F.C. Li, Incompressible limit of the compressible magnetohydrodynamic equations with vanishing viscosity coefficients, SIAM J. Math. Anal. 42 (2010), 2539-2553. * [30] S. Jiang, Q.C. Ju and F.C. Li, Low Mach number limit for the multi-dimensional full magnetohydrodynamic equations, Nonlinearity, 15 (2012), 1351-1365. * [31] S. Jiang, Q. C. Ju and F. C. Li, Incompressible limit of the non-isentropic ideal magnetohydrodynamic equations, Preprint. * [32] S. Jiang and Y.B. Ou, Incompressible limit of the non-isentropic Navier-Stokes equations with well-prepared initial data in three-dimensional bounded domains, J. Math. Pures Appl. 96 (2011), 1-28. * [33] H. Kim, J. Lee, The incompressible limits of viscous polytropic fluids with zero thermal conductivity coefficient, Comm. Partial Differential Equations 30 (2005) 1169-1189. * [34] S. Klainerman, A. Majda, Singular perturbations of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids, Comm. Pure Appl. Math. 34 (1981), 481-524. * [35] P. Kuku$\rm\check{c}$ka, Singular limits of the equations of magnetohydrodynamics, J. Math. Fluid Mech., 13 (2011), 173-189. * [36] Y.-S. Kwon and K. Trivisa, On the incompressible limits for the full magnetohydrodynamics flows, J. Differential Equations 251 (2011), 1990-2023. * [37] A. G. Kulikovskiy and G. A. Lyubimov, Magnetohydrodynamics, Addison-Wesley, Reading, Massachusetts, 1965. * [38] C. D. Levermore, W. Sun, K. Trivisa, A low Mach number limit of a dispersive Navier-Stokes system, SIAM J. Math. Anal. 44 (2012), 176-1807. * [39] F. C. Li, and H.Y. Yu, Optimal decay rate of classical solutions to the compressible magnetohydrodynamic equations, Proc. Roy. Soc. Edinburgh Sect. A, 141 (2011), 109–126. * [40] G. Métivier, S. Schochet, The incompressible limit of the non-isentropic Euler equations, Arch. Ration. Mech. Anal. 158 (2001), 61-90. * [41] G. Métivier, S. Schochet, Averaging theorems for conservative systems and the weakly compressible Euler equations, J. Differential Equations 187 (2003), 106-183. * [42] A. Novotny, M. Ruzicka and G. Thäter, Singular limit of the equations of magnetohydrodynamics in the presence of strong stratification, Math. Models Methods Appl. Sci. 21 (2011), 115-147. * [43] L. D. Laudau and E. M. Lifshitz, Electrodynamics of Continuous Media, 2nd ed., Pergamon, New York, 1984. * [44] P.-L. Lions and N. Masmoudi, Incompressible limit for a viscous compressible fluid, J. Math. Pures Appl., 77 (1998), 585–627. * [45] N. Masmoudi, Examples of singular limits in hydrodynamics, In: Handbook of Differential Equations: Evolutionary equations, Vol. III, , 195-275, Elsevier/North-Holland, Amsterdam, 2007\. * [46] R. V. Polovin and V. P. Demutskii, Fundamentals of Magnetohydrodynamics, Consultants, Bureau, New York, 1990. * [47] S. Schochet, Fast singular limits of hyperbolic PDEs, J. Differential Equations 114 (1994), 476-512. * [48] S. Schochet, The compressible Euler equations in a bounded domain: existence of solutions and the incompressible limit, Comm. Math. Phys. 104 (1986), 49-75. * [49] S. Schochet, The mathematical theory of the incompressible limit in fluid dynamics, in: Handbook of mathematical fluid dynamics, Vol. IV, 123-157, Elsevier/North-Holland, Amsterdam, 2007. * [50] S. Ukai, The incompressible limit and the initial layer of the compressible Euler equation, J. Math. Kyoto Univ. 26 (1986) 323-331. * [51] A.I. Vol’pert, S. I. Hudiaev, On the Cauchy problem for composite systems of nonlinear equations, Mat. Sbornik 87 (1972), 504-528. * [52] J.W. Zhang, S. Jiang, F. Xie, Global weak solutions of an initial boundary value problem for screw pinches in plasma physics, Math. Models Methods Appl. Sci. 19 (2009), 833-875.	arxiv-papers	2011-11-12T13:36:54	2024-09-04T02:49:24.247703	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Song Jiang, Qiangchang Ju, Fucai Li, and Zhouping Xin", "submitter": "Fucai Li", "url": "https://arxiv.org/abs/1111.2925" }
1111.2926	# Incompressible limit of the compressible non-isentropic magnetohydrodynamic equations with zero magnetic diffusivity Song Jiang LCP, Institute of Applied Physics and Computational Mathematics, P.O. Box 8009, Beijing 100088, P.R. China jiang@iapcm.ac.cn , Qiangchang Ju Institute of Applied Physics and Computational Mathematics, P.O. Box 8009-28, Beijing 100088, P.R. China qiangchang_ju@yahoo.com and Fucai Li Department of Mathematics, Nanjing University, Nanjing 210093, P.R. China fli@nju.edu.cn ###### Abstract. We study the incompressible limit of the compressible non- isentropic magnetohydrodynamic equations with zero magnetic diffusivity and general initial data in the whole space $\mathbb{R}^{d}$ ($d=2,3$). We first establish the existence of classic solutions on a time interval independent of the Mach number. Then, by deriving uniform a priori estimates, we obtain the convergence of the solution to that of the incompressible magnetohydrodynamic equations as the Mach number tends to zero. ###### Key words and phrases: Compressible MHD equations, non-isentropic, zero magnetic diffusivity, incompressible limit ###### 2000 Mathematics Subject Classification: 76W05, 35B40 ## 1\. Introduction This paper is concerned with the incompressible limit to the compressible non- isentropic magnetohydrodynamic (MHD) equations with zero magnetic diffusivity and general initial data in the whole space $\mathbb{R}^{d}$ ($d=2,3$). In the study of a highly conducting fluid, for example, the magnetic fusion, it is rational to ignore the magnetic diffusion term in the MHD equations since the magnetic diffusion coefficient (resistivity coefficient) is inversely proportional to the electrical conductivity coefficient, see [10]. In this situation, the system, describing the motion of the fluid in ${\mathbb{R}^{d}}$, can be described by the following compressible non- isentropic MHD equations with zero magnetic diffusivity: $\displaystyle\partial_{t}\rho+{\rm div}(\rho{\mathbf{u}})=0,$ (1.1) $\displaystyle\partial_{t}(\rho{\mathbf{u}})+{\rm div}\left(\rho{\mathbf{u}}\otimes{\mathbf{u}}\right)+{\nabla p}=(\nabla\times\mathbf{H})\times\mathbf{H}+{\rm div}\Psi,$ (1.2) $\displaystyle\partial_{t}\mathbf{H}-\nabla\times({\mathbf{u}}\times\mathbf{H})=0,\quad{\rm div}\mathbf{H}=0,$ (1.3) $\displaystyle\partial_{t}{\mathcal{E}}+{\rm div}\left({\mathbf{u}}({\mathcal{E}}^{\prime}+p)\right)={\rm div}(({\mathbf{u}}\times\mathbf{H})\times\mathbf{H})+{\rm div}({\mathbf{u}}\Psi+\kappa\nabla\theta).$ (1.4) Here $\rho$ denotes the density, ${\mathbf{u}}\in{\mathbb{R}}^{d}$ the velocity, $\mathbf{H}\in{\mathbb{R}}^{d}$ the magnetic field, and $\theta$ the temperature, respectively; $\Psi$ is the viscous stress tensor given by $\Psi=2\mu\mathbb{D}({\mathbf{u}})+\lambda{\rm div}{\mathbf{u}}\;\mathbf{I}_{d}$ with $\mathbb{D}({\mathbf{u}})=(\nabla{\mathbf{u}}+\nabla{\mathbf{u}}^{\top})/2$, and $\mathbf{I}_{d}$ being the $d\times d$ identity matrix, and $\nabla{\mathbf{u}}^{\top}$ the transpose of the matrix $\nabla{\mathbf{u}}$; ${\mathcal{E}}$ is the total energy given by ${\mathcal{E}}={\mathcal{E}}^{\prime}+\|\mathbf{H}\|^{2}/2$ and ${\mathcal{E}}^{\prime}=\rho\left(e+\|{\mathbf{u}}\|^{2}/2\right)$ with $e$ being the internal energy, $\rho\|{\mathbf{u}}\|^{2}/2$ the kinetic energy, and $\|\mathbf{H}\|^{2}/2$ the magnetic energy. The viscosity coefficients $\lambda$ and $\mu$ of the fluid satisfy $2\mu+d\lambda>0$ and $\mu>0$; $\kappa>0$ is the heat conductivity. For simplicity, we assume that $\mu,\lambda$ and $\kappa$ are constants. The equations of state $p=p(\rho,\theta)$ and $e=e(\rho,\theta)$ relate the pressure $p$ and the internal energy $e$ to the density $\rho$ and the temperature $\theta$ of the flow. For the smooth solution to the system (1.1)–(1.4), we can rewrite the total energy equation (1.4) in the form of the internal energy. In fact, multiplying (1.2) by ${\mathbf{u}}$ and (1.3) by $\mathbf{H}$, and summing the resulting equations together, we obtain $\displaystyle\frac{d}{dt}\Big{(}\frac{1}{2}\rho\|{\mathbf{u}}\|^{2}+\frac{1}{2}\|\mathbf{H}\|^{2}\Big{)}$ $\displaystyle+\frac{1}{2}{\rm div}\big{(}\rho\|{\mathbf{u}}\|^{2}{\mathbf{u}}\big{)}+\nabla p\cdot{\mathbf{u}}$ $\displaystyle={\rm div}\Psi\cdot{\mathbf{u}}+(\nabla\times\mathbf{H})\times\mathbf{H}\cdot{\mathbf{u}}+\nabla\times({\mathbf{u}}\times\mathbf{H})\cdot\mathbf{H}.$ (1.5) Using the identities $\displaystyle{\rm div}(\mathbf{H}\times(\nabla\times\mathbf{H}))=\|\nabla\times\mathbf{H}\|^{2}-\nabla\times(\nabla\times\mathbf{H})\cdot\mathbf{H},$ (1.6) $\displaystyle{\rm div}(({\mathbf{u}}\times\mathbf{H})\times\mathbf{H})=(\nabla\times\mathbf{H})\times\mathbf{H}\cdot{\mathbf{u}}+\nabla\times({\mathbf{u}}\times\mathbf{H})\cdot\mathbf{H},$ (1.7) and subtracting (1) from (1.4), we thus obtain the internal energy equation $\partial_{t}(\rho e)+{\rm div}(\rho{\mathbf{u}}e)+({\rm div}{\mathbf{u}})p=\Psi:\nabla{\mathbf{u}}+\kappa\Delta\theta,$ (1.8) where $\Psi:\nabla{\mathbf{u}}$ denotes the scalar product of two matrices: $\Psi:\nabla{\mathbf{u}}=\sum^{3}_{i,j=1}\frac{\mu}{2}\left(\frac{\partial u^{i}}{\partial x_{j}}+\frac{\partial u^{j}}{\partial x_{i}}\right)^{2}+\lambda\|{\rm div}{\mathbf{u}}\|^{2}=2\mu\|\mathbb{D}({\mathbf{u}})\|^{2}+\lambda(\mbox{tr}\mathbb{D}({\mathbf{u}}))^{2}.$ Using the Gibbs relation $\theta\mathrm{d}S=\mathrm{d}e+p\,\mathrm{d}\left(\frac{1}{\rho}\right),$ (1.9) we can further replace the equation (1.8) by $\partial_{t}(\rho S)+{\rm div}(\rho S{\mathbf{u}})=\Psi:\nabla{\mathbf{u}}+\kappa\Delta\theta,$ (1.10) where $S$ denotes the entropy. In the present paper, we assume that $\kappa=0$ in (1.10). Now, as in [27], we reconsider the equations of state as functions of $S$ and $p$, i.e., $\rho=R(S,p)$ and $\theta=\Theta(S,p)$ for some positive smooth functions $R$ and $\Theta$ defined for all $S$ and $p>0$, and satisfying $\partial R/\partial p>0$. For instance, we have $\rho=p^{1/\gamma}e^{-S/\gamma}$ for ideal fluids. Then, by utilizing (1.1) together with the constraint ${\rm div}{\mathbf{H}}=0$, the system (1.1), (1.2), (1.4) and (1.10) can be rewritten as $\displaystyle A(S,p)(\partial_{t}p+({\mathbf{u}}\cdot\nabla)p)+{\rm div}{\mathbf{u}}=0,$ (1.11) $\displaystyle R(S,p)(\partial_{t}{\mathbf{u}}+({\mathbf{u}}\cdot\nabla){\mathbf{u}})+\nabla p=(\nabla\times\mathbf{H})\times\mathbf{H}+{\rm div}\Psi,$ (1.12) $\displaystyle\partial_{t}{\mathbf{H}}-{\rm curl\,}({\mathbf{u}}\times\mathbf{H})=0,\quad{\rm div}\mathbf{H}=0,$ (1.13) $\displaystyle R(S,p)\Theta(S,p)(\partial_{t}S+({\mathbf{u}}\cdot\nabla)S)=\Psi:\nabla{\mathbf{u}},$ (1.14) where $\displaystyle A(S,p)=\frac{1}{R(S,p)}\frac{\partial R(S,p)}{\partial p}.$ (1.15) Considering the physical explanation of the incompressible limit, we introduce the dimensionless parameter $\epsilon$, the Mach number, and make the following changes of variables: $\displaystyle p(x,t)=p^{\epsilon}(x,\epsilon t),\quad S(x,t)=S^{\epsilon}(x,\epsilon t),$ $\displaystyle{{\mathbf{u}}}(x,t)=\epsilon{\mathbf{u}}^{\epsilon}(x,\epsilon t),\;\;\;{\mathbf{H}}(x,t)=\epsilon\mathbf{H}^{\epsilon}(x,\epsilon t),$ and $\displaystyle\mu=\epsilon\,\mu^{\epsilon},\;\;\;\lambda=\epsilon\,\lambda^{\epsilon}.$ As the analysis in [27], we use the transformation $p^{\epsilon}(x,\epsilon t)=\underline{p}e^{\epsilon q^{\epsilon}(x,\epsilon t)}$ for some positive constant $\underline{p}$. Under these changes of variables, the system (1.11)–(1.14) becomes $\displaystyle a^{\epsilon}(S^{\epsilon},\epsilon q^{\epsilon})(\partial_{t}q^{\epsilon}+({\mathbf{u}}^{\epsilon}\cdot\nabla)q^{\epsilon})+\frac{1}{\epsilon}{\rm div}{\mathbf{u}}^{\epsilon}=0,$ (1.16) $\displaystyle r^{\epsilon}(S^{\epsilon},\epsilon q^{\epsilon})(\partial_{t}{\mathbf{u}}^{\epsilon}+({\mathbf{u}}^{\epsilon}\cdot\nabla){\mathbf{u}}^{\epsilon})+\frac{1}{\epsilon}\nabla q^{\epsilon}=({\rm curl\,}{\mathbf{H}^{\epsilon}})\times{\mathbf{H}^{\epsilon}}+{\rm div}\Psi^{\epsilon},$ (1.17) $\displaystyle\partial_{t}{\mathbf{H}}^{\epsilon}-{\rm curl\,}({\mathbf{u}}^{\epsilon}\times\mathbf{H}^{\epsilon})=0,\quad{\rm div}\mathbf{H}^{\epsilon}=0,$ (1.18) $\displaystyle b^{\epsilon}(S^{\epsilon},\epsilon q^{\epsilon})(\partial_{t}S^{\epsilon}+({\mathbf{u}}^{\epsilon}\cdot\nabla)S^{\epsilon})=\epsilon^{2}\Psi^{\epsilon}:\nabla{\mathbf{u}}^{\epsilon},$ (1.19) where we have used the abbreviations $\Psi^{\epsilon}=2\mu^{\epsilon}\mathbb{D}({\mathbf{u}}^{\epsilon})+\lambda^{\epsilon}{\rm div}{\mathbf{u}}^{\epsilon}\;\mathbf{I}_{d}$ and $\displaystyle a^{\epsilon}(S^{\epsilon},\epsilon q^{\epsilon})$ $\displaystyle:=A(S^{\epsilon},\underline{p}e^{\epsilon q^{\epsilon}})\underline{p}e^{\epsilon q^{\epsilon}}=\frac{\underline{p}e^{\epsilon q^{\epsilon}}}{R(S^{\epsilon},\underline{p}e^{\epsilon q^{\epsilon}})}\cdot\frac{\partial R(S^{\epsilon},s)}{\partial s}\Big{\|}_{s=\underline{p}e^{\epsilon q^{\epsilon}}},$ (1.20) $\displaystyle r^{\epsilon}(S^{\epsilon},\epsilon q^{\epsilon})$ $\displaystyle:=\frac{R(S^{\epsilon},\underline{p}e^{\epsilon q^{\epsilon}})}{\underline{p}e^{\epsilon q^{\epsilon}}},\quad b^{\epsilon}(S^{\epsilon},\epsilon q^{\epsilon}):=R(S^{\epsilon},\epsilon q^{\epsilon})\Theta(S^{\epsilon},\epsilon q^{\epsilon}).$ (1.21) Formally, we obtain from (1.16) and (1.17) that $\nabla q^{\epsilon}\rightarrow 0$ and ${\rm div}{\mathbf{u}}^{\epsilon}=0$ as $\epsilon\rightarrow 0$. Applying the operator _curl_ to (1.17), using the fact that ${\rm curl\,}\nabla=0$, and letting $\epsilon\rightarrow 0$ and $\mu^{\epsilon}\rightarrow\mu>0$, we obtain $\displaystyle{\rm curl\,}\big{(}r(\bar{S},0)(\partial_{t}{\mathbf{v}}+{\mathbf{v}}\cdot\nabla{\mathbf{v}})-({\rm curl\,}\bar{\mathbf{H}})\times\bar{\mathbf{H}}-\mu\Delta{\mathbf{v}}\big{)}=0,$ where we have assumed that $(S^{\epsilon},q^{\epsilon},{\mathbf{u}}^{\epsilon},\mathbf{H}^{\epsilon})$ and $r^{\epsilon}(S^{\epsilon},\epsilon q^{\epsilon})$ converge to $(\bar{S},0,{\mathbf{v}},\bar{\mathbf{H}})$ and $r(\bar{S},0)$ in some sense, respectively. Finally, applying the identity $\displaystyle{\rm curl\,}({{\mathbf{u}}}\times{\mathbf{H}})={{\mathbf{u}}}({\rm div}{\mathbf{H}})-{\mathbf{H}}({\rm div}{{\mathbf{u}}})+({\mathbf{H}}\cdot\nabla){{\mathbf{u}}}-({{\mathbf{u}}}\cdot\nabla){\mathbf{H}},$ (1.22) we expect to get the following incompressible non-isentropic MHD equations $\displaystyle r(\bar{S},0)(\partial_{t}{\mathbf{v}}+({\mathbf{v}}\cdot\nabla){\mathbf{v}})-({\rm curl\,}\bar{\mathbf{H}})\times\bar{\mathbf{H}}+\nabla\pi=\mu\Delta{\mathbf{v}},$ (1.23) $\displaystyle\partial_{t}\bar{\mathbf{H}}+({{\mathbf{v}}}\cdot\nabla)\bar{\mathbf{H}}-(\bar{\mathbf{H}}\cdot\nabla){{\mathbf{v}}}=0,$ (1.24) $\displaystyle\partial_{t}\bar{S}+({\mathbf{v}}\cdot\nabla)\bar{S}=0,$ (1.25) $\displaystyle{\rm div}\,{\mathbf{v}}=0,\quad{\rm div}\bar{\mathbf{H}}=0$ (1.26) for some function $\pi$. The aim of this paper is to establish the above limit process rigorously in the whole space $\mathbb{R}^{d}$. Before stating our main results, we review the previous related works. We begin with the results for the Euler and Navier-Stokes equations. For well- prepared initial data, Schochet [33] obtained the convergence of the compressible non-isentropic Euler equations to the incompressible non- isentropic Euler equations in a bounded domain for local smooth solutions. For general initial data, Métivier and Schochet [27] proved rigorously the incompressible limit of the compressible non-isentropic Euler equations in the whole space ${\mathbb{R}}^{d}$. There are two key points in the article [27]. First, they obtained the uniform estimates in Sobolev norms for the acoustic component of the solutions, which are propagated by a wave equation with unknown variable coefficients. Second, they proved that the local energy of the acoustic wave decays to zero in the whole space case. This approach was extended to the non-isentropic Euler equations in the exterior domain and the full Navier-Stokes equations in the whole space by Alazard in [1] and [2], respectively, and to the dispersive Navier-Stokes equations by Levermore, Sun and Trivisa [26]. For the spatially periodic case, Métivier and Schochet [28] showed the incompressible limit of the one-dimensional non-isentropic Euler equations with general data. Compared to the non-isentropic case, the treatment of the propagation of oscillations in the isentropic case is simpler and there are many works on this topic. For example, see Ukai [35], Asano [3], Desjardins and Grenier [7] in the whole space; Isozaki [16, 17] on the exterior domain; Iguchi [15] on the half space; Schochet [32] and Gallagher [11] in a periodic domain; and Lions and Masmoudi [30], and Desjardins, et al. [8] in a bounded domain. Recently, Jiang and Ou [22] investigated the incompressible limit of the non-isentropic Navier-Stokes equations with zero heat conductivity and well-prepared initial data in three-dimensional bounded domains. The justification of the incompressible limit of the non-isentropic Euler or Navier-Stokes equations with general initial data in a bounded domain or a multi-dimensional periodic domain is still open. The interested reader can refer to [5] for formal computations on the case of viscous polytropic gases and [28, 4] for some analysis on the non-isentropic Euler equations in a multi-dimensional periodic domain. For more results on the incompressible limit of the Euler and Navier-Stokes equations, please see the monograph [9] and the survey articles [6, 31, 34]. For the isentropic compressible MHD equations, the justification of the low Mach limit was given in several aspects. In [23], Klainerman and Majda first studied the incompressible limit of the isentropic compressible ideal MHD equations in the spatially periodic case with well-prepared initial data. Recently, the incompressible limit of the isentropic viscous (including both viscosity and magnetic diffusivity) of compressible MHD equations with general data was studied in [14, 18, 19]. In [14], Hu and Wang obtained the convergence of weak solutions of the compressible viscous MHD equations in bounded, spatially periodic domains and the whole space, respectively. In [18], the authors employed the modulated energy method to verify the limit of weak solutions of the compressible MHD equations in the torus to the strong solution of the incompressible viscous or partial viscous MHD equations (the shear viscosity coefficient is zero but the magnetic diffusion coefficient is a positive constant). In [19], the authors obtained the convergence of weak solutions of the viscous compressible MHD equations to the strong solution of the ideal incompressible MHD equations in the whole space by using the dispersion property of the wave equation if both the shear viscosity and the magnetic diffusion coefficients go to zero. For the full compressible MHD equations, the incompressible limit in the framework of the so-called variational solutions was studied in [24, 25, 29]. Recently, the authors [20] justified rigourously the low Mach number limit of classical solutions to the ideal or full compressible non-isentropic MHD equations with small entropy or temperature variations. When the heat conductivity and large temperature variations are present, the low Mach number limit for the full compressible non-isentropic MHD equations justified in [21]. We emphasize here that the arguments in [21] are different from the present paper and depend essentially on positivity of fluid viscosity and magnetic diffusivity coefficients. As aforementioned, in this paper we want to establish rigorously the limit as $\epsilon\to 0$ to the system (1.16)–(1.19) for $\mu_{\epsilon}\to\mu>0$. In this case, the magnetic equation is purely hyperbolic due to the lack of magnetic diffusivity. The first-order derivatives of $\mathbf{H}^{\epsilon}$ in the momentum equation and magnetic equation cannot be controlled. It is very hard to study the system (1.16)–(1.19). To the author’s knowledge, there is very few mathematical analysis on the system (1.16)–(1.19) with fixed or unfixed $\epsilon$, even for the isentropic case. Our main idea is trying to make full use of the fluid viscosities to control the higher order derivatives of the magnetic field. Now, we supplement the system (1.16)–(1.19) with initial conditions $\displaystyle(S^{\epsilon},q^{\epsilon},{\mathbf{u}}^{\epsilon},\mathbf{H}^{\epsilon})\|_{t=0}=(S^{\epsilon}_{0},q^{\epsilon}_{0},{\mathbf{u}}^{\epsilon}_{0},\mathbf{H}^{\epsilon}_{0})$ (1.27) and state the main results as follows. ###### Theorem 1.1. Let $s>d/2+2$ be an integer. Assume that $\mu^{\epsilon}\rightarrow\mu>0$ and $\lambda^{\epsilon}\rightarrow\lambda$ as $\epsilon\to 0$. Suppose that the initial data $(S^{\epsilon}_{0},q^{\epsilon}_{0},{\mathbf{u}}^{\epsilon}_{0},\mathbf{H}^{\epsilon}_{0})$ satisfy $\displaystyle\\|(S^{\epsilon}_{0},q^{\epsilon}_{0},{\mathbf{u}}^{\epsilon}_{0},\mathbf{H}^{\epsilon}_{0})\\|_{H^{s}({\mathbb{R}}^{d})}\leq M_{0}.$ (1.28) Then there exists a $T>0$ such that for any $\epsilon\in(0,1]$, the Cauchy problem (1.16)–(1.19), (1.27) has a unique solution $(S^{\epsilon},q^{\epsilon},{\mathbf{u}}^{\epsilon},\mathbf{H}^{\epsilon})\in C^{0}([0,T],H^{s}({\mathbb{R}}^{d}))$, and there exists a positive constant $N$, depending only on $T$ and $M_{0}$, such that $\displaystyle\\|\big{(}S^{\epsilon},q^{\epsilon},{\mathbf{u}}^{\epsilon},\mathbf{H}^{\epsilon}\big{)}(t)\\|_{H^{s}({\mathbb{R}}^{d})}\leq N,\quad\forall\,t\in[0,T].$ (1.29) Furthermore, if there exist positive constants $\underline{S}$, $N_{0}$ and $\delta$ such that $S^{\epsilon}_{0}(x)$ satisfies $\|S^{\epsilon}_{0}(x)-\underline{S}\,\,\|\leq{N}_{0}\|x\|^{-1-\delta},\quad\|\nabla S^{\epsilon}_{0}(x)\|\leq N_{0}\|x\|^{-2-\delta},$ (1.30) then the sequence of solutions $(S^{\epsilon},q^{\epsilon},{\mathbf{u}}^{\epsilon},\mathbf{H}^{\epsilon})$ converges weakly in $L^{\infty}(0,T;H^{s}({\mathbb{R}}^{d}))$ and strongly in $L^{2}(0,T;H^{s^{\prime}}_{\mathrm{loc}}({\mathbb{R}}^{d}))$ for all $s^{\prime}<s$ to a limit $(\bar{S},0,{\mathbf{v}},\bar{\mathbf{H}})$, where $(\bar{S},{\mathbf{v}},\bar{\mathbf{H}})$ is the unique solution in $C([0,T],H^{s}({\mathbb{R}}^{d}))$ of (1.23)–(1.26) with initial data $(\bar{S},{\mathbf{v}},\bar{\mathbf{H}})\|_{t=0}=(S_{0},{\mathbf{w}}_{0},\mathbf{H}_{0})$, where ${\mathbf{w}}_{0}\in H^{s}({\mathbb{R}}^{d})$ is determined by ${\rm div}\,{\mathbf{w}}_{0}=0,\,\;{\rm curl\,}(r(S_{0},0){\mathbf{w}}_{0})={\rm curl\,}(r(S_{0},0){\mathbf{v}}_{0}),\;\,r(S_{0},0):=\lim_{\epsilon\rightarrow 0}r^{\epsilon}(S^{\epsilon}_{0},0).$ (1.31) The function $\pi\in C([0,T]\times{\mathbb{R}}^{d})$ satisfies $\nabla\pi\in C([0,T],H^{s-1}({\mathbb{R}}^{d}))$. We briefly describe the strategy of the proof. The proof of Theorems 1.1 includes two main steps: the uniform estimates of the solutions, and the convergence from the original scaling equations to the limiting ones. Once we have established the uniform estimates (1.29) of the solutions in Theorems 1.1, the convergence of solutions is easily proved by using the local energy decay theorem for fast waves in the whole space, which is shown by Métivier and Schochet in [27]. Thus, the main task in the present paper is to obtain the uniform estimates (1.29). For this purpose, we shall modify the approach developed in [27]. In fact, due to the strong coupling of hydrodynamic motion and magnetic field, and the lack of magnetic diffusivity, new difficulties arise in obtaining the uniform estimates for the solutions to (1.16)–(1.19), (1.27). First of all, when we perform the operator $(\\{E^{\epsilon}\\}^{-1}L(\partial_{x}))^{\sigma}$ to the continuity and momentum equations, or the operator curl to the momentum equations, one order more spatial derivatives arise for the magnetic field, and this prevents us from closing the energy estimates. Second, since the coefficients of the acoustic wave equations depend on the entropy, we could not get the estimates of $\\|{\mathbf{u}}^{\epsilon}\\|_{L^{2}(0,T;H^{s+1})}$ directly from the system. The ideas to overcome these difficulties here are the following: We transfer one spatial derivative from the magnetic field to the velocity with the help of the special coupled way between magnetic field and fluid velocity. Then, to control $\\|{\mathbf{u}}^{\epsilon}\\|_{L^{2}(0,T;H^{s+1})}$, we employ a kind of Helmholtz decomposition of the velocity. Third, we make full use the special structure of the magnetic field equation and the estimates on ${\mathbf{u}}$ to control $\\|\mathbf{H}^{\epsilon}\\|_{L^{\infty}(0,T;H^{s})}$. We point out that our arguments in this paper can be modified slightly to the case of the the compressible non-isentropic MHD equations with infinite Reynolds number. We shall give a brief discussion in Section 5. This paper is arranged as follows. In Section 2, we give notations, recall basic facts, and present commutators estimates. In Section 3 we establish the uniform boundeness of the solutions and prove the existence part of Theorem 1.1. In Section 4, we use the decay of the local energy to the acoustic wave equations to prove the convergent part of Theorem 1.1. In the last section, we consider the incompressible limit to the compressible non-isentropic MHD equations with infinite Reynolds number. ## 2\. Preliminary We give notations and recall basic facts which will be used frequently in the proofs. (1) We denote $\langle\cdot,\cdot\rangle$ the standard inner product in $L^{2}({\mathbb{R}}^{d})$ with $\langle f,f\rangle=\\|f\\|^{2}$ and $H^{k}$ the usual Sobolev space $W^{k,2}$ with norm $\\|\cdot\\|_{k}$, in particular, $\\|\cdot\\|_{0}=\\|\cdot\\|$. The notation $\\|(A_{1},\dots,A_{k})\\|$ means the summation of $\\|A_{i}\\|$ ($i=1,\cdots,k$), and it also applies to other norms. For a multi-index $\alpha=(\alpha_{1},\dots,\alpha_{d})$, we denote $D^{\alpha}=\partial^{\alpha_{1}}_{x_{1}}\dots\partial^{\alpha_{d}}_{x_{d}}$ and $\|\alpha\|=\|\alpha_{1}\|+\cdots+\|\alpha_{d}\|$. We also omit the spatial domain ${\mathbb{R}}^{d}$ in integrals for convenience. We use the symbols $K$ or $C_{0}$ to denote the generic positive constants, and $C(\cdot)$ and $\tilde{C}(\cdot)$ to denote the smooth functions, which may vary from line to line. (2) For a scalar function $f$ and vector functions $\mathbf{a}$, $\mathbf{b}$ and $\mathbf{c}$, we have the following basic vector identities: $\displaystyle{\rm div}(\mathbf{a}\times\mathbf{b})$ $\displaystyle=\mathbf{b}\cdot{\rm curl\,}\mathbf{a}-\mathbf{a}\cdot{\rm curl\,}\mathbf{b},$ (2.1) $\displaystyle\nabla(\|\mathbf{a}\|^{2})$ $\displaystyle=2(\mathbf{a}\cdot\nabla)\mathbf{a}+2\mathbf{a}\times{\rm curl\,}\mathbf{a},$ (2.2) $\displaystyle{\rm curl\,}(f\mathbf{a})$ $\displaystyle=f\cdot{\rm curl\,}\mathbf{a}-\nabla f\times\mathbf{a},$ (2.3) $\displaystyle{\rm curl\,}(\mathbf{a}\times\mathbf{b})$ $\displaystyle=(\mathbf{b}\cdot\nabla)\mathbf{a}-(\mathbf{a}\cdot\nabla)\mathbf{b}+\mathbf{a}({\rm div}\mathbf{b})-\mathbf{b}({\rm div}\mathbf{a}),$ (2.4) $\displaystyle{\rm div}\big{(}(\mathbf{a}\times\mathbf{b})\times\mathbf{c}\big{)}$ $\displaystyle=\mathbf{c}\cdot{\rm curl\,}(\mathbf{a}\times\mathbf{b})-(\mathbf{a}\times\mathbf{b})\cdot{\rm curl\,}\mathbf{c}.$ (2.5) (3) We have the following well-known nonlinear estimates [12]. (i) Let $\alpha=(\alpha_{1},\alpha_{2},\alpha_{d})$ be a multi-index such that $\|\alpha\|=k$. Then, for all $\sigma\geq 0$, and $f,g\in H^{k+\sigma}({\mathbb{R}}^{d})$, there exists a generic constant $C_{0}$ such that $\displaystyle\\|[f,\partial^{\alpha}]g\\|_{H^{\sigma}}\leq$ $\displaystyle C_{0}(\\|f\\|_{W^{1,\infty}}\\|g\\|_{H^{\sigma+k-1}}+\\|f\\|_{H^{\sigma+k}}\\|g\\|_{L^{\infty}}).$ (2.6) (ii) For integers $k\geq 0$, $l\geq 0$, $k+l\leq\sigma$ and $\sigma>d/2$, the product maps continuously $H^{\sigma-k}({\mathbb{R}}^{d})\times H^{\sigma-l}({\mathbb{R}}^{d})$ to $H^{\sigma-k-l}({\mathbb{R}}^{d})$ and $\displaystyle\\|uv\\|_{\sigma-k-l}\leq K\\|u\\|_{\sigma-k}\\|v\\|_{\sigma-l}.$ (2.7) (iii) Let $\sigma>d/2$ be an integer. Assume that $F(u)$ is a smooth function such that $F(0)=0$ and $u\in H^{\sigma}({\mathbb{R}}^{d})$, then $F(u)\in H^{\sigma}({\mathbb{R}}^{d})$ and its norm is bounded by $\displaystyle\\|F(u)\\|_{\sigma}\leq C(\\|u\\|_{\sigma})\\|u\\|_{\sigma},$ (2.8) where $C(\cdot)$ is independent of $u$ and maps $[0,\infty)$ into $[0,\infty)$. ## 3\. Uniform estimates In this section and the first part of the next section we assume that $\mu^{\epsilon}\equiv\mu>0$ and $\lambda^{\epsilon}\equiv\lambda$ for simplicity of the presentation. The general case can be treated by a slight modification in the arguments presented here. In view of [27] and the classical local existence results obtained by Vol’pert and Khudiaev [36] for hyperbolic-parabolic systems, the key point in the proof of the existence part of Theorem 1.1 is to establish the uniform estimate (1.29), which can be deduced from the following _a priori_ estimate. ###### Theorem 3.1. For any $\epsilon>0$, let $(S^{\epsilon},q^{\epsilon},{\mathbf{u}}^{\epsilon},\mathbf{H}^{\epsilon})\in C([0,T],H^{s}({\mathbb{R}}^{d}))$ be the solution to (1.16)–(1.19). Then there exists an increasing $C(\cdot)$ from $[0,\infty)$ to $[0,\infty)$, such that $\displaystyle\mathcal{M}_{\epsilon}(T)\leq C_{0}+(T+\epsilon)C(\mathcal{M}_{\epsilon}(T)),$ (3.1) where $\displaystyle\mathcal{M}_{\epsilon}(T):=\,$ $\displaystyle{\mathcal{N}_{\epsilon}(T)}^{2}+\int_{0}^{T}\\|{\mathbf{u}}^{\epsilon}\\|_{s+1}^{2}d\tau,$ (3.2) with $\displaystyle\mathcal{N}_{\epsilon}(T):=\,$ $\displaystyle\sup_{t\in[0,T]}\\|(S^{\epsilon},q^{\epsilon},{\mathbf{u}}^{\epsilon},\mathbf{H}^{\epsilon})(t)\\|_{s}.$ (3.3) The remainder of this section is devoted to establishing (3.1). In the calculations that follow, we always suppose that the assumptions in Theorem 1.1 hold. We consider a solution $(S^{\epsilon},q^{\epsilon},{\mathbf{u}}^{\epsilon},\mathbf{H}^{\epsilon})$ to the problem (1.16)–(1.19), (1.27) on $C([0,T],H^{s}({\mathbb{R}}^{d}))$ with initial data satisfying (1.28). The main idea for proving the uniform estimate (3.1) is motivated by the work [27] where the operator $(\\{E^{\epsilon}\\}^{-1}L(\partial_{x}))^{m}$ is introduced to control the acoustic components of velocity for the Euler equations. When the strong coupling of the fluid and magnetic filed is present, however, the arguments in [27] cannot be directly applied to get a uniform estimate of the acoustic parts due to lack of magnetic diffusion in the magnetic equation. Instead, here we transfer one order spatial derivative from $\mathbf{H}^{\epsilon}$ to ${\mathbf{u}}^{\epsilon}$, and then employ the fluid viscosity to control higher derivatives. We remark that the reason that these techniques work is due to the special structure of coupling between the fluid and magnetic fields. We begin with the estimate on the entropy $S^{\epsilon}$. ###### Lemma 3.2. There exist a constant $C_{0}>0$ and a function $C(\cdot)$, independent of $\epsilon$, such that for all $t\in(0,T]$, $\displaystyle\\|S^{\epsilon}(t)\\|^{2}_{s}\leq C_{0}+tC(\mathcal{M}_{\epsilon}(T))+\epsilon^{2}C(\mathcal{M}_{\epsilon}(T)).$ (3.4) ###### Proof. For the multi-index $\alpha$ satisfying $\|\alpha\|\leq s-1$, denote $f_{\alpha}=\partial_{x}^{\alpha}S^{\epsilon}$. In view of the positivity of $b^{\epsilon}(S^{\epsilon},\epsilon q^{\epsilon})$, we deduce from (1.19) that, $\displaystyle\partial_{t}f_{\alpha}+({\mathbf{u}}^{\epsilon}\cdot\nabla)f_{\alpha}=g_{\alpha}+\epsilon^{2}h_{\alpha},$ (3.5) where $\displaystyle g_{\alpha}=-[\partial_{x}^{\alpha},{\mathbf{u}}^{\epsilon}]\cdot\nabla S^{\epsilon},\;\;h_{\alpha}=\epsilon^{2}\partial^{\alpha}_{x}\left(\frac{\Psi^{\epsilon}:\nabla{\mathbf{u}}^{\epsilon}}{b^{\epsilon}(S^{\epsilon},\epsilon q^{\epsilon})}\right).$ The commutator inequality (2.6) and Sobolev embedding theorem imply that $\\|g_{\alpha}\\|\leq C(M_{\epsilon}(T))$. On the other hand, from the Sobolev embedding theorem and the Moser-type inequality [23] we get $\displaystyle\\|h_{\alpha}\\|$ $\displaystyle\leq K\Big{(}\\|(\Psi^{\epsilon}:\nabla{\mathbf{u}}^{\epsilon})\\|_{L^{\infty}}\Big{\\|}D^{s}\Big{(}\frac{1}{b^{\epsilon}}\Big{)}\Big{\\|}+\\|D^{s}(\Psi^{\epsilon}:\nabla{\mathbf{u}}^{\epsilon})\\|\Big{\\|}\frac{1}{b^{\epsilon}}\Big{\\|}_{L^{\infty}}\Big{)}$ $\displaystyle\leq C(\mathcal{N}_{\epsilon}(T))+C(\mathcal{N}_{\epsilon}(T))\\|{\mathbf{u}}^{\epsilon}\\|_{s+1}.$ Multiplying (3.5) by $f_{\alpha}$ and integrating over $[0,t]\times{\mathbb{R}}^{d}$ with $t\leq T$, we obtain $\displaystyle\\|f_{\alpha}(t)\\|^{2}\leq$ $\displaystyle\\|f_{\alpha}(0)\\|^{2}+\\|\partial_{x}{\mathbf{u}}^{\epsilon}\\|_{L^{\infty}((0,t)\times{\mathbb{R}}^{d})}\int^{t}_{0}\\|f_{\alpha}(\tau)\\|^{2}d\tau$ $\displaystyle+2\int^{t}_{0}\\|g_{\alpha}(\tau)\\|\,\\|f_{\alpha}(\tau)\\|d\tau+2\epsilon^{2}\int^{t}_{0}\\|h_{\alpha}(\tau)\\|\,\\|f_{\alpha}(\tau)\\|d\tau$ $\displaystyle\leq$ $\displaystyle C_{0}+tC(\mathcal{M}_{\epsilon}(T))+\epsilon^{2}C(\mathcal{M}_{\epsilon}(T)),$ where we have used Young’s inequality and the embedding $H^{\sigma}\hookrightarrow L^{\infty}$ for $\sigma>d/2$. The conclusion then follows by adding up these estimates for all $\|\alpha\|\leq s-1$. ∎ The following $L^{2}$-bound of $(q^{\epsilon},{\mathbf{u}}^{\epsilon},\mathbf{H}^{\epsilon})$ can be obtained directly using the energy method due to the skew-symmetry of the singular term in the system and the special structure of coupling between the magnetic field and fluid velocity. This $L^{2}$-bound is very important in our arguments, since the induction analysis will be used to get the desired Sobolev estimates. ###### Lemma 3.3. There exist constants $C_{0}>0$ and $0<\xi_{1}<\mu$, and a function $C(\cdot)$ independent of $\epsilon$, such that for all $t\in[0,T]$, $\displaystyle\\|(q^{\epsilon},{\mathbf{u}}^{\epsilon},\mathbf{H}^{\epsilon})(t)\\|^{2}+\xi_{1}\int^{t}_{0}\\|\nabla{\mathbf{u}}^{\epsilon}(\tau)\\|^{2}d\tau\leq C_{0}+tC(\mathcal{M}_{\epsilon}(T)).$ (3.6) ###### Proof. Multiplying (1.16) by $q^{\epsilon}$, (1.17) by ${\mathbf{u}}^{\epsilon}$, and (1.18) by $\mathbf{H}^{\epsilon}$, respectively, integrating over ${\mathbb{R}}^{d}$, and adding the resulting equations together, we obtain $\displaystyle\langle a^{\epsilon}\partial_{t}q^{\epsilon},q^{\epsilon}\rangle+\langle r^{\epsilon}\partial_{t}{\mathbf{u}}^{\epsilon},{\mathbf{u}}^{\epsilon}\rangle+\langle\partial_{t}\mathbf{H}^{\epsilon},\mathbf{H}^{\epsilon}\rangle+\mu\\|\nabla{\mathbf{u}}^{\epsilon}\\|^{2}+(\mu+\lambda)\\|{\rm div}{\mathbf{u}}^{\epsilon}\\|^{2}$ $\displaystyle\qquad+\langle a^{\epsilon}({\mathbf{u}}^{\epsilon}\cdot\nabla)q^{\epsilon},q^{\epsilon}\rangle+\langle r^{\epsilon}({\mathbf{u}}^{\epsilon}\cdot\nabla){\mathbf{u}}^{\epsilon},{\mathbf{u}}^{\epsilon}\rangle$ $\displaystyle=\int[({\rm curl\,}{\mathbf{H}^{\epsilon}})\times{\mathbf{H}^{\epsilon}}]\cdot{\mathbf{u}}^{\epsilon}\;dx+\int{\rm curl\,}({\mathbf{u}}^{\epsilon}\times\mathbf{H}^{\epsilon})\cdot\mathbf{H}^{\epsilon}\;dx.$ (3.7) Here the singular terms involving $1/\epsilon$ are canceled. Using the identity (1.7) and integrating by parts, we immediately obtain that $\displaystyle\int[({\rm curl\,}{\mathbf{H}^{\epsilon}})\times{\mathbf{H}^{\epsilon}}]\cdot{\mathbf{u}}^{\epsilon}\;dx+\int{\rm curl\,}({\mathbf{u}}^{\epsilon}\times\mathbf{H}^{\epsilon})\cdot\mathbf{H}^{\epsilon}\;dx=0.$ In view of the positivity and smoothness of $a^{\epsilon}(S^{\epsilon},\epsilon q^{\epsilon})$ and $r^{\epsilon}(S^{\epsilon},\epsilon q^{\epsilon})$, we get directly from (1.16), (1.19) and (2.8) that $\displaystyle\\|\partial_{t}S^{\epsilon}\\|_{s-1}\leq C(\mathcal{N}_{\epsilon}(T)),\quad\\|\epsilon\partial_{t}q^{\epsilon}\\|_{s-1}\leq C(\mathcal{N}_{\epsilon}(T)),$ (3.8) while by the Sobolev embedding theorem, we find that $\displaystyle\\|(\partial_{t}a^{\epsilon},\partial_{t}r^{\epsilon})\\|_{L^{\infty}}\leq\\|(\partial_{t}a^{\epsilon},\partial_{t}r^{\epsilon})\\|_{s-2}\leq C(\mathcal{N}_{\epsilon}(T)).$ (3.9) By the definition of $\mathcal{N}_{\epsilon}(T)$ and the Sobolev embedding theorem, it is easy to see that $\displaystyle\\|(\nabla a^{\epsilon},\nabla r^{\epsilon})\\|_{L^{\infty}}\leq C(\mathcal{N}_{\epsilon}(T)).$ Since $\mu>0,2\mu+d\lambda>0$, there exists a positive constant $\kappa_{1}$ such that $\displaystyle\mu\\|\nabla{\mathbf{u}}^{\epsilon}\\|^{2}+(\mu+\lambda)\\|{\rm div}{\mathbf{u}}^{\epsilon}\\|^{2}\geq\kappa_{1}\\|\nabla{\mathbf{u}}^{\epsilon}\\|^{2}.$ Thus, from (3.7) we get that $\displaystyle\langle a^{\epsilon}q^{\epsilon},q^{\epsilon}\rangle+$ $\displaystyle\langle r^{\epsilon}{\mathbf{u}}^{\epsilon},{\mathbf{u}}^{\epsilon}\rangle+\langle\mathbf{H}^{\epsilon},\mathbf{H}^{\epsilon}\rangle+\kappa_{1}\int^{t}_{0}\\|\nabla{\mathbf{u}}^{\epsilon}(\tau)\\|^{2}d\tau$ $\displaystyle\leq$ $\displaystyle\big{\\{}\langle a^{\epsilon}q^{\epsilon},q^{\epsilon}\rangle+\langle r^{\epsilon}{\mathbf{u}}^{\epsilon},{\mathbf{u}}^{\epsilon}\rangle+\langle\mathbf{H}^{\epsilon},\mathbf{H}^{\epsilon}\rangle\big{\\}}\big{\|}_{t=0}$ $\displaystyle+C(\mathcal{M}_{\epsilon}(T))\int^{t}_{0}\big{\\{}\|q^{\epsilon}(\tau)\|^{2}+\|{\mathbf{u}}^{\epsilon}(\tau)\|^{2}+\|\mathbf{H}^{\epsilon}(\tau)\|^{2}\big{\\}}d\tau.$ (3.10) Moreover, we have $\displaystyle\\|q^{\epsilon}\\|^{2}+\\|{\mathbf{u}}^{\epsilon}\\|^{2}$ $\displaystyle\leq\\|(a^{\epsilon})^{-1}\\|_{L^{\infty}}\langle a^{\epsilon}q^{\epsilon},q^{\epsilon}\rangle+\\|(r^{\epsilon})^{-1}\\|_{L^{\infty}}\langle r^{\epsilon}{\mathbf{u}}^{\epsilon},{\mathbf{u}}^{\epsilon}\rangle$ $\displaystyle\leq C_{0}(\langle a^{\epsilon}q^{\epsilon},q^{\epsilon}\rangle+\langle r^{\epsilon}{\mathbf{u}}^{\epsilon},{\mathbf{u}}^{\epsilon}\rangle),$ (3.11) since $a^{\epsilon}$ and $r^{\epsilon}$ are uniformly bounded away from zero. Applying Gronwall’s Lemma to (3), we conclude $\displaystyle\\|(q^{\epsilon},{\mathbf{u}}^{\epsilon},\mathbf{H}^{\epsilon})(t)\\|^{2}\leq C_{0}\\|(q^{\epsilon}_{0},{\mathbf{u}}^{\epsilon}_{0},\mathbf{H}^{\epsilon}_{0})\\|^{2}\exp\\{tC(\mathcal{M}_{\epsilon}(T))\\}.$ Therefore, the estimate (3.6) follows from an elementary inequality $\displaystyle e^{Ct}\leq 1+\tilde{C}t,\quad 0\leq t\leq T_{0},$ (3.12) where $T_{0}$ is some fixed constant. ∎ Concerning the desired higher order estimates, we cannot directly get them by differentiating the system as done in [27], since the coefficients $a^{\epsilon}(S^{\epsilon},\epsilon q^{\epsilon}),r^{\epsilon}(S^{\epsilon},\epsilon q^{\epsilon})$ and $b^{\epsilon}(S^{\epsilon},\epsilon q^{\epsilon})$ contain two scales $S^{\epsilon}$ and $\epsilon q^{\epsilon}$. We shall adapt and modify the techniques developed in [27] to derive the higher order estimates. Set $\displaystyle E^{\epsilon}(S^{\epsilon},\epsilon q^{\epsilon})=\left(\begin{array}[]{cc}a^{\epsilon}(S^{\epsilon},\epsilon q^{\epsilon})&0\\\ 0&r^{\epsilon}(S^{\epsilon},\epsilon q^{\epsilon})\mathbf{I}_{d}\\\ \end{array}\right),\quad{\mathbf{U}}^{\epsilon}=\left(\begin{array}[]{c}q^{\epsilon}\\\ {\mathbf{u}}^{\epsilon}\end{array}\right),$ $\displaystyle L(\partial_{x})=\left(\begin{array}[]{cc}0&{\rm div}\\\ \nabla&0\\\ \end{array}\right),$ where $\mathbf{I}_{d}$ denotes the $d\times d$ unit matrix. Let $L_{E^{\epsilon}}(\partial_{x})=\\{E^{\epsilon}\\}^{-1}L(\partial_{x})$ and $r_{0}(S^{\epsilon})=r^{\epsilon}(S^{\epsilon},0)$. Note that $r_{0}(S^{\epsilon})$ is smooth, positive, and bounded away from zero with respect to each $\epsilon$. First, using Lemma 3.2 and employing the same analysis as in [27], we have ###### Lemma 3.4. There exist constants $C_{1}>0$, $K>0$, and a function $C(\cdot)$, depending only on $M_{0}$, such that for all $\sigma\in[1,\dots,s]$ and $t\in[0,T]$, $\displaystyle\\|{\mathbf{U}}^{\epsilon}\\|_{\sigma}\leq K\\|L(\partial_{x}){{\mathbf{U}}}^{\epsilon}\\|_{\sigma-1}+\tilde{C}\big{(}\\|{\rm curl\,}(r_{0}{\mathbf{u}}^{\epsilon})\\|_{\sigma-1}+\\|{\mathbf{U}}^{\epsilon}\\|_{\sigma-1}\big{)}$ (3.13) and $\displaystyle\\|{\mathbf{U}}^{\epsilon}\\|_{\sigma}\leq\tilde{C}\big{\\{}\\|\\{L_{E^{\epsilon}}(\partial_{x})\\}^{\sigma}{{\mathbf{U}}}^{\epsilon}\\|_{0}+\\|{\rm curl\,}(r_{0}{\mathbf{u}}^{\epsilon})\\|_{\sigma-1}+\\|{\mathbf{U}}^{\epsilon}\\|_{\sigma-1}\big{\\}},$ (3.14) where $\tilde{C}:=C_{1}+tC(\mathcal{M}_{\epsilon}(T))+\epsilon C(\mathcal{M}_{\epsilon}(T))$. We remark that the inequalities (3.13) and (3.14) are similar to the well known Helmholtz decomposition, and the estimate on $\\|S^{\epsilon}(t)\\|^{2}_{s}$ in Lemma 3.2 plays a key role in the proof of Lemma 3.4. Our next task is to derive a bound on $\\|\\{L_{E^{\epsilon}}(\partial_{x})\\}^{\sigma}{{\mathbf{U}}}^{\epsilon}\\|_{0}$ and $\\|{\rm curl\,}(r_{0}{\mathbf{u}}^{\epsilon})\\|_{\sigma-1}$ by induction arguments. Let ${\mathbf{W}}_{\sigma}^{\epsilon}=\\{L_{E^{\epsilon}}(\partial_{x})\\}^{\sigma}(0,{\mathbf{u}}^{\epsilon})^{\top}$. We first show the following estimate. ###### Lemma 3.5. There exist a sufficiently small constant $\eta_{1}>0$ and two constants $C_{0}>0$, $0<\xi_{2}<\mu$, and a function $C(\cdot)$ from $[0,\infty)$ to $[0,\infty)$, independent of $\epsilon$, such that for all $\sigma\in[1,\dots,s]$ and $t\in[0,T]$, $\displaystyle\\|\\{L_{E^{\epsilon}}(\partial_{x})\\}^{\sigma}{{\mathbf{U}}}^{\epsilon}(t)\\|^{2}$ $\displaystyle+\frac{\xi_{2}}{2}\int_{0}^{t}\\|\nabla{\mathbf{W}}_{\sigma}^{\epsilon}\\|^{2}(\tau)d\tau$ $\displaystyle\leq C_{0}+tC(\mathcal{N}_{\epsilon}(T))+\eta_{1}\int_{0}^{t}\\|{\mathbf{u}}^{\epsilon}(\tau)\\|_{s+1}^{2}d\tau.$ (3.15) ###### Proof. Let ${\mathbf{U}}^{\epsilon}_{\sigma}:=\\{L_{E^{\epsilon}}(\partial_{x})\\}^{\sigma}{\mathbf{U}}^{\epsilon}$, $\sigma\in\\{0,\dots,s\\}.$ For simplicity, we set $\mathcal{M}:=\mathcal{M}_{\epsilon}(T)$, $\mathcal{N}:=\mathcal{N}_{\epsilon}(T)$, and $E:=E^{\epsilon}(S^{\epsilon},\epsilon q^{\epsilon})$. The case $k=0$ is an immediate consequence of Lemma 3.3. It is easy to verify that the operator $L_{E}(\partial_{x})$ is bounded from $H^{k}$ to $H^{k-1}$ for $k\in\\{1,\dots,s+1\\}$. Note that the equations (1.16), (1.17) can be written as $\displaystyle(\partial_{t}+{\mathbf{u}}^{\epsilon}\cdot\nabla){\mathbf{U}}^{\epsilon}+\frac{1}{\epsilon}E^{-1}L(\partial_{x}){\mathbf{U}}^{\epsilon}=E^{-1}(\mathbf{J}^{\epsilon}+\mathbf{V}^{\epsilon})$ (3.16) with $\displaystyle\mathbf{J}^{\epsilon}=\left(\begin{array}[]{c}0\\\ ({\rm curl\,}{\mathbf{H}^{\epsilon}})\times{\mathbf{H}^{\epsilon}}\end{array}\right),\quad\mathbf{V}^{\epsilon}=\left(\begin{array}[]{c}0\\\ {\rm div}\Psi^{\epsilon}\end{array}\right).$ For $\sigma\geq 1$, we commute the operator $\\{L_{E}\\}^{\sigma}$ with (3.16) and multiply the resulting equation by $E$ to infer that $\displaystyle E(\partial_{t}+{\mathbf{u}}^{\epsilon}\cdot\nabla){\mathbf{U}}^{\epsilon}_{\sigma}+\frac{1}{\epsilon}L(\partial_{x}){\mathbf{U}}^{\epsilon}_{\sigma}=E(\mathbf{f}_{\sigma}+\mathbf{g}_{\sigma}+\mathbf{h}_{\sigma}),$ (3.17) where $\displaystyle\mathbf{f}_{\sigma}:=$ $\displaystyle[\partial_{t}+{\mathbf{u}}^{\epsilon}\cdot\nabla,\\{L_{E}\\}^{\sigma}]{\mathbf{U}}^{\epsilon},$ $\displaystyle\mathbf{g}_{\sigma}:=$ $\displaystyle\\{L_{E}\\}^{\sigma}(E^{-1}\mathbf{J}^{\epsilon}),$ $\displaystyle\mathbf{h}_{\sigma}:=$ $\displaystyle\\{L_{E}\\}^{\sigma}(E^{-1}\mathbf{V}^{\epsilon}).$ Multiplying (3.17) by ${\mathbf{U}}^{\epsilon}_{\sigma}$ and integrating over $[0,t]\times{\mathbb{R}}^{d}$ with $t\leq T$, noticing that the singular terms cancel out since $L(\partial_{x})$ is skew-adjoint, we use the inequalities (3.8) and (3.9), and Cauchy-Schwarz’s inequality to deduce that $\displaystyle\langle E(t){\mathbf{U}}_{\sigma}^{\epsilon}(t),{\mathbf{U}}_{\sigma}^{\epsilon}(t)\rangle\leq$ $\displaystyle\langle E(0){\mathbf{U}}_{\sigma}^{\epsilon}(0),{\mathbf{U}}_{\sigma}^{\epsilon}(0)\rangle+C(\mathcal{M})\int^{t}_{0}\\|{\mathbf{U}}^{\epsilon}_{\sigma}(\tau)\\|^{2}d\tau$ $\displaystyle+\int^{t}_{0}\\|\mathbf{f}_{\sigma}(\tau)\\|^{2}d\tau+2\int^{t}_{0}\int_{{\mathbb{R}}^{d}}(E(\mathbf{g}_{\sigma}+\mathbf{h}_{\sigma}){\mathbf{U}}_{\sigma}^{\epsilon})(\tau)d\tau.$ (3.18) Following the proof process of Lemma 2.4 in [27], we obtain that $\displaystyle\\|\mathbf{f}_{k}(t)\\|\leq C(\mathcal{N}_{\epsilon}(t)).$ (3.19) Now we estimate the nonlinear term in (3) involving $\mathbf{g}_{\sigma}$. We expand $\mathbf{g}_{\sigma}$ as follows $\displaystyle\mathbf{g}_{\sigma}=$ $\displaystyle\,\sum^{d}_{i,j=1}\sum_{\|\alpha\|=\sigma+1}\\{E^{-1}\\}^{k+1}\partial_{x}^{\alpha}H_{i}^{\epsilon}H^{\epsilon}_{j}$ $\displaystyle+\sum^{d}_{i,j=1}\sum_{\Lambda_{1}}\sum_{\Lambda_{2}}\\{E^{-1}\\}^{l}\partial_{x}^{\beta_{1}}\\{E^{-1}\\}\cdots\partial_{x}^{\beta_{k}}\\{E^{-1}\\}\partial_{x}^{\gamma}H_{i}^{\epsilon}\partial_{x}^{\delta}H^{\epsilon}_{j}$ $\displaystyle:=$ $\displaystyle\,B_{1}+B_{2},$ where $\displaystyle\Lambda_{1}=\\{(\beta_{1},\cdots,\beta_{k},\gamma,\delta)\big{\|}\|\beta_{1}\|+\cdots+\|\beta_{k}\|+\|\gamma\|+\|\delta\|\leq k+1,0<\|\gamma\|\leq k,\|\delta\|\leq k\\},$ $\displaystyle\Lambda_{2}=\\{l\big{\|}l=k+1-(\|\beta_{1}\|+\cdots+\|\beta_{k}\|),(\beta_{1},\cdots,\beta_{k},0,0)\in\Lambda_{1}\\}.$ Since there is no magnetic diffusion in the system, we cannot deal with directly the terms involving $B_{1}$. Instead, we transform one spatial derivative to ${\mathbf{U}}_{\sigma}^{\epsilon}$. Integrating by parts, we have $\displaystyle\int_{{\mathbb{R}}^{d}}(EB_{1}{\mathbf{U}}_{\sigma}^{\epsilon})(\tau)dx=$ $\displaystyle-\sum^{d}_{i,j=1}\sum_{\|\alpha\|=\sigma}\int_{{\mathbb{R}}^{d}}\\{E^{-1}\\}^{k}\partial_{x}^{\alpha}H_{i}^{\epsilon}\partial_{x}H^{\epsilon}_{j}{\mathbf{U}}_{\sigma}^{\epsilon}dx$ $\displaystyle-\sum^{d}_{i,j=1}\sum_{\|\alpha\|=\sigma}\int_{{\mathbb{R}}^{d}}\partial_{x}\\{E^{-1}\\}^{k}\partial_{x}^{\alpha}H_{i}^{\epsilon}H^{\epsilon}_{j}{\mathbf{U}}_{\sigma}^{\epsilon}dx$ $\displaystyle-\sum^{d}_{i,j=1}\sum_{\|\alpha\|=\sigma}\int_{{\mathbb{R}}^{d}}\\{E^{-1}\\}^{k}\partial_{x}^{\alpha}H_{i}^{\epsilon}H^{\epsilon}_{j}\partial_{x}({\mathbf{U}}_{\sigma}^{\epsilon})dx$ $\displaystyle\leq$ $\displaystyle C(\mathcal{N})+\eta_{1}\\|{\mathbf{u}}^{\epsilon}(\tau)\\|_{s+1}^{2}$ (3.20) for sufficiently small constant $\eta_{1}>0$. By virtue of Cauchy-Schwarz’s and Sobolev’s inequalities, and (3.9), a direct computation implies that $\displaystyle\int_{{\mathbb{R}}^{d}}\|(EB_{2}{\mathbf{U}}_{\sigma}^{\epsilon})(\tau)\|d\tau\leq C(\mathcal{N}).$ (3.21) Next, we deal with the term involving the viscosity. Recall that $L(\partial_{x}){\mathbf{U}}^{\epsilon}=({\rm div}{\mathbf{u}}^{\epsilon},\nabla q^{\epsilon})$. Denote $\displaystyle L_{1}:=\\{a^{\epsilon}\\}^{-1}{\rm div},\;\;\;\;L_{2}:=\\{r^{\epsilon}\\}^{-1}\nabla.$ A straightforward computation implies that $\displaystyle{\mathbf{U}}^{\epsilon}_{k}=\left\\{\begin{array}[]{ll}\left(\begin{array}[]{c}\\{L_{1}L_{2}\\}^{\frac{k-1}{2}}L_{1}{\mathbf{u}}^{\epsilon}\\\ \\{L_{2}L_{1}\\}^{\frac{k-1}{2}}L_{2}q^{\epsilon}\end{array}\right),&\text{if}\ k\ \text{is odd};\\\\[-4.30554pt] \\\ \left(\begin{array}[]{c}\\{L_{1}L_{2}\\}^{k/2}q^{\epsilon}\\\ \\{L_{2}L_{1}\\}^{k/2}{\mathbf{u}}^{\epsilon}\end{array}\right),&\hbox{\text{if}}\ k\ \text{is even}.\end{array}\right.$ Thus, we induce that $\displaystyle\int^{t}_{0}\int_{{\mathbb{R}}^{d}}(E\mathbf{h}_{\sigma}{\mathbf{U}}_{\sigma}^{\epsilon})(\tau)dxd\tau=\int^{t}_{0}\int_{{\mathbb{R}}^{d}}E\\{L_{E}\\}^{\sigma}(E^{-1}\mathbf{V}^{\epsilon}){\mathbf{W}}_{\sigma}^{\epsilon}dxd\tau.$ An integration by parts gives $\displaystyle\int^{t}_{0}\int_{{\mathbb{R}}^{d}}EL^{\sigma}_{E}(E^{-1}\mathbf{V}^{\epsilon}){\mathbf{W}}_{\sigma}^{\epsilon}dxd\tau$ $\displaystyle=$ $\displaystyle-\int^{t}_{0}\int_{{\mathbb{R}}^{d}}\mu\|\nabla{\mathbf{W}}_{\sigma}^{\epsilon}\|^{2}+(\mu+\lambda)\|{\rm div}{\mathbf{W}}_{\sigma}^{\epsilon}\|^{2}dxd\tau$ $\displaystyle+\int^{t}_{0}\int_{{\mathbb{R}}^{d}}E[\mu E^{-1}\Delta+(\mu+\lambda)E^{-1}\nabla{\rm div},\\{L_{E}\\}^{\sigma}](0,{\mathbf{u}}^{\epsilon})^{T}{\mathbf{W}}_{\sigma}^{\epsilon}dxd\tau,$ where it is easy to verify that $\displaystyle[E^{-1}\Delta,\\{L_{E}\\}^{\sigma}]=\sum_{i=0}^{k-1}\\{L_{E}\\}^{i}[E^{-1}\Delta,L_{E}]\\{L_{E}\\}^{\sigma-i-1}.$ (3.22) Noting that $L_{E}(\partial_{x})={E}^{-1}L(\partial_{x})$, we find that $\displaystyle[E^{-1}\Delta,L_{E}]=-E^{-1}\Delta E^{-1}L(\partial_{x})+\sum_{i,j=1}^{d}B_{ij}\partial_{x_{ij}},$ where $B_{ij}$ $(i,j=1,\cdots,d$) are the sums of bilinear functions of $E^{-1}$ and $\partial_{x}\\{E^{-1}\\}$, and Sobolev’s inequalities imply that $\displaystyle\\|B_{ij}\\|_{s-1}\leq C(\mathcal{N}).$ Thus, we integrate by parts to infer that $\displaystyle\mu\int^{t}_{0}\int_{{\mathbb{R}}^{d}}E[E^{-1}\Delta,\\{L_{E}\\}^{\sigma}](0,{\mathbf{u}}^{\epsilon})^{\top}{\mathbf{W}}_{\sigma}^{\epsilon}dxd\tau$ $\displaystyle\leq$ $\displaystyle\frac{\mu}{2}\int_{0}^{t}\\|\nabla{\mathbf{W}}_{\sigma}^{\epsilon}\\|^{2}d\tau+C(\mathcal{N})\int_{0}^{t}\\|{\mathbf{W}}_{\sigma}^{\epsilon}(\tau)\\|^{2}d\tau$ $\displaystyle+\int_{0}^{t}\\|\tilde{H}_{2}^{\sigma}(\tau)\\|^{2}+\\|\tilde{H}_{1}^{\sigma}(\tau)\\|^{2}d\tau.$ Here $\tilde{H}_{1}^{\sigma}$ is a finite sum of terms of the form $(\partial^{\alpha_{1}}_{x}e_{1})\cdots(\partial^{\alpha_{l}}_{x}e_{l})\,(\partial^{\beta}_{x}w)\,(\partial^{\gamma}_{x}u^{\epsilon}_{m})$ with $\|\alpha_{1}\|+\cdots+\|\alpha_{l}\|+\|\beta\|+\|\gamma\|\leq\sigma\leq s$, $\|\gamma\|>0$, and thus $\|\beta\|\leq k-1\leq s-1$, where $(e_{1},\dots,e_{l})$, $w$ and $u^{\epsilon}_{m}$ denote the coefficients of $E^{-1}$, $C_{j}$ and ${\mathbf{u}}^{\epsilon}$ respectively, with $C_{j}$ taking a form similar to that of $B_{ij}$. $\tilde{H}_{2}^{\sigma}$ is a finite sum of terms of the form $(\partial^{\alpha_{1}}_{x}e_{1})\cdots(\partial^{\alpha_{l}}_{x}e_{l})\,(\partial^{\beta}_{x}w)\,(\partial^{\gamma}_{x}u^{\epsilon}_{m})$ with $\|\alpha_{1}\|+\cdots+\|\alpha_{l}\|+\|\beta\|+\|\gamma\|\leq\sigma+1\leq s+1$, $\|\gamma\|>1$, and thus $\|\beta\|\leq\sigma-1\leq s-1$, where $(e_{1},\dots,e_{l})$, $w$ and $u^{\epsilon}_{m}$ denote the coefficients of $E^{-1}$, $B_{ij}$ and ${\mathbf{u}}^{\epsilon}$ respectively. Hence, we have $\displaystyle\\|\tilde{H}_{1}^{\sigma}\\|^{2}+\\|\tilde{H}_{2}^{\sigma}\\|^{2}\leq C(\mathcal{M}).$ Similarly, we can show that $\displaystyle(\mu+\lambda)\int^{t}_{0}\int_{{\mathbb{R}}^{d}}E[E^{-1}\nabla{\rm div},\\{L_{E}\\}^{\sigma}](0,{\mathbf{u}}^{\epsilon})^{T}{\mathbf{W}}_{\sigma}^{\epsilon}dxd\tau$ $\displaystyle\qquad\quad\leq\frac{\mu+\lambda}{2}\int_{0}^{t}\\|{\rm div}{\mathbf{W}}_{\sigma}^{\epsilon}\\|^{2}d\tau+C(\mathcal{N})\int_{0}^{t}\\|{\mathbf{W}}_{\sigma}^{\epsilon}\\|^{2}d\tau+tC(\mathcal{M}).$ Finally, the above estimates (3.19)–(3.21) and the positivity of $E$ imply (3.5). ∎ Next, we derive an estimate for $\\|{\rm curl\,}(r_{0}{\mathbf{u}}^{\epsilon})\\|_{\sigma-1}$. Define $\displaystyle f^{\epsilon}(S^{\epsilon},\epsilon q^{\epsilon}):=1-\frac{r_{0}(S^{\epsilon})}{r^{\epsilon}(S^{\epsilon},\epsilon q^{\epsilon})}.$ (3.23) Hereafter we denote $r_{0}(t):=r_{0}(S^{\epsilon}(t))$ and $f^{\epsilon}(t):=f^{\epsilon}(S^{\epsilon}(t),\epsilon q^{\epsilon}(t))$ for notational simplicity. One can factor out $\epsilon q^{\epsilon}$ in $f^{\epsilon}(t)$. In fact, using Taylor’s expansion, one obtains that there exists a smooth function $g^{\epsilon}(t)$, such that $\displaystyle f^{\epsilon}(t)=\epsilon g^{\epsilon}(t):=\epsilon g^{\epsilon}(S^{\epsilon}(t),\epsilon q^{\epsilon}(t)),\quad\\|g^{\epsilon}(t)\\|_{s}\leq C(\mathcal{M}_{\epsilon}(T)).$ (3.24) Since $\partial_{t}S^{\epsilon}+({\mathbf{u}}^{\epsilon}\cdot\nabla)S^{\epsilon}=\epsilon^{2}\frac{\Psi^{\epsilon}:\nabla{\mathbf{u}}^{\epsilon}}{b^{\epsilon}(S^{\epsilon},\epsilon q^{\epsilon})},$ the equations for ${\mathbf{u}}^{\epsilon}$ are equivalent to $\displaystyle[\partial_{t}+({\mathbf{u}}^{\epsilon}\cdot\nabla)](r_{0}{\mathbf{u}}^{\epsilon})+\frac{1}{\epsilon}\nabla q^{\epsilon}=$ $\displaystyle g^{\epsilon}\nabla q^{\epsilon}+(1-\epsilon g^{\epsilon})({\rm curl\,}{\mathbf{H}^{\epsilon}})\times{\mathbf{H}^{\epsilon}}$ $\displaystyle+(1-\epsilon g^{\epsilon}){\rm div}\Psi^{\epsilon}+\epsilon^{2}\,r^{\prime}_{0}(S^{\epsilon}){\mathbf{u}}^{\epsilon}\frac{\Psi^{\epsilon}:\nabla{\mathbf{u}}^{\epsilon}}{b^{\epsilon}(S^{\epsilon},\epsilon q^{\epsilon})}.$ (3.25) We perform the operator _curl_ to the equation (3) to obtain that $\displaystyle[\partial_{t}+({\mathbf{u}}^{\epsilon}\cdot\nabla)]({\rm curl\,}(r_{0}{\mathbf{u}}^{\epsilon}))$ $\displaystyle=$ $\displaystyle[{\mathbf{u}}^{\epsilon}\cdot\nabla,{\rm curl\,}](r_{0}{\mathbf{u}}^{\epsilon})+{\rm curl\,}[(1-\epsilon g^{\epsilon})({\rm curl\,}{\mathbf{H}^{\epsilon}})\times{\mathbf{H}^{\epsilon}}]$ $\displaystyle+{\rm curl\,}[(1-\epsilon g^{\epsilon}){\rm div}\Psi^{\epsilon}]+[{\rm curl\,},g^{\epsilon}]\nabla q^{\epsilon}$ $\displaystyle+\epsilon^{2}{\rm curl\,}\left\\{r^{\prime}_{0}(S^{\epsilon}){\mathbf{u}}^{\epsilon}\frac{\Psi^{\epsilon}:\nabla{\mathbf{u}}^{\epsilon}}{b^{\epsilon}(S^{\epsilon},\epsilon q^{\epsilon})}\right\\}.$ (3.26) ###### Lemma 3.6. There exist constants $C_{0}>0$, $0<\xi_{3}<\mu$, a function $C(\cdot)$ from $[0,\infty)$ to $[0,\infty)$ and a sufficiently small constant $\eta_{2}>0$, such that for all $\epsilon\in(0,1]$ and all $t\in[0,T]$, $\displaystyle\\|\\{{\rm curl\,}(r_{0}{\mathbf{u}}^{\epsilon}),{\rm curl\,}\mathbf{H}^{\epsilon}\\}(t)\\|^{2}_{s-1}$ $\displaystyle+\xi_{3}\int_{0}^{t}\\|\nabla{\rm curl\,}{\mathbf{u}}^{\epsilon}\\|_{s-1}^{2}d\tau$ $\displaystyle\leq$ $\displaystyle C_{0}+tC(\mathcal{N}_{\epsilon}(T))+\eta_{2}\int_{0}^{t}\\|{\mathbf{u}}^{\epsilon}\\|^{2}_{s+1}d\tau.$ (3.27) ###### Proof. Set $\mathcal{N}:=\mathcal{N}_{\epsilon}(T)$ and $\omega={\rm curl\,}(r_{0}{\mathbf{u}}^{\epsilon})$. Taking $\partial^{\alpha}_{x}$ $(\|\alpha\|\leq s-1)$ on (3), multiplying the resulting equations by $\partial^{\alpha}_{x}\omega$, and integrating over $[0,t]\times{\mathbb{R}}^{d}$ with $t\leq T$, we obtain $\displaystyle\frac{1}{2}\\|\partial^{\alpha}\omega(t)\\|^{2}=\,$ $\displaystyle\frac{1}{2}\\|\partial^{\alpha}\omega(0)\\|^{2}-\int^{t}_{0}\langle({\mathbf{u}}^{\epsilon}\cdot\nabla)\partial^{\alpha}\omega,\partial^{\alpha}\omega\rangle(\tau)d\tau$ $\displaystyle+\int^{t}_{0}\langle[{\mathbf{u}}^{\epsilon}\cdot\nabla,\partial_{x}^{\alpha}]\omega,\partial^{\alpha}\omega\rangle(\tau)d\tau$ $\displaystyle+\int^{t}_{0}\langle\partial^{\alpha}\\{[{\rm curl\,},g^{\epsilon}]\nabla q^{\epsilon}\\},\partial^{\alpha}\omega\rangle(\tau)d\tau$ $\displaystyle+\int^{t}_{0}\langle\partial^{\alpha}\\{[{\mathbf{u}}^{\epsilon}\cdot\nabla,{\rm curl\,}](r_{0}{\mathbf{u}}^{\epsilon})\\},\partial^{\alpha}\omega\rangle(\tau)d\tau$ $\displaystyle+\int^{t}_{0}\left\langle\partial^{\alpha}\left\\{\epsilon^{2}\,{\rm curl\,}\left\\{r^{\prime}_{0}(S^{\epsilon}){\mathbf{u}}^{\epsilon}\frac{\Psi^{\epsilon}:\nabla{\mathbf{u}}^{\epsilon}}{b^{\epsilon}(S^{\epsilon},\epsilon q^{\epsilon})}\right\\}\right\\},\partial^{\alpha}\omega\right\rangle(\tau)d\tau$ $\displaystyle+\int^{t}_{0}\langle\partial^{\alpha}\\{{\rm curl\,}[(1-\epsilon g^{\epsilon})({\rm curl\,}{\mathbf{H}^{\epsilon}})\times{\mathbf{H}^{\epsilon}}]\\},\partial^{\alpha}\omega\rangle(\tau)d\tau$ $\displaystyle+\int^{t}_{0}\langle\partial^{\alpha}\\{{\rm curl\,}[(1-\epsilon g^{\epsilon}){\rm div}\Psi^{\epsilon}]\\},\partial^{\alpha}\omega\rangle(\tau)d\tau$ $\displaystyle=$ $\displaystyle:\frac{1}{2}\\|\partial^{\alpha}\omega(0)\\|^{2}+\int^{t}_{0}\sum_{i=1}^{7}I_{i}(\tau).$ (3.28) We have to estimate the terms $I_{i}(\tau)$ ($1\leq i\leq 7$) on the right- hand side of (3). Applying partial integrations, we have $\displaystyle I_{1}(\tau)=\int_{\mathbb{R}^{d}}\|\partial^{\alpha}\omega\|^{2}{\rm div}{\mathbf{u}}^{\epsilon}\leq\\|{\rm div}{\mathbf{u}}^{\epsilon}(\tau)\\|_{L^{\infty}}\\|\partial^{\alpha}\omega(\tau)\\|^{2}\leq C(\mathcal{N})\\|\partial^{\alpha}\omega(\tau)\\|^{2},$ (3.29) while for the term $I_{2}(\tau)$, an application of Cauchy-Schwarz’s inequality gives $\displaystyle\|I_{2}(\tau)\|\leq\\|\partial^{\alpha}\omega\\|\,\\|\mathbf{h}_{\alpha}(\tau)\\|,\quad\mathbf{h}_{\alpha}(\tau):=[{\mathbf{u}}^{\epsilon}\cdot\nabla,\partial_{x}^{\alpha}]\omega.$ The commutator $\mathbf{h}_{\alpha}$ is a sum of terms $\partial^{\beta}_{x}{\mathbf{u}}^{\epsilon}\partial^{\gamma}_{x}\omega$ with multi-indices $\beta$ and $\gamma$ satisfying $\|\beta\|+\|\gamma\|\leq s$, $\|\beta\|>0$, and $\|\gamma\|>0$. Thus, the inequality (2.7) with $\sigma=s-1>d/2$ implies that $\\|\mathbf{h}_{\alpha}(\tau)\\|\leq C(\mathcal{N})$. Hence, we have $\displaystyle\|I_{2}(\tau)\|\leq C(\mathcal{N})+\\|\partial^{\alpha}\omega(\tau)\\|^{2}.$ (3.30) Noting that $([{\rm curl\,},g^{\epsilon}]\,\mathbf{a}\,)_{i,j}=a_{i}\partial_{x_{j}}g^{\epsilon}-a_{j}\partial_{x_{i}}g^{\epsilon}$ for $\mathbf{a}=(a_{1},\cdots,a_{d})$, the inequality (2.7), and the estimate (3.24), we can control the term $I_{3}(\tau)$ as follows $\displaystyle\|I_{3}(\tau)\|$ $\displaystyle\leq\\|\partial^{\alpha}\\{[{\rm curl\,},g^{\epsilon}]\nabla q^{\epsilon}\\}\\|\;\\|\partial^{\alpha}\omega\\|$ $\displaystyle\leq K\\|[{\rm curl\,},g^{\epsilon}]\nabla q^{\epsilon}\\|_{s-1}\,\\|\partial^{\alpha}\omega\\|$ $\displaystyle\leq K\\|\nabla g^{\epsilon}(\tau)\\|_{s-1}\\|\nabla q^{\epsilon}(\tau)\\|_{s-1}\,\\|\partial^{\alpha}\omega\\|$ $\displaystyle\leq C(\mathcal{N})+\\|\partial^{\alpha}\omega(\tau)\\|^{2}.$ (3.31) Similarly, the term $I_{4}(\tau)$ can be bounded as follows. $\displaystyle\|I_{4}(\tau)\|$ $\displaystyle\leq K\\|\partial^{\alpha}\\{[{\mathbf{u}}^{\epsilon}\cdot\nabla,{\rm curl\,}](r_{0}{\mathbf{u}}^{\epsilon})\\}\\|\;\\|\partial^{\alpha}\omega\\|$ $\displaystyle\leq K\\|[{\mathbf{u}}^{\epsilon}\cdot\nabla,{\rm curl\,}](r_{0}{\mathbf{u}}^{\epsilon})\\|_{s-1}\,\\|\partial^{\alpha}\omega\\|$ $\displaystyle\leq K\\|[{\mathbf{u}}^{\epsilon}_{j},{\rm curl\,}]\partial_{x_{j}}(r_{0}{\mathbf{u}}^{\epsilon})\\|_{s-1}\,\\|\partial^{\alpha}\omega\\|$ $\displaystyle\leq C(\mathcal{N})+\\|\partial^{\alpha}\omega(\tau)\\|^{2}.$ (3.32) To bound the term $I_{5}(\tau)$, we use the Moser-type inequality (see [23]) to deduce $\displaystyle\|I_{5}(\tau)\|\leq$ $\displaystyle\epsilon^{2}K\left\\|\partial^{\alpha}\left\\{{\rm curl\,}\left\\{r^{\prime}_{0}(S^{\epsilon}){\mathbf{u}}^{\epsilon}\frac{\Psi^{\epsilon}:\nabla{\mathbf{u}}^{\epsilon}}{b^{\epsilon}(S^{\epsilon},\epsilon q^{\epsilon})}\right\\}\right\\}\right\\|\cdot\\|\partial^{\alpha}\omega\\|$ $\displaystyle=$ $\displaystyle\epsilon^{2}K\left\\|\partial^{\alpha}\left[\left(\frac{r^{\prime}_{0}(S^{\epsilon})\Psi^{\epsilon}:\nabla{\mathbf{u}}^{\epsilon}}{b^{\epsilon}(S^{\epsilon},\epsilon q^{\epsilon})}\right){\rm curl\,}{\mathbf{u}}^{\epsilon}\right]\right\\|\cdot\\|\partial^{\alpha}\omega\\|$ $\displaystyle+\epsilon^{2}K\left\\|\partial^{\alpha}\left[\nabla\left(\frac{r^{\prime}_{0}(S^{\epsilon})\Psi^{\epsilon}:\nabla{\mathbf{u}}^{\epsilon}}{b^{\epsilon}(S^{\epsilon},\epsilon q^{\epsilon})}\right)\times{\mathbf{u}}^{\epsilon}\right]\right\\|\cdot\\|\partial^{\alpha}\omega\\|$ $\displaystyle\leq$ $\displaystyle\epsilon^{2}K\\|{\rm curl\,}{\mathbf{u}}^{\epsilon}\\|_{L^{\infty}}\left\\|D^{s-1}\Big{(}\frac{r^{\prime}_{0}(S^{\epsilon})\Psi^{\epsilon}:\nabla{\mathbf{u}}^{\epsilon}}{b^{\epsilon}(S^{\epsilon},\epsilon q^{\epsilon})}\Big{)}\right\\|\cdot\\|\partial^{\alpha}\omega\\|$ $\displaystyle+\epsilon^{2}K\\|D^{s-1}({\rm curl\,}{\mathbf{u}}^{\epsilon})\\|\left\\|\frac{r^{\prime}_{0}(S^{\epsilon})\Psi^{\epsilon}:\nabla{\mathbf{u}}^{\epsilon}}{b^{\epsilon}(S^{\epsilon},\epsilon^{2}q^{\epsilon})}\right\\|_{L^{\infty}}\cdot\\|\partial^{\alpha}\omega\\|$ $\displaystyle+\epsilon^{2}K\\|{\mathbf{u}}^{\epsilon}\\|_{L^{\infty}}\left\\|D^{s}\Big{(}\frac{r^{\prime}_{0}(S^{\epsilon})\Psi^{\epsilon}:\nabla{\mathbf{u}}^{\epsilon}}{b^{\epsilon}(S^{\epsilon},\epsilon^{2}q^{\epsilon})}\Big{)}\right\\|\cdot\\|\partial^{\alpha}\omega\\|$ $\displaystyle+\epsilon^{2}K\\|D^{s-1}{\mathbf{u}}^{\epsilon}\\|\left\\|\nabla\left(\frac{r^{\prime}_{0}(S^{\epsilon})\Psi^{\epsilon}:\nabla{\mathbf{u}}^{\epsilon}}{b^{\epsilon}(S^{\epsilon},\epsilon^{2}q^{\epsilon})}\right)\right\\|_{L^{\infty}}\cdot\\|\partial^{\alpha}\omega\\|$ $\displaystyle\leq$ $\displaystyle C(\mathcal{N})+\epsilon^{2}C(\mathcal{N})\\|{\mathbf{u}}^{\epsilon}\\|_{s+1}^{2}+\\|\partial^{\alpha}\omega(\tau)\\|^{2},$ (3.33) where the condition $s>2+d/2$ and the inequality (2.8) have been used. For the term $I_{6}(\tau)$, by virtue of (2.1), ${\rm curl\,}{\rm curl\,}\mathbf{a}=\nabla\,{\rm div}\,\mathbf{a}-\Delta\mathbf{a}$. Thus, we integrate by parts to see that $\displaystyle I_{6}(\tau)=$ $\displaystyle\langle\partial^{\alpha}\\{(1-\epsilon g^{\epsilon})({\rm curl\,}{\mathbf{H}^{\epsilon}})\times{\mathbf{H}^{\epsilon}}\\},\partial^{\alpha}\\{{\rm curl\,}{\rm curl\,}(r_{0}{\mathbf{u}}^{\epsilon})\\}\rangle$ $\displaystyle=$ $\displaystyle\langle\partial^{\alpha}\\{(1-\epsilon g^{\epsilon})({\rm curl\,}{\mathbf{H}^{\epsilon}})\times{\mathbf{H}^{\epsilon}}\\},\partial^{\alpha}\\{\nabla{\rm div}(r_{0}{\mathbf{u}}^{\epsilon})-\Delta(r_{0}{\mathbf{u}}^{\epsilon})\\}\rangle,$ and use Cauchy-Schwarz’s inequality and (2.8) to conclude $\displaystyle\|I_{6}(\tau)\|\leq C(\mathcal{N})+\theta_{1}\\|{\mathbf{u}}^{\epsilon}(\tau)\\|^{2}_{s+1},$ (3.34) where $\theta_{1}>0$ is a sufficiently small constant independent of $\epsilon$. Next, we deal with the term $I_{7}(\tau)$. By the vector identities and integration by parts, we see that there exists a sufficiently small $\theta_{2}$, such that $\displaystyle I_{7}(\tau)\leq$ $\displaystyle-\inf\\{r_{0}(S^{\epsilon})\\}\\|\nabla{\rm curl\,}{\mathbf{u}}^{\epsilon}(\tau)\\|_{\sigma-1}+C(\mathcal{N})$ $\displaystyle+\theta_{2}\\|{\mathbf{u}}^{\epsilon}(\tau)\\|_{s+1}+\epsilon C(\mathcal{N})\\|{\mathbf{u}}^{\epsilon}(\tau)\\|_{s+1}.$ (3.35) Finally, to estimate $\\|{\rm curl\,}\mathbf{H}^{\epsilon}\\|_{s-1}$, we apply the operator _curl_ to (1.18) and use the vector identity (1.22) to obtain $\displaystyle\partial_{t}({\rm curl\,}\mathbf{H}^{\epsilon})+{\mathbf{u}}^{\epsilon}\cdot\nabla({\rm curl\,}\mathbf{H}^{\epsilon})$ $\displaystyle=-[{\rm curl\,},{\mathbf{u}}^{\epsilon}]\cdot\nabla\mathbf{H}^{\epsilon}+{\rm curl\,}((\mathbf{H}^{\epsilon}\cdot\nabla){\mathbf{u}}^{\epsilon}-\mathbf{H}^{\epsilon}{\rm div}{\mathbf{u}}^{\epsilon}).$ (3.36) By the commutator inequality and Sobolev’s inequalities, we find that $\displaystyle\\|[{\rm curl\,},{\mathbf{u}}^{\epsilon}]\cdot\nabla\mathbf{H}^{\epsilon}\\|\leq C(\mathcal{N})$ and $\displaystyle\\|{\rm curl\,}((\mathbf{H}^{\epsilon}\cdot\nabla){\mathbf{u}}^{\epsilon}-\mathbf{H}^{\epsilon}{\rm div}{\mathbf{u}}^{\epsilon})\\|\leq C(\mathcal{N})+\theta_{3}\\|{\mathbf{u}}^{\epsilon}\\|_{s+1}^{2}$ for sufficiently small constant $\theta_{3}>0$. Then, by arguments similar to those used in Lemma 3.2, we derive that $\displaystyle\\|{\rm curl\,}\mathbf{H}^{\epsilon}\\|_{s-1}^{2}\leq C_{0}+tC(\mathcal{N})+\theta_{3}\int_{0}^{t}\\|{\mathbf{u}}^{\epsilon}(\tau)\\|_{s+1}^{2}d\tau.$ (3.37) Thus, the lemma follows from adding up the estimates (3)–(3.37) for all $\|\alpha\|\leq s-1$ and choosing constants $\theta_{1}$, $\theta_{2}$ and $\theta_{3}$ appropriately small. ∎ Next we complete the proof of Theorem 3.1 by the following estimate. ###### Lemma 3.7. There exist constants $C_{0}>0$, $0<\xi_{4}<\mu$, and a function $C(\cdot)$ from $[0,\infty)$ to $[0,\infty)$, such that for all $\epsilon\in(0,1]$ and $t\in[0,T]$, $\displaystyle\\|(S^{\epsilon},q^{\epsilon},{\mathbf{u}}^{\epsilon},\mathbf{H}^{\epsilon})(t)\\|^{2}_{s}+\xi_{4}\int_{0}^{t}\\|{\mathbf{u}}^{\epsilon}\\|^{2}d\tau\leq C_{0}+(t+\epsilon)C(\mathcal{M}_{\epsilon}(T)).$ (3.38) ###### Proof. First, from (3.14) we get $\displaystyle\\|{\mathbf{u}}^{\epsilon}\\|_{\sigma+1}^{2}\leq\tilde{C}\Big{\\{}\\|\nabla(\\{L_{E^{\epsilon}}(\partial_{x})\\}^{\sigma}{{\mathbf{W}}}^{\epsilon}\\|_{0}^{2}+\\|\nabla{\rm curl\,}(r_{0}{\mathbf{u}}^{\epsilon})\\|_{\sigma-1}^{2}+\\|{\mathbf{u}}^{\epsilon}\\|_{\sigma}^{2}\Big{\\}},$ (3.39) where $\tilde{C}:=C_{1}+tC(\mathcal{M}_{\epsilon}(T))+\epsilon C(\mathcal{M}_{\epsilon}(T))$. Moreover, using Lemma 3.2, we obtain $\displaystyle\\|{\mathbf{u}}^{\epsilon}\\|_{\sigma+1}^{2}\leq\tilde{C}\Big{\\{}\\|\nabla\\{L_{E^{\epsilon}}(\partial_{x})\\}^{\sigma}{{\mathbf{W}}}^{\epsilon}\\|_{0}^{2}+\\|\nabla({\rm curl\,}{\mathbf{u}}^{\epsilon})\\|_{\sigma-1}^{2}+\\|{\mathbf{u}}^{\epsilon}\\|_{\sigma}^{2}\Big{\\}}.$ (3.40) In view of (3.40), there exists a constant $\kappa_{2}$ such that $\displaystyle\frac{\xi_{2}}{2}\\|\nabla{\mathbf{W}}_{\sigma}^{\epsilon}\\|_{0}^{2}+\xi_{3}\\|\nabla{\rm curl\,}{\mathbf{u}}^{\epsilon}\\|_{\sigma-1}^{2}\geq$ $\displaystyle\kappa_{2}\\|{\mathbf{u}}^{\epsilon}\\|_{\sigma+1}^{2}-\tilde{C}_{1}\Big{\\{}\\|\nabla\\{L_{E^{\epsilon}}(\partial_{x})\\}^{\sigma}{{\mathbf{W}}}^{\epsilon}\\|_{0}^{2}$ $\displaystyle+\\|\nabla({\rm curl\,}{\mathbf{u}}^{\epsilon})\\|_{\sigma-1}^{2}\Big{\\}}-\tilde{C}\\|{\mathbf{u}}^{\epsilon}\\|_{\sigma}^{2},$ (3.41) where $\tilde{C}_{1}=tC(\mathcal{M}_{\epsilon}(T))+\epsilon C(\mathcal{M}_{\epsilon}(T))$. Now, we combine the estimates (3.5) and (3.6) with (3.41), and use the fact that ${\rm div}\mathbf{H}=0$ to conclude that there exists a positive constant $\kappa_{3}$, such that $\displaystyle\\|(L_{E}(\partial_{x}))^{\sigma}{\mathbf{U}}^{\epsilon}\\|_{0}^{2}+\\|{\rm curl\,}(r_{0}{\mathbf{u}}^{\epsilon})\\|_{\sigma-1}^{2}+\kappa_{3}\int_{0}^{t}\\|{\mathbf{u}}^{\epsilon}\\|_{\sigma+1}^{2}dx$ $\displaystyle\leq$ $\displaystyle C_{0}+tC(\mathcal{M}_{\epsilon}(T))+\tilde{C}\int_{0}^{t}\\|{\mathbf{u}}^{\epsilon}\\|_{\sigma}^{2}d\tau$ $\displaystyle+\tilde{C}_{1}\int_{0}^{t}\Big{\\{}\\|\nabla\\{L_{E^{\epsilon}}(\partial_{x})\\}^{\sigma}{{\mathbf{W}}}^{\epsilon}\\|_{0}^{2}+\\|\nabla({\rm curl\,}{\mathbf{u}}^{\epsilon})\\|_{\sigma-1}^{2}\Big{\\}}d\tau$ $\displaystyle\leq$ $\displaystyle C_{0}+(t+\epsilon)C(\mathcal{M}_{\epsilon}(T))+\tilde{C}\int_{0}^{t}\\|{\mathbf{u}}^{\epsilon}\\|_{\sigma}^{2}d\tau$ for sufficiently small $\eta_{1}$ and $\eta_{2}$. Thus by induction, we conclude that $\displaystyle\\|\\{L_{E^{\epsilon}}(\partial_{x})\\}^{\sigma}{\mathbf{U}}^{\epsilon}\\|_{0}^{2}$ $\displaystyle+\\|{\rm curl\,}(r_{0}{\mathbf{u}}^{\epsilon})\\|_{\sigma-1}^{2}$ $\displaystyle+\kappa_{3}\int_{0}^{t}\\|{\mathbf{u}}^{\epsilon}\\|_{\sigma+1}^{2}d\tau\leq C_{0}+(t+\epsilon)C(\mathcal{M}_{\epsilon}(T)).$ Using (3.14) again, we obtain the estimate (3.38) by induction on $\sigma\in\\{0,\dots,s\\}$. ∎ ## 4\. Incompressible limit In this section, we shall prove the convergence part of Theorem 1.1 by modifying the method developed by Métivier and Schochet [27], see also some extensions [1, 2, 26]. ###### Proof of the convergence part of Theorem 1.1. The uniform bound (1.29) implies that, after extracting a subsequence, one gets the following limits: $\displaystyle(q^{\epsilon},{\mathbf{u}}^{\epsilon},\mathbf{H}^{\epsilon})\rightharpoonup(q,{\mathbf{v}},\bar{\mathbf{H}})\quad\text{weakly-}\ast\ \text{in}\quad L^{\infty}(0,T;H^{s}(\mathbb{R}^{d})).$ (4.1) The equations (1.18) and (1.19) imply that $\partial_{t}S^{\epsilon}$ and $\partial_{t}\mathbf{H}^{\epsilon}\in C([0,T],H^{s-1}(\mathbb{R}^{d}))$. Thus, after further extracting a subsequence, we obtain that, for all $s^{\prime}<s$, $\displaystyle S^{\epsilon}\rightarrow\bar{S}\quad\text{strongly in}\quad C([0,T],H^{s^{\prime}}_{\mathrm{loc}}(\mathbb{R}^{d})),$ (4.2) $\displaystyle\mathbf{H}^{\epsilon}\rightarrow\bar{\mathbf{H}}\quad\text{strongly in}\quad C([0,T],H^{s^{\prime}}_{\mathrm{loc}}(\mathbb{R}^{d})),$ (4.3) where the limit $\bar{\mathbf{H}}\in C([0,T],H^{s^{\prime}}_{\mathrm{loc}}(\mathbb{R}^{d}))\cap L^{\infty}(0,T;H^{s}_{\mathrm{loc}}(\mathbb{R}^{d}))$. Similarly, by (3) and the uniform bound (1.29), we have $\displaystyle{\rm curl\,}(r_{0}(S^{\epsilon}){\mathbf{u}}^{\epsilon})\rightarrow{\rm curl\,}(r_{0}(\bar{S}){\mathbf{v}})\quad\text{strongly in}\quad C([0,T],H^{s^{\prime}-1}_{\mathrm{loc}}(\mathbb{R}^{d}))$ (4.4) for all $s^{\prime}<s$, where $r_{0}(\bar{S})=\lim_{\epsilon\rightarrow 0}r_{0}(S^{\epsilon}):=\lim_{\epsilon\rightarrow 0}r^{\epsilon}(S^{\epsilon},0)$. In order to obtain the limit system, we need to prove that the convergence in (4.1) holds in the strong topology of $L^{2}(0,T;H^{s^{\prime}}_{\mathrm{loc}}(\mathbb{R}^{d}))$ for all $s^{\prime}<s$. To this end, we first show that $q=0$ and ${\rm div}{\mathbf{v}}=0$. In fact, from (3.16) we get $\displaystyle\epsilon E^{\epsilon}(S^{\epsilon},\epsilon q^{\epsilon})\partial_{t}{\mathbf{U}}^{\epsilon}+L(\partial_{x}){\mathbf{U}}^{\epsilon}=-\epsilon E^{\epsilon}(S^{\epsilon},\epsilon q^{\epsilon}){\mathbf{u}}^{\epsilon}\cdot\nabla{\mathbf{U}}^{\epsilon}+\epsilon(\mathbf{J}^{\epsilon}+\mathbf{V}^{\epsilon}).$ (4.5) Since $\displaystyle E^{\epsilon}(S^{\epsilon},\epsilon q^{\epsilon})-E^{\epsilon}(S^{\epsilon},0)=O(\epsilon),$ we have $\displaystyle\epsilon E^{\epsilon}(S^{\epsilon},0)\partial_{t}{\mathbf{U}}^{\epsilon}+L(\partial_{x}){\mathbf{U}}^{\epsilon}=\epsilon\mathbf{h}^{\epsilon},$ (4.6) where $\mathbf{h}^{\epsilon}$ is uniformly bounded in $C([0,T],H^{s-2}(\mathbb{R}^{d}))$ in view of (1.29). Passing to the weak limit to (4.6), we obtain $\nabla q=0$ and ${\rm div}{\mathbf{v}}=0$. Since $q\in L^{\infty}(0,T;H^{s}(\mathbb{R}^{d}))$, we infer that $q=0$. Noticing that the strong compactness for the incompressible components by (4.4), it is sufficient to prove the following proposition on the acoustic components. ###### Proposition 4.1. Suppose that the assumptions in Theorem 1.1 hold, then $q^{\epsilon}$ converges strongly to $0$ in $L^{2}(0,T;H^{s^{\prime}}_{\mathrm{loc}}(\mathbb{R}^{d}))$ for all $s^{\prime}<s$, and ${\rm div}{\mathbf{u}}^{\epsilon}$ converges strongly to $0$ in $L^{2}(0,T;H^{s^{\prime}-1}_{\mathrm{loc}}(\mathbb{R}^{d}))$ for all $s^{\prime}<s$. The proof of Proposition 4.1 is built on the the following dispersive estimates on the wave equations obtained by Métivier and Schochet [27] and reformulated in [2]. ###### Lemma 4.2 ([27, 2]). Let $T>0$ and $w^{\epsilon}$ be a bounded sequence in $C([0,T],H^{2}(\mathbb{R}^{d}))$, such that $\displaystyle\epsilon^{2}\partial_{t}(a^{\epsilon}\partial_{t}w^{\epsilon})-\nabla\cdot(b^{\epsilon}\nabla w^{\epsilon})=c^{\epsilon},$ where $c^{\epsilon}$ converges to $0$ strongly in $L^{2}(0,T;L^{2}(\mathbb{R}^{d}))$. Assume further that, for some $s>d/2+1$, the coefficients $(a^{\epsilon},b^{\epsilon})$ are uniformly bounded in $C([0,T];H^{s}(\mathbb{R}^{d}))$ and converges in $C([0,T];H^{s}_{\mathrm{loc}}(\mathbb{R}^{d}))$ to a limit $(a,b)$ satisfying the decay estimate $\displaystyle\|a(x,t)-\underline{a}\|\leq C_{0}\|x\|^{-1-\delta},\quad\|\nabla_{x}a(x,t)\|\leq C_{0}\|x\|^{-2-\delta},$ $\displaystyle\|b(x,t)-\underline{b}\|\leq C_{0}\|x\|^{-1-\delta},\quad\|\nabla_{x}b(x,t)\|\leq C_{0}\|x\|^{-2-\delta},$ for some given positive constants $\underline{a}$, $\underline{b}$, $C_{0}$ and $\delta$. Then the sequence $w^{\epsilon}$ converges to $0$ in $L^{2}(0,T;L^{2}_{\mathrm{loc}}(\mathbb{R}^{d}))$. ###### Proof of Proposition 4.1. We first show that $q^{\epsilon}$ converges strongly to $0$ in $L^{2}(0,T;\linebreak H^{s^{\prime}}_{\mathrm{loc}}(\mathbb{R}^{d}))$ for all $s^{\prime}<s$. An application of the operator $\epsilon^{2}\partial_{t}$ to (1.16) gives $\displaystyle\epsilon^{2}\partial_{t}(a^{\epsilon}(S^{\epsilon},\epsilon q^{\epsilon})\partial_{t}q^{\epsilon})+\epsilon\partial_{t}{\rm div}{\mathbf{u}}^{\epsilon}=-\epsilon^{2}\partial_{t}\\{a^{\epsilon}(S^{\epsilon},\epsilon q^{\epsilon})({\mathbf{u}}^{\epsilon}\cdot\nabla)q^{\epsilon}\\}.$ (4.7) Dividing (1.17) by $r^{\epsilon}(S^{\epsilon},\epsilon q^{\epsilon})$ and then taking the operator _div_ to the resulting equation, one has $\displaystyle\partial_{t}{\rm div}{\mathbf{u}}^{\epsilon}+\frac{1}{\epsilon}{\rm div}\Big{(}\frac{1}{r^{\epsilon}(S^{\epsilon},\epsilon q^{\epsilon})}\nabla q^{\epsilon}\Big{)}$ $\displaystyle\qquad\qquad=-{\rm div}(({\mathbf{u}}^{\epsilon}\cdot\nabla){\mathbf{u}}^{\epsilon})+{\rm div}\Big{(}\frac{1}{r^{\epsilon}(S^{\epsilon},\epsilon q^{\epsilon})}({\rm curl\,}\mathbf{H}^{\epsilon})\times\mathbf{H}^{\epsilon}\Big{)}$ $\displaystyle\qquad\qquad\quad+{\rm div}\Big{(}\frac{1}{r^{\epsilon}(S^{\epsilon},\epsilon q^{\epsilon})}{\rm div}\Psi({\mathbf{u}}^{\epsilon})\Big{)}.$ (4.8) Subtracting (4) from (4.7), we obtain $\displaystyle\epsilon^{2}\partial_{t}(a^{\epsilon}(S^{\epsilon},\epsilon q^{\epsilon})\partial_{t}q^{\epsilon})-{\rm div}\Big{(}\frac{1}{r^{\epsilon}(S^{\epsilon},\epsilon q^{\epsilon})}\nabla q^{\epsilon}\Big{)}=F^{\epsilon}(S^{\epsilon},q^{\epsilon},{\mathbf{u}}^{\epsilon},\mathbf{H}^{\epsilon}),$ (4.9) where $\displaystyle F^{\epsilon}(S^{\epsilon},q^{\epsilon},{\mathbf{u}}^{\epsilon},\mathbf{H}^{\epsilon})=\,$ $\displaystyle\epsilon{\rm div}\Big{(}\frac{1}{r^{\epsilon}(S^{\epsilon},\epsilon q^{\epsilon})}({\rm curl\,}\mathbf{H}^{\epsilon})\times\mathbf{H}^{\epsilon}\Big{)}$ $\displaystyle+\epsilon{\rm div}\Big{(}\frac{1}{r^{\epsilon}(S^{\epsilon},\epsilon q^{\epsilon})}{\rm div}\Psi({\mathbf{u}}^{\epsilon})\Big{)}-\epsilon{\rm div}(({\mathbf{u}}^{\epsilon}\cdot\nabla){\mathbf{u}}^{\epsilon})$ $\displaystyle-\epsilon^{2}\partial_{t}\\{a^{\epsilon}(S^{\epsilon},\epsilon q^{\epsilon})({\mathbf{u}}^{\epsilon}\cdot\nabla)q^{\epsilon}\\}.$ In view of the uniform boundedness of $(S^{\epsilon},q^{\epsilon},{\mathbf{u}}^{\epsilon},\mathbf{H}^{\epsilon})$, the smoothness and positivity assumptions on $a^{\epsilon}$ and $r^{\epsilon}$, and the convergence of $S^{\epsilon}$, we find that $\displaystyle F^{\epsilon}(S^{\epsilon},q^{\epsilon},{\mathbf{u}}^{\epsilon},\mathbf{H}^{\epsilon})\rightarrow 0\quad\text{strongly in}\quad L^{2}(0,T;L^{2}(\mathbb{R}^{d})),$ and the coefficients in (4.9) satisfy the requirements in Lemma 4.2. Therefore, by virtue of Lemma 4.2, $\displaystyle q^{\epsilon}\rightarrow 0\quad\text{strongly in}\quad L^{2}(0,T;L^{2}_{\mathrm{loc}}(\mathbb{R}^{d})).$ On the other hand, the uniform boundedness of $q^{\epsilon}$ in $C([0,T],H^{s}(\mathbb{R}^{d}))$ and an interpolation argument yield that $\displaystyle q^{\epsilon}\rightarrow 0\quad\text{strongly in}\quad L^{2}(0,T;H^{s^{\prime}}_{\mathrm{loc}}(\mathbb{R}^{d}))\ \ \text{for all}\ \ s^{\prime}<s.$ Similarly, we can obtain the convergence of ${\rm div}u^{\epsilon}$. ∎ We continue our proof of Theorem 1.1. From Proposition 4.1, we know that $\displaystyle{\rm div}\,{\mathbf{u}}^{\epsilon}\rightarrow{\rm div}\,{\mathbf{v}}\quad\mathrm{in}\quad L^{2}(0,T;H^{s^{\prime}-1}_{\mathrm{loc}}(\mathbb{R}^{d})).$ Hence, from (4.4) it follows that $\displaystyle{\mathbf{u}}^{\epsilon}\rightarrow{\mathbf{v}}\quad\mathrm{in}\quad L^{2}(0,T;H^{s^{\prime}}_{\mathrm{loc}}(\mathbb{R}^{d}))\qquad\mbox{for all }s^{\prime}<s.$ By (4.2), (4.3) and Proposition 4.2, we obtain $\begin{array}[]{ccl}r^{\epsilon}(S^{\epsilon},\epsilon q^{\epsilon})\rightarrow r_{0}(\bar{S})&\mathrm{in}&L^{\infty}(0,T;L^{\infty}(\mathbb{R}^{d}));\\\ \nabla{\mathbf{u}}^{\epsilon}\rightarrow\nabla{\mathbf{v}}&\mathrm{in}&L^{2}(0,T;H^{s^{\prime}-1}_{\mathrm{loc}}(\mathbb{R}^{d}));\\\ \nabla\mathbf{H}^{\epsilon}\rightarrow\nabla\bar{\mathbf{H}}&\mathrm{in}&L^{2}(0,T;H^{s^{\prime}-1}_{\mathrm{loc}}(\mathbb{R}^{d})).\end{array}$ Passing to the limit in the equations for $S^{\epsilon}$ and $\mathbf{H}^{\epsilon}$, we see that the limits $\bar{S}$ and $\bar{\mathbf{H}}$ satisfy $\displaystyle\partial_{t}\bar{S}+({\mathbf{v}}\cdot\nabla)\bar{S}=0,\quad\partial_{t}\bar{\mathbf{H}}+({{\mathbf{v}}}\cdot\nabla)\bar{\mathbf{H}}-(\bar{\mathbf{H}}\cdot\nabla){{\mathbf{v}}}=0$ in the sense of distributions. Since $r^{\epsilon}(S^{\epsilon},\epsilon q^{\epsilon})-r_{0}(S^{\epsilon})=O(\epsilon)$, we have $\displaystyle(r^{\epsilon}(S^{\epsilon},\epsilon q^{\epsilon})-r_{0}(S^{\epsilon}))(\partial_{t}{\mathbf{u}}^{\epsilon}+({\mathbf{u}}^{\epsilon}\cdot\nabla){\mathbf{u}}^{\epsilon})\rightarrow 0,$ whence, $\displaystyle r^{\epsilon}(S^{\epsilon},\epsilon q^{\epsilon})(\partial_{t}{\mathbf{u}}^{\epsilon}+({\mathbf{u}}^{\epsilon}\cdot\nabla){\mathbf{u}}^{\epsilon})=\,$ $\displaystyle(r^{\epsilon}(S^{\epsilon},\epsilon q^{\epsilon})-r_{0}(S^{\epsilon}))(\partial_{t}{\mathbf{u}}^{\epsilon}+({\mathbf{u}}^{\epsilon}\cdot\nabla){\mathbf{u}}^{\epsilon})$ $\displaystyle+\partial_{t}(r_{0}(S^{\epsilon}){\mathbf{u}}^{\epsilon})+({\mathbf{u}}^{\epsilon}\cdot\nabla)(r_{0}(S^{\epsilon}){\mathbf{u}}^{\epsilon})$ $\displaystyle\rightarrow$ $\displaystyle\,r_{0}(\bar{S})(\partial_{t}{\mathbf{v}}+({\mathbf{v}}\cdot\nabla){\mathbf{v}})$ in the sense of distributions. Applying the operator _curl_ to the momentum equation (1.17) and taking to the limit, we find that $\displaystyle{\rm curl\,}\big{(}r_{0}(\bar{S})(\partial_{t}{\mathbf{v}}+{\mathbf{v}}\cdot\nabla{\mathbf{v}})-({\rm curl\,}\bar{\mathbf{H}})\times\bar{\mathbf{H}}-\mu\Delta{\mathbf{v}}\big{)}=0.$ Therefore, by the fact that ${\rm curl\,}\nabla=0$, the limit $(\bar{S},{\mathbf{v}},\bar{\mathbf{H}})$ satisfies $\displaystyle r(\bar{S},0)(\partial_{t}{\mathbf{v}}+({\mathbf{v}}\cdot\nabla){\mathbf{v}})-({\rm curl\,}\bar{\mathbf{H}})\times\bar{\mathbf{H}}-\mu\Delta{\mathbf{v}}+\nabla\pi=0,$ (4.10) $\displaystyle\partial_{t}\bar{\mathbf{H}}+({{\mathbf{v}}}\cdot\nabla)\bar{\mathbf{H}}-(\bar{\mathbf{H}}\cdot\nabla){{\mathbf{v}}}=0,$ (4.11) $\displaystyle\partial_{t}\bar{S}+({\mathbf{v}}\cdot\nabla)\bar{S}=0,$ (4.12) $\displaystyle{\rm div}{\mathbf{v}}=0,\quad{\rm div}\bar{\mathbf{H}}=0$ (4.13) for some function $\pi$. If we employ the same arguments as in the proof of Theorem 1.5 in [27], we find that $(\bar{S},{\mathbf{v}},\bar{\mathbf{H}})$ satisfies the initial conditions (1.31). Moreover, the standard iterative method shows that the system (4.10)–(4.13) with initial data (1.31) has a unique solution $(S^{},{\mathbf{v}}^{},\mathbf{H}^{})\in C([0,T],H^{s}(\mathbb{R}^{d})).$ Thus, the uniqueness of solutions to the limit system (4.10)–(4.13) implies that the above convergence results hold for the full sequence of $(S^{\epsilon},q^{\epsilon},{\mathbf{u}}^{\epsilon},\mathbf{H}^{\epsilon})$. Thus, the proof is completed. ∎ ## 5\. Compressible non-isentropic MHD equations with infinite Reynolds number In the study of magnetohydrodynamics, for some local processes in the cosmic system, the effect of the magnetic diffusion will become very important, see [13]. Moreover, when the Reynolds number of a fluid is very high and the temperature changes very slowly, we can ignore the viscosity and the heat conductivity of the fluid in the MHD equations. In such situation, the compressible MHD equations in the non-isentropic case take the form: $\displaystyle\partial_{t}\rho+{\rm div}(\rho{\mathbf{u}})=0,$ (5.1) $\displaystyle\partial_{t}(\rho{\mathbf{u}})+{\rm div}\left(\rho{\mathbf{u}}\otimes{\mathbf{u}}\right)+\nabla p=({\rm curl\,}\mathbf{H})\times\mathbf{H},$ (5.2) $\displaystyle\partial_{t}{\mathcal{E}}+{\rm div}\left({\mathbf{u}}({\mathcal{E}}^{\prime}+p)\right)={\rm div}\big{[}({\mathbf{u}}\times\mathbf{H})\times\mathbf{H}+\nu\mathbf{H}\times({\rm curl\,}\mathbf{H})\big{]},$ (5.3) $\displaystyle\partial_{t}\mathbf{H}-{\rm curl\,}({\mathbf{u}}\times\mathbf{H})=-{\rm curl\,}(\nu\,{\rm curl\,}\mathbf{H}),\quad{\rm div}\mathbf{H}=0.$ (5.4) As before here $\rho$ denotes the density, ${\mathbf{u}}\in{\mathbb{R}}^{d}$ ($d=2,3$) the velocity, $\mathbf{H}\in{\mathbb{R}}^{d}$ the magnetic field; ${\mathcal{E}}$ the total energy given by ${\mathcal{E}}={\mathcal{E}}^{\prime}+\|\mathbf{H}\|^{2}/2$ and ${\mathcal{E}}^{\prime}=\rho\left(e+\|{\mathbf{u}}\|^{2}/2\right)$ with $e$ being the internal energy, $\rho\|{\mathbf{u}}\|^{2}/2$ the kinetic energy, and $\|\mathbf{H}\|^{2}/2$ the magnetic energy. The equations of state $p=p(\rho,\theta)$ and $e=e(\rho,\theta)$ relate the pressure $p$ and the internal energy $e$ to the density $\rho$ and the temperature $\theta$. The constant $\nu>0$ is the magnetic diffusivity acting as a magnetic diffusion coefficient of the magnetic field. Using the Gibbs relation (1.9) and the identities (1.6) and (1.7), the equation of energy conservation (5.3) can be replaced by $\partial_{t}(\rho S)+{\rm div}(\rho S{\mathbf{u}})=\frac{\nu}{\theta}\|{\rm curl\,}\mathbf{H}\|^{2},$ (5.5) where $S$ denotes the entropy. As in Section 1, we reconsider the equations of state as functions of $S$ and $p$, i.e., $\rho=R(S,p)$ and $\theta=\Theta(S,p)$ for some positive smooth functions $R$ and $\Theta$ defined for all $S$ and $p>0$, and satisfying $\partial R/\partial p>0$. Then, by utilizing (1.1) together with the constraint ${\rm div}{\mathbf{H}}=0$, the system (5.1), (5.2), (5.4) and (5.5) can be rewritten as $\displaystyle A(S,p)(\partial_{t}p+({\mathbf{u}}\cdot\nabla)p)+{\rm div}{\mathbf{u}}=0,$ (5.6) $\displaystyle R(S,p)(\partial_{t}{\mathbf{u}}+({\mathbf{u}}\cdot\nabla){\mathbf{u}})+\nabla p=({\rm curl\,}\mathbf{H})\times\mathbf{H},$ (5.7) $\displaystyle\partial_{t}{\mathbf{H}}-{\rm curl\,}({\mathbf{u}}\times\mathbf{H})=-{\rm curl\,}(\nu\,{\rm curl\,}\mathbf{H}),\quad{\rm div}\mathbf{H}=0,$ (5.8) $\displaystyle R(S,p)\Theta(S,p)(\partial_{t}S+({\mathbf{u}}\cdot\nabla)S)={\nu}\|{\rm curl\,}\mathbf{H}\|^{2},$ (5.9) where $A(S,p)$ is defined by (1.15). By introducing the dimensionless parameter $\epsilon$, and making the following changes of variables: $\displaystyle p(x,t)=p^{\epsilon}(x,\epsilon t),\quad S(x,t)=S^{\epsilon}(x,\epsilon t),$ $\displaystyle{{\mathbf{u}}}(x,t)=\epsilon{\mathbf{u}}^{\epsilon}(x,\epsilon t),\;\;\;{\mathbf{H}}(x,t)=\epsilon\mathbf{H}^{\epsilon}(x,\epsilon t),\;\;\;\nu=\epsilon\,\mu^{\epsilon},$ and $p^{\epsilon}(x,\epsilon t)=\underline{p}e^{\epsilon q^{\epsilon}(x,\epsilon t)}$ for some positive constant $\underline{p}$, the system (5.6)–(5.9) can be rewritten as $\displaystyle a^{\epsilon}(S^{\epsilon},\epsilon q^{\epsilon})(\partial_{t}q^{\epsilon}+({\mathbf{u}}^{\epsilon}\cdot\nabla)q^{\epsilon})+\frac{1}{\epsilon}{\rm div}{\mathbf{u}}^{\epsilon}=0,$ (5.10) $\displaystyle r^{\epsilon}(S^{\epsilon},\epsilon q^{\epsilon})(\partial_{t}{\mathbf{u}}^{\epsilon}+({\mathbf{u}}^{\epsilon}\cdot\nabla){\mathbf{u}}^{\epsilon})+\frac{1}{\epsilon}\nabla q^{\epsilon}=({\rm curl\,}{\mathbf{H}^{\epsilon}})\times{\mathbf{H}^{\epsilon}},$ (5.11) $\displaystyle\partial_{t}{\mathbf{H}}^{\epsilon}-{\rm curl\,}({\mathbf{u}}^{\epsilon}\times\mathbf{H}^{\epsilon})-\mu^{\epsilon}\Delta\mathbf{H}^{\epsilon}=0,\quad{\rm div}\mathbf{H}^{\epsilon}=0,$ (5.12) $\displaystyle b^{\epsilon}(S^{\epsilon},\epsilon q^{\epsilon})(\partial_{t}S^{\epsilon}+({\mathbf{u}}^{\epsilon}\cdot\nabla)S^{\epsilon})=\epsilon^{2}{\mu}^{\epsilon}\|{\rm curl\,}\mathbf{H}^{\epsilon}\|^{2},$ (5.13) where we have used the identity ${\rm curl\,}{\rm curl\,}\mathbf{H}^{\epsilon}=\nabla{\rm div}\mathbf{H}^{\epsilon}-\Delta\mathbf{H}^{\epsilon},$ the constraint ${\rm div}\mathbf{H}^{\epsilon}=0$, and the abbreviations (1.20) and (1.21). Formally, we obtain from (5.10) and (5.11) that $\nabla q^{\epsilon}\rightarrow 0$ and ${\rm div}{\mathbf{u}}^{\epsilon}=0$ as $\epsilon\rightarrow 0$. Applying the operator _curl_ to (5.11), using the fact that ${\rm curl\,}\nabla=0$, and letting $\epsilon\rightarrow 0$, we obtain $\displaystyle{\rm curl\,}\big{(}r(\bar{S},0)(\partial_{t}{\mathbf{v}}+{\mathbf{v}}\cdot\nabla{\mathbf{v}})-({\rm curl\,}\bar{\mathbf{H}})\times\bar{\mathbf{H}}\big{)}=0,$ where we have assumed that $(S^{\epsilon},q^{\epsilon},{\mathbf{u}}^{\epsilon},\mathbf{H}^{\epsilon})$ and $r^{\epsilon}(S^{\epsilon},\epsilon q^{\epsilon})$ converge to $(\bar{S},0,{\mathbf{v}},\bar{\mathbf{H}})$ and $r(\bar{S},0)$ in some sense, respectively. Finally, Letting $\mu^{\epsilon}\rightarrow\mu>0$ and applying the identity (1.22), we expect to get the following incompressible MHD equations: $\displaystyle r(\bar{S},0)(\partial_{t}{\mathbf{v}}+({\mathbf{v}}\cdot\nabla){\mathbf{v}})-({\rm curl\,}\bar{\mathbf{H}})\times\bar{\mathbf{H}}+\nabla\hat{\pi}=0,$ (5.14) $\displaystyle\partial_{t}\bar{\mathbf{H}}+({{\mathbf{v}}}\cdot\nabla)\bar{\mathbf{H}}-(\bar{\mathbf{H}}\cdot\nabla){{\mathbf{v}}}-\mu\Delta\bar{\mathbf{H}}=0,$ (5.15) $\displaystyle\partial_{t}\bar{S}+({\mathbf{v}}\cdot\nabla)\bar{S}=0,$ (5.16) $\displaystyle{\rm div}\,{\mathbf{v}}=0,\quad{\rm div}\bar{\mathbf{H}}=0$ (5.17) for some function $\hat{\pi}$. We supplement the system (1.16)–(1.19) with initial conditions $\displaystyle(S^{\epsilon},q^{\epsilon},{\mathbf{u}}^{\epsilon},\mathbf{H}^{\epsilon})\|_{t=0}=(S^{\epsilon}_{0},q^{\epsilon}_{0},{\mathbf{u}}^{\epsilon}_{0},\mathbf{H}^{\epsilon}_{0}).$ (5.18) The main result of this section reads as follows. ###### Theorem 5.1. Let $s>d/2+2$ be an integer and $\mu^{\epsilon}\rightarrow\mu>0$. For any constant $M_{0}>0$, there is a positive constant $T=T(M_{0})$, such that for all $\epsilon\in(0,1]$ and any initial data $(S^{\epsilon}_{0},q^{\epsilon}_{0},{\mathbf{u}}^{\epsilon}_{0},\mathbf{H}^{\epsilon}_{0})$ satisfying $\displaystyle\\|(S^{\epsilon}_{0},q^{\epsilon}_{0},{\mathbf{u}}^{\epsilon}_{0},\mathbf{H}^{\epsilon}_{0})\\|_{H^{s}({\mathbb{R}}^{d})}\leq M_{0},$ (5.19) the Cauchy problem (5.10)–(5.13), (5.18) has a unique solution $(S^{\epsilon},q^{\epsilon},{\mathbf{u}}^{\epsilon},\mathbf{H}^{\epsilon})\in C^{0}([0,T],H^{s}({\mathbb{R}}^{d}))$, satisfying that for all $\epsilon\in(0,1]$ and $t\in[0,T]$, $\displaystyle\\|\big{(}S^{\epsilon},q^{\epsilon},{\mathbf{u}}^{\epsilon},\mathbf{H}^{\epsilon}\big{)}(t)\\|_{H^{s}({\mathbb{R}}^{d})}\leq N\quad\mbox{for some constant }N=N(M_{0})>0.$ (5.20) Moreover, suppose further that the initial data $(S^{\epsilon}_{0},q^{\epsilon}_{0},{\mathbf{u}}^{\epsilon}_{0},\mathbf{H}^{\epsilon}_{0})$ converge strongly in $H^{s}({\mathbb{R}}^{d})$ to $(S_{0},0,{\mathbf{v}}_{0},\mathbf{H}_{0})$ and $S^{\epsilon}_{0}$ decays sufficiently rapidly at infinity in the sense that $\|S^{\epsilon}_{0}(x)-\underline{S}\,\,\|\leq{N}_{0}\|x\|^{-1-\iota},\quad\|\nabla S^{\epsilon}_{0}(x)\|\leq N_{0}\|x\|^{-2-\iota}$ (5.21) for all $\epsilon\in(0,1]$ and some positive constants $\underline{S}$, $N_{0}$ and $\iota$. Then, $(S^{\epsilon},q^{\epsilon},{\mathbf{u}}^{\epsilon},\mathbf{H}^{\epsilon})$ converges weakly in $L^{\infty}(0,T;H^{s}({\mathbb{R}}^{d}))$ and strongly in $L^{2}(0,T;H^{s^{\prime}}_{\mathrm{loc}}({\mathbb{R}}^{d}))$ to a limit $(\bar{S},0,{\mathbf{v}},\bar{\mathbf{H}})$ for all $s^{\prime}<s$. Moreover, $(\bar{S},{\mathbf{v}},\bar{\mathbf{H}})$ is the unique solution in $C([0,T],$ $H^{s}({\mathbb{R}}^{d}))$ to the system (5.14)–(5.17) with initial data $(\bar{S},{\mathbf{v}},\bar{\mathbf{H}})\|_{t=0}=(S_{0},{\mathbf{w}}_{0},\mathbf{H}_{0})$, where ${\mathbf{w}}_{0}\in H^{s}({\mathbb{R}}^{d})$ is determined by ${\rm div}\,{\mathbf{w}}_{0}=0,\,\;{\rm curl\,}(r(S_{0},0){\mathbf{w}}_{0})={\rm curl\,}(r(S_{0},0){\mathbf{v}}_{0}),\;\,r(S_{0},0):=\lim_{\epsilon\rightarrow 0}r^{\epsilon}(S^{\epsilon}_{0},0).$ (5.22) The function $\hat{\pi}\in C([0,T]\times{\mathbb{R}}^{d})$ satisfies $\nabla\hat{\pi}\in C([0,T],H^{s-1}({\mathbb{R}}^{d}))$. ###### Sketch of the proof of Theorem 5.1. As explained before, the main step is to establish the uniform estimate (5.20). For this purpose, we define $\mathcal{M}_{\epsilon}(T)$ as follows $\displaystyle\mathcal{M}_{\epsilon}(T):=\,$ $\displaystyle{\mathcal{N}_{\epsilon}(T)}^{2}+\int_{0}^{T}\\|\mathbf{H}^{\epsilon}\\|_{s+1}^{2}d\tau,$ (5.23) where $\mathcal{N}_{\epsilon}(T)$ is defined by (3.3). By arguments similar to those used in the proof of Theorem 3.1, one can obtain the desired estimate. Indeed, the arguments are easier since one can use the magnetic diffusion term to control the terms involving $\mathbf{H}$ in the momentum equations, and therefore we omit the details here. ∎ Acknowledgements: The authors would like to thank Prof. Fanghua Lin for suggesting this problem and for helpful discussions. This work was partially done when Li visited the Institute of Applied Physics and Computational Mathematics in Beijing. He would like to thank the institute for hospitality. Jiang was supported by the National Basic Research Program under the Grant 2011CB309705 and NSFC (Grant No. 40890154). Ju was supported by NSFC (Grant No. 40890154, 11171035). Li was supported by NSFC (Grant No. 10971094), PAPD, and the Fundamental Research Funds for the Central Universities. ## References [1] T. Alazard, Incompressible limit of the nonisentropic Euler equations with solid wall boundary conditions, Adv. Differential Equations, 10 (2005), 19-44. * [2] T. Alazard, Low Mach number limit of the full Navier-Stokes equations, Arch. Ration. Mech. Anal. 180 (2006), 1-73. * [3] K. Asano, On the incompressible limit of the compressible Euler equations, Japan J. Appl. Math., 4 (1987), 455-488. * [4] D. Bresch, B. Desjardins and E. Grenier, Oscillatory Limit with Changing Eigenvalues: A Formal Study. In: New Directions in Mathematical Fluid Mechanics, A. V. Fursikov, G. P. Galdi, V. V. Pukhnachev (eds.), 91-105, Birkhäuser Verlag, Basel, 2010. * [5] D. Bresch, B. Desjardins, E. Grenier and C.-K. Lin, Low Mach number limit of viscous polytropic flows: formal asymptotics in the periodic case, Stud. Appl. Math. 109 (2002), 125-149. * [6] R. Danchin, Low Mach number limit for viscous compressible flows, M2AN Math. Model. Numer. Anal. 39 (2005), 459-475. * [7] B. Desjardins and E. Grenier, Low Mach number limit of viscous compressible flows in the whole space, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 455 (1999), 2271-2279. * [8] B. Desjardins, E. Grenier, P.-L. Lions and N. Masmoudi, Incompressible limit for solutions of the isentropic Navier-Stokes equations with Dirichlet boundary conditions, J. Math. Pures Appl. (9) 78 (1999), 461-471. * [9] E. Feireisl znd A. Novotný, Singular Limits in Thermodynamics of Viscous Fluids, Birkhauser Basel, 2009. * [10] J.P. Freidberg, Ideal magnetohydrodynamic theory of magnetic fusion systems, Rev. Modern Phys. 54 (1982), 801-903. * [11] I. Gallagher, A remark on smooth solutions of the weakly compressible periodic Navier-Stokes equations, J. Math. Kyoto Univ. 40 (2000), 525-540. * [12] L. Hömander, Lectures on Nonlinear Hyperbolic Differential Equations, Springer-Verlag, 1997. * [13] W.R. Hu, Cosmic Magnetohydrodynamics, Science Press, Beijing, 1987 (in Chinese). * [14] X.P. Hu and D.H. Wang, Low Mach number limit of viscous compressible magnetohydrodynamic flows, SIAM J. Math. Anal. 41 (2009), 1272-1294. * [15] T. Iguchi, The incompressible limit and the initial layer of the compressible Euler equation in $R_{n}^{+}$, Math. Methods Appl. Sci. 20 (1997), 945-958. * [16] H. Isozaki, Singular limits for the compressible Euler equation in an exterior domain, J. Reine Angew. Math. 381 (1987), 1-36. * [17] H. Isozaki, Singular limits for the compressible Euler equation in an exterior domain, I Bodies in an uniform flow, Osaka J. Math. 26 (1989), 399-410. * [18] S. Jiang, Q.C. Ju and F.C. Li, Incompressible limit of the compressible Magnetohydrodynamic equations with periodic boundary conditions, Comm. Math. Phys. 297 (2010), 371-400. * [19] S. Jiang, Q.C. Ju and F.C. Li, Incompressible limit of the compressible magnetohydrodynamic equations with vanishing viscosity coefficients, SIAM J. Math. Anal. 42 (2010), 2539-2553. * [20] S. Jiang, Q.C. Ju and F.C. Li, Low Mach number limit for the multi-dimensional full magnetohydrodynamic equations, Preprint, available at arXiv:1105.0729v1 [math.AP]. * [21] S. Jiang, Q.C. Ju, F.C. Li and Z.P. Xin, Low Mach number limit for the full compressible magnetohydrodynamic equations with general initial data, submitted, 2011. * [22] S. Jiang and Y.B. Ou, Incompressible limit of the non-isentropic Navier-Stokes equations with well-prepared initial data in three-dimensional bounded domains, J. Math. Pures Appl. 96 (2011), 1-28. * [23] S. Klainerman and A. Majda, Singular perturbations of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids, Comm. Pure Appl. Math. 34 (1981), 481-524. * [24] P. Kuku${\rm\check{c}}$ka, Singular Limits of the Equations of Magnetohydrodynamics, J. Math. Fluid Mech. 13 (2011), 173-189. * [25] Y.-S. Kwon and K. Trivisa, On the incompressible limits for the full magnetohydrodynamics flows, J. Differential Equations 251 (2011), 1990-2023. * [26] C.D. Levermore, W. Sun and K. Trivisa, A low Mach number limit of a dispersive Navier-Stokes system, Preprint, 2011. * [27] G. Métivier and S. Schochet, The incompressible limit of the non-isentropic Euler equations, Arch. Ration. Mech. Anal. 158 (2001), 61-90. * [28] G. Métivier and S. Schochet, Averaging theorems for conservative systems and the weakly compressible Euler equations, J. Differential Equations 187 (2003), 106-183. * [29] A. Novotný, M. Ruzicka and G. Thäter, Singular limit of the equations of magnetohydrodynamics in the presence of strong stratification, Math. Models Methods Appl. Sci. 21 (2011), 115-147. * [30] P.-L. Lions and N. Masmoudi, Incompressible limit for a viscous compressible fluid, J. Math. Pures Appl. 77 (1998), 585-627. * [31] N. Masmoudi, Examples of singular limits in hydrodynamics, In: Handbook of Differential Equations: Evolutionary equations, Vol. III, 195-275, Elsevier/North-Holland, Amsterdam, 2007. * [32] S. Schochet, Fast singular limits of hyperbolic PDEs, J. Differential Equations 114 (1994), 476-512. * [33] S. Schochet, The compressible Euler equations in a bounded domain: existence of solutions and the incompressible limit, Comm. Math. Phys. 104 (1986), 49-75. * [34] S. Schochet, The mathematical theory of the incompressible limit in fluid dynamics, in: Handbook of Mathematical Fluid Dynamics, Vol. IV, pp.123-157, Elsevier/North-Holland, Amsterdam, 2007. * [35] S. Ukai, The incompressible limit and the initial layer of the compressible Euler equation, J. Math. Kyoto Univ., 26 (1986), 323-331. * [36] A.I. Vol’pert and S.I. Khudiaev, On the Cauchy problem for composite systems of nonlinear equations, Mat. Sbornik 87 (1972), 504-528.	arxiv-papers	2011-11-12T13:45:42	2024-09-04T02:49:24.264027	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Song Jiang, Qiangchang Ju, and Fucai Li", "submitter": "Fucai Li", "url": "https://arxiv.org/abs/1111.2926" }
1111.3083	11institutetext: Université de Lyon, Lyon, F-69003, France; Université Lyon 1, Villeurbanne, F-69622, France; CNRS, UMR 5574, Centre de Recherche Astrophysique de Lyon; École normale supérieure de Lyon, 46, allée d’Italie, F-69364 Lyon Cedex 07, France 11email: [Guillaume.Laibe;Jean-Francois.Gonzalez]@ens-lyon.fr 22institutetext: Centre for Stellar and Planetary Astrophysics, School of Mathematical Sciences, Monash University, Clayton Vic 3168, Australia 22email: guillaume.laibe@monash.edu 33institutetext: Centre for Astrophysics and Supercomputing, Swinburne University, PO Box 218, Hawthorn, VIC 3122, Australia 33email: smaddison@swin.edu.au # Revisiting the “radial-drift barrier” of planet formation and its relevance in observed protoplanetary discs G. Laibe 1122 J.-F. Gonzalez 11 S.T. Maddison 33 (Received 7 July 2010; Accepted ??) ###### Abstract Context. To form metre-sized pre-planetesimals in protoplanetary discs, growing grains have to decouple from the gas before they are accreted onto the central star during their phase of fast radial migration and thus overcome the so-called “radial-drift barrier” (often inaccurately referred to as the “metre-size barrier”). Aims. To predict the outcome of the radial motion of dust grains in protoplanetary discs whose surface density and temperature follow power-law profiles, with exponent $p$ and $q$ respectively. We investigate both the Epstein and the Stokes drag regimes which govern the motion of the dust. Methods. We analytically integrate the equations of motion obtained from perturbation analysis. We compare these results with those from direct numerical integration of the equations of motion. Then, using data from observed discs, we predict the fate of dust grains in real discs. Results. When a dust grain reaches the inner regions of the disc, the acceleration due to the increase of the pressure gradient is counterbalanced by the increase of the gas drag. We find that most grains in the Epstein (resp. the Stokes) regime survive their radial migration if $-p+q+\frac{1}{2}\leq 0$ (resp. if $q\leq\frac{2}{3}$). The majority of observed discs satisfies both $-p+q+\frac{1}{2}\leq 0$ and $q\leq\frac{2}{3}$: a large fraction of both their small and large grains remain in the disc, for them the radial drift barrier does not exist. ###### Key Words.: planetary systems: protoplanetary discs — methods: analytical ## 1 Introduction Much of the information about the gas structure of protoplanetary discs is inferred from the emission by the dust component and an assumed dust-to-gas ratio. Interpretations of recent observations in the (sub)millimetre domain (Andrews & Williams 2005, 2007; Lommen et al. 2007) show that observed discs typically have masses between $10^{-4}$ and $10^{-1}\ M_{\odot}$ and a spatial extent of a few hundred AU. Their radial surface density and temperature profiles are approximated by power laws ($\Sigma\propto r^{-p}$, $\mathcal{T}\propto r^{-q}$), whose respective exponents $p$ and $q$ have positive values typically of order unity. Seminal studies describe the dust motion in protoplanetary discs, which depends strongly on the gas structure. Weidenschilling (1977a, hereafter W77) and Nakagawa et al. (1986, hereafter NSH86) demonstrated that dust grains from micron sizes to pre-planetesimals (a few metres in size) experience a radial motion through protoplanetary discs. This motion is called radial drift or migration. Due to its pressure gradient, the gas orbits the central star at a sub-Keplerian velocity. Grains therefore have a differential velocity with respect to the gas. The ensuing drag transfers linear and angular momentum from the dust to the gas. Thus, dust particles can not sustain the Keplerian motion they would have without the presence of gas and as a result migrate toward the central star. This migration motion depends strongly on the grain size, which sets the magnitude of the drag, as well as the nature of the drag regime. Specifically, as shown by W77 and NSH86, grains of a critical size pass through the disc in a fraction of the disc lifetime. This catastrophic outcome is called the “radial-drift barrier” of planet formation. More precisely, we will adopt the subsequent definition for the “radial-drift barrier” in this study: “the ability of grains of be accreted onto the central star/depleted from the disc within its lifetime”. Historically, this process was first studied in a Minimum Mass Solar Nebula (MMSN, see Weidenschilling 1977b; Hayashi 1981; Desch 2007; Crida 2009), in which the critical size corresponds to metre-sized bodies and thus was called the “metre-size barrier”. However, planets are frequently observed (besides the 8 planets in our solar system, more than 700 extra-solar planets have been discovered to date111http://exoplanet.eu): some solid material must therefore have overcome this barrier and stayed in the disc to form larger bodies. Moreover, if the small grains of every disc were submitted to the radial-drift barrier, we would barely detect them since their emission via optical/IR scattering and IR thermal radiation is due to small grains. As discs are frequently observed, the grains from which the emission is detected cannot be strongly depleted for a substantial fraction of discs. From a theoretical point of view, such a discrepancy between the observations and the theoretical predictions imply that the seminal theory has to be extended (some physical element is lacking) or that it has not been fully exploited. This second option has been investigated by Youdin & Shu (2002, hereafter YS02). They highlight the fact that, contrary to the primary hypothesis of W77, observed dusty discs are drastically different from the MMSN prototype. As the radial surface density and temperature profiles fix both the radial pressure gradient and the magnitude of the gas drag, different values for the power-law exponents $p$ and $q$ affect the optimal grain size of migration and thus induce different radial motions for the dust through the disc. Specifically, YS02 showed that for steep surface density profiles and smooth temperature profiles, the grains radial velocity decreases when the grains reach the inner discs regions. Grains in such discs therefore experience a “pile-up”. However, while important, the work of YS02 does not provide a precise conclusion on the outcome of the grains nor any quantitative criterion for the “pile-up” process to be efficient enough to avoid the radial-drift barrier. Furthermore, YS02 restricts their study to the special case of a gas phase with a low density (e.g. the grain size smaller than the gas mean free path, called the Epstein regime). This hypothesis is not valid anymore when considering the radial drift of pre-planetesimals, whose grain sizes are larger than the gas mean free path and are submitted to the Stokes drag regime. Although the radial drift of pre-planetesimals has already been studied in different situations with numerical or semi-analytical methods — see e.g. Haghighipour & Boss (2003); Birnstiel et al. (2009); Youdin (2011) — its rigorous theory for the standard case of a simple disc has not yet been derived. Within this context, we see that (i) the seminal theory describing the radial motion of dust grains has been developed within the limits of the Epstein regime but does not treat the Stokes regime ; (ii) here exists no clear theoretical criterion to predict the impact of the “pile-up effect” on the outcome of the dust radial motion ; (iii) there exists no criterion to predict whether a given disc will be submitted to the “radial-drift barrier” phenomenon. To answer these three points, we re-visit in this study the work of W77 and NSH86 and extend the developments of YS02 for both the Epstein and the Stokes regime. Performing rigorous perturbative expansions, we find two theoretical criteria (one for each regime) which predict when the “pile-up” effect is sufficient for the grains not to be accreted onto the central star. We then test when these theoretical criteria can be applied in real discs. Additionally, our work is motivated by the recent observational results of Ricci et al. (2010a, b). From their observations they claim that “a mechanism halting or slowing down the inward radial drift of solid particles is required to explain the data”. In this work we aim to show that contrary to what is usually invoked, local pressure maxima due to turbulent vortices or spiral density waves may help but are not necessarily required to explain the observations. Ricci et al. (2010a) also mention that “the observed flux of the fainter discs are instead typically overpredicted even by more than an order of magnitude”. Here, we also aim to provide a quantitative criterion to determine which discs are faint and which one are not. Thus, revisiting the seminal theory of the radial drift is timely, all the more so than an important quantity of new data is soon to be provided by ALMA, the Atacama Large Millimeter/submillimeter Array. In this paper, we first recall some general properties of grain motion in protoplanetary discs for both the Epstein and Stokes regime in Sect. 2. We then focus on the radial motion of non-growing grains in the Epstein regime. We expand the radial motion equations assuming a weak pressure gradient in Sect. 3 and detail the two different modes of migration which grains may experience in Sects. 3.1 and 3.2. This allows us to derive an analytic criterion which determines the asymptotic dust behaviour in the Epstein regime in Sect. 3.3. We transpose these derivations for the Stokes regime at low Reynolds number in Sect. 4 and obtain a similar criterion for this regime. We also discuss the grains outcome for large Reynolds numbers. In Sect. 5, we discuss the relevance of these criteria and study their implications for observed protoplanetary discs and planet formation in Sect. 6. Our conclusions are presented in Sect. 7. ## 2 Dynamics of dust grains To reduce the parameter space for this study, we assume the following: 1. 1. The disc is a thin, non-magnetic, non-self-graviting, inviscid perfect gas disc which is vertically isothermal. Its radial surface density and temperature are described by power-law profiles. Notations are described in Appendix A. The flow is laminar and in stationary equilibrium. Consequently, the gas velocity and density are described by well-known relations, which we present in Appendix B. 2. 2. The grains are compact homogeneous spheres of fixed radius. The collisions between grains and the collective effects due to large dust concentrations are neglected. When the grains are small compared to the mean free path of the gas ($\lambda_{\mathrm{g}}>{4s}/{9}$, where $s$ is the grain size), their interactions with the gas are treated by the Epstein drag force for diluted media (Epstein 1924; Baines et al. 1965; Stepinski & Valageas 1996). This drag is caused by the transfer of momentum by individual collisions with gas molecules at the grains surface. Assuming specular reflections on the grain and when the differential velocity with the gas is negligible compared to the gas sound speed, the now common expression of the drag force is $\left\\{\begin{array}[]{rcl}\mathbf{F}_{\mathrm{D}}&=&-\displaystyle\frac{m_{\mathrm{d}}}{t_{\mathrm{s}}}\,\Delta\mathbf{v}\\\\[10.00002pt] t_{\mathrm{s}}&=&\displaystyle\frac{\rho_{\mathrm{d}}s}{\rho_{\mathrm{g}}c_{\mathrm{s}}}\,,\end{array}\right.$ (1) where $m_{\mathrm{d}}$ is the dust grain’s mass, $t_{\mathrm{s}}$ the stopping time, $\rho_{\mathrm{g}}$ the gas density, $c_{\mathrm{s}}$ the local gas sound speed, $\rho_{\mathrm{d}}$ the intrinsic dust density, and $\Delta\mathbf{v}=\mathbf{v}-\mathbf{v}_{\mathrm{g}}$ the differential velocity between dust and the mean gas motion. In classical T Tauri star (CTTS) protoplanetary discs, drag forces for particles smaller than $\sim 10$ m are well described by the Epstein regime (Garaud et al. 2004, see also Sect. 6.1). Small grains which produce the emission of observed protoplanetary discs satisfy this criterion. The interactions between large dust particles ($\lambda_{\mathrm{g}}<{4s}/{9}$) and the gas are treated by the Stokes drag force (Whipple 1972; Stepinski & Valageas 1996). In this case, the gas mean free path is small and the dust particle is locally surrounded by a viscous fluid. Depending on the local Reynolds number of the flow around the grains $R_{\mathrm{g}}\leavevmode\nobreak\ =\leavevmode\nobreak\ \frac{2s\|\Delta\mathbf{v}\|}{\nu}$, where $\nu$ is the microscopic kinematic viscosity of the gas, the drag force takes the following expression: $\textbf{F}_{\mathrm{D}}=-\frac{1}{2}C_{\mathrm{D}}\pi s^{2}\rho_{\mathrm{g}}\left\|\Delta\mathbf{v}\right\|\Delta\mathbf{v},$ (2) where the drag coefficient $C_{\mathrm{D}}$ is given by $C_{\mathrm{D}}=\left\\{\begin{array}[]{ll}24R_{\mathrm{g}}^{-1}&\mathrm{for}\ R_{\mathrm{g}}<1\\\\[10.00002pt] 24R_{\mathrm{g}}^{-0.6}&\mathrm{for}\ 1<R_{\mathrm{g}}<800\\\\[10.00002pt] 0.44&\mathrm{for}\ 800<R_{\mathrm{g}}\,.\end{array}\right.$ (3) If $R_{\mathrm{g}}<1$, the drag force remains linear in $\Delta\mathbf{v}$. In this work, the physical relations are written in cylindrical coordinates ($r$, $\theta$, $z$). The related unit vector system is given by $\left(\textbf{e}_{r},\textbf{e}_{\theta},\textbf{e}_{z}\right)$. As the system is invariant by rotation around the vertical axis $\textbf{e}_{z}$, the physical quantities depend only on $r$ and $z$. The physical quantities of the gas, designated by subscript g, are first determined in a general way. Then, the limit $z=0$ is taken to study the restricted radial motion. Dust dynamics depends on both the magnitude of the drag (driven by the differential velocity) and on its relative contribution with respect to the gravity of the central star. Seminal studies of dust dynamics were conducted by Whipple (1972), W77, Weidenschilling (1980) and NSH86, and extended by others (YS02; Takeuchi & Lin 2002; Haghighipour & Boss 2003; Garaud et al. 2004; Youdin & Chiang 2004). Here we recall the major points of those studies. We consider two forces acting on the grain: the gravity of the central star and gas drag. (We assume that the momentum transferred by drag from a single grain on the gas phase is negligible.) Thus $m_{\mathrm{d}}\frac{\mathrm{d}\textbf{v}}{\mathrm{d}t}=-\textbf{F}_{\mathrm{D}}+m_{\mathrm{d}}\textbf{g},$ (4) where $\textbf{F}_{\mathrm{D}}$ is the drag force. As shown by Eqs. (1)–(2), a general expression of the ratio $\frac{\textbf{F}_{\mathrm{D}}}{m_{\mathrm{d}}}$ is of the form $\frac{\textbf{F}_{\mathrm{D}}}{m_{\mathrm{d}}}=-\frac{\tilde{\mathcal{C}}\left(r,z\right)}{s^{y}}\|\textbf{v}-\textbf{v}_{\mathrm{g}}\|^{\lambda}\left(\textbf{v}-\textbf{v}_{\mathrm{g}}\right),$ (5) where the quantities $\tilde{\mathcal{C}}$, $y$ and $\lambda$ are defined for both the Epstein and the Stokes regime in Appendix C. ## 3 Radial motion in the Epstein regime: perturbation analysis at small pressure gradients Considering the Epstein (small grains) regime, Eq. (4) reduces to: $m_{\mathrm{d}}\frac{\mathrm{d}\textbf{v}}{\mathrm{d}t}=-\frac{m_{\mathrm{d}}}{t_{\mathrm{s}}}\left(\textbf{v}-\textbf{v}_{\mathrm{g}}\right)+m_{\mathrm{d}}\textbf{g}.$ (6) Writing Eq. (6) in $\left(r,\theta,z\right)$ coordinates leads to $\left\\{\begin{array}[]{rcl}\displaystyle\frac{\mathrm{d}v_{r}}{\mathrm{d}t}-\frac{v_{\theta}^{2}}{r}+\frac{\left(v_{r}-v_{\mathrm{g}r}\right)}{t_{\mathrm{s}}}+\frac{\mathcal{G}Mr}{\left(z^{2}+r^{2}\right)^{3/2}}&=&\displaystyle 0\\\ \displaystyle\frac{\mathrm{d}v_{\theta}}{\mathrm{d}t}+\frac{v_{r}v_{\theta}}{r}+\frac{\left(v_{\theta}-v_{\mathrm{g}\theta}\right)}{t_{\mathrm{s}}}&=&\displaystyle 0\\\ \displaystyle\frac{\mathrm{d}v_{z}}{\mathrm{d}t}+\frac{\left(v_{z}-v_{\mathrm{g}z}\right)}{t_{\mathrm{s}}}+\frac{\mathcal{G}Mz}{\left(z^{2}+r^{2}\right)^{3/2}}&=&\displaystyle 0.\end{array}\right.$ (7) To highlight the important parameters involved in the grains dynamics, we introduce dimensionless quantities (see Appendix C). It is crucial to note that the ratio $\frac{t_{\mathrm{s}}}{t_{\mathrm{k}}}$ of the two timescales related to the physical processes acting on the grain is given by $\frac{t_{\mathrm{s}}}{t_{\mathrm{k}}}=\frac{t_{\mathrm{s}0}}{t_{\mathrm{k}0}}R^{p}\mathrm{e}^{\frac{Z^{2}}{2R^{3-q}}}=S_{0}R^{p}\mathrm{e}^{\frac{Z^{2}}{2R^{3-q}}}.$ (8) With Eq. (1) and noting $\Omega_{\mathrm{k}}$ the Keplerian angular velocity, this ratio can be written as $\frac{t_{\mathrm{s}}}{t_{\mathrm{k}}}=\frac{s}{\left(\frac{\rho_{\mathrm{g}}c_{\mathrm{s}}}{\rho_{\mathrm{d}}\Omega_{\mathrm{k}}}\right)}=\frac{s}{s_{\mathrm{opt}}}=S,$ (9) where $s_{\mathrm{opt}}=\frac{\rho_{\mathrm{g}}c_{\mathrm{s}}}{\rho_{\mathrm{d}}\Omega_{\mathrm{k}}}$. This timescale ratio therefore corresponds to a dimensionless size $S=S_{0}R^{p}e^{\frac{Z^{2}}{2R^{3-q}}}$ for the grain. If $S\ll 1$ (resp. $S\gg 1$), the effects of drag will occur much faster (resp. slower) than gravitational effects. If $S\simeq 1$, both gravity and drag will act on the same timescale. Interestingly, $s_{\mathrm{opt}}$ varies in the disc midplane as $r^{-p}$, as does surface density. Then using Eqs. (8), (51) and (81), we obtain for ($\textbf{e}_{r},\textbf{e}_{\theta},\textbf{e}_{z}$): $\left\\{\begin{array}[]{rcl}\displaystyle\frac{\mathrm{d}\tilde{v}_{r}}{\mathrm{d}T}-\frac{\tilde{v}_{\theta}^{2}}{R}+\frac{\tilde{v}_{r}}{S_{0}}R^{-\left(p+\frac{3}{2}\right)}\mathrm{e}^{-\frac{Z^{2}}{2R^{3-q}}}+\frac{R}{\left(R^{2}+\phi_{0}Z^{2}\right)^{3/2}}&=&0\\\\[8.61108pt] \lx@intercol\displaystyle\frac{\mathrm{d}\tilde{v}_{\theta}}{\mathrm{d}T}+\frac{\tilde{v}_{\theta}\tilde{v}_{r}}{R}+\hfil\lx@intercol&&\\\ \frac{\left(\tilde{v}_{\theta}-\sqrt{\frac{1}{R}-\eta_{0}R^{-q}-q\left(\frac{1}{R}-\frac{1}{\sqrt{R^{2}+\phi_{0}Z^{2}}}\right)}\right)}{S_{0}}R^{-\left(p+\frac{3}{2}\right)}\mathrm{e}^{-\frac{Z^{2}}{2R^{3-q}}}&=&0\\\\[8.61108pt] \displaystyle\frac{\mathrm{d}^{2}Z}{\mathrm{d}T^{2}}+\frac{1}{S_{0}}\frac{\mathrm{d}Z}{\mathrm{d}T}R^{-\left(p+\frac{3}{2}\right)}\mathrm{e}^{-\frac{Z^{2}}{2R^{3-q}}}+\frac{Z}{\left(R^{2}+\phi_{0}Z^{2}\right)^{3/2}}&=&0.\end{array}\right.$ (10) These equations depend on five control parameters: the initial dimensionless grain size, $S_{0}$, the radial surface density and temperature exponents, $p$ and $q$, the square of the disc aspect ratio, $\phi_{0}=\left(H_{0}/r_{0}\right)^{2}$, and the subkeplerian parameter, $\eta_{0}$, given by Eq. (82). The equations can be simplified in some cases, e.g. if the vertical motion is considered to occur faster than the radial motion, $R\simeq 1$ and $\frac{\mathrm{d}^{2}Z}{\mathrm{d}T^{2}}$ simplifies to the damped harmonic oscillator equation. If we consider only the radial motion (for a 2D disc), we have $Z=0$, and $\left\\{\begin{array}[]{l}\displaystyle\frac{\mathrm{d}\tilde{v}_{r}}{\mathrm{d}T}=\displaystyle\frac{\tilde{v}_{\theta}^{2}}{R}-\frac{\tilde{v}_{r}}{S_{0}}R^{-\left(p+\frac{3}{2}\right)}-\frac{1}{R^{2}}\\\ \displaystyle\frac{\mathrm{d}\tilde{v}_{\theta}}{\mathrm{d}T}=\displaystyle-\frac{\tilde{v}_{\theta}\tilde{v}_{r}}{R}-\frac{\left(\tilde{v}_{\theta}-\sqrt{\frac{1}{R}-\eta_{0}R^{-q}}\right)}{S_{0}}R^{-\left(p+\frac{3}{2}\right)}.\end{array}\right.$ (11) Even for discs in two dimensions, Eq. (11) is not analytically tractable. However, as some of the parameters involved in the equation are small, approximations of the solution can be found by performing perturbative expansions. Some of the classical results detailed below have been studied in W77 and NSH86, but are here properly justified. The principle of those expansions is described on Fig. 1. Figure 1: Principle of the various perturbative expansions for the grain radial motion. Expanding first with respect to the small pressure gradient ($\eta_{0}R^{-q+1}\ll 1$) leads to NSH86 equations. Expanding first with respect to the grain sizes ($S_{0}R^{p}\ll 1$ or $S_{0}R^{p}\gg 1$) leads to W77 expressions for the particular case $p=0$. Combining both leads to the A- and B- mode, respectively for small and large grains. At $t=0$, $r=r_{0}$ which implies that $R\left(T=0\right)=1$. Because of gas drag, a grain dissipates both its energy and angular momentum and therefore, experiences a radial inward motion, i.e. $R<1$. The first parameter with respect to which a perturbative expansion can be performed is $\eta_{0}$ (linked to the pressure gradient by Eq. (83)) as it takes values of approximately $10^{-3}$–$10^{-2}$ in real protoplanetary discs (see NSH86), and thus $\eta_{0}\ll 1$. We consider that this inequality also implies that $\eta_{0}R^{-q}\ll\frac{1}{R}.$ (12) This inequality is always verified when $q\leq 1$ and thus applies to observed discs (see Sect. 6). For $q>1$, there is a region where this inequality is not verified. However, in this case, the pressure gradient has the same order of magnitude as the gravity of the central star and the model of a power-law profile for the radial temperature is not accurate enough to model realistic discs. We thus consider that for real discs Eq. (12) is always justified. Then, following NSH86, we consider the system of equations given by Eq. (11). We set $\left\\{\begin{array}[]{l}\displaystyle\tilde{v}_{r}=\displaystyle\tilde{v}_{r0}+\eta_{0}\tilde{v}_{r1}+\mathcal{O}\left(\eta_{0}^{2}\right)\\\ \displaystyle\tilde{v}_{\theta}=\displaystyle\tilde{v}_{\theta 0}+\eta_{0}\tilde{v}_{\theta 1}+\mathcal{O}\left(\eta_{0}^{2}\right)\end{array}\right.$ (13) and look at the orders $\mathcal{O}\left(1\right)$, $\mathcal{O}\left(\eta_{0}\right)$,… of the expansion – see Appendix E. We find that: $\tilde{v}_{r}=\eta_{0}\tilde{v}_{r1}+\mathcal{O}\left(\eta_{0}^{2}\right)=-\frac{2S_{0}R^{p-\frac{1}{2}}\left(1-\sqrt{1-\eta_{0}R^{-q+1}}\right)}{1+R^{2p}S_{0}^{2}}+\mathcal{O}\left(\eta_{0}^{2}\right).$ (14) The pressure gradient term has been retained to keep the generality, however since we assume that $\eta_{0}\ll 1$, we equivalently have $\sqrt{\frac{1}{R}-\eta_{0}R^{-q}}-\sqrt{\frac{1}{R}}=-\frac{\eta_{0}}{2}R^{-q+\frac{1}{2}}+\mathcal{O}\left(\eta_{0}^{2}\right).$ (15) Thus, to order $\mathcal{O}\left(\eta_{0}\right)$, $\tilde{v}_{r}=-\frac{\eta_{0}S_{0}R^{p-q+\frac{1}{2}}}{1+R^{2p}S_{0}^{2}}+\mathcal{O}\left(\eta_{0}^{2}\right),$ (16) or equivalently, using Eqs. (84) and (81), $v_{r}=\frac{rc_{\mathrm{s}}^{2}}{v_{\mathrm{k}}}\frac{\mathrm{d}\,\mathrm{ln}\bar{P}}{\mathrm{d}r}\frac{\left(t_{\mathrm{s}}/t_{\mathrm{k}}\right)}{1+\left(t_{\mathrm{s}}/t_{\mathrm{k}}\right)^{2}}.$ (17) $S_{0}R^{p}$ is the dimensionless expression of the ratio $t_{\mathrm{s}}/t_{\mathrm{k}}$. Eq. (16) shows that $R^{2p}S_{0}^{2}\ll 1$ or $R^{2p}S_{0}^{2}\gg 1$, and thus $R^{p}S_{0}\ll 1$ or $R^{p}S_{0}\gg 1$, resulting in asymptotic behaviours for the radial grain motion. These asymptotic regimes were first described by W77 for the particular case $p=0$. They correspond physically to two limiting cases: where the gas drag dominates, which we call the A-mode, and where gravity dominates, which we call the B-mode. In the next sections, we study and describe these two so- called “regimes of migration” or “modes of migration” before treating the global evolution of grains given by Eq. (16). ### 3.1 A-mode (Radial differential migration) The A-mode corresponds to the regime $R^{p}S_{0}\ll 1$ (or equivalently $t_{\mathrm{s}}/t_{\mathrm{k}}\ll 1$). In the A-mode, Eq. (16) reduces to $\tilde{v}_{r}=\frac{\mathrm{d}r}{\mathrm{d}t}=\frac{\mathrm{d}R}{\mathrm{d}T}=-\eta_{0}S_{0}R^{p-q+\frac{1}{2}},$ (18) or equivalently $v_{r}=\frac{rc_{\mathrm{s}}^{2}}{v_{\mathrm{k}}}\frac{\mathrm{d}\,\mathrm{ln}\bar{P}}{\mathrm{d}r}\frac{t_{\mathrm{s}}}{t_{\mathrm{k}}},$ (19) where the $\mathcal{O}\left(R^{p}S_{0}\right)$ has been neglected. In this mode of migration, the stopping time is much smaller than the Keplerian time scale. Considering one grain’s orbit around the central star, its orbital velocity is forced by the gas drag to become sub-Keplerian in just a few stopping times, i.e. almost instantaneously. Thus, the centrifugal acceleration is not efficient enough to counterbalance the gravitational attraction of the central star, and the grain feels an inward radial differential acceleration. The gas drag counterbalances this radial motion and the grain reaches a local limit velocity in a few stopping times. We call the physical process of the A-mode of migration “Radial Differential Migration”. The A-mode of migration originates first from a perturbative expansion for $\eta_{0}\ll 1$ (rigorously for $\eta_{0}R^{-q+1}\ll 1$) and second from a perturbative expansion for $S_{0}\ll 1$ (rigorously for $S_{0}R^{p}\ll 1$). Formally, we have performed: $\lim\limits_{\begin{subarray}{c}S_{0}\ll 1\end{subarray}}\lim\limits_{\begin{subarray}{c}\eta_{0}\ll 1\end{subarray}}\left[...\right]$. Historically, the A-mode had been derived by W77 to explain the radial motion of small grains. In his study, he neglected the radial dependence of the stopping time and assumed $S_{0}\ll 1$ (this approximation also implies that $R^{2p}S_{0}^{2}\ll 1$, as $R<1$, see Appendix F). It is straightforward to integrate the differential equation Eq. (18) by separating the $R$ and $T$ variables. Noting that $R\left(T=0\right)=1$, we have: * • If $-p+q+\frac{1}{2}\neq 0$: $\left\\{\begin{array}[]{l}\displaystyle R=\displaystyle\left[1-\left(-p+q+\frac{1}{2}\right)\eta_{0}S_{0}T\right]^{\frac{1}{-p+q+\frac{1}{2}}}\\\ \displaystyle T=\displaystyle\frac{1-R^{-p+q+\frac{1}{2}}}{\left(-p+q+\frac{1}{2}\right)\eta_{0}S_{0}}.\end{array}\right.$ (20) * • If $-p+q+\frac{1}{2}=0$: $\left\\{\begin{array}[]{l}\displaystyle R=\displaystyle\mathrm{e}^{-\eta_{0}S_{0}T}\\\ \displaystyle T=\displaystyle-\frac{\mathrm{ln}\left(R\right)}{\eta_{0}S_{0}}.\end{array}\right.$ (21) The outcome of the dust radial motion comes from a competition between two effects. As the grain reaches smaller radii, (1) gas drag increases, which slows down the radial drift, and (2) the differential acceleration due to the pressure gradient increases which enhances the migration efficiency. Point (1) is related to $s_{\mathrm{opt}}$, which scales as the surface density profile, while the acceleration due to the pressure gradient in (2) is related to the temperature profile (see Eq. (81)). Depending on which process is dominant, the grain’s dynamics can lead to two regimes: * • If $-p+q+\frac{1}{2}\leq 0$: then as $T(R)\sim\displaystyle\frac{-R^{-p+q+\frac{1}{2}}}{\left(-p+q+\frac{1}{2}\right)\eta_{0}S_{0}}=\mathcal{O}\left(R^{-p+q+\frac{1}{2}}\right),$ (22) the time it takes the grain to reach smaller and smaller radii increases drastically, according to the diverging power-law. Importantly, this behaviour constitutes our definition of the grain “pile-up”. Mathematically speaking, accretion onto the central star occurs in an infinite time, i.e. $\lim\limits_{\begin{subarray}{c}T\to+\infty\end{subarray}}R=0.$ (23) * • If $-p+q+\frac{1}{2}>0$: the grain is accreted onto the central star in a finite migration time given by $T_{\mathrm{m}}=\frac{1}{\eta_{0}S_{0}\left(-p+q+\frac{1}{2}\right)},$ (24) which increases as $S_{0}$ and $\eta_{0}$ decrease, so that $\lim\limits_{\begin{subarray}{c}T\to T_{\mathrm{m}}\end{subarray}}R=0.$ (25) The presence or absence of a physical grain pile-up is therefore demonstrated considering the asymptotic behaviour of $R(T)$ at large times. It is important to realise that the pile-up is a cumulative effect that can not arise from velocities only (which however provide qualitative information on the grain’s motion) but can only be found by integrating the equation of motion. This rigorously allows us to distinguish two different behaviours for the outcome of the grain’s radial motion, and thus two classes of discs with respect to the A-mode. ### 3.2 B-mode (Drift forced by a resistive torque) Returning to Eq. (16), the B-mode corresponds to the other asymptotic regime, where $R^{p}S_{0}\gg 1$ (or equivalently $t_{\mathrm{s}}/t_{\mathrm{k}}\gg 1$). In this case, $\tilde{v}_{r}=\frac{\mathrm{d}R}{\mathrm{d}T}=-\frac{\eta_{0}}{S_{0}}R^{-p-q+\frac{1}{2}},$ (26) or equivalently $v_{r}=\frac{rc_{\mathrm{s}}^{2}}{v_{\mathrm{k}}}\frac{\mathrm{d}\,\mathrm{ln}\bar{P}}{\mathrm{d}r}\frac{t_{\mathrm{k}}}{t_{\mathrm{s}}}.$ (27) In this mode of migration, the stopping time is much larger than the Keplerian time scale. Hence, the orbital velocity of a grain around the central star is almost the Keplerian velocity. However, because of the pressure gradient, the gas orbits around the central star at a sub-Keplerian velocity. Thus, the azimuthal differential velocity between the gas and the grain generates an azimuthal drag force whose torque extracts angular momentum from the grain. Given that the Keplerian angular momentum increases with radius ($l\propto\sqrt{r}$), this torque results in the inward migration of the grain. We call the physical process of this B-mode of migration “Drift Forced by a Resistive Torque”. As for the A-mode, the B-mode of migration can also be derived first from an expansion in $\left(S_{0}R^{p}\right)^{-2}$ and then from an expansion in $\eta_{0}$. Historically, W77 found an expression while only assuming that $S_{0}\gg 1$ since he considered a flat density profile. To find the expression derived by W77 for large grains, we must assume that $S_{0}R^{p}\gg 1$. It is crucial to see that this expression does not imply that $S_{0}\gg 1$ when $p>0$ and $R\to 0$. It is straightforward to integrate the differential equation Eq. (26) by separating the $R$ and $T$ variables. Noting that $R\left(T=0\right)=1$, we have: * • If $p+q+\frac{1}{2}\neq 0$: $\left\\{\begin{array}[]{l}\displaystyle R=\displaystyle\left[1-\left(p+q+\frac{1}{2}\right)\frac{\eta_{0}}{S_{0}}T\right]^{\frac{1}{p+q+\frac{1}{2}}}\\\ \displaystyle T=\displaystyle\frac{S_{0}}{\eta_{0}}\frac{1-R^{p+q+\frac{1}{2}}}{\left(p+q+\frac{1}{2}\right)}.\end{array}\right.$ (28) * • If $p+q+\frac{1}{2}=0$: $\left\\{\begin{array}[]{l}\displaystyle R=\displaystyle\mathrm{e}^{-\frac{\eta_{0}}{S_{0}}T}\\\ \displaystyle T=\displaystyle-\frac{S_{0}}{\eta_{0}}\mathrm{ln}\left(R\right).\end{array}\right.$ (29) As for the A-mode, two kinds of behaviours appear, depending on the $p$ and $q$ exponents: * • If $p+q+\frac{1}{2}\leq 0$: The grain migrates inwards, piles-up in the disc’s inner regions and falls onto the star in an infinite time: $\lim\limits_{\begin{subarray}{c}T\to+\infty\end{subarray}}R=0.$ (30) However, the negative exponents required to be in this regime do not correspond to physical discs. Therefore, the grain dynamics in the B-mode in real discs belong to the second case: * • If $p+q+\frac{1}{2}>0$: The grain is accreted onto the central star in a finite time $T_{\mathrm{m}}=\frac{S_{0}}{\eta_{0}\left(p+q+\frac{1}{2}\right)},$ (31) which increases as $S_{0}$ increases and $\eta_{0}$ decreases and so that $\lim\limits_{\begin{subarray}{c}T\to T_{\mathrm{m}}\end{subarray}}R=0.$ (32) As for the A-mode, considering the limit of $R(T)$ at large times also proves the existence of two classes of discs with respect to the B-mode of migration. The radial motion of a grain in the B-mode of migration is also driven by a competition between the increase of both the drag and the acceleration due to the pressure gradient. However, for real discs, $p>0$ and $q>0$, and therefore, $p+q+\frac{1}{2}>0$. Grains migrating in B-mode in such discs fall in a finite time onto the central star. ### 3.3 Radial evolution and asymptotic behaviour of single grains As we have seen, the grains behaviour is divided into two asymptotic regimes, called the A-mode and the B-mode, which come from two different physical origins. However, the two criteria determining if the grains are accreted onto the central object in a finite or infinite time differ for the A- and the B-mode. It is thus crucial to determine in which mode a dust grain ends its motion to predict if the grain is ultimately accreted or not. Returning again to Eq. (16), we have $\frac{\mathrm{d}R}{\mathrm{d}T}=\frac{-\eta_{0}S_{0}R^{p-q+\frac{1}{2}}}{1+R^{2p}S_{0}^{2}}.$ (33) We can separate the $R$ and $T$ variables and integrate to obtain an expression for $T(R)$: $T=\frac{1}{\eta_{0}S_{0}}\left[T_{\mathrm{A}}\left(R\right)+T_{\mathrm{B}}\left(R\right)\right],$ (34) where $T_{\mathrm{A}}\left(R\right)=\left\\{\begin{array}[]{ll}\displaystyle\frac{1-R^{-p+q+\frac{1}{2}}}{-p+q+\frac{1}{2}}&\mathrm{if}\ -p+q+\frac{1}{2}\neq 0\\\\[10.00002pt] -\mathrm{ln}\left(R\right)&\mathrm{if}\ -p+q+\frac{1}{2}=0.\end{array}\right.$ (35) and $T_{\mathrm{B}}\left(R\right)=\left\\{\begin{array}[]{ll}\displaystyle S_{0}^{2}\frac{1-R^{p+q+\frac{1}{2}}}{p+q+\frac{1}{2}}&\mathrm{if}\ p+q+\frac{1}{2}\neq 0\\\\[10.00002pt] \displaystyle- S_{0}^{2}\mathrm{ln}\left(R\right)&\mathrm{if}\ p+q+\frac{1}{2}=0.\end{array}\right.$ (36) Eq. (34) provides the asymptotic behaviour of the grains at large times. Interestingly, as $p\geq 0$ for realistic discs, the contribution of the B-mode becomes negligible when $R\ll S_{0}^{-1/p}$. Hence, grains initially migrating in the A-mode stay in the A-mode, but grains initially migrating in the B-mode end their radial motion in the A-mode. This behaviour is summarized on Fig. 2 and detailed in Appendix G. This result was not predicted by W77, as he neglected the radial dependence of the stopping time. Mathematically speaking, it comes from the fact that the perturbative expansion of W77 has been performed with respect to powers of $S_{0}$ and not powers of $S_{0}R^{p}$. Figure 2: Radial evolution of grains in the ($R$,$S_{0}$) plane showing that a grain in the Epstein drag regime ends its radial motion in the A-mode. The solid curves represent $R^{-p}$ for various values of $p$, they separate the A-mode (below) from the B-mode (above) regions. The horizontal dashed lines show trajectories of grains as they migrate inwards from $R=1$. The shaded area is a forbidden zone. To illustrate the radial motion of dust grains in protoplanetary discs, we numerically integrate the equations of motion for different values of the parameters $\eta_{0}$, $S_{0}$, $p$, $q$. We set $\eta_{0}=10^{-2}$ to mimic a realistic disc and vary the order of magnitude of $S_{0}$ from $10^{-4}$ to $10^{2}$ for two sets of ($p$,$q$) values. First, we choose ($p=0$, $q=\frac{3}{4}$); according to the NSH86 expansion, the grains are accreted by the central star in a time $T_{\mathrm{m}}$. Second, we set ($p=\frac{3}{2}$, $q=\frac{3}{4}$); the grains fall onto the central star in an infinite time from the same approximation. This set of ($p$,$q$) values is taken to mimic discs profiles that are commonly used and for which $-p+q+\frac{1}{2}$ can take a positive or a negative value. Consequently, we interpret the radial motion $R\left(T\right)$ of dust grains plotted in Fig. 3. * • The top panel of Fig. 3 shows the results for ($p=0$, $q=\frac{3}{4}$): Grains fall onto the central star, initially in the A-mode for the small grains and in the B-mode for the large ones. The radial-drift process is long for small and large grains but is optimal for grains with $S=S_{\mathrm{m}}=1$ for which the accretion time is $T_{\mathrm{m}}=1.6/\eta_{0}$, or $T_{\mathrm{m}}=160$ with $\eta_{0}=10^{-2}$. Figure 3: Radial motion $R\left(T\right)$ of dust grains in the Epstein regime for $\eta_{0}=10^{-2}$. $S_{0}$ varies from $10^{-4}$ to $10^{2}$. Top: $p=0$, $q=\frac{3}{4}$, here $-p+q+\frac{1}{2}>0$ and the grain is accreted onto the central star in a finite time. Bottom: $p=\frac{3}{2}$, $q=\frac{3}{4}$, here $-p+q+\frac{1}{2}<0$ and the grain piles up and is consequently accreted onto the central star in an infinite time. * • The bottom panel of Fig. 3 shows the results for ($p=\frac{3}{2}$, $q=\frac{3}{4}$): In this case, the radial density profile is steep enough to ensure that the grains are not accreted onto the central star. To reach a given radius (for example $R_{\mathrm{f}}=0.1)$, the optimal size is $S_{\mathrm{m,f}}\simeq 2.9=\mathcal{O}\left(1\right)$ (see Eq. (112)). Hence, in this case grains efficiently reach the disc inner regions without ever being accreted onto the central star. The transition from the B-mode to the A-mode (for which $R\propto T^{-4}$ in this case) for the large grains is visible in this plot. ## 4 Radial motion in the Stokes regime Radial migration of large particles occurs in the Stokes drag regime, which depends on the dynamical viscosity $\mu$ of the gas. For hydrogen molecules: $\mu=\frac{5m\sqrt{\pi}}{64\sigma_{\mathrm{s}}}\sqrt{\frac{k_{\mathrm{B}}T}{m}},$ (37) where $m=2m_{\mathrm{H}}=3.347446922\times 10^{-27}$ kg and $\sigma_{\mathrm{s}}=2.367\times 10^{-19}$ m2 is the molecular cross section of the molecule (Chapman & Cowling 1970). The kinematic viscosity $\nu$ is then defined by $\mu=\rho_{\mathrm{g}}\nu$ and the gas collisional mean free path is given by $\lambda_{g}=\sqrt{\frac{\pi}{2}}\frac{\nu}{c_{\mathrm{s}}}.$ (38) We now generalise the procedure outlined in Sects. 2 and 3 to the three Stokes regimes of Eq. (3). Using the dimensionless coordinates described above, we have $\mu=\mu_{0}R^{-\frac{q}{2}},$ (39) $\mu_{0}=\frac{5m\sqrt{\pi}}{64\sigma_{\mathrm{s}}}c_{\mathrm{s}0},$ (40) where $c_{\mathrm{s}0}$ is given by Eq. (72). Thus, the expression of the kinematic viscosity $\nu$ is $\nu=\displaystyle\nu_{0}R^{\frac{3}{2}+p-q}e^{\frac{Z^{2}}{2R^{3-q}}},$ (41) $\nu_{0}=\displaystyle\frac{\mu_{0}}{\rho_{\mathrm{g}0}}\,.$ (42) First, if $R_{\mathrm{g}}<1$, the drag force is linear in $\textbf{v}-\textbf{v}_{\mathrm{g}}$ and thus has the same structure as for the Epstein regime. Comparing the expressions of $\mathcal{C}\left(R,0\right)$ for the Epstein and the linear Stokes regime (see Appendix C), all the results found for the radial motion in the Epstein regime can therefore be directly transposed by setting $q^{\prime}=q$ and $p^{\prime}=\frac{q-3}{2}$. In this case, the grain radial motion does not depend on $p$ anymore and the NSH86 expansion of the radial motion for small pressure gradients provides (see Eq. (33)) $\frac{\mathrm{d}R}{\mathrm{d}T}=\frac{-\eta_{0}S_{0}^{2}R^{-\frac{q}{2}-1}}{1+R^{q-3}S_{0}^{4}}.$ (43) These crucial results follow: * • In the A-mode ($S_{0}\ll R^{\frac{3-q}{4}}$), grains experience a pile-up and migrate onto the central star in an infinite time if $-p^{\prime}+q^{\prime}+\frac{1}{2}\leq 0$, i.e. if $q\leq-4$ (which never occurs in real discs). * • In the B-mode ($S_{0}\gg R^{\frac{3-q}{4}}$), grains migrate onto the central star in an infinite time if $p^{\prime}+q^{\prime}+\frac{1}{2}\leq 0$, i.e. if $q\leq\frac{2}{3}$. Figure 4: Radial evolution of grains in the ($R$,$S_{0}$) plane showing that a grain in the linear Stokes drag regime ends its radial motion in the B-mode. The solid curves represent $R^{\frac{3-q}{4}}$ for various values of $q$, they separate the A-mode (below) from the B-mode (above) regions. The horizontal dashed lines show trajectories of grains as they migrate inwards from $R=1$. Shaded area: forbidden. Thus, similar to the Epstein regime, we derived one criterion for each mode and need to determine in which mode the grain ends its motion. For observed discs, $q-3<0$ (see Sect. 6), and as the particle migrates inward, $R$ becomes smaller than $S_{0}^{\frac{4}{3-q}}$ and grains end their radial motion in the B-mode (see Fig. 4). This result is fundamentally different to the one we obtained for the Epstein regime. Indeed, for grains migrating in the A-mode in the Stokes regime at low Reynolds numbers, the criterion obtained for a pile- up in the A-mode is never satisfied for real discs. However, after migrating inside a critical radius, grains switch to the B-mode, for which the pile-up can potentially occur, depending on the value of $q$. The corollary is that in discs having $q\leq\frac{2}{3}$, i.e. a shallow enough temperature profile, large grains in the Stokes regime at small Reynolds numbers remain in the disc. Such a criterion is applicable for real protoplanetary discs. Second, if $R_{\mathrm{g}}>800$, the drag force is quadratic in $\textbf{v}-\textbf{v}_{\mathrm{g}}$. Assuming that the radial motion is decoupled from the vertical motion, we perform the NSH expansion at small pressure gradient (cf. Eq. (13)). We find that whatever the integer $j$, $\left(\eta_{0}R^{-p}\right)^{j}\left(\textbf{v}-\textbf{v}_{\mathrm{g}}\right)\to 0$ at the limit $\eta_{0}\to 0$. This means that both $v_{r}$ and $v_{\theta}-\sqrt{1/R}$ are flat functions as their Taylor series expansion equals zero at each order. Consequently, they can not be determined by perturbation analysis. This property comes from the quadratic dependency of the drag with respect to the differential velocity and thus is not related to the grain size. Consequently, in this drag regime, the drag force is extremely efficient and the corrections to the Keplerian motion are negligible at every order of the perturbative expansion. The particles are very well coupled to the gas and do not migrate significantly. Third, for the intermediate case, we could not manage to perform the expansion at small pressure gradients. However, we expect an intermediate behaviour between the two Stokes regime at small and large Reynolds numbers. Consequently, if $R_{\mathrm{g}}>1$, the migration motion becomes less efficient as the drag force is no longer linear with respect to the differential velocity between the gas and the dust particles. Thus, the main constraint for the radial-drift barrier due to the Stokes drag comes from the low Reynolds number regime for which the migration motion is the most efficient. Finally, confusion often arises when defining the “radial-drift barrier” as the difficulty a grain has of “overcoming $s=s_{\mathrm{opt}}$” (i.e.) reaching the B-mode. Indeed, as we have shown, grains can survive their migration motion in the Epstein regime when they are in the A-mode whenever $-p+q+1/2<0$, and grains can start their migration motion in the B-mode but be accreted in a finite time if $-p+q+1/2>0$. This study also shows for the Stokes regime that a grain ends its migration motion in the B-mode. However, as demonstrated in this work, the ability of the grain to overcome the radial drift barrier is only linked to the value of $q$. If $q>2/3$, the grain will be accreted onto the central star in a finite time, even if it has $s>s_{\mathrm{opt}}$. Thus, we would argue that the definition of the radial- drift barrier has to remain the ability of the grain to be accreted onto the central star or depleted from the disc within its lifetime . ## 5 Limitations of the model We have demonstrated that the time it takes for grains to reach smaller and smaller radii increases dramatically under certain conditions. Specifically, if $-p+q+\frac{1}{2}<0$ (resp. $q<\frac{2}{3}$), grains experience a pile-up in the Epstein (resp. Stokes) regime. However, the model developed for the radial evolution of dust grains in this paper remains simple in that we neglect several important physical processes: turbulence, grain growth, collective motion of dust grains, dust feedback on the gas surface density and temperature profiles. We now discuss how those processes can modify the criteria derived above. 1. 1. The local pressure maxima created by turbulence (Cuzzi et al. 2001, 2008) and the collective effects due to the dust drag onto the gas phase (Youdin & Goodman 2005) are known to slow down the dust particles. However, the efficiency of these processes — such as the non linearity of the streaming instability in global disc models and the life time of the pressure maxima — in real discs remains difficult to quantify. Omitting these phenomena constitutes therefore an upper limit for the grain migration efficiency, which will be slowed by these additional processes. 2. 2. In this study, we assume that changing the dust distribution does not change the thermal profile of the disc. We also neglect the viscous evolution of the disc, assuming that the viscous timescales are larger than the characteristic timescales of the initial dust evolution. It implies that we assume that $p$ is constant during the whole grains evolution. We can expect that for long term evolution, the surface density profile will flatten, leading to a smaller value of $p$. This makes the Epstein criterion harder to be met, while the Stokes criterion is not affected. 3. 3. We have shown that even if the velocity of the grain’s inward motion depends on their sizes, their outcome only depends on the surface density and temperature profiles. Thus, if we now consider growing (or fragmenting) grains, we expect that (i) the intensity of the inward motion depends on the growth efficiency (this point will be discussed in detail in a forthcoming paper), but that (ii) the grains outcome remains determined by our criteria, regardless of the growth regime. Following this discussion, our simple model deals with processes that are optimized for the grains to be depleted on the central object. Consequently, our Epstein and Stokes criteria for the radial-drift barrier constitute the least favourable limit for grain survival. Thus, we are confident when claiming that the radial-drift barrier does not occur in some classes of discs. One may however expect that more discs are retaining their grains due to the complementary processes mentioned above. It should be noted that the criterion $-p+q+1/2$ quantifies the outcome of the radial-drift motion of the grains, but not their kinematics (which depends on the grain size, the grain density, etc…). Thus, we provide predictions for which discs will retain the largest mass of solid particles, but do not predict for which discs the radial migration to the inner disc regions is the fastest. Full simulations like those developed in Brauer et al. (2008) are required to make predictions of the dust kinetics, even more so when the grain size evolution is driven by a complex model of growth and fragmentation. However, this study suggests that while complex simulations are useful to study the details of the dust dynamics, they are not required to determine the grains outcome. As a conclusion of this section, we have mentioned that the physics treated by our model is not exhaustive. In real discs, the limits $-p+q+\frac{1}{2}=0$ and $q=\frac{2}{3}$ may be softened by the effects of additional physical phenomena. However, these neglected processes (such as turbulence and grain growth) tend to decrease the efficiency of the dust radial motion. Our predictions of when the “radial-draft barrier” does not occur therefore remain valid. Our model represents a powerful indicator for predicting the dust behaviour in discs with given power law profiles: we expect that (i) discs satisfying $-p+q+\frac{1}{2}\leq 0$ retain their small grain population and that (ii) discs satisfying $q\leq\frac{2}{3}$ keep their large solids. On the contrary, discs for which $-p+q+\frac{1}{2}>0$ (resp. $q>\frac{2}{3}$) likely lose their small (resp. large) particles. ## 6 Application to observed discs and planet formation ### 6.1 Validity of the criteria in real protoplanetary discs We now study how the criteria we derived can be applied when considering the physical evolution of grains in observed discs, with finite inner radii and finite lifetimes. The analytic expressions of the previous sections have been derived using dimensionless quantities. We now provide the physical timescales of the radial dust motion estimating the parameters involved in real protoplanetary discs. We consider in this section a typical CTTS disc, of mass $M_{\mathrm{disc}}=10^{-2}\ M_{\odot}$ around a 1 $M_{\odot}$ star, extending from $r_{\mathrm{in}}=10^{-2}$ AU to $r_{\mathrm{out}}=10^{3}$ AU. The disc inner edge is chosen to correspond to the dust sublimation radius for a 1 $M_{\odot}$ star, whereas its outer boundary is representative of the largest observed discs. Its vertical extent is set by the choice of the temperature scale. We take $T(1$ AU$)=150$ K, a typical value obtained by Andrews & Williams (2007) in their disc observations. The transition from the Epstein to the Stokes regime occurs when $\lambda_{\mathrm{g}}=\frac{4s}{9}$, or $s=\frac{9}{4}\lambda_{\mathrm{g}}=\frac{45\pi^{\frac{3}{2}}}{256}\frac{mH^{\prime}_{0}}{\sigma_{\mathrm{s}}\Sigma^{\prime}_{0}}\,r^{p-\frac{q}{2}+\frac{3}{2}},$ (44) and is represented in the $(r,s)$ plane in Fig. 5 for this typical disc for different values of the surface density and temperature power-law exponents $p$ and $q$. The Stokes regime is seen to apply to large bodies in the disc inner regions and for large values of both $p$ and $q$. Figure 5: Transition between the Epstein and the Stokes regimes in a protoplanetary disc of $M_{\rm{disc}}=10^{-2}M_{\odot}$ extending from $r_{\rm{in}}=10^{-2}$ AU to $r_{\rm{out}}=10^{3}$ AU for several values of $p$ and $q$. Grains with sizes below (resp. above) than the curve experience the Epstein (resp. Stokes) drag regime. Disc lifetimes are generally thought to be a few Myr (Haisch et al. 2001; Carpenter et al. 2005), and thus we take $t_{\mathrm{disc}}\sim 10^{6}$ yr. For a grain starting at a distance $r_{0}$ from a 1 $M_{\odot}$ star, the dimensionless value $T_{\mathrm{disc}}$ is therefore $T_{\mathrm{disc}}=\frac{t_{\mathrm{disc}}}{t_{\mathrm{k}0}}=\frac{\sqrt{\mathcal{G}M_{\odot}}\,t_{\mathrm{disc}}}{r_{0}^{3/2}}\sim\frac{6\times 10^{6}}{[r_{0}\ \mathrm{(AU)}]^{3/2}}.$ (45) The dimensionless value of the dust disc inner radius ($r_{\mathrm{in}}\sim 0.01$ AU) for a grain starting at $r_{0}$ is $R_{\mathrm{in}}=\frac{r_{\mathrm{in}}}{r_{0}}\sim\frac{10^{-2}}{r_{0}\ \mathrm{(AU)}}.$ (46) The link between dimensionless and real grain sizes is made through the optimal size for radial migration. Considering first the Epstein regime, its midplane value has a radial dependence given by $s_{\mathrm{opt}}(r,0)=\frac{\Sigma(r)}{\sqrt{2\pi}\,\rho_{\mathrm{d}}}=\frac{\Sigma^{\prime}_{0}\,r^{-p}}{\sqrt{2\pi}\,\rho_{\mathrm{d}}}.$ (47) The dimensionless size of a particle of size $s$ starting its migration at position $r_{0}$ is therefore $S_{0}=\frac{s}{s_{\mathrm{opt},0}}=\frac{\sqrt{2\pi}\,\rho_{\mathrm{d}}}{\Sigma^{\prime}_{0}}\,s\,r_{0}^{p}.$ (48) Values of $S_{0}$ are plotted in the $(r_{0},s)$ plane in Fig. 6 for a disc of total mass $M_{\mathrm{disc}}=0.01\ M_{\odot}$ extending from $r_{\mathrm{in}}=10^{-2}$ AU to $r_{\mathrm{out}}=10^{3}$ AU, with grains of intrinsic density $\rho_{\mathrm{d}}=1000$ kg m-3 and $\eta_{0}=10^{-2}$, for both $p=0$ and $p=\frac{3}{2}$. Figure 6: Values of $S_{0}$ as a function of grain size $s$ and initial position $r_{0}$ for a disc of mass $M_{\mathrm{disc}}=0.01\ M_{\odot}$ extending from $r_{\mathrm{in}}=10^{-2}$ AU to $r_{\mathrm{out}}=10^{3}$ AU, with grains of intrinsic density $\rho_{\mathrm{d}}=1000$ kg m-3 and $\eta_{0}=10^{-2}$, for $p=0$ and $q=\frac{3}{4}$ (top) and $p=\frac{3}{2}$ and $q=\frac{3}{4}$ (bottom). The thick line shows the limit between grains that are accreted onto the star ($t_{\mathrm{in}}<t_{\mathrm{disc}}$) and those that survive in the disc ($t_{\mathrm{in}}>t_{\mathrm{disc}}$), i.e. the survival limit, in the Epstein regime. The dimensionless time $T_{\mathrm{in}}$ for a grain to reach $R_{\mathrm{in}}$ is given by Eq. (34). In combination with Eq. (46), this gives an expression of $T_{\mathrm{in}}$ as a function of $S_{0}$ and $r_{0}$, whereas Eq. (45) gives an expression of $T_{\mathrm{disc}}$ as a function of $r_{0}$. Equating them yields a second order equation in $S_{0}$ as a function of $r_{0}$, which can be solved to determine under which conditions a grain reaches the disc inner edge at the end of its lifetime. Using Eq. (48) gives the corresponding relationship between the grain size and its initial position: $\begin{array}[]{l}s=\displaystyle\frac{p+q+\frac{1}{2}}{2\sqrt{2\pi}\,\rho_{\mathrm{d}}}\frac{\Sigma^{\prime}_{0}\,r_{0}^{-p}}{1-\left(\frac{r_{\mathrm{in}}}{r_{0}}\right)^{p+q+\frac{1}{2}}}\left[\frac{\sqrt{\mathcal{G}M}\,t_{\mathrm{disc}}\,\eta_{0}}{r_{0}^{3/2}}\right.\\\\[21.52771pt] \displaystyle\left.\pm\sqrt{\frac{\mathcal{G}M\,t_{\mathrm{disc}}^{2}\,\eta_{0}^{2}}{r_{0}^{3}}-4\frac{\left(1-\left(\frac{r_{\mathrm{in}}}{r_{0}}\right)^{-p+q+\frac{1}{2}}\right)\left(1-\left(\frac{r_{\mathrm{in}}}{r_{0}}\right)^{p+q+\frac{1}{2}}\right)}{\left(-p+q+\frac{1}{2}\right)\left(p+q+\frac{1}{2}\right)}}\right],\end{array}$ (49) which is plotted as a thick line in Fig. 6. It separates the $(r_{0},s)$ plane into regions in which grains reach $r_{\mathrm{in}}$ and leave the disc before it dissipates ($t_{\mathrm{in}}<t_{\mathrm{disc}}$) or survive in the disc throughout its lifetime ($t_{\mathrm{in}}>t_{\mathrm{disc}}$). We call this curve the survival limit. For $-p+q+\frac{1}{2}>0$, illustrated by the case $(p=0,q=\frac{3}{4})$, Fig. 6 shows as expected that most of the grains are lost during the disc lifetime. However, small and large grains initially in the outer disc survive, therefore even with this profile, the disc retains a fraction of its grain population before it dissipates. Moreover, one may expect growing grains that reach $S_{0}=1$ to be inevitably accreted onto the star (unless the growth process is fast enough for grains to outgrow the fast migrating sizes before they leave the disc, see Laibe et al. 2008). Such discs may not form planets, but their remaining dust content may still make them observable, although they would likely be faint. For $-p+q+\frac{1}{2}\leq 0$, illustrated by the case $(p=\frac{3}{2},q=\frac{3}{4})$, even though all grains fall on the star in an infinite time, some of them reach the disc’s inner edge before it dissipates. On the contrary, for $r_{0}>350$ AU, grains of all sizes remain in the disc. This is also the case for all (sub)micron-sized grains, whatever their initial location, as well as most of the grains up to 0.1 mm. These grains likely make the disc bright and easy to observe, since they are the grains contributing most to the disc emission at IR and submm wavelengths. A large reservoir of grains is available to participate in the planet formation process, however a firm conclusion on their survival as their size evolves would require incorporating a treatment of grain growth, as discussed in Sect. 5. It should be noted that the disc used in these examples represents a lower limit, as it is low mass and very extended. A more massive disc with a smaller outer radius would have a larger surface density, and the corresponding survival limit in Fig. 6 would be shifted vertically towards larger sizes and more and more grains of larger sizes would survive. Figure 7: Survival limits of grains for different values of $p$ and $q$. Left: Epstein regime, right: Stokes regime. Grains to the left of the curves ($t_{\mathrm{in}}<t_{\mathrm{disc}}$) are accreted onto the star whereas those to the right ($t_{\mathrm{in}}>t_{\mathrm{disc}}$) survive in the disc. The left panels of Fig. 7 show the influence of the surface density and temperature profiles on the location and shape of the survival limit curve in the $(r_{0},s)$ plane in the Epstein regime. Increasing $p$ from 0 to 2 shifts the curve towards smaller radii and larger grain sizes, as well as slightly tilts it clockwise. The outer disc region in which grains of all sizes survive extends inwards, as well as the surviving population of small grains as the curve’s lower branch shifts upwards and becomes flatter. On the contrary, the steepening of the curve’s upper branch, confining the population of surviving large solids to the disc outer regions, is less dramatic. Increasing $q$ from $\frac{1}{4}$ to $\frac{3}{4}$ also tilts the curve clockwards, but shifts it towards larger radii and smaller grain sizes. However, its effect is more limited than that of changing $p$. A disc with a steeper surface density profile and a shallower temperature profile is therefore more efficient at retaining a larger quantity of small grains and up to larger sizes. Indeed, large $p$ and small $q$ values are required to meet the $-p+q+\frac{1}{2}<0$ criterion introduced in Sect. 3. An equation very similar to Eq. (49) can be obtained for the linear Stokes regime by replacing $p$ and $q$ by $p^{\prime}=\frac{q-3}{2}$ and $q^{\prime}=q$ (since the equation of motion has the same structure for both drag regimes, see Sect. 4), and using the expression of $s_{\mathrm{opt},0}$ for that regime, given in Table 2. Here $s_{\mathrm{opt},0}\propto T^{\frac{1}{4}}$: the weak temperature dependence results in very little change for a large range of temperatures below or above our adopted value of $T(1$ AU$)=150$ K. The right panels of Fig. 7 show the survival limit in the Stokes regime for the different values of $q$ (note that it no longer depends on $p$). When $q$ increases, the curve’s lower branch slides towards larger radii, making the survival of particles in the Stokes regime less and less favourable. Given the form of Eq. (49) and its different expressions for each drag regime, it is not possible to compute analytically the survival limit for a grain transitioning from the Epstein to the Stokes regime as it migrates inwards. However, large values of $p$ and $q$, for which the Stokes region is the largest, are not observed in protoplanetary discs (see Sect. 6.2), and in practical cases the Stokes regime only applies to a small area of the $(r_{0},s)$ plane. Small grains, which are detected in disc observations at IR and sub-millimetre (submm) wavelengths, are mostly subject to the Epstein drag. We therefore focus on that regime in the following. Figure 8: Isocontours of the survival time (i.e. the time needed to reach the disc inner edge at $r_{\mathrm{in}}=0.01$ AU) of grains of size $s$ and initial position $r_{0}$ for $q=\frac{3}{4}$ and different values of $p$. Equation (49) can give quantitative information about the outcome of the grain population. Replacing $t_{\mathrm{disc}}$ by any time $t$ gives the location in the $(r_{0},s)$ plane of grains reaching the disc inner edge (at $r=r_{\mathrm{in}}$) in that time $t$, which is therefore the survival time of those grains. Its isocontours are shown in Fig. 8 for different values of $p$ and for $q=\frac{3}{4}$. Only one value of $q$ is shown as the $q$ dependence is moderate, as can be seen from the left panels of Fig. 7. The fate of particular dust grains can easily be obtained from these figures. For example, in the context of disc observations, 1 mm grains initially at 100 AU fall on the $10^{5}$ yr contour for $p=0$. Their survival time decreases to a few $10^{4}$ yr for $p\sim 0.8$, and increases again to values larger than $10^{6}$ yr as $p$ increases. At an initial position of a few hundred AU, 1 mm grains survive longer than $10^{6}$ yr for any $p$, therefore long enough to contribute to the disc emission over its entire lifetime. As noted above, such grains have longer survival times for higher $p$ and lower $q$ values. As another example, in the context of planetesimal formation, the survival time of a 1 m particle initially at 1 AU is $\sim 10^{5}$ yr for $p=0$, decreases to $\sim 10^{2}$ yr for $p\sim 1$, and increases again to $\sim 10^{3}-10^{4}$ yr for $p=2$. The ability of such particles to remain for long enough in the disc to grow to larger sizes therefore strongly depends on the surface density profile. As a general rule, the survival of pre-planetesimals in the inner disc is favoured by small values of $p$. Figure 9: Time evolution (in nine snapshots from $t=10^{-2}$ to $10^{6}$ yr) of isocontours of the initial position of grains in the $(r,s)$ plane for the disc with $p=0$ and $q=\frac{3}{4}$. The label for each contour can be deduced from its abscissa in the upper left panel at $t=10^{-2}$ yr. Figure 10: Same as Fig. 9 for $p=\frac{3}{2}$ and $q=\frac{3}{4}$. Similarly, replacing now $r_{\mathrm{in}}$ with any radius $r$ in Eq. (49) gives the locus in the $(r_{0},s)$ plane of grains reaching that radius $r$ at any time $t$. Alternatively, one can plot isocontours of the initial position $r_{0}$ of grains in the $(r,s)$ plane at various times, thus showing the radial evolution of grains with the same initial position but different sizes. This is shown in Fig. 9 for ($p=0$, $q=\frac{3}{4}$) and Fig. 10 for ($p=\frac{3}{2}$, $q=\frac{3}{4}$). These plots make it easy to compare the radial evolution of any particle to any physical timescale of interest in the disc. In particular, they show that the disc still contains particles at all radii at the end of its evolution ($t=10^{6}$ yr). No grains are found to the right of the $r_{0}=10^{3}$ AU contour, since this was the initial outer disc radius. In the disc with $(p=0,q=\frac{3}{4})$, no grains between $\sim 0.06$ and $\sim 0.2$ mm remain, and grains of other sizes still present were initially in the outer disc. Given that the grains of sizes which contribute to IR and submm emission have come from a small fraction of the initial disc, this disc is likely faint. In the disc with $(p=\frac{3}{2},q=\frac{3}{4})$, only grains with $s\sim 0.1$ mm are absent from the very outer regions, and the observable grains come from a larger portion of the disc, likely making the disc brighter than in the previous case. As a conclusion, the analytic criteria derived above apply even when taking into account the finite lifetime (or inner radius) of the disc. For most CTTS discs, the dust is in the Epstein drag regime (except for some extreme values for the grains sizes and discs profiles). Therefore, the grain’s radial outcome is given by the value of $-p+q+\frac{1}{2}$. However, the transition between discs for which the radial-drift barrier occurs or not consists more of a continuum around the value $-p+q+\frac{1}{2}=0$ than in the sharp transition predicted by the analytic model. Therefore, the radial motion of the grains has to be studied on a case-by-case basis for discs close to the transition $-p+q+\frac{1}{2}=0$, using the figures shown above in this section. ### 6.2 Constraining physical systems We now turn to observed discs and check if they meet our Epstein and Stokes criteria to determine whether the radial-drift barrier is constraining for planet formation. To estimate the values of the $p$ and $q$ exponents for real discs, we use the results of disc modeling obtained by Andrews & Williams (2005, 2007) from data on 63 discs in $\rho$ Ophiuchi, Taurus and Aurigae. Using sub-millimetre fluxes measured at several wavelengths, they fit a range of disc parameters assuming a geometrically thin irradiated disc with opacities from Beckwith et al. (1990), a gas-to-dust ratio of 100, a disc radius of 100 AU and zero disc inclination. The temperature exponent $q$ is well constrained by the observational data set: the histogram of most probable $q$ values is shown in Fig. 11. Figure 11: Histogramm of the $q$ parameters obtained from Andrews & Williams (2005, 2007) data of 63 observed discs. The distribution is roughly comprised between 0.4 and 0.8 and, centred around 0.55. Approximately 90 % of the discs satisfy $q<2/3$. However, $p$ is not well constrained and is usually assumed to be $\frac{3}{2}$. Very flat profiles with $p<\frac{1}{2}$ and very steep profiles with $p>\frac{3}{2}$ seem to be excluded (Dutrey et al. 1996; Wilner et al. 2000; Kitamura et al. 2002; Testi et al. 2003; Isella et al. 2009; Andrews & Williams 2007; Andrews et al. 2009). We represent the disc distribution modeled by Andrews & Williams from observations in the $\left(p,q\right)$ diagram of Fig. 12: the histogram of Fig. 11 is represented by the gray-shaded area and spread over a range of $p$ values, taking into account that extreme values of $p$ are less probable. The dashed line ($-p+q+1/2=0$) represents the border between migration in an infinite time and accretion onto the central star for the A-mode of migration in the Epstein regime, while the thick dotted line ($q=3/2$) represents that same border for the B-mode of migration in the Stokes regime at low Reynolds number. The two black circles indicate the discs used as examples in Sect. 3.3. Figure 12: Location of the different outcomes of radial migration in the ($p$,$q$) plane. Dashed (resp. dotted) line: limit between accretion without or with grains pile-up resulting in a finite or infinite time in the A-mode of the Epstein regime (resp. B-mode of the Stokes regime at small Reynolds numbers). Shaded area: location of observed discs. Black dots: discs used as examples in Sect. 3.3. We have split the disc distribution in four regions in the $\left(p,q\right)$ plane: 1. 1. region 1: $-p+q+\frac{1}{2}\leq 0$ and $q\leq\frac{2}{3}$: both small and large grains experience the pile-up effect. Those discs are potentially observable and may favour planet formation. 2. 2. region 2: $-p+q+\frac{1}{2}\leq 0$ and $q>\frac{2}{3}$: only small grains experience the pile-up effect: even though such discs retain their small grains, the population of pre-planetesimals in the disc inner regions may efficiently be accreted onto the central star (at least until they reach the high-$R_{\mathrm{g}}$ Stokes regime). 3. 3. region 3: $-p+q+\frac{1}{2}>0$ and $q\leq\frac{2}{3}$: if the pre- planetesimals can form before the entire distribution of small grains has been accreted onto the central object, they will remain in the disc and may constitute planet embryos. 4. 4. region 4: $-p+q+\frac{1}{2}>0$ and $q>\frac{2}{3}$: both small and large grains are accreted onto the central star. The Epstein criterion indicates that for $q$ values in the range constrained by observations, discs which keep their small grain population, and are therefore likely to be bright in the IR and submm, should have $p$ values approximately in the $[1;\frac{3}{2}]$ range. This is indeed what is found in most disc surveys (Ricci et al. 2010a, b). On the contrary, smaller $p$ values should correspond to discs which lose most of their small grains, and are therefore more difficult to detect. This is what is found by Andrews et al. (2010), who pushed their previous observations of the Ophiuchus star forming region (Andrews et al. 2009) down to fainter discs, finding for this new sample a median $p$ value of 0.9, lower than for brighter discs. The criterion we derive in this paper for small grains in the Epstein regime provides therefore the correct behaviour for explaining the range of $p$ values of observed discs. However, this result has to be considered carefully for two reasons. Firstly, the $p$ and the $q$ exponents determined from the observations have to be considered with their respective errors. Given these uncertainties, one may not be able to distinguish between a strict negative or positive value for $-p+q+\frac{1}{2}$. Second, the boundary between the different zones of the $(p,q)$ plane consists more of a continuum rather than a strict limit due to the finite lifetime/inner radii of the discs. The outcome of the grains may thus not be predicted when the value of $-p+q+\frac{1}{2}$ is close to zero. Now turning to the Stokes criterion for large solids, Figs. 11 and 12 show that the vast majority of observed protoplanetary discs have shallow temperature profiles ($q\leq\frac{2}{3}$) and are thus able to retain their population of pre-planetesimals. These discs are therefore relevant places to find evidence of planet formation, provided small grains can efficiently grow to form pre-planetesimals. For the remainder of the disc population, the outcome of pre-planetesimals will likely depend on their ability to reach the high Reynolds number Stokes regime. However, the case of a steep radial temperature profile can be encountered in at least one particular situation: circumplanetary discs which typically have temperature profiles with $q=1$ (Ayliffe & Bate 2009). In this environment, we predict from our Stokes criterion that planetesimals will be accreted onto the planet. The timescale of the planet formation by the core-accretion process, which usually corresponds to the time required to release the gravitational energy of the accreted bodies (Pollack et al. 1996), is thus increased as the drag from the gas onto the planetesimals releases an additional thermal contribution. ## 7 Conclusion and perspectives In this study, we have generalised the radial grain motion studies of W77, NSH86 and YS02 for both the Epstein and the Stokes regimes, taking into account the effects of both the surface density and temperature profiles in the disc. As observations do not provide direct information about the three dimensional structure of discs, radial profiles of surface density and temperature are often described by power laws: $\Sigma\left(r\right)=\Sigma_{0}^{\prime}r^{-p}$ and $\mathcal{T}\left(r\right)=\mathcal{T}_{0}^{\prime}r^{-q}$, where both $p$ and $q$ take positive values. The radial dust behaviour in those discs is governed by the competition between gravity and gas drag. The final outcome of the radial motion is set by two counterbalancing effects. First, the temperature increases when the radius decreases. Consequently, the deviation from the Keplerian velocity increases, which accelerates the dust’s radial inward motion. At the same time, the surface density also increases, which increases the gas drag efficiency and slows down the dust motion. The competition between these two effects fixes the ultimate mode of migration of the grain (A-mode, where the drag dominates or B-mode, where the gravity dominates) and thus the final outcome for the dust motion. In this work, we have shown that it can be represented by an analytical criterion which depends on the drag regime. For the Epstein drag regime (in which the ultimate radial motion is in the A-mode), if $-p+q+\frac{1}{2}>0$, the dust particle is accreted onto the central star in a finite time, and if $-p+q+\frac{1}{2}\leq 0$, the grain pile-up results in an infinite accretion time and small dust grains remain in the disc. We have shown that, as expected, these conclusions are somewhat mitigated when taking into account the finite disc lifetime and finite disc size. However, the outcomes still remain similar: the bulk of the small grain population is lost to the star in the first case, whereas in the second case the disc keeps most of its small grains. A similar criterion is found for the Stokes regime at low Reynolds number: if $q\leq\frac{2}{3}$, the accretion time is infinite and large pre-planetesimals remain in the disc and can constitute the primary material for planet formation. However, the Stokes radial motion differs from the Epstein regime as the ultimate radial motion occurs in the B-mode. The observational consequence is that discs with a large population of small grains should be strong emitters in the infrared and sub-millimetre and should be easier to observe, and that those having lost most of their small dust should be fainter and harder to detect. This is indeed what is found: a large fraction of the observed discs have large $p$ values whereas fainter discs tend to have lower $p$ values (Andrews et al. 2010), in agreement with this Epstein criterion. In addition, most of the observed discs have $q\leq\frac{2}{3}$, allowing them to retain also their large pre- planetesimals. As noted by Ricci et al. (2010a, b), explaining the data requires a mechanism halting or slowing down the radial migration of dust grains. We show here that local pressure maxima need not be invoked, but rather that the combination of adequate surface density and temperature profiles is sufficient. The $p$ and $q$ exponents used to reach our conclusions are of course strongly dependent on the model used to fit the data. However, even varying the fitting models, a large majority of discs still satisfy both conditions $-p+q+\frac{1}{2}\leq 0$ and $q\leq\frac{2}{3}$. Consequently, the radial-drift barrier (or the so-called metre-size barrier when considering an MMSN disc) does not appear to constitute a problem for planet formation for the discs that we do observe. Our conclusions presented in this study assumed that the grain size remains constant during its motion. However, observations tell us that grains do grow (Testi et al. 2003; Wilner et al. 2003; Apai et al. 2005; Lommen et al. 2007, 2009). Grain growth is studied in various theoretical studies (Schmitt et al. 1997; Stepinski & Valageas 1997; Suttner et al. 1999; Tanaka et al. 2005; Dullemond & Dominik 2005; Klahr & Bodenheimer 2006; Garaud 2007; Brauer et al. 2008; Laibe et al. 2008; Birnstiel et al. 2009). In a forthcoming paper, we will generalise the formalism developed here to explain the radial and vertical behaviour of growing dust grains. ###### Acknowledgements. This research was partially supported by the Programme National de Physique Stellaire and the Programme National de Planétologie of CNRS/INSU, France, and the Agence Nationale de la Recherche (ANR) of France through contract ANR-07-BLAN-0221. The authors want to thank C. Terquem, L. Fouchet, S. Arena and E. Crespe for useful comments and discussions. We also thank the referee for greatly improving the quality of this work by suggesting we include the important Stokes regime and present real numbers (time scales and grain sizes) for our criteria. ## References * Andrews & Williams (2005) Andrews, S. M. & Williams, J. P. 2005, ApJ, 631, 1134 * Andrews & Williams (2007) Andrews, S. M. & Williams, J. P. 2007, ApJ, 671, 1800 * Andrews et al. (2009) Andrews, S. M., Wilner, D. J., Hughes, A. M., Qi, C., & Dullemond, C. P. 2009, ApJ, 700, 1502 * Andrews et al. (2010) Andrews, S. M., Wilner, D. J., Hughes, A. M., Qi, C., & Dullemond, C. P. 2010, ApJ, 723, 1241 * Apai et al. (2005) Apai, D., Pascucci, I., Bouwman, J., et al. 2005, Science, 310, 834 * Ayliffe & Bate (2009) Ayliffe, B. A. & Bate, M. R. 2009, MNRAS, 393, 49 * Baines et al. (1965) Baines, M. J., Williams, I. P., & Asebiomo, A. S. 1965, MNRAS, 130, 63 * Beckwith et al. (1990) Beckwith, S. V. W., Sargent, A. I., Chini, R. S., & Guesten, R. 1990, AJ, 99, 924 * Birnstiel et al. (2009) Birnstiel, T., Dullemond, C. P., & Brauer, F. 2009, A&A, 503, L5 * Brauer et al. (2008) Brauer, F., Dullemond, C. P., & Henning, T. 2008, A&A, 480, 859 * Carpenter et al. (2005) Carpenter, J. M., Wolf, S., Schreyer, K., Launhardt, R., & Henning, T. 2005, AJ, 129, 1049 * Chapman & Cowling (1970) Chapman, C. & Cowling, T. 1970, The mathematical theory of non-uniform gases (Cambridge at the university press) * Crida (2009) Crida, A. 2009, ApJ, 698, 606 * Cuzzi et al. (2001) Cuzzi, J. N., Hogan, R. C., Paque, J. M., & Dobrovolskis, A. R. 2001, ApJ, 546, 496 * Cuzzi et al. (2008) Cuzzi, J. N., Hogan, R. C., & Shariff, K. 2008, ApJ, 687, 1432 * Desch (2007) Desch, S. J. 2007, ApJ, 671, 878 * Dullemond & Dominik (2005) Dullemond, C. P. & Dominik, C. 2005, A&A, 434, 971 * Dutrey et al. (1996) Dutrey, A., Guilloteau, S., Duvert, G., et al. 1996, A&A, 309, 493 * Epstein (1924) Epstein, P. S. 1924, Physical Review, 23, 710 * Garaud (2007) Garaud, P. 2007, ApJ, 671, 2091 * Garaud et al. (2004) Garaud, P., Barrière-Fouchet, L., & Lin, D. N. C. 2004, ApJ, 603, 292 * Haghighipour & Boss (2003) Haghighipour, N. & Boss, A. P. 2003, ApJ, 598, 1301 * Haisch et al. (2001) Haisch, Jr., K. E., Lada, E. A., & Lada, C. J. 2001, ApJ, 553, L153 * Hayashi (1981) Hayashi, C. 1981, Progress of Theoretical Physics Supplement, 70, 35 * Isella et al. (2009) Isella, A., Carpenter, J. M., & Sargent, A. I. 2009, ApJ, 701, 260 * Kitamura et al. (2002) Kitamura, Y., Momose, M., Yokogawa, S., et al. 2002, ApJ, 581, 357 * Klahr & Bodenheimer (2006) Klahr, H. & Bodenheimer, P. 2006, ApJ, 639, 432 * Laibe et al. (2008) Laibe, G., Gonzalez, J.-F., Fouchet, L., & Maddison, S. T. 2008, A&A, 487, 265 * Lommen et al. (2009) Lommen, D., Maddison, S. T., Wright, C. M., et al. 2009, A&A, 495, 869 * Lommen et al. (2007) Lommen, D., Wright, C. M., Maddison, S. T., et al. 2007, A&A, 462, 211 * Nakagawa et al. (1986) Nakagawa, Y., Sekiya, M., & Hayashi, C. 1986, Icarus, 67, 375 (NSH86) * Pollack et al. (1996) Pollack, J. B., Hubickyj, O., Bodenheimer, P., et al. 1996, Icarus, 124, 62 * Ricci et al. (2010a) Ricci, L., Testi, L., Natta, A., & Brooks, K. J. 2010a, A&A, 521, A66 * Ricci et al. (2010b) Ricci, L., Testi, L., Natta, A., et al. 2010b, A&A, 512, A15 * Schmitt et al. (1997) Schmitt, W., Henning, T., & Mucha, R. 1997, A&A, 325, 569 * Stepinski & Valageas (1996) Stepinski, T. F. & Valageas, P. 1996, A&A, 309, 301 * Stepinski & Valageas (1997) Stepinski, T. F. & Valageas, P. 1997, A&A, 319, 1007 (SV97) * Suttner et al. (1999) Suttner, G., Yorke, H. W., & Lin, D. N. C. 1999, ApJ, 524, 857 * Takeuchi & Lin (2002) Takeuchi, T. & Lin, D. N. C. 2002, ApJ, 581, 1344 * Tanaka et al. (2005) Tanaka, H., Himeno, Y., & Ida, S. 2005, ApJ, 625, 414 * Testi et al. (2003) Testi, L., Natta, A., Shepherd, D. S., & Wilner, D. J. 2003, A&A, 403, 323 * Weidenschilling (1977a) Weidenschilling, S. J. 1977a, MNRAS, 180, 57 (W77) * Weidenschilling (1977b) Weidenschilling, S. J. 1977b, Ap&SS, 51, 153 * Weidenschilling (1980) Weidenschilling, S. J. 1980, Icarus, 44, 172 * Whipple (1972) Whipple, F. L. 1972, in From Plasma to Planet, ed. A. Elvius, 211 * Wilner et al. (2003) Wilner, D. J., Bourke, T. L., Wright, C. M., et al. 2003, ApJ, 596, 597 * Wilner et al. (2000) Wilner, D. J., Ho, P. T. P., Kastner, J. H., & Rodríguez, L. F. 2000, ApJ, 534, L101 * Youdin (2011) Youdin, A. N. 2011, ApJ, 731, 99 * Youdin & Chiang (2004) Youdin, A. N. & Chiang, E. I. 2004, ApJ, 601, 1109 * Youdin & Goodman (2005) Youdin, A. N. & Goodman, J. 2005, ApJ, 620, 459 * Youdin & Shu (2002) Youdin, A. N. & Shu, F. H. 2002, ApJ, 580, 494 (YS02) ## Appendix A Notations The notations and conventions used throughout this paper are summarized in Table 1. Symbol \| Meaning ---\|--- $M$ \| Mass of the central star g \| Gravity field of the central star $r_{0}$ \| Initial distance to the central star $\rho_{\mathrm{g}}$ \| Gas density $\bar{\rho}_{\mathrm{g}}\left(r\right)$ \| $\rho_{\mathrm{g}}\left(r,z=0\right)$ $c_{\mathrm{s}}$ \| Gas sound speed $\bar{c}_{\mathrm{s}}\left(r\right)$ \| $c_{\mathrm{s}}\left(r,z=0\right)$ $c_{\mathrm{s}0}$ \| Gas sound speed at $r_{0}$ $T$ \| Dimensionless time $\mathcal{T}$ \| Gas temperature ($\mathcal{T}_{0}$: value at $r_{0}$) $\Sigma_{0}$ \| Gas surface density at $r_{0}$ $p$ \| Radial surface density exponent $q$ \| Radial temperature exponent $P$ \| Gas pressure $v_{\mathrm{k}}$ \| Keplerian velocity at $r$ $v_{\mathrm{k}0}$ \| Keplerian velocity at $r_{0}$ $H_{0}$ \| Gas scale height at $r_{0}$ $\phi_{0}$ \| Square of the aspect ratio $H_{0}/r_{0}$ at $r_{0}$ $\eta_{0}$ \| Sub-Keplerian parameter at $r_{0}$ $s$ \| Grain size $S$ \| Dimensionless grain size $S_{0}$ \| Initial dimensionless grain size $y$ \| Grain size exponent in the drag force $\textbf{v}_{\mathrm{g}}$ \| Gas velocity v \| Grain velocity $\rho_{\mathrm{d}}$ \| Dust intrinsic density $m_{\mathrm{d}}$ \| Mass of a dust grain $t_{\mathrm{s}}$ \| Drag stopping time $t_{\mathrm{s}0}$ \| Drag stopping time at $r_{0}$ Table 1: Notations used in the article. ## Appendix B disc structure ### B.1 Hydrostatic equilibrium At stationary equilibrium ($\frac{\partial}{\partial t}=0$), gas velocities $\textbf{v}_{\mathrm{g}r}$, $\textbf{v}_{\mathrm{g}\theta}$, $\textbf{v}_{\mathrm{g}z}$ and the gas density $\rho_{\mathrm{g}}$ obey mass conservation and the Euler equation: $\left\\{\begin{array}[]{rcl}\displaystyle\frac{1}{r}\frac{\rho_{\mathrm{g}}\partial rv_{\mathrm{g}r}}{\partial r}+\frac{\rho_{\mathrm{g}}\partial v_{\mathrm{g}z}}{\partial z}&=&\displaystyle 0,\\\ \displaystyle\rho_{\mathrm{g}}\textbf{v}_{\mathrm{g}}.\nabla\textbf{v}_{\mathrm{g}}&=&\displaystyle-\nabla P+\rho_{\mathrm{g}}\textbf{g}.\end{array}\right.$ (50) The solution of Eq. (50) requires: $v_{\mathrm{g}r}=v_{\mathrm{g}z}=0,$ (51) which ensures mass conservation. Projecting the Euler equation on $\textbf{e}_{z}$: $\frac{1}{\rho_{\mathrm{g}}}\frac{\partial P}{\partial z}=-\frac{\mathcal{G}Mz}{\left(z^{2}+r^{2}\right)^{3/2}}.$ (52) Assuming that: $P=c_{\mathrm{s}}^{2}\left(r,z\right)\rho_{\mathrm{g}},$ (53) and dividing both sides of Eq. (52) by $c_{\mathrm{s}}^{2}$, we have: $\frac{\partial\,\mathrm{ln}\left(c_{\mathrm{s}}^{2}\rho_{\mathrm{g}}\right)}{\partial z}=-\frac{\mathcal{G}Mz}{\left(z^{2}+r^{2}\right)^{3/2}c_{\mathrm{s}}^{2}}.$ (54) Integrating Eq. (54) between 0 and $z$ provides: $\rho_{\mathrm{g}}\left(r,z\right)=\frac{\bar{P}\left(r\right)}{c_{\mathrm{s}}^{2}\left(r,z\right)}\mathrm{e}^{\displaystyle-\int_{0}^{z}\frac{\mathcal{G}Mz^{\prime}\mathrm{d}z^{\prime}}{\left(r^{2}+z^{\prime 2}\right)^{3/2}c_{\mathrm{s}}^{2}\left(r,z^{\prime}\right)}}.$ (55) This expression can be simplified by the following approximations: * • In the special vertically isothermal case, where the sound speed depends only on the radial coordinate, Eq. (55) simplifies to: $\rho_{\mathrm{g}}\left(r,z\right)=\bar{\rho}_{\mathrm{g}}\left(r\right)\mathrm{e}^{\displaystyle-\frac{\mathcal{G}M}{\bar{c}_{\mathrm{s}}^{2}\left(r\right)}\left[\frac{1}{r}-\frac{1}{\sqrt{r^{2}+z^{2}}}\right]}.$ (56) * • Further, assuming a thin disc ($\left(\frac{z}{r}\right)^{2}\ll 1$), a Taylor series expansion of Eq. (56) leads to: $\rho_{\mathrm{g}}\left(r,z\right)=\bar{\rho}_{\mathrm{g}}\left(r\right)\mathrm{e}^{-\frac{z^{2}}{2H\left(r\right)^{2}}},$ (57) with: $H\left(r\right)=\frac{r\bar{c}_{\mathrm{s}}\left(r\right)}{v_{\mathrm{k}}\left(r\right)},$ (58) which is the classical scale height for vertically isothermal thin discs. ### B.2 Azimuthal velocity The radial component of the Euler equations is given by: $-\frac{v_{\mathrm{g}\theta}^{2}}{r}=-\frac{1}{\rho_{\mathrm{g}}}\frac{\partial P}{\partial r}-\frac{\mathcal{G}Mr}{\left(z^{2}+r^{2}\right)^{3/2}},$ (59) where $\rho_{\mathrm{g}}$ is given by Eq. (55). Thus: $\rho_{\mathrm{g}}\left(r,z\right)=\frac{\bar{P}\left(r\right)}{c_{\mathrm{s}}^{2}\left(r,z\right)}\mathrm{e}^{-I_{1}\left(r,z\right)},$ (60) with: $\left\\{\begin{array}[]{rcl}I_{1}\left(r,z\right)&=&\displaystyle\int_{0}^{z}\mathcal{G}Mc_{\mathrm{s}}^{-2}\left(r,z^{\prime}\right)\partial_{z^{\prime}}f\left(r,z^{\prime}\right)\mathrm{d}z^{\prime},\\\\[8.61108pt] f\left(r,z\right)&=&\displaystyle\frac{1}{r}-\frac{1}{\sqrt{r^{2}+z^{2}}}.\end{array}\right.$ (61) To simplify Eq. (59), we first use the following identity: $\frac{1}{\rho_{\mathrm{g}}}\frac{\,\partial P}{\partial r}=c_{\mathrm{s}}^{2}\frac{\partial\,\mathrm{ln}\left(c_{\mathrm{s}}^{2}\rho_{\mathrm{g}}\right)}{\partial r},$ (62) which becomes with Eq. (60): $\frac{1}{\rho_{\mathrm{g}}}\frac{\partial P}{\partial r}=c_{\mathrm{s}}^{2}\frac{\mathrm{d}\,\mathrm{ln}\left(\bar{P}\right)}{\mathrm{d}r}-c_{\mathrm{s}}^{2}\frac{\partial\,I_{1}}{\partial r}.$ (63) Noting that $f\left(r,z=0\right)=0$ and integrating $I_{1}$ by parts provides: $I_{1}\left(r,z\right)=\mathcal{G}Mc_{\mathrm{s}}^{-2}f\left(r,z\right)-\int_{0}^{z}\mathcal{G}Mf\left(r,z^{\prime}\right)\partial_{z^{\prime}}c_{\mathrm{s}}^{-2}\left(r,z^{\prime}\right)\mathrm{d}z^{\prime},$ (64) and: $\displaystyle\frac{1}{\rho_{\mathrm{g}}}\frac{\mathrm{d}P}{\mathrm{d}r}$ $\displaystyle=$ $\displaystyle c_{\mathrm{s}}^{2}\frac{\mathrm{d}\,\mathrm{ln}\left(\bar{P}\right)}{\mathrm{d}r}-\left(-\frac{\mathcal{G}M}{r^{2}}+\frac{\mathcal{G}Mr}{\left(z^{2}+r^{2}\right)^{3/2}}\right.$ $\displaystyle\left.+c_{\mathrm{s}}^{2}\mathcal{G}Mf\partial_{r}c_{\mathrm{s}}^{-2}-c_{\mathrm{s}}^{2}\frac{\partial}{\partial r}\int_{0}^{z}\mathcal{G}Mf\partial_{z^{\prime}}c_{\mathrm{s}}^{-2}\mathrm{d}z^{\prime}\right).$ Then, Eq. (59) becomes: $\frac{v_{\mathrm{g}\theta}^{2}}{r}=c_{\mathrm{s}}^{2}\frac{\mathrm{d}\,\mathrm{ln}\left(\bar{P}\right)}{\mathrm{d}r}+\frac{\mathcal{G}M}{r^{2}}+c_{\mathrm{s}}^{2}\frac{\partial}{\partial r}\int_{0}^{z}\mathcal{G}Mf\partial_{z^{\prime}}c_{\mathrm{s}}^{-2}\mathrm{d}z^{\prime}-c_{\mathrm{s}}^{2}\mathcal{G}Mf\partial_{r}c_{\mathrm{s}}^{-2}.$ (66) Noting that : $\frac{\partial}{\partial r}\int_{0}^{z}f\partial_{z^{\prime}}c_{\mathrm{s}}^{-2}\mathrm{d}z^{\prime}=\int_{0}^{z}\frac{\partial f}{\partial r}\partial_{z^{\prime}}c_{\mathrm{s}}^{-2}\mathrm{d}z^{\prime}+\int_{0}^{z}f\partial_{r}\partial_{z^{\prime}}c_{\mathrm{s}}^{-2}\mathrm{d}z^{\prime},$ (67) and integrating the last term of the right hand side of Eq. (67) by parts provides: $\int_{0}^{z}f\partial_{r}\partial_{z^{\prime}}c_{\mathrm{s}}^{-2}\mathrm{d}z^{\prime}=f\partial_{r}c_{\mathrm{s}}^{-2}-\int_{0}^{z}\partial_{z^{\prime}}f\partial_{r}c_{\mathrm{s}}^{-2}\mathrm{d}z^{\prime}.$ (68) Therefore, Eq. (66) reduces to: $\frac{v_{\mathrm{g}\theta}^{2}}{r}=\frac{\mathcal{G}M}{r^{2}}+c_{\mathrm{s}}^{2}\frac{\mathrm{d}\,\mathrm{ln}\bar{P}}{\mathrm{d}r}+\mathcal{G}Mc_{\mathrm{s}}^{2}\int_{0}^{z}\left(\partial_{r}f\partial_{z^{\prime}}c_{\mathrm{s}}^{-2}-\partial_{z^{\prime}}f\partial_{r}c_{\mathrm{s}}^{-2}\right)\mathrm{d}z^{\prime},$ (69) which can be more elegantly written as: $\frac{v_{\mathrm{g}\theta}^{2}}{r}=\frac{\mathcal{G}M}{r^{2}}+c_{\mathrm{s}}^{2}\frac{\mathrm{d}\,\mathrm{ln}\bar{P}}{\mathrm{d}r}+\mathcal{G}Mc_{\mathrm{s}}^{2}\int_{0}^{z}\left[\nabla f\times\nabla c_{\mathrm{s}}^{-2}\right].\textbf{e}_{\theta}\mathrm{d}z^{\prime}.$ (70) Thus, the expression of the azimuthal velocity of such a disc can be separated in three terms, called the Keplerian, the pressure gradient and the baroclinic terms respectively. This last term is neglected in most studies. For a three dimensional disc, this term rigorously cancels for $c_{\mathrm{s}}=$ constant. In this case, the flow is inviscid and derives from a potential, and isobars and isodensity surfaces coincide: thus, there is no source of vorticity and the azimuthal velocity depends only on the radial coordinate. This terms also cancels out for flat discs in two dimensions. If the disc is vertically isothermal, Eq. (70) becomes: $\frac{v_{\mathrm{g}\theta}^{2}}{r}=\frac{\mathcal{G}M}{r^{2}}+c_{\mathrm{s}}^{2}\frac{\mathrm{d}\,\mathrm{ln}\bar{P}}{\mathrm{d}r}-\mathcal{G}M\left[\frac{1}{r}-\frac{1}{\sqrt{r^{2}+z^{2}}}\right]\partial_{r}\mathrm{ln}c_{\mathrm{s}}^{-2}.$ (71) ### B.3 Radial profiles of surface density and temperature In this section, we consider that the disc surface density and the temperature (and thus the sound speed) depend only on the radial coordinate and are given by the following power-law profiles: $\left\\{\begin{array}[]{l}\Sigma\left(r\right)=\Sigma_{0}\left(\frac{r}{r_{0}}\right)^{-p}=\Sigma_{0}^{\prime}r^{-p},\\\ T\left(r\right)=T_{0}\left(\frac{r}{r_{0}}\right)^{-q}=T_{0}^{\prime}r^{-q},\\\ c_{\mathrm{s}}\left(r\right)=c_{\mathrm{s}0}\left(\frac{r}{r_{0}}\right)^{-q/2}=c^{\prime}_{\mathrm{s}0}r^{-q/2}.\end{array}\right.$ (72) For vertically isothermal thin discs, the vertical density is therefore given by Eq. (57) with the scale height given by Eq. (58), which can be expressed as: $H\left(r\right)=H_{0}\left(\frac{r}{r_{0}}\right)^{\frac{3}{2}-\frac{q}{2}}=H_{0}^{\prime}r^{\frac{3}{2}-\frac{q}{2}},$ (73) with: $H_{0}^{\prime}=\frac{c^{\prime}_{\mathrm{s}0}}{\sqrt{\mathcal{G}M}}.$ (74) The expression of $\rho_{\mathrm{g}}$ compatible with the vertical hydrostatic equilibrium and providing the power-law profile set by Eq. (72) is written as: $\rho_{\mathrm{g}}=\rho_{\mathrm{g}0}^{\prime}r^{-x}\mathrm{e}^{-\frac{z^{2}}{2H^{2}\left(r\right)}}.$ (75) Indeed: $\displaystyle\int_{-\infty}^{+\infty}\rho_{\mathrm{g0}}^{\prime}r^{-x}\mathrm{e}^{-\frac{z^{2}}{2H^{2}\left(r\right)}}\mathrm{d}z$ $\displaystyle=$ $\displaystyle\rho_{\mathrm{g}0}^{\prime}\sqrt{2\pi}H\left(r\right)r^{-x},$ $\displaystyle=$ $\displaystyle\Sigma_{0}^{\prime}r^{-p}.$ Hence, with $\Sigma_{0}^{\prime}=\sqrt{2\pi}\rho_{\rm{g}0}^{\prime}H_{0}^{\prime}$ and $x=p-\frac{q}{2}+\frac{3}{2}$, $\rho_{\mathrm{g}}=\frac{\Sigma_{0}^{\prime}}{\sqrt{2\pi}H_{0}^{\prime}}r^{-\left(p-\frac{q}{2}+\frac{3}{2}\right)}\mathrm{e}^{-\left[\frac{z^{2}}{2H_{0}^{\prime 2}r^{3-q}}\right]},$ (77) which gives the correct surface density profile when integrated with respect to $z$. With this expression of $\rho_{\mathrm{g}}$, $\bar{P}\left(r\right)$ is given by: $\bar{P}\left(r\right)=c_{\mathrm{s}}^{2}\left(r\right)\rho_{\mathrm{g}}\left(r,z=0\right)=c_{\mathrm{s}0}^{\prime 2}\frac{\Sigma_{0}^{\prime}}{\sqrt{2\pi}H_{0}^{\prime}}r^{-\left(p+\frac{q}{2}+\frac{3}{2}\right)},$ (78) which ensures that: $\frac{\mathrm{d}\,\mathrm{ln}\bar{P}}{\mathrm{d}r}=-\frac{\left(p+\frac{q}{2}+\frac{3}{2}\right)}{r},$ (79) and, using Eq. (71), we find: $v_{\mathrm{g}\theta}=\sqrt{\frac{\mathcal{G}M}{r}-\left(p+\frac{q}{2}+\frac{3}{2}\right)c_{\mathrm{s}}^{\prime 2}r^{-q}-\mathcal{G}Mq\left(\frac{1}{r}-\frac{1}{\sqrt{r^{2}+z^{2}}}\right)}.$ (80) ## Appendix C Dimensionless quantities and equations of motion To highlight the important physical parameters involved, we set $v_{\mathrm{k}0}=\sqrt{\frac{\mathcal{G}M}{r_{0}}}$ and introduce dimensionless quantities given by the following expressions: $\left\\{\begin{array}[]{rcl}r/r_{0}&=&R,\\\ \mathcal{T}/\mathcal{T}_{0}&=&R^{-q},\\\ \Sigma/\Sigma_{0}&=&R^{-p},\\\ v_{\mathrm{k}}/v_{\mathrm{k}0}&=&R^{-\frac{1}{2}},\\\ H/H_{0}&=&R^{\frac{3}{2}-\frac{q}{2}},\\\ z/H_{0}&=&Z,\\\ v_{\mathrm{g}\theta}/v_{\mathrm{k}0}&=&\displaystyle\sqrt{\frac{1}{R}-\eta_{0}R^{-q}-q\left(\frac{1}{R}-\frac{1}{\sqrt{R^{2}+\phi_{0}Z^{2}}}\right)},\\\ \rho_{\mathrm{g}}/\rho_{\mathrm{g}0}&=&R^{-\left(p-\frac{q}{2}+\frac{3}{2}\right)}\mathrm{e}^{-\frac{Z^{2}}{2R^{3-q}}}.\\\ \end{array}\right.$ (81) with: $\eta_{0}=\left(p+\frac{q}{2}+\frac{3}{2}\right)\left(\frac{c_{\mathrm{s}0}}{v_{\mathrm{k}0}}\right)^{2}.$ (82) The dimensionless parameter $\eta_{0}$ gives the order of magnitude of the relative discrepancy between the Keplerian motion and the gas azimuthal velocity. We note that: $\left.\frac{rc_{\mathrm{s}}^{2}}{v_{\mathrm{k}}}\frac{\mathrm{d}\,\mathrm{ln}\bar{P}}{\mathrm{d}r}\right/v_{\mathrm{k}0}=-\eta_{0}R^{-q+1/2}.$ (83) Then, we set $t_{\mathrm{k}0}=\sqrt{\frac{r_{0}^{3}}{\mathcal{G}M}}$ and define $\left\\{\begin{array}[]{rcl}\displaystyle\frac{t}{t_{\mathrm{k}0}}&=&\displaystyle T\\\\[8.61108pt] \displaystyle\frac{v_{r}}{v_{\mathrm{k}0}}&=&\displaystyle\frac{\mathrm{d}R}{\mathrm{d}T}=\tilde{v}_{r}\\\\[8.61108pt] \displaystyle\frac{v_{\theta}}{v_{\mathrm{k}0}}&=&\displaystyle R\frac{\mathrm{d}\theta}{\mathrm{d}T}=\tilde{v}_{\theta}\\\\[8.61108pt] \displaystyle\frac{z}{H_{0}}&=&\displaystyle Z\\\\[8.61108pt] \displaystyle\frac{v_{z}}{H_{0}/t_{\mathrm{k}0}}&=&\displaystyle\frac{\mathrm{d}Z}{\mathrm{d}T}.\end{array}\right.$ (84) Writing the coefficient $\tilde{\mathcal{C}}\left(r,z\right)$ of the drag force of Eq. (5) as: $\tilde{\mathcal{C}}\left(r,z\right)=\mathcal{C}_{0}\mathcal{C}\left(R,Z\right),$ (85) and using dimensionless coordinates, we have: $\frac{\textbf{F}_{\mathrm{D}}/m_{\mathrm{d}}}{v_{\mathrm{k}0}/t_{\mathrm{k}0}}=-\frac{\mathcal{C}\left(R,Z\right)}{\left[\frac{s}{\left(v_{\mathrm{k}0}^{\lambda}t_{\mathrm{k}0}\mathcal{C}_{0}\right)^{\frac{1}{y}}}\right]^{y}}\|\tilde{\textbf{v}}-\tilde{\textbf{v}}_{\mathrm{g}}\|^{\lambda}\left(\tilde{\textbf{v}}-\tilde{\textbf{v}}_{\mathrm{g}}\right).$ (86) We also introduce: $s_{\mathrm{opt},0}=\left(v_{\mathrm{k}0}^{\lambda}t_{\mathrm{k}0}\mathcal{C}_{0}\right)^{\frac{1}{y}},$ (87) and $S_{0}=\frac{s}{s_{\mathrm{opt},0}},$ (88) so that Eq. (86) becomes: $\frac{\textbf{F}_{\mathrm{D}}/m_{\mathrm{d}}}{v_{\mathrm{k}0}/t_{\mathrm{k}0}}=-\frac{\mathcal{C}\left(R,Z\right)}{S_{0}^{y}}\|\tilde{\textbf{v}}-\tilde{\textbf{v}}_{\mathrm{g}}\|^{\lambda}\left(\tilde{\textbf{v}}-\tilde{\textbf{v}}_{\mathrm{g}}\right).$ (89) Physically, $s_{\mathrm{opt},0}$ corresponds to the grain size at which the drag stopping time equals to the Keplerian time at $r_{0}$. In Table 2, we give the expressions of $y$, $\lambda$, $s_{\mathrm{opt},0}$, $\mathcal{C}\left(R,Z\right)$ for the Epstein and the three Stokes drag regimes. The dimensionless equations of motion for a dust grain are then: $\left\\{\begin{array}[]{rcl}\displaystyle\frac{\mathrm{d}\tilde{v}_{r}}{\mathrm{d}T}-\frac{\tilde{v}_{\theta}^{2}}{R}+\frac{\tilde{v}_{r}}{S_{0}^{y}}\mathcal{C}\left(R,Z\right)\|\tilde{\textbf{v}}-\tilde{\textbf{v}}_{\mathrm{g}}\|^{\lambda}+\frac{R}{\left(R^{2}+\phi_{0}Z^{2}\right)^{3/2}}&=&0\\\\[8.61108pt] \lx@intercol\displaystyle\frac{\mathrm{d}\tilde{v}_{\theta}}{\mathrm{d}T}+\frac{\tilde{v}_{\theta}\tilde{v}_{r}}{R}+\hfil\lx@intercol&&\\\ \frac{\left(\tilde{v}_{\theta}-\sqrt{\frac{1}{R}-\eta_{0}R^{-q}-q\left(\frac{1}{R}-\frac{1}{\sqrt{R^{2}+\phi_{0}Z^{2}}}\right)}\right)}{S_{0}^{y}}\mathcal{C}\left(R,Z\right)\|\tilde{\textbf{v}}-\tilde{\textbf{v}}_{\mathrm{g}}\|^{\lambda}&=&0\\\\[8.61108pt] \displaystyle\frac{\mathrm{d}^{2}Z}{\mathrm{d}T^{2}}+\frac{1}{S_{0}^{y}}\frac{\mathrm{d}Z}{\mathrm{d}T}\mathcal{C}\left(R,Z\right)\|\tilde{\textbf{v}}-\tilde{\textbf{v}}_{\mathrm{g}}\|^{\lambda}+\frac{Z}{\left(R^{2}+\phi_{0}Z^{2}\right)^{3/2}}&=&0.\end{array}\right.$ (90) Table 2: Expressions of the coefficients $y$, $\lambda$, $s_{\mathrm{opt},0}$ and $\mathcal{C}\left(R,Z\right)$ for different drag regimes. Drag regime \| $y$ \| $\lambda$ \| $s_{\mathrm{opt},0}$ \| $\mathcal{C}\left(R,Z\right)$ ---\|---\|---\|---\|--- Epstein \| 1 \| 0 \| $\displaystyle\frac{\Sigma_{0}}{\sqrt{2\pi}\rho_{\mathrm{d}}}$ \| $\displaystyle R^{-\left(p+\frac{3}{2}\right)}e^{-\frac{Z^{2}}{2R^{3-q}}}$ Stokes ($R_{\mathrm{g}}<1$) \| 2 \| 0 \| $\displaystyle\sqrt{\frac{9t_{\mathrm{k}0}\mu_{0}}{2\rho_{\mathrm{d}}}}$ \| $\displaystyle R^{-\frac{q}{2}}$ Stokes ($1<R_{\mathrm{g}}<800$) \| 1.6 \| 0.4 \| $\displaystyle\left(\frac{18r_{0}\nu_{0}^{0.6}\Sigma_{0}}{v_{\mathrm{k}0}^{0.6}\sqrt{2\pi}2^{1.6}\rho_{\mathrm{d}}H_{0}}\right)^{\frac{1}{1.6}}$ \| $\displaystyle R^{-\left(\frac{2p}{5}+\frac{q}{10}+\frac{3}{5}\right)}e^{-\frac{2}{5}\frac{Z^{2}}{2R^{3-q}}}$ Stokes ($800<R_{\mathrm{g}}$) \| 1 \| 1 \| $\displaystyle\frac{1.32r_{0}\Sigma_{0}}{8\rho_{\mathrm{d}}\sqrt{2\pi}H_{0}}$ \| $\displaystyle R^{-\left(p-\frac{q}{2}+\frac{3}{2}\right)}e^{-\frac{Z^{2}}{2R^{3-q}}}$ ## Appendix D Lemma for the different expansions Lemma: Let $x$ be either $r$ or $\theta$ and $i$ the order of the perturbative expansion. If: * • $\tilde{v}_{r0}=0$, and * • $\tilde{v}_{xi}$ can be written as a function of $R$ ($\tilde{v}_{xi}=f\left(R\right)$) with $f=\mathcal{O}\left(1\right)$ of the expansion in $\eta_{0}$, then, $\displaystyle\frac{\mathrm{d}\tilde{v}_{xi}}{\mathrm{d}T}$ is of order $\mathcal{O}\left(\eta_{0}\right)$. Proof: $\frac{\mathrm{d}\tilde{v}_{xi}}{\mathrm{d}T}=\frac{\mathrm{d}\tilde{v}_{xi}}{\mathrm{d}R}\frac{\mathrm{d}R}{\mathrm{d}T}=\tilde{v}_{r}f^{\prime}\left(R\right)=\eta_{0}\tilde{v}_{r1}f^{\prime}\left(R\right)+\mathcal{O}\left(\eta_{0}^{2}\right)=\mathcal{O}\left(\eta_{0}\right).$ (91) ## Appendix E Epstein regime: perturbation analysis * • Order $\mathcal{O}\left(1\right)$: At this order of expansion, $\eta_{0}R^{-q}$ is negligible compared to $\frac{1}{R}$. Thus, substituting Eq. (13) into Eq. (11) provides $\left\\{\begin{array}[]{l}\displaystyle\frac{\mathrm{d}\tilde{v}_{r0}}{\mathrm{d}T}=\displaystyle\frac{\tilde{v}_{\theta 0}^{2}}{R}-\frac{\tilde{v}_{r0}}{S_{0}}R^{-\left(p+\frac{3}{2}\right)}-\frac{1}{R^{2}}\\\ \displaystyle\frac{\mathrm{d}\tilde{v}_{\theta 0}}{\mathrm{d}T}=\displaystyle-\frac{\tilde{v}_{\theta 0}\tilde{v}_{r0}}{R}-\frac{\left(\tilde{v}_{\theta 0}-\sqrt{\frac{1}{R}}\right)}{S_{0}}R^{-\left(p+\frac{3}{2}\right)}.\end{array}\right.$ (92) At this stage, we do not know the order of $\frac{\mathrm{d}\tilde{v}_{\theta 0}}{\mathrm{d}T}$. We show in the lemma of Appendix D that $\frac{\mathrm{d}\tilde{v}_{\theta 0}}{\mathrm{d}T}=\mathcal{O}\left(\eta_{0}\right)$. Applying this lemma, we see that at the order $\mathcal{O}\left(1\right)$, taking $\tilde{v}_{r0}=0$ and $\tilde{v}_{\theta 0}=\sqrt{\frac{1}{R}}$ (which ensures that $\frac{\mathrm{d}\tilde{v}_{\theta 0}}{\mathrm{d}T}=\mathcal{O}\left(\eta_{0}\right)$) is a relevant solution for the equations of motion (which corresponds to circular Keplerian motion). Thus, $\left\\{\begin{array}[]{l}\displaystyle\tilde{v}_{r0}=\displaystyle 0\\\ \displaystyle\tilde{v}_{\theta 0}=\displaystyle\sqrt{\frac{1}{R}}.\end{array}\right.$ (93) * • Order $\mathcal{O}\left(\eta_{0}\right)$: Applying the lemma in this order of expansion and noting that $\frac{\mathrm{d}\tilde{v}_{\theta 0}}{\mathrm{d}T}=-\frac{1}{2}R^{-3/2}\tilde{v}_{r}=-\frac{\eta_{0}}{2}R^{-3/2}\tilde{v}_{r1}+\mathcal{O}\left(\eta_{0}^{2}\right),$ (94) Eq. (11) becomes $\left\\{\begin{array}[]{l}\displaystyle 0=-\frac{R^{-\left(p+\frac{3}{2}\right)}}{S_{0}}\tilde{v}_{r1}+\displaystyle 2R^{-3/2}\tilde{v}_{\theta 1}\\\ \displaystyle 0=\displaystyle-\frac{1}{2}R^{-3/2}\tilde{v}_{r1}-\frac{\left(\tilde{v}_{\theta 1}-\frac{1}{\eta_{0}}\left(\sqrt{\frac{1}{R}-\eta_{0}R^{-q}}-\sqrt{\frac{1}{R}}\right)\right)}{S_{0}}R^{-\left(p+\frac{3}{2}\right)}.\end{array}\right.$ (95) Solving the linear system Eq. (95) for $\left(\tilde{v}_{r1},\tilde{v}_{\theta 1}\right)$ provides $\left\\{\begin{array}[]{l}\displaystyle\tilde{v}_{r1}=\displaystyle-\frac{1}{\eta_{0}}\frac{2S_{0}R^{p-\frac{1}{2}}\left(1-\sqrt{1-\eta_{0}R^{-q+1}}\right)}{1+R^{2p}S_{0}^{2}}\\\ \displaystyle\tilde{v}_{\theta 1}=\displaystyle-\frac{1}{\eta_{0}}\frac{R^{-\frac{1}{2}}\left(1-\sqrt{1-\eta_{0}R^{-q+1}}\right)}{1+R^{2p}S_{0}^{2}}.\end{array}\right.$ (96) In addition to the expression of $\tilde{v}_{r}$ given in Sect. 3, we also note that $\begin{array}[]{rcl}\tilde{v}_{\theta}&=&\displaystyle\sqrt{\frac{1}{R}}+\eta_{0}\tilde{v}_{\theta 1}+\mathcal{O}\left(\eta_{0}^{2}\right)\\\ &=&\displaystyle\sqrt{\frac{1}{R}}-\frac{R^{-\frac{1}{2}}\left(1-\sqrt{1-\eta_{0}R^{-q+1}}\right)}{1+R^{2p}S_{0}^{2}}+\mathcal{O}\left(\eta_{0}^{2}\right),\end{array}$ (97) which provides with Eq. (15) $\tilde{v}_{\theta}=\sqrt{\frac{1}{R}}-\frac{\eta_{0}R^{-q+\frac{1}{2}}}{2\left(1+R^{2p}S_{0}^{2}\right)}.$ (98) ## Appendix F Link with W77’s original derivation Following W77’s historic reasoning for small grains (see Sect. 3.1), we perform a perturbative expansion of the radial equation of motion with the $S_{0}$ variable. We verify that taking the limit at small $\eta_{0}$ provides the expression found for the A-mode in the NSH86 expansion. (Formally, we will show that $\lim\limits_{\begin{subarray}{c}S_{0}\ll 1\end{subarray}}\lim\limits_{\begin{subarray}{c}\eta_{0}\ll 1\end{subarray}}\left[...\right]=\lim\limits_{\begin{subarray}{c}\eta_{0}\ll 1\end{subarray}}\lim\limits_{\begin{subarray}{c}S_{0}\ll 1\end{subarray}}\left[...\right]$ ). Hence, we set: $\left\\{\begin{array}[]{l}\displaystyle\tilde{v}_{r}=\displaystyle\tilde{v}_{r0}+S_{0}\tilde{v}_{r1}+S_{0}^{2}\tilde{v}_{r2}+\mathcal{O}\left(S_{0}^{3}\right),\\\ \displaystyle\tilde{v}_{\theta}=\displaystyle\tilde{v}_{\theta 0}+S_{0}\tilde{v}_{\theta 1}+S_{0}^{2}\tilde{v}_{\theta 2}+\mathcal{O}\left(S_{0}^{3}\right),\end{array}\right.$ (99) where we have used for convenience the same formalism as for the expansion in $\eta_{0}$ – see Eq. (13) (noting of course that $\tilde{v}$ represents different functions). An important point is that the lemma of Appendix D holds when substituting $S_{0}$ to $\eta_{0}$. Therefore, substituting Eq. (99) into Eq. (11) provides the equations of motion for different orders of $\mathcal{O}\left(S_{0}\right)$: * • Order $\mathcal{O}\left(\frac{1}{S_{0}}\right)$: Eq. (11) provides $\mathcal{O}\left(1\right)$ expressions for the velocities: $\left\\{\begin{array}[]{r@{\ }l}\displaystyle\tilde{v}_{r0}=&\displaystyle 0,\\\ \displaystyle\tilde{v}_{\theta 0}=&\displaystyle\sqrt{\frac{1}{R}-\eta_{0}R^{-q}}.\end{array}\right.$ (100) In this order of expansion, the azimuthal velocity corresponds to the sub- Keplerian velocity of the gas. There is no radial motion. * • Order $\mathcal{O}\left(1\right)$: $\left\\{\begin{array}[]{r@{\ }l}\displaystyle\tilde{v}_{r1}=&\displaystyle-2R^{p-\frac{1}{2}}\left(1-\sqrt{1-\eta_{0}R^{-q+1}}\right),\\\ \displaystyle\tilde{v}_{\theta 1}=&\displaystyle 0.\end{array}\right.$ (101) * • Order $\mathcal{O}\left(S_{0}\right)$: $\left\\{\begin{array}[]{r@{\ }l}\displaystyle\tilde{v}_{r2}=&\displaystyle 0,\\\ \displaystyle\tilde{v}_{\theta 2}=&\displaystyle R^{p+\frac{3}{2}}\left[\eta_{0}\sqrt{\frac{1}{R}-\eta_{0}R^{-q}}R^{p-q-\frac{1}{2}}\right.\\\ &\qquad\quad\left.\displaystyle+\frac{1}{2}\frac{-\frac{1}{R^{2}}+\eta_{0}qR^{-q-1}}{\sqrt{\frac{1}{R}-\eta_{0}R^{-q}}\eta_{0}R^{p-q+\frac{1}{2}}}\right].\end{array}\right.$ (102) Finally, we obtain expressions for $\tilde{v}_{r}$ and $\tilde{v}_{\theta}$: $\left\\{\begin{array}[]{r@{\ }l}\displaystyle\tilde{v}_{r}=&\displaystyle-2S_{0}R^{p-\frac{1}{2}}\left(1-\sqrt{1-\eta_{0}R^{-q+1}}\right)+\mathcal{O}\left(S_{0}^{3}\right),\\\\[8.61108pt] \displaystyle\tilde{v}_{\theta}=&\displaystyle\sqrt{\frac{1}{R}-\eta_{0}R^{-q}}+S_{0}^{2}R^{p+\frac{3}{2}}\left[\eta_{0}\sqrt{\frac{1}{R}-\eta_{0}R^{-q}}R^{p-q-\frac{1}{2}}\right.\\\ &\qquad\quad\left.\displaystyle+\frac{1}{2}\frac{-\frac{1}{R^{2}}+\eta_{0}qR^{-q-1}}{\sqrt{\frac{1}{R}-\eta_{0}R^{-q}}\eta_{0}R^{p-q+\frac{1}{2}}}\right]+\mathcal{O}\left(S_{0}^{3}\right).\end{array}\right.$ (103) We now compare the NSH86 expansion at $R^{p}S_{0}\ll 1$ (A-mode) and the W77 expansion at $\eta_{0}\ll 1$. * • NSH86: From Eqs. (16) and Eq. (98): $\left\\{\begin{array}[]{r@{\ }l}\displaystyle\tilde{v}_{r}=&\displaystyle-\frac{\eta_{0}S_{0}R^{p-q+\frac{1}{2}}}{1+R^{2p}S_{0}^{2}}+\mathcal{O}\left(\eta_{0}^{2}\right)\\\ =&\displaystyle-\eta_{0}S_{0}R^{p-q+\frac{1}{2}}+\mathcal{O}\left(\eta_{0}^{2}\right)+\mathcal{O}\left(S_{0}^{2}\right),\\\\[8.61108pt] \displaystyle\tilde{v}_{\theta}=&\displaystyle\sqrt{\frac{1}{R}}-\frac{\eta_{0}}{2}\frac{R^{-q+\frac{1}{2}}}{1+S_{0}^{2}R^{2p}}+\mathcal{O}\left(\eta_{0}^{2}\right)\\\ =&\displaystyle\sqrt{\frac{1}{R}}-\frac{\eta_{0}}{2}R^{-q+\frac{1}{2}}+\frac{\eta_{0}S_{0}^{2}}{2}R^{2p-q+\frac{1}{2}}+\mathcal{O}\left(\eta_{0}^{2}\right)+\mathcal{O}\left(S_{0}^{3}\right).\end{array}\right.$ (104) * • W77 small grains: From Eq. (103): $\left\\{\begin{array}[]{r@{\ }l}\displaystyle\tilde{v}_{r}=&\displaystyle-2S_{0}R^{p-\frac{1}{2}}\left(1-\sqrt{1-\eta_{0}R^{-q+1}}\right)+\mathcal{O}\left(S_{0}^{2}\right)\\\ =&\displaystyle-\eta_{0}S_{0}R^{p-q+\frac{1}{2}}+\mathcal{O}\left(S_{0}^{2}\right)+\mathcal{O}\left(\eta_{0}^{2}\right),\\\\[8.61108pt] \displaystyle\tilde{v}_{\theta}=&\displaystyle\sqrt{\frac{1}{R}-\eta_{0}R^{-q}}+S_{0}^{2}R^{p+\frac{3}{2}}\Big{[}\eta_{0}\sqrt{\frac{1}{R}-\eta_{0}R^{-q}}R^{p-q-\frac{1}{2}}\\\ &\displaystyle+\frac{1}{2}\frac{\eta_{0}R^{p-q+\frac{1}{2}}\left(-\frac{1}{R^{2}}+\eta_{0}qR^{-q-1}\right)}{\sqrt{\frac{1}{R}-\eta_{0}R^{-q}}}\Big{]}+\mathcal{O}\left(S_{0}^{3}\right)\\\ =&\displaystyle\sqrt{\frac{1}{R}}-\frac{\eta_{0}}{2}R^{-q+\frac{1}{2}}+\frac{\eta_{0}S_{0}^{2}}{2}R^{2p-q+\frac{1}{2}}+\mathcal{O}\left(S_{0}^{3}\right)+\mathcal{O}\left(\eta_{0}^{2}\right).\end{array}\right.$ (105) Clearly, Eqs. (104) and (105) are identical, demonstrating that the theories of W77 and NSH86 are consistent. We also note that if the simplification of Eq. (15) is not performed, the two W77 expansions directly appear as the expansion of NSH86 in $\mathcal{O}\left(S_{0}\right)$ or $\mathcal{O}\left(S_{0}^{-1}\right)$. Now, in the case of large grains, we perform a perturbative expansion of the radial equation of motion with respect to $\frac{1}{S_{0}}$ while assuming that $S_{0}R^{p}\gg 1$, and verify that taking the limit at small $\eta_{0}$ provides the expression found for the B-mode in the NSH86 expansion. Taking the same precautions as for the previous expansions, we write: $\left\\{\begin{array}[]{r@{\ }l}\displaystyle\tilde{v}_{r}=&\displaystyle\tilde{v}_{r0}+\frac{1}{S_{0}}\tilde{v}_{r1}+\mathcal{O}\left(\frac{1}{S_{0}^{2}}\right),\\\ \displaystyle\tilde{v}_{\theta}=&\displaystyle\tilde{v}_{\theta 0}+\frac{1}{S_{0}}\tilde{v}_{\theta 1}+\mathcal{O}\left(\frac{1}{S_{0}^{2}}\right).\end{array}\right.$ (106) Following the same method as for the small grain sizes expansion, we obtain: * • Order $\mathcal{O}\left(1\right)$: $\left\\{\begin{array}[]{l}\tilde{v}_{r0}=0,\\\ \tilde{v}_{\theta 0}=\displaystyle\sqrt{\frac{1}{R}}.\end{array}\right.$ (107) It this order of expansion, the azimuthal velocity of the grain is the standard Keplerian velocity. * • Order $\mathcal{O}\left(\frac{1}{S_{0}}\right)$: $\left\\{\begin{array}[]{l}\tilde{v}_{r1}=\displaystyle-2\left(\sqrt{\frac{1}{R}}-\sqrt{\frac{1}{R}-\eta_{0}R^{-q}}\right)R^{-p},\\\ \tilde{v}_{\theta 1}=0.\end{array}\right.$ (108) The expansion at higher order is more complicated and will not be used for further developments. At the order $\mathcal{O}\left(\frac{1}{S_{0}^{2}}\right)$, we have for $\tilde{v}_{r}$ and $\tilde{v}_{\theta}$: $\left\\{\begin{array}[]{l}\displaystyle\tilde{v}_{r}=\displaystyle-\frac{2}{S_{0}}\left(\sqrt{\frac{1}{R}}-\sqrt{\frac{1}{R}-\eta_{0}R^{-q}}\right)R^{-p}+\mathcal{O}\left(\frac{1}{S_{0}^{2}}\right),\\\ \displaystyle\tilde{v}_{\theta}=\displaystyle\sqrt{\frac{1}{R}}+\mathcal{O}\left(\frac{1}{S_{0}^{2}}\right).\end{array}\right.$ (109) We now compare the expressions provided by the NSH86 expansion at $R^{2p}S_{0}^{2}\gg 1$ (B-mode) and the W77 expansion at $\eta_{0}\ll 1$ for the radial velocity: * • NSH86: $\displaystyle\tilde{v}_{r}$ $\displaystyle=$ $\displaystyle-\frac{\eta_{0}S_{0}R^{p-q+\frac{1}{2}}}{1+R^{2p}S_{0}^{2}}+\mathcal{O}\left(\eta_{0}^{2}\right),$ $\displaystyle=$ $\displaystyle-\frac{\eta_{0}}{S_{0}}R^{-p-q+\frac{1}{2}}+\mathcal{O}\left(\eta_{0}^{2}\right)+\mathcal{O}\left(\frac{1}{S_{0}^{2}}\right).$ * • W77 large grains: $\displaystyle\tilde{v}_{r}$ $\displaystyle=$ $\displaystyle-\frac{2}{S_{0}}R^{-p-\frac{1}{2}}\left(1-\sqrt{1-\eta_{0}R^{-q+1}}\right)+\mathcal{O}\left(\frac{1}{S_{0}^{2}}\right),$ $\displaystyle=$ $\displaystyle-\frac{\eta_{0}}{S_{0}}R^{-p-q+\frac{1}{2}}+\mathcal{O}\left(\frac{1}{S_{0}^{2}}\right)+\mathcal{O}\left(\eta_{0}^{2}\right).$ Once again, we show that the W77 and NSH86 theories are consistent. ## Appendix G Asymptotic radial behaviour of a single grain Figure 13: The discrepancy (bottom panel) between the exact motion (top panel) and its NSH86 approximation (central panel) is negligible. This is illustrated plotting the radial motion of dust grains for $S_{0}=10^{-2}$, $\eta_{0}=10^{-2}$ and for $p=0,q=\frac{3}{4}$ (solid) and $p=\frac{3}{2},q=\frac{3}{4}$ (dashed). Top: $R_{1}\left(T\right)$, middle: $R_{2}\left(T\right)$, bottom: relative difference $\left(R_{2}\left(T\right)-R_{1}\left(T\right)\right)/R_{1}\left(T\right)$. Noting $R_{1}\left(T\right)$ the position of a grain integrated directly from the equation of motion (Eq. (11)) and $R_{2}\left(T\right)$ the position integrated from the NSH86 approximation (Eq. (34)), we highlight (Fig. 13) that the discrepancy between the motion from the exact equations and its NSH86 approximation is negligible (the relative error is lower than $10^{-3}$ for all the considered sizes). It is therefore justified to use the analytical results derived in Sect. 3 to interpret the grain behaviour. Figure 14: Values of $S_{\mathrm{m}}$ (left) and $\eta_{0}T_{\mathrm{m}}\left(S_{\mathrm{m}}\right)$ (right) in the ($p$,$q$) plane. Thus, from Eq. (34), we see that the time for a grain starting at $R=1$ to reach some final radius $R_{\mathrm{f}}$ is minimized for an optimal grain size $S_{\mathrm{m,f}}$ given by $S_{\mathrm{m,f}}=\sqrt{\frac{p+q+\frac{1}{2}}{-p+q+\frac{1}{2}}\times k_{\mathrm{f}}\left(R_{\mathrm{f}}\right)}\,,$ (112) with $k_{\mathrm{f}}\left(R_{\mathrm{f}}\right)=\left\\{\begin{array}[]{ll}\frac{\left(1-R_{\mathrm{f}}^{-p+q+\frac{1}{2}}\right)}{\left(1-R_{\mathrm{f}}^{p+q+\frac{1}{2}}\right)}&\mathrm{if}\leavevmode\nobreak\ p+q+\frac{1}{2}\neq 0\\\\[4.30554pt] -\mathrm{ln}\left(R_{\mathrm{f}}\right)&\mathrm{if}\leavevmode\nobreak\ p+q+\frac{1}{2}=0.\\\ \end{array}\right.$ (113) As shown in Eq. (34), the outcome of the grain radial motion depends on the value of $-p+q+\frac{1}{2}$: * • If $-p+q+\frac{1}{2}\leq 0$: $\lim\limits_{\begin{subarray}{c}T\to+\infty\end{subarray}}R=0.$ (114) For such disc profiles, all grains pile-up and fall onto the central star in an infinite time. Indeed, the surface density profile given by $p\geq q+\frac{1}{2}$ is steep enough to counterbalance the increase of the acceleration due to the pressure gradient. Therefore, grains fall onto the central star in an infinite time, whatever their initial size. Such an evolution happens because the grains always end migrating in the A-mode when they reach the disc’s inner regions. A crucial consequence is that grains are not depleted on the central star and therefore stay in the disc where they can potentially form planet embryos. * • If $-p+q+\frac{1}{2}>0$: $\lim\limits_{\begin{subarray}{c}T\to T_{\mathrm{m}}\end{subarray}}R=0,$ (115) where $T_{\mathrm{m}}=\frac{1}{\eta_{0}S_{0}}\left(\frac{1}{-p+q+\frac{1}{2}}+\frac{S_{0}^{2}}{p+q+\frac{1}{2}}\right).$ (116) In this case, grains fall onto the central star in a finite time. The surface density profile given by $p<q+\frac{1}{2}$ is now too flat to counterbalance the increasing acceleration due to pressure gradient. We note that: * – For small sizes ($S_{0}\ll 1$), $T_{\mathrm{m}}=\mathcal{O}\left(S_{0}\eta_{0}\right)$. * – For large sizes ($S_{0}\gg 1$), $T_{\mathrm{m}}=\mathcal{O}\left(\frac{S_{0}}{\eta_{0}}\right)$. * – $T_{\mathrm{m}}$ reaches a minimal value for a size $S_{\mathrm{m}}$ given by $S_{\mathrm{m}}=\sqrt{\frac{p+q+\frac{1}{2}}{-p+q+\frac{1}{2}}}.$ (117) Therefore $T_{\mathrm{m}}\left(S_{\mathrm{m}}\right)=\frac{2}{\eta_{0}\sqrt{\left(p+q+\frac{1}{2}\right)\left(-p+q+\frac{1}{2}\right)}}.$ (118) $S_{\mathrm{m}}$ is of order unity and corresponds to an optimal size of migration. Values of $S_{\mathrm{m}}$ and $\eta_{0}T_{\mathrm{m}}\left(S_{\mathrm{m}}\right)$ in the $(p,q)$ plane are shown in Fig. 14. When $S\simeq S_{\mathrm{m}}$, both the A- and B-modes contribute in an optimal way to the grains radial motion. In this case, grains can be efficiently accreted by the central star if $S\simeq S_{\mathrm{m}}=\mathcal{O}\left(1\right)$. Thus, they can not contribute to the formation of pre-planetesimals. This process is called the radial-drift barrier for planet formation.	arxiv-papers	2011-11-14T01:49:47	2024-09-04T02:49:24.281474	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Guillaume Laibe (Monash), Jean-Fran\\c{c}ois Gonzalez (CRAL), and Sarah\n T. Maddison (Swinburne)", "submitter": "Guillaume Laibe", "url": "https://arxiv.org/abs/1111.3083" }
1111.3089	# Dusty gas with SPH — II. Implicit timestepping and astrophysical drag regimes Guillaume Laibe, Daniel J. Price Centre for Stellar and Planetary Astrophysics and School of Mathematical Sciences, Monash University, Clayton, Vic 3800, Australia ###### Abstract In a companion paper (Laibe & Price, 2011b), we have presented an algorithm for simulating two-fluid gas and dust mixtures in Smoothed Particle Hydrodynamics (SPH). In this paper, we develop an implicit timestepping method that preserves the exact conservation of the both linear and angular momentum in the underlying SPH algorithm, but unlike previous schemes, allows the iterations to converge to arbitrary accuracy and is suited to the treatment of non-linear drag regimes. The algorithm presented in Paper I is also extended to deal with realistic astrophysical drag regimes, including both linear and non-linear Epstein and Stokes drag. The scheme is benchmarked against the test suite presented in Paper I, including i) the analytic solutions of the dustybox problem and ii) solutions of the dustywave, dustyshock, dustysedov and dustydisc obtained with explicit timestepping. We find that the implicit method is 1–10 times faster than the explicit temporal integration when the ratio $r$ between the the timestep and the drag stopping time is $1\lesssim r\lesssim 1000$. ###### keywords: hydrodynamics — methods: numerical — ISM: dust, extinction — protoplanetary discs — planets and satellites: formation ††pagerange: Dusty gas with SPH — II. Implicit timestepping and astrophysical drag regimes–References††pubyear: 2011 ## 1 Introduction Dust in cold astrophysical systems spans a huge range of sizes from sub-micron sized grains in the interstellar medium to kilometre sized planetesimals involved in planet formation. Moreover, the ratio of dust to gas as well as the density and temperature of the gaseous environment in which dust is embedded can also vary strongly. Handling this full range of physical parameters presents a challenge to numerical schemes designed to simulate dusty gas in astrophysics. The main challenges are i) that at high drag (e.g. small grains), the small timestep required means that purely explicit timestepping methods become prohibitive and ii) that a wide range of physical drag prescriptions, including non-linear drag regimes, need to be handled by the code. Two main prescriptions for drag between gas and solid particles are applicable to astrophysics: The Epstein regime — where the gas surrounding a grain can be treated as a dilute medium — and the Stokes regime — where the grains can be treated as solid bodies surrounded by a fluid —(see e.g. Baines et al. 1965; Stepinski & Valageas 1996). The dependance of the drag term on local parameters of the gas (density, temperature) and the dust (typical grain size, mass) differ between the two regimes, in turn implying very different dynamics for the dust grains. For example, in a protoplanetary disc, both of these regimes may be applicable in different regions of the disc. In a companion paper (Laibe & Price 2011b, hereafter Paper I), we have developed a new algorithm for treating two-fluid gas-dust astrophysical mixtures in Smoothed Particle Hydrodynamics (SPH). Benchmarking of the method demonstrated that the algorithm gives accurate solutions on a range of test problems relevant to astrophysics and substantially improve previous algorithms (Monaghan & Kocharyan, 1995). However, in Paper I, we used only a simple explicit time stepping and considered only linear drag regimes with a constant drag coefficient. In this paper, we present an implicit timestepping method that can be applied to both linear and non-linear drag regimes, which is both more accurate and more general than the scheme proposed by Monaghan (1997). We also discuss the SPH implementation of both the Epstein and the Stokes regimes in their full generality. The paper is organised as follows: The equations of motion and the characteristics of the different astrophysical drag regimes for gas and dust mixtures are given in Sec. 2. We summarise the SPH formalism used for integrating these equations (derived in detail in Paper I) in Sec. 3 and extend it to deal with the drag regimes encountered in astrophysics. Particular attention is paid in Sec. 4 to improving the Monaghan (1997) implicit timestepping scheme, including its generalisation for non-linear drag regimes. Finally, the algorithm for non-linear drag regimes is tested against the analytic solutions of the dustybox problem, as well as the dustywave, dustysedov, dustyshock and dustydisc tests, in Sec. 5. ## 2 Gas and dust evolution in astrophysical systems ### 2.1 Evolution equations The equations describing the evolution of astrophysical gas and dust mixtures, where dust is treated as a pressureless, inviscid, continuous fluid have been described in detail in Paper I. The equations in the continuum limit are given by: $\displaystyle\frac{\partial\hat{\rho}_{\mathrm{g}}}{\partial t}+\nabla.\left(\hat{\rho}_{\mathrm{g}}\textbf{v}_{\mathrm{g}}\right)$ $\displaystyle=$ $\displaystyle 0,$ (1) $\displaystyle\frac{\partial\hat{\rho}_{\mathrm{d}}}{\partial t}+\nabla.\left(\hat{\rho}_{\mathrm{d}}\textbf{v}_{\mathrm{d}}\right)$ $\displaystyle=$ $\displaystyle 0,$ (2) $\displaystyle\hat{\rho}_{\mathrm{g}}\left(\frac{\partial\textbf{v}_{\mathrm{g}}}{\partial t}+\textbf{v}_{\mathrm{g}}.\nabla\textbf{v}_{\mathrm{g}}\right)$ $\displaystyle=$ $\displaystyle-\theta\phantom{.}\nabla P_{\rm g}+\hat{\rho}_{\mathrm{g}}\textbf{f}-F_{\rm drag}^{\rm V},$ (3) $\displaystyle\hat{\rho}_{\mathrm{d}}\left(\frac{\partial\textbf{v}_{\mathrm{d}}}{\partial t}+\textbf{v}_{\mathrm{d}}.\nabla\textbf{v}_{\mathrm{d}}\right)$ $\displaystyle=$ $\displaystyle-\left(1-\theta\right)\nabla P_{\rm g}+\hat{\rho}_{\mathrm{d}}\textbf{f}+F_{\rm drag}^{\rm V},$ (4) $\displaystyle\frac{{\rm d}u_{{\rm g}}}{{\rm d}t}$ $\displaystyle=$ $\displaystyle-\frac{P_{\mathrm{g}}}{\hat{\rho}_{\mathrm{g}}}\left[\theta\nabla\cdot{\bf v}_{\rm g}+(1-\theta)\nabla\cdot{\bf v}_{\rm d}\right]+\Lambda_{\rm drag}.$ (5) where the subscripts ${\rm g}$ and ${\rm d}$ refer to the gas and dust, respectively such that $P_{\rm g}$ is the gas pressure, $\textbf{v}_{\mathrm{g}}$ and $\textbf{v}_{\mathrm{d}}$ are the fluid velocities and $u$ is the specific internal energy of gas. The volume densities of gas and dust ($\hat{\rho}_{\mathrm{g}}$ and $\hat{\rho}_{\mathrm{d}}$, respectively) are related to the corresponding intrinsic densities ($\rho_{\rm{g}}$ and $\rho_{\rm{d}}$, respectively) according to $\displaystyle\hat{\rho}_{\mathrm{d}}$ $\displaystyle=$ $\displaystyle(1-\theta)\rho_{\mathrm{d}},$ (6) $\displaystyle\hat{\rho}_{\mathrm{g}}$ $\displaystyle=$ $\displaystyle\theta\rho_{\mathrm{g}},$ (7) where $\theta$ is the volume filling fraction of the dust. Finally, the drag force and heating terms are given by: $F_{\rm drag}^{\rm V}=K(\textbf{v}_{\mathrm{g}}-\textbf{v}_{\mathrm{d}}),$ (8) and $\Lambda_{\rm drag}=K(\textbf{v}_{\mathrm{g}}-\textbf{v}_{\mathrm{d}})^{2}.$ (9) The drag coefficient K has dimensions of mass per unit volume per unit time and is generally a function of the relative velocity between the two fluids $\Delta v\equiv\|\textbf{v}_{\mathrm{g}}-\textbf{v}_{\mathrm{d}}\|$, implying a non-linear drag regime with respect to the differential velocity between the gas and the dust. In most of the astrophysical systems, the dust is diluted enough into the gas so that the gas filling fraction is $\theta\simeq 1$ to a very good level of approximation, such that the dust buoyancy term $\left(1-\theta\right)\nabla P_{\rm g}$ is negligible. ### 2.2 Astrophysical drag regimes Microscopic collisions of gas molecules on a single dust grain result in a net exchange of momentum which is equivalent to a drag force $\bf{F}_{\rm{drag}}$ between the two phases. Two limiting cases occur when comparing the typical geometrical size $s$ of a dust grain to the mean free path $\lambda_{\rm{g}}$ of the gas. When the typical grain size is negligible compared to the collisional mean free path of the gas particles ($s\ll\lambda_{\rm{g}}$), the grains are surrounded by a dilute gas phase and may be treated using the Epstein drag prescription. In this limit, the analytic expression of the resulting drag force has been derived, assuming spherical, compact grains with homogeneous composition, for both specular and diffuse reflections on the grain surface (see Baines et al. 1965 for the complete derivation). These expressions have been widely used in astrophysical studies (see e.g. Chiang & Youdin 2010 for references), sometimes incorrectly where the grains are known to be porous and have fractal structures (Blum & Wurm, 2008) (thus breaking the assumptions of the Epstein prescription). For grain sizes larger than the collisional mean free path ($s\gg\lambda_{\rm{g}}$), grains experience a local differential velocity with respect to a uniform viscous flow and should be treated using the Stokes drag prescription (Fan & Zhu, 1998). In this case, the momentum is diffused by viscosity into the fluid, which implies that the drag expression strongly depends on the local Reynolds number defined according to $R_{\mathrm{d}}=\frac{2s\left\|\textbf{v}_{\mathrm{d}}-\textbf{v}_{\mathrm{g}}\right\|}{\nu},$ (10) where $\nu$ is the kinematic viscosity of the gas. Analytic expressions for the drag force can be derived at small Reynolds numbers. At higher Reynolds number, the drag law is inferred from experiments. Rigorously, additional contributions to the drag should arise from the grain acceleration (carried mass and Basset contribution), grain rotation (Magnus) and in the presence of strong local shear, pressure and temperature gradients (e.g. Fan & Zhu 1998). These corrections are negligible in nearly all astrophysical contexts. No current analytic theory describes how both the gas and the dust fluid exchange momentum in the intermediate regime (i.e. $s\simeq\lambda_{\rm{g}}$). Generally, an asymptotic continuous interpolation between the two limiting Epstein and Stokes regimes is used. Stepinski & Valageas (1996) suggest adopting $s=\frac{4}{9}\lambda_{\rm{g}}$ as a means of obtaining a smooth transition. It should be noted that although this approach is convenient, there is no clear measure of the physical accuracy of this assumption. #### 2.2.1 Epstein regime for dilute media In a dilute medium ($\lambda_{\mathrm{g}}>{4s}/{9}$), grains are small enough not to disturb the Maxwellian distribution of the gas velocity. Assuming grains are spherical, that the mass of a gas molecule is negligible compared to the mass of a dust grain and that the reflection of gas particles from collisions with dust grains are specular, the expression of the drag force on a single grain $\bf{F}_{\rm{drag}}$ (which differs from the volume force $F_{\rm drag}^{\rm V}$ by a factor $\hat{\rho}_{\mathrm{d}}/m_{\rm d}$, see Paper I) for the Epstein regime is given by $\displaystyle\mathbf{F}_{\rm{drag}}=\displaystyle-2\pi s^{2}\rho_{\mathrm{g}}\Delta v^{2}$ $\displaystyle\left[\frac{1}{2\sqrt{\pi}}\left\\{\left(\frac{1}{\psi}+\frac{1}{2\psi^{3}}\right)e^{-\psi^{2}}+\right.\right.$ (11) $\displaystyle\left.\left.\left(1+\frac{1}{\psi^{2}}-\frac{1}{4\psi^{4}}\right)\sqrt{\pi}\,\mathrm{erf}\left(\psi\right)\right\\}\right]\textbf{x},$ where $s$ corresponds to the grain radius and $m$, $\rho_{\mathrm{g}}$, $T$ denote the mass of the gas molecules, the intrinsic gas density and the local temperature of the mixture (the gas and the dust are supposed to have the same temperature). The thermal sound speed of the gas is thus $c_{\mathrm{s}}=\sqrt{\gamma k_{\mathrm{B}}T/m}$ and the mean thermal velocity of the gas, $c_{\mathrm{s}}\sqrt{8/\pi\gamma}$. The dimensionless quantity $\psi$ is defined according to $\psi\equiv\displaystyle\sqrt{\frac{\gamma}{2}}\frac{\Delta v}{c_{\mathrm{s}}},$ (12) where $\Delta\textbf{v}=\textbf{v}_{\mathrm{d}}-\textbf{v}_{\mathrm{g}}=\Delta v$ x is the differential velocity (x being a unit vector). However, depending on the characteristics of the problem (i.e. low or high Mach numbers, or both), simpler and computationally less expensive approximations may be used. For $\psi\ll 1$, i.e. low Mach numbers, Eq. 11 can be expanded to third order in $\psi$, giving $\mathbf{F}_{\rm{drag}}=\displaystyle-\frac{4\pi}{3}\rho_{\mathrm{g}}s^{2}\sqrt{\frac{8}{\pi\gamma}}c_{\mathrm{s}}\Delta v\left[1+\frac{\psi^{2}}{5}+\mathcal{O}\left(\psi^{4}\right)\right]\textbf{x},$ (13) which is usually simplified to its linear term, $\mathbf{F}_{\rm{drag}}=-\frac{4\pi}{3}\rho_{\mathrm{g}}s^{2}\sqrt{\frac{8}{\pi\gamma}}c_{\mathrm{s}}\Delta\mathbf{v}.$ (14) For $\psi\gg 1$, i.e. high Mach numbers, the Taylor expansion in $1/\psi$ of Eq. 11 gives $\mathbf{F}_{\rm{drag}}=\displaystyle-\left[\pi\rho_{\mathrm{g}}s^{2}\Delta v^{2}\left(1+\frac{1}{\psi^{2}}-\frac{1}{4\psi^{4}}\right)+\mathcal{O}\left(e^{-\psi^{2}}\right)\right]\textbf{x},$ (15) which is usually reduced to its quadratic term, $\mathbf{F}_{\rm{drag}}=-\pi\rho_{\mathrm{g}}s^{2}\Delta v\Delta\mathbf{v}.$ (16) A convenient way to handle Epstein drag at both low and high Mach numbers is to use an interpolation between the two asymptotic regimes given by Eqs. 14 and 16 as derived in Kwok (1975) (cf. Paardekooper & Mellema 2006), giving $\mathbf{F}_{\rm{drag}}=-\frac{4\pi}{3}\rho_{\mathrm{g}}s^{2}\sqrt{\frac{8}{\pi\gamma}}c_{\mathrm{s}}\sqrt{1+\frac{9\pi}{128}\frac{\Delta v^{2}}{c_{\rm{s}}^{2}}}\Delta\mathbf{v}.$ (17) The deviation of Eq. 17 from the full expression (Eq. 11) is $\lesssim 1\%$ (Kwok, 1975). Thus, in general, we adopt Eq. 17 for the Epstein regime. We compare the differences between the various Epstein expressions in Sec. 5. #### 2.2.2 Stokes regime for dense media In a dense medium ($\lambda_{\mathrm{g}}>{4s}/{9}$), grains should be treated with the Stokes drag regime, for which the expression of the drag force $\bf{F}_{\rm{drag}}$ is: $\mathbf{F}_{\rm{drag}}=-\frac{1}{2}C_{\mathrm{D}}\pi s^{2}\rho_{\mathrm{g}}\Delta v\Delta\textbf{v},$ (18) where the coefficient $C_{\mathrm{D}}$ is a piecewise function of the local Reynolds number: $C_{\mathrm{D}}=\begin{cases}24R_{\mathrm{d}}^{-1},&R_{\mathrm{d}}<1;\\\ 24R_{\mathrm{d}}^{-0.6},&1<R_{\mathrm{d}}<800;\\\ 0.44,&800<R_{\mathrm{d}},\end{cases}$ (19) where $R_{\mathrm{d}}$ is defined in Eq. 10. Equation 19 indicates that at small Reynolds numbers ($R_{\mathrm{d}}<1$), the drag force is linear with respect to the local differential velocity between the grain and the gas. The relation transitions to a power-law regime ($\mathbf{F}_{\rm{drag}}\propto\Delta v^{0.4}\Delta\textbf{v}$) at intermediate Reynolds numbers ($1<R_{\mathrm{d}}<800$) and becomes quadratic at large Reynolds numbers ($R_{\mathrm{d}}>800$). When the local concentration of dust grains becomes very large (i.e., average distance between the particles comparable to the grain size), the coefficient $C_{\mathrm{D}}$ should also depend on the local concentration of particles. However, this extreme situation is not encountered in astrophysical situations. Assuming gas molecules interact as hard spheres, the dynamic viscosity of the gas can be computed according to (Chapman & Cowling, 1970): $\mu=\frac{5m}{64\sigma_{\mathrm{s}}}\sqrt{\frac{\pi}{\gamma}}c_{\mathrm{s}},$ (20) where $m=2m_{\mathrm{H}}$ and $\sigma_{\mathrm{s}}$ is the geometric cross section of the molecule ($\sigma_{\mathrm{s}}=2.367\times 10^{-15}$ cm2 for H2). The gas mean free path $\lambda_{\mathrm{g}}$ and the kinematic viscosity $\nu$ of the gas are deduced from $\mu$ using $\lambda_{\mathrm{g}}=\displaystyle\sqrt{\frac{\pi\gamma}{2}}\frac{\mu}{\rho_{\mathrm{g}}c_{\mathrm{s}}},$ (21) and $\nu=\displaystyle\frac{\mu}{\rho_{\mathrm{g}}}.$ (22) ## 3 Asytrophysical dust and gas mixtures in SPH ### 3.1 SPH evolution equations The SPH version of the continuity equations Eqs. 1 – 2 are given by the density summations for both the gas and the dust phase, computed according to: $\displaystyle\hat{\rho}_{a}=\sum_{b}m_{b}W_{ab}(h_{a});$ $\displaystyle h_{a}=\eta\left(\frac{m_{a}}{\hat{\rho}_{a}}\right)^{1/\nu},$ (23) $\displaystyle\hat{\rho}_{i}=\sum_{j}m_{j}W_{ij}(h_{i});$ $\displaystyle h_{i}=\eta\left(\frac{m_{j}}{\hat{\rho}_{i}}\right)^{1/\nu},$ (24) where as in Paper I, the indices $a,b,c$ refer to quantities computed on gas particles and $i,j,k$ refer to quantities computed on dust particles. The volume filling fraction $\theta$, is defined on a _gas_ particle, $a$, according to $\theta_{a}=1-\frac{\hat{\rho}_{{\rm d},a}}{\rho_{\rm d}},$ (25) where $\hat{\rho}_{{\rm d},a}$ is the density of _dust_ at the _gas_ particle location, calculated using $\hat{\rho}_{{\rm d},a}=\sum^{N_{neigh,dust}}_{j=1}m_{j}W_{aj}(h_{a}),$ (26) where $h_{a}$ is the smoothing length of the _gas_ particle computed using gas neighbours. The local density of dust at the gas location can thus be zero (giving $\theta=1$) if no dust particles are found within the kernel radius computed with the gas smoothing length. Importantly, as $\hat{\rho}$ and $h$ are mutually dependent, they have to be simultaneously calculated for each type of particle, e.g. by the iterative procedure described in Price & Monaghan (2007). The SPH equations of motion for the gas and the dust particles, corresponding to the SPH translation of Eqs. 3 and 4, are given by $\displaystyle\frac{\mathrm{d}\textbf{v}_{a}}{\mathrm{d}t}=$ $\displaystyle-\sum_{b}m_{b}\left[\frac{P_{a}\tilde{\theta}_{a}}{\Omega_{a}\hat{\rho}_{a}^{2}}\nabla_{\\!\\!\ a}W_{\\!\\!\ a\\!\\!\ b}\left(h_{a}\right)+\frac{P_{b}\tilde{\theta}_{b}}{\Omega_{b}\hat{\rho}_{b}^{2}}\nabla_{\\!\\!\ a}W_{\\!\\!\ a\\!\\!\ b}\left(h_{b}\right)\right]$ $\displaystyle-\sum_{j}m_{j}\frac{P_{a}\left(1-\theta_{a}\right)}{\hat{\rho}_{a}\hat{\rho}_{{\rm d},a}}\nabla_{\\!\\!\ a}W_{\\!\\!\ a\\!\\!\ j}\left(h_{a}\right)$ $\displaystyle+\nu\sum_{j}m_{j}\frac{K_{aj}}{\hat{\rho}_{a}\hat{\rho}_{j}}\left({\bf v}_{aj}\cdot\hat{\textbf{r}}_{aj}\right)\hat{\textbf{r}}_{aj}D_{aj}(h_{a}),$ (27) for an SPH gas particle and $\displaystyle\frac{\mathrm{d}{\bf v}_{i}}{\mathrm{d}t}=$ $\displaystyle\sum_{b}m_{b}\frac{P_{b}\left(1-\theta_{b}\right)}{\hat{\rho}_{b}\hat{\rho}_{{\mathrm{d},b}}}\nabla_{\\!\\!\ i}W_{\\!\\!\ b\\!\\!\ i}\left(h_{b}\right)$ (28) $\displaystyle-\nu\sum_{b}m_{b}\frac{K_{bi}}{\hat{\rho}_{b}\hat{\rho}_{i}}\left({\bf v}_{bi}\cdot\hat{\textbf{r}}_{bi}\right)\hat{\textbf{r}}_{bi}D_{ib}(h_{i}),$ for an SPH dust particle. $\Omega$ is the usual variable smoothing length term $\Omega_{b}\equiv 1-\frac{\partial h_{b}}{\partial\hat{\rho}_{b}}\sum_{c}m_{c}\frac{\partial W_{\\!\\!\ b\\!\\!\ c}\left(h_{b}\right)}{\partial h_{b}}.$ (29) It should be noted that $\Omega_{\rm{d}}$ is computed only using dust particle neighbours according to: $\Omega_{\mathrm{d},b}=1-\frac{\partial h_{b}}{\partial\hat{\rho}_{{\mathrm{d},b}}}\sum_{j}m_{j}\frac{\partial W_{\\!\\!\ b\\!\\!\ j}\left(h_{b}\right)}{\partial h_{b}}.$ (30) $\tilde{\theta}$ is defined according to $\tilde{\theta}\equiv\theta+\frac{\hat{\rho}_{\rm g}}{\hat{\rho}_{\mathrm{d}}}(1-\theta)(1-\Omega_{\rm d}).$ (31) At this stage, no assumptions are made with respect to the functional form of the drag coefficient $K$. The evolution of the internal energy for an SPH gas particle is given by $\displaystyle\frac{{\rm d}u_{a}}{{\rm d}t}$ $\displaystyle=\frac{\tilde{\theta}_{a}P_{a}}{\Omega_{a}\hat{\rho}_{a}^{2}}\sum_{b}m_{b}\left(\textbf{v}_{a}-\textbf{v}_{b}\right)\cdot\nabla_{a}W_{ab}(h_{a})$ (32) $\displaystyle+\frac{(1-\theta_{a})P_{a}}{\hat{\rho}_{a}\hat{\rho}_{{\rm d},a}}\sum^{N_{neigh,dust}}_{j=1}m_{j}\left({\bf v}_{a}-{\bf v}_{j}\right)\cdot\nabla_{a}W_{aj}(h_{a})$ $\displaystyle+\nu\sum_{k}m_{k}\frac{K_{ak}}{\hat{\rho}_{a}\hat{\rho}_{k}}\left(\textbf{v}_{ak}\cdot\hat{\textbf{r}}_{ak}\right)^{2}D_{ak}(h_{a}).$ In Paper I, we showed that the total linear and angular momentum as well as the total energy are exactly conserved. Thermal coupling terms have been neglected in this paper. ### 3.2 Kernel functions Two different kernels are employed to perform the SPH interpolations. First, a standard bell-shaped kernel $W$: $W\left(r,h\right)=\frac{\sigma}{h^{\nu}}f\left(q\right),$ (33) where $h$ denotes the smoothing lengths of each phases, $\nu$ the number of spatial dimensions and $q\equiv\|{\bf r}-{\bf r}^{\prime}\|/h$ is the dimensionless variable used to calculate the densities and the buoyancy terms. The function $f$ is usually the $M_{4}$ cubic spline kernel (Monaghan, 2005). The drag interpolation is performed using a second kernel $D$. As shown in Paper I, double-hump shaped kernels given by $D\left(r,h\right)=\frac{\tilde{\sigma}}{h^{\nu}}q^{2}f(q),$ (34) significantly improve the accuracy of the drag interpolation — for the same computational cost — compared to bell-shaped kernels. The normalisation constants $\tilde{\sigma}$ for various double hump kernels are given in Paper I. We adopt the double hump cubic for the drag terms in this paper. ### 3.3 Astrophysical drag regimes in SPH #### 3.3.1 Gas viscosity and mean free path The drag coefficients $K_{ak}$ involved in Eqs. 3.1 – 28 and 32 are computed independently for each pair of any gas particle $a$ and dust particle $k$. We first use the sound speed $c_{\rm{s},a}$ to estimate the viscosity $\mu_{a}$ on the gas particle $a$ using (see Eq. 20) $\mu_{a}=\frac{5m}{64\sigma_{\mathrm{s}}}\sqrt{\frac{\pi}{\gamma}}c_{\rm{s},a}.$ (35) The mean free path is then computed according to Eq. 21, giving $\lambda_{\mathrm{g},a}=\displaystyle\sqrt{\frac{\pi\gamma}{2}}\frac{\mu_{\rm{a}}}{\hat{\rho}_{a}c_{\rm{s},a}}.$ (36) Finally, $\lambda_{\mathrm{g},a}$ is compared to the quantity $4s_{k}/9$ — $s_{k}$ being the grain size of the dust particle — to determine whether the drag coefficient of the SPH pair $K_{ak}$ is calculated using the Epstein or the Stokes drag regimes. #### 3.3.2 Epstein regime If $4s_{k}/9\leq\lambda_{\mathrm{g},a}$, the drag coefficient $K_{ak}$ is calculated using the Epstein prescription. Introducing the SPH quantity $\psi_{ak}$ calculated on a pair of gas and dust SPH particles and defined by $\psi_{ak}\equiv\sqrt{\frac{\gamma}{2}}\frac{\|{\bf v}_{ak}\|}{c_{\mathrm{s},a}},$ (37) Eq. 11 can be straightforwardly translated to get the drag coefficient $K_{ak}$ involved in the SPH drag force $\displaystyle K_{ak}=\displaystyle-\sqrt{\pi}s^{2}\rho_{\mathrm{g}}\frac{\hat{\rho}_{\mathrm{d}}}{m_{\rm{d}}}\|\textbf{v}_{ak}\|$ $\displaystyle\left[\frac{1}{2\sqrt{\pi}}\left\\{\left(\frac{1}{\psi_{ak}}+\frac{1}{2\psi_{ak}^{3}}\right)e^{-\psi_{ak}^{2}}+\right.\right.$ (38) $\displaystyle\left.\left.\left(1+\frac{1}{\psi_{ak}^{2}}-\frac{1}{4\psi_{ak}^{4}}\right)\sqrt{\pi}\,\mathrm{erf}\left(\psi_{ak}\right)\right\\}\right]\textbf{x},$ where $s$ is the grain radius, $m_{\rm{d}}$ is the grain mass and $\gamma$ is the adiabatic index. Eq. 38 is computationally expensive as it involves exponential and error functions. The SPH equivalent of Eq.17 is given by $K_{ak}=\frac{4}{3}\pi\sqrt{\frac{8}{\pi\gamma}}\frac{\hat{\rho}_{k}}{m_{\rm{d}}}\frac{\hat{\rho}_{a}}{\theta_{a}}s^{2}c_{\mathrm{s},a}\sqrt{1+\frac{9\pi}{128}\frac{v_{ak}^{2}}{c_{\mathrm{s},a}^{2}}}.$ (39) Both Eqs. 38 and 39 reduce to the linear Epstein regime at low Mach numbers (equivalent of Eq. 14) for which the coefficient $K_{ak}$ is $K_{ak}=\frac{4}{3}\pi\sqrt{\frac{8}{\pi\gamma}}\frac{\hat{\rho}_{k}}{m_{\rm{d}}}\frac{\hat{\rho}_{a}}{\theta_{a}}s^{2}c_{\mathrm{s},a},$ (40) and to the quadratic drag regime at high Mach numbers (equivalent of Eq. 16), for which the coefficient $K_{ak}$ is $K_{ak}=\pi\rho_{\mathrm{g}}s^{2}\frac{\hat{\rho}_{\mathrm{d}}}{m_{\rm{d}}}\|{\bf v}_{ak}\|.$ (41) #### 3.3.3 Stokes regime If $4s_{k}/9>\lambda_{\mathrm{g},a}$, the drag coefficient $K_{ak}$ is calculated using the Stokes prescription (see Eqs. 18–19). The local Reynolds number $R_{\mathrm{d},ak}$ is computed for each pair of gas and dust particles using $R_{\mathrm{d},ak}\equiv\frac{2s\hat{\rho}_{a}\left\|\textbf{v}_{ak}\right\|}{\mu_{a}\theta_{a}},$ (42) such that the drag coefficient $K_{ak}$ can be computed according to $K_{ak}=\begin{cases}\displaystyle 6\pi\frac{\hat{\rho}_{k}}{m_{\rm{d}}}\mu_{a}s&R_{\mathrm{d},ak}<1,\\\ \displaystyle\frac{12\pi}{2^{0.6}}\frac{\hat{\rho}_{k}}{m_{\rm{d}}}\frac{\mu_{a}^{0.6}}{\theta_{a}^{0.4}\hat{\rho}_{a}^{0.6}}s^{1.4}\left\|\textbf{v}_{ak}\right\|^{0.4}&1<R_{\mathrm{d},ak}<800,\\\ \displaystyle 0.22\pi\frac{\hat{\rho}_{k}}{m_{\rm{d}}}\frac{\hat{\rho}_{a}}{\theta_{a}}s^{2}\left\|\textbf{v}_{ak}\right\|&R_{\mathrm{d},ak}>800.\\\ \end{cases}$ (43) These expressions have been used by Ayliffe et al. (2011) to compute the drag on planetesimals in a protoplanetary disc. ## 4 Timesteping ### 4.1 Explicit timesteping The simplest method to evolve the evolution equations for the SPH particles is to use an explicit integrator (e.g. the standard Leapfrog). The stability of the system is guaranteed provided the timestep remains smaller than a critical value $\Delta t_{\rm{c}}$. In Paper I, we performed a Von Neumann analysis of the continuous equations, deriving the explicit timestepping criterion $\Delta t_{\rm{c},a}=\min_{k}\left[\frac{\hat{\rho}_{a}\hat{\rho}_{k}}{K_{ak}(\hat{\rho}_{a}+\hat{\rho}_{k})}\right];\hskip 14.22636pt\Delta t_{\rm{c},i}=\min_{b}\left[\frac{\hat{\rho}_{b}\hat{\rho}_{i}}{K_{bi}(\hat{\rho}_{b}+\hat{\rho}_{i})}\right];$ (44) for gas and dust particles, respectively, with the minimum being taken over all the particle’s neighbours. Although this criterion was derived in Paper I for linear drag regimes only, it remains valid even for non-linear drag regimes where the drag coefficients depend on the differential velocity between the particles, i.e. $K_{ak}=K_{ak}\left(\|\mathbf{v}_{ak}\|\right)$. ### 4.2 Implicit timestepping When the drag timescale becomes smaller than other time scales in the system (e.g. the Courant condition or the orbital timescale), the timestep restriction of the explicit methods may become prohibitive and implicit methods are required. Monaghan (1997) considered the application of two implicit schemes (the first-order Backward-Euler and second-order Tischer scheme) to SPH dust-gas mixtures. Both schemes are unconditionally stable, but a higher accuracy is achieved with second-order schemes. #### 4.2.1 Backward-Euler method The Backward-Euler scheme applied to the drag interaction between SPH dust and gas particles is given by $\displaystyle\frac{\mathbf{v}^{n+1}_{a}-\mathbf{v}^{n}_{a}}{\Delta t}$ $\displaystyle=$ $\displaystyle-\nu\sum_{k}m_{k}\frac{K_{ak}^{n+1}}{\hat{\rho}_{a}\hat{\rho}_{k}}\left(\mathbf{v}^{n+1}_{ak}\cdot\hat{\mathbf{r}}_{ak}\right)\hat{\mathbf{r}}_{ak}D_{ak},$ (45) $\displaystyle\frac{\mathbf{v}^{n+1}_{i}-\mathbf{v}^{n}_{i}}{\Delta t}$ $\displaystyle=$ $\displaystyle+\nu\sum_{b}m_{b}\frac{K_{bi}^{n+1}}{\hat{\rho}_{b}\hat{\rho}_{i}}\left(\mathbf{v}^{n+1}_{bi}\cdot\hat{\mathbf{r}}_{bi}\right)\hat{\mathbf{r}}_{bi}D_{bi}.$ (46) Although the scheme is unconditionally stable, the implicit equation with respect to the velocities $\textbf{v}^{n+1}$ must be solved at each time step. Direct numerical inversion of this linear system would be prohibitive given the typical number of neighbour interactions for each SPH particle. Thus, approximate or iterative solutions to Eqs. 45 – 46 are required. #### 4.2.2 Monaghan (1997) scheme Monaghan (1997) suggested approximating the velocities $\textbf{v}^{n+1}$ of Eqs. 45 – 46 using a pairwise treatment in order to preserve the exact conservation of linear and angular momentum in the SPH formalism. Considering the interaction between the SPH gas particle $a$ the dust particle $i$, Monaghan (1997) introduced pairwise auxiliary velocities $\tilde{\bf{v}}$ defined by: $\displaystyle\tilde{\textbf{v}}_{a}$ $\displaystyle=$ $\displaystyle{\bf v}_{a}^{n}-m_{i}~{}\Delta t\frac{\nu K_{ai}D_{ai}}{\hat{\rho}_{a}\hat{\rho}_{i}}\left(\tilde{\textbf{v}}_{ai}\cdot\hat{\bf{r}}_{ai}\right)\hat{\bf{r}}_{ai},$ (47) $\displaystyle\tilde{\textbf{v}}_{i}$ $\displaystyle=$ $\displaystyle{\bf v}_{i}^{n}+m_{a}~{}\Delta t\frac{\nu K_{ai}D_{ai}}{\hat{\rho}_{a}\hat{\rho}_{i}}\left(\tilde{\textbf{v}}_{ai}\cdot\hat{\bf{r}}_{ai}\right)\hat{\bf{r}}_{ai},$ (48) Eqs. 47 and 48 are solved, for a given pair of particles, by taking the scalar product by $\hat{\bf{r}}_{ai}$ of the difference of the two equations, giving $\tilde{\textbf{v}}_{ai}\cdot\hat{\bf{r}}_{ai}=\frac{\textbf{v}_{ai}^{n}\cdot\hat{\bf{r}}_{ai}}{1+\Delta t\frac{\nu K_{ai}D_{ai}}{\hat{\rho}_{a}\hat{\rho}_{i}}\left(m_{a}+m_{i}\right)}.$ (49) Substituting this expression into Eq. 47 and 48 gives expressions for $\tilde{\textbf{v}}_{a}$ and $\tilde{\textbf{v}}_{i}$. Iterating this pairwise process by looping over all the SPH particles provides an approximate solution for the velocities $\bf{v}^{n+1}$, i.e. $\displaystyle\frac{\mathbf{v}^{n+1}_{a}-\mathbf{v}^{n}_{a}}{\Delta t}$ $\displaystyle\simeq$ $\displaystyle-\nu\sum_{k}m_{k}\frac{K_{ak}}{\hat{\rho}_{a}\hat{\rho}_{k}}\left(\tilde{\mathbf{v}}_{ak}\cdot\hat{\mathbf{r}}_{ak}\right)\hat{\mathbf{r}}_{ak}D_{ak},$ (50) $\displaystyle\frac{\mathbf{v}^{n+1}_{i}-\mathbf{v}^{n}_{i}}{\Delta t}$ $\displaystyle\simeq$ $\displaystyle+\nu\sum_{b}m_{b}\frac{K_{bi}}{\hat{\rho}_{b}\hat{\rho}_{i}}\left(\tilde{\mathbf{v}}_{bi}\cdot\hat{\mathbf{r}}_{bi}\right)\hat{\mathbf{r}}_{bi}D_{bi}.$ (51) The main drawback of this method is that the approximation given by Eqs. 50 and 51 is inexact – that is, it provides only an approximate solution to Eqs. 45 and 46. Furthermore the accuracy of the approximation is not known _a priori_ and there is no possibility of performing repeated sweeps in order to converge to a more accurate solution. In practice, we find that the velocities obtained by this scheme (for example on the dustybox test) can be significantly in error, with no possibility of improving the convergence (for example, by doing several iterations/sweeps). #### 4.2.3 Alternative pairwise treatment for linear drag regimes We propose a more consistent method for solving Eqs. 45–46 on a given gas or dust particle ($a$ and $i$, respectively) by sweeping over all particle pairs and updating the velocities iteratively according to $\displaystyle{\textbf{v}}_{a}^{*}$ $\displaystyle=$ $\displaystyle{\bf v}_{a}^{n}+\Delta t\textbf{F}_{a,\mathrm{drag}}^{}$ (52) $\displaystyle-$ $\displaystyle m_{i}~{}\Delta t\frac{\nu K_{ai}D_{ai}}{\hat{\rho}_{a}\hat{\rho}_{i}}\left[\left({\textbf{v}}_{ai}^{*}-\textbf{v}_{ai}^{}\right)\cdot\hat{\bf{r}}_{ai}\right]\hat{\bf{r}}_{ai},$ $\displaystyle{\textbf{v}}_{i}^{*}$ $\displaystyle=$ $\displaystyle{\bf v}_{i}^{n}+\Delta t\textbf{F}_{i,\mathrm{drag}}^{}$ (53) $\displaystyle+$ $\displaystyle m_{a}~{}\Delta t\frac{\nu K_{ai}D_{ai}}{\hat{\rho}_{a}\hat{\rho}_{i}}\left[\left({\textbf{v}}_{ai}^{*}-\textbf{v}_{ai}^{}\right)\cdot\hat{\bf{r}}_{ai}\right]\hat{\bf{r}}_{ai},$ where ${\bf v}^{*}$ refers to the improved approximation to ${\bf v}^{n+1}$ obtained after updating each pair and ${\bf v}^{}$ to the previous iteration value of ${\bf v}^{}$. Eqs. 52 and 53 are solved for each pair of particles by taking the dot product of $\hat{\bf{r}}_{ai}$ with the difference of the two equations, giving $\left({\textbf{v}}_{ai}\cdot\hat{\bf{r}}_{ai}\right)^{}=\frac{\left(\textbf{v}_{ai}^{n}+\Delta t\textbf{F}_{ai,\mathrm{drag}}^{}+\Delta t\frac{\nu K_{ai}D_{ai}}{\hat{\rho}_{a}\hat{\rho}_{i}}\left(m_{a}+m_{i}\right)\textbf{v}_{ai}^{}\right)\cdot\hat{\bf{r}}_{ai}}{1+\Delta t\frac{\nu K_{ai}D_{ai}}{\hat{\rho}_{a}\hat{\rho}_{i}}\left(m_{a}+m_{i}\right)}.$ (54) Substituting Eq. 54 in Eqs. 52 and 53 gives the updated velocities for the pair. Note that during the global sweep over particle pairs ${\bf v}^{}$ begins as ${\bf v}^{n}$ at the first iteration but is updated as soon as new values become available. This pairwise correction ensures that i) both the linear and the angular momentum are exactly conserved and ii) the velocities converge to the correct solution of the implicit scheme given by Eqs. 45 and 46, since the last term of Eqs. 52 and 53 tends to zero as the number of iterations increases. We thus refine our approximation to the solution by performing as many successive iterations as are required to reach a suitable convergence criterion. #### 4.2.4 Convergence criterion We consider that the approximation we obtain from the implicit scheme described above is accurate enough when $\frac{\|\bf{v}^{k+1}-\bf{v}^{k}\|}{\min c_{\rm{s}}}<\varepsilon,$ (55) is satisfied for each particle. Typically, we adopt $\varepsilon=10^{-4}$, which ensures that the approximation we make on the time stepping is negligible compared to the $\mathcal{O}(h^{2})$ truncation error of the underlying SPH scheme. #### 4.2.5 Implementation into Leapfrog The Leapfrog scheme is well suited to the evolution of particle methods because, for position-dependant forces, it preserves geometric properties of particle orbits and requires only one evaluation per timestep to give second order accuracy. In the standard formulation, the evolution is computed according to $\begin{array}[]{lrrcll}{\rm Kick}&\left[\right.&{\bf v}^{1/2}&=&{\bf v}^{0}+\frac{\Delta t}{2}{\bf f}^{0}\left({\bf x}^{0},{\bf v}^{0}\right),&\left.\right]\\\\[5.0pt] {\rm Drift}&\left[\right.&{\bf x}^{1}&=&{\bf x}^{0}+\Delta t{\bf v}^{1/2},&\left.\right]\\\\[5.0pt] {\rm Kick}&\left[\right.&{\bf v}^{1}&=&{\bf v}^{1/2}+\frac{\Delta t}{2}{\bf f}^{1}\left({\bf x}^{1},{\bf v}^{1}\right),&\left.\right]\end{array}$ (56) corresponding to Kick, Drift and Kick steps respectively. Adapting Leapfrog to deal with velocity dependent forces (e.g. drag) is a priori more difficult since for velocity-dependent forces, the last Kick is implicit in ${\bf v}^{1}$. For our present purposes, this does not present a major problem since the drag is already computed implicitly. Splitting the forces into position- dependent (${\bf f}_{\rm SPH}$) and drag (${\bf f}_{\rm drag}$) contributions, the scheme becomes $\begin{array}[]{lrrcll}{\rm Kick}&\left[\right.&{\bf\tilde{v}}^{1/2}&=&{\bf v}^{0}+\frac{\Delta t}{2}{\bf f}^{0}_{\rm SPH}\left({\bf x}^{0}\right),&\left.\right]\\\\[5.0pt] {\rm Drift}&\left[\right.&{\bf x}^{1/2}&=&{\bf x}^{0}+\frac{\Delta t}{2}\tilde{{\bf v}}^{1/2},&\left.\right]\\\\[5.0pt] {\rm Drag}&\left[\right.&{\bf v}^{1/2}&=&{\bf\tilde{v}}^{1/2}+\frac{\Delta t}{2}{\bf f}^{1/2}_{\rm drag}\left({\bf x}^{1/2},{\bf v}^{1/2}\right)&\left.\right]\\\\[5.0pt] {\rm Drift}&\left[\right.&{\bf x}^{1}&=&{\bf x}^{0}+\Delta t{\bf v}^{1/2},&\left.\right]\\\\[5.0pt] {\rm Kick}&\left[\right.&{\bf\tilde{v}}^{1}&=&{\bf v}^{1/2}+\frac{\Delta t}{2}{\bf f}^{1}_{\rm SPH}\left({\bf x}^{1}\right),&\left.\right]\\\\[5.0pt] {\rm Drag}&\left[\right.&{\bf v}^{1}&=&{\bf\tilde{v}}^{1}+\frac{\Delta t}{2}{\bf f}^{1}_{\rm drag}\left({\bf x}^{1},{\bf v}^{1}\right)&\left.\right]\end{array}$ (57) where the Drag steps represent the implicit updates computed as described in Sec. 4.2.3. The disadvantage of Eq. 57 is that two drag force evaluations are required, removing one of the advantages of the Leapfrog integrator. Inspection of 57 reveals that an alternative version that requires only one Drag step can be constructed according to $\begin{array}[]{lrlr}{\rm Kick}&\bigl{[}&\begin{array}[]{rcl}{\bf v}^{1/2}&=&{\bf v}^{0}+\frac{\Delta t_{0}}{2}{\bf\tilde{f}},\end{array}&\bigr{]}\\\\[15.00002pt] {\rm Drift}&\bigl{[}&\begin{array}[]{rcl}{\bf x}^{1}&=&{\bf x}^{0}+\Delta t_{0}{\bf v}^{1/2},\end{array}&\bigr{]}\\\\[10.00002pt] {\rm Drag}&&\left\\{\begin{array}[]{rcl}{\tilde{\bf v}}^{3/2}&=&{\bf v}^{1/2}+\frac{\Delta t_{0}+\Delta t_{1}}{2}{\bf f}^{1}_{\rm SPH}\left({\bf x}^{1}\right),\\\\[5.0pt] {\bf v}^{3/2}&=&{\tilde{\bf v}}^{3/2}+\frac{\Delta t_{0}+\Delta t_{1}}{2}{\bf f}^{1}_{\rm drag}\left({\bf x}^{1},{\bf v}^{3/2}\right),\\\\[5.0pt] {\tilde{\bf f}}&=&2\left({\tilde{\bf v}}^{3/2}-{\tilde{\bf v}}^{1/2}\right)/\left(\Delta t_{0}+\Delta t_{1}\right),\end{array}\right.&\\\\[30.00005pt] {\rm Kick}&\bigl{[}&\begin{array}[]{rcl}{\bf v}^{1}&=&{\bf v}^{1/2}+\frac{\Delta t_{0}}{2}{\bf\tilde{f}}.\end{array}&\bigr{]}\end{array}$ (58) where we have combined the drag steps by predicting the velocity ${\bf v}^{3/2}$. Note that strictly, the Drag step in this method is semi-implicit since the force is evaluated using ${\bf x}^{1}$ rather than ${\bf x}^{3/2}$. However, we expect this approximation to be reasonable as at high drag (for which the implicit method is designed), the drag mainly changes the differential velocity between the fluids and has less of an effect on the positions. Care is also required when the timestep changes between the steps. We have indicated the correct procedure by specifying $\Delta t_{0}$ and $\Delta t_{1}$ where $\Delta t_{1}$ is the timestep computed based on ${\bf x}^{1}$. Finally, Eq. 58 requires that ${\tilde{\bf f}}$ is known at the beginning of the integration. This can be easily achieved by performing the Drag step in Eq. 58 with ${\bf v}^{1/2}={\bf v}^{0}$, $\Delta t_{0}=0$ and $\Delta t_{1}$ equal to the timestep calculated using the initial particle positions. #### 4.2.6 Generalisation to non-linear drag regimes To extend this alternative pairwise treatment to any non-linear drag regime, two additional points have to be considered. Firstly, although in principle six quantities ($v_{x,y,z}$ for each particle) have to be determined for each pair, this can be reduced to a single unknown quantity since the drag coefficient depends only on the modulus of the differential velocity and the exchange of momentum is directed along the line of sight joining the particles. The system of equations for a single pair thus reduces to $\displaystyle{\textbf{v}}_{a}^{}$ $\displaystyle=$ $\displaystyle{\bf v}_{a}^{n}+\Delta t\textbf{F}_{a,\mathrm{drag}}^{}$ (59) $\displaystyle-$ $\displaystyle m_{i}~{}\Delta t\frac{\nu D_{ai}}{\hat{\rho}_{a}\hat{\rho}_{i}}K_{ai}\sqrt{\left[\left({\textbf{v}}_{ai}^{*}-\textbf{v}_{ai}^{}\right)\cdot\hat{\bf{r}}_{ai}\right]^{2}+V^{2,n}_{\rm orth}},$ $\displaystyle{\textbf{v}}_{i}^{*}$ $\displaystyle=$ $\displaystyle{\bf v}_{i}^{n}+\Delta t\textbf{F}_{i,\mathrm{drag}}^{}$ (60) $\displaystyle+$ $\displaystyle m_{a}~{}\Delta t\frac{\nu D_{ai}}{\hat{\rho}_{a}\hat{\rho}_{i}}K_{ai}\sqrt{\left[\left({\textbf{v}}_{ai}^{*}-\textbf{v}_{ai}^{}\right)\cdot\hat{\bf{r}}_{ai}\right]^{2}+V^{2,n}_{\rm orth}},$ where $V^{2,n}_{\rm orth}={\textbf{v}}_{ai}^{n}\cdot{\textbf{v}}_{ai}^{n}-\left({\textbf{v}}_{ai}^{n}\cdot\hat{\bf{r}}_{ai}\right)^{2}.$ (61) Secondly, taking the dot product of $\hat{\bf{r}}_{ai}$ with the difference of the two equations Eqns. 59 and 60 does not lead in general to an equation which can be solved analytically. The values of ${\textbf{v}}_{ai}\cdot\hat{\bf{r}}_{ai}$ must therefore be determined using a numerical rootfinding procedure (we use a Newton-Raphson scheme) before being substituted in Eqns. 59 and 60 to determine the velocities for both the gas and the dust particles. #### 4.2.7 Performance of the implicit scheme The computational cost of a timestep with the implicit pairwise treatment is more expensive than an explicit timestep since at least two iterations have to be performed to ensure that the scheme is converged. However, the implicit pairwise treatment will be more efficient provided that the number of iterations is much smaller than the number of explicit timesteps that would otherwise be required. We find in practice that the efficiency of the pairwise treatment is mainly determined by the number of iterations required to satisfy Eq. 55 (this aspect was not addressed in the Monaghan (1997) scheme where only one iteration is ever taken in the hope that the approximation is sufficiently accurate). The rapidity of the convergence depends primarily on the ratio $r=\Delta t/t_{\rm s}$ of the timestep over the drag stopping time (defined in Eq. (96) of Paper I) and on the value of $\varepsilon$. Empirically, we have found that, for $\varepsilon=10^{-4}$ and $1\lesssim r\lesssim 10$, the implicit pairwise treatment converges efficiently, the ratio $\|\bf{v}^{k+1}-\bf{v}^{k}\|/(\min c_{\rm{s}})$ decreasing by $\sim$ two orders of magnitude at each iterations. Thus, the implicit pairwise treatment improves the computational time by a factor of $\sim 1$–$10$. However, this rapidity of convergence decreases as $r$ increases. At very high drag ($r\gtrsim 1000$), we find that the implicit scheme becomes less efficient than explicit timestepping due to the large number of iterations required. A similar behaviour has been found using the Gauss-Seidel iterative scheme developed by Whitehouse et al. (2005) to treat SPH radiative transfer in the flux-limited diffusion approximation (Bate 2011, private communication), so this issue is not specific to the pairwise treatment. It is important to note that the computational gain obtained with the pairwise scheme does not solve the resolution issue at high drag extensively discussed in Paper I. Both of these problems suggest that a more efficient method for handling high drag regimes is required. Such a method is beyond the scope of the present paper. #### 4.2.8 Higher order implicit schemes Higher temporal accuracy may be achieved by using second instead of first order implicit schemes. The gain in accuracy is obtained by dividing the drag timestep $\Delta t$ into two half timesteps. Monaghan (1997) suggested the ‘Tischer’ scheme, where the two half timesteps are given by $\displaystyle\frac{\mathbf{v}^{n+\frac{1}{2}}_{a}-\mathbf{v}^{n}_{a}}{\Delta t/2}$ $\displaystyle=$ $\displaystyle-0.6\nu\sum_{k}m_{k}\frac{K_{ak}}{\hat{\rho}_{a}\hat{\rho}_{k}}\left(\mathbf{v}^{n+\frac{1}{2}}_{ak}\cdot\hat{\mathbf{r}}_{ak}\right)\hat{\mathbf{r}}_{ak}D_{ak},$ (62) $\displaystyle-0.4\nu\sum_{k}m_{k}\frac{K_{ak}}{\hat{\rho}_{a}\hat{\rho}_{k}}\left(\mathbf{v}^{n}_{ak}\cdot\hat{\mathbf{r}}_{ak}\right)\hat{\mathbf{r}}_{ak}D_{ak},$ $\displaystyle\frac{\mathbf{v}^{n+\frac{1}{2}}_{i}-\mathbf{v}^{n}_{i}}{\Delta t/2}$ $\displaystyle=$ $\displaystyle+0.6\nu\sum_{b}m_{b}\frac{K_{bi}}{\hat{\rho}_{b}\hat{\rho}_{i}}\left(\mathbf{v}^{n+\frac{1}{2}}_{bi}\cdot\hat{\mathbf{r}}_{bi}\right)\hat{\mathbf{r}}_{bi}D_{bi},$ (63) $\displaystyle+0.4\nu\sum_{b}m_{b}\frac{K_{bi}}{\hat{\rho}_{b}\hat{\rho}_{i}}\left(\mathbf{v}^{n}_{bi}\cdot\hat{\mathbf{r}}_{bi}\right)\hat{\mathbf{r}}_{bi}D_{bi},$ and then $\displaystyle\frac{\mathbf{v}^{n+1}_{a}-\left[1.4\mathbf{v}^{n+\frac{1}{2}}_{a}-0.4\mathbf{v}^{n}_{a}\right]}{\Delta t/2}$ $\displaystyle=$ $\displaystyle-0.6\nu\sum_{k}m_{k}\frac{K_{ak}}{\hat{\rho}_{a}\hat{\rho}_{k}}\left(\mathbf{v}^{n+1}_{ak}\cdot\hat{\mathbf{r}}_{ak}\right)\hat{\mathbf{r}}_{ak}D_{ak},$ $\displaystyle\frac{\mathbf{v}^{n+1}_{i}-\left[1.4\mathbf{v}^{n+\frac{1}{2}}_{i}-0.4\mathbf{v}^{n}_{i}\right]}{\Delta t/2}$ $\displaystyle=$ $\displaystyle+0.6\nu\sum_{b}m_{b}\frac{K_{bi}}{\hat{\rho}_{b}\hat{\rho}_{i}}\left(\mathbf{v}^{n+1}_{bi}\cdot\hat{\mathbf{r}}_{bi}\right)\hat{\mathbf{r}}_{bi}D_{bi}.$ The last terms of Eqs. 62–63 correspond to the explicit drag force involved in the Forward-Euler scheme. These quantities are computed form the velocities at the timestep $n$ at the beginning of the scheme, as described in Sec. 4.1. Then, the successive determination of $\bf{v}^{n+\frac{1}{2}}$ and $\bf{v}^{n}$ as given by Eqs. LABEL:eq:Tischer2_scheme1–LABEL:eq:Tischer2_scheme2 consists of two Backward- Euler steps with step size $\frac{\Delta t}{2}$. They are therefore computed using our alternative pairwise scheme, until the iterations for each half time step have converged. Eqs. 62 and 63 concerns the specific case of a linear drag regime, but this scheme can easily be extended to non-linear drag regimes as in Sec. 4.2.6. ## 5 Numerical tests ### 5.1 dustybox: Two fluid drag in a periodic box The dustybox problem presented by Laibe & Price (2011a) and described in detail in Paper I consists of two fluids in a periodic box moving with a differential velocity ($\Delta v_{0}=v_{d,0}-v_{g,0}$). This is the only test where analytic solutions are known for several functional forms corresponding to non-linear drag regimes (see Laibe & Price 2011a). These represent the functional forms of the Epstein and Stokes prescription. We thus use the dustybox problem to benchmark the accuracy of our algorithm for non-linear drag regimes using both explicit and implicit timestepping. #### 5.1.1 dustybox: setup We set up constant densities $\hat{\rho}_{\mathrm{g}}$ and $\hat{\rho}_{\mathrm{d}}$ and gas pressure $P_{\mathrm{g}}$ in a 3D periodic domain $x,y,z\in[0,1]$ filled by $20^{3}$ gas particles set up on a regular cubic lattice and $20^{3}$ dust particles shifted by half of the lattice spacing in each direction (as in Paper I, we verified that the results are independent of the offset of the dust lattice). The gas sound speed, the gas and the dust densities are set to unity in code units and no artificial viscosity is applied. The intrinsic dust volume is neglected by assuming $\theta=1$. Simulations have been performed using both the explicit timestepping presented in Paper I and the implicit pairwise timestepping described in Sec. 4. For the latter, we verified that both the total linear and angular momentum are exactly conserved as expected. #### 5.1.2 dustybox: different drag regimes Figure 1: Dust velocity (solid lines) as a function of time in the dustybox test, using $2\times 20^{3}$ particles, a dust-to-gas ratio of unity and five different linear and non-linear drag regimes — quadratic, power-law, linear, third order expansion and mixed, from top to bottom — compared to the exact solution for each case (long dashed/red lines). The initial velocities are set to $v_{\mathrm{d},i}=1$, $v_{\mathrm{g},i}=0$ and the time integration is performed using using the pairwise implicit treatment described in Sec. 4. The accuracy ($\lesssim 0.1\%$) of the SPH treatment for dust-gas mixtures is obtained by using the double-hump cubic kernel. Fig. 1 shows the results of the dustybox test for the five different regimes given in Table 1 of Laibe & Price (2011a): linear ($K=K_{0}$), quadratic ($K=K_{0}\|\Delta v\|$), power-law ($K=K_{0}\|\Delta v\|^{a}$, with $a=0.4$), third order expansion ($K=K_{0}[1+a_{3}\|\Delta v\|^{2}]$, with $a_{3}=0.5$) and mixed ($K=K_{0}\sqrt{1+a_{2}\|\Delta v\|^{2}}$, with $a_{2}=5$) where we have used $K_{0}=1$ in each case. The analytic solutions are reproduced within an accuracy comprised between $0.1\%$ and $1\%$ in every case — both linear and non-linear. The implicit scheme was find to converge quickly for this problem, requiring no more than two iterations at every stage of the evolution in each case. The efficiency of the damping for the dustybox problem decreases when the exponent of the drag regime increases, since $\|\|\Delta v\|\|<1$. On the contrary, additional non-linear terms give an additional contribution to the drag for the mixed and third order drag regimes, leading to a differential velocity that is more efficiently damped compared to the linear case. ### 5.2 dustywave: Sound waves in a dust-gas mixture The exact solution for linear waves propagating in a dust-gas mixture (dustywave) was derived in Laibe & Price (2011a) assuming a linear drag regime. Unfortunately, exact solutions prove difficult to obtain for the case of non-linear drag. Instead, we have verified simply that our simulations of the dustywave problem for non-linear drag regimes are converged in both space and time with an explicit timestepping scheme. We have then used these results to benchmark our simulations using implicit timestepping. We run the dustyywave problem for the same five non-linear drag regimes used above. Strictly speaking, It should be noted that assuming $K_{0}$ is constant (which we assume for this test problem) corresponds to Epstein and Stokes drag only to first order for the dustywave problem. #### 5.2.1 dustywave: Setup The dustywave test is performed in a 1D periodic box, placing equally spaced particles in the periodic domain $x\in[0,1]$ such that the gas and dust densities are unity in code units. We do not apply any form of viscosity and the gas sound speed is set to unity. To remain in the linear acoustic regime, the relative amplitude of the perturbation of both velocity and density are set to $10^{-4}$. #### 5.2.2 dustywave: Different drag regimes Figure 2: Solution of the 1D dustywave problem showing the SPH gas (solid black lines) and dust (dashed black) velocities using three different drag regimes: linear (top panel), power-law with an exponent of $0.4$ (center panel) and quadratic (bottom panel). The damping is strongly reduced for non- linear drag regimes compared to the linear case. Results are shown after 5 wave periods and the linear case (top panel) may be compared to the exact analytic solution (red lines). The dust to gas ratio and $K_{0}$ are set to 1 in code units. Fig. 2 shows the velocity profiles after 5 periods for three drag regimes — linear, power law and quadratic — in the 1D dustywave problem using $K_{0}=1$ and a dust-to-gas ratio of unity. The solution obtained for the linear drag regime shows an efficiently damped perturbation at $t=5$, consistent with a stopping time of order unity. By comparison, the perturbation is only weakly damped for the case of the power-law drag regime at the same time. The drag is weaker still in the quadratic regime, for which both the dust and the gas are mostly decoupled. Indeed, since the drag stopping time is a decreasing function of the differential velocity for non-linear drag, and the differential velocity is small with respect to the sound speed, the damping is inefficient. We have also performed dustywave simulations using the third order expansion and mixed drag regimes. However, for these cases, non-linear terms represent only negligible corrections compared to the linear term, thus giving the same results as for the linear case. ### 5.3 dustyshock: Two fluid dust-gas shocks The dustyshock problem (Paper I) is a two fluid version of the standard Sod (1978) shock tube problem. Results were presented in Paper I using a constant drag coefficient $K$ and no heat transfer between the gas and the dust phase. Here we extend the test to non-linear drag regimes. While the evolution during the transient stage is dependent on the drag regime, the solution during the stationary stage remains unchanged, being a fixed function of the gas sound speed and the dust-to-gas ratio. To facilitate the comparison between a physical Epstein drag and the case of a constant drag coefficient (Paper I), we fix the ratio $s^{2}/m_{\rm{d}}$ — $s$ being the grain size and $m_{\rm{d}}$ the grain mass— to unity in code units, such that the Epstein drag coefficient is unity in regions where $\hat{\rho}_{\mathrm{g}}=1$. #### 5.3.1 dustyshock: Setup Equal mass particles are placed in the 1D domain $x\in[-0.5,0.5]$, where for $x<0$ we use $\rho_{\mathrm{g}}=\rho_{\mathrm{d}}=1$, $v_{\rm{g}}=v_{\rm{d}}=0$ and $P_{\rm{g}}=1$, while for $x>0$ $\rho_{\mathrm{g}}=\rho_{\mathrm{d}}=0.125$, $v_{\rm{g}}=v_{\rm{d}}=0$ and $P_{\rm{g}}=0.1$. We use an ideal gas equation of state $P=(\gamma-1)\rho u$ with $\gamma=5/3$. Initial particle spacing to the left of the shock in both fluids is $\Delta x=0.001$ while to the right it is $\Delta x=0.008$, giving $569$ equal mass particles in each phase. Standard SPH artificial viscosity and conductivity terms are applied as in Paper I. #### 5.3.2 dustyshock: Different drag regimes Figure 3: Results of the dustyshock problem with a linear constant drag coefficient ($K=1$) (left panel) and a non-linear Epstein drag (right panel) where the drag coefficients are initially the same ahead of the shock. The dust-to-gas ratio is set to unity. At $t=0.2$, the solutions are in the transient stage where the analytic solution is not known. As an indication, the solution for the later stationary stage is shown by the dotted/red lines. The profiles differ as the damping is less efficient using the non-linear Epstein regime. Fig. 3 illustrates how the transient regime of the dustyshock is affected when treating the drag with an astrophysical prescription where the drag coefficient depends on the local density and the gas sound speed (right panel) rather than a constant coefficient (left panel). Specifically, we use the non- linear Epstein drag regime given by Eq. 39. In this dustyshock test, the non- linear terms constitute a small correction to the linear Epstein drag regime given by Eq. 40. Fig. 3 shows that in the case of the Epstein regime, the drag is less efficient than for the constant coefficient case, leading to a larger ($\sim$ by a factor $7$) differential velocity between the gas and the dust after $t=0.2$ in code units. The dust velocity profile is also smoother than in the constant coefficient case. As a result, the kinetic energy is less efficiently dissipated by the drag, leading to a less sharp peak in the internal energy of the gas. The density profile of the dust is also closer to its initial profile behind and ahead of the shock. ### 5.4 dustysedov: Two fluid dust-gas blast wave The dustysedov problem (Paper I) involves the propagation of a blast wave in an astrophysical mixture of dust and gas. We adopt physical units for this problem, assuming a box size of $1$ pc, an ambient sound speed of $2\times 10^{4}$ cm/s and a gas density of $\rho_{0}=6\times 10^{-23}$ g/cm3 the energy of the blast is $2\times 10^{51}$ erg and time is measured in units of $100$ years, roughly corresponding to a supernova blast wave propagating into the interstellar medium. We these units, we choose the grain size, $0.1\mu$m and the dust-to-gas ratio, $0.01$, to be typical of the interstellar medium. In code units, this corresponds to an initial drag coefficient of $K=1$ outside the blast radius. As for the dustyshock, we compare the results using a non- linear Epstein drag prescription with with the constant coefficient case described in Paper I. #### 5.4.1 dustysedov: Setup The problem is set up in a 3D periodic box ($x,y,z\in[-0.5,0.5]$), filled by $50^{3}$ particles for both the gas and the dust. Gas particles are set up on a regular cubic lattice, with the dust particles also on a cubic lattice but shifted by half of the lattice step in each direction. For shock-capturing, we set $\alpha_{\rm SPH}=1$ and $\beta_{\rm SPH}=2$ for the artificial viscosity terms and $\alpha_{u}=1$ for the artificial conductivity term. An ideal gas equation of state $P=(\gamma-1)\rho u$ is adopted with $\gamma=5/3$. The internal energy is distributed of the gas over the particles located inside a radius $r<r_{\rm{b}}$ where $r_{\rm b}$ is set to 2h (i.e., the radius of the smoothing kernel which for $50^{3}$ particles and $\eta=1.2$ is $0.048$). In code units the total blast energy is $E=1$, with $\hat{\rho}_{\mathrm{g}}=1$ and $\hat{\rho}_{\mathrm{d}}=0.01$. For $r>r_{\rm{b}}$, the gas sound speed is set to be $2\times 10^{-5}$ in code units. #### 5.4.2 dustysedov: Different drag regimes Figure 4: Results of the 3D dustysedov test, showing the density in the gas (left figure) and dust (right figure) from a Sedov blast wave propagating in an astrophysical ($1\%$ dust-to-gas ratio) mixture of gas and $0.1\mu$m dust grains in a $1$ pc box. The drag coefficient is constant ($K=1$, top panels) or given by the Epstein regime (bottom panels). The low dust-to-gas ratio means that the gas is only weakly affected by the drag from the dust, and is thus close to the self-similar Sedov solution (dotted/red line). In the Epstein case, the drag is much higher inside the blast radius and the dust particles are efficiently piled up by the passage of the gas over-density. Figure 5: Cross-section slice showing density in the midplane in the 3D dustysedov problem, for both the gas (left panel) and the dust (right panel) at $t=0.1$. Initially, the dust-to-gas ratio is $0.01$ and the drag coefficient is given by the Epstein regime for grains of $0.1\mu$m in size. $50^{3}$ SPH particles have been used in each phase. Figs. 4 and 5 show the evolution of the gas and dust mixture where a constant drag coefficient is used (top panels of Fig. 4) compared to a drag prescribed by the Epstein regime (bottom panels of Fig. 4, Fig. 5). The gas profiles are similar in both cases since the gas is poorly affected by the dust given the low dust-to-gas ratio. However, the dust density profiles differ, essentially due to the fact that the drag coefficient scales with the sound speed and is thus higher in the inner blast region for the Epstein case. Thus, the dust is efficiently piled up and accumulates in the gas over-density. As a result, the dust is cleaned up by the gas in the inner regions of the blast, but is more concentrated (by $\sim 10\%$) close to the gas over-density than for the constant drag coefficient case. The results using either explicit or implicit timestepping were found to be indistinguishable. For the Epstein case, we found that roughly ten iterations were required for the implicit scheme to converge on this problem. ### 5.5 dustydisc The dustydisc problem concerns the evolution of a dusty gas mixture in a protoplanetary disc (see Paper I for details). For our test case, we study how the dust distribution is affected when considering a general non-linear Epstein drag instead of the standard linear regime. The results obtained when implicitly integrating the non-linear drag regime have been found to be similar to benchmark tests performed with explicit integration. #### 5.5.1 dustydisc: Setup We setup $10^{5}$ gas particles and $10^{5}$ dust particles in a $0.01M_{\odot}$ gas disc (with $0.0001M_{\odot}$ of dust) surrounding a $1M_{\odot}$ star. The disc extends from 10 to 400 AU. Both gas and dust particles are placed using a Monte-Carlo setup such that the surface density profiles of both phases are $\Sigma\left(r\right)\propto r^{-1}$. The radial profile of the gas temperature is taken to be $T\left(r\right)\propto r^{-0.6}$ with a flaring $H/r=0.05$ at 100 AU. One code unit of time corresponds to $10^{3}$ yrs. #### 5.5.2 dustydisc: Evolution of the particles Figure 6: Rendering of the density for the dust of a typical T-Tauri Star protoplanerary disc using $2\times 10^{5}$ SPH particles, using an explicit time integration in the linear Epstein regime (left panel) and an implicit integrator in the full non-linear Epstein drag regimes (right panel). Fig. 6 shows a face-on view of a protoplanetary disc, integrating the linear Epstein regime (left panel) and the full non-linear Epstein drag (right panel). The dust distributions are not found to exhibit significant discrepancies. In the non-linear drag regime case however, the dust distribution is slightly smoother since the drag (and thus, the coupling with the gas phase) is more efficient. Figure 7: Vertical settling of a dust grain ($1$cm in size) initially located at $r_{0}=100$ AU and $z_{0}=2$ AU (solid/black), integrating implicitly the non-linear Epstein regime. SPH results are compared to the explicit integration of the linear Epstein regime (dashed/red) and the estimation given by the damped harmonic oscillator approximation (pointed/red). In the full non-linear drag regime, the settling is more efficient than for the linear case since the vertical oscillations in the dust motion reaches a fraction $z_{0}/H$ of the sound speed. Fig. 7 compares the vertical motion of a dust grain initially located at $z=z_{0}$ using the linear (explicit integration) and the full non linear (implicit integration) Epstein regimes. In the full non-linear case, the settling is more efficient since the vertical differential velocity between the dust grains and the gas in the mid plane of the disc reaches a fraction $z_{0}/H$ of the sound speed, meaning that the non-linear terms are no longer negligible. ## 6 Conclusions We have extended the SPH formalism for two-fluid dust and gas mixtures developed in Paper I to handle the drag regimes usually encountered in a large range of astrophysical contexts. Specifically, our algorithm is now designed to treat the dynamics of grains surrounded by a dilute medium (Epstein regime) or dense fluid (Stokes regime), for which the drag force can be either linear or non-linear with respect to the differential velocity between the gas and the dust. Particular attention has been paid to developing an implicit timestepping scheme to efficiently simulate the case of high drag, extending the scheme proposed by Monaghan (1997) which we found to be unsatisfactory. We have presented a new pairwise implicit scheme that, like the Monaghan (1997) scheme, preserves the exact conservation of linear and angular momentum but, unlike the Monaghan (1997) scheme, i) provides control over the accuracy of the iterative procedure and ii) can incorporate non-linear terms for both Epstein and Stokes drag. We found that when the ratio $r$ between the the timestep and the drag stopping time is $1\lesssim r\lesssim 1000$, the implicit timestepping is faster than a standard explicit integration. However, at higher values of $r$, the algorithm is less efficient. The accuracy of the generalised algorithm is benchmarked against the suite of test problems presented in Paper I. In particular, the solutions obtained for the dustybox problem are compared to their known analytic solutions for a large range of non-linear drag regimes and the solutions of the dustywave, dustyshock, dustysedov and dustydisc problems are benchmarked against converged results obtained with explicit timestepping. The two key issues addressed in this paper complete the study of our algorithm developed in Paper I. Our intention is to apply it to various astrophysical problems involving gas and dust mixtures in star and planet formation. A first application is given in Ayliffe et al. (2011). ## Acknowledgments We thank Ben Ayliffe and Matthew Bate and Joe Monaghan for useful discussions and comments. Figures have been produced using splash (Price, 2007) with the new giza backend by DP and James Wetter. We are grateful to the Australian Research Council for funding via Discovery project grant DP1094585. ## References * Ayliffe et al. (2011) Ayliffe B., Laibe G., Price D. J., Bate M. R., 2011, MNRAS, submitted * Baines et al. (1965) Baines M. J., Williams I. P., Asebiomo A. S., 1965, MNRAS, 130, 63 * Blum & Wurm (2008) Blum J., Wurm G., 2008, ARA&A, 46, 21 * Chapman & Cowling (1970) Chapman C., Cowling T., 1970, The mathematical theory of non-uniform gases. Cambridge at the university press * Chiang & Youdin (2010) Chiang E., Youdin A. N., 2010, Annual Review of Earth and Planetary Sciences, 38, 493 * Fan & Zhu (1998) Fan L.-S., Zhu C., 1998, Principles of Gas-Solid Flows. Cambridge University Press * Kwok (1975) Kwok S., 1975, ApJ, 198, 583 * Laibe & Price (2011b) Laibe G., Price D. J., 2011b, MNRAS, submitted * Laibe & Price (2011a) Laibe G., Price D. J., 2011a, MNRAS, in press * Monaghan (1997) Monaghan J. J., 1997, Journal of Computational Physics, 138, 801 * Monaghan (2005) Monaghan J. J., 2005, Reports on Progress in Physics, 68, 1703 * Monaghan & Kocharyan (1995) Monaghan J. J., Kocharyan A., 1995, Computer Physics Communications, 87, 225 * Paardekooper & Mellema (2006) Paardekooper S.-J., Mellema G., 2006, A&A, 453, 1129 * Price (2007) Price D. J., 2007, Publications of the Astronomical Society of Australia, 24, 159 * Price & Monaghan (2007) Price D. J., Monaghan J. J., 2007, MNRAS, 374, 1347 * Sod (1978) Sod G. A., 1978, Journal of Computational Physics, 27, 1 * Stepinski & Valageas (1996) Stepinski T. F., Valageas P., 1996, A&A, 309, 301 * Whitehouse et al. (2005) Whitehouse S. C., Bate M. R., Monaghan J. J., 2005, MNRAS, 364, 1367	arxiv-papers	2011-11-14T02:28:49	2024-09-04T02:49:24.297384	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Guillaume Laibe (Monash), Daniel J. Price (Monash)", "submitter": "Guillaume Laibe", "url": "https://arxiv.org/abs/1111.3089" }
1111.3090	# Dusty gas with SPH — I. Algorithm and test suite Guillaume Laibe, Daniel J. Price Monash Centre for Astrophysics (MoCA) and School of Mathematical Sciences, Monash University, Clayton, Vic 3800, Australia ###### Abstract We present a new algorithm for simulating two-fluid gas and dust mixtures in Smoothed Particle Hydrodynamics (SPH), systematically addressing a number of key issues including the generalised SPH density estimate in multi-fluid systems, the consistent treatment of variable smoothing length terms, finite particle size, time step stability, thermal coupling terms and the choice of kernel and smoothing length used in the drag operator. We find that using double-hump shaped kernels improves the accuracy of the drag interpolation by a factor of several hundred compared to the use of standard SPH bell-shaped kernels, at no additional computational expense. In order to benchmark our algorithm, we have developed a comprehensive suite of standardised, simple test problems for gas and dust mixtures: dustybox, dustywave, dustyshock, dustysedov and dustydisc, the first three of which have known analytic solutions. We present the validation of our algorithm against all of these tests. In doing so, we show that the spatial resolution criterion $\Delta\lesssim c_{\rm s}t_{\rm s}$ is a necessary condition in all gas+dust codes that becomes critical at high drag (i.e. small stopping time $t_{\rm s}$) in order to correctly predict the dynamics. Implicit timestepping and the implementation of realistic astrophysical drag regimes are addressed in a companion paper. ###### keywords: hydrodynamics — methods: numerical — ISM: dust, extinction — protoplanetary discs — planets and satellites: formation ††pagerange: Dusty gas with SPH — I. Algorithm and test suite–References††pubyear: 2011 ## 1 Introduction Most of our observational information regarding the interstellar medium comes to us via dust. Over the last few years, observations using the _Spitzer_ and _Herschel_ space telescopes have substantially improved our observational sensitivity to and resolution of dust emission in a wide range of astrophysical environments. Dust grains provide the materials from which the solid cores required for the planet formation process are built (see e.g. Chiang & Youdin 2010a). They also modify the dynamical evolution of the surrounding gas by exchanging momentum and energy via microscopic collisions (Epstein, 1924; Baines et al., 1965). Dust grains are also the main sources of the opacities in star-forming molecular clouds, thus determining their evolution by controlling the thermodynamics. Accurate determination of both the dynamics of the star and planet formation process and its observational signature thus require modelling the coupled evolution of gas and dust. Given that the N-body evolution of solid particles in a mixture of gas and solid material would be prohibitive in terms of both physical complexity and computational cost, the usual approach is to regard the solid phase as a continuum and the mixture as a two-fluid system coupled by a drag term. This requires averaging physical quantities over a control volume $V$ that is large enough to be statistically meaningful but sufficiently small compared to the macroscopic scale to allow a continuum description. In diluted astrophysical media, the frequency of collisions between dust particles are infrequent enough that the intrinsic pressure of the dust phase can be regarded as negligible to a very good level of approximation, leaving the dust as a free- streaming collisionless fluid whose motion is controlled solely by gravitational forces and the drag-term interaction with the gas. However, even with a continuous description of the mixture, the equations can be solved analytically only for a few simple cases (the solutions to two specific problems, dustybox and dustywave, corresponding to mutually interpenetrating fluids and acoustic waves propagating in a dusty gas, respectively, are derived in Laibe & Price 2011a). As a result, numerical codes have been developed in order to model more realistic systems based either on $N$-body dust particles in Eulerian grid-based hydrodynamics (e.g. Fromang & Papaloizou 2006; Paardekooper & Mellema 2006; Johansen et al. 2007; Miniati 2010; Bai & Stone 2010) or with a two-fluid Smoothed Particle Hydrodynamics (SPH) approach. SPH methods for simulating two-fluid mixtures were first developed by Monaghan & Kocharyan (1995), improved (via an implicit treatment of the drag terms) in Monaghan (1997) and applied in an astrophysical context to the dynamics of dust grains in protoplanetary disks (Maddison et al., 2003; Rice et al., 2004; Barrière-Fouchet et al., 2005). The particle-based nature of the SPH formalism means that the addition of a dusty fluid is natural. More importantly, the drag term that couples the two phases can be implemented such that the total linear and angular momentum of the system are exactly (and simultaneously) conserved, in line with the Hamiltonian and exactly conservative nature of the core SPH method (e.g. Price, 2011). However, the standard methods for treating dusty gas in SPH were developed over 15 years ago and our initial attempts to simply apply the existing formulations uncovered several issues that needed to be addressed. Specifically: 1) the original formulations assumed a spatially constant SPH smoothing length; 2) the SPH terms for the conservative part of the equations should be derived from a Lagrangian; 3) we found that the use of the standard cubic spline kernel for drag terms could be significantly inaccurate; 4) we encountered several previously unexplored resolution issues in simulating two- fluid mixtures; 5) aspects of the implicit timestepping scheme suggested by Monaghan (1997) were found to be problematic; 6) that treatments of drag have to date generally limited to linear drag regimes; and finally 7) that the existing schemes — having been developed with both astrophysical and geophysical dust problems in mind — have not been widely benchmarked on problems appropriate to astrophysics; Indeed there is a general lack of standardised test problems for two-fluid dust/gas codes, a problem partially addressed by our first paper (Laibe & Price, 2011a). In this and a companion paper (Laibe & Price 2011c, hereafter Paper II), we set out to systematically address issues 1)–7) in order to develop a robust and accurate code for simulating the dynamics of dust in star and planet formation. The importance of modelling the dust-gas interaction has been highlighted by recent studies showing that instabilities in dust-gas mixtures are good candidates for triggering the concentration of dust during planetesimal formation (Goodman & Pindor, 2000; Youdin & Goodman, 2005). The continuum equations and the relevant parameters describing the evolution of dust-gas mixtures are given in Section 2.1. Section 2 describes the two- fluid SPH algorithm, addressing issues 1)-3). The code is benchmarked against a suite of test problems that we have specifically designed in order to provide standardised benchmarks for other two-fluid gas/dust codes, addressing issues 4) and 7) (Sec. 4). The implicit timestepping scheme and treatment of non-linear drag (issues 5 and 6) are discussed in Paper II. ## 2 Two-fluid mixtures in SPH ### 2.1 Two-fluid gas and dust mixtures #### 2.1.1 Densities The fact that dust grains of finite size occupy a finite volume is accounted for by defining the volume fraction available to the gas according to (e.g. Marble, 1970; Harlow & Amsden, 1975) $\theta=1-\frac{\hat{\rho}_{\mathrm{d}}}{\rho_{\mathrm{d}}}.$ (1) This means that the volume densities of gas and dust $\hat{\rho}_{\mathrm{g}}$ and $\hat{\rho}_{\mathrm{d}}$, respectively, are distinguished from the intrinsic densities denoted $\rho_{\mathrm{g}}$ and $\rho_{\mathrm{d}}$, respectively, according to $\displaystyle\hat{\rho}_{\mathrm{d}}$ $\displaystyle=$ $\displaystyle(1-\theta)\rho_{\mathrm{d}},$ (2) $\displaystyle\hat{\rho}_{\mathrm{g}}$ $\displaystyle=$ $\displaystyle\theta\rho_{\mathrm{g}}.$ (3) The effects associated with finite dust particle size are mostly negligible in astrophysical problems since typically the intrinsic dust density $\rho_{\mathrm{d}}$ is much higher than the volume density $\hat{\rho}_{\mathrm{d}}$ and thus $\theta\approx 1$. We retain these terms, as in earlier SPH formulations (c.f. Monaghan & Kocharyan, 1995) in order to retain a general algorithm that can be applied both within and outside of astrophysics. The conservation of mass in a two-fluid mixture is thus expressed by the continuity equations $\displaystyle\frac{\partial\hat{\rho}_{\mathrm{g}}}{\partial t}+\nabla.\left(\hat{\rho}_{\mathrm{g}}\textbf{v}_{\mathrm{g}}\right)$ $\displaystyle=$ $\displaystyle 0,$ (4) $\displaystyle\frac{\partial\hat{\rho}_{\mathrm{d}}}{\partial t}+\nabla.\left(\hat{\rho}_{\mathrm{d}}\textbf{v}_{\mathrm{d}}\right)$ $\displaystyle=$ $\displaystyle 0,$ (5) where ${\bf v}_{\rm g}$ and ${\bf v}_{\rm d}$ are the gas and dust fluid velocities, respectively. #### 2.1.2 Equations of motion The equations of motion, expressing momentum conservation in a continuous, inviscid, two-fluid mixture of gas and dust are given by $\displaystyle\hat{\rho}_{\mathrm{g}}\left(\frac{\partial\textbf{v}_{\mathrm{g}}}{\partial t}+\textbf{v}_{\mathrm{g}}.\nabla\textbf{v}_{\mathrm{g}}\right)$ $\displaystyle=$ $\displaystyle-\theta\phantom{.}\nabla P_{\rm g}+\hat{\rho}_{\mathrm{g}}\textbf{f}-\textbf{F}^{\rm V}_{\mathrm{drag}},$ (6) $\displaystyle\hat{\rho}_{\mathrm{d}}\left(\frac{\partial\textbf{v}_{\mathrm{d}}}{\partial t}+\textbf{v}_{\mathrm{d}}.\nabla\textbf{v}_{\mathrm{d}}\right)$ $\displaystyle=$ $\displaystyle-\nabla P_{\rm d}-\left(1-\theta\right)\nabla P_{\rm g}+\hat{\rho}_{\mathrm{d}}\textbf{f}+\textbf{F}^{\rm V}_{\mathrm{drag}},$ (7) where $P_{\rm g}$ and $P_{\rm d}$ are the intrinsic pressures. Any intrinsic viscosities have been neglected. For astrophysical purposes it may be assumed that the dust is pressureless, i.e. $P_{\rm d}=0$. Similarly, the term $\left(1-\theta\right)\nabla P_{\rm g}$ in the momentum equation for the dust phase — a buoyancy term related to the finite size of the dust particles — is in general negligibly small. The reader should note that the definitions of physical quantities in a two fluid medium require the local fluid volume over which the averaging is performed to be defined (see, e.g. Marble, 1970; Fan & Zhu, 1998). The two fluids exchange momentum $\textbf{F}^{\rm V}_{\mathrm{drag}}$, the drag force per unit volume, the expression for which is obtained by averaging the local drag stress tensor (denoted $\mathbf{\epsilon}^{ij}_{\rm drag}$) over the surface area of the dust grains: $F_{\rm drag}^{{\rm V},i}=\frac{1}{V}\int_{A_{\mathrm{d}}}\mathbf{\epsilon}^{ij}_{\rm drag}\mathrm{d}A^{j}.$ (8) In the case where the local distribution of dust particles is homogeneous (i.e., dust particles have the same mass, size and intrinsic density), Eq. 8 simplifies to $\textbf{F}^{\rm V}_{\mathrm{drag}}=K(\textbf{v}_{\mathrm{g}}-\textbf{v}_{\mathrm{d}}).$ (9) Note that since $\textbf{F}^{\rm V}_{\mathrm{drag}}$ is a force per unit volume, the drag coefficient $K$, has dimensions of mass per unit volume per unit time. This coefficient is related to the drag coefficient on a single grain (denoted $K_{\rm s}$) by $K=\frac{\hat{\rho}_{\mathrm{d}}}{m_{\mathrm{d}}}K_{\rm s}$ (10) where $m_{\mathrm{d}}$ is the mass per grain. The drag force (not per unit volume) on a single grain is given by ${\bf F}_{\rm drag}=K_{\rm s}(\textbf{v}_{\mathrm{g}}-\textbf{v}_{\mathrm{d}}).$ (11) In general $K$ (or equivalently $K_{s}$) can itself be a function of the relative velocity between the two fluids $\Delta v\equiv\|\textbf{v}_{\mathrm{g}}-\textbf{v}_{\mathrm{d}}\|$, resulting in a non- linear drag regime. In this, Paper I we consider the simplest case of linear drag, where $K$ is constant with respect to $\Delta v$. Extension of our scheme to the main non-linear regimes applicable to astrophysics are considered in Paper II. Finally, it should be noted that in general additional forces (e.g. the carried mass, Basset and Saffman forces, Fan & Zhu 1998) may be present in two-fluid systems. We have assumed in adopting Eqs. 6–7 that these forces can be neglected for astrophysical applications. Figure 1: Computing density in SPH gas (solid points) and dust (hollow circles) mixtures. Standard bell-shaped, Gaussian-like, kernels are adopted (weighting indicated by the shading), with a single smoothing length on each particle related to the local number density of particles of the _same type_. This provides good density estimates in both extremes — where dust is concentrated below the gas scale (left panel) and where gas is concentrated below the dust scale (right panel). The density of another fluid at the position of a reference fluid (e.g. dust density at the location of a gas particle) is computed using the same smoothing length but only neighbours of the desired type. This density is thus allowed to be identically zero, as would be the case for the density of gas-at-dust in the left panel (top), or dust-at-gas in the right panel. #### 2.1.3 Energy equation The evolution equation for the specific internal energy of the gas, $u_{\rm g}$, is given by $\hat{\rho}_{\mathrm{g}}\frac{{\rm d}u_{\rm g}}{{\rm d}t}=-P_{\mathrm{g}}\left[\theta\nabla\cdot{\bf v}_{\rm g}+(1-\theta)\nabla\cdot{\bf v}_{\rm d}\right]+\Lambda_{\rm drag}+\Lambda_{\rm therm},$ (12) where the first term corresponds to the usual compressive ($P{\rm d}V$) term with the volume reduced by the dust filling factor $\theta$. The second term is the work done by the gas in triggering buoyancy effects. The third term is the frictional heating due to the drag force, given by $\Lambda_{\rm drag}=\hat{\rho}_{\mathrm{g}}K(\textbf{v}_{\mathrm{g}}-\textbf{v}_{\mathrm{d}})^{2}.$ (13) The fourth, thermal coupling, term arises when the internal temperature of the grains differs from the gas temperature (c.f. Marble, 1970; Harlow & Amsden, 1975), and in general consists of terms related to heat transfer due to conduction ($\Lambda_{\rm cond}$) and radiation ($\Lambda_{\rm rad}$), given by $\Lambda_{\rm therm}\equiv\Lambda_{\rm cond}+\Lambda_{\rm rad}=Q(T_{\rm g}-T_{\rm d})+R(aT_{\rm g}^{4}-aT_{\rm d}^{4}),$ (14) where $T_{\rm g}$ and $T_{\rm d}$ are the temperatures of the gas and dust, respectively, $a$ is the radiation constant and $Q$ and $R$ are coefficients, dependent on gas and dust properties, that characterise the heat transfer. The thermal energy of the dust evolves according to $\hat{\rho}_{\mathrm{d}}\frac{{\rm d}u_{\rm d}}{{\rm d}t}=-\Lambda_{\rm therm}.$ (15) ### 2.2 Densities for two-fluid mixtures in SPH #### 2.2.1 Computing densities in two-fluid SPH For two-fluid mixtures, we require a density estimate _for each phase_ , corresponding to the exact solution of Eqs. 4 and 5 in SPH. The main complication arises from the fact that the local particle spacing can be different for each fluid, implying that the two fluids should have different resolution lengths calculated based on the local particle number density of their own type. Figure 1 illustrates the two limiting cases, i.e. a high concentration of dust in a diluted gas (left panel) and conversely a high concentration of gas in a low density fluid of dust (right panel). In each case the smoothing length for each type is determined by the local number density of particles of _the same type_. That is, the SPH translation of Eqs. 4 and 5 correspond to $\displaystyle\hat{\rho}_{a}=\sum_{b}m_{b}W_{ab}(h_{a});$ $\displaystyle h_{a}=\eta\left(\frac{m_{a}}{\hat{\rho}_{a}}\right)^{1/\nu},$ (16) $\displaystyle\hat{\rho}_{i}=\sum_{j}m_{j}W_{ij}(h_{i});$ $\displaystyle h_{i}=\eta\left(\frac{m_{j}}{\hat{\rho}_{i}}\right)^{1/\nu},$ (17) where $\nu$ is the number of spatial dimensions and $\eta$ is a constant determining the resolution length as a function of the local particle spacing (typically $\eta=1.2$ is a good choice for the standard cubic spline kernel, see Price 2011). We adopt the convention that the indices $a,b,c$ refer to quantities computed on gas particles while $i,j,k$ refer to quantities computed on dust particles. Note that the densities and smoothing lengths are independently computed for each fluid and are thus — so far — only defined on particles of the same type. The numerical solution of Eqs. 16 and 17 involves determining both $\hat{\rho}$ and $h$ for each type simultaneously, since they are mutually dependent, thus requiring an iterative procedure. The procedure is identical to that adopted in standard variable smoothing length SPH formulations (see e.g. Price & Monaghan 2007 for details). An additional complication arises from the need to compute the volume filling fraction $\theta$ (Eq. 1), defined on a _gas_ particle, $a$, according to $\theta_{a}=1-\frac{\hat{\rho}_{{\rm d},a}}{\rho_{\rm d}},$ (18) which depends on the density of _dust_ at the gas particle location. Initially we considered computing this density using a second smoothing length for each particle based on neighbours of the _other_ type (this would in turn lead to multiple smoothing lengths on each particle if more types were present). However, the key point, illustrated by the right hand panel of Fig. 1, is that it should be possible for the local density of dust at the gas location to be identically zero (giving $\theta=1$) if no dust particles are found within the kernel radius computed with the gas smoothing length. Thus, the density of dust-at-gas should be calculated according to $\hat{\rho}_{{\rm d},a}=\sum^{N_{neigh,dust}}_{j=1}m_{j}W_{aj}(h_{a}),$ (19) where $h_{a}$ is the smoothing length of the _gas_ particle computed using gas neighbours as in Eq. 16. In the case where no dust neighbours fall within the kernel radius, $\rho_{{\rm d},a}=0$. This is a very simple and efficient method that can easily be generalised to multiple fluids, requires only one smoothing length per particle and does not require any significant additional computational expense. The discussion above resolves the first issue highlighted in Sec. 1, namely how to deal with variable resolution in multi-fluid SPH, generalising the earlier fixed-smoothing-length formulation of Monaghan & Kocharyan (1995). A similar discussion to the above applies to gravitational force softening on multiple fluids in $N$-body/SPH codes where the softening formulation is derived from a kernel density estimate (Price & Monaghan, 2007), in particular for the case of a mixture of dark matter and baryonic gas (e.g. Merlin et al., 2010; Iannuzzi & Dolag, 2011). #### 2.2.2 Kernel function The kernel function itself can be written as a function of the smoothing length $h$ and the dimensionless variable $q=\|{\bf r}-{\bf r}^{\prime}\|/h$ in the form $W\left(r,h\right)=\frac{\sigma}{h^{\nu}}f\left(q\right),$ (20) where $\sigma$ is a normalisation constant. The standard Gaussian kernel is given by $f(q)=e^{-q^{2}},$ (21) where $\sigma=\pi^{-\nu/2}$. The Gaussian is infinitely smooth (differentiable) but has the practical disadvantage of infinite range. A standard alternative (providing a Gaussian-like kernel but truncated at $2h$) is the $M_{4}$ cubic spline kernel (Monaghan, 1992) $f(q)=\begin{cases}1-\frac{3}{2}q^{2}+\frac{3}{4}q^{3},&0\leq q<1;\\\ \frac{1}{4}\left(2-q\right)^{3},&1\leq q<2;\\\ 0,&q\geq 2,\end{cases}$ (22) where $\sigma=\left[2/3,10/\left(7\pi\right),1/\pi\right]$ in $\left[1,2,3\right]$ dimensions. An error analysis of the SPH density estimate (e.g. Price, 2011) shows that in general the measure of a good density kernel is that the normalisation condition $\sum_{b}\frac{m_{b}}{\rho_{b}}W_{ab}\approx 1,$ (23) is well satisfied for typical SPH particle distributions, corresponding to $\int W{\rm d}V=1$ in the continuum limit. In general most bell-shaped (Gaussian-like) kernels, such as the cubic spline, fulfil this criterion (Fulk & Quinn, 1996). More accurate density estimates can be obtained — at the price of additional computational expense — by using kernels with extended range that form a better approximation to the Gaussian (see Price, 2011). In particular the $M_{6}$ quintic kernel, truncated at $3h$, gives results that are in practice largely indistinguishable from the Gaussian, with the functional form $f(q)=\begin{cases}(3-q)^{5}-6(2-q)^{5}+15(1-q)^{5},&\text{$0\leq q<1$;}\\\ (3-q)^{5}-6(2-q)^{5},&\text{$1\leq q<2$;}\\\ (3-q)^{5},&\text{$2\leq q<3$;}\\\ 0,&\text{$q\geq 3$}.\end{cases}$ (24) where $\sigma=[1/24,96/(1199\pi),1/(20\pi)]$. Use of the quintic is a factor of $(3/2)^{3}\approx 3.4$ times more expensive than the cubic spline (or other $2h$-truncated kernels) in three dimensions. The functional form of the $M_{4}$ cubic and $M_{6}$ quintic spline kernels are shown in the top row of Fig. 2, showing the kernel function $f(q)$ (solid/black lines) and its first (dashed/red lines) and second (short dashed/green line) derivatives. ### 2.3 Equations of motion As discussed by Price (2011), specifying the manner in which the density is calculated in SPH can be used to self-consistently determine the equations of motion and energy from a variational principle, using only the additional constraint of the first law of thermodynamics. For a two-fluid system, only the dissipationless part of the algorithm can be derived in this manner — that is, not including the drag terms. #### 2.3.1 Lagrangian For a system consisting of gas and dust, the Lagrangian is given by $L=\sum_{b}m_{b}\left[\frac{1}{2}\textbf{v}_{b}^{2}-u_{b}\left(\rho_{b},s_{b}\right)\right]+\sum_{k}m_{k}\left(\frac{1}{2}\textbf{v}_{k}^{2}\right)$ (25) where $u_{b}$ is the thermal energy per unit mass of the gas (in general a function of the entropy $s$ and _intrinsic_ density $\rho$). The equations of motion can be derived from the Euler-Lagrange equations, $\frac{\mathrm{d}}{\mathrm{d}t}\left(\frac{\partial L}{\partial{\bf v}}\right)=\frac{\partial L}{\partial{\bf r}}.$ (26) #### 2.3.2 Equations of motion for the gas We first consider the evolution of the gas particles. The partial derivative of the Lagrangian with respect to the velocity $\textbf{v}_{a}$ of a given gas particle $a$ provides: $\frac{\mathrm{d}}{\mathrm{d}t}\left(\frac{\partial L}{\partial\textbf{v}_{a}}\right)=m_{a}\frac{\mathrm{d}\textbf{v}_{a}}{\mathrm{d}t}.$ (27) The partial derivative of the Lagrangian with respect to the position $\textbf{r}_{a}$ of the gas particle $a$ is given by $\frac{\partial L}{\partial\textbf{r}_{a}}=-\sum_{b}m_{b}\left.\frac{\partial u_{b}}{\partial\rho_{b}}\right\|_{s}\frac{\partial\rho_{b}}{\partial\textbf{r}_{a}},$ (28) where the entropy is constant for a non-dissipative system. Eq. 28 differs from the usual expression for single fluids because of the distinction between the intrinsic and volume density of the gas caused by the finite volume occupied by dust. That is, the thermal energy depends on the _intrinsic_ density rather than the volume density, giving $\left.\frac{\partial u_{b}}{\partial\rho_{b}}\right\|_{s}=\frac{P_{b}}{\rho_{b}^{2}}=\frac{\theta_{b}^{2}P_{b}}{\hat{\rho}_{b}^{2}}.$ (29) The derivative of the intrinsic density with respect to the particle coordinates is given by $\frac{\partial\rho_{b}}{\partial\textbf{r}_{a}}=\left.\frac{\partial\rho_{b}}{\partial\hat{\rho}_{b}}\right\|_{\theta_{b}}\frac{\partial\hat{\rho}_{b}}{\partial\textbf{r}_{a}}+\left.\frac{\partial\rho_{b}}{\partial\theta_{b}}\right\|_{\hat{\rho}_{b}}\frac{\partial\theta_{b}}{\partial\textbf{r}_{a}},$ (30) where, from 18, we have $\left.\frac{\partial\rho_{b}}{\partial\hat{\rho}_{b}}\right\|_{\theta_{b}}=\displaystyle\frac{1}{\theta_{b}};\hskip 28.45274pt\left.\frac{\partial\rho_{b}}{\partial\theta_{b}}\right\|_{\hat{\rho}_{b}}=-\frac{\hat{\rho}_{b}}{\theta_{b}^{2}}.$ (31) The spatial derivative of the density sum for the gas (Eq. 16) is given by $\frac{\partial\hat{\rho}_{b}}{\partial\textbf{r}_{a}}=\frac{1}{\Omega_{b}}\sum_{c}m_{c}\left(\delta_{ba}-\delta_{ca}\right)\nabla_{a}W_{bc}(h_{b}),$ (32) where $\Omega$ is the usual variable smoothing length term $\Omega_{b}\equiv 1-\frac{\partial h_{b}}{\partial\hat{\rho}_{b}}\sum_{c}m_{c}\frac{\partial W_{\\!\\!\ b\\!\\!\ c}\left(h_{b}\right)}{\partial h_{b}}.$ (33) The spatial derivative of the volume filling fraction $\theta$ is given, from Eq. 18 by $\frac{\partial\theta_{b}}{\partial\textbf{r}_{a}}=-\frac{1}{\rho_{\rm d}}\frac{\partial\hat{\rho}_{{\rm d},b}}{\partial\textbf{r}_{a}},$ (34) where $\displaystyle\frac{\partial\hat{\rho}_{{\rm d},b}}{\partial\textbf{r}_{a}}=$ $\displaystyle\sum_{j}m_{j}\left(\delta_{ba}-\delta_{ja}\right)\nabla_{a}W_{bj}(h_{b})$ $\displaystyle+\frac{1-\Omega_{\mathrm{d},b}}{\Omega_{b}}\sum_{c}m_{c}\left(\delta_{ba}-\delta_{ca}\right)\nabla_{a}W_{bc}(h_{b}),$ (35) where $\Omega_{\rm d}$ is $\Omega$ computed only using dust particle neighbours, i.e. $\Omega_{\mathrm{d},b}=1-\frac{\partial h_{b}}{\partial\hat{\rho}_{{\mathrm{d},b}}}\sum_{j}m_{j}\frac{\partial W_{\\!\\!\ b\\!\\!\ j}\left(h_{b}\right)}{\partial h_{b}}.$ (36) Collecting Eqs. 28–36, noting that $\delta_{ja}=0$ (since a gas and dust index can never refer to the same particle) and using the fact that $\nabla_{\\!\\!\ a}W_{\\!\\!\ b\\!\\!\ c}=-\nabla_{\\!\\!\ a}W_{\\!\\!\ c\\!\\!\ b}$, gives $\displaystyle\frac{\partial L}{\partial\textbf{r}_{a}}=$ $\displaystyle- m_{a}\sum_{b}m_{b}\left[\frac{P_{a}\tilde{\theta}_{a}}{\Omega_{a}\hat{\rho}_{a}^{2}}\nabla_{\\!\\!\ a}W_{\\!\\!\ a\\!\\!\ b}\left(h_{a}\right)+\frac{P_{b}\tilde{\theta}_{b}}{\Omega_{b}\hat{\rho}_{b}^{2}}\nabla_{\\!\\!\ a}W_{\\!\\!\ a\\!\\!\ b}\left(h_{b}\right)\right]$ $\displaystyle- m_{a}\sum_{j}m_{j}\frac{P_{a}\left(1-\theta_{a}\right)}{\hat{\rho}_{a}\hat{\rho}_{{\rm d},a}}\nabla_{\\!\\!\ a}W_{\\!\\!\ a\\!\\!\ j}\left(h_{a}\right),$ (37) where we have defined $\tilde{\theta}$ to include the correction terms for a variable smoothing length, i.e. $\tilde{\theta}\equiv\theta+\frac{\hat{\rho}_{\rm g}}{\hat{\rho}_{\mathrm{d}}}(1-\theta)(1-\Omega_{\rm d}).$ (38) Although this correction is necessary for strict energy conservation, it is expected to be negligibly small in practice, since $(1-\theta)$ is negligible for small grains and $(1-\Omega_{\rm d})$ is $\mathcal{O}(h^{2})$. Finally, the equations of motion for a gas particle, from the Euler-Lagrange equations, are given by $\displaystyle\frac{\mathrm{d}\textbf{v}_{a}}{\mathrm{d}t}=$ $\displaystyle-\sum_{b}m_{b}\left[\frac{P_{a}\tilde{\theta}_{a}}{\Omega_{a}\hat{\rho}_{a}^{2}}\nabla_{\\!\\!\ a}W_{\\!\\!\ a\\!\\!\ b}\left(h_{a}\right)+\frac{P_{b}\tilde{\theta}_{b}}{\Omega_{b}\hat{\rho}_{b}^{2}}\nabla_{\\!\\!\ a}W_{\\!\\!\ a\\!\\!\ b}\left(h_{b}\right)\right]$ $\displaystyle-\sum_{j}m_{j}\frac{P_{a}\left(1-\theta_{a}\right)}{\hat{\rho}_{a}\hat{\rho}_{{\rm d},a}}\nabla_{\\!\\!\ a}W_{\\!\\!\ a\\!\\!\ j}\left(h_{a}\right).$ (39) The reader should note that while the first term is a summation over gas particle neighbours, the second is summed over dust particle neighbours. Eq. 39 may be straightforwardly shown to be a direct translation of Eq. 6 into SPH form. Note that the summation over dust particles (the buoyancy term) does not involve $\Omega$ since the smoothing length is independent of the dust particle positions. #### 2.3.3 Equations of motion for the dust The partial derivative of the Lagrangian with respect to the velocity ${\bf v}_{i}$ of a given dust particle $i$ gives $\frac{\mathrm{d}}{\mathrm{d}t}\left(\frac{\partial L}{\partial{\bf v}_{i}}\right)=m_{i}\frac{\mathrm{d}{\bf v}_{i}}{\mathrm{d}t}.$ (40) A buoyancy term arises in the dust because of the dependence of the gas internal energy on $\theta$, which in turn depends on the positions of dust particles. That is, $\frac{\partial L}{\partial{\bf r}_{i}}=-\sum_{b}m_{b}\left.\frac{\partial u_{b}}{\partial\rho_{b}}\right\|_{s}\frac{\partial\rho_{b}}{\partial{\bf r}_{i}},$ (41) where $\frac{\partial\rho_{b}}{\partial{\bf r}_{i}}=\frac{\partial\rho_{b}}{\partial\theta_{b}}\frac{\partial\theta_{b}}{\partial{\bf r}_{i}},$ (42) and in turn, $\displaystyle\frac{\partial\theta_{b}}{\partial{\bf r}_{i}}=$ $\displaystyle-\frac{1}{\rho_{\rm d}}\frac{\partial\hat{\rho}_{{\rm d},b}}{\partial{\bf r}_{i}}$ $\displaystyle=$ $\displaystyle-\frac{1}{\rho_{\rm d}}\sum_{j}m_{j}\left(\delta_{bi}-\delta_{ji}\right)\nabla_{i}W_{bj}(h_{b})$ $\displaystyle-\frac{1-\Omega_{\mathrm{d},b}}{\Omega_{b}}\sum_{c}m_{c}\left(\delta_{bi}-\delta_{ci}\right)\nabla_{i}W_{bc}(h_{b})$ (43) Collecting Eqs. 27–43 and noting that $\delta_{bi}=\delta_{ci}=0$, we obtain the equations of motion for a dust particle in the form $\frac{\mathrm{d}{\bf v}_{i}}{\mathrm{d}t}=\sum_{b}m_{b}\frac{P_{b}\left(1-\theta_{b}\right)}{\hat{\rho}_{b}\hat{\rho}_{{\mathrm{d},b}}}\nabla_{\\!\\!\ i}W_{\\!\\!\ b\\!\\!\ i}\left(h_{b}\right).$ (44) where we have written the kernel using $\nabla_{\\!\\!\ i}W_{\\!\\!\ i\\!\\!\ b}=-\nabla_{\\!\\!\ i}W_{\\!\\!\ b\\!\\!\ i}$ to show that the force is equal and opposite to that in the gas (Eq. 39). It may be straightforwardly verified that Eq. 44 is indeed a direct translation of Eq. 7 in SPH form. Equations 39 and 44 may be combined to show that the total momentum is exactly conserved, i.e. $\frac{\rm d}{{\rm d}t}\left(\sum_{a}m_{a}{\bf v}_{a}+\sum_{i}m_{i}{\bf v}_{i}\right)=0.$ (45) #### 2.3.4 Internal energy equation for the gas The SPH form of the non-dissipative terms in the internal energy equation for the gas (Eq. 12) can similarly be derived from the SPH density estimates. In the absence of dissipation the evolution equation for a given gas particle $a$ is given by $\frac{{\rm d}u_{a}}{{\rm d}t}=\frac{P_{a}}{\rho_{a}^{2}}\frac{{\rm d}\rho_{a}}{{\rm d}t}=\frac{P_{a}}{\rho_{a}^{2}}\left[\left.\frac{\partial\rho_{a}}{\partial\hat{\rho}_{a}}\right\|_{\theta_{a}}\frac{{\rm d}\hat{\rho}_{a}}{{\rm d}t}+\left.\frac{\partial\rho_{a}}{\partial\theta_{a}}\right\|_{\hat{\rho}_{a}}\frac{{\rm d}\theta_{a}}{{\rm d}t}\right].$ (46) Using the expressions (31) and simplifying using (18), we have $\frac{{\rm d}u_{a}}{{\rm d}t}=\frac{\theta_{a}P_{a}}{\hat{\rho}_{a}^{2}}\frac{{\rm d}\hat{\rho}_{a}}{{\rm d}t}+\frac{(1-\theta_{a})P_{a}}{\hat{\rho}_{a}\hat{\rho}_{{\rm d},a}}\frac{{\rm d}\hat{\rho}_{{\rm d},a}}{{\rm d}t}.$ (47) Taking the time derivative of the density sums (16) and (19) we have $\displaystyle\frac{{\rm d}\hat{\rho}_{a}}{{\rm d}t}$ $\displaystyle=$ $\displaystyle\frac{1}{\Omega_{a}}\sum_{b}m_{b}\left(\textbf{v}_{a}-\textbf{v}_{b}\right)\cdot\nabla_{a}W_{ab}(h_{a}),$ (48) $\displaystyle\frac{{\rm d}\hat{\rho}_{{\rm d},a}}{{\rm d}t}$ $\displaystyle=$ $\displaystyle\sum^{N_{neigh,dust}}_{j=1}m_{j}\left({\bf v}_{a}-{\bf v}_{j}\right)\cdot\nabla_{a}W_{aj}(h_{a}),$ (49) $\displaystyle+\frac{(1-\Omega_{{\rm d},a})}{\Omega_{a}}\sum_{b}m_{b}\left(\textbf{v}_{a}-\textbf{v}_{b}\right)\cdot\nabla_{a}W_{ab}(h_{a})$ giving the SPH internal energy equation in the form $\displaystyle\frac{{\rm d}u_{a}}{{\rm d}t}$ $\displaystyle=\frac{\tilde{\theta}_{a}P_{a}}{\Omega_{a}\hat{\rho}_{a}^{2}}\sum_{b}m_{b}\left(\textbf{v}_{a}-\textbf{v}_{b}\right)\cdot\nabla_{a}W_{ab}(h_{a})$ $\displaystyle+\frac{(1-\theta_{a})P_{a}}{\hat{\rho}_{a}\hat{\rho}_{{\rm d},a}}\sum^{N_{neigh,dust}}_{j=1}m_{j}\left({\bf v}_{a}-{\bf v}_{j}\right)\cdot\nabla_{a}W_{aj}(h_{a}),$ (50) which indeed can be shown to be an SPH translation of the first two terms in Eq. 12. ### 2.4 SPH representation of drag terms #### 2.4.1 Drag interpolation The remaining aspect is to provide an SPH representation of Eq. 9, specifying the drag term $\textbf{F}^{\rm V}_{\mathrm{drag}}$ involved in Eqs. 6–7. Monaghan & Kocharyan (1995) proposed an SPH interpolation of the drag term given by $\left<K\Delta\mathbf{v}\right>=\nu\int K\left(\mathbf{x},\mathbf{x}^{\prime}\right)\left\\{\left[\textbf{v}_{\mathrm{g}}(\mathbf{x})-\textbf{v}_{\mathrm{d}}(\mathbf{x}^{\prime})\right]\cdot\hat{\mathbf{r}}\right\\}\hat{\mathbf{r}}D\left(\mathbf{x}-\mathbf{x}^{\prime},h\right)\mathrm{d}\mathbf{x}^{\prime},$ (51) where $\hat{\mathbf{r}}$ is the unit vector defined by: $\hat{\mathbf{r}}=\frac{\mathbf{x}-\mathbf{x}^{\prime}}{\|\mathbf{x}-\mathbf{x}^{\prime}\|},$ (52) and $\nu$ is the number of spatial dimensions of the system (and _not_ the inverse of the number of spatial dimensions as one might intuitively guess — see below). Monaghan & Kocharyan (1995) proposed this formulation — with velocity difference projected along the line of sight joining the particles — mainly because it gives exact conservation of both linear and angular momentum in the resulting drag terms. As the SPH interpolation of the drag term does not come from the Euler- Lagrange equations derived for non-dissipative term form the SPH Lagrangian, the kernel function used in the drag term is not constrained to be the same function $W$ used for the density (as assumed by Monaghan & Kocharyan 1995). Indeed one of our findings from this paper (discussed below) is that use of a standard (bell-shaped) density kernel for drag computations can be significantly inaccurate. We thus use $D$ to denote the kernel employed for the drag interpolation. #### 2.4.2 Choice of smoothing length in the drag terms A key issue is the choice of smoothing length involved in the interpolation term (51) when the gas and dust have different spatial resolutions, as illustrated in Fig. 1. We have found from experiment that it is very important to smooth the drag term using the maximum smoothing length of the two fluids, rather than using an average (c.f. Sec. 4.6 and also Ayliffe et al. 2011). Otherwise, unphysical resolution-dependent clumping of one fluid below the scale of the other can occur. For gas this presents less of a problem because there remain pressure gradients that prevent such clumping. However, for dust it is crucial since there are no forces that can otherwise counterbalance any artificial over-concentration. Since most astrophysical problems involve the concentration of dust in a flow of gas, a straightforward approach is to simply use the gas smoothing length when computing the drag interaction. Unless otherwise specified (Sec. 4.6) this is the approach we adopt in this paper. #### 2.4.3 Errors in the integral drag interpolant The origin of Eq. 51 can be understood by considering the projection of $\left<\Delta\mathbf{v}\right>$ onto $\hat{\mathbf{r}}_{\alpha\alpha^{\prime}}$, the projection of $\hat{\mathbf{r}}$ onto the coordinate $\alpha$ (which equivalently denotes the coordinates $x$, $y$ or $z$ as the system is invariant by rotation) and use a Taylor expansion of $K$ and $\textbf{v}_{\mathrm{d}}(\mathbf{x}^{\prime})$ around their values on $\mathbf{x}$: $\displaystyle\left<K\Delta\mathbf{v}\right>^{\alpha}=$ $\displaystyle\nu\int\mathrm{d}\mathbf{x}^{\prime}\hat{\bf r}^{\alpha}$ $\displaystyle\left\\{K\Delta\mathbf{v}(\mathbf{x})+\frac{\partial(K\Delta\mathbf{v})}{\partial\mathbf{x}}\cdot(\mathbf{x}-\mathbf{x}^{\prime})+\mathcal{O}\left((\mathbf{x}-\mathbf{x}^{\prime})^{2}\right)\right\\}$ $\displaystyle\cdot\hat{\mathbf{r}}D\left(\mathbf{x}-\mathbf{x}^{\prime},h\right).$ (53) giving $\left<K\Delta\mathbf{v}\right>^{\alpha}=\nu K\Delta\mathbf{v}(\mathbf{x})^{\beta}I^{\alpha\beta}+\nu\frac{\partial(K\Delta\mathbf{v}^{\alpha})}{\partial\mathbf{x}^{\gamma}}J^{\alpha\beta\gamma}+\mathcal{O}\left(h^{2}\right),$ (54) where $\displaystyle I^{\alpha\beta}$ $\displaystyle\equiv$ $\displaystyle\int\mathrm{d}\mathbf{x}^{\prime}\hat{r}^{\alpha}~{}\hat{r}^{\beta}D\left(\mathbf{x}-\mathbf{x}^{\prime},h\right),$ (55) $\displaystyle J^{\alpha\beta\gamma}$ $\displaystyle\equiv$ $\displaystyle\int\mathrm{d}\mathbf{x}^{\prime}\hat{r}^{\alpha}\hat{r}^{\beta}\hat{r}^{\gamma}D\left(\mathbf{x}-\mathbf{x}^{\prime},h\right).$ (56) This shows that Eq. (51) is a second-order approximation to the drag term, that is, $\left<K\Delta\mathbf{v}\right>^{\alpha}=K\Delta\mathbf{v}^{\alpha}+\mathcal{O}\left(h^{2}\right),$ (57) provided the normalisation conditions $\displaystyle I^{\alpha\beta}$ $\displaystyle=$ $\displaystyle\frac{\delta^{\alpha\beta}}{\nu},$ (58) $\displaystyle J^{\alpha\beta\gamma}$ $\displaystyle=$ $\displaystyle 0,$ (59) hold. Condition (59) and the zeroing of the off-diagonal terms in Eq. 58 may be proved straightforwardly by the fact that the integrals in (55)–(56) are odd. The normalisation condition of the diagonal terms in Eq. 58 arises because in 3D we have $\displaystyle I^{xx}+I^{yy}+I^{zz}$ $\displaystyle=$ $\displaystyle 1,$ (60) $\displaystyle I^{xx}=I^{yy}=I^{zz},$ (61) giving $I^{xx}=I^{yy}=I^{zz}=1/\nu$. This explains the factor of $\nu$ in front of the drag summation term. #### 2.4.4 Discretisation of drag term Discretising Eq. 51 provides the SPH translation of the acceleration due to the drag term for both the gas and the dust. Replacing the integral by a summation (over particles of the opposing type) and $\rho{\rm d}V$ with the particle mass, we have $\left(\frac{\mathrm{d}\textbf{v}_{a}}{\mathrm{d}t}\right)_{\mathrm{drag}}=\frac{1}{\hat{\rho}_{\mathrm{g}}}\left<K\Delta\mathbf{v}\right>=\nu\sum_{j}m_{j}\frac{K_{aj}}{\hat{\rho}_{a}\hat{\rho}_{j}}\left({\bf v}_{aj}\cdot\hat{\textbf{r}}_{aj}\right)\hat{\textbf{r}}_{aj}D_{aj}(h_{a}),$ (62) for a gas particle and $\left(\frac{\mathrm{d}{\bf v}_{i}}{\mathrm{d}t}\right)_{\mathrm{drag}}=\frac{1}{\hat{\rho}_{\mathrm{g}}}\left<K\Delta\mathbf{v}\right>=-\nu\sum_{b}m_{b}\frac{K_{bi}}{\hat{\rho}_{b}\hat{\rho}_{i}}\left({\bf v}_{bi}\cdot\hat{\textbf{r}}_{bi}\right)\hat{\textbf{r}}_{bi}D_{ib}(h_{b}),$ (63) for a dust particle, where we have defined ${\bf v}_{aj}\equiv{\bf v}^{\rm g}_{a}-{\bf v}^{\rm d}_{j}$ and $\hat{\bf r}_{aj}\equiv({\bf r}_{a}-{\bf r}_{j})/\|{\bf r}_{a}-{\bf r}_{j}\|$. Importantly, from Eqs. 62–63, we have: $\sum_{a}m_{a}\left(\frac{\mathrm{d}\textbf{v}_{a}}{\mathrm{d}t}\right)_{\mathrm{drag}}+\sum_{i}m_{i}\left(\frac{\mathrm{d}{\bf v}_{i}}{\mathrm{d}t}\right)_{\mathrm{drag}}=0,$ (64) which ensures that the momentum is exactly exchanged between the gas and the dust phase by the SPH formalism. Similarly $\sum_{a}m_{a}{\bf r}_{a}\times\left(\frac{\mathrm{d}\textbf{v}_{a}}{\mathrm{d}t}\right)_{\mathrm{drag}}+\sum_{i}m_{i}{\bf r}_{i}\times\left(\frac{\mathrm{d}{\bf v}_{i}}{\mathrm{d}t}\right)_{\mathrm{drag}}=0,$ (65) showing that the total angular momentum is conserved. #### 2.4.5 Errors in the SPH drag interpolation A key point to note in the formulation of drag terms is that the criterion for an accurate kernel drag estimate is different from that required for an accurate density estimate (Eq. 23). Taking Eq. 62 and expanding the velocities and the drag coefficient $K$ around the position of the gas particle ${\bf r}_{a}$, to lowest order, i.e. $K_{aj}\left({\bf v}^{g}_{a}-{\bf v}^{d}_{j}\right)=K_{a}\left({\bf v}^{g}_{a}-{\bf v}^{d}_{a}\right)+\mathcal{O}(h),$ (66) we find $-\nu\frac{K_{a}}{\hat{\rho}_{a}}\left({\bf v}^{g}_{a}-{\bf v}^{d}_{a}\right)\cdot\sum_{j}\frac{m_{j}}{\hat{\rho}_{j}}\hat{\textbf{r}}_{aj}\hat{\textbf{r}}_{aj}D_{aj}+\mathcal{O}(h),$ (67) implying a discrete normalisation condition on the drag kernel of the form $\nu\sum_{j}\frac{m_{j}}{\hat{\rho}_{j}}\hat{\textbf{r}}_{aj}^{\alpha}\hat{\textbf{r}}_{aj}^{\beta}D_{aj}\approx\delta^{\alpha\beta}.$ (68) The condition (68) implies that the summation on the diagonal terms ($xx$, $yy$, $zz$) are equal to unity, while the summations on off-diagonal terms ($xy$, $xz$, $yz$) should be zero. The accuracy with which this normalisation condition is satisfied depends on the particle arrangement. While we find that the diagonal terms are well computed using standard (bell-shaped) kernels, we find that — apart from the special case where the dust particles lie on top of the gas particles — the off-diagonal terms can be very poorly normalised. Fig. 3 shows the $xy$ component of Eq. 68 as a function of the smoothing length (in units of the particle spacing, $\Delta x$) computed for a dust particle offset by $\Delta x/4$ in the x-direction from a cubic lattice of gas particles in 3D. Using the cubic spline kernel (top left) results in errors of order 5-10% of the diagonal terms for reasonable neighbour numbers ($h/\Delta x\approx 1.1-1.5$). Furthermore, improving the smoothness of the kernel by using the $M_{6}$ quintic or even the Gaussian (top right) does _not_ significantly reduce the error. In numerical tests (Sec. 4.2) this manifests as a large error in the drag between the two fluids, implying that a more suitable kernel is highly desirable. Figure 2: Functional form of the standard bell-shaped cubic spline (top left) and quintic (top right) kernels, compared to the double-hump versions of these kernels (bottom row). Kernel functions are shown by the solid/black lines, while the long-dashed/red and short-dashed/green lines correspond to the first and second derivatives, respectively. We find that double-hump kernels are significantly more accurate than bell-shaped kernels when computing SPH drag terms (see Fig. 3). Figure 3: Accuracy with which the normalisation condition for the drag force is computed using standard bell-shaped kernels (top row) and double-hump kernels (bottom row). The plots show the $xy$ component of Eq. 68 computed on a dust particle offset from a regular cubic lattice of gas particles, as a function of the smoothing length in units of the particle spacing ($h/\Delta x$). With the bell-shaped kernels (top row) the errors are of order $5-10\%$ (the off-diagonal terms should sum to zero). Changing to double-hump shaped kernels (bottom row) gives errors $\lesssim 0.5\%$. #### 2.4.6 Drag kernel function After conducting a search for suitable alternative kernels, we found that the so-called “double-hump” shaped kernels (Fulk & Quinn, 1996) gave a substantial improvement in accuracy — that is, giving errors in the computation of Eq. 68 of similar order to the bell-shaped kernels in computing Eq. 23. Defining the kernel function as previously $D\left(r,h\right)=\frac{\sigma}{h^{\nu}}g\left(q\right),$ (69) we construct double-hump kernels from the $M_{4}$ cubic and $M_{6}$ quintic kernels using $g(q)=q^{2}f(q),$ (70) giving, for example, the “double $M_{4}$ cubic” (bottom left panel of Fig. 2), the “double $M_{6}$ quintic” (bottom right panel of Fig. 2) and similarly the “double Gaussian”. The normalisation constants are found in the usual manner by enforcing $\int D{\rm d}V=1$, i.e., $\sigma\int g(q){\rm d}V=1,$ (71) where ${\rm d}V$ corresponds to ${\rm d}q$, $2\pi q{\rm d}q$ and $4\pi q^{2}{\rm d}q$ in one, two and three dimensions, respectively. The normalisation constants for the double cubic, double quintic and double Gaussian are given by $\displaystyle\sigma_{\textrm{double M}_{4}}$ $\displaystyle=$ $\displaystyle\left[2,\frac{70}{31\pi},\frac{10}{9\pi}\right];$ (72) $\displaystyle\sigma_{\textrm{double M}_{6}}$ $\displaystyle=$ $\displaystyle\left[\frac{1}{60},\frac{42}{2771\pi},\frac{1}{168\pi}\right];$ (73) $\displaystyle\sigma_{\textrm{double Gaussian}}$ $\displaystyle=$ $\displaystyle\frac{2}{\nu}\pi^{-\nu/2},$ (74) in [1,2,3] dimensions. The computation of the off-diagonal term in (68) for the double-cubic and double-Gaussian kernels are shown in the bottom row of Fig. 3 and indeed show a substantial improvement, giving errors $\lesssim 0.5\%$ compared to the $5-10\%$ errors obtained using the standard kernels (top row). This improvement in accuracy is also reflected in our numerical tests (c.f. Sec. 4.2). It is also possible to physically understand the reason why the double hump kernel is suited to deal with drag computation. In treating multi-fluid interactions, one requires the information of one type of particle at the location of a particle of the opposing type. Assuming that the number of dimensions of the space is three, the SPH smoothing of the physical quantity $A$ corresponds approximately to $\begin{array}[]{rcl}\displaystyle 4\pi^{2}\\!\\!\int_{0}^{1}\\!\\!\\!A\left(q\right)D(q)\mathrm{d}q&\simeq&\displaystyle 4\pi^{2}\\!\\!\int_{0}^{1}\\!\\!\\!A\left(q\right)\frac{\delta\left(q+q_{\mathrm{M}}\right)+\delta\left(q-q_{\mathrm{M}}\right)}{2}\mathrm{d}q\\\\[13.00005pt] &&=\displaystyle\frac{A\left(-q_{\mathrm{M}}\right)+A\left(q_{\mathrm{M}}\right)}{2},\end{array}$ (75) showing that — for a given particle — the double hump kernel provides an average value of a physical quantity stored in the neighbours of the other species and located at a distance $q=q_{\mathrm{M}}$ of the particle. For the same reason, the poor accuracy of the bell-shaped kernels can be understood because the maximum weight corresponds to $q=0$, where in general, no particle of the other type is present. #### 2.4.7 Frictional heating terms due to drag When the system is made of a single gas fluid, the _specific_ thermal energy $u$ is a function of state whose total derivative is expressed by: $\rm{d}u=T\rm{d}s+\frac{P}{\rho^{2}}\rm{d}\rho.$ (76) To generalise this relation with two-fluids interacting with a drag term, we derive an additional term for Eq. 76 arising from exchange of momentum (all the other quantities fixed) considering a closed thermally isolated system made of gas and dust SPH particles, whose energy exchange arises only because of momentum exchange (i.e. drag) between two states (denoted _i_ and _f_ , respectively). Applying the first law of thermodynamics to an infinitesimal transformation of the system, we have: $\rm{d}u+\rm{d}e_{\rm{k}}=\delta w_{\rm{i}\to\rm{f}}+\delta q_{\rm{i}\to\rm{f}},$ (77) where $u$ is the total specific internal energy, $e_{\rm{k}}$ is the macroscopic kinetic energy of the system and $w$ is the total work and $q$ is the total heat exchanged during the transformation. Assuming that the transformation occurs slowly enough for the gas to remain in thermodynamic equilibrium, Eq. 77 reduces to: $\rm{d}u\|_{s,\rho}+\rm{d}e_{\rm{k}}=T\rm{d}s+\frac{P}{\rho^{2}}\rm{d}\rho=0.$ (78) Consequently, $\displaystyle\rm{d}u\|_{s,\rho}=-\rm{d}e_{\rm{k}}=$ $\displaystyle-\sum_{a}\frac{\left(\bf{v}_{a}+\left.\rm{d}\bf{v}_{a}\right\|_{s_{a},\rho_{a}}\right)^{2}}{2}+\sum_{a}\frac{\bf{v}_{a}^{2}}{2}$ (79) $\displaystyle-\sum_{k}\frac{\left(\bf{v}_{k}+\left.\rm{d}\bf{v}_{k}\right\|_{s_{k},\rho_{k}}\right)^{2}}{2}+\sum_{k}\frac{\bf{v}_{k}^{2}}{2}.$ As the conservation of the momentum during the transformation ensures that: $\sum_{k}\left.\rm{d}\bf{v}_{k}\right\|_{s_{k},\rho_{k}}=-\sum_{a}\left.\rm{d}\bf{v}_{a}\right\|_{s_{a},\rho_{a}},$ (80) we obtain: $\rm{d}u\|_{s,\rho}=\sum_{a}\bf{v}_{ka}\cdot\left.\rm{d}\bf{v}_{a}\right\|_{s_{a},\rho_{a}}.$ (81) Using the expression Eq. 62 which gives the evolution of the velocity of a gas particle due to drag term (i.e. at constant specific entropy and density): $\frac{\rm{d}u\|_{s,\rho}}{\rm{d}t}=\sum_{a}\frac{\rm{d}u_{a}}{\rm{d}t}=\sum_{a}\mathbf{v}_{ka}\cdot\left[\nu\sum_{k}m_{k}\frac{K_{ak}}{\hat{\rho}_{a}\hat{\rho}_{k}}\left(\textbf{v}_{ak}\cdot\hat{\textbf{r}}_{ak}\right)\hat{\textbf{r}}_{ak}D_{ak}(h_{a})\right].$ (82) which implies that the evolution of the specific internal energy for each gas particle is given by: $\left(\frac{\rm{d}u_{a}}{\rm{d}t}\right)_{\rm drag}=\frac{\Lambda_{\rm drag}}{\hat{\rho}_{a}}=\nu\sum_{k}m_{k}\frac{K_{ak}}{\hat{\rho}_{a}\hat{\rho}_{k}}\left(\textbf{v}_{ak}\cdot\hat{\textbf{r}}_{ak}\right)^{2}D_{ak}(h_{a}).$ (83) The positive value of ${\rm d}u_{a}/{\rm{d}t}$ (and the fact that the kernel function is always positive) ensures a positive definite contribution to the specific internal energy of each SPH gas particle from frictional drag heating. Eq. 83 provides the SPH translation of the drag heating term (Eq. 13). It is straightforward to show that with the above expression the total energy is exactly conserved, i.e. $\sum_{a}m_{a}\frac{{\rm d}u_{a}}{{\rm d}t}+\sum_{a}m_{a}{\bf v}_{a}\cdot\frac{{\rm d}{\bf v}_{a}}{{\rm d}t}+\sum_{i}m_{i}{\bf v}_{i}\cdot\frac{{\rm d}{\bf v}_{i}}{{\rm d}t}=0.$ (84) #### 2.4.8 Thermal coupling terms The thermal coupling terms can be expressed in SPH form using (Monaghan & Kocharyan, 1995) $\Lambda_{{\rm therm},a}=\sum_{j}m_{a}\frac{Q_{aj}}{\hat{\rho}_{a}\hat{\rho}_{j}}(T_{a}-T_{j})W_{aj}+\sum_{j}m_{a}\frac{R_{aj}}{\hat{\rho}_{a}\hat{\rho}_{j}}a(T_{a}^{4}-T_{j}^{4})W_{aj},$ (85) for a gas particle, and $\Lambda_{{\rm therm},i}=-\sum_{b}m_{i}\frac{Q_{bi}}{\hat{\rho}_{i}\hat{\rho}_{b}}(T_{b}-T_{j})W_{bi}-\sum_{b}m_{b}\frac{R_{bi}}{\hat{\rho}_{b}\hat{\rho}_{i}}a(T_{b}^{4}-T_{i}^{4})W_{bi},$ (86) for a dust particle. Note that we use the standard SPH kernel for the thermal coupling terms. Detailed study of the effect of the thermal coupling terms, or specific expressions for $Q$ and $R$, are beyond the scope of this paper, and thus for the tests in Sec. 4 we simply set $T_{\rm d}=T_{\rm g}$. #### 2.4.9 Drag coefficient In general the drag coefficient $K$ is a function of the properties of both the gas and dust. For example in the linear Epstein regime relevant to dilute gases in the limit of low Mach numbers, the coefficient $K_{ak}$ is given by $K_{ak}=\frac{4}{3}\pi\sqrt{\frac{8}{\pi\gamma}}\frac{\hat{\rho}_{k}}{m_{\rm{d}}}\frac{\hat{\rho}_{a}}{\theta_{a}}s^{2}c_{\mathrm{s},a},$ (87) where $s$ is the grain radius, $m_{\rm{d}}$ is the grain mass, $\gamma$ is the adiabatic index, $c_{\mathrm{s},a}$ is the gas sound speed. Since the basic dust-gas algorithm described above is insensitive to the specific form of the drag, we consider only constant drag coefficients in this paper in order to benchmark the method. The detailed implementation of a full range of both linear and non-linear physical drag formulations is considered in Paper II. ## 3 Timestepping ### 3.1 Empirical timestep criterion The drag terms impose an additional constraint on the timestep $\Delta t$, such that it has to be smaller than a critical value $\Delta t_{\rm{c}}$ for an explicit scheme (e.g. Leapfrog) to remain stable. Empirically, Monaghan & Kocharyan (1995) use the criterion: $\Delta t<\min\left(\frac{\rho}{K}\right),$ (88) which is essentially the minimum of the drag stopping time taken over all of the SPH particles. ### 3.2 Von Neumann stability analysis A more precise criterion can be derived by considering the stability of a simple explicit scheme such as the Forward Euler method. We consider the evolution of the drag terms over a single drag timetstep $\Delta t$, calculating the velocities at the timestep $n+1$ from the velocities at the timestep $n$. Considering only the time-discretisation of the equations, we have $\displaystyle\frac{{\bf v}^{n+1}_{\rm g}-\mathbf{v}^{n}_{\rm g}}{\Delta t}$ $\displaystyle=$ $\displaystyle-\frac{K}{\hat{\rho}_{\mathrm{g}}}\left({\bf v}_{\rm g}-{\bf v}_{\rm d}\right),$ (89) $\displaystyle\frac{{\bf v}^{n+1}_{\rm d}-\mathbf{v}^{n}_{\rm d}}{\Delta t}$ $\displaystyle=$ $\displaystyle+\frac{K}{\hat{\rho}_{\mathrm{d}}}\left({\bf v}_{\rm g}-{\bf v}_{\rm d}\right),$ (90) We then perform a standard Von Neumann analysis, considering a perturbation of the velocity field with respect to equilibrium at the timestep $m$ corresponding to a monochromatic plane wave, i.e. $\displaystyle{\bf v}^{m}_{\rm g}$ $\displaystyle=$ $\displaystyle{\bf V}^{m}_{\rm{g}}e^{ikx},$ (91) $\displaystyle{\bf v}^{m}_{\rm d}$ $\displaystyle=$ $\displaystyle{\bf V}^{m}_{\rm{d}}e^{ikx},$ (92) where ${\bf v}^{m}_{\rm{g}}$ and ${\bf v}^{m}_{\rm{d}}$ are complex constants and $k$ is the wavenumber. Substituting Eqs. 91 and 92 into Eqs. 89 and 90 leads to the linear system $\left(\begin{array}[]{c}{\bf V}_{\rm{g}}\\\ {\bf V}_{\rm{d}}\end{array}\right)^{n+1}=\left(\begin{array}[]{cc}1-\Delta t~{}\frac{K}{\hat{\rho}_{\mathrm{g}}}&\Delta t~{}\frac{K}{\hat{\rho}_{\mathrm{g}}}\\\ \Delta t~{}\frac{K}{\hat{\rho}_{\mathrm{d}}}&1-\Delta t~{}\frac{K}{\hat{\rho}_{\mathrm{d}}}\end{array}\right)\left(\begin{array}[]{c}{\bf V}_{\rm{g}}\\\ {\bf V}_{\rm{d}}\end{array}\right)^{n}.$ (93) The two complex eigenvalues $\Lambda_{\pm,aj}$ of the matrix $\mathcal{M}$ are given by: $\Lambda_{\pm,ai}=1-\frac{\Delta t}{2}\left(\frac{K}{\hat{\rho}_{\mathrm{g}}}+\frac{K}{\hat{\rho}_{\mathrm{d}}}\right)\pm\frac{\Delta t}{2}\left(\frac{K}{\hat{\rho}_{\mathrm{g}}}+\frac{K}{\hat{\rho}_{\mathrm{d}}}\right).$ (94) The condition for the numerical scheme to remain stable ($\|\Lambda_{-}\|<1$) implies a minimum timestep given by $\Delta t<\Delta t_{\rm{c}}=\frac{\hat{\rho}_{\mathrm{g}}\hat{\rho}_{\mathrm{d}}}{K(\hat{\rho}_{\mathrm{g}}+\hat{\rho}_{\mathrm{d}})}(\equiv t_{\rm s}).$ (95) We note that this expression differs slightly to the one suggested by Monaghan & Kocharyan (1995) (Eq. 88) as it involves the _physical_ drag stopping time $t_{\rm s}$ — i.e. the typical time to damp the differential velocity between the gas and the dust fluids — rather than $\frac{K}{\rho}$. The timestep of Monaghan & Kocharyan (1995) is thus correct in the limit where the density of one phase is negligible compared to the density of the other phase, but becomes erroneous in the case of two fluids having densities of the same order of magnitude. Note this would apply to grid codes also. ### 3.3 SPH explicit timestep The stability criterion for the full SPH system (and also for other explicit schemes) is expected to be similar to that derived for the continuum case (Eq. 95). The main difference is that the drag coefficient $K$ is in general only defined on particle _pairs_ rather than individual particles. We thus take the minimum of Eq. 95 over a particle’s neighbours, i.e. $\Delta t_{\rm{c},a}=\min_{k}\left[\frac{\hat{\rho}_{a}\hat{\rho}_{k}}{K_{ak}(\hat{\rho}_{a}+\hat{\rho}_{k})}\right];\hskip 14.22636pt\Delta t_{\rm{c},i}=\min_{b}\left[\frac{\hat{\rho}_{b}\hat{\rho}_{i}}{K_{bi}(\hat{\rho}_{b}+\hat{\rho}_{i})}\right];$ (96) for gas and dust particles, respectively. ### 3.4 Implicit timestepping For strong drag regimes, the timestep restriction imposed by Eq. (96) becomes prohibitive, and an implicit timestepping algorithm is required, as proposed by Monaghan (1997). We use only explicit timestepping for the tests shown in this paper, with implicit timestepping methods discussed in detail in Paper II. ## 4 Numerical tests Despite a number of codes having already been developed for simulating astrophysical gas-dust mixtures, none have been benchmarked against a wide range of test problems relevant to astrophysics. For example, while Monaghan & Kocharyan (1995), Maddison et al. (2003) consider drag on a single dust particle in a box of gas (similar to our dustybox test below), no waves or shocks are considered. Similarly Paardekooper & Mellema (2006) benchmark their algorithm against a single dust-gas shock problem with only a qualitative solution. Other authors simply check that the timescale for settling in an accretion disc is roughly consistent (Barrière-Fouchet et al., 2005) or provide no tests at all (Rice et al., 2004; Fromang & Nelson, 2005; Fromang & Papaloizou, 2006). In the absence of known analytic solutions for simple problems, Johansen et al. (2007), Miniati (2010) and Bai & Stone (2010) use the linear growth rates for the streaming instability (Youdin & Goodman, 2005) as a test problem, though this is already a complicated problem. In this paper we present a comprehensive suite of test problems designed to investigate all aspects of our algorithm relevant to astrophysics. These we refer to as dustybox, dustywave, dustyshock, dustysedov and dustydisc. Analytic solutions for the dustybox and dustywave problems have been derived in Laibe & Price (2011a), while the solution for dustyshock is known in the limit of high drag. The solutions for dustysedov and dustydisc are more qualitative but are important reference problems for astrophysical dust-gas mixtures. We consider simulation of the streaming instability to be of sufficient importance and complexity to be covered in detail in a separate paper. Note that all of the tests considered in this paper are performed, for simplicity, using a constant drag coefficient $K$. Realistic drag regimes are considered in Paper II. Figure 4: Dust velocity as a function of time in the dustybox problem, using $2\times 20^{3}$ particles, a dust-to-gas ratio of unity and a constant drag coefficient $K=1$. Use of the standard cubic spline kernel for the drag terms (short dashed/green line) results in errors of order $10\%$ in the velocities compared to the exact solution (long dashed/red line). Using a double-hump kernel (solid black) improves the accuracy — for the same computational cost — by a factor of several hundred, to $\lesssim 0.1\%$. ### 4.1 Code implementation Implementation of the algorithm into a standard SPH code with explicit timestepping is relatively straightforward. The main changes are i) to store a particle type allowing setup and simulation of multiple fluids; ii) to compute the densities and smoothing lengths on each fluid as described in Sec. 2.2.1; iii) compute the drag term between each fluid according to Eqs. 62–63 and the heating term given by Eq. 83 and iv) (optional for astrophysics) to implement the modifications to the equations of motion due to the volume-filling fraction of the dust. We have implemented the two-fluid algorithm into both the $N-$dimensional ndspmhd test code (Price, 2011) and into the parallel phantom code for 3D problems (Price & Federrath, 2010; Lodato & Price, 2010). ### 4.2 dustybox: Two fluid drag in a periodic box The dustybox problem presented by Laibe & Price (2011a) involves two fluids in a periodic box moving with a differential velocity ($\Delta v_{0}=v_{d,0}-v_{g,0}$). It is similar to the test performed by Monaghan & Kocharyan (1995) showing the drag on a single dust grain in a box of gas, except that here we consider the dust as a fluid, meaning that the densities and smoothing lengths of both phases are computed self-consistently. #### 4.2.1 dustybox: Setup We setup the particles in a 3D periodic domain $x,y,z\in[0,1]$ such that the densities $\hat{\rho}_{\mathrm{g}}$ and $\hat{\rho}_{\mathrm{d}}$ and the gas pressure $P_{\mathrm{g}}$ are constant, and neglect the dust intrinsic volume by fixing the volume fraction $\theta=1$. The box is filled by $20^{3}$ SPH gas particles set up on a regular cubic lattice and $20^{3}$ dust particles set up on a cubic lattice shifted by half of the lattice step in each direction. The gas sound speed, the gas and the dust densities are set to unity in code units and no artificial viscosity terms are applied. We give the fluids initial velocities $v_{\rm d}=1$ and $v_{\rm g}=0$. During the simulation, we verified that both the total linear and angular momentum are exactly conserved as expected (Eqs. 64–65). We have also verified that 1) the offset of the dust lattice with respect to the gas lattice and 2) the timestepping scheme do not affect the results. #### 4.2.2 dustybox: Choice of drag kernel Fig. 4 shows the dust velocity as a function of time in the dustybox test using $\hat{\rho}_{\mathrm{g}}=\hat{\rho}_{\mathrm{d}}=1$ (i.e., a dust to gas ratio of unity) and $K=1$, with the exact solution from Laibe & Price (2011a) shown by the long-dashed/red line. Using the cubic spline $M_{4}$ kernel (short-dashed/green line), the errors are of order 10%. Since these errors are due to intrinsic bias in the kernel interpolation of the drag terms (Fig. 3), they are independent of resolution, though can be improved – at considerable cost – by increasing the ratio of smoothing length to particle spacing (i.e., the neighbour number). By comparison, use of the double-hump cubic spline kernel gives errors $\lesssim 0.1\%$ (solid/black line) with no additional overhead in terms of cost. #### 4.2.3 dustybox: Effect of drag coefficient and dust-to-gas ratio Fig. 5 is identical to Fig. 4 but for a range of drag coefficients $K=0.01,0.1,1,10,100$, compared to the exact solution in each case given by a solid/black line. Irrespective of the value of $K$, both gas and dust velocities relax to the barycentric velocity ($\textbf{v}_{\mathrm{g}}=\textbf{v}_{\mathrm{d}}=0.5$) in a few stopping times $t_{\rm{s}}=(\hat{\rho}_{\mathrm{g}}\hat{\rho}_{\mathrm{d}})/[K(\hat{\rho}_{\mathrm{g}}+\hat{\rho}_{\mathrm{d}})]$. Using the double-hump cubic, an accuracy between 0.1 and 1% is achieved in all cases (long dashed/red lines). Fig. 6 is similar, but varying the dust-to-gas ratio using $\hat{\rho}_{\mathrm{d}}/\hat{\rho}_{\mathrm{d}}=0.01,0.1,1,10,100$ (achieved by varying $\hat{\rho}_{\mathrm{d}}$ with $\hat{\rho}_{\mathrm{g}}=1$) and using $K=1$. This changes both the drag stopping time and the barycentric velocity towards which the system relaxes. Here again, an accuracy between 0.1 and 1% is achieved in all cases. Figure 5: As in Fig. 4 but using only the double hump cubic kernel with a range of drag coefficients $K=0.01,0.1,1,10,100$ (top-to-bottom, solid/black lines), compared with the exact solution in each case given by the long- dashed/red lines. Figure 6: As in Figs. 4 and 5 but varying the dust-to-gas ratio $\hat{\rho}_{\mathrm{d}}/\hat{\rho}_{\mathrm{d}}=0.01,0.1,1,10,100$ (top-to-bottom, solid/black lines) and a fixed drag coefficient $K=1$ using the double hump cubic kernel. Exact solutions for each case are given by the long-dashed/red lines. ### 4.3 dustywave: Sound waves in a dust-gas mixture The exact solution for linear waves propagating in a dust-gas mixture (dustywave) has been presented by Laibe & Price (2011a). We have performed a series of tests involving the propagation of a sound wave along the $x-$axis in both one and three dimensions in a periodic box, adopting the setup described in Table 2 of Laibe & Price (2011a). The dustywave problem is more complex than the dustybox problem as the motion of the mixture is driven by both the drag and the gas pressure. Specifically, Laibe & Price (2011a) derive the dispersion relation: $\omega^{3}+iK\left(\frac{1}{\hat{\rho}_{\mathrm{g}}}+\frac{1}{\hat{\rho}_{\mathrm{d}}}\right)\omega^{2}-k^{2}c_{\rm s}^{2}\omega-iK\frac{k^{2}c_{\rm s}^{2}}{\hat{\rho}_{\mathrm{d}}}=0,$ (97) for solutions in the form $e^{i\left(kx-\omega t\right)}$. At high drag, Eq. 97 can be expanded in a Taylor series, which to first order gives: $\omega=\pm k\tilde{c}_{\rm s}-i\frac{\hat{\rho}_{\mathrm{g}}\hat{\rho}_{\mathrm{d}}}{K\left(\hat{\rho}_{\mathrm{g}}+\hat{\rho}_{\mathrm{d}}\right)}k^{2}c_{\mathrm{s}}^{2}\left(\frac{1-A^{2}}{2}\right)$ (98) where the effective sound speed is defined according to $\tilde{c}_{\rm s}\equiv c_{\rm s}A=c_{\rm s}\left(1+\frac{\hat{\rho}_{\mathrm{d}}}{\hat{\rho}_{\mathrm{g}}}\right)^{-\frac{1}{2}}.$ (99) The first term of Eq. 98 gives the propagation of the centre of mass of the mixture at the effective sound speed $\tilde{c}_{\rm s}$. The second term corresponds to a corrective dissipative term since $A\in\left[0,1\right]$. #### 4.3.1 dustywave: Setup The equilibrium state is characterised by the two phases at rest where the gas sound speed, and both gas and dust densities are set to unity in code units. In 1D this is achieved by placing equally spaced particles in the periodic domain $x\in[0,1]$. For the 3D simulations, the tests are run in a periodic box $x,y,z\in[0,1]$ with gas particles set up on a regular cubic lattice and dust particles set up on a cubic lattice shifted by half of the lattice step in each directions. As previously, no artificial viscosity is applied. We set the relative amplitude of the perturbation to $10^{-4}$ in both velocity and density in order to remain in the linear acoustic regime for which the solution in Laibe & Price (2011a) is derived (we have verified that running the same simulations setting the relative amplitudes to $10^{-8}$ gives the same results). The density perturbation is applied to the particles as described in Appendix B of Price & Monaghan (2004). We adopt an isothermal equation of state $P=c_{\rm s}^{2}\rho$ with $c_{\rm s}=1$. Figure 7: Results of the dustywave test in 3D at $t=0$ (top row) and after 1 and 2 wave periods (middle and bottom rows) using $2\times 32^{3}$ particles, $K=1$ and a dust-to-gas ratio of unity. The analytic solution is given by the solid/red (gas) and long-dashed/red (dust) lines. The standard cubic spline kernel (green points, left panel) performs poorly on this test. Using the double hump cubic kernel (black points, left panel) both the amplitude and frequency are correct but there remains a small phase error due to the kernel bias which can be corrected by using the smoother double-hump quintic kernel (black points, right panel). #### 4.3.2 dustywave: Effect of the smoothing kernel The results of the dustywave test in 3D using $2\times 32^{3}$ particles with $K=1$ and a dust-to-gas ratio of unity is shown in Fig. 7, using the standard bell-shaped cubic (green points, lower amplitude) and double-hump cubic kernel (black points, correct amplitudes) for the drag (left panel) and the double- hump quintic kernel (c.f. Sec. 2.4.6) (right panel), at $t=0$ (top) and after 1 and 2 periods (middle and bottom panels). The numerical solutions (green and black markers) may be compared to the analytic solutions given by the solid/red (gas) and long-dashed/red (dust) lines. The amplitude and frequency of the solution are only correctly captured using double-hump kernels. With the double-hump cubic employed (left panel, black points) there remains a slight (few %) phase error in the numerical solution caused by the remaining kernel bias, independent of resolution. A similar error is found generically in multidimensional SPH simulations of linear waves (see e.g. Fig. 6 of Price & Monaghan 2005) and in standard SPH is improved by using a smoother kernel such as the quintic spline. Indeed, using the double-hump version of the quintic kernel for the drag terms and the standard quintic for the SPH terms (right panel) we find the phase error is smaller by a factor of $\sim 5$. Longer multidimensional simulations of linear waves are more complicated in SPH because placement of the gas particles on regular lattices are unstable to low-amplitude transverse modes that cause the particles to rearrange towards a “glass-like” configuration (Morris, 1996a, b). Furthermore, we find that with large drag coefficients we require extremely high resolutions to match the analytic solutions (see below), which becomes prohibitive in 3D. In one dimension however, the numerical stability of a sound wave is achieved simply by satisfying the courant condition (i.e. $\Delta t\leq 0.3c_{\rm{s}}/h$) and the timestep constraint from the drag (Eq. 95). We thus turn to 1D to investigate the full parameter range of the dustywave solution. Figure 8: Resolution study for the dustywave test in 1D using a high drag coefficient ($K=100$) and a dust-to-gas ratio of unity using 32, 64, 128, 256, 512 and 1024 particles from bottom to top. At large drag high resolution is required to resolve the small differential motions between the fluids and thus prevent over-damping of the numerical solution, corresponding to the criterion $h\lesssim c_{\rm s}t_{\rm s}$, here implying $\gtrsim 240$ particles. See also Fig. 9. Figure 9: As in Fig. 8 but showing the kinetic energy as a function of time in the numerical solution at progressively increasing resolution, compared to the analytic solution given by the solid black line. The kinetic energy decay converges to the analytic solution at $\sim 256-512$ particles per wavelength, implying a demanding resolution criterion ($h\lesssim c_{\rm s}t_{\rm s}$) for high drag. Figure 10: Parameter study of the 1D dustywave problem, varying the drag coefficient from $0.01$ to $100$ (top to bottom) and using two different dust- to-gas ratios ($\hat{\rho}_{\mathrm{d}}/\hat{\rho}_{\mathrm{g}}=1$, left figure and $\hat{\rho}_{\mathrm{d}}/\hat{\rho}_{\mathrm{g}}=0.01$, right figure). Results are shown after 5 wave periods using 128 particles except for the $K=100$ case where $512$ particles are used to satisfy the resolution criterion $h\lesssim c_{\rm s}t_{\rm s}$. The double hump cubic kernel is used for the drag terms. The results obtained at these resolutions are indistinguishable from the analytic solutions (red solid [gas] and dashed [dust] lines) for both the gas (dots) and the dust (circle) particles. #### 4.3.3 dustywave: Resolution requirements at high drag Fig. 8 shows the velocity profiles after 10 periods in the 1D dustywave problem for a large drag coefficient ($K=100$) and a dust-to-gas ratio of unity, with numerical resolution as indicated. At low resolutions ($\lesssim 256$ particles per wavelength) and high drag, the amplitude of the wave in the numerical simulations (black solid [gas] and open [dust] circles, on top of each other) is severely overdamped compared to the analytic solution (red solid and long-dashed lines, also on top of each other). This is further illustrated in Fig. 9 which shows the kinetic energy as a function of time for simulations at different resolutions, compared to the analytic solution given by the solid red line. Figs. 8 and 9 illustrate a key difficulty that arises when considering high drag coefficients, i.e., where the drag stopping time $t_{\rm{s}}$ defined in Eq. 95 is much smaller than the period $T$ of the wave. In this case, the drag term efficiently damps the initial differential velocity between the gas and the dust in a few $t_{\rm{s}}$. However, as the pressure continues to drive the propagation of the wave in the gas, a small residual de-phasing of order $\sim c_{\rm s}t_{\rm s}$ occurs, which is simply the distance travelled by the gas before it is damped by the dust. This de-phasing induces a small differential velocity which in turn be damped by the drag. This small differential effect dissipates the kinetic energy on a timescale $\sim t_{\rm{s}}$. The spatial de-phasing between the gas and the dust represents the smallest length of the problem that must be resolved numerically in order to capture the physics of the process. If the spatial de-phasing between the gas and the dust is under-resolved, the differential velocity between the gas and the dust is artificially larger than the theoretical one, leading to a non-physical over-dissipation of the kinetic energy of the system, as observed in Figs. 8 and 9. We thus propose a resolution criterion for resolving the differential drag of the form $\Delta\lesssim c_{\rm s}t_{\rm s},$ (100) where $\Delta$ is the resolution length. For SPH, this becomes $h\lesssim c_{\rm s}t_{\rm s}.$ (101) For $K=100$ and $c_{\rm s}=\hat{\rho}_{\mathrm{g}}=\hat{\rho}_{\mathrm{d}}=1$ in code units this implies $h<0.02$, i.e. a minimum of $\sim 240$ particles (assuming $\eta=1.2$ in Eqs. 16 and 17), which is consistent with Figs. 8 and 9. Simulating dust-gas interactions at high drag therefore requires a high spatial resolution in order to accurately resolve the propagation without over-dissipating the energy of the system. This can lead to a prohibitive computational cost, somewhat counterintuitively since the drag simply tends to make the dust stick to the gas. Most importantly, this requirement is not unique to SPH and is a critical issue for any numerical method. Indeed, Bai & Stone (2010) find similarly high resolution requirements at short stopping times in their simulations of the streaming instability. #### 4.3.4 dustywave: Parameter study Fig. 10 shows the results of 1D simulations of the dustywave problem for 5 drag coefficients (from $K=10^{-2}$ to $K=10^{2}$) and two different dust to gas ratio relevant for astrophysical systems ($1$ and $0.01$ for left and right figures, respectively), showing the velocities after five periods compared with the analytic solutions in each case. The simulations employ 128 particles except for the $K=100$ case where $512$ particles have been used in order to satisfy the criterion (101). For this set of parameters, our method provides results with an excellent accuracy (better than one per cent) on the frequencies, the amplitudes and the phases of both the gas and the dust velocities (and consequently the energy of the system). For equal dust to gas ratios ($\hat{\rho}_{\mathrm{d}}/\hat{\rho}_{\mathrm{g}}=1$, left figure), both phases are equally affected by the drag. At low drag ($K=0.01$, top panel of left figure), the damping is not efficient enough for gas or the dust to be damped as the stopping time is $\sim$ hundreds of periods. At intermediate drag ($K=1$, middle panel of left figure), the damping is the most efficient for the two phases. At large drag regimes ($K=100$), the damping of the differential velocity occurs quickly, but the dust density is large enough to distort the gas propagation: the wave is de-phased by a half-period compared to the gas-only solution. With more typical astrophysical dust to gas ratios ($\hat{\rho}_{\mathrm{d}}/\hat{\rho}_{\mathrm{g}}=0.01$, right figure), the gas remains essentially unaffected by the dust. It thus propagates almost freely in the box at a velocity close to the sound speed. By contrast, the dust phase is strongly affected by the drag as shown by the $K=0.01$ case (top panel, right figure), where the damping time for the dust phase is $\sim$ one period. The differential velocity between the two phases becomes more and more efficiently damped as the drag coefficient increases (right panel, from top to bottom), making the dust phase stick to the gas. ### 4.4 dustyshock: shock tube in a dust-gas mixture Propagation of a shock in a two-fluid dust and gas mixture (the dustyshock problem hereafter) has been studied both analytically (see e.g. Rudinger 1964) and numerically (see e.g. Miura & Glass 1982; Saito et al. 2003), using grid based methods. The dustyshock occurs in two stages: a transient stage (for which no analytic solution is known and therefore studied numerically) followed by a stationary stage which consists of the solution for a pure gas solution propagating at a modified $\gamma$ and the modified sound speed (Eq. 99, see also Miura & Glass 1982). In an astrophysical context, simulations of a dusty shock were used by Paardekooper & Mellema (2006) to test their Godunov-type scheme using a Roe Solver developed to simulate astrophysical dust and gas mixtures. The hypothesis for the dust phase in these seminal studies are essentially the same as the ones used in this paper. However, unnecessary additional complications arise from their choice of the Stokes drag regime (a function of the local Reynolds number for the particles), the addition of a heat transfer term (depending on the dust conductivity and the Nusselt number of the system) and a temperature-dependent gas viscosity. For the purposes of benchmarking of our numerical scheme, we instead simulate a simplified problem: using a linear drag regime with constant drag term $K$, no heat transfer between the phases and no viscosity other than the standard shock-capturing terms used in SPH. While the evolution during the transient stage may be different from those considered in previous studies, the solution during the stationary stage remains unchanged. #### 4.4.1 dustyshock: setup We setup the dustyshock problem as a two fluid version of the standard Sod (1978) problem. Equal mass particles are placed in the 1D domain $x\in[-0.5,0.5]$, where for $x<0$ we use $\rho_{\mathrm{g}}=\rho_{\mathrm{d}}=1$, $v_{\rm{g}}=v_{\rm{d}}=0$ and $P_{\rm{g}}=1$, while for $x>0$ $\rho_{\mathrm{g}}=\rho_{\mathrm{d}}=0.125$, $v_{\rm{g}}=v_{\rm{d}}=0$ and $P_{\rm{g}}=0.1$. We use an ideal gas equation of state $P=(\gamma-1)\rho u$ with $\gamma=5/3$. The density jump means that for SPH the resolution is 8 times higher to the left of the shock than to the right. We adopt the same initial resolution in both the gas and the dust. This differs slightly from the setup used by Miura & Glass (1982) and Saito et al. (2003) where the dust is only placed in the right half of the box. Standard artificial viscosity and conductivity terms are employed for shock-capturing in SPH as described in Price (2011) with constant coefficients $\alpha_{\rm SPH}=1$, $\beta_{\rm SPH}=2$ and $\alpha_{\rm u}=1$. #### 4.4.2 dustyshock: transient evolution Figure 11: Results of the dustyshock problem with a moderate drag coefficient $K=1$ and a dust-to-gas ratio of unity. This shows the dustyshock solution during the transient stage where the analytic solution is not known (the solution for the later stationary stage is shown by the dotted red line for comparison). Results are similar to those obtained in previous studies (Miura & Glass, 1982; Saito et al., 2003). Top panels show velocity and density in both gas (solid points) and dust (open circles), while bottom panels show thermal energy and pressure in the gas. Initial particle spacing to the left of the shock in both fluids is $\Delta x=0.001$ while to the right it is $\Delta x=0.008$, giving $569$ equal mass particles in each phase. Figure 12: Results of the dustyshock problem with a high drag coefficient ($K=1000$) and a dust-to-gas ratio of unity, thus being in the stationary phase where the analytic solution is known (solid/red lines). At low resolution (left figure, same resolution as Fig. 11) the results are incorrect due to the failure to satisfy the resolution criterion at high drag (Eq. 101). With this criterion satisfied (right panel, using $2\times 11255$ particles) the numerical solution faithfully reproduces the analytic result. Thus, as in the dustywave test, extremely high resolution is required to obtain the correct solution at high drag. Fig. 11 shows the results of a simulation using $2\times 569$ particles (i.e. a particle spacing of $\Delta x=0.001$ for $x<0$) with a moderate drag coefficient ($K=1$) and a dust-to-gas ratio of unity, showing velocity and density for both the gas and dust (top panels) and the thermal energy and pressure in the gas (lower panels). With this choice of drag coefficient the system remains in the transient regime at the time shown (since $t<t_{\rm s}$). It should be noted that while there is no known analytic solution for this stage of the problem, the shock profile we obtain is similar to those found previously (see e.g. Miura & Glass 1982; Saito et al. 2003). Initially as the shock propagates in the mixture, the dust (initially at rest) dissipates the momentum and kinetic energy from the gas, lowering the propagation velocity compared to the ideal (gas only) case (dotted red line). The dust density ramps up roughly linearly behind the shock, reaching a density near the contact discontinuity roughly twice the unshocked dust density. Saito et al. (2003); Miura & Glass (1982) and Paardekooper & Mellema (2006) also found a similar behaviour, also with a factor of 2 increase in the dust density behind the shock. The gas-dust interaction to the left of the contact discontinuity and in the rarefaction wave has not been previously studied since the above authors place the dust only downwards of the shock front. We find that the dust density decreases to near zero upstream of the contact discontinuity, increasing sharply at the head of the rarefaction wave, transitioning smoothly through the rarefaction wave to match the undisturbed value. We have checked that increasing the resolution further does not change the solution significantly for this choice of drag parameters. #### 4.4.3 dustyshock: stationary regime Fig. 12 shows the results of simulations with a high drag coefficient ($K=1000$) and a dust-to-gas ratio of unity. In this case, since $t>t_{\rm s}$, the mixture quickly reaches the stationary regime. We are thus able to compare the SPH results to the analytic solution given by the solid red line (this corresponds to the standard hydrodynamic shock solution with modified sound speeds given by Eq. 99). The left figure shows the results at low resolution ($2\times 569$ particles, as in Fig. 11, while the right figure shows the results at $20\times$ higher resolution ($2\times 11255$ particles). As in the dustywave test, we find that at high drag an extremely high resolution is required to obtain the correct solution, consistent with our resolution criterion derived above (Eq. 101). If this criterion is not satisfied the numerical shock solution is strongly inaccurate (left panel). Figure 13: Cross-section slice showing density in the midplane in the 3D dustysedov problem, for both the gas (left panel) and the dust (right panel) at $t=0.1$. A dust-to-gas ratio of $0.01$ and a drag coefficient of $K=1$ have been used with $100^{3}$ SPH particles in each phase. Note the slight difference in the blast radius between the dust and the gas, consistent with the response time ($t_{\rm s}$) of the dust to the gas drag. Figure 14: Results of the 3D dustysedov test, showing the density in the gas (left figure) and dust (right figure) from a Sedov blast wave propagating in an astrophysical ($1\%$ dust-to-gas ratio) mixture of gas and dust with a constant drag coefficient $K=1$. The low dust-to-gas ratio means that the gas is only weakly affected by the drag from the dust, and is thus close to the self-similar Sedov solution (dotted/red line). The dust density is affected by the propagation of the blast, resulting in an overdensity that closely mirrors the gas overdensity. Results are shown using $50^{3}$ (top panels) and $100^{3}$ particles (bottom panels). ### 4.5 dustysedov: Sedov blast waves in a dust-gas mixture The dustysedov test concerns the propagation of a Sedov blast wave in a dust- gas mixture. Although the self-similar Sedov solution is known for the propagation of a blast wave in a gas phase, the solution for a two fluid dust- gas mixture is unknown (though at high drag as previously it may be expected that the solution should revert to the gas-only solution using the modified sound speed). We do not attempt to simulate this problem at high drag as it would involve a prohibitive computational expense, instead adopting an “astrophysical” dust-to-gas ratio of $1\%$ and a moderate drag coefficient such that the presence of dust represents only a small perturbation to the gas evolution. Results of purely hydrodynamic SPH solutions for this test can be found e.g. in Rosswog & Price (2007) and Springel & Hernquist (2002). #### 4.5.1 dustysedov: Setup We setup the dustysedov problem in a 3D periodic box (the boundary conditions are irrelevant for the times shown) at two different resolutions, filling the box $x,y,z\in[-0.5,0.5]$ by $50^{3}$ and $100^{3}$ SPH particles for both the gas and the dust. Gas particles are set up on a regular cubic lattice, with the dust particles also on a cubic lattice but shifted by half of the lattice step in each direction. We use $\alpha_{\rm SPH}=1$ and $\beta_{\rm SPH}=2$ in the artificial viscosity terms, and $\alpha_{\rm u}=1$ in the artificial conductivity term. An ideal gas equation of state $P=(\gamma-1)\rho u$ is adopted with $\gamma=5/3$. In the self-similar Sedov solution, the thermal energy of the gas is initially concentrated at $r=0$. In the SPH simulation we distribute the internal energy of the gas over the particles located inside a radius $r<r_{\rm{b}}$ where $r_{\rm b}$ is set to 2h (i.e., the radius of the smoothing kernel). In code units the total blast energy is $E=1$, with $\hat{\rho}_{\mathrm{g}}=1$ and $\hat{\rho}_{\mathrm{d}}=0.01$. For $r>r_{\rm{b}}$, the gas sound speed is set to be $2\times 10^{-5}$ in code units. The dust-to-gas ratio is set to $0.01$ to be consistent with the value measured for the interstellar medium. The drag coefficient is set to $K=1$. Translated to physical units, assuming a box size of $1$ pc, an ambient sound speed of $2\times 10^{4}$ cm/s and a gas density of $\rho_{0}=6\times 10^{-23}$ g/cm3 the energy of the blast is $2\times 10^{51}$ erg and time is measured in units of $100$ years, roughly corresponding to a supernova blast wave propagating into the interstellar medium. Obviously in a real supernova the temperature inside the blast would be much higher than the sublimation temperature of the dust, meaning that it would be quickly evaporated, so the dustysedov test is mainly useful as a benchmarking problem. #### 4.5.2 dustysedov: Results Fig. 14 shows the densities of both the gas (left figure) and the dust phase (right figure) at $t=0.1$ using $50^{3}$ (top) and $100^{3}$ (bottom) particles for both fluids. Fig. 13 shows a cross section of the density in the midplane in the high resolution ($100^{3}$) simulation, showing the gas (left) and dust (right). As the gas and dust densities are $1$ and $0.01$, respectively and the drag coefficient is $K=1$ in code units, the stopping time is $t_{\rm{s}}=0.01$, which represents $10\%$ of the time required for the blast to fill the box. The response of the dust to the forcing by the gas drag is therefore of order $10\%$. Consequently, an overdensity in the dust phase forms due to the passage of the overdensity in the gas. The overdensities in the gas and the dust phases are dephased slightly (seen by comparing the position of the peak densities in the gas and dust in Fig. 14), consistent with the finite time ($t_{\rm{s}}$) required for the dust to respond to the gas forcing. Figure 15: Rendering of the gas density of a typical T-Tauri Star protoplanetary disc at two different resolutions: $10^{5}$ (left) and $10^{6}$ (right) gas particles. Increasing the resolution smoothens the gas phase. The initial dust to gas ratio is $0.01$ so that the dust only slightly affects the gas. Figure 16: Rendering of the dust surface density in a typical T-Tauri Star protoplanetary disc with four different configurations: $2\times 10^{5}$ particles using a mixed smoothing length (top left) and $2\times 10^{5}$ particles (top right), $10^{5}$ dust and $10^{5}$ dust particles (bottom left) and $2\times 10^{6}$ particles (bottom right) using the gas smoothing length. Using the mixed smoothing length results in artificial structures in the dust density. Smoother dust profiles are achieved by i) using the gas smoothing length and ii) increasing the gas resolution. More accurate results are obtained by increasing the number of both the gas and the dust particles. ### 4.6 dustydisc: Settling and migration in an accretion disc Our final test, dustydisc, concerns the evolution of the dust and gas mixture in a protoplanetary disc, where vertical settling and radial drift of the dust particles (see e.g. Chiang & Youdin 2010b) are known to be crucial processes in the early stages of planet formation. SPH is well suited to this problem since i) free boundaries are trivial to implement and ii) the exact conservation of angular momentum by both the gas and dust parts of the algorithm means that the problem can be simulated for many dynamical times. We used the standard linear Epstein regime given by Eq. 87. The drag force is integrated explicitly. The key features — namely both vertical settling and radial migration — are expected to occur. We focus here on the vertical settling of the grains since the migration is extensively discussed in Ayliffe et al. (2011). For this specific test, the ’artificial viscosity for a disc’ described in Lodato & Price (2010) and implemented in phantom is used. #### 4.6.1 dustydisc: Setup We setup $10^{5}$ gas particles and $10^{5}$ dust particles in a $0.01M_{\odot}$ gas disc (with $0.0001M_{\odot}$ of dust) surrounding a $1M_{\odot}$ star. The disc extends from 10 to 400 AU. Both gas and dust particles are placed using a Monte-Carlo setup such that the surface density profiles of both phases are $\Sigma\left(r\right)\propto r^{-1}$. The radial profile of the gas temperature is taken to be $T\left(r\right)\propto r^{-0.6}$ with a flaring $H/r=0.05$ at 100 AU. One code unit of time corresponds to $10^{3}$ yrs. A uniform grain size of 1 cm is used. #### 4.6.2 dustydisc: Resolution issue Fig. 15 compares the evolution of the gas phase, varying the number of gas particles from $10^{5}$ (left panel) to $10^{6}$ (right panel). The number of dust particles does not affect the density profile of the gas given the small initial dust to gas ratio. As expected, a smoother gas profile is achieved by using a higher resolution. Fig. 16 compares the evolution of the dust phase varying i) the smoothing length used to compute the drag term and ii) the number of particles in each phase. When the drag term is computed with a mixed smoothing length ($h=[h_{\rm g}+h_{\rm d}]/2$, top left panel), artificial structures develop in the dust phase due to over-concentration of dust particles below the resolution of the gas. These numerical artefacts are removed using instead the gas smoothing length (top right panel). Indeed, the gas smoothing length is larger than the dust smoothing length since the dust grains concentrate when they reach the disc mid plane (see discussion in Sec. 2.4.2). Smoother dust density profiles are achieved increasing the gas resolution ($10^{6}$ gas particles keeping $10^{5}$ dust particles, bottom left panel). Increasing the number of gas particles thus reduces the numerical noise in the dust phase. Finally, the smoothest dust density profile is naturally obtained when the resolution in the two phases is the highest ($2\times 10^{6}$ particles, bottom right panel). #### 4.6.3 dustydisc: Vertical settling of the particles Figure 17: Vertical settling of a centimetre dust grain initially located at $r_{0}=100$ AU and $z_{0}=2$ AU (solid/black). SPH results are compared to the estimate given by the damped harmonic oscillator approximation (pointed/red). The agreement between the numerical and the analytic solutions indicates that the vertical settling of the dust grain is correctly reproduced by the SPH algorithm. The analytic estimate neglects the radial drift of the grain and the vertical stratification of the disc. Fig. 17 compares the vertical settling of a dust particle obtained with the SPH simulation and the analytic estimation given by the evolution of a damped harmonic oscillator (see e.g. Garaud & Lin 2004). The particle is initially located at $r=100$ AU and $z=2$ AU, and the evolution is computed for $50$ code units. The agreement between the numerical and the analytic solutions indicates that the vertical settling of the dust grain is accurately reproduced by the SPH algorithm. It should be noted that the analytic estimation neglects the radial drift of the grain and the vertical stratification of the disc. More precisely, this model assumes an expansion to order zero in $(\frac{\partial P_{\rm g}}{\partial r}/\hat{\rho}_{\mathrm{g}})/(\mathcal{G}M/r^{2})$ (meaning that the radial and the vertical motion of the dust particles are decoupled) and to first order in $z_{0}/H$ (the vertical stratification is neglected). The SPH results are thus expected to slightly differ from the analytic approximation in both amplitude and phase. ## 5 Discussion In astrophysics, gas and dust mixtures have been predominantly studied with grid-based codes. The gas phase is computed as usual whereas the dust is treated by using superparticles (e.g. Youdin & Johansen 2007, Bai & Stone 2010). Computing the drag is usually divided into three steps: i) interpolation of the gas velocities at the particle positions, ii) calculation of the drag force on the particles and iii) attribution of the back-reaction from the particles onto the nearby cells. Our algorithm has two key advantages compared to the current grid-based algorithms. First, the procedure used for interpolating the gas velocities at the dust position conserves angular momentum exactly, avoiding artificial local torques. Moreover, the interpolation in current grid-based codes is performed with a standard bell-shaped kernel, regardless of the nature of the error in the drag terms. We expect that these aspects of the drag computation in grid-based codes may be improved by generalising the techniques involved in our SPH algorithm. Second, the equations of motion of the mixture (considering the drag terms only) in SPH are invariant when permuting the dust and the gas indices, i.e. $\rm{g}\leftrightarrow\rm{d}$. This symmetry is broken in grid- based schemes where the two phases are treated with two different methods (super-particles superimposed to a grid). The SPH algorithm provides rigorously identical results when interchanging the dust and the gas properties (this has been verified on the dustybox problem which involves only the drag, see above). ## 6 Conclusions We have developed a new general SPH formalism for two-fluid dust and gas mixtures, with the aim of simulating the dynamics of dusty gas systems in a range of astrophysical contexts. In doing so we have generalised the standard methods developed over 15 years ago by Monaghan & Kocharyan (1995) and Monaghan (1997) for treating dusty gas in SPH. In Sec. 1, we highlighted seven key issues. In this, Paper I, we have addressed five of these issues as follows: 1) we have introduced a simple way to compute SPH densities on two fluids with variable smoothing lengths; 2) the conservative part of the SPH equations have been derived from a Lagrangian; 3) we have demonstrated how the use of “double-hump” shaped kernels significantly improve the accuracy of the SPH interpolation of drag terms; 4) we find a necessary criterion $h\lesssim c_{\rm s}t_{\rm s}$ in order to correctly resolve differential motion between gas and dust that becomes critical at high drag; we also find it important to ensure $h_{\rm gas}\lesssim h_{\rm dust}$ to avoid artificial over- concentration of dust particles, implying a higher resolution should be employed in the gas phase relative to the dust. Finally, to address issue 7), we have presented a comprehensive suite of simple test problems that can be used to benchmark astrophysical dusty gas codes. These consist of the dustybox, dustywave, dustyshock, dustysedov and dustydisc problems. The first three of these have known (or partially known in the case of dustyshock) analytic solutions and can be easily setup in any code with standard boundary conditions. We have used these tests to explore the issues raised above and have demonstrated that with the appropriate resolution criteria satisfied, our formalism is robust and provides accurate results. The two remaining issues — namely implicit timestepping and treatments of astrophysical drag regimes —- are addressed in a companion paper (Paper II). While this paper concentrates on two-fluid gas and dust mixtures, the algorithm is general and can be applied easily to the treatment of other multi-fluid systems in SPH (e.g. ambipolar diffusion). ## Acknowledgments We thank Ben Ayliffe, Matthew Bate, Joe Monaghan and Laure Fouchet for useful discussions and comments. Figures have been produced using splash (Price, 2007) with the new giza backend by DJP and James Wetter. We are grateful to the Australian Research Council for funding via Discovery project grant DP1094585. ## References * Ayliffe et al. (2011) Ayliffe B., Laibe G., Price D. J., Bate M. R., 2011, MNRAS, submitted * Bai & Stone (2010) Bai X.-N., Stone J. M., 2010, ApJS, 190, 297 * Baines et al. (1965) Baines M. J., Williams I. P., Asebiomo A. S., 1965, MNRAS, 130, 63 * Barrière-Fouchet et al. (2005) Barrière-Fouchet L., Gonzalez J.-F., Murray J. R., Humble R. J., Maddison S. T., 2005, A&A, 443, 185 * Chiang & Youdin (2010a) Chiang E., Youdin A. N., 2010a, Annual Review of Earth and Planetary Sciences, 38, 493 * Chiang & Youdin (2010b) Chiang E., Youdin A. N., 2010b, Annual Review of Earth and Planetary Sciences, 38, 493 * Epstein (1924) Epstein P. S., 1924, Physical Review, 23, 710 * Fan & Zhu (1998) Fan L.-S., Zhu C., 1998, Principles of Gas-Solid Flows. Cambridge University Press * Fromang & Nelson (2005) Fromang S., Nelson R. P., 2005, MNRAS, 364, L81 * Fromang & Papaloizou (2006) Fromang S., Papaloizou J., 2006, A&A, 452, 751 * Fulk & Quinn (1996) Fulk D. A., Quinn D. W., 1996, J. Comp. Phys., 126, 165 * Garaud & Lin (2004) Garaud P., Lin D. N. C., 2004, ApJ, 608, 1050 * Goodman & Pindor (2000) Goodman J., Pindor B., 2000, Icarus, 148, 537 * Harlow & Amsden (1975) Harlow F. H., Amsden A. A., 1975, J. Comp. Phys., 17, 19 * Iannuzzi & Dolag (2011) Iannuzzi F., Dolag K., 2011, MNRAS, submitted preprint * Johansen et al. (2007) Johansen A., Oishi J. S., Low M., Klahr H., Henning T., Youdin A., 2007, Nature, 448, 1022 * Laibe & Price (2011c) Laibe G., Price D. J., 2011c, MNRAS, in press (Paper II) * Laibe & Price (2011a) Laibe G., Price D. J., 2011a, MNRAS, in press * Lodato & Price (2010) Lodato G., Price D. J., 2010, MNRAS, 405, 1212 * Maddison et al. (2003) Maddison S. T., Humble R. J., Murray J. R., 2003, in D. Deming & S. Seager ed., Scientific Frontiers in Research on Extrasolar Planets Vol. 294 of Astronomical Society of the Pacific Conference Series, Building Planets with Dusty Gas. pp 307–310 * Marble (1970) Marble F. E., 1970, Ann. Rev. Fluid Mech., 2, 397 * Merlin et al. (2010) Merlin E., Buonomo U., Grassi T., Piovan L., Chiosi C., 2010, A&A, 513, A36 * Miniati (2010) Miniati F., 2010, Journal of Computational Physics, 229, 3916 * Miura & Glass (1982) Miura H., Glass I. I., 1982, Roy. Soc. Lon. Proc. Ser. A, 382, 373 * Monaghan (1992) Monaghan J. J., 1992, Ann. Rev. Astron. Astrophys., 30, 543 * Monaghan (1997) Monaghan J. J., 1997, Journal of Computational Physics, 138, 801 * Monaghan & Kocharyan (1995) Monaghan J. J., Kocharyan A., 1995, Computer Physics Communications, 87, 225 * Morris (1996a) Morris J. P., 1996a, PASA, 13, 97 * Morris (1996b) Morris J. P., 1996b, PhD thesis, Monash University, Melbourne, Australia * Paardekooper & Mellema (2006) Paardekooper S.-J., Mellema G., 2006, A&A, 453, 1129 * Price (2007) Price D. J., 2007, Publ. Astron. Soc. Aust., 24, 159 * Price (2011) Price D. J., 2011, J. Comp. Phys., in press, doi:10.1016/j.jcp.2010.12.011 * Price & Federrath (2010) Price D. J., Federrath C., 2010, MNRAS, 406, 1659 * Price & Monaghan (2004) Price D. J., Monaghan J. J., 2004, MNRAS, 348, 139 * Price & Monaghan (2005) Price D. J., Monaghan J. J., 2005, MNRAS, 364, 384 * Price & Monaghan (2007) Price D. J., Monaghan J. J., 2007, MNRAS, 374, 1347 * Rice et al. (2004) Rice W. K. M., Lodato G., Pringle J. E., Armitage P. J., Bonnell I. A., 2004, MNRAS, 355, 543 * Rosswog & Price (2007) Rosswog S., Price D., 2007, MNRAS, 379, 915 * Rudinger (1964) Rudinger 1964, Physics of Fluids, 7, 658 * Saito et al. (2003) Saito T., Marumoto M., Takayama K., 2003, Shock Waves, 13, 299 * Sod (1978) Sod G. A., 1978, J. Comp. Phys., 27, 1 * Springel & Hernquist (2002) Springel V., Hernquist L., 2002, MNRAS, 333, 649 * Youdin & Johansen (2007) Youdin A., Johansen A., 2007, ApJ, 662, 613 * Youdin & Goodman (2005) Youdin A. N., Goodman J., 2005, ApJ, 620, 459	arxiv-papers	2011-11-14T02:29:04	2024-09-04T02:49:24.308319	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Guillaume Laibe (Monash), Daniel J. Price (Monash)", "submitter": "Guillaume Laibe", "url": "https://arxiv.org/abs/1111.3090" }
1111.3356	# A note about the relation between fixed point theory on cone metric spaces and fixed point theory on metric spaces Ion Marian Olaru ###### Abstract Let $Y$ be a locally convex Hausdorff space, $K\subset E$ a cone and $\leq_{K}$ the partial order defined by $K$. Let $(X,p)$ be a $TVS-$ cone metric space, $\varphi:K\rightarrow K$ a vectorial comparison function and $f:X\rightarrow X$ such that $p(f(x),f(y))\leq_{K}\varphi(p(x,y)),$ for all $x,y\in X$. We shall show that there exists a scalar comparison function $\psi:\mathbb{R}_{+}\rightarrow\mathbb{R}_{+}$ and a metric $d_{p}$(in usual sense) on $X$ such that $d_{p}(f(x),f(y))\leq\psi(d_{p}(x,y)),$ for all $x,y\in X$. Our results extend the results of Du (2010) [Wei-Shih Du, A note on cone metric fixed point theory and its equivalence, Nonlinear Anal. 72 (2010), 2259-2261]. 2010 Mathematical Subject Classification: 47H10, 54H25 Keywords: K-metric spaces, cone metric space, TVS- cone metric spaces, comparison function ## 1 Introduction and preliminaries Fixed point theory in K-metric and K-normated spaces was developed by A.I. Perov and his consortiums ([7], [8], [9]). The main idea consists to use an ordered Banach space instead of the set of real numbers, as the codomain for a metric. For more details on fixed point theory in K-metric and K-normed spaces, we refer the reader to [15]. Without mentioning these previous works, Huang and Zhang [6] reintroduced such spaces under the name of cone metric spaces but went further, defining convergent and Cauchy sequences in the terms of interior points of the underlying cone. They also proved some fixed point theorems in such spaces in the same work. After that, fixed point results in cone metric spaces have been studied by many other authors. References [1],[3], [10], [11], [12], [13] are some works in this line of research. However, very recently Wei-Shih Du in [5] used the scalarization function and investigated the equivalence of vectorial versions of fixed point theorems in cone metric spaces and scalar versions of fixed point theorems in metric spaces. He showed that many of the fixed point results in ordered K-metric spaces for maps satisfying contractive conditions of a linear type in K-metric spaces can be considered as the corollaries of corresponding theorems in metric spaces. Let $E$ be a topological vector space (for short $t.v.s$) with its zero vector $\theta_{E}$. ###### Definition 1.1. ( [5], [6]) A subset K of E is called a cone if: * (i) K is closed, nonempty and $K\neq\\{\theta_{E}\\}$; * (ii) $a,b\in\mathbb{R}$, $a,b\geq 0$ and $x,y\in K$ imply $ax+by\in K$; * (iii) $K\cap-K=\\{\theta_{E}\\}.$ For a given cone $K\subset E$, we can define a partial ordering $\leq_{K}$ with respect to $K$ by (1.1) $x\leq_{K}y\ if\ and\ only\ if\ y-x\in K.$ We shall write $x<_{K}y$ to indicate that $x\leq_{K}y$ but $x\neq y$, while $x\ll y$ will stand for $y-x\in intK$ (interior of $K$). In the following, unless otherwise specified, we always suppose that $Y$ is a locally convex Hausdorff with its zero vector $\theta$, $K$ a cone in $Y$ with $intK\neq\emptyset$ , $e\in intK$ and $\leq_{K}$ a partial ordering with respect to $K$. ###### Definition 1.2. ( [5]) Let $X$ be a nonempty set. Suppose that a mapping $d:X\times X\rightarrow Y$ satisfies: * (i) $\theta\leq_{K}d(x,y)$ for all $x,y\in X$ and $d(x,y)=\theta$ if and only if $x=y$; * (ii) $d(x,y)=d(y,x)$, for all $x,y\in X$ ; * (iii) $d(x,y)\leq_{K}d(x,z)+d(z,y)$ for all $x,y,z\in X$. Then $d$ is called a TVS-cone metric on $X$ and $(X,d)$ is called a TVS-cone metric space. The nonlinear scalarization function $\xi_{e}:Y\rightarrow\mathbb{R}$ is defined as follows $\xi_{e}(y)=\inf\\{r\in\mathbb{R}\mid y\in r\cdot e-K\\}.$ ###### Lemma 1.1. ( [4]) For each $r\in\mathbb{R}$ and $y\in Y$, the following statements are satisfied: * (i) $\xi_{e}(y)\leq r$ if and only if $y\in r\cdot e-K$; * (ii) $\xi_{e}(y)>r$ if and only if $y\notin r\cdot e-K$; * (iii) $\xi_{e}(y)\geq r$ if and only if $y\notin r\cdot e-intK$; * (iv) $\xi_{e}(y)<r$ if and only if $y\in r\cdot e-intK$; * (vi) $\xi_{e}(\cdot)$ is positively homogeneous and continuous on Y; * (vii) if $y_{1}\in y_{2}+K$ then $\xi_{e}(y_{2})\leq\xi_{e}(y_{1})$; * (viii) $\xi_{e}(y_{1}+y_{2})\leq\xi_{e}(y_{1})+\xi_{e}(y_{2})$, for all $y_{1},y_{2}\in Y$. ###### Theorem 1.1. ( [5]) Let $(X,p)$ be a $TVS-$cone metric space. Then $d_{p}:X\times X\rightarrow[0,\infty)$ defined by $d_{p}=\xi_{e}\circ d$ is a metric. ## 2 Main results ###### Definition 2.1. Let $K\subset Y$ be a cone. A function $\varphi:K\rightarrow K$ is called a vectorial comparison function if * (i) $k_{1}\leq_{P}k_{2}$ implies $\varphi(k_{1})\leq_{P}\varphi(k_{2})$; * (ii) $\varphi(0)=0$ and $0<_{P}\varphi(k)<_{P}k$ for $k\in K-\\{0\\}$; * (iii) $k\in intK$ implies $k-\varphi(k)\in intK$; * (iv) if $t_{0}\geq 0$ then $\lim\limits_{t\rightarrow t_{0}^{+}}\varphi(t\cdot e)=\varphi(t_{0}\cdot e)$. ###### Example 1. * (i) if K is an arbitrary cone in a Banach space E and $\lambda\in(0,1)$, then $\varphi:K\rightarrow K$, defined by $\varphi(k)=\lambda k$ is a vectorial comparison function; * (ii) Let $E=\mathbb{R}^{2}$, $K=\\{(x,y)\mid x,y\geq 0\\}$ and let $\varphi_{1},\varphi_{2}:[0,\infty)\rightarrow[0,\infty)$ be such that * (a) $\varphi_{1},\varphi_{2}$ are increasing functions; * (b) if $t>0$ then $\varphi_{i}(t)<t$ for $i=\overline{1,2}$; * (c) $\varphi_{1},\varphi_{2}$ are right continuous. Then $\varphi:K\rightarrow K$, defined by $\varphi(x,y)=(\varphi_{1}(x),\varphi_{2}(y))$ is a vectorial comparison function; ###### Definition 2.2. ( [14]) A function $\varphi:\mathbb{R}_{+}\rightarrow\mathbb{R}_{+}$ is called a scalar comparison function if * (i) $t_{1}\leq t_{2}$ implies $\varphi(t_{1})\leq\varphi(t_{2})$; * (ii) $\varphi^{n}(t)\stackrel{{\scriptstyle n\rightarrow\infty}}{{\rightarrow}}0$ for all $t>0$ The following lemma will be useful in the sequel ###### Lemma 2.1. ( [14]) If $\varphi:\mathbb{R}_{+}\rightarrow\mathbb{R}_{+}$ is increasing and right upper semicontinuous then the following assertions are equivalent: * (a) $\varphi^{n}(t)\stackrel{{\scriptstyle n\rightarrow\infty}}{{\rightarrow}}0$ for all $t>0$; * (b) $\varphi(t)<t$ for all $t>0$. ###### Lemma 2.2. We consider $M:\mathbb{R}\rightarrow Y,M(r)=r\cdot e$. Then we have * (i) $M(0)=\theta$; * (ii) if $r_{1}\leq r_{2}$ then $M(r_{1})\leq_{K}M(r_{2})$; * (iii) $y\leq_{K}M\circ\xi_{e}(y)$ for all $y\in Y$; * (iv) $\xi\circ M(r)\leq r$ for all $r\in\mathbb{R}$; * (v) if $y_{1}\ll y_{2}$ then $\xi_{e}(y_{1})<\xi_{e}(y_{2})$. Proof: $(i)$ It is obvious; $(ii)$ Let be $r_{1}\leq r_{2}$. Then $(r_{2}-r_{1})\cdot e\in K$. Thus $M(r_{1})\leq_{K}M(r_{2})$; $(iii)$ Since $\xi_{e}(y)=\inf\\{r\in\mathbb{R}\mid y\leq_{K}r\cdot e\\}$ it follows that $y\leq_{K}\xi_{e}(y)\cdot e=M\circ\xi_{e}(y)$ for all $y\in Y$; $(iv)$ Let be $r\in\mathbb{R}$. Since $\\{r^{\prime}\in\mathbb{R}\mid r\cdot e\leq_{K}r^{\prime}\cdot e\\}\supseteq\\{r^{\prime}\in\mathbb{R}\mid r\leq r^{\prime}\\}$ we get $\xi_{e}(M(r))=\xi_{e}(r\cdot e)=\inf\\{r^{\prime}\in\mathbb{R}\mid r\cdot e\leq_{K}r^{\prime}\cdot e\\}\leq\inf\\{r^{\prime}\in\mathbb{R}\mid r\leq r^{\prime}\\}=r.$ $(v)$ Let be $y_{1}\ll y_{2}$. We remark that $y_{1}\ll y_{2}\leq_{K}\xi_{e}(y_{2})\cdot e$. Then, via Remark 1.3 of Radenović and Kadelburg [11], it follows that $y_{1}\ll\xi_{e}(y_{2})\cdot e$. Hence $y_{1}\in\xi_{e}(y_{2})\cdot e-intK$. By using Lemma 1.1 (iv) we get $\xi_{e}(y_{1})<\xi_{e}(y_{2})$. ###### Theorem 2.1. Let $(X,p)$ be a TVS-cone metric and $\varphi:K\rightarrow K$ be a vectorial comparison function such that $p(f(x),f(y))\leq_{K}\varphi(p(x,y)),$ for all $x,y\in X$. Then there exists a scalar comparison function $\psi:\mathbb{R}_{+}\rightarrow\mathbb{R}_{+}$ such that $d_{p}(f(x),f(y))\leq\psi(d_{p}(x,y)),$ for all $x,y\in X$. Proof: Let be $t\in\mathbb{R}_{+}$. Then $\theta\leq_{K}M(t)$. It follows that $M(t)\in K$ for all $t\in\mathbb{R}_{+}$. We define $\psi:\mathbb{R}_{+}\rightarrow\mathbb{R}_{+},$ $\psi(t)=\xi_{e}\circ\varphi\circ M(t)$ First, we note that for all $t\in\mathbb{R}_{+}$ we have $0\leq\xi_{e}\circ\varphi\circ M(t)\leq\xi_{e}\circ M(t)\leq t.$ Now, we remark that for each $x,y\in X$ we have $d_{p}(f(x),f(y))\leq\xi_{e}\circ\varphi(p(x,y))\leq\xi_{e}\circ\varphi(M(\xi_{e}(p(x,y))))=\psi(d_{p}(x,y)).$ We claim that $\psi$ is a scalar comparison function. Since $\xi_{e}$, $\varphi$ and $M$ are increasing functions, it follows that $\psi$ is increasing function. In order to prove that $\psi^{n}(t)\stackrel{{\scriptstyle n\rightarrow\infty}}{{\rightarrow}}0$ for all $t>0$, we shall use Lemma 2.1. Next we show that $\psi(t)<t$ for all $t>0$. Let be $t_{0}>0$. Then $t_{0}\cdot e\in intK$. Therefore $\varphi(t_{0}\cdot e)\ll t_{0}\cdot e$. It follows that $\psi(t_{0})=\xi_{e}\circ\varphi(t_{0}\cdot e)<\xi_{e}\circ M(t_{0})\leq t_{0}.$ Since $\lim\limits_{t\rightarrow t_{0}^{+}}\psi(t)=\lim\limits_{t\rightarrow t_{0}^{+}}\xi_{e}\circ\varphi(t\cdot e)=\xi_{e}(\lim\limits_{t\rightarrow t_{0}^{+}}\varphi(t\cdot e))=\xi_{e}\circ\varphi(t_{0}\cdot e)=\psi(t_{0})$ it follows that $\psi$ is right upper semicontinuous. Hence $\psi^{n}(t)\stackrel{{\scriptstyle n\rightarrow\infty}}{{\rightarrow}}0$. ###### Corollary 2.1. Let $(X,p)$ be a complete TVS cone metric space and $\varphi:K\rightarrow K$ a vectorial comparison function such that $p(fx,fy)\leq_{K}\varphi(p(x,y)),$ for all $x,y\in X$. Then, f has a unique fixed point $x_{0}$. Proof: We apply Theorem 2.1 and Theorem 1 pp 459 of Boyd and Wong ([2]). ###### Remark 2.1. For $\varphi(k)=\lambda\cdot k$, $\lambda\in[0,1)$ we obtain, via Lemma 2.2 $(iv)$ and Corollary 2.1, the results of W.S. Du [5]. ###### Remark 2.2. Let $(X,p)$ a cone metric space. For $\varphi(k)=\lambda\cdot k$, $\lambda\in[0,1)$ we obtain, via Remark 2.1, the results of L.G. Huang and Zhang Xian [6]. Let $(X,d)$ be a TVS cone-metric space and let $\varphi:K\rightarrow K$ be a vectorial comparison function. For a pair $(f,g)$ of self-mappings on $X$ consider the following conditions: * (C) for arbitrary $x,y\in X$ there exists $u\in\\{d(gx,gy),d(gx,fx),d(gy,fy)\\}$ such that $d(fx,fy)\leq_{P}\varphi(u)$. * $(C_{1})$ for arbitrary $x,y\in X$ there exists $w\in\\{d_{p}(gx,gy),d_{p}(gx,fx),d_{p}(gy,fy)\\}$ such that $d_{p}(fx,fy)\leq\psi(u)$. ###### Remark 2.3. The condition (C) imply the condition $(C_{1})$. Indeed since the condition $(C)$ hold, it follows that at least one of the following three cases holds: * Case 1: $u=d(gx,gy)$. Then $\xi_{e}(p(fx,fy))\leq\xi_{e}\circ\varphi(u)\leq\xi_{e}\circ\varphi\circ M(\xi_{e}(u))=\psi(d_{p}(gx,gy))$ * Case 2: $u=d(gx,fx)$. Then $\xi_{e}(p(fx,fy))\leq\xi_{e}\circ\varphi(u)\leq\xi_{e}\circ\varphi\circ M(\xi_{e}(u))=\psi(d_{p}(gx,fx))$ * Case 3: $u=d(gy,fy)$. Then $\xi_{e}(p(fx,fy))\leq\xi_{e}\circ\varphi(u)\leq\xi_{e}\circ\varphi\circ M(\xi_{e}(u))=\psi(d_{p}(gy,fy))$ ## References * [1] M. Abbas, G. Jungck, Common fixed point results for noncommuting mappings without continuity in cone metric spaces, J. Math.Anal.Appl. 341 (2008) * [2] D.W. Boyd, J.S.W. Wong, On nonlinear contraction, Proceed. A.M.S., 20(1969). * [3] Arandjelović I., Kadelburg Z, Radenović S., Boyd-Wong-type common fixed point results in cone metric spaces, Applied Mathematics and Computation 217(2011), 7167-7171. * [4] G.Y. Chen, X.X. Huang, X.Q. Yang, Vector Optimization, Springer-Verlag, Berlin, Heidelberg, Germany, 2005. * [5] Wei-Shih Du, A note on cone metric fixed point theory and its equivalence, Nonlinear Anal. 72 (2010), 2259-2261. * [6] L.G. Huang, X. Zhang, Cone metric spaces and fixed Point theorems of contractive mappings , J. Math. Anal. Appl., 332(2007), 1468-1476. * [7] B.V. Kvedaras, A.V. Kibenko and A.I. Perov, On some boundary value problems, Litov. matem. sbornik, 5 (1) (1965), 69-84. * [8] A.I. Perov, The Cauchy problem for systems of ordinary diferential equations, Approximate methods of solving diferential equations, Kiev, Naukova Dumka, 1964, 115- 134 [Russian]. * [9] A.I. Perov and A.V. Kibenko, An approach to studying boundary value problems, Izvestija AN SSSR, Seria Math. 30 (2) (1966), 249-264 [Russian]. * [10] G. Jungck, S. Radenović, S. Radojević and V. Rakočević, Common fixed point theorems for weakly compatible pairs on cone metric spaces, Fixed Point Theory and Applications, 2009, Article ID 643640, 12 pages, doi: 10.1155/2009/643640. * [11] Stojan Radenović, Zoran Kadelburg, Quasi-Contractions on symetric and cone symetric spaces, Banach J. Math. Anal.(2011), No.1 pp 38-50. * [12] Raja P., Vaezpour S.M, Some extensions of Banach’s contraction principle in complete cone metric spaces, Fixed Point Theory and Applications Vol. 2008, article ID 768294, 11 pages, doi:101155/2008/768924. * [13] Rezapour Sh., Hamblbarani R., Some notes on the paper ”Cone metric spaces and fixed point thorems of contractive mappings”, J. Math., Anal. Appl., 345(2008), 719-724. * [14] I.A. Rus, Generalized contractions, Seminar on Fixed Point Theory, Preprint No. 3, 1983, pp 1-130. * [15] P.P. Zabrejko, K-metric and K-normed linear spaces: survey, Collect. Math. 48 (4-6) (1997), 825-859. Departament of Mathematics, Faculty of Sciences, University ”Lucian Blaga” of Sibiu, Dr. Ion Ratiu 5-7, Sibiu, 550012, Romania E-mail: marian.olaru@ulbsibiu.ro	arxiv-papers	2011-11-14T20:55:36	2024-09-04T02:49:24.329657	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Ion Olaru", "submitter": "Ion Olaru", "url": "https://arxiv.org/abs/1111.3356" }
1111.3425	On leave of absence from Center for Nuclear Physics, ] Institute of Physics, Hanoi, Vietnam # Pairing reentrance in hot rotating nuclei N. Quang Hung1 [ hung.nguyen@ttu.edu.vn N. Dinh Dang2,3 dang@riken.jp 1) School of Engineering, TanTao University, TanTao University Avenue, TanDuc Ecity, Duc Hoa, Long An Province, Vietnam 2) Theoretical Nuclear Physics Laboratory, RIKEN Nishina Center for Accelerator-Based Science, 2-1 Hirosawa, Wako City, 351-0198 Saitama, Japan 3) Institute for Nuclear Science and Technique, Hanoi, Vietnam ###### Abstract The pairing gaps, heat capacities and level densities are calculated within the BCS-based quasiparticle approach including the effect of thermal fluctuations on the pairing field within the pairing model plus noncollective rotation along the $z$ axis for 60Ni and 72Ge nuclei. The analysis of the numerical results obtained shows that, in addition to the pairing gap, the heat capacity can also serve as a good observable to detect the appearance of the pairing reentrance in hot rotating nuclei, whereas such signature in the level density is rather weak. Suggested keywords ###### pacs: 21.60.-n, 21.60.Jz, 24.60.-k, 24.10.Pa ## I Introduction In the collective rotation of a deformed nucleus, the rotation axis is perpendicular to the symmetry axis, and the Coriolis force, which breaks the Cooper pairs, increases with the total angular momentum so that at a certain critical angular momentum all Cooper pairs are broken. The nucleus undergoes then a phase transition from the superfluid phase to the normal one (the SN phase transition). This is called the Mottelson-Valatin effect Mottelson . A similar effect is expected in spherical nuclei, although the rotation is no longer collective. The total angular momentum is made up by those of the nucleons from the broken pairs, which occupy the single-particle levels around the Fermi surface and block them against the scattered pairs. The pairing correlations decrease until a sufficiently large total angular momentum $M_{c}$, where the pairing gap $\Delta$ completely vanishes. At finite temperature ($T\neq$ 0), the increase of $T$ relaxes the tight packing of quasiparticles around the Fermi surface, which is caused by a large angular momentum $M\geq M_{c}$, and spreads them farther away from the Fermi level. This makes some levels become partially unoccupied, therefore, available for scattered pairs. As the result, when $T$ increases up to some critical value $T_{1}$, the pairing correlations are energetically favored, and the pairing gap reappears. As $T$ goes higher, the increase of a large number of quasiparticles eventually breaks down the pairing gap at $T_{2}$ $(>T_{1})$. This phenomenon, predicted by Kamuri Kammuri and Moretto Moretto , is called thermally assisted pairing or anomalous pairing, and later as pairing reentrance by Balian, Flocard and V$\acute{\rm e}$n$\acute{\rm e}$roni Balian . However, it has been shown already in the 1960s that the sharp SN-phase transition at $M=M_{c}$ in the Mottelson-Valatin effect is an artifact of the BCS method. As a matter of fact, a proper particle-number projection before variation has removed the discontinuity in the pairing gap as a decreasing function of the angular momentum Mang . Similarly, by taking the effect of thermal fluctuations in the pairing field into account, the SN phase transition predicted by the BCS theory is smoothed out. The gap $\Delta(T)$ of a non-rotating nucleus does not collapse at $T_{c}\simeq 0.568\Delta(T=0)$, but monotonically decreases with increasing $T$, remaining finite even at $T\gg T_{c}$ Moretto2 ; MBCS . This result is reconfirmed by shell-model calculations of pairing energy as a function of excitation energy Zele , and by embedding the exact eigenvalues of the pairing problem into the canonical ensemble Exact . By considering an exactly solvable cranked deformed shell model Hamiltonian it has also been shown that the pairing gap, quenched at $T=$ 0 and high rotational frequency, reappears at $T_{1}$ ($\ll T_{c}$) at $M\geq M_{c}$ Frauendorf . However, different from the prediction by the BCS theory, the pairing gap does not vanish at $T>T_{1}$. The behavior of hot rotating nuclei can be put in correspondence with superconductors in the presence of an external magnetic field, where the magnetic field plays the role as that of the nuclear rotation. The reentrance of superconducting correlations, which is also known as the unconventional superconductivity, has been the subject of recent theoretical and experimental studies in condensed matter. The Grenoble High Magnetic Field Laboratory has recently discovered that URhGe becomes superconducting at low temperature in the presence of a strong magnetic field (between 8 and 13 T), well above the field value of 2 T, at which superconductivity is first destroyed Sci . The reentrance of superconductivity under the magnetic field is interpreted as to be caused by spin reorientation reent , which bears some similarity with the reappearance of the scattered pairs in rotating hot nuclei discussed above. Recently, we have developed an approach based on the finite-temperature BCS (FTBCS) that includes the effects due to quasiparticle-number fluctuations in the pairing field and the $z$ projection of angular momentum at $T\neq$ 0, which we call as the FTBCS1 (with ”1” denoting the effect due to quasiparticle-number fluctuations) SCQRPA . This approach reproduces well the effect of smoothing out the SN phase transition at $T\neq$ 0 as well as the pairing reentrance in hot (noncollectively) rotating nuclei. In the latter case, for $M\geq M_{c}$ the pairing gap also reappears at $T_{1}$ and remains finite at $T>T_{1}$. It has also been pointed out that the microscopic mechanism of the nonvanishing gap at high $T$ is the quasiparticle-number fluctuations, which are ignored in the conventional BCS theory. A refined version of the FTBCS1 also includes the contribution of coupling to pair vibrations within the self-consistent quasiparticle random-phase approximation SCQRPA . That the pairing reentrance is not an artifact of the mean field analysis, but a robust physical effect, has been obtained by exact diagonalization of the 2D attractive Hubbard model, where a nonmonotonic filed dependence of the pair susceptibility in the presence of the external magnetic field was found for various cluster sizes both in the weak and strong coupling limit Gorczyca . Nonetheless, the experimental extraction of the pairing gap in hot nuclei is not simple because one has to properly exclude the admixture with the contribution of uncorrelated single-particle configurations from the odd-even mass difference Ensemble . Therefore the detection of the pairing reentrance effect by using the experimentally extracted pairing gaps seems to be elusive, especially when the formally derived pairing gap has a value smaller than the average spacing between the single-particle levels. Meanwhile, the heat capacity has been extracted from the experimental level densities EXP . The existence of a bump or an $S$ shape on the curve of the heat capacity at $T\sim T_{c}$ allows one to discuss about the smoothing of the SN phase transition in finite nuclei. In a recent calculation of the heat capacity in 72Ge within the Shell Model Monte Carlo (SMMC) approach, by reconfirming the pairing reentrance effect, the authors of Ref. Dean claimed that they found a local dip in the heat capacity at rotation frequency of 0.5 MeV at $T\sim$ 0.45 MeV, and a corresponding local maximum on the temperature dependence of the logarithm of level density. They associated such irregularities in the heat capacity and level density as the signatures of the pairing reentrance. There are, however, two concerns regarding these results. The first one is that, as well-known, at such low temperature the SMMC approach produces quite large error bars. As a consequence, instead of approaching zero as it should be to fulfill the third law of thermodynamics, the SMMC heat capacity at $T<$ 0.5 MeV jumps to 25 $\sim$ 30 (Fig. 4 of Dean ), which makes the statement on the signature of the pairing reentrance ambiguous. The second one is that the SMMC in Ref. Dean used the same Fock- space single-particle energies of the shells ($0f1p-0g1d2s$) for both neutron and proton spectra. Hence, the difference between neutron and proton spectra came solely from the difference in the valence particle numbers outside the closed-shell core of 40Ca, which ignored altogether the Coulomb barrier in the proton spectrum. Our FTBCS1 theory is free from such deficiency at low $T$, and it works well with the schematic as well as single-particle spectra, which are obtained from the realistic Woods-Saxon potential. Therefore, in the present paper we will calculate the heat capacity as well as the level density within the FTBCS1 theory to see if these quantities can be used to identify the pairing reentrance phenomenon in realistic nuclei at finite temperature and angular momentum. The paper is organized as follows. The formalism for the calculations of thermodynamic quantities such as pairing gap, heat capacity, and level density in hot non-collectively rotating nuclei within the FTBCS and FTBCS1 theories is presented in Sec. II. The results of numerical calculations are analyzed in Sec. III. The paper is summarized in the last section, where conclusions are drawn. ## II Formalism We consider the pairing Hamiltonian describing a spherical system rotating about the symmetry $z$ axis Moretto : $H=H_{P}-\lambda\hat{N}-\gamma\hat{M}~{},$ (1) where $\lambda$ and $\gamma$ are the chemical potential and rotation frequency, respectively. $H_{P}$ is the standard pairing Hamiltonian of a system, which consists $N$ particles interacting via a monopole pairing force with the constant parameter $G$ (the BCS pairing Hamiltonian), namely $H_{P}=\sum_{k}\epsilon_{k}(a_{+k}^{\dagger}a_{+k}+a_{-k}^{\dagger}a_{-k})-G\sum_{kk^{\prime}}{a_{k}^{\dagger}a_{-k}^{\dagger}a_{-k^{\prime}}a_{k^{\prime}}}~{},$ (2) with $a_{\pm k}^{\dagger}(a_{\pm k})$ being the creation (annihilation) operators of a particle (neutron or proton) with angular momentum $k$, projection $\pm m_{k}$, and energy $\epsilon_{k}$. The particle-number operator $\hat{N}$ and total angular momentum $\hat{M}$, which coincides with its $z$ projection, are given as $\hat{N}=\sum_{k}(a_{+k}^{\dagger}a_{+k}+a_{-k}^{\dagger}a_{-k})~{},\hskip 14.22636pt\hat{M}=\sum_{k}m_{k}(a_{+k}^{\dagger}a_{+k}-a_{-k}^{\dagger}a_{-k})~{}.$ (3) After the Bogoliubov transformation from the particle operators, $a_{k}^{\dagger}$ and $a_{k}$, to the quasiparticle ones, $\alpha_{k}^{\dagger}$ and $\alpha_{k}$, $a_{k}^{\dagger}=u_{k}\alpha_{k}^{\dagger}+v_{k}\alpha_{-k}~{},\hskip 14.22636pta_{-k}=u_{k}\alpha_{-k}-v_{k}\alpha_{k}^{\dagger}~{},$ (4) the Hamiltonian (1) is transformed into the quasiparticle one $\cal H$, whose explicit form can be found, e.g., in Refs. SCQRPA ; SCQRPA1 . As has been discussed in Refs. Kammuri ; Moretto ; SCQRPA , for a spherically symmetric system, the laboratory-frame $z$ axis, which is taken as the axis of quantization, can always be made coincide with the body-fixed one, which is aligned with the direction of the total angular momentum within the quantum mechanical uncertainty. Therefore the total angular momentum is completely determined by its $z$-projection $M$ alone. For systems of an axially symmetric oblate shape rotating about the symmetry axis, which in this case is the principal body-fixed one, this noncollective motion is known as “single- particle” rotation. The pairing reentrance effect was originally obtained within the BCS theory in Refs. Kammuri ; Moretto by considering such systems described by Hamiltonian (1). Its physical interpretation based on the thermal effect, which relaxes the tight packing of quasiparticles around the Fermi surface due to a large angle momentum $M\geq M_{c}$, and spreads them farther away from the Fermi level, fits well in the framework of this “single- particle” rotation. However, as has been pointed out in Ref. Moretto , for non-spherical nuclei, and specifically in the case of axially symmetric ones, the spin and angular momentum projections on the symmetry axis are not good quantum numbers. In this case the formalism used here is not completed because it does not include the angular momentum’s component perpendicular to the symmetry axis. Cranking model might serve as a better solution of the problem in this situation. This remains to be investigated because the results obtained within the Lipkin model with $J_{x}$ cranking did not reveal any pairing reentrance so far Civi . On the other hand, in the region of high level densities (at high excitation energies and/or high $T$) the values of angular momentum projection on the symmetry axis will be mixed among the levels, which worsen the axial symmetry. The melting of shell structure will also eventually drive nuclei to their average spherical shape. ### II.1 FTBCS1 equations at finite angular momentum The FTBCS1 includes a set of FTBCS-based equations, corrected by the effects of quasiparticle-number fluctuations, for the level-dependent pairing gap $\Delta_{k}$, average particle number $N$, and average angular momentum $M$. The derivation of the FTBCS1 equations was reported in detail in Ref. SCQRPA , so we do present here only the final equations. The FTBCS1 equation for the pairing gap is written as a sum of two parts, the level-independent part $\Delta$ and the level-dependent part $\delta\Delta_{k}$, namely $\Delta_{k}=\Delta+\delta\Delta_{k}~{},$ (5) where $\Delta=G\sum_{k^{\prime}}{u_{k^{\prime}}v_{k^{\prime}}(1-n_{k^{\prime}}^{+}-n_{k^{\prime}}^{-})}~{},\hskip 14.22636pt\delta\Delta_{k}=G\frac{\delta{\cal N}_{k}^{2}}{1-n_{k}^{+}-n_{k}^{-}}u_{k}v_{k}~{},$ (6) where $\displaystyle u_{k}^{2}$ $\displaystyle=$ $\displaystyle\frac{1}{2}\left(1+\frac{\epsilon_{k}-Gv_{k}^{2}-\lambda}{E_{k}}\right)~{},\hskip 14.22636ptv_{k}^{2}=\frac{1}{2}\left(1-\frac{\epsilon_{k}-Gv_{k}^{2}-\lambda}{E_{k}}\right)~{},$ $\displaystyle E_{k}$ $\displaystyle=$ $\displaystyle\sqrt{(\epsilon_{k}-Gv_{k}^{2}-\lambda)^{2}+\Delta_{k}^{2}}~{},\hskip 14.22636ptn_{k}^{\pm}=\frac{1}{1+e^{\beta(E_{k}\mp\gamma m_{k})}}~{},\hskip 14.22636pt\beta=1/T~{}.$ (7) with the quasiparticle-number fluctuations $\delta{\cal N}_{k}^{2}$ at nonzero angular momentum $\delta{\cal N}_{k}^{2}=(\delta{\cal N}_{k}^{+})^{2}+(\delta{\cal N}_{k}^{-})^{2}=n_{k}^{+}(1-n_{k}^{+})+n_{k}^{-}(1-n_{k}^{-})~{}.$ (8) The corrections due to coupling to pair vibration beyond the quasiparticle mean field at finite temperature and angular momentum are significant only in light nuclei like oxygen or neon isotopes, whereas they are negligible for medium and heavy nuclei (See Figs. 6 - 8 of Ref. SCQRPA ). As the present paper considers two medium nuclei, 60Ni and 72Ge, these corrections on the FTBCS1 equations are neglected in the numerical calculations. The equations for the particle number and total angular momentum are the same as Eqs. (25) and (26) of Ref. SCQRPA , namely $N=2\sum_{k}\left[v_{k}^{2}(1-n_{k}^{+}-n_{k}^{-})+\frac{1}{2}(n_{k}^{\dagger}+n_{k}^{-})\right],\hskip 14.22636ptM=\sum_{k}m_{k}(n_{k}^{+}-n_{k}^{-})~{}.$ (9) The system of coupled equations (5) – (9) are called the FTBCS1 equations at finite angular momentum. Once the FTBCS1 equations are solved, the total energy ${\cal E}$, heat capacity $C$ and entropy $S$ of the system are calculated $\displaystyle{\cal E}$ $\displaystyle=$ $\displaystyle\langle{\cal H}\rangle~{},\hskip 14.22636ptC=\frac{\partial{\cal E}}{\partial T}~{},$ $\displaystyle S$ $\displaystyle=$ $\displaystyle-\sum_{k}[n_{k}^{+}{\rm ln}n_{k}^{+}+(1-n_{k}^{+}){\rm ln}(1-n_{k}^{+})+n_{k}^{-}{\rm ln}n_{k}^{-}+(1-n_{k}^{-}){\rm ln}(1-n_{k}^{-})]~{}.$ (10) ### II.2 Level density Within the conventional FTBCS, the level density is calculated as the invert Laplace transformation of the grand partition function Moretto $\rho(E,N,M)=\frac{1}{(2\pi i)^{3}}\oint d\beta\oint d\alpha\oint d\mu e^{S}~{},$ (11) where $\alpha=\beta\lambda$, $\mu=\beta\gamma$, and $S$ is entropy of the system. The saddle-point approximation gives a good evaluation of the integral (11). As the result, the total level density of a system with $N$ neutrons and $Z$ protons is given as $\rho(E,N,M)=\frac{e^{S}}{(2\pi)^{2}\sqrt{D}}~{},$ (12) where $S=S_{N}+S_{Z}$ and $D=\left\|\begin{array}[]{cccc}\frac{\partial^{2}\Omega}{\partial\alpha_{N}^{2}}&\frac{\partial^{2}\Omega}{\partial\alpha_{N}\partial\alpha_{Z}}&\frac{\partial^{2}\Omega}{\partial\alpha_{N}\partial\mu}&\frac{\partial^{2}\Omega}{\partial\alpha_{N}\partial\beta}\\\ \frac{\partial^{2}\Omega}{\partial\alpha_{Z}\partial\alpha_{N}}&\frac{\partial^{2}\Omega}{\partial\alpha_{Z}^{2}}&\frac{\partial^{2}\Omega}{\partial\alpha_{Z}\partial\mu}&\frac{\partial^{2}\Omega}{\partial\alpha_{Z}\partial\beta}\\\ \frac{\partial^{2}\Omega}{\partial\mu\partial\alpha_{N}}&\frac{\partial^{2}\Omega}{\partial\mu\partial\alpha_{Z}}&\frac{\partial^{2}\Omega}{\partial\mu^{2}}&\frac{\partial^{2}\Omega}{\partial\mu\partial\beta}\\\ \frac{\partial^{2}\Omega}{\partial\beta\partial\alpha_{N}}&\frac{\partial^{2}\Omega}{\partial\beta\partial\alpha_{Z}}&\frac{\partial^{2}\Omega}{\partial\beta\partial\mu}&\frac{\partial^{2}\Omega}{\partial\beta^{2}}\end{array}\right\|~{}.$ (13) The logarithm of the grand-partition function of the systems is given as $\Omega=\Omega_{N}+\Omega_{Z}=S+\alpha_{N}N+\alpha_{Z}Z+\mu M-\beta{\cal E}~{}.$ (14) The derivation of $\Omega$ with respect to $\alpha$ and $\mu$ can be seen explicitly in Eqs. (25)-(35) of Ref. Moretto . Within the FTBCS1, the grand- partition function has the same form as that given by Eq. (14) of the FTBCS. Therefore, the first and second derivatives of the FTBCS1 grand-partition function are the same as those of the FTBCS ones. The only difference comes from the first derivatives of the pairing gap with respect to $\alpha$, $\mu$, and $\beta$ because of the quasiparticle-number fluctuations in the FTBCS1 gap equation (6). In this case, instead of the simple Eqs. [(33)-(35)] of Ref. Moretto , the derivatives become rather complicate expressions, which are obtained by taking the first derivatives of the left and right-hand sides of Eq. (6) with respect to $\alpha$, $\mu$, and $\beta$. Amongst the three derivatives of the FTBCS1 gap, ${\partial\Delta_{k}}/{\partial\beta}$ can be obtained by using its definition, namely $\frac{\partial\Delta_{k}}{\partial\beta}=-T^{2}\frac{\partial\Delta_{k}}{\partial T}=-T^{2}\frac{\Delta_{k}(T+\delta T)-\Delta_{k}(T)}{\delta T}~{},$ (15) and can be easily calculated numerically by choosing an appropriate value of $\delta T$ as the input parameter. The other two derivatives, ${\partial\Delta_{k}}/{\partial\alpha}$ and ${\partial\Delta_{k}}/{\partial\mu}$, must be calculated from their explicit analytic expressions because $\alpha$ and $\beta$ are two Lagrange multipliers, which are obtained by solving the FTBCS1 equations. The final equations for ${\partial\Delta_{k}}/{\partial\alpha}$ and ${\partial\Delta_{k}}/{\partial\mu}$ are derived as $\sum_{k}\left(A_{k}\beta\frac{\partial\Delta_{k}}{\partial\alpha}+B_{k}\right)+\left(C_{k}^{+}+C_{k}^{-}-\frac{2}{G}\right)\beta\frac{\partial\Delta_{k}}{\partial\alpha}+(D_{k}^{+}+D_{k}^{-})=0~{},$ (16) $\sum_{k}\left(A_{k}^{\prime}\beta\frac{\partial\Delta_{k}}{\partial\mu}+B_{k}^{\prime}\right)+\left(C_{k}^{\prime}+D_{k}^{\prime}-\frac{2}{G}\right)\beta\frac{\partial\Delta_{k}}{\partial\mu}+E_{k}^{\prime}=0~{},$ (17) where $A_{k}=\frac{1}{E_{k}^{3}}\left\\{(\lambda-\epsilon_{k})^{2}(1-n_{k}^{+}-n_{k}^{-})+\beta\Delta_{k}^{2}E_{k}^{2}[n_{k}^{+}+n_{k}^{-}-2(n_{k}^{+})^{2}-2(n_{k}^{-})^{2}]\right\\}~{},$ (18) $B_{k}=-\frac{\Delta_{k}(\lambda-\epsilon_{k})}{E_{k}^{3}}\left\\{1-n_{k}^{+}-n_{k}^{-}-\beta E_{k}^{2}(n_{k}^{+}+n_{k}^{-}-2(n_{k}^{+})^{2}-2(n_{k}^{-})^{2}\right\\}~{},$ (19) $C_{k}^{+}=\frac{n_{k}^{+}(1-n_{k}^{+})}{(1-n_{k}^{+}-n_{k}^{-})^{2}E_{k}^{3}}\left\\{(\lambda-\epsilon_{k})^{2}(1-n_{k}^{+}-n_{k}^{-})-\beta\Delta_{k}^{2}E_{k}[(n_{k}^{+}-1)^{2}+n_{k}^{-}(2n_{k}^{+}-n_{k}^{-})]\right\\}~{},$ (20) $C_{k}^{-}=\frac{n_{k}^{-}(1-n_{k}^{-})}{(1-n_{k}^{+}-n_{k}^{-})^{2}E_{k}^{3}}\left\\{(\lambda-\epsilon_{k})^{2}(1-n_{k}^{+}-n_{k}^{-})-\beta\Delta_{k}^{2}E_{k}[(n_{k}^{-}-1)^{2}+n_{k}^{+}(2n_{k}^{-}-n_{k}^{+})]\right\\}~{},$ (21) $D_{k}^{+}=-\frac{n_{k}^{+}(1-n_{k}^{+})\Delta_{k}(\lambda-\epsilon_{k})}{(1-n_{k}^{+}-n_{k}^{-})^{2}E_{k}^{3}}\\{1-n_{k}^{+}-n_{k}^{-}+\beta E_{k}[(n_{k}^{+}-1)^{2}+n_{k}^{-}(2n_{k}^{+}-n_{k}^{-})]\\}~{},$ (22) $D_{k}^{-}=-\frac{n_{k}^{-}(1-n_{k}^{-})\Delta_{k}(\lambda-\epsilon_{k})}{(1-n_{k}^{+}-n_{k}^{-})^{2}E_{k}^{3}}\\{1-n_{k}^{+}-n_{k}^{-}+\beta E_{k}[(n_{k}^{-}-1)^{2}+n_{k}^{+}(2n_{k}^{-}-n_{k}^{+})]\\}~{},$ (23) $A_{k}^{\prime}=\frac{1}{E_{k}^{3}}\\{(\lambda-\epsilon_{k})^{2}(1-n_{k}^{+}-n_{k}^{-})+\beta\Delta_{k}^{2}E_{k}[n_{k}^{+}(1-n_{k}^{+})+n_{k}^{-}(1-n_{k}^{-})]\\}~{},$ (24) $B_{k}^{\prime}=\frac{\beta m_{k}\Delta_{k}}{E_{k}}[n_{k}^{+}(1-n_{k}^{+})+n_{k}^{-}(1-n_{k}^{-})]~{},$ (25) $C_{k}^{\prime}=\frac{(\lambda-\epsilon_{k})^{2}}{(1-n_{k}^{+}-n_{k}^{-})E_{k}^{3}}[n_{k}^{+}(1-n_{k}^{+})+n_{k}^{-}(1-n_{k}^{-})]~{},$ (26) $D_{k}^{\prime}=-\frac{\beta\Delta_{k}^{2}}{(1-n_{k}^{+}-n_{k}^{-})^{2}E_{k}^{2}}\\{n_{k}^{+}(1-n_{k}^{+})[(n_{k}^{+}-1)^{2}+n_{k}^{-}(2n_{k}^{+}-n_{k}^{-})]$ $+n_{k}^{-}(1-n_{k}^{-})[(n_{k}^{-}-1)^{2}+n_{k}^{+}(2n_{k}^{-}-n_{k}^{+})]\\}~{},$ (27) $E_{k}^{\prime}=-\frac{\beta m_{k}\Delta_{k}}{E_{k}}[n_{k}^{+}(1-n_{k}^{+})+n_{k}^{-}(1-n_{k}^{-})]~{}.$ (28) By solving first the FTBCS1 equations, then Eqs. (16) and (17), one obtains ${\partial\Delta_{k}}/{\partial\alpha}$ and ${\partial\Delta_{k}}/{\partial\alpha}$ as functions of $T$ at a given value of the total angular momentum $M$. ## III Analysis of numerical results The numerical calculations are carried out for two realistic 60Ni and 72Ge nuclei. The latter is considered in order to have a comparison with the results obtained within the SMMC approach in Ref. Dean . The single-particle spectra for these two nuclei are obtained within the axially deformed Woods- Saxon potential WS , whose parameters are chosen to be the same as those given in Ref. leveldens . All the bound (negative energy) single-particle states are used in the calculations. The quadrupole deformation parameters $\beta_{2}$ are equal to 0 and -0.224 for 60Ni and 72Ge, respectively. The pairing interaction parameters are adjusted so that the pairing gaps at $T=0$ fit the experimental values obtained from the odd-even mass differences. These values are $G_{N}=$ 0.347 MeV, which gives $\Delta_{N}=$ 1.7 MeV for neutrons in 60Ni and $G_{N}=$ 0.291 MeV, which gives $\Delta_{N}=$ 1.7 MeV for neutrons in 72Ge. For protons in 72Ge, the value $G_{Z}=$ 0.34 MeV is chosen to give $\Delta_{Z}=$ 1.5 MeV, whereas there is no pairing gap for the closed-shell protons ($Z=$ 28) in 60Ni. Because the pairing gap (5) is level-dependent, the level-weighted gap $\bar{\Delta}$ is considered, which is defined as $\bar{\Delta}=\sum_{k}\Delta_{k}/\Omega$ with the total number $\Omega$ of levels in the deformed basis [In the case of spherical basis it becomes $\bar{\Delta}=\sum_{j}(2j+1)\Delta_{j}/\sum_{j}(2j+1)$ SCQRPA ]. Figure 1: (Color online) Level-weighted neutron pairing gap $\bar{\Delta}$ [(a), (d)], heat capacity $C$ [(b), (e)], and heat capacity divided by temperature $C/T$ [(c), (f)] for 60Ni obtained at different values of angular momentum $M$ as functions of $T$. Panels (a) - (c) show the FTBCS results, whereas the predictions by the FTBCS1 are displayed in (d) - (f). Shown in Fig. 1 are the neutron level-weighted pairing gap $\bar{\Delta}$, the heat capacity $C$, and the ratio $C/T$ obtained as functions of $T$ at several values of the total angular momentum $M$. The left column represents the predictions by the standard FTBCS, whereas the results obtained within the FTBCS1 are displayed in the right column. Both approaches show the pairing reentrance in the gaps at $M=$ 4 and 6 $\hbar$, namely the gap increases with $T$ up to $T\simeq$ 0.3 MeV, then decreases as $T$ increases further. Because of the quasiparticle-number fluctuations, the FTBCS1 gap does not collapse at $T_{c}$ as the FTBCS one, but decreases monotonically at high $T$. At $M=$ 14 $\hbar$, while the FTBCS gap completely vanishes at all $T$, the FTBCS1 gap shows a spectacular reentrance effect, namely it increases from the zero value at $T=$ 0 up to around 0.3 MeV at $T\simeq$ 0.8 MeV, and then slowly decreases as $T$ further increases. The heat capacities obtained within the FTBCS and FTBCS1 look alike, except for the region around $T_{c}$, where the quasiparticle-number fluctuations smooth out the sharp SN phase transition so that the sharp local maximum is depleted to a broad bump. In the region, where the pairing reentrance takes place, namely at $T\simeq$ 0.3 MeV and $M=$ 4 or 6 $\hbar$, a weak local minimum is seen on the curve representing the temperature dependence of the heat capacity similarly to the feature reported in Ref. Dean . This local minimum is magnified by using the ratio $C/T$ so that the latter might be useful in experiments as a quantity to identify the pairing reentrance. However, when the gap is too small as in the pairing reentrance at $M=$ 14 $\hbar$, the heat capacity $C$ ($C/T$) obtained within FTBCS1 is almost identical to that predicted by the FTBCS, where the gap is zero. Figure 2: (Color online) Level-weighted pairing gaps for neutrons [(a), (e)], protons [(b), (f)], heat capacity $C$ [(c), (g)], and heat capacity divided by temperature $C/T$ [(d), (h)] for 72Ge obtained at different values of angular momentum $M$ as functions of $T$. Panels (a) - (d) show the FTBCS results, whereas the predictions by the FTBCS1 are displayed in (e) - (h). For 72Ge, both neutron and proton gaps exist, which cause two peaks in the temperature dependence of the heat capacity obtained within the FTBCS, as shown in Fig. 2 (c). The overall features of $C$ and $C/T$ for 72Ge are similar to those obtained for 60Ni. As compared with the results of Ref. Dean , where the same single-particle energies in the ($0f1p-0g1d2s$) shells were used for both neutrons and protons, and where the pairing reentrance was predicted for neutrons, no pairing reentrance effect for neutrons is seen in the results of our calculations. On the other hand, the pairing reentrance takes place for protons at $M\geq$ 6 $\hbar$, as shown in Fig. 2 (b) and 2 (f). Figure 3: (Color online) Level densities as functions of $T$ at several values of total angular momentum $M$ obtained for 60Ni [(a), (b)] and 72Ge [(c), (d)] within the FTBCS (left panels) and FTBCS1 (right panels). Another experimentally measurable quantity, which may help to identify the pairing reentrance effect, is the level density. In fact, the authors of Ref. Dean claimed that the pairing reentrance causes an irregularity in a shape of a small local maximum at low $T$ on the curve, which describes the temperature dependence of the level density. The level densities obtained at several values of the total angular momentum $M$ for 60Ni and 72Ge are displayed in Fig. 3 as functions of $T$. These results show a trend of transition of the level density from a convex function of $T$ to a concave function after the pairing reentrance occurs. This is particularly clear for 60Ni by comparing the FTBCS1 predictions for the level density at $M<$ 14 $\hbar$, which are convex functions of $T$, with that obtained at $M=$ 14 $\hbar$, which is a concave function of $T$. For 72Ge this trend is less obvious because of the existence of proton and neutron pairing gaps with different values of $T_{c}$ within the FTBCS. However, contrary to the result shown in the inset of Fig. 4 in Ref. Dean , no pronounced local maximum that might correspond to the pairing reentrance is seen here in the temperature dependence of the level density for 72Ge. Since the results for the pairing gap, heat capacity, and level density strongly depend on the selected single-particle energies, the irregularity seen in the temperature dependence of the level density at $\omega=$ 0.5 MeV in the inset of Fig. 4 of Ref. Dean might well be an artifact caused by using the same single-particle energies for both neutron and proton spectra. Figure 4: (Color online) Level averaged neutron and proton gaps $\bar{\Delta}_{N,Z}$, heat capacity $C$, $C/T$, excitation energy $E^{}$ and level density for 72Ge as functions of $T$ at several values of $M$ [shown in (a)] obtained within the FTBCS1 in the test calculations by using the shells $(0f1p-0g1d2s)$ atop the 40Ca core with the values of Woods-Saxon neutron single-particle energies adopted for both neutron and proton spectra. To show that it is indeed the case, we carried out the test calculations by using only the $(0f1p-0g1d2s)$ shells on top of the 40Ca core with the values of Woods-Saxon neutron single-particle energies adopted for both neutron and proton spectra of 72Ge, as in Ref. Dean . The difference now solely comes from that between the numbers of valence neutrons and protons (20 valence neutrons and 12 valence protons). The results of these test calculations within the FTBCS1 are shown in Fig. 4. They show the pairing reentrance in the neutron pairing gap instead of the proton one, at low $M$. This is in qualitative agreement with the pairing reentrance predicted for neutrons in Fig. 2 of Ref. Dean , where no proton pairing reentrance is seen up to $\omega=$ 0.5 MeV. In our calculations, however, the proton pairing reentrance takes place at rather high $M=$ 22 $\hbar$. In either case, the pairing reentrance is so strong that causes the excitation energy $E^{}$ to decrease slightly with increasing $T$ at low $T$. This violates the second law of thermodynamics. As a consequence, at $M\geq$ 6 $\hbar$, the heat capacity becomes negative at $T<$ 0.4 MeV. The results for the level density remain essentially the same as compared to those previously obtained by using all proton and neutron single-particle levels, but with only bound-state single particle energies [Fig. 3 (d)], and no irregularities such as a pronounced local maximum are found. In our opinion, the pick on the dotted curve in the inset of Fig. 4 in Ref. Dean emerges because of the two lower values of ${\rm ln}\rho$ at $T\simeq$ 0.42 and 0.45 MeV. These two lower values are the results obtained by calculating the level density in the canonical ensemble $\rho(E)=\beta e^{S}/\sqrt{2\pi C}$, making use of the two large values of $C$ equal to around 16 and 28 (with large error bars). However, these large values of the heat capacity at low $T$ are the artifacts of the SMMC calculations because the heat capacity must be zero (or very small) at $T=$ 0 (or very low $T$) to avoid an infinite (or very large) entropy, which would violate the third law of thermodynamics. Therefore, we conclude that the neutron pairing reentrance effect, reported in Ref. Dean , is caused by the use of the same single-particle spectrum for both protons and neutrons, whereas the irregularity seen on the curve of ${\rm ln}\rho$ in the inset of Fig. 4 of Ref. Dean is caused by unphysically large values of the heat capacity at low $T$ in the SMMC technique. The present calculations within the FTBCS and FTBCS1 do not take into account the effects of residual interactions beyond the monopole pairing one. It is well known that these effects are responsible for strong collective motion in finite nuclei, which leads to the increase of nuclear level density. In spherical nuclei the collective enhancement of level density is caused by vibrational excitations, whereas in deformed nuclei it comes from the collective rotation. The contribution of collective motion to the increase of nuclear level density has been studied in detail by Ignatyuk and collaborators starting from the early 1970s Ignatyuk ; Junghans . Because the Hamiltonian used in the SMMC calculations of Ref. Dean included the quadrupole-quadrupole interaction, let us estimate the effect of the collective quadrupole vibration on the increase of level density. By using the adiabatic approximation (3) of Ref. Junghans for the enhancement coefficient $K_{vib}$ due to quadrupole vibration, and the experimental energies $E(2^{+}_{1})$ of the lowest quadrupole excitation in Ref. 1st2 , we found $K_{vib}\simeq$ 1.06 and 1.94 for 60Ni at $T=$ 0.3 MeV and 72Ge at $T=$ 0.4 MeV, respectively. These values of T are those, at which the pairing reentrance starts to show up in these nuclei. With the deformation parameter $\beta_{2}=$ -0.224 adopted in the present calculations for 72Ge and by using Eq. (9) of Ref. Zagreb , we found the enhancement coefficient $K_{coll}(\beta_{2})\simeq$ 1.96 for 72Ge at $T=$ 0.4 MeV, whereas for the spherical nucleus, 60Ni, $K_{coll}=K_{vib}$ = 1.06 at $T=$ 0.3 MeV. Therefore, for both nuclei, 60Ni and 72Ge, one can expect that the collective quadrupole enhancement of level density is not dramatic at the value of temperature, where the pairing reentrance is supposed to take place. The contribution of collective motion generated by higher multipolarities to the increase of level density is expected to be much smaller. In Ref. combi the quasiparticle Tamm-Dancoff Approximation, which includes the isoscalar quadrupole-quadrupole interaction and $J_{x}$ cranking, was used to calculate the level density within the microcanonical ensemble. The authors of Ref. combi found 4 $\leq K_{rot}\leq$ 6 and 1.002 $\leq K_{vib}\leq$ 1.012 at excitation energy 3 $\leq E\leq$ 8 MeV (i.e. at around 0.38 $\leq T\leq$ 0.63 MeV) for 162Dy. They also found a monotonic decrease of the average pairing gap with increasing the excitation energy up to $E=$ 8.5 MeV ($T\simeq$ 0.65 MeV), i.e. much higher than $T_{c}\simeq$ 0.34 MeV. These results are in good qualitative agreement with our estimations. Finally, it is worth noticing that the inclusion of the approximate particle- number projection within the Lipkin-Nogami method does not significantly alter the behavior of the paring reentrance obtained within the FTBCS1 theory (See Fig. 6 of Ref. SCQRPA ). This does not diminish the value of an approach based on exact particle-number and angular momentum projections. In Ref. Horoi the exact solution of the nuclear shell model is used to study the SN phase transition including residual interactions other than the pairing one. The results of Ref. Horoi , which fully respect the particle number and angular momentum conservations, confirm the presence of a long tail of pair correlations far beyond the BCS phase transition region in agreement with the prediction by the FTBCS1. The approach of Ref. Horoi does not use any external heat bath, which determines the temperature of thermal equilibrium. Therefore the nuclear temperature can only be extracted from the level density by using the Clausius definition of thermodynamic entropy. This task is not easy because of the discrete and finite nuclear spectra (See, e.g. Ref. Ensemble and references therein). Nonetheless, instead of using temperature, it would be interesting to see if the pairing reentrance takes place in the pair correlator as a function of excitation energy at various values of angular momentum within the method of Ref. Horoi . ## IV Conclusions The present paper studies the temperature dependences of the heat capacity and level density in hot medium-mass nuclei, which undergo a noncollective rotation about the symmetry axis. The numerical calculations, carried out by using the realistic Woods-Saxon single-particle energies for 60Ni and 72Ge within the FTBCS and FTBCS1 theories, have shown the pairing reentrance in the pairing gap at finite angular momentum $M$ and temperature $T$. Instead of decreasing with increasing $T$, the gap first increases with $T$ then decreases at higher $T$. It is demonstrated that the heat capacity $C$, or rather $C/T$, and level density $\rho$ can be used to experimentally identify the pairing reentrance effect. The pairing reentrance, when it occurs, leads to a clear depletion in the temperature dependence of the heat capacity, whereas the level density weakly changes from a convex function of $T$ to a concave one. Regarding the appearance of the local minimum in the heat capacity because of the pairing reentrance, the results of the present paper agree with that of the SMMC calculations in Ref. Dean . However, the present results show no pronounced local maximum in the temperature dependence of the level density. The pairing reentrance is seen in the proton pairing gap of 72Ge at low $M$, whereas Ref. Dean reported this effect in the neutron pairing energy. The test calculations by using the same single-particle configuration as that used in Ref. Dean , but obtained within the Woods-Saxon potential, reveals that the neutron pairing reentrance in 72Ge is an artifact, which is caused by the use of the same single-particle spectrum for both protons and neutrons, whereas the irregularity on the curve for the logarithm of level density, reported in Ref. Dean , is caused by unphysically large values of the heat capacity at low $T$ in the SMMC approach. ###### Acknowledgements. The numerical calculations were carried out using the FORTRAN IMSL Library by Visual Numerics on the RIKEN Integrated Cluster of Clusters (RICC) system. NQH acknowledges the support by the National Foundation for Science and Technology Development(NAFOSTED) of Vietnam through Grant No. 103.04-2010.02. He also thanks the Theoretical Nuclear Physics Laboratory of RIKEN Nishina Center for its hospitality during his visit in RIKEN. ## References * (1) B.R. Mottelson and J.G. Valatin, Phys. Rev. Lett. 5, 511 (1960). * (2) T. Kammuri, Prog. Theor. Phys. 31, 595 (1964). * (3) L. G. Moretto, Nucl. Phys. A 185, 145 (1972). * (4) R. Balian, H. Flocard and M. Vénéroni, Phys. Rep. 317, 251 (1999). * (5) H.J. Mang, O. Rasmussen, and M. Rho, Phys. Rev. 141, 941 (1966). * (6) L.G. Moretto, Phys. Lett. B 40, 1 (1972); A.L. Goodman, Phys. Rev. C 29, 1887 (1984). * (7) N. Dinh Dang and V. Zelevinsky, Phys. Rev. C 64, 064319 (2001); N. Dinh Dang and A. Arima, Phys. Rev. C 67, 014304 (2003); N.D. Dang and A. Arima, Phys. Rev. C 68, 014318 (2003); N.D. Dang, Nucl. Phys. A 784, 147 (2007). * (8) V. Zelevinsky, B.A. Brown, N. Fazier, and M. Horoi, Phys. Rep. 276, 85 (1996). * (9) R.W. Richardson, Phys. Lett. 3, 277 (1963); Ibid. 14, 325 (1965); A. Volya, B.A. Brown, and V. Zelevinsky, Phys. Lett. B 509, 37 (2001). * (10) S. Frauendorf, N. K. Kuzmenko, V. M. Mikhajlov, and J. A. Sheikh, Phys. Rev. B 68, 024518 (2003), J. A. Sheikh, R. Pa lit, and S. Frauendorf, Phys. Rev. C 72, 041301(R) (2005). * (11) F. L$\acute{\rm e}$vy, I. Sheikin, B. Grenier, and A. D. Huxley, Science 309, 1343 (2005). * (12) A. Miyake, D. Aoki, G. Knebel, V. Taufour, and J. Flouquet, J. Phys: Conf. Series 200, 012122 (2010). * (13) A. Gorczyca and M. Mierzejewski, Phys. Stat. Sol. (b) 244, 2503 (2007). * (14) N.Q. Hung and N.D. Dang, Phys. Rev. C 78, 064315 (2008). * (15) N.Q. Hung and N.D. Dang, Phys. Rev. C 79, 054328 (2009). * (16) E. Algin et al., Phys. Rev. C 78, 054321 (2008); A. Schiller et al., Phys. Atom. Nucl. 64, 1186 (2001); K. Kaneko et al., Phys. Rev. C 74, 024325 (2006). * (17) D.J. Dean, K. Langanke, H.A. Nam, and W. Nazarewicz, Phys. Rev. Lett. 105, 212504 (2010). * (18) N.Q. Hung and N.D. Dang, Phys. Rev. C 81, 057302 (2010); N.Q. Hung and N.D. Dang, Phys. Rev. C 82, 044316 (2010). * (19) N. Dinh Dang, Z. Phys. A 335, 253 (1990). * (20) N.Q. Hung and N.D. Dang, Phys. Rev. C 76, 054302 (2007), ibid. 77 029905(E) (2008); N. Dinh Dang and N. Quang Hung, Phys. Rev. C 77, 064315 (2008). * (21) O. Civitarese, A. Plastino, and A. Faessler, Z. Phys. A 313, 197 (1983). * (22) S. Cwiok, J. Dudek, W. Nazarewicz, J. Skalski, and T. Werner, Comput. Phys. Commun. 46, 379 (1987). * (23) A.V. Ignatyuk, Statistical properties of excited atomic nuclei (Moscow, Energoatomizdat, 1983). * (24) A.R. Junghans, M. de Jong, H.-G. Clerc, A.V. Ignatyuk, G.A. Kudryaev, amd K.-H. Schmidt, Nucl. Phys. A 629, 635 (1998). * (25) S. Raman, C.W. Nestor, Jr., S.K. Kahane, and K.H. Bhatt, At. Data and Nucl. Data Tables, 42, 1 (1989). * (26) V.I. Zagrebaev, Y. Aritomo, M.G. Itkis, Yu. Ts. Oganessian, and M. Ohta, Phys. Rev. C 65, 014607 (2001). * (27) H. Uhrenholt, S. Åberg, P. Möller, and T. Ichikawa, arXiv: 0901.10872v2 [nucl-th] (2009). * (28) M. Horoi and V. Zelevinsky, Phys. Rev. C 75, 054303 (2007).	arxiv-papers	2011-11-15T03:57:41	2024-09-04T02:49:24.336753	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "N. Quang Hung and N. Dinh Dang", "submitter": "Nguyen Quang Hung", "url": "https://arxiv.org/abs/1111.3425" }
1111.3511	# Polygons of the Lorentzian plane and spherical simplexes François Fillastre University of Cergy-Pontoise UMR CNRS 8088 Departement of Mathematics F-95000 Cergy-Pontoise FRANCE francois.fillastre@u-cergy.fr ((v3) ) ## 1 Introduction It is a common occurence that sets of geometric objects themselves carry some kind of geometric structure. A classical example for this is the set of all conformal structures on a given compact surface. Riemann discovered that this set, the “space” of conformal structures, can be described by a finite number of parameters called moduli. The corresponding parameter or moduli space turned out to be a very interesting geometric object in itself whose study is the subject of Teichmüller theory. On a more basic level, one can consider spaces consisting of objects of elementary geometry like (shapes of) polyhedra in Euclidean space. Thurston [Thu98] found that in this case, the corresponding moduli space carries the structure of a complex hyperbolic manifold, and he established a link with sets of triangulations of the 2-sphere. Bavard and Ghys [BG92] considered sets of polygons in the Euclidean plane. Fix a compact convex polygon $P$ with $n\geq 3$ edges and let ${\cal P}(P)$ be the space of convex polygons with $n$ edges parallel to those of $P$. The elements of ${\cal P}(P)$ are then determined by the distances of the lines containing the edges from the origin, which gives $n$ parameters. Following [Thu98], Bavard and Ghys proved that on the space of parameters, the area of the polygons in ${\cal P}(P)$ is a quadratic form, and they computed its signature. The kernel of the corresponding bilinear form has dimension 2 (due to the fact that area is invariant under translations), and there is only one positive direction. Hence, up to the kernel, one gets a Lorentzian signature. As a consequence, the set of elements of ${\cal P}(P)$ with area equal to one, considered up to translations, can be identified with a subset of the hyperbolic space $\mathbb{H}^{n-3}$. This subset turns out to be a hyperbolic convex polyhedron of a special kind: it is a simplex with the property that each hyperplane containing a facet meets orthogonally all but two hyperplanes containing the other facets. Such simplices are called hyperbolic orthoschemes. The dihedral angles of the orthoscheme can be computed from the angles of $P$, and [BG92] contains a list of convex polygons $P$ such that the orthoscheme obtained from $P$ is of Coxeter type, i.e. has acute angles of the form $\pi/k$, $k\in\mathbb{N}$. This list was previously known [IH85, IH90], but it appeared it was incomplete [Fil11]. In this paper we consider a class of non-compact plane polygons whose moduli space is a spherical orthoscheme. These polygons, the $t$-convex polygons introduced in Section 3, are best described not in terms of the Euclidean geometry on $\mathbb{R}^{2}$, but as subsets of the Lorentz plane. Instead of the area we will consider a suitably defined coarea that turns out to be a positive definite quadratic form on the parameter space, an $n$-dimensional vector space. Restricting to coarea one we obtain a subset of the unit sphere in that parameter space, and this subset is shown to be a spherical orthoscheme. Moreover, any spherical orthoschem can be obtained in this way. It is amusing that in [BG92] Euclidean polygons led to Lorentz metrics and hyperbolic orthoschemes, while in the present paper Lorentzian polygons give rise to Euclidean metrics and spherical orthoschemes. The author does not know if there is a way to obtain Euclidean orthoschemes from spaces of plane convex polygons. ## 2 Background on the Lorentz plane Recall that the Lorentz plane is $\mathbb{R}^{2}$ equipped with the Lorentz inner product, that is the bilinear form $\langle\binom{x_{1}}{x_{2}},\binom{y_{1}}{y_{2}}\rangle_{1}=x_{1}y_{1}-x_{2}y_{2}.$ A non-zero vector $v$ can be _space-like_ ($\langle v,v\rangle_{1}>0$), _time- like_ ($\langle v,v\rangle_{1}<0$) or _light-like_ ($\langle v,v\rangle_{1}=0$). The set of time-like vectors has two connected components, and we denote the upper one, the set of _future_ time-like vectors, by ${\mathcal{F}}:=\\{x\in{\mathbb{R}}^{2}\|\langle x,x\rangle_{1}<0,x_{2}>0\\}.$ The set of unit future time-like vectors is ${\mathbb{H}}:=\\{x\in{\mathbb{R}}^{2}\|\langle x,x\rangle_{1}=-1,x_{2}>0\\},$ which will be the analog of the circle in the Euclidean plane, see Figure 1. In higher dimension, the generalization of ${\mathbb{H}}$ together with its induced metric is a model of the hyperbolic space, in the same way that the unit sphere for the Euclidean metric with its induced metric is a model of the round sphere. In particular, if the angle between two unit vectors in the Euclidean plane is seen as the distance between the two corresponding points on the circle, the _(Lorentzian) angle_ between two future time-like vectors $x$ and $y$ is the unique $\varphi>0$ such that $\cosh\varphi=-\frac{\langle x,y\rangle_{1}}{\sqrt{\langle x,x\rangle_{1}\langle y,y\rangle_{1}}}$ (1) (see [Rat06, (3.1.7)] for the existence of $\varphi$). The angle $\varphi$ is the distance on ${\mathbb{H}}$ (for the induced metric) between $x/\sqrt{-\langle x,x\rangle_{1}}$ and $y/\sqrt{-\langle y,y\rangle_{1}}$. Figure 1: The cone ${\mathcal{F}}$ of future time-like vectors and the curve ${\mathbb{H}}$ of unit future time-like vectors. ${\mathcal{F}}$ and ${\mathbb{H}}$ are globally invariant under the action of the linear isometries of the Lorentzian plane, called _hyperbolic translations_ : $H_{t}:=\left(\begin{array}[]{cc}\cosh t&\sinh t\\\ \sinh t&\cosh t\end{array}\right),t\in{\mathbb{R}}.$ (2) In all the paper we fix a positive $t$. We denote by $<H_{t}>$ the free group spanned by $H_{t}$. ## 3 $t$-convex polygons Let $a\in{\mathcal{F}}$. We will denote by $a^{\bot}:=\\{x\in{\mathbb{R}}^{2}\|\langle x,a\rangle_{1}=\langle a,a\rangle_{1}\\}$ the line that passes through $a$ and is parallel to the $1$-dimensional subspace orthogonal to $a$ under $\langle\cdot,\cdot\rangle_{1}$. ###### Definition 3.1. Let $(\eta_{1},\ldots,\eta_{n})$, $n\geq 1$, be pairwise distinct unit future time-like vectors in the Lorentzian plane (i.e. $\eta_{i}\in{\mathbb{H}}$), and let $h_{1},\ldots,h_{n}$ be positive numbers. A _$t$ -convex polygon_ $P$ is the intersection of the half-planes bounded by the lines $(H_{t}^{k}(h_{i}\eta_{i}))^{\bot},\forall k\in\mathbb{Z},\forall i=1,\ldots,n.$ The half-planes are chosen such that the vectors $\eta_{i}$ are inward pointing. The positive numbers $h_{i}$ are the _support numbers_ of $P$. A $t$-convex polygon is called _elementary_ if it is defined by a single future time-like vector $\eta$ and a positive number $h$. Note that for each $k$, $(H_{t}^{k}(h\eta))^{\bot}$ is tangent to $h{\mathbb{H}}$ (the upper hyperbola with radius $h$). Hence a $t$-convex polygon is the intersection of a finite number of elementary $t$-convex polygons. ###### Example 3.2. Let $t_{0}=\sinh^{-1}(1)$, so $H_{t_{0}}:=\left(\begin{array}[]{cc}\sqrt{2}&1\\\ 1&\sqrt{2}\end{array}\right).$ Let us denote by $P_{1}$ the elementary $t_{0}$-convex polygon defined by the vector $\eta=\binom{0}{1}$ and the number $h=1$, see Figure 2a. The elementary $t_{0}$-convex polygon $P_{2}$ of Figure 2b is obtained from $p_{1}$ by a slightly change of $\eta$ and $h$. Their intersection forms the $t_{0}$-convex polygon of Figure 2c. (a) A part of the $t_{0}$-convex polygon $P_{1}$. For the Lorentzian metric, all the edges have equal length and all the angles between edges are equal. (b) A part of the $t_{0}$-convex polygon $P_{2}$. For the Lorentzian metric, all the edges have equal length and all the angles between edges are equal. (c) A part of the $t_{0}$-convex polygon obtained as the intersection of $P_{1}$ and $P_{2}$. Figure 2: To Example 3.2. ###### Lemma 3.3. A $t$-convex polygon $P$ is a proper convex subset of ${\mathbb{R}}^{2}$ contained in ${\mathcal{F}}$, bounded by a polygonal line with a countable number of sides, and globally invariant under the action of $<H_{t}>$. ###### Proof. The group invariance is clear from the definition. $P$ is the intersection of a finite number of elementary $t$-convex polygons, so we only have to check the other properties in the elementary case. Actually the only non-immediate one is that an elementary $t$-convex polygon is contained in ${\mathcal{F}}$. Let us consider an elementary $t$-convex polygon made from a single future time-like vector $\eta$ and a number $h$. Without loss of generality, consider that $h=1$. Let $u=H^{k}_{t}(\eta)$ and $v=H^{k^{\prime}}_{t}(\eta)$ and let $x$ be the intersection between $u^{\bot}$ and $v^{\bot}$. As $\langle x,u\rangle_{1}=\langle x,v\rangle_{1}=-1$, $x$ is orthogonal to $u-v$, which is a space-like vector (compute its norm with the help of (1)). Hence $x$ is time-like, and as $u^{\bot}$ and $v^{\bot}$ never meet the past cone, $x$ is future. It is easy to deduce that the $t$-convex polygon is contained in ${\mathcal{F}}$. ∎ Note that as a convex surface, a $t$-convex polygon can also be a $t^{\prime}$-convex polygon (for example it is also invariant under the action of any subgroup of $<H_{t}>$), but we will only consider the action of a given $<H_{t}>$. Given a $t$-convex polygon $P$, we will require that the set of elementary $t$-convex polygons such that their intersection gives $P$ is minimal, i.e each $\eta_{i}$ is the inward unit normal of a genuine edge $e_{i}$ of $P$. The edge at the left (resp. right) of $e_{i}$ is denoted by $e_{i-1}$ (resp. $e_{i+1}$). Let $p_{i}$ be the foot of the perpendicular from the origin to the line containing $e_{i}$ (in particular, $p_{i}=h_{i}\eta_{i}$). Let $p_{ii+1}$ be the vertex between $e_{i}$ and $e_{i+1}$. We denote by $h_{ii+1}$ (resp. $h_{ii-1}$) the signed distance from $p_{i}$ to $p_{ii+1}$ (resp. from $p_{i}$ to $p_{i-1i}$): it is non negative if $p_{i}$ is on the same side of $e_{i+1}$ (resp. $e_{i-1}$) as $P$. The angle between $\eta_{i}$ and $\eta_{i+1}$ is denoted by $\varphi_{i}$. See Figure 3. Figure 3: Notations for a $t$-convex polygon. ###### Lemma 3.4. With the notations introduced above, $h_{ii+1}=\frac{h_{i}\cosh\varphi_{i}-h_{i+1}}{\sinh\varphi_{i}},h_{ii-1}=\frac{h_{i}\cosh\varphi_{i-1}-h_{i-1}}{\sinh\varphi_{i-1}}.$ (3) ###### Proof. By definition, $h_{ii+1}$ is non negative when $\langle p_{i}-p_{i+1},\eta_{i+1}\rangle_{1}\leq 0$, i.e. $-(h_{i+1}-h_{i}\cosh\varphi_{i})\geq 0.$ Hence $h_{ii+1}=-\frac{h_{i+1}-h_{i}\cosh\varphi_{i}}{\|h_{i+1}-h_{i}\cosh\varphi_{i}\|}\sqrt{\langle p_{ii+1}-p_{i},p_{ii+1}-p_{i}\rangle_{1}}.$ Up to an orientation and time orientation preserving linear isometry, one can take $\eta_{i}=\left(0\atop 1\right)$. In particular $p_{i}=\left(0\atop h_{i}\right)$ and $(p_{ii+1})_{2}=h_{i}$ hence $\langle p_{ii+1}-p_{i},p_{ii+1}-p_{i}\rangle_{1}=(p_{ii+1})_{1}^{2}.$ We also have $\eta_{i+1}=\left(\sinh\varphi_{i}\atop\cosh\varphi_{i}\right)$, and as $\langle p_{ii+1},\eta_{i+1}\rangle_{1}=-h_{i+1}$ we get $(p_{ii+1})_{1}=\frac{-h_{i+1}+h_{i}\cosh\varphi_{i}}{\sinh\varphi_{i}}.$ The proof for $h_{ii-1}$ is similar, considering $\eta_{i-1}=\left(-\sinh\varphi_{i}\atop\cosh\varphi_{i}\right)$. ∎ ## 4 The cone of support vectors Let $P$ be a $t$-convex polygon. Choose an edge and denote its inward unit normal by $\eta_{1}$. We denote the inward unit normal of the edge on the right by $\eta_{2}$, and so on until $\eta_{n+1}=H_{t}(\eta_{1})$. The edges with normals $\eta_{1},\ldots,\eta_{n}$ are the _fundamental edges_ of $P$. Note that with this labeling, if $\varphi_{i}$ is the angle between $\eta_{i}$ and $\eta_{i+1}$, we have $\varphi_{1}+\varphi_{2}+\cdots+\varphi_{n}=t.$ (4) The number $h_{i}(P)$ is the support number of the edge with normal $\eta_{i}$, and $h(P)=(h_{1}(P),\ldots,h_{n}(P))$ is the _support vector_ of $P$. So $P$ is identified with a vector of ${\mathbb{R}}^{n}$, in such a way that $\eta_{1},\ldots,\eta_{n}$ are in bijection with the standard basis of $\mathbb{R}^{n}$. Of course $P$ is uniquely determined by its support vector. ###### Definition 4.1. Choose $\eta\in{\mathbb{H}}$ and let $\varphi_{1},\varphi_{2},\cdots,\varphi_{n}$ be positive numbers satisfying (4). The _cone of support vectors_ $\overline{\mathcal{P}}(\varphi_{1},\varphi_{2},\ldots,\varphi_{n})$ is the set of support vectors of $t$-convex polygons with inward unit normals $\eta_{1}=\eta$, $\eta_{i+1}=H_{\varphi_{i}}(\eta_{i})$. A priori the definition of $\overline{\mathcal{P}}$ depends not only on the angles $\varphi_{i}$ but also on the choice of $\eta$. Actually choosing another starting $\eta^{\prime}\in{\mathbb{H}}$, the hyperbolic translation from $\eta$ to $\eta^{\prime}$ gives a linear isomorphism between the two resulting sets of support vectors. Hence $\overline{\mathcal{P}}(\varphi_{1},\varphi_{2},\ldots,\varphi_{n})$ could be defined as the set of $t$-convex polygons with ordered angles $(\varphi_{1},\varphi_{2},\ldots,\varphi_{n})$ up to hyperbolic translations. Note also that if $s$ is a cyclic permutation, then $\overline{\mathcal{P}}(\varphi_{s(1)},\ldots,\varphi_{s(n)})$ is the same as $\overline{\mathcal{P}}(\varphi_{1},\ldots,\varphi_{n})$. It is possible to prove that $\overline{\mathcal{P}}(\varphi_{1},\varphi_{2},\ldots,\varphi_{n})$ is a convex polyhedral cone with non-empty interior in ${\mathbb{R}}^{n}$, but this will be easier after a suitable metrization of ${\mathbb{R}}^{n}$, that is the subject of the next section. ## 5 Coarea ###### Definition 5.1. Let $P\in\overline{\mathcal{P}}(\varphi_{1},\varphi_{2},\ldots,\varphi_{n})$. The _coarea_ of $P$ is $\operatorname{coarea}(P)=\frac{1}{2}\sum_{i=1}^{n}h_{i}(P)\ell_{i}(P)$ where the sum is on the fundamental edges, and $\ell_{i}(P)=h_{ii-1}(P)+h_{ii+1}(P)$ is the length of the $i$th fundamental edge (hence positive). Geometrically $\operatorname{coarea}(P)$ is the area (in the sense of the Lebesgue measure) of a fundamental domain for the action of $H_{t}$ on the complement of $P$ in ${\mathcal{F}}$. The main point is that hyperbolic translations (2) have determinant $1$, so they preserve the area, which is then independent of the choice of the fundamental domain, see Figure 4. Moreover the area of a triangle with a space-like edge $e$ of length $l$ and $0$ as a vertex has area $\frac{1}{2}lh$, if $h$ is the Lorentzian distance between $0$ and the line containing $e$. (To see this, perform a hyperbolic translation such that $e$ is horizontal and compute the area.) Note that the coarea depends not only on the polygonal line $P$ but also on the group $<H_{t}>$, so it would be more precise to speak about “$t$-coarea”, but as the group is fixed from the beginning, no confusion is possible. Figure 4: The two shaded regions have the same area. This area is the coarea of the polygon. For a given cone of support vectors, the coarea can be formally extended to ${\mathbb{R}}^{n}$ with the help of (3): for $h\in{\mathbb{R}}^{n}$, $\operatorname{coarea}(h)=\frac{1}{2}\sum_{i=1}^{n}h_{i}\ell_{i}(h)$ with $\ell_{i}(h):=\frac{h_{i}\cosh\varphi_{i-1}-h_{i-1}}{\sinh\varphi_{i-1}}+h_{i}\frac{h_{i}\cosh\varphi_{i}-h_{i+1}}{\sinh\varphi_{i}}.$ (5) If $n=1$, there is only one angle between the unit inward normal $\eta$ and its image under $H_{t}$, which is equal to $t$, and $\operatorname{coarea}(h)=h^{2}\frac{\cosh t-1}{\sinh t}.$ If $n\geq 2$, we introduce the _mixed-coarea_ $\operatorname{coarea}(h,k)=\frac{1}{2}\sum_{i=1}^{n}h_{i}\frac{k_{i}\cosh\varphi_{i-1}-k_{i-1}}{\sinh\varphi_{i-1}}+h_{i}\frac{k_{i}\cosh\varphi_{i}-k_{i+1}}{\sinh\varphi_{i}},$ which is the polarization of the $\operatorname{coarea}$. Actually, it is clearly a bilinear form, and $\operatorname{coarea}(\eta_{k},\eta_{j})=\left\\{\begin{array}[]{ccc}0&\mbox{if }&2\leq\|j-k\|\leq n+1\\\ \displaystyle{-\frac{1}{2}\frac{1}{\sinh\varphi_{k-1}}}&\mbox{if}&j=k-1\\\ \displaystyle{-\frac{1}{2}\frac{1}{\sinh\varphi_{k}}}&\mbox{if}&j=k+1\\\ \displaystyle{\frac{1}{2}\left(\frac{\cosh\varphi_{k-1}}{\sinh\varphi_{k-1}}+\frac{\cosh\varphi_{k}}{\sinh\varphi_{k}}\right)}&\mbox{if}&j=k\end{array}\right.$ (6) so $\operatorname{coarea}$ is symmetric. We also obtain the following key result. ###### Proposition 5.2. The symmetric bilinear form $\operatorname{coarea}$ is positive definite. ###### Proof. As $\cosh\varphi_{k}>1$, the matrix $(\operatorname{coarea}(u_{k},u_{j}))_{kj}$ is strictly diagonally dominant, and symmetric with positive diagonal entries, hence positive definite, see for example [Var00, 1.22]. ∎ The Cauchy–Schwarz inequality applied to support vectors of $t$-convex polygons gives the following _reversed Minkowski inequality_ : ###### Corollary 5.3. Let $P,Q$ be $t$-convex polygons with parallel edges. Then $\operatorname{coarea}(P,Q)^{2}\leq\operatorname{coarea}(P)\operatorname{coarea}(Q),$ with equality if and only if $P$ and $Q$ are homothetic: $\exists\lambda>0,\forall i,h_{i}(P)=\lambda h_{i}(Q)$. ## 6 Spherical orthoschemes $\overline{\mathcal{P}}(\varphi_{1},\varphi_{2},\ldots,\varphi_{n})$ is clearly a cone in ${\mathbb{R}}^{n}$. Moreover it is the set of vectors of positive edge lengths, for the edge lengths defined by (5). From the definition of the coaera, for $h\in{\mathbb{R}}^{n}$, $2\operatorname{coarea}(\eta_{i},h)=\ell_{i}(h)$, so $\eta_{i}$ is an inward normal vector to the facet of $\overline{\mathcal{P}}$ defined by $\ell_{i}=0$. So $\overline{\mathcal{P}}$ is polyhedral, and it is convex because the $\eta_{i}$ form a basis of ${\mathbb{R}}^{n}$. Let us denote by $\mathcal{P}(\varphi_{1},\varphi_{2},\ldots,\varphi_{n})$ the intersection of $\overline{\mathcal{P}}(\varphi_{1},\varphi_{2},\ldots,\varphi_{n})$ with the unit sphere of $({\mathbb{R}}^{n},\operatorname{coarea})$ (i.e. the set of support vectors of $t$-convex polygons with coarea one). It follows that $\mathcal{P}$ is a spherical simplex. If $n=1$, $\mathcal{P}$ is a point on a line, so from now on assume that $n>1$. When $n=2$, $\mathcal{P}$ is an arc on the unit circle with length $\theta$ satisfying $\cos\theta=\frac{\sinh\varphi_{2}}{\sinh(\varphi_{1}+\varphi_{2})}.$ When $n=3$, $\mathcal{P}$ is a spherical triangle with acute inner angles, whose cosines are given by: $-\frac{\operatorname{coarea}(\eta_{k},\eta_{k+1})}{\sqrt{\operatorname{coarea}(\eta_{k},\eta_{k})}\sqrt{\operatorname{coarea}(\eta_{k+1},\eta_{k+1})}}=\sqrt{\frac{\sinh\varphi_{k-1}\sinh\varphi_{k+1}}{\sinh(\varphi_{k-1}+\varphi_{k})\sinh(\varphi_{k}+\varphi_{k+1})}}.$ (7) When $n\geq 3$, from (6) we see that each facet has an acute interior dihedral angle with exactly two other facets, and is orthogonal to the other facets. Such spherical simplexes are called _acute spherical orthoschemes_. See [Deb90, 5] for the history and main properties of these very particular simplexes. Note that there are no spherical Coxeter orthoschemes, because the Coxeter diagram of a spherical orthoscheme must be a cycle, and there is no cycle in the list of Coxeter diagrams of spherical Coxeter simplexes. The list can be found for example in [Rat06]. Let us denote by $U_{k}$ the line through $p_{k}$ (so the angle between $U_{k}$ and $U_{k+1}$ is $\varphi_{k}$), and by $\lambda$ the cross ratio $[U_{k-1},U_{k},U_{k+1},U_{k+2}]$, namely if $u_{k-1},u_{k},u_{k+1},u_{k+2}$ are the intersections of the lines $U_{i}$ with any line not passing through zero and endowed with coordinates then (see [Ber94]) $\lambda=[U_{k-1},U_{k},U_{k+1},U_{k+2}]=\frac{u_{k+1}-u_{k-1}}{u_{k+1}-u_{k}}\frac{u_{k+2}-u_{k}}{u_{k+2}-u_{k-1}}.$ We have the formula (see [PY12]) $\frac{\sinh\varphi_{k-1}\sinh\varphi_{k+1}}{\sinh(\varphi_{k-1}+\varphi_{k})\sinh(\varphi_{k}+\varphi_{k+1})}=\frac{\lambda-1}{\lambda}=[U_{k-1},U_{k+2},U_{k},U_{k+1}].$ From a given $n$-dimensional acute spherical orthoscheme $O$ we can find angles (positive real numbers) $(\varphi_{1},\varphi_{2},\ldots,\varphi_{n})$ such that $\mathcal{P}(\varphi_{1},\varphi_{2},\ldots,\varphi_{n})$ is isometric to $O$. Let $0<A<1$ be the square of the cosine of an acute dihedral angle of $O$. We have first to find ordered time-like lines $U_{1},U_{2},U_{3},U_{4}$ such that $[U_{1},U_{2},U_{3},U_{4}]=\frac{1}{1-A}$, i.e. we have to prove that the cross-ratio of the lines can reach any value $>1$. Choose arbitrary distinct ordered time-like $U_{1},U_{2},U_{4}$. If $U_{3}=U_{4}$ then $[U_{1},U_{2},U_{3},U_{4}]=1$, and if $U_{3}=U_{2}$ then $[U_{1},U_{2},U_{3},U_{4}]=+\infty$, so by continuity any given value $>1$ can be reached for a suitable $U_{3}$ between $U_{2}$ and $U_{4}$. $U_{1},U_{2},U_{3},U_{4}$ give angles $\varphi_{1},\varphi_{2},\varphi_{3}$. Now the other $\varphi_{k}$ are easily obtained as follows. Given the next dihedral angle of $O$ (they can be ordered by ordering the unit normals to $O$, see [Deb90]), the square of its cosine should be equal to $\frac{\sinh\varphi_{2}\sinh\varphi_{4}}{\sinh(\varphi_{2}+\varphi_{3})\sinh(\varphi_{3}+\varphi_{4})}$ and $\varphi_{2},\varphi_{3}$ are known, so we get $\varphi_{4}$. And so on. ## 7 Spherical cone-manifolds Let $n>2$ and consider the orthoscheme $\mathcal{P}=\mathcal{P}(\varphi_{1},\ldots,\varphi_{n})$. A facet of $\mathcal{P}$ is isometric to the space of $t$-convex polygons with $\eta_{1},\ldots,\hat{\eta_{i}},\ldots,\eta_{n}$ ($\hat{\eta_{i}}$ means that $\eta_{i}$ is deleted from the list) as normals to the fundamental edges. The angles between the normals are $\varphi_{1},\ldots,\varphi_{i-2},\varphi_{i-1}+\varphi_{i},\varphi_{i+1},\ldots,\varphi_{n}$. This orthoscheme is also isometric to a facet of the orthoscheme $\mathcal{P}^{\prime}$ obtained by permuting $\varphi_{i-1}$ and $\varphi_{i}$ in the list of angles. Hence we can glue $\mathcal{P}$ and $\mathcal{P}^{\prime}$ isometrically along this common facet. We denote by ${\mathcal{C}}(\varphi_{1},\ldots,\varphi_{n})$ the $(n-1)$-dimensional spherical cone-manifold obtained by gluing in this way all the $(n-1)!$ orthoschemes obtained by permutations of the list $\varphi_{1},\ldots,\varphi_{n}$, up to cyclic permutations. When $n=3$, ${\mathcal{C}}(\varphi_{1},\varphi_{2},\varphi_{3})$ is isometric to a spherical cone-metric on the sphere with three conical singularities, with cone-angles $<\pi$, obtained by gluing two isometric spherical triangles along corresponding edges. Let $n\geq 4$. Around the codimension 2 face of ${\mathcal{C}}$ isometric to $N:=\mathcal{C}(\varphi_{1},\ldots,\varphi_{k}+\varphi_{k+1},\ldots,\varphi_{j}+\varphi_{j+1},\ldots,\varphi_{n+3})$ are glued four orthoschemes, corresponding to the four ways of ordering $(\varphi_{k},\varphi_{k+1})$ and $(\varphi_{j},\varphi_{j+1}).$ As the dihedral angle of each orthoscheme at such codimension $2$ face is $\pi/2$, the total angle around $N$ in ${\mathcal{C}}$ is $2\pi$. Hence metrically $N$ is actually not a singular set. Around the codimension 2 face of ${\mathcal{C}}$ isometric to $S:={\mathcal{C}}(\varphi_{1},\ldots,\varphi_{k}+\varphi_{k+1}+\varphi_{k+2},\ldots,\varphi_{n+3})$ are glued six orthoschemes corresponding to the six ways of ordering $(\varphi_{k},\varphi_{k+1},\varphi_{k+2})$. Let $\Theta$ be the cone-angle around $S$. It is the sum of the dihedral angles of the six orthoschemes glued around it. As formula (7) is symmetric for two variables, $\Theta$ is two times the sum of three different dihedral angles. A direct computation gives ($k=1$ in the formula) $\cos(\Theta/2)=\textstyle\frac{\sinh\varphi_{1}\sinh\varphi_{2}\sinh\varphi_{3}-\sinh(\varphi_{1}+\varphi_{2}+\varphi_{3})(\sinh\varphi_{1}\sinh\varphi_{2}+\sinh\varphi_{2}\sinh\varphi_{3}+\sinh\varphi_{3}\sinh\varphi_{1})}{\sinh(\varphi_{1}+\varphi_{2})\sinh(\varphi_{2}+\varphi_{3})\sinh(\varphi_{3}+\varphi_{1})}.$ During the computation we used that $\sinh(a+b)\sinh(b+c)-\sinh a\sinh c=\sinh b\sinh(a+b+c)$ which can be checked with $\frac{1}{2}\left(\cosh(x+y)-\cosh(x-y)\right)=\sinh x\sinh y$. The analogous formula in the Euclidean convex polygons case was obtained in [KNY99]. For example when $\varphi_{i}=\varphi\;\forall i$, we have $\cos(\Theta/2)=-\frac{2\cosh(\varphi)^{2}+\sinh(\varphi)^{2}}{2\cosh(\varphi)^{3}}.$ The function on the right-hand side is a bijection from the positive numbers to $]-1,0[$, hence all the $\Theta\in]2\pi,3\pi[$ (the dihedral angle $\theta\in]\pi/3,\pi/2[$) are uniquely reached. In particular ${\mathcal{C}}$ is not an orbifold. The cone-manifold ${\mathcal{C}}$ comes with an isometric involution which consists of reversing the order of the angles $(\varphi_{1},\ldots,\varphi_{n})$. ## 8 Higher dimensional generalization The generalization of $t$-convex polygons to higher dimensional Minkowski spaces is as follows. Let us consider the $d$-dimensional hyperbolic space ${\mathbb{H}}^{d}$ as a pseudo-sphere in the $d+1$-dimensional Minkowski space $M^{d+1}$, and let $\Gamma$ be a discrete group of linear isometry of $M^{d+1}$ such that ${\mathbb{H}}^{d}/\Gamma$ is a compact hyperbolic manifold. A $\Gamma$-convex polyhedron is, given $\eta_{1},\ldots,\eta_{n}\in{\mathbb{H}}^{d}$ and positive numbers $h_{1},\ldots,h_{n}$, the intersection of the future sides of the space-like hyperplanes $(\gamma(h_{i}\eta_{i}))^{\bot}$ $\forall i,\forall\gamma\in\Gamma$. The mixed-coarea is generalized as a “mixed covolume”. For details and computation of the signature, see [Fil13]. Actually for a given set of $\eta_{i}$, many combinatorial types may appear, and one has to restrict to type cones (cones of polyhedra with parallel facets and same combinatorics). It should be interesting to investigate the kind of spherical polytopes that appear. Another related question is to look at the quadratic form given by the face area of the polyhedra (in a fundamental domain) and its relations with the moduli spaces of flat metric with conical singularities of negative curvature on compact surfaces of genus $>1$ (the quotient of the boundary of a $\Gamma$-convex polyhedron is isometric to such a metric). The analogous questions in the convex polytopes case are the subject of [FI13]. The moduli space of flat metrics on the sphere was studied in [Thu98]. ## Acknowledgement The author thanks anonymous referee and Haruko Nishi who helped to imporve the redaction of the present text. Up to trivial changes, the introduction was written by an anonymous referee. The polygons introduced in the present paper are very particular cases of objects studied in [Fil13] and [FV13]. Work supported by the ANR GR Analysis-Geometry. ## References * [Ber94] M. Berger. Geometry. I. Universitext. Springer-Verlag, Berlin, 1994. Translated from the 1977 French original by M. Cole and S. Levy, Corrected reprint of the 1987 translation. * [BG92] C. Bavard and É. Ghys. Polygones du plan et polyèdres hyperboliques. Geom. Dedicata, 43(2):207–224, 1992. * [Deb90] H. E. Debrunner. Dissecting orthoschemes into orthoschemes. Geom. Dedicata, 33(2):123–152, 1990. * [FI13] F. Fillastre and I. Izmestiev. Shapes of polyhedra, mixed volumes, and hyperbolic geometry. In preparation, 2013. * [Fil11] F. Fillastre. From spaces of polygons to spaces of polyhedra following Bavard, Ghys and Thurston. Enseign. Math. (2), 57(1-2):23–56, 2011. * [Fil13] F. Fillastre. Fuchsian convex bodies: basics of Brunn–Minkowski theory. To appear _Geometric and Functional Analysis_ , 2013. * [FV13] F. Fillastre and G. Veronelli. Lorentzian area measures and the Christoffel problem. 2013\. * [IH85] H.-C. Im Hof. A class of hyperbolic Coxeter groups. Exposition. Math., 3(2):179–186, 1985. * [IH90] H.-C. Im Hof. Napier cycles and hyperbolic Coxeter groups. Bull. Soc. Math. Belg. Sér. A, 42(3):523–545, 1990. Algebra, groups and geometry. * [KNY99] S. Kojima, H. Nishi, and Y. Yamashita. Configuration spaces of points on the circle and hyperbolic Dehn fillings. Topology, 38(3):497–516, 1999. * [PY12] A. Papadopoulos and S. Yamada. A Remark on the Projective Geometry of Constant Curvature Spaces. 2012\. * [Rat06] J. Ratcliffe. Foundations of hyperbolic manifolds, volume 149 of Graduate Texts in Mathematics. Springer, New York, second edition, 2006. * [Thu98] W. P. Thurston. Shapes of polyhedra and triangulations of the sphere. In The Epstein birthday schrift, volume 1 of Geom. Topol. Monogr., pages 511–549 (electronic). Geom. Topol. Publ., Coventry, 1998. Circulated as a preprint since 1987. * [Var00] R. Varga. Matrix iterative analysis, volume 27 of Springer Series in Computational Mathematics. Springer-Verlag, Berlin, expanded edition, 2000.	arxiv-papers	2011-11-15T12:01:06	2024-09-04T02:49:24.346896	{ "license": "Public Domain", "authors": "Fran\\c{c}ois Fillastre", "submitter": "Fran\\c{c}ois Fillastre", "url": "https://arxiv.org/abs/1111.3511" }
1111.3569	# Dynamics of Hot Accretion Flow with Thermal Conduction Kazem Faghei School of Physics, Damghan University, Damghan, Iran E-mail:kfaghei@du.ac.ir ###### Abstract The purpose of this paper is to explore the dynamical behaviour of hot accretion flow with thermal conduction. The importance of thermal conduction on hot accretion flow is confirmed by observations of the hot gas that surrounds Sgr A∗ and a few other nearby galactic nuclei. In this research, the effect of thermal conduction is studied by a saturated form of it, as is appropriate for weakly collisional systems. The angular momentum transport is assumed to be a result of viscous turbulence and the $\alpha$-prescription is used for the kinematic coefficient of viscosity. The equations of accretion flow are solved in a simplified one-dimensional model that neglects the latitudinal dependence of the flow. To solve the integrated equations that govern the dynamical behaviour of the accretion flow, we have used an unsteady self-similar solution. The solution provides some insights into the dynamics of quasi-spherical accretion flow and avoids from limits of the steady self- similar solution. In comparison to accretion flows without thermal conduction, the disc generally becomes cooler and denser. These properties are qualitatively consistent with performed simulations in hot accretion flows. Moreover, the angular velocity increases with the magnitude of conduction, while the radial infall velocity decreases. The mass accretion rate onto the central object is reduced in the presence of thermal conduction. We found that the viscosity and thermal conduction have the opposite effects on the physical variables. Furthermore, the flow represents a transonic point that moves inward with the magnitude of conduction or viscosity. ###### keywords: accretion, accretion discs, conduction – hydrodynamics. ††pubyear: 2011 ## 1 Introduction Accretion is one of the most important physical processes in astrophysics. It is widely accepted that the accreting matter toward the central object is a source of power active galactic nuclei (AGN), and galactic X-ray sources (see for a review e.g. Frank et al. 2002). This idea is also well-applicable to interpret many observations of astrophysical phenomena, such as, prototype stellar objects (Chang & Choi 2002), symbiotic stars (Lee & Park 1999), gamma- ray bursts (Brown et al. 2000). There are some types of accretion flows that a significant fraction of the generated heat by dissipation processes retains in the fluid rather than being radiated away, have been the subject of considerable attention in recent years (Ogilvie 1999; Chang 2005, Akizuki & Fukue 2006, Khesali & Faghei 2009, Faghei 2011). These advection dominated accretion flows (ADAF) place an intermediate position between the spherically symmetric accretion flow of non-rotating fluid (Bondi 1952) and the cool thin disc of classical accretion disc theory (e. g. Pringle 1981). These types of accretion flows have been widely applied to explain observations of the galactic black hole candidate (e. g. Narayan et al. 1996, Hameury et al. 1997), the spectral transition of Cyg x-1 (Esin 1996) and multi-wavelength spectral properties of Sgr A∗ (Narayan & Yi 1995; Manmoto et al. 2000; Narayan et al. 1997). The X-ray observations of black holes imply that they are capable of accreting gas under a variety of flow configurations. In particular, observational evidences confirm existence of hot accretion flow, contrasted with the classical cold and thin accretion disc scenario (Shakura & Sunyaev 1973). Hot accretion flow can be found in the population of supermassive black holes in galactic nuclei and during quiescent of accretion onto stellar-mass black holes in X-ray transients (e.g., Narayan et al. 1998a; Lasota et al. 1996; Di Matteo et al. 2000; Esin et al. 1997, 2001; Menou et al. 1999; see Narayan et al. 1998b; Melia & Falcke 2001; Narayan 2002; Narayan & Quataert 2005 for reviews). Chandra observations provide tight constraints on the density and temperature of gas at or near the Bondi capture radius in Sgr A∗ and several other nearby galactic nuclei. Tanaka & Menou (2006) used these constraints (Loewenstein et al. 2001; Baganoff et al. 2003; Di Matteo et al. 2003; Ho et al. 2003) to calculate the mean free path for the observed gas. They suggested that accretion in these systems will be proceeded under the weakly collisional condition. Furthermore, they suggested that thermal conduction can be as a possible mechanism by which the sufficient extra heating is provided in hot advection dominated accretion flows. Generally, semi-analytical studies of hot accretion flows with thermal conduction have been related to steady state models (e. g. Tanaka & Menou 2006; Johnson & Quataert 2007; Shadmehri 2008; Abbassi et al. 2008, 2010; Ghanbari et al. 2009), and dynamics of such systems have been studied in simulation models (e. g. Sharma et al. 2008; Wu et al. 2010). For example, Tanaka & Menou (2006) have carried out a related analysis and found the accretion flow can spontaneously produce thermal outflows driven in part by conduction. Their analysis is two-dimensional but self-similar in radius. Their assumption of self-similarity enforces a density profile that varies as $r^{-3/2}$, whereas simulations of ADAFs consistently find density profiles shallower than this (e.g., Stone et al. 1999; Igumenshchev & Abramowicz 1999; Stone & Pringle 2001; Hawley & Balbus 2002; Igumenshchev et al. 2003). Johnson & Quataert (2007) studied the effects of electron thermal conduction on the properties of hot accretion flows, under the assumption of spherical symmetry. Since, electron heat conduction is important for low accretion rate systems, thus their model is applicable for Sgr A∗ in the Galactic centre. They show that heat conduction leads to supervirial temperatures, implying that conduction significantly modifies the structure of the accretion flow. Their model similar to Tanaka & Menou (2006) was the steady state, but they solved their equations numerically. As mentioned, semi-analytical studies of hot accretion flows with thermal conduction have been in a steady state. Thus, it will be interesting to study dynamics of such systems. Ogilvie (1999) by the unsteady self-similar method studied time-dependence of quasi-spherical accretion flow without thermal conduction. The solutions of Ogilvie (1999) provided some insight into the dynamics of quasi-spherical accretion and avoided many of the limits of the steady self-similar solution. In this research, we want to explore how thermal conduction can affect the dynamics of a rotating and accreting viscous gas. We answer this question by solving Ogilvie (1999) model that is affected by thermal conduction. This paper is organized as follows. In Section 2, we define the general problem of constructing a model for hot quasi-spherical accretion flow. In Section 3, we use the unsteady self-similar method to solve the integrated equations that govern the dynamical behaviour of the accreting gas, and numerical study of the model is brought in this section, too. We will present a summary of the model in Section 4. ## 2 Basic Equations We start with the approach adopted by Ogilvie (1999), who studied quasi- spherical accretion flows without thermal conduction. Thus, we derive the basic equations that describe the physics of accretion flow with thermal conduction. We use the spherical coordinates $(r,\theta,\phi)$ centred on the accreting object and make the following standard assumptions: 1. 1. The gravitational force on a fluid element is characterized by the Newtonian potential of a point mass, $\Psi=-GM_{}/r$, with $G$ representing the gravitational constant and $M_{}$ standing for the mass of the central star. 2. 2. The written equations in spherical coordinates are considered in the equatorial plane $\theta=\pi/2$ and terms with any $\theta$ and $\phi$ dependence are neglected, hence all quantities will be expressed in terms of spherical radius $r$ and time $t$. 3. 3. For simplicity, self-gravity and general relativistic effects have been neglected. Under these assumptions, the dynamics of accretion flow describes by the following equations: the continuity equation $\frac{\partial\rho}{\partial t}+\frac{1}{r^{2}}\frac{\partial}{\partial r}(r^{2}\rho v_{r})=0,$ (1) the radial force equation $\frac{\partial v_{r}}{\partial t}+v_{r}\frac{\partial v_{r}}{\partial r}=r(\Omega^{2}-\Omega_{K}^{2})-\frac{1}{\rho}\frac{\partial p}{\partial r},$ (2) the azimuthal force equation $\rho\left[\frac{\partial}{\partial t}(r^{2}\Omega)+v_{r}\frac{\partial}{\partial r}(r^{2}\Omega)\right]=\frac{1}{r^{2}}\frac{\partial}{\partial r}\left[\nu\rho r^{4}\frac{\partial\Omega}{\partial r}\right],$ (3) the energy equation $\displaystyle\frac{1}{\gamma-1}\left[\frac{\partial p}{\partial t}+v_{r}\frac{\partial p}{\partial r}\right]+\frac{\gamma}{\gamma-1}\frac{p}{r^{2}}\frac{\partial}{\partial r}\left(r^{2}v_{r}\right)=$ $\displaystyle Q_{vis}-Q_{rad}+Q_{cond}.$ (4) Here $\rho$ the density, $v_{r}$ the radial velocity, $\Omega$ the angular velocity, $\Omega_{K}[=(GM_{}/r^{3})^{1/2}]$ Keplerian angular velocity, $p$ the gas pressure, $\gamma$ is the adiabatic index, $\nu$ the kinematic viscosity coefficient and it is given as in Narayan & Yi (1995a) by an $\alpha$-model $\nu=\alpha\frac{p_{gas}}{\rho\Omega_{K}}.$ (5) The parameter of $\alpha$ is assumed to be a constant less than unity. The terms on the right-hand side of the energy equation, $Q_{vis}$ is the heating rate of the gas by the viscous dissipation, $Q_{rad}$ represents the energy loss through radiative cooling, and $Q_{cond}$ is the transported energy by thermal conduction. For the right-hand side of the energy equation, we can write $Q_{adv}=Q_{vis}-Q_{rad}+Q_{cond}$ (6) where $Q_{adv}$ is the advective transport of energy. We employ the advection factor, $f=1-Q_{rad}/Q_{vis}$, that describes the fraction of the dissipation energy which is stored in the accretion flow and advected into the central object rather than being radiated away. The advection factor of $f$ in general depends on the details of the heating and radiative cooling mechanism and will vary with position (e.g. Watari 2006, 2007; Sinha et al. 2009). However, we assume a constant $f$ for simplicity. Clearly, the case $f=1$ corresponds to the extreme limit of no radiative cooling and in the limit of efficient radiative cooling, we have $f=0$. In a collisional plasma, mean free path for electron energy exchange, $\lambda$, is shorter than temperature scale height, $L_{T}=T/\|\nabla T\|$, and thus the heat flux due to thermal conduction can be written as $F_{cond}=-\kappa\nabla T,$ (7) where $\kappa$ is the thermal conductivity coefficient. Thermal conductivity in a dense, fully ionized gas is given by the Spitzer (1962) formula, $\kappa=\frac{1.84\times 10^{-5}T_{e}^{5/2}}{\ln\Lambda},$ (8) where $T_{e}$ is the electron temperature ($T_{e}=T$ for a one-temperature plasma) and $\ln\Lambda$ is Coulomb logarithm that for $T>4.2\times 10^{5}K$ is $\ln\Lambda=29.7+\ln n^{-1/2}(T_{e}/10^{6}K).$ (9) The heat is conducted by the electron, and equation (8) includes the effect of the self-consistent electric required to maintain the electric current at zero; this reduces $\kappa$ by a factor of about $0.4$ from the value it would otherwise have (Cowie & McKee 1977, hereafter CM77). As noted in the introduction, the inner regions of hot accretion flows are, in many cases, collisionless with electron mean free path due to Coulomb collision larger than the radius (e. g. Tanaka & Menou 2006). When the mean free path of an electron becomes comparable to or larger than the temperature gradient scale $\lambda\gtrsim T/\|\nabla T\|$, equation (7) for the heat flux is no longer valid; CM77 described this effect as saturation. The maximum heat flux in a plasma can be expressed as $(3/2)n_{e}kT_{e}v_{char}$, where $v_{char}$ is a characteristic velocity which one might expect to be the order of the electron thermal velocity (Parker 1963). Assuming Maxwellian distribution for heat source, the characteristic velocity can be written as (Williams 1971; CM77) $V_{char}=(\frac{8}{9\,\pi})^{1/2}\,(\frac{k\,T}{m_{e}})^{1/2}.$ (10) Similar to CM77, we assume that the heat flux is reduced by the same factor of $0.4$ in the saturated case as in the classical (collisional) case so that the saturated heat flux is $F_{sat}=0.4\,n_{e}kT_{e}\,\sqrt{\frac{2kT_{e}}{\pi m_{e}}}.$ (11) CM77 showed that the saturated heat flux is significantly less than conjectured by Parker (1963). Thus, in order to explicitly allow for uncertainty in the estimate of $F_{sat}$, they introduced a factor of $\phi_{s}$, which was less than unity and rewrote equation (11) as $F_{sat}=5\phi_{s}\rho c_{s}^{3}=5\phi_{s}p\sqrt{\frac{p}{\rho}},$ (12) where $c_{s}$ is sound speed, which is defined as $c_{s}^{2}=p/\rho$. The factor of $\phi_{s}$ is called as saturation constant (CM77). Now, the viscous heating rate and the energy transport by thermal conduction are expressed as $Q_{vis}=\nu\rho r^{2}\left(\frac{\partial\Omega}{\partial r}\right)^{2}$ (13) $Q_{cond}=-\frac{1}{r^{2}}\frac{\partial}{\partial r}\left(r^{2}F_{sat}\right)$ (14) By using equations (13) and (14) for the advective transport of energy, we can write $Q_{adv}=f\nu\rho r^{2}\left(\frac{\partial\Omega}{\partial r}\right)^{2}-\frac{1}{r^{2}}\frac{\partial}{\partial r}\left(r^{2}F_{sat}\right)$ (15) The mass accretion rate in a qausi-spherical accretion flow can be written as $\dot{M}(r,t)=-4\pi r^{2}\rho v_{r}.$ (16) We will use this quantity in the next section, and will investigate effects of saturation constant and viscous parameter on it. ## 3 Self-Similar Solutions ### 3.1 analysis Tanaka & Menou (2006) solved essentially equations (1)-(4) for the case of a steady, radially self-similar flow. Here, we will try to find unsteady self- similar solutions for these equations. Thus, we introduce a similarity variable $\xi$ and assume that each physical quantity is given by the following form: $\xi=r(GM_{}t^{2})^{-1/3},$ (17) $\rho(r,t)=R(\xi)(\dot{M}_{0}/GM_{})t^{-1},$ (18) $p(r,t)=\Pi(\xi)(\dot{M}_{0}/(GM_{})^{1/3})t^{-5/3},$ (19) $v_{r}(r,t)=V(\xi)(GM_{})^{1/3}t^{-1/3},$ (20) $\Omega(r,t)=\omega(\xi)t^{-1},$ (21) $\dot{M}(r,t)=\dot{M}_{0}\dot{m}(\xi),$ (22) where $\dot{M}_{0}$ is a constant and its value can be obtained by typical values of the system. In addition, we assumed that $\dot{M}(r,t)$ under similarity transformations is a function of $\xi$ only (Khesali & Faghei 2008, 2009). Substitution of above transformations into the basic equations (1)-(4), yields dimensionless equations below, $\left(V-\frac{2\xi}{3}\right)\frac{dR}{d\xi}-R=-\frac{R}{\xi^{2}}\frac{d}{d\xi}\left(\xi^{2}V\right),$ (23) $\displaystyle\left(V-\frac{2\xi}{3}\right)\frac{dV}{d\xi}-\frac{V}{3}=\xi(\omega^{2}-\xi^{-3})-\frac{1}{R}\frac{d\Pi}{d\xi},$ (24) $\displaystyle R\left[\left(V-\frac{2\xi}{3}\right)\frac{d}{d\xi}\left(\xi^{2}\omega\right)+\frac{1}{3}\left(\xi^{2}\omega\right)\right]~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$ $\displaystyle=\frac{\alpha}{\xi^{2}}\frac{d}{d\xi}\left[\Pi\xi^{11/2}\frac{d\omega}{d\xi}\right],$ (25) $\displaystyle\frac{1}{\gamma-1}\left[\left(V-\frac{2\xi}{3}\right)\frac{d\Pi}{d\xi}-\frac{5}{3}\Pi\right]+\frac{\gamma}{\gamma-1}\frac{\Pi}{\xi^{2}}\frac{d}{d\xi}\left(\xi^{2}V\right)$ $\displaystyle=\alpha f\Pi\xi^{7/2}\left(\frac{d\omega}{d\xi}\right)^{2}-\frac{5\phi_{s}}{\xi^{2}}\frac{d}{d\xi}\left(\xi^{2}\Pi\sqrt{\frac{\Pi}{R}}\right).$ (26) These equations provide a fourth-order system of non-linear ordinary differential equations that must be solved numerically. ### 3.2 Inner limit An appropriate asymptotic solution as $\xi\rightarrow 0$ is the form as $R(\xi)\sim\xi^{-3/2}(R_{0}+R_{1}\xi+\cdot\cdot\cdot),$ (27) $\Pi(\xi)\sim\xi^{-5/2}(\Pi_{0}+\Pi_{1}\xi+\cdot\cdot\cdot),$ (28) $V(\xi)\sim\xi^{-1/2}(V_{0}+V_{1}\xi+\cdot\cdot\cdot),$ (29) $\omega(\xi)\sim\xi^{-3/2}(\omega_{0}+\omega_{1}\xi+\cdot\cdot\cdot),$ (30) in which undetermined coefficients of $R_{0}$, $R_{1}$, $\Pi_{0}$, and etc must be specified. By substituting above relations in equations (23)-(26) and choosing the significant sentences, we can write $V_{0}^{2}+\frac{5\Pi_{0}}{R_{0}}-2+2\,\omega_{0}^{2}\approx 0,$ (31) $2\,R_{0}\,V_{0}+3\,\alpha\,\Pi_{0}\approx 0,$ (32) $6\,V_{0}\,(\gamma-\frac{5}{3})+(\gamma-1)\left[20\phi_{s}\sqrt{\frac{\Pi_{0}}{R_{0}}}-9\alpha f\omega_{0}^{2}\right]\approx 0.$ (33) Also, the dimensionless mass accretion rate, $\dot{m}(\xi)$, under the above asymptotic solution becomes $\dot{m}_{in}\approx-4\pi R_{0}V_{0},$ (34) where $\dot{m}_{in}$ is the value of $\dot{m}$ at $\xi_{in}$, where $\xi_{in}$ is a point near to the centre. After algebraic manipulations for equations (31)-(34), we obtain an algebraic equation for $R_{0}$: $\displaystyle R_{0}^{2}-\frac{10}{27}\frac{\phi_{s}}{\alpha f}\sqrt{\frac{6\dot{m}_{in}}{\alpha\pi}}\,R_{0}^{3/2}-\frac{5}{12}\frac{\dot{m}_{in}}{\alpha\pi}\times$ $\displaystyle\left(1-\frac{2}{5f}\frac{\gamma-5/3}{\gamma-1}\right)R_{0}-\frac{1}{32}\left(\frac{\dot{m}_{in}}{\pi}\right)^{2}\approx 0,$ (35) and the rest of the physical variables are $\Pi_{0}\approx\frac{\dot{m}_{in}}{6\pi\alpha},$ (36) $V_{{0}}\approx-\frac{\dot{m}_{in}}{4\pi R_{0}},$ (37) $\omega_{0}^{2}\approx 1-\frac{5}{12}\frac{\dot{m}_{in}}{\pi\alpha R_{0}}\left(1+\frac{3}{40}\frac{\alpha\dot{m}_{in}}{\pi R_{0}}\right).$ (38) Without thermal conduction, $\phi_{s}=0$, equation (35) can be solved analytically. Since, we want to consider systems with non-zero saturation constant, $\phi_{s}\neq 0$, we will solve this equation numerically. Figure 1: Time-dependent self-similar solution for $\gamma=1.3$, $\alpha=0.1$, $f=1.0$, and $\dot{m}_{in}=0.001$. The solid, the dashed, and the short-dashed lines represent $\phi_{s}=0$, $0.05$, and $0.1$, respectively. Figure 2: Same as Figure 1, but $\phi_{s}=0.03$. The solid, dahsed, and short- dashed lines represent $\alpha=0.05$, $0.1$, and $0.2$, respectively. ### 3.3 Numerical solution If the value of $\xi_{in}$ is guessed, i.e. we take a point very near to the centre, the equations (23)-(26) by Runge-Kutta-Fehlberg fourth-fifth order method can be integrated from this point outward by the above expansion [(27)-(30)]. Examples of such solutions are presented in Figs 1-5. #### 3.3.1 The influences of saturation constant and viscous parameter on physical quantities The delineated quantity of $\Pi/R$ in Figs 1 and 2 is the sound speed square in self-similar flow, which is rescaled in the course of time and represents the flow temperature. The profiles of $\Pi/R$ in Fig. 1 show the flow temperature decreased by adding saturation constant, $\phi_{s}$. Because, the generated heat by viscous dissipation can be transfered by thermal conduction. Furthermore, this temperature decrease is qualitatively consistent with simulation results of Sharma et al. (2008) and Wu et al. (2010). We know the viscous dissipation of the flow increases by adding $\alpha$ parameter. Thus, the temperature increased by viscous parameter confirmed by the temperature profiles in Fig 2. The density profiles show the gas density increased by adding the $\phi_{s}$ parameter. It can be due to temperature fall of fluid. Increasing density by adding saturation constant is another consistency of our results with simulations of Wu et al. (2010). Also, the density profiles show that it decreases by adding the viscous parameter. This also could be result of the temperature rising. The viscous turbulence in this paper is proportional to the gas temperature ($\nu\propto c_{s}^{2}\propto T$). Thus, increase or decrease of temperature will affect dynamics of the accreting gas. As we said the temperature decreased by saturation constant implies that the viscous turbulence decreases, too. The decreasing of viscous turbulence reduces the effect of negative viscous torque in angular momentum equation. Thus, we expect the flow rotates faster by adding saturation constant that confirmed by the angular velocity profiles in Fig. 1. Since, the efficiency of the angular momentum transport decreases by adding the saturation constant, we expect the decrease of radial infall velocity that the radial velocity profiles in Fig. 1 confirm it. The efficiency of angular momentum transport increases by adding the viscous parameter of $\alpha$. Thus, we expect the flow rotates slower and accretes faster by adding the viscous parameter. The profiles of radial and angular velocities in Fig. 2 confirm them. Figure 3: Time-dependent self-similar solution of mass accretion rate. The input parameters in left panel are same as Figure 1, but the solid, the dashed, and the short-dashed lines represent $\phi_{s}=0.01$, $0.02$, and $0.03$, respectively. The input parameters in right panel are same as Figure 2. but $\phi_{s}=0.01$. #### 3.3.2 Mass accretion rate The behaviour of mass accretion rate as a function of similarity variable $\xi$ for several values of the viscous parameter and saturation constant are plotted in Fig. 3. In the present model, the mass accretion rate is reduced by radius. While, the mass accretion rate in steady hot accretion flows is a constant (Tanaka & Menou 2006). There are some researches in steady hot accretion flows that have studied power-law function of mass accretion rate (Shadmehri 2008; Abbassi et al. 2008). However, the mass accretion rate in their models is not dependent on important parameters such as saturation constant and viscous parameter. The profiles of mass accretion rate in Fig. 3 show that it is reduced by adding the saturation constant. This property is qualitatively consistent with numerical results of Johnson & Quataert (2007). Also, the profiles of mass accretion rate imply that it increases by adding the viscous parameter of $\alpha$. This property is qualitatively consistent with previous works in accretion flows (e. g. Park 2009). Figure 4: Time-dependent self-similar solution of Mach number. The input parameters in left panel are same as Figure 1, but $\alpha=0.5$ and the solid, the dashed, and the short-dashed lines represent $\phi_{s}=0.0$, $0.2$, and $0.3$, respectively. The input parameters in right panel are same as Figure 1, but $\phi_{s}=0.05$ and the solid, the dashed, and the short-dashed lines represent $\alpha=0.2$, $0.3$, $0.4$, respectively. #### 3.3.3 Mach number Here, it will be interesting to investigate the existence of the transonic point in hot accretion flow. The transonic point occurs in place that the amount of Mach number becomes equal to unity. The Mach number referring to the reference frame is defined as (Gaffet & Fukue 1983; Fukue 1984) $\mu\equiv\frac{v_{r}-v_{F}}{c_{s}}=\frac{V-n\xi}{S}$ (39) where $v_{F}=\frac{dr}{dt}=n\frac{r}{t}$ (40) is the velocity of the reference frame which is moving outward as time goes by, which the sound speed can be subsequently expressed as $c_{s}^{2}\equiv\frac{p}{\rho}=S^{2}(GM_{}/t)^{2/3}$ (41) and, $S=\left(\Pi/R\right)^{1/2}$ the sound speed in self-similar flow is rescaled in the course of time. The Mach number introduced so far, represents the _instantaneous_ and _local_ Mach number of the unsteady self-similar flow. In steady self-similar solution (e. g. Tanaka & Menou 2006), the Mach number does not vary by radii and is a constant. While, the Mach number in unsteady self-similar varies by radii (see Fig. 4). As seen in Fig. 4, there is a transonic point ($\|\mu\|=1$). The dependency of transonic point to saturation constant shows that this point moves inward by adding the parameter of $\phi_{s}$. Because, thermal conduction transfers the heat to larger radii, so the sound speed/temperature decreases by adding the saturation constant. In other words, whatever the radii smaller, the radial velocity relative to the sound speed larger. Also, the Mach number profiles show the transonic point decreasing by adding the viscous parameter which can be due to increase of radial velocity along with adding the viscous parameter. Figure 5: Same as Figure 1, but $\phi_{s}=0.1$. The solid, the dashed, and the short-dashed lines represent $\gamma=1.3$, $1.4$, and $1.5$, respectively. #### 3.3.4 Comparison of steady and unsteady self-similar solutions As a comparison between steady and unsteady self-similar solution, the physical quantities in unsteady self-similar solutions are divided into their radial dependence in steady self-similar solution then plotted in terms of $\xi$ in Fig. 5. Also, the effect of adiabatic index on hot accretion flow is investigated in Fig. 5. The delineated quantities ($R\,\xi^{3/2}$, $V\,\xi^{1/2}$, $\cdot$ $\cdot$ $\cdot$) in Fig 5 are constant in steady self- similar solutions of hot accretion flows (Tanaka & Menou 2006; Shadmehri 2008; Abbassi et al. 2008; Ghanbari et al. 2009). While they vary with position in this research. Fig. 5 represents the density profile varies shallower than $\xi^{-3/2}$, that this property is qualitatively consistent with simulation results (e.g., Stone et al. 1999; Igumenshchev & Abramowicz 1999; Stone & Pringle 2001; Hawley & Balbus 2002; Igumenshchev et al. 2003). The radial dependency of the temperature and radial velocity in unsteady self-similar solution show that they vary deeper than the steady self-similar solution. Also, the study of the angular velocity show that it varies shallower than $\xi^{-3/2}$. Thus, the physical quantities in the present model avoid the limits of the steady self-similar solution. The physical quantities profiles in Fig. 5 show that their radial dependency in the unsteady self-similar limited to the steady self-similar solution by adding adiabatic index $\gamma$. ## 4 Summary and Discussion In hot accretion flows, the collision timescale between ions and electrons is longer than the inflow timescale. Thus, the inflow plasma is collisionless, and the transfer of energy by thermal conduction can be dynamically important. The low collisional rate of the gas is confirmed by direct observation, particularly in the case of the Galactic centre (Quataert 2004; Tanaka & Menou 2006) and in the intracluster medium of galaxy clusters (Sarazin 1986). Here, we have investigated how thermal conduction affects dynamics of hot quasi-spherical accretion flows. We adopted the presented solutions by Ogilvie (1999) and Tanaka & Menou (2006). Thus, we assumed that angular momentum transport is due to viscous turbulence and the $\alpha$-prescription is used for the kinematic coefficient of viscosity. We also assumed the flow does not have a good cooling efficiency and so a fraction of energy accretes along with matter on to the central object. The effect of thermal conduction is studied by a saturation form of it introduced by Cowie & McKee (1977). To solve the equations that govern the dynamical behaviour of hot accretion flow, we have used unsteady self-similar solution. The effect of saturation constant and the viscous parameter on the present model is investigated. The solutions show that with the increase of conductivity, the equatorial density becomes denser and the temperature becomes lower. These results are qualitatively consistent with simulation results of Wu et al. (2010). Furthermore, the solutions show that by adding the saturation constant, the angular velocity becomes larger and the radial velocity decreases. The mass accretion rate is reduced by adding the saturation constant that is qualitatively consistent with the result of Johnson & Quataert (2007). The solutions imply that the viscous parameter has opposite effects in comparison to saturation constant on physical quantities of the system. Also, the study of physical quantities of the present model in comparison to steady self-similar solution show that our results deviate from steady self-similar solution and do not have its limits. Here, we studied dynamical behaviour of hot accretion flow in one-dimensional approach ignored by latitudinal dependence of physical quantities. Although, some authors have shown that latitudinal dependence of physical quantities is important for structure and dynamics of hot accretion flow (Tanaka & Menou 2006; Ghanbari et al. 2009; Wu et al. 2010). Thus, latitudinal behaviour of the present model can be investigated in other studies. ## Acknowledgments I would like to thank the anonymous referee for very useful comments that helped me to improve the initial version of the paper. ## References * (1) Abbassi S., Ghanbari G., Najjar S., 2008, MNRAS, 388, 663 * (2) * (3) Abbassi S., Ghanbari G., Ghasemnezhad M., 2010, MNRAS, 409, 1113 * (4) * (5) Akizuki C., Fukue J., 2006, PASJ, 58, 469 * (6) * (7) Baganoff, F. K., et al. 2003, ApJ, 591, 891 * (8) * (9) Bondi, H. 1952, MNRAS, 112, 195 * (10) * (11) Brown G. E., Lee C.-H., Wijers R. A. M. J., Lee H. K., Israelian G., Bethe H. A., 2000, New Astron., 5, 191 * (12) * (13) Chang H.-Y, 2005, JA&SS, 22, 139 * (14) * (15) Chang H.-Y, Choi C.-S, 2002, JA&SS, 19, 181 * (16) * (17) Cowie L. L., McKee C. F., 1977, ApJ, 211, 135 (CM77) * (18) * (19) Di Matteo T., Allen S. W., Fabian A. C., Wilson A. S., Young, A. J. 2003, ApJ, 582, 133 * (20) * (21) Di Matteo T., Quataert, E., Allen S. W., Narayan R., Fabian A. C., 2000, MNRAS, 311, 507 * (22) * (23) Esin A. A., Narayan R., Ostriker E., Yi H., 1996, ApJ, 465, 312 * (24) * (25) Esin A. A., McClintock J.E., Narayan R. 1997, ApJ, 489, 865 * (26) * (27) Esin A. A., McClintock J. E., Drake J. J., Garcia M. R., Haswell C. A., Hynes R. I., Muno M. P., 2001, ApJ, 555, 483 * (28) * (29) Faghei, K., 2011, J. Astrophys. Astr., accepted * (30) * (31) Frank J., King A. R., Raine D., 2002, Accretion Power in Astrophysics (Cambridge: Cambridge University Press) * (32) * (33) Fukue J., 1984, PASJ, 36, 87 * (34) * (35) Gaffet B., Fukue J., 1983, PASJ, 35, 365 * (36) * (37) Ghanbari J., Abbassi S., Ghasemnezhad M., 2009, MNRAS, 400, 422 * (38) * (39) Hameury J. M., Lasota J. P., MaClintock J. E., Narayan R., 1997, ApJ, 489, 234 * (40) * (41) Hawley J. F., Balbus S. A., 2002, ApJ, 573, 738 * (42) * (43) Ho L. C., Terashima Y., Ulvestad J. S., 2003, ApJ, 589, 783 * (44) * (45) Igumenshchev I. V., Abramowicz M. A., 1999, MNRAS, 303, 309 * (46) * (47) Igumenshchev I. V., Narayan R., Abramowicz M. A., 2003, ApJ, 592, 1042 * (48) * (49) Johnson B. M., Quataert E., 2007, ApJ, 660, 1273 * (50) * (51) Khesali A., Faghei K., 2008, MNRAS, 389, 1218 * (52) * (53) Khesali A., Faghei K., 2009, MNRAS, 398, 1361 * (54) * (55) Lasota J.-P., Abramowicz M. A., Chen X., Krolik J., Narayan R., Yi I., 1996, ApJ, 462, 142L * (56) * (57) Lee H.-W., Park M.-G, 1999, ApJ, 515, L89 * (58) * (59) Loewenstein M., Mushotzky R. F., Angelini L., Arnaud K. A., Quataert, E., 2001, ApJ, 555, L21 * (60) * (61) Manmoto T., Kato S., Nakamura K., Narayan R., 2000, ApJ, 529, 127 * (62) * (63) Melia F., Falcke H. 2001, ARA&A, 39, 309 * (64) * (65) Menou K., Esin, A. A., Narayan R., Garcia M. R., Lasota J.-P., McClintock J. E., 1999, ApJ, 520, 276 * (66) * (67) Narayan R. 2002, in Lighthouses of the Universe, ed. Gilfanov M., Sunyaev R. A., Churazov E. ( Berlin: Springer), 405 * (68) * (69) Narayan R., Igumenshchev I. V., Abramowicz M. A., 2000, ApJ, 539, 798 * (70) * (71) Narayan R., Kato S., Honma F., 1997, ApJ, 476, 49 * (72) * (73) Narayan R., Mahadevan R., Grindlay J. E., Popham R. G., Gammie C., 1998a, ApJ, 492, 554 * (74) * (75) Narayan, R., Mahadevan, R., Quataert, E. 1998b, in Theory of Black Hole Accretion Disks, ed. Abramowicz M., Bjornsson G., Pringle J. (Cambridge: Cambridge Univ. Press), 148 * (76) * (77) Narayan R., Quataert E., 2005, Science, 307, 77 * (78) * (79) Narayan R., Yi I., 1995a, ApJ, 444, 231 * (80) * (81) Narayan R., Yi I., 1995b, ApJ, 452, 710 * (82) * (83) Ogilvie G. I., 1999, MNRAS, 306, L90 * (84) * (85) Park M.-G., 2009, ApJ, 706, 637 * (86) * (87) Parker E. N., 1963, Interplanetary Dynamical Processes (New York: Interscience) * (88) * (89) Pringle J. E., 1981, ARA&A, 19, 137 * (90) * (91) Quataert E., 2004, ApJ, 613, 322 * (92) * (93) Sarazin C. L., 1986, Rev. Mod. Phys., 58, 1 * (94) * (95) Shadmehri M., 2008, Ap&SS, 317, 201 * (96) * (97) Sharma P., Quataert E., Stone J. M., 2008, MNRAS, 389, 1815 * (98) * (99) Shakura N. I., Sunyaev R. A., 1973, A&A, 24, 337 * (100) * (101) Sinha M., Rajesh S. R., Mukhopadhyay B., 2009, RAA, 9, 1331 * (102) * (103) Spitzer L., 1962, Physics of Fully Ionized Gases (New York: Interscience) * (104) * (105) Stone J. M., Pringle J. E. 2001, MNRAS, 322, 461 * (106) * (107) Stone J. M., Pringle J. E., Begelman M. C. 1999, MNRAS, 310, 1002 * (108) * (109) Tanaka T., Menou K., 2006, ApJ, 649, 345 * (110) * (111) Watarai, K. 2006, ApJ, 648, 523 * (112) * (113) Watarai, K. Y. 2007, PASJ, 59, 443 * (114) * (115) Williams M. M. R., 1971, Mathematical Methods in Particle Transport Theory (New York: Wiley-Interscience) * (116) * (117) Wu M., Yuan F., Bu D., 2010, ScChG, 53S, 168 * (118)	arxiv-papers	2011-11-15T16:27:39	2024-09-04T02:49:24.355053	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Kazem Faghei", "submitter": "Kazem Faghei", "url": "https://arxiv.org/abs/1111.3569" }
1111.3618	# Linking to Data - Effect on Citation Rates in Astronomy Edwin A. Henneken1, Alberto Accomazzi1 ###### Abstract Is there a difference in citation rates between articles that were published with links to data and articles that were not? Besides being interesting from a purely academic point of view, this question is also highly relevant for the process of furthering science. Data sharing not only helps the process of verification of claims, but also the discovery of new findings in archival data. However, linking to data still is a far cry away from being a “practice”, especially where it comes to authors providing these links during the writing and submission process. You need to have both a willingness and a publication mechanism in order to create such a practice. Showing that articles with links to data get higher citation rates might increase the willingness of scientists to take the extra steps of linking data sources to their publications. In this presentation we will show this is indeed the case: articles with links to data result in higher citation rates than articles without such links. The ADS is funded by NASA Grant NNX09AB39G. 1Smithsonian Astrophysical Observatory, 60 Garden Street, Cambridge, MA 02138 ## 1 Introduction Furthering science depends to a large degree on knowledge and information transfer. Therefore it critically relies on discoverability. This applies to findings in publications and to the underlying data that led to these findings. Therefore, significant amounts of energy (and funds) should be invested in improving discoverability, of both publications and data. Major progress has been made on the level of publications by improved visibility and more sophisticated techniques for information discovery. The adoption of faceted filtering, recommender systems and semantic interlinking of resources are good examples of this (Accomazzi & Dave (2011), Henneken et al. (2011)). It is time that exposure of data becomes common practice. A publication based on a data set is just one expression of the potential of that data set. It totally depends on the background and the interests of the researchers which representation of that potential will be selected. However, there are many other representations. The scientific community would also benefit greatly from the ability to combine a data set with other available data sets. Also, having data available publicly would greatly facilitate the verification of claims (Fischer & Zigmond (2010)). The special session “The Literature-Data Connection: Meaning, Infrastructure and Impact” at the 218th Meeting of the American Astronomical Society (Boston, May 2011) was dedicated to this discussion. As part of the discussion of how to create a practice of linking data to publications, the question was raised whether such publications would see a citation advantage. That would be like getting a tax benefit for “being green”. Everybody agrees that “being green” is a sensible thing to do, but having some kind of incentive definitely helps as additional motivation. Motivation is an essential ingredient for creating a practice. Since citations are a measure used for scientific impact, it is logical to ask whether investing energy into making data available publicly results in a citation advantage. In this presentation we address the question whether there is a citation advantage. We explore the question using the holdings and citation data of the SAO/NASA Astrophysics Data System (ADS). ## 2 Results With every record in the ADS holdings a number of possible attributes (“links”) can be associated, giving access to information related to that record. The attribute used for this analysis is the “D” link, associated with access to on-line data. Currently these links point to data hosted at data centers (like CDS, HEASARC and MAST). The following set of records was chosen for this study: articles published in _The Astrophysical Journal_ (including _Letters_ and _Supplement_), _The Astronomical Journal_ , _The Monthly Notices of the R.A.S._ and _Astronomy & Astrophysics_ including _Supplement_), during the period 1995 through 2000. Comparing publications with a “D” link to those without such a link would, to a large degree, be comparing apples with oranges, because of the range in subject matter. In order for the comparison to make sense, the subject matter of the publications needs to be restricted. We decided to use keywords as filter. We determined the set of 50 most frequently used keywords in articles with data links. The articles to be used for the analysis were obtained by requiring that they have at least 3 keywords in common with that set of 50 keywords. This resulted in a set of 3814 articles with data links and 7218 articles without data links. The box diagram in figure 1 characterizes the distribution of citations in the sets with and without data links, for respectively 2 and 4 years after publication. Figure 1.: Distribution of citations of articles published in _The Astrophysical Journal_ (including _Letters_ and _Supplement_), _The Astronomical Journal_ , _The Monthly Notices of the R.A.S._ and _Astronomy & Astrophysics_ including _Supplement_), during the period 1995 through 2000. The extent of the box corresponds with the interquartile range of the citations and whiskers extend to 1.5 times the interquartile range. The horizontal lines within the boxes correspond with the medians. From left to right, the boxes correspond respectively with the citation distributions for the article set with and without data links 2 years after publication, and 4 years after publication. The medians are respectively at 10, 8, 17 and 13 citations. For this analysis, a random selection of 3814 articles was extracted from the set of 7218 articles (without links to data). For both sets the citation accumulation was determined for each article. From now on, we will refer to the set with data links as $\emph{D}_{d}$ and the one without data links as $\emph{D}_{n}$. These citation distributions were used to calculate the mean citation accumulation for each set, normalized by the total number of citations in the entire set of publications. The results are shown in figure 2. Figure 2.: The normalized number of citations for data sets $\emph{D}_{d}$ and $\emph{D}_{n}$. The citations have been normalized by the total number of citations. Figure 2 indicates that publications with a data link have a larger citation rate than publications that do not. To get get an indication of how much more citations a publication with a data link accumulates, on average, figure 3 shows the cumulative citation distribution, normalized by the total number of citations for articles without data links, 10 years after publication. Figure 3.: The cumulative citation distributions for data sets $\emph{D}_{d}$ and $\emph{D}_{n}$. The citation counts have been normalized by the total number of citations for articles without data links, 10 years after publication. Figure 3 indicates that for this data set, articles with data links on average acquired 20% more citations (compared to articles without these links) over a period of 10 years. The fact that this increase is statistically significant follows from a regression analysis performed on the entire data set. This confirmed the increase of 20% in citation count (at a 95% confidence level). ## 3 Discussion Our study seems to indicate that publications with links to on-line data seem to have a higher citation rate than publications that do not. Could this effect be attributed to another systematic effect? For example, studies have shown that e-printing results in higher citation rates (see for example Henneken et al. (2006)). However, both sets used to construct figures 2 and 3 turn out to be homogeneous in other publication attributes. For example, in each set about 20% of the publications have e-prints associated with them. So, the increased citation rates associated with e-printing contribute similarly in both sets. Also, both sets are homogenous in links to object information (NED and SIMBAD links). Lastly, could data centers, in attributing data links to articles, have cherry-picked important (i.e. more citable) data sets? Both sets of publications turn out to be homogenous in citation distributions as well. This leads us to believe that the effect observed is real. In a study of medical literature on cancer microarray clinical trials, Piwowar et al. (2007) found that “publicly available data was significantly associated with a 69% increase in citations”. Even though citation rates are different for different disciplines, the qualitative observation still holds. Studies and discussions in other disciplines show that data sharing is viewed as important and highly relevant for the integrity and furthering of science, and that the hurdles encountered have much in common between various disciplines (Bruna (2010), Delamothe (1996), Kansa et al. (2010), Pisani et al. (2010), South & Duke (2010), Vickers (2011), Vandewalle et al. (2009)). ## References * Accomazzi & Dave (2011) Accomazzi, A., & Dave, R. 2011, in Astronomical Data Analysis Software and Systems XX, edited by I. N. Evans, A. Accomazzi, D. J. Mink, & A. H. Rots, vol. 442 of Astronomical Society of the Pacific Conference Series, 415 * Bruna (2010) Bruna, E. M. 2010, Biotropica, 42, 399 * Delamothe (1996) Delamothe, T. 1996, British Medical Journal, 312, 1241 * Fischer & Zigmond (2010) Fischer, B. A., & Zigmond, M. J. 2010, Science and Engineering Ethics, 16, 783 * Henneken et al. (2011) Henneken, E. A., Kurtz, M. J., Accomazzi, A., Grant, C., Thompson, D., Bohlen, E., Milia, G. D., Luker, J., & Murray, S. S. 2011, in Future Professional Communication in Astronomy II, Edited by Alberto Accomazzi, Astrophysics and Space Science Proceedings, Volume 1. ISBN 978-1-4419-8368-8. Springer Science+Business Media, LLC, 2011, p. 125, edited by A. Accomazzi, 125. arXiv:1005.2308 * Henneken et al. (2006) Henneken, E. A., Kurtz, M. J., Eichhorn, G., Accomazzi, A., Grant, C., Thompson, D., & Murray, S. S. 2006, Journal of Electronic Publishing, 9, 2. arXiv:cs/0604061 * Kansa et al. (2010) Kansa, E. C., Kansa, S. W., Burton, M. M., & Stankowski, C. 2010, Archaeologies, 6, 301 * Pisani et al. (2010) Pisani, E., Whitworth, J., Zaba, B., & Abou-Zahr, C. 2010, The Lancet, 375, 703 * Piwowar et al. (2007) Piwowar, H. A., Day, R. S., & Fridsma, D. B. 2007, PLoS One, 2 * R Development Core Team (2011) R Development Core Team 2011, R: A Language and Environment for Statistical Computing, R Foundation for Statistical Computing, Vienna, Austria. ISBN 3-900051-07-0, URL http://www.R-project.org * South & Duke (2010) South, D. B., & Duke, C. S. 2010, Journal of Forestry, 108, 370 * Vandewalle et al. (2009) Vandewalle, P., Kovacevic, J., & Vetterli, M. 2009, IEEE Signal Processing Magazine, 26, 37 * Vickers (2011) Vickers, A. J. 2011, British Medical Journal, 342	arxiv-papers	2011-11-15T19:40:32	2024-09-04T02:49:24.363394	{ "license": "Public Domain", "authors": "Edwin A. Henneken, Alberto Accomazzi", "submitter": "Edwin Henneken", "url": "https://arxiv.org/abs/1111.3618" }
1111.3647	# Flavor SU(4) breaking between effective couplings Bruno El-Bennich Universidade Cruzeiro do Sul, Rua Galvão Bueno, 868, 01506-000 São Paulo, SP, Brazil Instituto de Física Teórica, Universidade Estadual Paulista, Rua Dr. Bento Teobaldo Ferraz, 271, 01140-070 São Paulo, SP, Brazil Gastão Krein Instituto de Física Teórica, Universidade Estadual Paulista, Rua Dr. Bento Teobaldo Ferraz, 271, 01140-070 São Paulo, SP, Brazil Lei Chang Physics Division, Argonne National Laboratory, Argonne, Illinois 60439, USA Craig D. Roberts Physics Division, Argonne National Laboratory, Argonne, Illinois 60439, USA Institut für Kernphysik, Forschungszentrum Jülich, D-52425 Jülich, Germany Department of Physics, Illinois Institute of Technology, Chicago, Illinois 60616-3793, USA David J. Wilson Physics Division, Argonne National Laboratory, Argonne, Illinois 60439, USA (4 November 2011) ###### Abstract Using a framework in which all elements are constrained by Dyson-Schwinger equation studies in QCD, and therefore incorporates a consistent, direct and simultaneous description of light- and heavy-quarks and the states they constitute, we analyze the accuracy of $SU(4)$-flavor symmetry relations between $\pi\rho\pi$, $K\rho K$ and $D\rho D$ couplings. Such relations are widely used in phenomenological analyses of the interactions between matter and charmed mesons. We find that whilst $SU(3)$-flavor symmetry is accurate to 20%, $SU(4)$ relations underestimate the $D\rho D$ coupling by a factor of five. ###### pacs: 14.40.Lb, 11.15.Tk, 12.39.Ki, 24.85.+p I. Introduction. Hadrons in-medium are the focus of intense theoretical and experimental activity. The chief motivation in heavy-ion collisions is a better understanding of QCD’s deconfined phase, viz. the putative quark-gluon plasma, its chiral restoration phase transition and associated order parameters. Whilst an enhancement of charm and strangeness in the quark-gluon phase is predicted to lead to the copious production of $D_{(s)}$ mesons Cacciari:2005rkKuznetsova:2006bh at the large hadron collider, $J/\psi$ suppression has long been suggested as an unambiguous signature for quark- gluon plasma formation Matsui:1986dk . Notwithstanding ongoing debates about charmonia production mechanisms and a wide range of suppression effects, much effort is sensibly dedicated to understanding the complicated final-state interactions which occur after hadronization of the plasma; see, e.g., Ref. Bracco:2011pg . Charmed-meson interactions with nuclear matter will also be studied at the future Facility for Antiproton and Ion Research (FAIR) and possibly at Jefferson Laboratory (JLab). Low-momentum charmonia, such as $J/\psi$ and $\psi$, and $D^{()}$ mesons can be produced by annihilation of antiprotons on nuclei (FAIR) or by scattering electrons from nuclei (JLab). Since charmonia do not share valence quarks in common with the surrounding nuclear medium, proposed interaction mechanisms include: QCD van der Waals forces, arising from the exchange of two or more gluons between color-singlet states Peskin:1979vaBrodsky:1989jd ; and intermediate charmed hadron states Brodsky:1997ghKo:2000jx , such that $\bar{D}^{(\ast)}D^{(\ast)}$ hadronic vacuum polarization components of the $J/\psi$ interact with the medium via meson exchanges Krein:2010vp . A kindred approach is applied to low-energy interactions of open-charm mesons with nuclei, which may create a path to the production of charmed nuclear bound states ($D$-mesic nuclei) Tsushima:1998ru ; Haidenbauer:2007jq ; Haidenbauer:2010ch ; Yamaguchi:2011xb . These studies rely on model Lagrangians, within which effective interactions are expressed through couplings between $D^{(\ast)}$\- and light-pseudoscalar- and vector-mesons. The models are typically an $SU(4)$ extension of light-flavor chirally- symmetric Lagrangians. Most recently, exotic states formed by heavy mesons and a nucleon were investigated, based upon heavy-meson chiral perturbation theory Yamaguchi:2011xb . In that study a universal coupling, $g_{\pi}$, between a heavy quark and a light pseudoscalar or vector meson was inferred from the strong decay $D^{}\to D\pi$, cf. Ref. ElBennich:2010ha . In the context of chiral Lagrangians, it is natural to question the reliability of couplings based on $SU(4)$ symmetry. Flavor breaking effects are already known to occur in the strange sector and should only be expected to increase when including charm quarks. The order of magnitude of this larger symmetry breaking is signalled by the compilation of charmed couplings in Ref. Bracco:2011pg , where $SU(4)$ relations are shown to be violated at various degrees (ranging from 7% to 70%) in couplings between two heavy mesons and one light meson. No states containing a $s$-quark were considered. Herein, we study a different quantitative measure, based upon ratios between the $D\rho D$, $K\rho K$ and $\pi\rho\pi$ couplings; namely, a difference between the same coupling involving either a $c$-, $s$\- or light-quark. We are motivated by the notion that the $K\rho K$ and $D\rho D$ systems are dynamically equivalent in the sense that the heavier quark acts as a spectator and contributes predominantly to the static properties of the mesons, whereas the exchange dynamics is mediated by the light quarks. In practice, the symmetry idea is expressed by implementing $g_{D\rho D}\simeq g_{K\rho K}$ in the meson-exchange models Haidenbauer:2007jq ; Haidenbauer:2010ch . The $\pi\rho\pi$ coupling provides a well-constrained benchmark. II. DSE Framework. Our primary object of interest is a phenomenological coupling that relates the transition amplitude of an initial pseudoscalar $H=Qf$-meson, $Q=c,s$ and $f=u,d$, to an identical meson via emission of an off-shell $\rho$. The matrix element for this transition is $\langle H(p_{2})\|\,\rho(P,\lambda)\,\|H(p_{1})\rangle=g_{H\rho H}\ \bm{\epsilon}_{\lambda}\\!\cdot P\,,$ (1) an expression which defines the dimensionless coupling of the two pseudoscalar mesons to a vector meson with momentum $P=p_{2}-p_{1}$ and polarization state $\lambda$. The decay $\rho\to\pi\pi$ is also described by such a matrix element. However, there is no associated physical process when $m_{\rho}^{2}<4m_{H}^{2}$ and $p_{1}^{2}=p_{2}^{2}=-m_{H}^{2}$. (N.B. A Euclidean metric is used: $\\{\gamma_{\mu},\gamma_{\nu}\\}=2\,\delta_{\gamma\nu};\,\gamma_{\mu}^{\dagger}=\gamma_{\mu};\;a\\!\cdot\\!b=\sum^{4}_{i=1}a_{i}b_{i}$; and $\mathrm{tr}[\gamma_{5}\gamma_{\mu}\gamma_{\nu}\gamma_{\rho}\gamma\sigma]=-4\epsilon_{\mu\nu\rho\sigma},\,\epsilon_{1234}=1$. For a space-like vector $P_{\mu},P^{2}>0$.) Nevertheless, a coupling of this sort is employed in defining $\rho$-meson-mediated exchange-interactions between a nucleon and pseudoscalar strange- or charm-mesons. In such applications: the off-shell $\rho$-meson’s momentum is necessarily spacelike; and a coupling and form factor may be defined once one settles on a definition of the off-shell $\rho$-meson. Symmetry-preserving models built upon predictions of QCD’s Dyson-Schwinger equations (DSEs) provide a sound framework within which to examine heavy-meson observables ElBennich:2010ha ; Ivanov:1997yg ; Ivanov:1998ms ; Ivanov:2007cw ; ElBennich:2009vx . Such studies describe quark propagation via fully dressed Schwinger functions, which has a material impact on light-quark characteristics Chang:2011vu . At leading-order in a systematic, symmetry-preserving truncation scheme Bender:1996bb , one may express Eq. (1) as $\displaystyle g_{H\\!\rho H}\ \bm{\epsilon}^{\lambda}\\!\cdot P$ $\displaystyle=$ $\displaystyle\mathrm{tr}_{\mathrm{CD}}\\!\ \\!\int\\!\frac{d^{4}k}{(2\pi)^{4}}\,\Gamma_{H}(k;k_{1})S_{Q}(k_{Q})$ (2) $\displaystyle\times\ \bar{\Gamma}_{H}(k;-k_{2})S_{f}(k_{f}^{\prime})\,\bm{\epsilon}^{\lambda}\\!\cdot\bar{\Gamma}_{\rho}(k;-P)S_{f}(k_{f})\;,$ where $S$ represent dressed-quark propagators for the indicated flavor and $\Gamma_{H}$ are meson Bethe-Salpeter amplitudes (BSAs), with $H=\pi,K,D$. In Eq. (2): the trace is over color and spinor indices; $k_{Q}=k+w_{1}p_{1},k_{f}^{\prime}=k+w_{1}p_{1}-p_{2}$, $k_{f}=k-w_{2}p_{1}$, where the relative- momentum partitioning parameters satisfy $w_{1}+w_{2}=1$; and $\bm{\epsilon}^{\lambda}_{\mu}$ is the vector-meson polarization four- vector. This approximation has been employed successfully; see, for instance, applications in Refs. Ivanov:2007cw ; Chang:2011vu ; Roberts:1994hh ; Tandy:1997qf ; Jarecke:2002xd ; Maris:2003vk ; Roberts:2007jh . We simultaneously calculate the $D$-, $K$\- and $\rho$-meson leptonic decay constants via Ivanov:1998ms : $\displaystyle P_{\mu}f_{H}$ $\displaystyle=$ $\displaystyle\mathrm{tr}_{\mathrm{CD}}\\!\ \int\\!\frac{d^{4}k}{(2\pi)^{4}}\,\gamma_{5}\gamma_{\mu}\,\chi_{H}(k;P)\,,$ (3) $\displaystyle M_{\rho}f_{\rho}$ $\displaystyle=$ $\displaystyle\frac{1}{3}\mathrm{tr}_{\mathrm{CD}}\\!\ \int\\!\frac{d^{4}k}{(2\pi)^{4}}\,\gamma_{\mu}\,\chi_{\mu}^{\rho}(k;P)\,,$ (4) where $\chi(k;P)=S_{f_{1}}(k+w_{1}P)\Gamma(k;P)S_{f_{2}}(k-w_{2}P)$. The BSAs are canonically normalized; viz., for pseudoscalars $\displaystyle 2\,P_{\mu}$ $\displaystyle=$ $\displaystyle\left[\frac{\partial}{\partial K_{\mu}}\Pi(P,K)\right]_{K=P}^{P^{2}=-m^{2}_{0^{-}}}\ ,$ (5) $\displaystyle\Pi(P,K)$ $\displaystyle=$ $\displaystyle\mathrm{tr}_{\mathrm{CD}}\\!\ \int\\!\frac{d^{4}k}{(2\pi)^{4}}\,\bar{\Gamma}_{0^{-}}(k;-P)S_{f_{1}}(k+w_{1}K)$ (6) $\displaystyle\times\ \Gamma_{0^{-}}(k;P)S_{f_{2}}(k-w_{2}K)\,,$ with an analogous expression for the $\rho$ Ivanov:1998ms . The solution of QCD’s gap equation is the dressed-quark propagator, which has the general form $S(p)=-i\gamma\cdot p\,\sigma_{V}(p^{2})+\sigma_{S}(p^{2})=1/[i\gamma\cdot p\,A(p^{2})+B(p^{2})]\,.$ (7) For light-quarks, it is a longstanding DSE prediction that both the wave- function renormalization, $Z(p^{2})=1/A(p^{2})$, and dressed-quark mass- function, $M(p^{2})=B(p^{2})/A(p^{2})=\sigma_{S}(p^{2})/\sigma_{V}(p^{2})$, receive strong momentum-dependent modifications at infrared momenta: $Z(p^{2})$ is suppressed and $M(p^{2})$ enhanced. These features are characteristic of dynamical chiral symmetry breaking (DCSB) and, plausibly, of confinement. (N.B. Eqs. (8), (9) represent the quark propagator $S(p)$ as an entire function, which entails the absence of a Lehmann representation and is a sufficient condition for confinement Krein:1990sf ; Roberts:2007ji .) The significance of this infrared dressing has long been emphasized Roberts:1994hh ; e.g., it is intimately connected with the appearance of Goldstone modes Chang:2011vu . The predicted behavior of $Z(p^{2})$, $M(p^{2})$ has been confirmed in numerical simulations of lattice-regularized QCD Roberts:2007ji ; Bowman:2005vxBhagwat:2006tu . Whilst numerical solutions of the quark DSE are readily obtained, the utility of an algebraic form for $S(p)$, when calculations require the evaluation of numerous integrals, is self-evident. An efficacious parametrization, exhibiting the aforementioned features and used extensively Ivanov:1998ms ; Ivanov:2007cw ; Roberts:1994hh ; Cloet:2008re , is expressed via $\displaystyle\bar{\sigma}_{S}(x)$ $\displaystyle=$ $\displaystyle 2\,\bar{m}\,{\cal F}(2(x+\bar{m}^{2}))$ (8) $\displaystyle+{\cal F}(b_{1}x)\,{\cal F}(b_{3}x)\,\left[b_{0}+b_{2}{\cal F}(\epsilon x)\right]\,,$ $\displaystyle\bar{\sigma}_{V}(x)$ $\displaystyle=$ $\displaystyle\frac{1}{x+\bar{m}^{2}}\,\left[1-{\cal F}(2(x+\bar{m}^{2}))\right]\,,$ (9) with $x=p^{2}/\lambda^{2}$, $\bar{m}$ = $m/\lambda$, ${\cal F}(x)=[1-\exp(-x)]/x$, $\bar{\sigma}_{S}(x)=\lambda\,\sigma_{S}(p^{2})$ and $\bar{\sigma}_{V}(x)=\lambda^{2}\,\sigma_{V}(p^{2})$. The parameter values were fixed Ivanov:1998ms by requiring a least-squares fit to a wide range of light- and heavy-meson observables, and take the values: $\begin{array}[]{llcccc}f&\bar{m}_{f}&b_{0}^{f}&b_{1}^{f}&b_{2}^{f}&b_{3}^{f}\\\ \hline\cr u=d&0.00948&0.131&2.94&0.733&0.185\\\ s&0.210&0.105&3.18&0.858&0.185\end{array}\,.$ (10) At a scale $\lambda=0.566\,$GeV, the current-quark masses take the values $m_{u}=5.4\,$MeV and $m_{s}=119\,$MeV, and one obtains the following Euclidean constituent-quark masses Maris:1997tm : $\hat{M}_{u}^{E}=0.36\,$GeV and $\hat{M}_{s}^{E}=0.49\,$GeV. (N.B. $\epsilon=10^{-4}$ in Eq. (8) acts only to decouple the large- and intermediate-$p^{2}$ domains Roberts:1994hh .) We note that studies which do not or cannot implement light-quark dressing in this QCD-consistent manner invariably encounter problems arising from the need to employ large constituent-quark masses and the associated poles in the light-quark propagators ElBennich:2008xyElBennich:2008qa . This typically translates into considerable model sensitivity for computed observables ElBennich:2009vx . Whereas the impact of DCSB on light-quark propagators is significant, the effect diminishes with increasing current-quark mass (see, e.g., Fig. 1 in Ref. Ivanov:1998ms ). This can be explicated by considering the dimensionless and renormalization-group-invariant ratio $\varsigma_{f}:=\sigma_{f}/M^{E}_{f}$, where $\sigma_{f}$ is a constituent- quark $\sigma$-term: $\varsigma_{f}$ measures the effect of explicit chiral symmetry breaking on the dressed-quark mass-function compared with the sum of the effects of explicit and dynamical chiral symmetry breaking. Calculation reveals Roberts:2007jh : $\varsigma_{u}=0.02$, $\varsigma_{s}=0.23$, $\varsigma_{c}=0.65$, $\varsigma_{b}=0.8$. Plainly, $\varsigma_{f}$ vanishes in the chiral limit and remains small for light quarks, since the magnitude of their constituent mass owes primarily to DCSB. On the other hand, for heavy quarks, $\varsigma_{f}\to 1$ because explicit chiral symmetry breaking is the dominant source of their mass. Notwithstanding this, confinement remains important for the heavy-quarks. These considerations are balanced in the following simple form for the $c$-quark propagator: $S_{c}(k)=\frac{-i\gamma\cdot k+\hat{M}_{c}}{\hat{M}_{c}^{2}}{\cal F}(k^{2}/\hat{M}_{c}^{2})\,,$ (11) which implements confinement but produces a momentum-independent c-quark mass- function; namely, $\sigma_{V}^{c}(k^{2})/\sigma_{S}^{c}(k^{2})=\hat{M}_{c}$. We use $\hat{M}_{c}=1.32\,{\rm GeV}$ Ivanov:1998ms . A meson is described by the amplitude obtained from a homogeneous Bethe- Salpeter equation. In solving that equation the simultaneous solution of the gap equation is required. Since we have already chosen to simplify the calculations by parametrizing $S(p)$, we follow Refs. ElBennich:2010ha ; Ivanov:1998ms ; Ivanov:2007cw ; ElBennich:2009vx and also employ that expedient with $\Gamma_{H(\rho)}$. In this connection, the quark-level Goldberger-Treiman relations derived in Ref. Maris:1997hd motivate and support the following parametrization of the $\pi$ and $K$ BSAs: $\Gamma_{\pi,K}(k;P)=i\gamma_{5}\,\frac{\surd 2}{f_{\pi,K}}\,B_{\pi,K}(k^{2})\,,\\\ $ (12) where $B_{\pi,K}:=\left.B_{u}\right\|_{m_{u}\to 0}^{b_{0}^{u}\to b_{0}^{\pi,K}}$ and are obtained from Eqs. (7) – (9) through the replacements $b_{0}^{u}\rightarrow b_{0}^{\pi}=0.204$, $b_{0}^{u}\rightarrow b_{0}^{K}=0.319$, which yield computed values $f_{\pi}=146\,$MeV, $f_{K}=178\,$MeV Ivanov:1998ms . Equation (12) expresses the fact that the dominant invariant function in a pseudoscalar meson’s BSA is closely related to the scalar piece of the dressed-quark self energy owing to the axial-vector Ward-Takahashi identity and DCSB. Regarding the $\rho$ meson, DSE studies Jarecke:2002xd ; Pichowsky:1999mu indicate that, in applications such as ours, one may effectively use $\Gamma^{\mu}_{\rho}(k;P)=\left(\gamma^{\mu}-P^{\mu}\,\frac{\gamma\cdot P}{P^{2}}\right)\frac{\exp(-k^{2}/\omega_{\rho}^{2})}{\mathcal{N}_{\rho}}\,,$ (13) namely, a function whose support is greatest in the infrared. Similarly, for the $D$ meson we choose: $\Gamma_{D}(k;P)=i\gamma_{5}\,\frac{\exp(-k^{2}/\omega_{D}^{2})}{\mathcal{N}_{D}}\;.$ (14) The normalizations, $\mathcal{N}_{\rho}$, $\mathcal{N}_{D}$, are obtained from Eqs. (5), (6) and simultaneous calculation of the weak decay constant in Eqs. (3), (4). In the expression for the coupling, Eq. (1), as well as in Eqs. (3)–(5), we follow the momentum-partitioning prescription of Ref. ElBennich:2010ha , which leads to $w_{1}^{c}=0.79$; viz., most but not all the heavy-light-meson’s momentum is carried by the $c$-quark. We note that Poincaré covariance is a hallmark of the direct application of DSEs to the calculation of hadron properties. In such an approach, no physical observable can depend on the choice of momentum partitioning. However, that feature is compromised if, as herein, one does not retain the complete structure of hadron bound-state amplitudes Maris:1997tm . Any sensitivity to the partitioning is an artifact arising from our simplifications Ivanov:2007cw ; ElBennich:2010ha . III. Results. The $D$-meson’s width parameter is determined via analysis of relevant leptonic and strong decays: $\omega_{D}=1.63\pm 0.10\,$GeV for $m_{D}=1.865\,$GeV yields $f_{D}=206\pm 9\,$MeV Eisenstein:2008sq and $g_{D^{\ast}D\pi}=18.7^{+2.5}_{-1.4}$ _cf_. $17.9\pm 1.9$ Anastassov:2001cw . For the $\rho$, we use $\omega_{\rho}=0.56\pm 0.01\,$GeV and $w_{2}^{\rho}=0.38$, both determined Ivanov:2007cw via a least-squares fit to an array of light-light- and heavy-light-meson observables with $m_{\rho}=0.77\,$GeV. Using Eqs. (3), (5) and (6), one therewith obtains $f_{\rho}=209\,$MeV, _cf_. experiment $216\,$MeV, which follows from the $e^{+}e^{-}$ decay width Nakamura:2010zzi . With the width parameters fixed, we computed the $D\rho D$, $K\rho K$ and $\pi\rho\pi$ couplings in impulse approximation, following Eq. (2). Our results are depicted in Fig. 1. Notably, we compute the amplitude directly: at all values of $P^{2}$ and current-quark mass. We do not need to resort to extrapolations, neither from spacelike$\,\to\,$timelike momenta nor in current-quark mass, expedients which are necessary in some other approaches Bracco:2011pg ; Becirevic:2009xp . Figure 1: _Upper panel_ – Dimensionless couplings: $g_{D\rho D}$ (solid curve); $g_{K\rho K}$ (dashed curve); and $g_{\pi\rho\pi}$ (dotted curve) – all computed as a function of the $\rho$-meson’s off-shell four-momentum- squared, with the pseudoscalar mesons on-shell. Recall that with our Euclidean metric, $P^{2}>0$ is spacelike. _Lower panel_ – Ratios of couplings: $g_{K\rho K}/g_{D\rho D}$ (solid curve); and $g_{K\rho K}/g_{\pi\rho\pi}$ (dashed curve). In the case of exact $SU(4)$ symmetry, these ratios take the values, respectively, $1$ (dot-dashed line) and $(1/2)$ (dotted line). The vertical dotted line marks the $\rho$-meson’s on-shell point in both panels. (N.B. In GeV: $m_{D}=1.865$, $m_{\rho}=0.77$, $m_{K}=0.494$, $m_{\pi}=0.138$.) The behavior of $g_{\pi\rho\pi}(P^{2})$ provides a context for our results. Experimentally Nakamura:2010zzi , $g_{\pi\rho\pi}(-m_{\rho}^{2})=6.0$; and the best numerically-intensive DSE computation available produces Jarecke:2002xd $g_{\pi\rho\pi}(-m_{\rho}^{2})=5.2$. Our algebraically-simplified framework produces $g_{\pi\rho\pi}(-m_{\rho}^{2})=4.8$, just 8% smaller than the latter, and a $P^{2}$-dependence for the coupling which closely resembles that in Ref. Mitchell:1996dn ; e.g., both are smooth, monotonically decreasing functions and our value of $g_{\pi\rho\pi}(-m_{\rho}^{2})/g_{\pi\rho\pi}(m_{\rho}^{2})=0.14$ is just 10% smaller. On the domain $P^{2}\in[-m_{\rho}^{2},m_{\rho}^{2}]$ $g_{\pi\rho\pi}(s=P^{2})=\frac{1.84-1.45s}{1+0.75s+0.085s^{2}}$ (15) provides an accurate interpolation of our result. If one insists on a monopole parametrization at spacelike-$P^{2}$, then a monopole mass of $\Lambda_{\pi\rho\pi}=0.61\,$GeV provides a fit with relative-error-standard- deviation$\,=5$%. In the case of exact $SU(3)$ symmetry, one would have $g_{K\rho K}=g_{\pi\rho\pi}/2$. It is clear from the figure that the assumption provides a fair approximation to our result on a domain which one can reasonably consider as relevant to meson-exchange model phenomenology; viz., on $P^{2}\in[-m_{\rho}^{2},m_{\rho}^{2}]$ the error ranges from $(-10)\,$–$40\,$%. On this domain an accurate interpolation is provided by $g_{K\rho K}(s)=\frac{0.94-0.62s}{1+0.55s-0.16s^{2}}.$ (16) If one insists on a monopole parametrization at spacelike-$P^{2}$, then a monopole mass of $\Lambda_{K\rho K}=0.77\,$GeV provides a fit with relative- error-standard-deviation$\,=4$%. With $SU(4)$ symmetry, the picture is different. We have a numerical result that is reliably interpolated via $g_{D\rho D}(s)=\frac{5.05-4.26s}{1+0.36s-0.060s^{2}}.$ (17) A monopole parametrization at spacelike-$P^{2}$, with mass-scale $\Lambda_{D\rho D}=0.69\,$GeV, provides a fit with relative-error-standard- deviation$\,=5$%. Our computed value $g_{D\rho D}(0)=5.05$ is 75% larger than an estimate obtained using QCD sum rules ($3.0\pm 0.02$ Bracco:2011pg ) and 100% larger than a vector-meson-dominance estimate ($2.52$ Lin:1999ad ). Moreover, if $SU(4)$ symmetry were exact, then $g_{D\rho D}=g_{K\rho K}=g_{\pi\rho\pi}/2$, but it is plain from Eq. (16) that $g_{K\rho K}(0)=0.92$, a result which exposes a symmetry violation of $440$% at $P^{2}=0$. Furthermore, on the entire domain $P^{2}\in[-m_{\rho}^{2},m_{\rho}^{2}]$, the symmetry-based expectation $g_{D\rho D}=g_{K\rho K}$ is always violated, at a level of between $360\,$–$\,440$%. The second identity, $g_{D\rho D}=g_{\pi\rho\pi}/2$, is violated at the level of $320\,$–$\,540$%. (N.B. In connection with heavy- quark symmetry, corrections of this order have also been encountered $c\to d$ transitions Ivanov:1998ms .) These conclusions are dramatic, so it is important to explain why we judge them to be robust. The computations of $g_{\pi\rho\pi}$ and $g_{K\rho K}$ are considered reliable because we can smoothly take the limit $s$-quark$\,\to\,u$-quark and thereby recover a unique function that agrees with earlier computations by other groups. This leaves the possibility of uncertainties connected with $S_{c}(k)$, Eq. (11); $\Gamma_{D}(k;P)$, Eq. (14); and the momentum partitioning parameter, $w_{1}^{c}$. To explore sensitivity to the $c$-quark propagator we used an even simpler, non-confining constituent-like form; viz., $S_{C}(k)=1/(i\gamma\cdot k+\hat{M}_{c})$. The effect at spacelike-$P^{2}$ is modest. However, the impact is large at timelike-$P^{2}$ because thereupon the $\rho$-meson momentum-squared begins to explore a neighborhood of the spurious pole in $S_{C}(k)$. Thus, the simpler propagator serves to _increase_ the violation of $SU(4)$ symmetry. Regarding $\Gamma_{D}(k;P)$, uncertainty is implicit in the value of $\omega_{D}=1.63\pm 0.10\,$GeV, constrained by the weak decay constant $f_{D^{+}}=206\pm 9\,$MeV Eisenstein:2008sq . However, variations of even 20% in $\omega_{D}$ have no material impact on our results. Connected with that, a 20% change in $w_{1}^{c}$ produces only a 4% variation in $\omega_{D}$ via the fit to $f_{D^{+}}$, hence any possibility of an effect from $w_{1}^{c}$ can be discounted owing to the previous consideration. IV. Discussion. Predictions for bound-states and resonances derived from meson-exchange models are sensitive to the values of couplings in their Lagrangians. In these non-relativistic models the couplings are commonly fixed to reproduce some known experimental data, e.g. the scattering length of a physical system. The most prominent such coupling, namely $g_{\pi\\!N}$, has long been used in nucleon-nucleon potentials and serves to define the strength of the pion’s coupling to a nucleon. It also determines the scale of the long- range force in the nucleon-nucleon interaction and associated scattering cross sections. Analogously, the strength of the couplings $D^{(\ast)}\\!D\pi$, $D^{(\ast)}D^{(\ast)}\rho$ between $D$ mesons and a light pion or $\rho$-meson plays a crucial role in the formation of charmed-nuclei. However, whereas $g_{\pi\\!N}$ can be extracted from $\pi N$-scattering data Ericson:2000md , no such information is available for charmed-meson interactions with nucleons. In our approach, which is based on an internally consistent use of impulse approximation and unifies the description of light- and heavy-mesons, we compute these couplings from the transition amplitude between two $D$ mesons and an off-shell light meson. We find that $SU(4)$ symmetry is a very poor guide to the couplings. On the other hand, in relation to such models it provides a constructive suggestion that one might reasonably employ $F^{\rm ME}_{D\rho D}(\|\vec{q}\|^{2})=g^{\rm ME}_{D\rho D}\frac{\Lambda_{D\rho D}^{{\rm ME}\,2}}{\Lambda_{D\rho D}^{{\rm ME}\,2}+\|\vec{q}\|^{2}},$ (18) with $g^{\rm ME}_{D\rho D}\approx 5$, $\Lambda^{\rm ME}_{D\rho D}\approx 0.7\,$GeV, to describe $D\,D$ scattering via $\rho(\vec{q})$-meson exchange. This might be compared with the parametrization Haidenbauer:2007jq : $F^{H}_{D\rho D}(\|\vec{q}\|^{2})=g^{H}_{D\rho D}\frac{\Lambda^{H\,2}_{D\rho D}}{\Lambda_{D\rho D}^{H\,2}+\|\vec{q}\|^{2}},$ (19) $\Lambda^{H}_{D\rho D}=1.4\,$GeV, $g^{H}_{D\rho D}\approx 2$, based on the notion of $SU(4)$ symmetry, which our analysis has discredited. The coupling in Eq. (19) is smaller than that in Eq. (18) but the evolution is harder. These effects cancel to some degree, but here the magnitudes are such that our result, Eq. (18), provides an integrated interaction $V_{0}=\int d^{3}\vec{q}\;F^{H}_{D\rho D}(\|\vec{q}\|^{2})^{2}\frac{1}{\|\vec{q}\|^{2}+m_{\rho}^{2}}$ (20) that is roughly 40% greater. (N.B. If $g^{H}_{D\rho D}\to 2.6\approx(1/2)g^{\rm ME}_{D\rho D}$, then $V_{0}^{H}\approx V_{0}^{ME}$.) By the same measure, our $D\rho D$ interaction is 20% stronger than that in Ref. Yamaguchi:2011xb , which uses $\Lambda^{Y}_{D\rho D}=1.14\,$GeV, $g_{V}=5.8$ and hence $g_{D\rho D}^{Y}=0.9g_{V}[1-m_{\rho}^{2}/\Lambda^{Y\,2}_{D\rho D}]=2.85\,.$ (21) Whilst our results argue against hard form factors, the interaction enhancement they produce is abundantly clear. Notably, a large value for the interaction strength entails an inflated cross-section in $DN$ scattering. In particular, in the meson-exchange model of Ref. Haidenbauer:2007jq (single- meson exchange version), the $I=1$ $\bar{D}N$ cross-section is inflated by a factor of $\sim 5$, when using the our result, Eq. (18), for $\omega$ and $\rho$, instead of Eq. (19). Hence, implementation of our results could have material consequences on, e.g., the possibility for formation of charmed- resonances or -bound-states in nuclei. ###### Acknowledgements. We acknowledge useful input from A. Hosaka and S. M. Schmidt. This work was supported by: Conselho Nacional de Desenvolvimento Científico e Tecnológico, grant no. 305894/2009-9, Fundação de Amparo à Pesquisa do Estado de São Paulo, grant nos. 2009/50180-0, 2009/51296-1 and 2010/05772-3; United States Department of Energy, Office of Nuclear Physics, contract no. DE- AC02-06CH11357; and Forschungszentrum Jülich GmbH. ## References (1) M. Cacciari, P. Nason and R. Vogt, Phys. Rev. Lett. 95, 122001 (2005); I. Kuznetsova and J. Rafelski, Eur. Phys. J. C 51, 113 (2007). * (2) T. Matsui and H. Satz, Phys. Lett. B178, 416 (1986). * (3) M. E. Bracco, M. Chiapparini, F. S. Navarra and M. Nielsen, arXiv:1104.2864 [hep-ph]. * (4) M. E. Peskin, Nucl. Phys. B 156, 365 (1979); S. J. Brodsky, I. A. Schmidt and G. F. de Teramond, Phys. Rev. Lett. 64, 1011 (1990). * (5) S. J. Brodsky and G. A. Miller, Phys. Lett. B 412, 125 (1997); S. H. Lee and C. M. Ko, Phys. Rev. C 67, 038202 (2003). * (6) G. Krein, A. W. Thomas and K. Tsushima, Phys. Lett. B 697, 136 (2011). * (7) K. Tsushima _et al_., Phys. Rev. C 59, 2824 (1999); A. Sibirtsev, K. Tsushima and A. W. Thomas, Eur. Phys. J. A 6, 351 (1999); T. Mizutani and A. Ramos, Phys. Rev. C 74, 065201 (2006); C. Garcia-Recio, J. Nieves and L. Tolos, Phys. Lett. B 690, 369 (2010). * (8) J. Haidenbauer, G. Krein, U. G. Meissner and A. Sibirtsev, Eur. Phys. J. A 33, 107 (2007). * (9) J. Haidenbauer, G. Krein, U. G. Meissner and L. Tolos, Eur. Phys. J. A 47, 18 (2011). * (10) Y. Yamaguchi, S. Ohkoda, S. Yasui and A. Hosaka, Phys. Rev. D 84, 014032 (2011). * (11) B. El-Bennich, M. A. Ivanov and C. D. Roberts, Phys. Rev. C 83, 025205 (2011). * (12) M. A. Ivanov, Yu. L. Kalinovsky, P. Maris and C. D. Roberts, Phys. Lett. B 416, 29 (1998). * (13) M. A. Ivanov, Yu. L. Kalinovsky and C. D. Roberts, Phys. Rev. D 60, 034018 (1999). * (14) M. A. Ivanov, J. G. Körner, S. G. Kovalenko and C. D. Roberts, Phys. Rev. D 76, 034018 (2007). * (15) B. El-Bennich, M. A. Ivanov and C. D. Roberts, Nucl. Phys. Proc. Suppl. 199, 184 (2010). * (16) L. Chang, C. D. Roberts and P. C. Tandy, Chin. J. Phys. 49, 955 (2011). * (17) A. Bender, C. D. Roberts and L. Von Smekal, Phys. Lett. B 380, 7 (1996). * (18) C. D. Roberts, Nucl. Phys. A 605, 475 (1996). * (19) P. C. Tandy, Prog. Part. Nucl. Phys. 39, 117-199 (1997). * (20) D. Jarecke, P. Maris and P. C. Tandy, Phys. Rev. C 67, 035202 (2003). * (21) P. Maris and C. D. Roberts, Int. J. Mod. Phys. E 12, 297 (2003). * (22) C. D. Roberts, M. S. Bhagwat, A. Höll and S. V. Wright, Eur. Phys. J. ST 140, 53 (2007). * (23) C. D. Roberts, A. G. Williams and G. Krein, Int. J. Mod. Phys. A 7, 5624 (1992). * (24) C. D. Roberts, Prog. Part. Nucl. Phys. 61, 50 (2008). * (25) P. O. Bowman _et al_., Phys. Rev. D 71, 054507 (2005); M. S. Bhagwat and P. C. Tandy, AIP Conf. Proc. 842, pp. 225-227 (2006). * (26) I. C. Cloët, _et al_. Few Body Syst. 46, 1 (2009). * (27) B. El-Bennich, O. Leitner, J. P. Dedonder and B. Loiseau, Phys. Rev. D 79, 076004 (2009); B. El-Bennich _et al_., Braz. J. Phys. 38, 465 (2008). * (28) P. Maris, C. D. Roberts and P. C. Tandy, Phys. Lett. B 420, 267 (1998). * (29) M. A. Pichowsky, S. Walawalkar and S. Capstick, Phys. Rev. D 60, 054030 (1999); P. Maris and P. C. Tandy, _ibid_. C 60, 055214 (1999); M. S. Bhagwat and P. Maris, _ibid_. C 77, 025203 (2008). * (30) P. Maris and C. D. Roberts, Phys. Rev. C 56, 3369 (1997). * (31) B. I. Eisenstein et al., Phys. Rev. D78, 052003 (2008). * (32) A. Anastassov et al., Phys. Rev. D 65, 032003 (2002). * (33) K. Nakamura et al., J. Phys. G 37 075021 (2010). * (34) D. Becirevic and B. Haas, Eur. Phys. J. C71, 1734 (2011). * (35) K. L. Mitchell and P. C. Tandy, Phys. Rev. C 55, 1477 (1997). * (36) Z. w. Lin and C. M. Ko, Phys. Rev. C 62, 034903 (2000) * (37) T. E. O. Ericson, B. Loiseau, A. W. Thomas, Phys. Rev. C66, 014005 (2002).	arxiv-papers	2011-11-15T21:01:08	2024-09-04T02:49:24.369314	{ "license": "Public Domain", "authors": "Bruno El-Bennich, Gast\\~ao Krein, Lei Chang, Craig D. Roberts and\n David J. Wilson", "submitter": "Bruno El-Bennich", "url": "https://arxiv.org/abs/1111.3647" }
1111.3812	# On exterior moduli of quadrilaterals and special functions Matti Vuorinen and Xiaohui Zhang Department of Mathematics and Statistics, University of Turku, 20014 Turku, Finland vuorinen@utu.fi, xiazha@utu.fi ###### Abstract. In this paper two identities involving a function defined by the complete elliptic integrals of the first and second kinds are proved. Some functional inequalities and elementary estimates for this function are also derived from the properties of monotonicity and convexity of this function. As applications, some functional inequalities and the growth of the exterior modulus of a rectangle are studied. ††footnotetext: File: vz20130411.tex, printed: 2024-8-27, 18.38 Keywords. Exterior modulus, complete elliptic integral, inequality 2010 Mathematics Subject Classification. 33E05, 31A15 ## 1\. Introduction ###### 1.1. Exterior modulus of a quadrilateral. For $h>0$ consider the rectangle $D$ with vertices $1+ih$, $ih$, $0$, $1$ in the upper half plane $\mathbb{H}^{2}=\\{x+iy:y>0\\}$ and a bounded harmonic function $u:\mathbb{C}\setminus D\to\mathbb{R}$ satisfying the Dirichlet-Neumann boundary value problem $u(z)=0$ for $z\in[0,1]$, $u(z)=1$ for $z\in[ih,1+ih]$, $\frac{\partial u}{\partial n}(z)=0$ for $z\in[1,1+ih]\cup[0,ih]$ where $n$ is the direction of the exterior normal to $\partial D$. The number $\mathcal{M}(1+ih,ih,0,1)=\int_{\mathbb{C}\setminus D}\|\nabla u\|^{2}dm$ is called the exterior modulus of the rectangle $D(1+ih,ih,0,1)$. This quantity also has an interpretation as the modulus of the family of all curves, joining the segments $[1+ih,ih]$ and $[0,1]$ in the complement of the rectangle $D$, which also is equal to $\mathcal{M}(1+ih,ih,0,1)$ (cf. [A]). For a polygonal quadrilateral $D(a,b,0,1)$ with vertices $a,b\in\mathbb{H}^{2}$ and base $[0,1]$, the exterior modulus $\mathcal{M}(a,b,0,1)$ can be defined in the same way. As far as we know there is no analytic formula for $\mathcal{M}(a,b,0,1)$. Numerical methods for the computation of $\mathcal{M}(a,b,0,1)$ were recently studied by H. Hakula, A. Rasila, and M. Vuorinen in [HRV2] which motivate the present study. They used numerical methods such as hp-FEM and the Schwarz- Christoffel mapping. Similar problems for the interior modulus have been studied in [HQR, HRV1]. The literature and software dealing with numerical conformal mapping problems are very wide, see e.g. [DT, PS]. Here we study the above problem for the case of a rectangle. In this case an explicit formula involving complete elliptic integrals was given by P. Duren and J. Pfaltzgraff [DP], and our goal is to analytically study the dependence of the formula on $h$. ###### 1.2. Complete elliptic integrals. Let $\mathcal{K}(r)$ and $\mathcal{E}(r)$ stand for the complete elliptic integrals of the first and second kind, respectively (see (2.1)). Let $r^{\prime}=\sqrt{1-r^{2}}$ for $r\in(0,1)$. We often denote $\mathcal{K}^{\prime}(r)=\mathcal{K}(r^{\prime}),\quad\mathcal{E}^{\prime}(r)=\mathcal{E}(r^{\prime})$. Define the function $\psi$ as follows (1.3) $\psi(r)=\frac{2(\mathcal{E}(r)-(1-r)\mathcal{K}(r))}{\mathcal{E}^{\prime}(r)-r\mathcal{K}^{\prime}(r)},\quad r\in(0,1).$ The function $\psi:(0,1)\to(0,\infty)$ is a homeomorphism, see Theorem 3.1 or [DP]. In particular, $\psi^{-1}:(0,\infty)\to(0,1)$ is well-defined. ###### 1.4. Duren-Pfaltzgraff formula for a rectangle. In [DP], P. Duren and J. Pfaltzgraff studied the modulus $\mathcal{M}(\Gamma)$ of the family of curves $\Gamma$ joining the opposite sides of length $b$ of the rectangle with sides $a$ and $b$, in the exterior of the rectangle, and gave the formula [DP, Theorem 5] (1.5) $\mathcal{M}(\Gamma)=\dfrac{\mathcal{K}^{\prime}(r)}{2\mathcal{K}(r)},\quad\mbox{where}\quad r=\psi^{-1}(a/b).$ The exterior modulus $\mathcal{M}(\Gamma)$ is a conformal invariant of a quadrilateral. In [ADV], the authors gave a sharp comparison between the function $\psi$ and Robin modulus of a given rectangle. Their result can be rewritten as the following inequality (1.6) $\dfrac{\pi r}{(1-r)^{2}}<\psi(r)<\dfrac{16r}{\pi(1-r)^{2}},\quad r\in(0,1).$ In this paper two identities involving the function $\psi$ are proved, and some functional inequalities and elementary estimates for the function $\psi$ are also derived from the monotonicity and convexity of the combinations of the function $\psi$ and some elementary functions. As applications, we will study the growth of the exterior modulus with respect to the length of one side of the rectangle. The main results are listed as follows. ###### Theorem 1.7. For $r\in(0,1)$, the function $\psi$ satisfies the identities $\psi(r^{2})\psi\left(\left(\dfrac{1-r}{1+r}\right)^{2}\right)=1,\quad\psi\left(\dfrac{1-r}{1+r}\right)\psi\left(\dfrac{1-r^{\prime}}{1+r^{\prime}}\right)=1.$ ###### Theorem 1.8. The function $f(r)=(1-\sqrt{r})^{2}\psi(r)/r$ is strictly decreasing from $(0,1)$ onto $(4/\pi,\pi)$. In particular, for all $r\in(0,1)$ $\dfrac{4r}{\pi(1-\sqrt{r})^{2}}<\psi(r)<\dfrac{\pi r}{(1-\sqrt{r})^{2}}.$ ###### Theorem 1.9. The function $f(x)=\psi(1/\operatorname{ch}(x))$ is decreasing and convex from $(0,\infty)$ onto $(0,\infty)$. In particular, for $r,s\in(0,1)$, (1.10) $2\psi\left(\frac{\sqrt{2rs}}{\sqrt{1+rs+r^{\prime}s^{\prime}}}\right)\leq\psi(r)+\psi(s)$ with equality in the above inequality if and only if $r=s$. ###### Theorem 1.11. For $x,y\in(0,1)$, $\psi\left(H_{p}(x,y)\right)\leq H_{p}\left(\psi(x),\psi(y)\right){\quad}\mbox{if}{\quad}p\geq 0,$ and $\psi\left(H_{p}(x,y)\right)\geq H_{p}\left(\psi(x),\psi(y)\right){\quad}\mbox{if}{\quad}p\leq-1.$ The equality holds in each case if and only if $x=y$. Here $H_{p}$ is the power mean defined as $H_{p}(x,y)=\left\\{\begin{array}[]{ll}\left(\dfrac{x^{p}+y^{p}}{2}\right)^{1/p},&p\neq 0\\\ \sqrt{xy},&p=0.\end{array}\right.$ ## 2\. Preliminaries For $0<r<1$, the functions (2.1) $\mathcal{K}(r)=\int_{0}^{\pi/2}\dfrac{dt}{\sqrt{1-r^{2}\sin^{2}t}},\quad\mathcal{E}(r)=\int_{0}^{\pi/2}\sqrt{1-r^{2}\sin^{2}t}\,dt$ with limiting values $\mathcal{K}(0)=\pi/2=\mathcal{E}(0)$, $\mathcal{K}(1-)=\infty$ and $\mathcal{E}(1)=1$ are known as Legendre’s complete elliptic integrals of the first and second kind, respectively. These two functions are connected by Legendre’s relation [BF, 110.10] (2.2) $\mathcal{E}\mathcal{K}^{\prime}+\mathcal{E}^{\prime}\mathcal{K}-\mathcal{K}\mathcal{K}^{\prime}=\dfrac{\pi}{2}.$ Some derivative formulas involving these elliptic integrals are as follows [Bo, p.21]: (2.3) $\left\\{\begin{array}[]{ll}\dfrac{d\mathcal{K}}{dr}=\dfrac{\mathcal{E}-r^{\prime 2}\mathcal{K}}{rr^{\prime 2}},&\dfrac{d\mathcal{E}}{dr}=\dfrac{\mathcal{E}-\mathcal{K}}{r},\vspace{1mm}\\\ \dfrac{d}{dr}\left(\mathcal{E}-r^{\prime 2}\mathcal{K}\right)=r\mathcal{K},&\dfrac{d}{dr}\left(\mathcal{K}-\mathcal{E}\right)=\dfrac{r\mathcal{E}}{r^{\prime 2}}.\\\ \end{array}\right.$ The functions $\mathcal{K}$ and $\mathcal{E}$ satisfy the following identities due to Landen [BF, 163.01, 164.02] (2.4) $\mathcal{K}\left(\dfrac{2\sqrt{r}}{1+r}\right)=(1+r)\mathcal{K}(r),$ (2.5) $\mathcal{K}\left(\dfrac{1-r}{1+r}\right)=\dfrac{1}{2}(1+r)\mathcal{K}^{\prime}(r),$ (2.6) $\mathcal{E}\left(\dfrac{2\sqrt{r}}{1+r}\right)=\dfrac{2\mathcal{E}(r)-r^{\prime 2}\mathcal{K}(r)}{1+r},$ (2.7) $\mathcal{E}\left(\dfrac{1-r}{1+r}\right)=\dfrac{\mathcal{E}^{\prime}(r)+r\mathcal{K}^{\prime}(r)}{1+r}.$ Using Landen’s transformation formulas, we have the following identities. ###### Lemma 2.8. For $r\in(0,1)$, let $t=(1-r)/(1+r)$. Then (2.9) $\mathcal{K}(t^{2})=\dfrac{(1+r)^{2}}{4}\mathcal{K}^{\prime}(r^{2}),$ (2.10) $\mathcal{K}^{\prime}(t^{2})={(1+r)^{2}}\mathcal{K}(r^{2}),$ (2.11) $\mathcal{E}(t^{2})=\dfrac{\mathcal{E}^{\prime}(r^{2})+(r+r^{2}+r^{3})\mathcal{K}^{\prime}(r^{2})}{(1+r)^{2}},$ (2.12) $\mathcal{E}^{\prime}(t^{2})=\dfrac{4\mathcal{E}(r^{2})-(3-2r^{2}-r^{4})\mathcal{K}(r^{2})}{(1+r)^{2}}.$ ###### Proof. By Landen’s transformations (2.4) and (2.5), we have $\dfrac{2(1+r^{2})}{(1+r)^{2}}\mathcal{K}(t^{2})=\mathcal{K}\left(\dfrac{1-r^{2}}{1+r^{2}}\right)=\dfrac{1}{2}(1+r^{2})\mathcal{K}^{\prime}(r^{2}).$ This implies (2.9). For (2.10), $\mathcal{K}^{\prime}(t^{2})=\dfrac{(1+r)^{2}}{1+r^{2}}\mathcal{K}\left(\dfrac{2r}{1+r^{2}}\right)=(1+r)^{2}\mathcal{K}(r^{2})$ where the first equality is Landen’s transformation (2.5) with the parameter $t^{2}$ and the second equality follows from (2.4) with the parameter $r^{2}$. Using Landen’s transformation (2.6) with the change of parameter $r\mapsto t^{2}$ and the formula (2.9), we get (2.13) $\mathcal{E}\left(\dfrac{1-r^{2}}{1+r^{2}}\right)=\dfrac{(1+r)^{2}\mathcal{E}(t^{2})-r(1+r^{2})\mathcal{K}^{\prime}(r^{2})}{1+r^{2}}.$ On the other hand, by (2.7) (2.14) $\mathcal{E}\left(\dfrac{1-r^{2}}{1+r^{2}}\right)=\dfrac{\mathcal{E}^{\prime}(r^{2})+r^{2}\mathcal{K}^{\prime}(r^{2})}{1+r^{2}}.$ Hence (2.11) follows from (2.13) and (2.14). For (2.12), by the change of parameter $r\mapsto t^{2}$ in Landen’s transformation (2.7) and the formula (2.10), we have (2.15) $\mathcal{E}\left(\dfrac{2r}{1+r^{2}}\right)=\dfrac{(1+r)^{2}\mathcal{E}^{\prime}(t^{2})+(1-r^{2})^{2}\mathcal{K}(r^{2})}{2(1+r^{2})}.$ On the other hand, by (2.6) (2.16) $\mathcal{E}\left(\dfrac{2r}{1+r^{2}}\right)=\dfrac{2\mathcal{E}(r^{2})-(1-r^{4})\mathcal{K}(r^{2})}{1+r^{2}}.$ Hence (2.12) follows from (2.15) and (2.16). ∎ The next lemma is a monotone form of l’Hôpital’s rule and will be useful in deriving monotonicity properties and obtaining inequalities [AVV1, Theorem 1.25]. ###### Lemma 2.17 (Monotone form of l’Hôpital’s Rule). Let $-\infty<a<b<\infty$, and let $f,g:[a,b]\to\mathbb{R}$ be continuous on $[a,b]$, differentiable on $(a,b)$. Let $g^{\prime}(x)\neq 0$ on $(a,b)$. Then, if $f^{\prime}(x)/g^{\prime}(x)$ is increasing (decreasing) on $(a,b)$, so are $\dfrac{f(x)-f(a)}{g(x)-g(a)}\qquad\mbox{and}\qquad\dfrac{f(x)-f(b)}{g(x)-g(b)}.$ If $f^{\prime}(x)/g^{\prime}(x)$ is strictly monotone, then the monotonicity on the conclusion is also strict. The following Lemma 2.18 is from [AVV1, Theorem 3.21 (1),(7)]. ###### Lemma 2.18. _(1)_ $r^{-2}(\mathcal{E}-r^{\prime 2}\mathcal{K})$ is strictly increasing and convex from $(0,1)$ onto $(\pi/4,1)$. _(2)_ For each $c\in[1/2,\infty)$, $r^{\prime c}\mathcal{K}$ is decreasing from $[0,1)$ onto $(0,\pi/2]$. ###### Lemma 2.19. _(1)_ $f_{1}(r)=\mathcal{E}-(1-r)\mathcal{K}$ is strictly increasing and concave from $(0,1)$ onto $(0,1)$. _(2)_ $f_{2}(r)=(\mathcal{E}-(1-r)\mathcal{K})/r$ is strictly decreasing from $(0,1)$ onto $(1,\pi/2)$. _(3)_ $f_{3}(r)=\mathcal{E}^{\prime}-r\mathcal{K}^{\prime}$ is strictly decreasing and convex from $(0,1)$ onto $(0,1)$. _(4)_ $f_{4}(r)=(\mathcal{E}^{\prime}-r\mathcal{K}^{\prime})/(1-r)$ is strictly decreasing from $(0,1)$ onto $(0,1)$. _(5)_ $f_{5}(r)=(\mathcal{E}-r^{\prime}\mathcal{K})/(1-\sqrt{r^{\prime}})^{2}$ is strictly decreasing from $(0,1)$ onto $(1,\pi/2)$. _(6)_ $f_{6}(r)=(3-r)\mathcal{E}^{\prime}-(1+r)\mathcal{K}^{\prime}$ is increasing form $(0,1)$ onto $(-\infty,0)$. _(7)_ $f_{7}(r)=(1+r)(\mathcal{E}^{\prime}-r\mathcal{K}^{\prime})/(1-r)$ is strictly deceasing from $(0,1)$ onto $(0,1)$. _(8)_ $f_{8}(r)=(\mathcal{E}-(1-r)\mathcal{K})/(\sqrt{r}(1-r)\mathcal{K})$ is strictly increasing from $(0,1)$ onto $(0,\infty)$. $f_{8}(0.479047\cdots)=1.$ ###### Proof. (1) By differentiation and the derivative formulas (2.3), $f_{1}^{\prime}(r)=\dfrac{\mathcal{E}}{1+r}$ which is positive and decreasing. Then the properties of monotonicity and concavity of $f_{1}$ follow. The limiting value $f_{1}(0)=0$ is clear and $f_{1}(1-)=\mathcal{E}(1)-\lim\limits_{r\to 1-}(r^{\prime 2}\mathcal{K}/(1+r))=1$ by Lemma 2.18(2). (2) Since $f_{1}$ is concave and $f_{2}(r)=f_{1}(r)/r$, $f_{2}$ is decreasing by the monotone form of l’Hôpital’s rule. By l’Hôpital’s rule $f_{2}(0)=f^{\prime}_{1}(0)=\pi/2$, and $f_{2}(1)=1$ is clear. (3) By (2.3), we have $f_{3}^{\prime}(r)=-\dfrac{\mathcal{K}^{\prime}-\mathcal{E}^{\prime}}{1+r},$ which is negative and increasing in $(0,1)$. Then $f_{3}$ is decreasing and convex in $(0,1)$. The limiting value $f_{3}(0)=1$ follows from Lemma 2.18(2), and $f_{3}(1)=0$ is clear. (4) Let $h(r)=1-r$. Since $f_{3}$ is convex, $f_{3}^{\prime}(r)/h^{\prime}(r)$ is decreasing. Thus $f_{4}(r)=f_{3}(r)/h(r)$ is also decreasing by the monotone form of l’Hôpital’s rule. By l’Hôpital’s rule $f_{4}(1)=-f^{\prime}_{3}(1)=0$, and $f_{4}(0)=f_{3}(0)=1$. (5) For the proof we first make the change of variable $r=2\sqrt{x}/(1+x)$. The Landen transformations (2.4) and (2.6) lead to $h(x)=f_{5}\left(\dfrac{2\sqrt{x}}{1+x}\right)=\dfrac{\mathcal{E}(x)-x^{\prime 2}\mathcal{K}(x)}{1-x^{\prime}}=\dfrac{h_{1}(x)}{h_{2}(x)},$ where $h_{1}(x)=\mathcal{E}(x)-x^{\prime 2}\mathcal{K}(x)$ and $h_{2}(x)=1-x^{\prime}$ with $h_{1}(0)=0=h_{2}(0)$. Then by (2.3) we have $\dfrac{h_{1}^{\prime}(x)}{h_{2}^{\prime}(x)}=\dfrac{x\mathcal{K}(x)}{x/x^{\prime}}=x^{\prime}\mathcal{K}(x),$ which is strictly decreasing by Lemma 2.18(2). This implies that $h$ is decreasing by the monotone form of l’Hôpital’s rule, and hence $f_{5}$ is also decreasing in $(0,1)$. (6) By differentiation, we have $f_{6}^{\prime}(r)=\dfrac{(1-r)(2r\mathcal{E}^{\prime}+\mathcal{E}^{\prime}-r\mathcal{K}^{\prime})}{r(1+r)}>0,$ and hence $f_{6}$ is increasing. The limiting values are clear. (7) By simple computation, $f_{7}^{\prime}(r)=f_{6}(r)/(1-r)^{2}<0$ and hence $f_{7}$ is decreasing. The limiting values follow from part (4). (8) Differentiation and simplification give that $f_{8}^{\prime}(r)=\dfrac{((1+r)\mathcal{K}-\mathcal{E})(2\mathcal{E}-r^{\prime 2}\mathcal{K})}{2r^{3/2}(1+r)(1-r)^{2}\mathcal{K}^{2}}>0,$ and hence $f_{8}$ is strictly increasing. The limit $f_{8}(0+)=\lim_{r\to 0+}\dfrac{\mathcal{E}-(1-r)\mathcal{K}}{\sqrt{r}\mathcal{K}}=\lim_{r\to 0+}\dfrac{\mathcal{E}-(1-r)\mathcal{K}}{r}\dfrac{\sqrt{r}}{\mathcal{K}}=0$ follows from the part (2). The limit $f_{8}(1-)=\infty$ is clear. ∎ Let (2.20) $\mu(r)=\frac{\pi}{2}\frac{\mathcal{K}^{\prime}(r)}{\mathcal{K}(r)}$ be the modulus of Grötzsch’s ring $\mathbb{B}^{2}\setminus[0,r]$ (see [LV],[AVV1]). ###### Lemma 2.21. The function $f(r)=\mu(r)\psi(r)$ is strictly increasing from $(0,1)$ onto $(0,\infty)$. ###### Proof. Since the function $f$ can be rewritten as $f(r)=\pi\sqrt{r}\mathcal{K}^{\prime}\dfrac{\mathcal{E}-(1-r)\mathcal{K}}{\sqrt{r}(1-r)\mathcal{K}}\dfrac{1-r}{\mathcal{E}^{\prime}-r\mathcal{K}^{\prime}},$ the conclusion follows from Lemma 2.18(2), Lemma 2.19(4) and (8). ∎ ## 3\. Proofs of Main Results In this section we will prove two identities involving the function $\psi$, and some functional inequalities and elementary estimates for the function $\psi$ are also derived from the monotonicity and convexity of the combinations of the function $\psi$ and some elementary functions. ###### Theorem 3.1. The function $\psi(r)$ is strictly increasing and convex from $(0,1)$ onto $(0,\infty)$, and the function $\psi(r)/r$ is strictly increasing from $(0,1)$ onto $(0,\infty)$. ###### Proof. By differentiation, and using (2.3) and Legendre’s identity (2.2), we have (3.2) $\dfrac{d\psi}{dr}=\dfrac{2(1-r)}{1+r}\dfrac{\mathcal{E}^{\prime}\mathcal{K}+\mathcal{E}\mathcal{K}^{\prime}-\mathcal{K}\mathcal{K}^{\prime}}{(\mathcal{E}^{\prime}-r\mathcal{K}^{\prime})^{2}}=\dfrac{\pi}{1-r^{2}}\left(\frac{1-r}{\mathcal{E}^{\prime}-r\mathcal{K}^{\prime}}\right)^{2},$ which is positive and strictly increasing by Lemma 2.19(4). Hence $\psi(r)$ is strictly increasing and convex, and consequently $\psi(r)/r$ is strictly increasing by the monotone form of l’Hôpital’s rule. ∎ ###### Proof of Theorem 1.7. By simple calculations, the first identity follows from the definition of $\psi$ and Lemma 2.8. The second identity follows from the first one with the change of parameter $r\mapsto\sqrt{(1-r)/(1+r)}$. ∎ ###### Corollary 3.3. $\psi(3-2\sqrt{2})=1.$ ###### Proof. Let $r=1/\sqrt{2}$. Then $(1-r)/(1+r)=3-2\sqrt{2}=(1-r^{\prime})/(1+r^{\prime})$, and the second identity in the Theorem 1.7 implies $\psi(3-2\sqrt{2})=1.$ ∎ ###### Remark 3.4. Let $\Delta$ be the family of curves lying outside the rectangle $R$ and joining the opposite sides of length $a$. Then a basic fact is $\mathcal{M}(\Gamma)=1/\mathcal{M}(\Delta).$ By (1.5) and (2.20), we have $\mathcal{M}(\Gamma)=\mu(r)/\pi,\quad\mbox{and}\quad\mathcal{M}(\Delta)=\mu(s)/\pi$ with $r=\psi^{-1}(a/b)$ and $s=\psi^{-1}(b/a).$ By the identity [AVV1, Exercises 5.68(2)] $\mu(r^{2})\mu\left(\left(\dfrac{1-r}{1+r}\right)^{2}\right)=\pi^{2},$ it is easy to see that $\mathcal{M}(\Gamma)=1/\mathcal{M}(\Delta)$ is equivalent to $s=((1-\sqrt{r})/(1+\sqrt{r}))^{2}.$ Since $s=\psi^{-1}(b/a)=\psi^{-1}(1/\psi(r))$, we have $s=((1-\sqrt{r})/(1+\sqrt{r}))^{2}$ which is equivalent to $\dfrac{1}{\psi(r)}=\psi\left(\left(\dfrac{1-\sqrt{r}}{1+\sqrt{r}}\right)^{2}\right).$ ###### Proof of Theorem 1.8. The theorem follows from $f(r)=\dfrac{(1-\sqrt{r})^{2}\psi(r)}{r}=2\dfrac{\mathcal{E}-(1-r)\mathcal{K}}{r}\dfrac{(1-\sqrt{r})^{2}}{\mathcal{E}^{\prime}-r\mathcal{K}^{\prime}},$ since $(\mathcal{E}-(1-r)\mathcal{K})/r$ and $(1-\sqrt{r})^{2}/(\mathcal{E}^{\prime}-r\mathcal{K}^{\prime})$ are both decreasing by Lemma 2.19(2), (5), respectively. The limiting values are clear by Lemma 2.19(2), (5). ∎ ###### Corollary 3.5. The function $g(r)=(1-\sqrt{r})\operatorname{arth}(1-\sqrt{r})\psi(r)/r$ is strictly decreasing from $(0,1)$ onto $(4/\pi,\infty)$. ###### Proof. This follows from Theorem 1.8, since $g(r)=f(r)\operatorname{arth}(1-\sqrt{r})/(1-\sqrt{r})$ where $f(r)$ is as in Theorem 1.8 and $\operatorname{arth}(1-\sqrt{r})/(1-\sqrt{r})$ is strictly decreasing from $(0,1)$ onto $(1,\infty)$. ∎ Since the bounds for $\psi$ in (1.6) and the Theorem 1.8 are not comparable in the whole interval $(0,1)$, we could combine them to get the following inequalities: ###### Corollary 3.6. For $0<r<1$, $\max\left\\{\dfrac{\pi r}{(1-r)^{2}},\dfrac{4r}{\pi(1-\sqrt{r})^{2}}\right\\}<\psi(r)<\min\left\\{\dfrac{16r}{\pi(1-r)^{2}},\dfrac{\pi r}{(1-\sqrt{r})^{2}}\right\\}.$ ###### Proof of Theorem 1.9. Let $r=1/\operatorname{ch}(x)$ and $s=1/\operatorname{ch}(y)$. Then $dr/dx=-\operatorname{sh}(x)/\operatorname{ch}^{2}(x)=-rr^{\prime}$ and $f^{\prime}(x)=-\pi rr^{\prime}\dfrac{1-r}{1+r}\dfrac{1}{(\mathcal{E}^{\prime}-r\mathcal{K}^{\prime})^{2}}=-\pi g(r),$ where $g(r)=rr^{\prime}(1-r)/((1+r)(\mathcal{E}^{\prime}-r\mathcal{K}^{\prime})^{2})$. By the change of variable $r=(1-t)/(1+t)$ and using Landen’s transformations (2.5) and (2.7), we have $g\left(\dfrac{1-t}{1+t}\right)=\dfrac{1-t}{2}\frac{t^{3/2}}{(\mathcal{E}-t^{\prime 2}\mathcal{K})^{2}}$ which is decreasing in $t$ by Lemma 2.18(1), and consequently, $f^{\prime}(x)$ is increasing in $x$. Therefore, $f$ is decreasing and convex on $(0,\infty)$. In particular, we have $f((x+y)/2)\leq(f(x)+f(y))/2$, with equality if and only if $x=y$. Now $\operatorname{ch}^{2}\left(\dfrac{x+y}{2}\right)=\dfrac{1+rs+r^{\prime}s^{\prime}}{2rs}.$ Hence $f\left(\frac{x+y}{2}\right)\leq\frac{f(x)+f(y)}{2}$ gives $\psi(r)+\psi(s)\geq 2\psi\left(\frac{\sqrt{2rs}}{\sqrt{1+rs+r^{\prime}s^{\prime}}}\right),$ with equality if and only if $r=s$. ∎ ###### Remark 3.7. It is clear that $f(x)$ is decreasing and $2f(x+y)\leq f(x)+f(y)$. Since $\operatorname{ch}(x+y)=\dfrac{1+r^{\prime}s^{\prime}}{rs},$ we have $2\psi\left(\dfrac{rs}{1+r^{\prime}s^{\prime}}\right)\leq\psi(r)+\psi(s)$ which is weaker than the inequality (1.10). A function $f:I\to J$ is called $H_{p,q}-$convex (concave) if it satisfies $f(H_{p}(x,y))\leq(\geq)H_{q}(f(x),f(y))$ for all $x,y\in I$ and strictly $H_{p,q}-$convex (concave) if the inequality is strict, except for $x=y$. Recently, many authors investigated the $H_{p,q}-$convexity (concavity) of special functions, see [AVV2, BalPV, Ba, BaPV, CWZQ, WZJ]. The following theorems give some functional inequalities by studying the generalized convexity (concavity) of the function $\psi$. ###### Theorem 3.8. The function $f(x)=\log(1/\psi(e^{-x}))$ is strictly increasing and concave from $(0,\infty)$ onto $(-\infty,\infty)$. In particular, for $r,s\in(0,1)$, $\psi(\sqrt{rs})\leq\sqrt{\psi(r)\psi(s)}$ with equality if and only if $r=s$. ###### Proof. Let $r=e^{-x}$ and $s=e^{-y}$. Then $dr/dx=-r$ and $f^{\prime}(x)=\dfrac{r\psi^{\prime}(r)}{\psi(r)}=\frac{\pi}{2}\dfrac{r}{\mathcal{E}-(1-r)\mathcal{K}}\dfrac{1-r}{(1+r)(\mathcal{E}^{\prime}-r\mathcal{K}^{\prime})}$ which is positive and increasing in $r$ by Lemma 2.19(2) and (7), hence decreasing in $x$. Therefore, $f$ is strictly increasing and concave on $(0,\infty)$. In particular, we have $f((x+y)/2)\geq(f(x)+f(y))/2$, with equality if and only if $x=y$. This gives $\psi(\sqrt{rs})\leq\sqrt{\psi(r)\psi(s)}$ with equality if and only if $r=s$. ∎ ###### Proof of Theorem 1.11. For $p=0$, the inequality is from Theorem 3.8. Now we assume that $p\neq 0$. Let $0<x<y<1$ and $t=((x^{p}+y^{p})/2)^{1/p}>x$. Define $f(x)=\psi(t)^{p}-\dfrac{\psi(x)^{p}+\psi(y)^{p}}{2}.$ By differentiation, we have $dt/dx=\frac{1}{2}(x/t)^{p-1}$ and $\displaystyle f^{\prime}(x)$ $\displaystyle=$ $\displaystyle\dfrac{1}{2}p\psi(t)^{p-1}\psi^{\prime}(t)\left(\dfrac{x}{t}\right)^{p-1}-\dfrac{1}{2}p\psi(x)^{p-1}\psi^{\prime}(x)$ $\displaystyle=$ $\displaystyle\dfrac{p}{2}x^{p-1}\left(\left(\dfrac{\psi(t)}{t}\right)^{p-1}\psi^{\prime}(t)-\left(\dfrac{\psi(x)}{x}\right)^{p-1}\psi^{\prime}(x)\right).$ We first consider the case of $p>0$. Previous calculation gives $\displaystyle f^{\prime}(x)$ $\displaystyle=$ $\displaystyle\dfrac{p}{2}x^{p-1}\left(\left(\dfrac{\psi(t)}{t}\right)^{-1}\psi^{\prime}(t)\left(\dfrac{\psi(t)}{t}\right)^{p}-\left(\dfrac{\psi(x)}{x}\right)^{-1}\psi^{\prime}(x)\left(\dfrac{\psi(x)}{x}\right)^{p}\right)$ $\displaystyle=$ $\displaystyle\frac{{\pi}p\,x^{p-1}}{4}\left(\dfrac{t}{\mathcal{E}(t)-(1-t)\mathcal{K}(t)}\dfrac{1-t}{(1+t)(\mathcal{E}^{\prime}(t)-t\mathcal{K}^{\prime}(t))}\left(\dfrac{\psi(t)}{t}\right)^{p}\right.$ $\displaystyle\qquad\left.-\dfrac{x}{\mathcal{E}(x)-(1-x)\mathcal{K}(x)}\dfrac{1-x}{(1+x)(\mathcal{E}^{\prime}(x)-x\mathcal{K}^{\prime}(x))}\left(\dfrac{\psi(x)}{x}\right)^{p}\right)$ which is positive by Lemma 2.19(2),(7) and Theorem 3.1 since $t>x$ and $p>0$. Hence $f$ is strictly increasing and $f(x)<f(y)=0$. This implies that $\psi\left(\left(\frac{x^{p}+y^{p}}{2}\right)^{1/p}\right)\leq\left(\frac{\psi(x)^{p}+\psi(y)^{p}}{2}\right)^{1/p}.$ For the case of $p\leq-1$, by previous calculation we have $f^{\prime}(x)=\dfrac{p}{2}x^{p-1}\left(\left(\dfrac{\psi(t)}{t}\right)^{-2}\psi^{\prime}(t)\left(\dfrac{\psi(t)}{t}\right)^{p+1}-\left(\dfrac{\psi(x)}{x}\right)^{-2}\psi^{\prime}(x)\left(\dfrac{\psi(x)}{x}\right)^{p+1}\right).$ Since $(\psi(x)/x)^{p+1}$ is decreasing, we only need to prove $(\psi(x)/x)^{-2}\psi^{\prime}(x)$ is strictly decreasing in $(0,1)$. In fact, with the change of variable $x\mapsto(1-t)/(1+t)$, $\left(\dfrac{\psi(x)}{x}\right)^{-2}\psi^{\prime}(x)=\dfrac{\pi}{4}\left(\dfrac{x(1-x)}{x^{\prime}(\mathcal{E}-(1-x)\mathcal{K})}\right)^{2}=\dfrac{\pi}{4}\left(\dfrac{t^{\prime 2}}{\mathcal{E}^{\prime}-t^{2}\mathcal{K}^{\prime}}\dfrac{\sqrt{t}}{1+t}\right)^{2}$ which is a product of two positive and strictly increasing functions of $t$ by Lemma 2.18(1). Hence $f^{\prime}(x)>0$ and $f$ is strictly increasing in $(0,1)$. Now we have $f(x)<f(y)=0$, and consequently $\psi\left(\left(\frac{x^{p}+y^{p}}{2}\right)^{1/p}\right)\geq\left(\frac{\psi(x)^{p}+\psi(y)^{p}}{2}\right)^{1/p}$ since $p$ is negative. The equality case is obvious. This completes the proof. ∎ ## 4\. Applications In this section we always denote $R=[0,1]\times[0,b]$. Let $\Gamma_{b}$ and $\Delta_{b}$ be the families of curves joining the opposite sides of length $b$ of the rectangle $R$, in the exterior and interior of the rectangle, respectively. It is well-known that $\mathcal{M}(\Delta_{b})=b$. By the formula of Duren and Pfaltzgraff (1.5), we have $\mathcal{M}(\Gamma_{b})=\dfrac{1}{\pi}\mu(\psi^{-1}(1/b)).$ Setting $r=\sqrt{2}-1$ in (2.14), we get $\mathcal{K}^{\prime}(3-2\sqrt{2})=2\mathcal{K}(3-2\sqrt{2})$. By Corollary 3.3, $\mathcal{M}(\Gamma_{1})=\dfrac{1}{\pi}\mu(\psi^{-1}(1))=1=\mathcal{M}(\Delta_{1}).$ Now we will study the behavior of the modulus $\mathcal{M}(\Gamma_{b})$ with respect to the sides of length $b$. The following Theorem 4.2 shows (4.1) $\left\\{\begin{array}[]{ll}\mathcal{M}(\Gamma_{b})>\mathcal{M}(\Delta_{b}),&\mbox{for}\quad 0<b<1,\\\ \mathcal{M}(\Gamma_{b})<\mathcal{M}(\Delta_{b}),&\mbox{for}\quad b>1.\end{array}\right.$ ###### Theorem 4.2. There exists a number $r_{0}=8.24639\ldots$ such that the function $f(r)=\dfrac{1}{\pi}\mu(\psi^{-1}(r))-\frac{1}{r}$ is strictly increasing in $(0,r_{0})$ and decreasing in $(r_{0},\infty)$, with the limiting value $f(\infty)=0$. In particular, $\dfrac{1}{\pi}\mu(\psi^{-1}(r))<\frac{1}{r},\quad\mbox{for}\quad 0<r<1,$ and $\dfrac{1}{\pi}\mu(\psi^{-1}(r))>\frac{1}{r},\quad\mbox{for}\quad r>1.$ ###### Proof. Let $s=\psi^{-1}(r)$. Then $r=\psi(s)$ and, by the derivative formula (3.2), $\dfrac{ds}{dr}=(\dfrac{dr}{ds})^{-1}=\dfrac{s^{\prime 2}}{\pi}\left(\dfrac{\mathcal{E}^{\prime}(s)-s\mathcal{K}^{\prime}(s)}{1-s}\right)^{2}.$ By differentiation and $\dfrac{d\mu(s)}{ds}=\dfrac{-\pi^{2}}{4ss^{\prime 2}\mathcal{K}(s)^{2}},$ we have $\displaystyle f^{\prime}(r)$ $\displaystyle=$ $\displaystyle\dfrac{1}{\pi}\dfrac{d\mu}{ds}\dfrac{ds}{dr}+\dfrac{1}{r^{2}}$ $\displaystyle=$ $\displaystyle\dfrac{1}{r^{2}}-\dfrac{1}{4s\mathcal{K}(s)^{2}}\left(\dfrac{\mathcal{E}^{\prime}(s)-s\mathcal{K}^{\prime}(s)}{1-s}\right)^{2}$ $\displaystyle=$ $\displaystyle\dfrac{1}{\psi(s)^{2}}\left(1-\left(\dfrac{\mathcal{E}(s)-(1-s)\mathcal{K}(s)}{\sqrt{s}(1-s)\mathcal{K}(s)}\right)^{2}\right).$ which is positive in $(0,r_{0})$ and negative in $(r_{0},\infty)$ with $r_{0}=\psi(0.479047\ldots)=8.24639\ldots$ by Lemma 2.19(8). Hence $f$ is strictly increasing in $(0,r_{0})$ and decreasing in $(r_{0},\infty)$. Since $f(1)=0$ and $f(\infty)=0$, we have $f(r)<0$ for $r\in(0,1)$ and $f(r)>0$ for $r\in(1,\infty)$. ∎ The next theorem shows that the modulus $\mathcal{M}(\Gamma_{b})$ has a logarithmic growth with respect to the length of side $b$. ###### Theorem 4.3. For $b\in(0,\infty)$, (4.4) $L(b)<\mathcal{M}(\Gamma_{b})<U(b),$ where (4.5) $\displaystyle L(b)$ $\displaystyle:=$ $\displaystyle\dfrac{2}{\pi}\left(1-\left(1+\sqrt{4b/\pi}\right)^{-4}\right)^{1/4}\,\log\left(2\left(1+\sqrt{{4b}/{\pi}}\right)\right)$ $\displaystyle>$ $\displaystyle\dfrac{2}{\pi}\left(1-\left(1+\sqrt{4b/\pi}\right)^{-1}\right)\,\log\left(2\left(1+\sqrt{{4b}/{\pi}}\right)\right),$ and (4.6) $\displaystyle U(b)$ $\displaystyle:=$ $\displaystyle\dfrac{1}{\pi}\log\left(2\left(1+\sqrt{\pi b}\right)^{2}\left(1+\sqrt{1-\left(1+\sqrt{\pi b}\right)^{-4}}\right)\right)$ $\displaystyle<$ $\displaystyle\dfrac{2}{\pi}\log\left(2\left(1+\sqrt{\pi b}\right)\right).$ ###### Proof. By Theorem 1.8 we have $\left(\dfrac{\sqrt{r}}{\sqrt{\pi}+\sqrt{r}}\right)^{2}<s=\psi^{-1}(r)<\left(\dfrac{\sqrt{r}}{\sqrt{4/\pi}+\sqrt{r}}\right)^{2},\quad r\in(0,\infty).$ By [AVV1, Theorem 5.13(4),(5)], $\sqrt{s^{\prime}}\log{\dfrac{4}{s}}<\mu(s)<\log{\dfrac{2(1+s^{\prime})}{s}},\quad s\in(0,1).$ Combining the above inequalities and replacing $r$ with $1/b$, we get the inequalities (4.4). The inequality (4.5) follows from the inequality $1-a^{x}>(1-a)^{x}$ for $a\in(0,1)$ and $x\in(1,\infty)$. The inequality (4.6) is obvious. ∎ ###### Theorem 4.7. For $a,b\in(0,\infty)$, 1. (1) $\mathcal{M}(\Gamma_{2ab/(a+b)})\leq\sqrt{\mathcal{M}(\Gamma_{a})\mathcal{M}(\Gamma_{b})}\leq\dfrac{\mathcal{M}(\Gamma_{a})+\mathcal{M}(\Gamma_{b})}{2}\leq\mathcal{M}(\Gamma_{(a+b)/2})$; 2. (2) $\left\\{\begin{array}[]{ll}\mathcal{M}(\Gamma_{H_{p}(a,b)})\leq H_{p}(\mathcal{M}(\Gamma_{a}),\mathcal{M}(\Gamma_{b})),&p\leq-1,\vspace{1mm}\\\ \mathcal{M}(\Gamma_{H_{p}(a,b)})\geq H_{p}(\mathcal{M}(\Gamma_{a}),\mathcal{M}(\Gamma_{b})),&p\geq 1.\\\ \end{array}\right.$ Equality holds in each case if and only if $a=b$. ###### Proof. In part (1), the second inequality is clear. For the third inequality, let $s=\psi^{-1}(1/a)$, $t=\psi^{-1}(1/b)$. Then $\displaystyle\dfrac{\mathcal{M}(\Gamma_{a})+\mathcal{M}(\Gamma_{b})}{2}$ $\displaystyle=$ $\displaystyle\dfrac{1}{\pi}\dfrac{\mu(s)+\mu(t)}{2}$ $\displaystyle\leq$ $\displaystyle\dfrac{1}{\pi}\mu(\sqrt{st})\leq\dfrac{1}{\pi}\mu(H_{-1}(s,t))$ $\displaystyle\leq$ $\displaystyle\dfrac{1}{\pi}\mu(\psi^{-1}(H_{-1}(\psi(s),\psi(t))))$ $\displaystyle=$ $\displaystyle\dfrac{1}{\pi}\mu(\psi^{-1}(H_{-1}(1/a,1/b)))$ $\displaystyle=$ $\displaystyle\mathcal{M}(\Gamma_{(a+b)/2}),$ where the first inequality follows from [AVV1, Theorem 5.12(1)] (also see [WZJ, Theorem]) and the third inequality follows from Theorem 1.11. Let $m(a)=\mu(\psi^{-1}(a))/\pi$ and $u=\psi^{-1}(a)$. By logarithmic differentiation, we have $\dfrac{d}{da}\log m(a)=-\dfrac{1}{2u\mathcal{K}^{\prime}(u)\mathcal{K}(u)}\left(\dfrac{\mathcal{E}^{\prime}(u)-u\mathcal{K}^{\prime}(u)}{1-u}\right)^{2},$ which is strictly increasing in $u$ by Lemma 2.18(2) and Lemma 2.19(4), and hence strictly increasing in $a$. Since $m(a)$ is logarithmic convex, we have $m\left(\dfrac{a+b}{2}\right)\leq\sqrt{m(a)m(b)},$ which implies the first inequality in part (1) by replacing $a,b$ with $1/a,1/b$, respectively. For the part (2), let $M(x):=\mathcal{M}(\Gamma_{x})$. Let $0<x<y<1$ and $t=((x^{p}+y^{p})/2)^{1/p}>x$. Define $f(x)=M(t)^{p}-\dfrac{M(x)^{p}+M(y)^{p}}{2}.$ By differentiation, we have $dt/dx=\frac{1}{2}(x/t)^{p-1}$ and (4.8) $f^{\prime}(x)=\dfrac{p}{2}x^{p-1}\left(\left(\dfrac{M(t)}{t}\right)^{p-1}M^{\prime}(t)-\left(\dfrac{M(x)}{x}\right)^{p-1}M^{\prime}(x)\right).$ Let $M(x)=m(a)$, then $a=1/x=\psi(u)$. Now we have $\displaystyle\left(\dfrac{M(x)}{x}\right)^{p-1}M^{\prime}(x)$ $\displaystyle=$ $\displaystyle(m(a)a)^{p-1}m^{\prime}(a)(-a^{2})$ $\displaystyle=$ $\displaystyle\left(\dfrac{\mu(u)\psi(u)}{\pi}\right)^{p-1}\psi(u)^{2}\dfrac{1}{4u\mathcal{K}(u)^{2}}\left(\dfrac{\mathcal{E}^{\prime}(u)-u\mathcal{K}^{\prime}(u)}{1-u}\right)^{2}$ $\displaystyle=$ $\displaystyle\left(\dfrac{\mu(u)\psi(u)}{\pi}\right)^{p-1}\left(\dfrac{\mathcal{E}(u)-(1-u)\mathcal{K}(u)}{\sqrt{u}(1-u)\mathcal{K}(u)}\right)^{2},$ which is strictly increasing in $u$ by Lemmas 2.21 and 2.19(8), and hence strictly decreasing in $x$ for each $p\geq 1$. This implies that $f^{\prime}(x)<0$ if $p\geq 1$. For the case of $p\leq-1$, we have $\displaystyle\left(\dfrac{M(x)}{x}\right)^{p-1}M^{\prime}(x)$ $\displaystyle=$ $\displaystyle(m(a)a)^{p-1}m^{\prime}(a)(-a^{2})=-(m(a)a)^{p+1}m(a)^{-2}m^{\prime}(a)$ $\displaystyle=$ $\displaystyle\left(\dfrac{\mu(u)\psi(u)}{\pi}\right)^{p+1}\dfrac{1}{u\mathcal{K}^{\prime}(u)^{2}}\left(\dfrac{\mathcal{E}^{\prime}(u)-u\mathcal{K}^{\prime}(u)}{1-u}\right)^{2},$ which is strictly decreasing in $u$ by Lemmas 2.21, 2.18(2) and 2.19(4), and hence strictly increasing in $x$ for each $p\leq-1$. Since $p$ is negative, this still implies that $f^{\prime}(x)<0$. It is easy to see that $f^{\prime}(x)<0$ implies the inequalities in the part (2). ∎ ###### Open problem 4.9. What is the exact domain of $p$ for which the function $\psi$ is $H_{p,p}$-convex (concave)? More generally, find the exact $(p,q)$ domain for which the function $\psi$ is $H_{p,q}$-convex (concave). The same questions can be asked for the modulus $\mathcal{M}(\Gamma_{b})$. ### Acknowledgments The research of Matti Vuorinen was supported by the Academy of Finland, Project 2600066611. Xiaohui Zhang is indebted to the CIMO (Grant TM-09-6629) and the Finnish National Graduate School of Mathematics and its Applications for financial support. Both authors wish to thank Árpád Baricz and the referee for their helpful comments on the manuscript. ## References * [1] * [A] L. V. Ahlfors: Conformal invariants: Topics in geometric function theory. AMS Chelsea Publishing, 2010. * [ADV] G. D. Anderson, P. Duren, and M. K. Vamanamurthy: An inequality for complete elliptic integrals. J. Math. Anal. Appl. 182 (1994), 257–259. * [AVV1] G. D. Anderson, M. K. Vamanamurthy, and M. Vuorinen: Conformal Invariants, Inequalities and Quasiconformal Maps. John Wiley & Sons, New York, 1997. * [AVV2] G. D. Anderson, M. K. Vamanamurthy, and M. Vuorinen: Generalized convexity and inequalities. J. Math. Anal. Appl. 335 (2007), 1294–1308. * [BalPV] R. Balasubramanian, S. Ponnusamy, and M. Vuorinen: Functional inequalities for the quotients of hypergeometric functions. J. Math. Anal. Appl. 218 (1998), 256–268. * [Ba] Á. Baricz: Functional inequalities involving special functions. II. J. Math. Anal. Appl. 327 (2007), 1202–1213. * [BaPV] Á. Baricz, S. Ponnusamy, and M. Vuorinen: Functional inequalities for modified Bessel functions. Expos. Math. 29 (2011), 399–414. * [Bo] F. Bowman: Introduction to Elliptic Functions with Applications. Dover, New York, 1961. * [BF] P. F. Byrd and M. D. Friedman: Handbook of Elliptic Integrals for Engineers and Scientists. 2nd ed., Die Grundlehren der Math. Wiss. 67. Springer-Verlag, Berlin, 1971. * [CWZQ] Y.-M. Chu, G.-D. Wang, X.-H. Zhang, and S.-L. Qiu: Generalized convexity and inequalities involving special functions. J. Math. Anal. Appl. 336 (2007), 768–776. * [DT] T. A. Driscoll and L. N. Trefethen: Schwarz-Christoffel mapping. Cambridge Monographs on Applied and Computational Mathematics 8. Cambridge University Press, Cambridge, 2002. * [DP] P. Duren and J. Pfaltzgraff: Robin capacity and extremal length. J. Math. Anal. Appl. 179 (1993), 110–119. * [HQR] H. Hakula, T. Quach, and A. Rasila: _Conjugate function method for numerical conformal mappings._ J. Comput. Appl. Math. 237 (2013), 340–353. * [HRV1] H. Hakula, A. Rasila, and M. Vuorinen: _On moduli of rings and quadrilaterals: algorithms and experiments._ SIAM J. Sci. Comput. 33 (2011), 279–302. * [HRV2] H. Hakula, A. Rasila, and M. Vuorinen: Computation of exterior moduli of quadrilateral. arXiv:1111.2146 [math.NA], 2011, 19 pp. * [LV] O. Lehto and K. I. Virtanen: Quasiconformal mappings in the plane. 2nd ed., Die Grundlehren der Math. Wiss. 126. Springer-Verlag, New York, 1973. * [PS] N. Papamichael and N. S. Stylianopoulos: Numerical conformal mapping: Domain decomposition and the mapping of quadrilaterals. World Scientific, 2010. * [WZJ] G.-D. Wang, X.-H. Zhang, and Y.-P. Jiang: Concavity with respect to Hölder means involving the generalized Grötzsch function. J. Math. Anal. Appl. 379 (2011), 200–204. * [2]	arxiv-papers	2011-11-16T14:26:14	2024-09-04T02:49:24.379551	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Matti Vuorinen and Xiaohui Zhang", "submitter": "Xiaohui Zhang", "url": "https://arxiv.org/abs/1111.3812" }
1111.3824	Rev. 10/III/12 JM Higher-order Erdős–Szekeres theorems ###### Abstract Let $P=(p_{1},p_{2},\ldots,p_{N})$ be a sequence of points in the plane, where $p_{i}=(x_{i},y_{i})$ and $x_{1}<x_{2}<\cdots<x_{N}$. A famous 1935 Erdős–Szekeres theorem asserts that every such $P$ contains a monotone subsequence $S$ of $\lceil\sqrt{N}\,\rceil$ points. Another, equally famous theorem from the same paper implies that every such $P$ contains a convex or concave subsequence of $\Omega(\log N)$ points. Monotonicity is a property determined by pairs of points, and convexity concerns triples of points. We propose a generalization making both of these theorems members of an infinite family of Ramsey-type results. First we define a $(k+1)$-tuple $K\subseteq P$ to be _positive_ if it lies on the graph of a function whose $k$th derivative is everywhere nonnegative, and similarly for a _negative_ $(k+1)$-tuple. Then we say that $S\subseteq P$ is _$k$ th-order monotone_ if its $(k+1)$-tuples are all positive or all negative. We investigate quantitative bound for the corresponding Ramsey-type result (i.e., how large $k$th-order monotone subsequence can be guaranteed in every $N$-point $P$). We obtain an $\Omega(\log^{(k-1)}N)$ lower bound ($(k-1)$-times iterated logarithm). This is based on a quantitative Ramsey- type theorem for _transitive colorings_ of the complete $(k+1)$-uniform hypergraph (these were recently considered by Pach, Fox, Sudakov, and Suk). For $k=3$, we construct a geometric example providing an $O(\log\log N)$ upper bound, tight up to a multiplicative constant. As a consequence, we obtain similar upper bounds for a Ramsey-type theorem for _order-type homogeneous_ subsets in ${\mathbb{R}}^{3}$, as well as for a Ramsey-type theorem for hyperplanes in ${\mathbb{R}}^{4}$ recently used by Dujmović and Langerman. ## 1 Introduction In this paper we mainly consider sets $P=\\{p_{1},p_{2},\ldots,p_{N}\\}$ of points in the plane, where $p_{i}=(x_{i},y_{i})$. We always assume that no two of the $x$-coordinates coincide, and unless stated otherwise, we also assume that the $p_{i}$ are numbered so that $x_{1}<x_{2}<\cdots<x_{N}$ (the same also applies to subsets of $P$, which we will enumerate in the order of increasing $x$-coordinates). Two theorems of Erdős and Szekeres. Among simple results in combinatorics, only few can compete with the following one in beauty and usefulness: ###### Theorem 1.1 (Erdős–Szekeres on monotone subsequences [ES35]) For every positive integer $n$, among every $N=(n-1)^{2}+1$ points $p_{1},\ldots,p_{N}\in{\mathbb{R}}^{2}$ as above, one can always choose a _monotone subset_ of at least $n$ points, i.e., indices $i_{1}<i_{2}<\cdots<i_{n}$ such that either $y_{i_{1}}\leq y_{i_{2}}\leq\cdots\leq y_{i_{n}}$ or $y_{i_{1}}\geq y_{i_{2}}\geq\cdots\geq y_{i_{n}}$. See, for example, Steele [Ste95] for a collection of six nice proofs and some applications. For many purposes, it is more natural to view the above theorem as a purely combinatorial result about permutations, but here we prefer the geometric formulation (which is also similar to the one in the original Erdős–Szekeres paper). Another result of the same paper of Erdős and Szekeres is the following well- known gem in discrete geometry:111Somewhat unfortunately, the name Erdős–Szekeres theorem refers to Theorem 1.1 in some sources and to Theorem 1.2 or similar statements in other sources. ###### Theorem 1.2 (Erdős–Szekeres on convex/concave configurations [ES35]) For every positive integer $n$, among every $N={2n-4\choose n-2}+1\approx 4^{n}/\sqrt{n}$ points $p_{1},\ldots,p_{N}\in{\mathbb{R}}^{2}$ as above, one can always choose a _convex configuration_ or a _concave configuration_ of $n$ points, i.e., indices $i_{1}<i_{2}<\cdots<i_{n}$ such that the slopes of the segments $p_{i_{j}}p_{i_{j+1}}$, $j=1,2,\ldots,n-1$, are either monotone nondecreasing or monotone nonincreasing. See, e.g., [MS00, Mat02] for proofs and surveys of developments around this result. $k$-general position. To simplify our forthcoming discussion, at some places it will be convenient to assume that the considered point sets are in a “sufficiently general” position. Namely, we define a set $P$ to be in _$k$ -general position_ if no $k+1$ points of $P$ lie on the graph of a polynomial of degree at most $k-1$. In particular, $1$-general position requires that no two $y$-coordinates coincide, and $2$-general position means the usual general position, i.e., no three points collinear. $k$th-order monotone subsets. Here we propose a view of Theorems 1.1 and 1.2 as the first two members in an infinite sequence of Ramsey-type results about planar point sets.222There is also a (trivial) 0th member, namely, the statement that in every $P$, at least half of the points either have all $y$-coordinates nonnegative or have or all $y$-coordinates nonpositive. In Theorem 1.1, monotonicity of a subset is a property of _pairs_ of points of the subset, and actually, it suffices to look at pairs of consecutive points. Similarly, convexity or concavity of a configuration in Theorem 1.2 is a property of triples, and again it is enough to look at consecutive triples. In the former case, we are considering the slope of the segment determined by a pair of points, which can be thought of as the first derivative. In the latter case, a triple is convex iff its points lie on the graph of a smooth convex function, i.e., one with nonnegative second derivative everywhere. With this point of view, it is natural to define a $(k+1)$-tuple $K\subseteq P$ to be _positive_ if it lies on the graph of a function whose $k$-th derivative (exists and) is everywhere nonnegative, and similarly for a _negative_ $(k+1)$-tuple (in Section 2, we will provide several other, equivalent characterizations of these properties). Then we say that an arbitrary subset $S\subseteq P$ is _$k$ th-order monotone_ if its $(k+1)$-tuples are all positive or all negative. First-order monotonicity is obviously equivalent to monotonicity as in Theorem 1.1, and second-order monotonicity is equivalent to convexity/concavity as in Theorem 1.2. We will also see (Lemma 2.5) that, to certify $k$th-order monotonicity, it is enough to consider all $(k+1)$-tuples of _consecutive_ points. Let us remark that every $(k+1)$-tuple $K$ is positive or negative, and moreover, if $K$ is in $k$-general position, it cannot be both positive and negative (Corollary 2.3). We will write $\mathop{\rm sgn}\nolimits(K)=+1$ if $K$ is positive and $\mathop{\rm sgn}\nolimits(K)=-1$ if $K$ is negative. Ramsey’s theorem, quantitative bounds, and transitive colorings. Using the just mentioned facts, one can immediately derive a Ramsey-type theorem for $k$th-order monotone subsets from Ramsey’s theorem. ###### Proposition 1.3 For every $k$ and $n$ there exists $N$ such that every $N$-point planar set in $k$-general position contains an $n$-point $k$th-order monotone subset. Proof. We recall Ramsey’s theorem (for two colors; see, e.g., Graham, Rothschild, and Spencer [GRS90]): for every $\ell$ and $n$ there exists $N$ such that for every coloring of the set ${X\choose\ell}$ of all $\ell$-element subsets of an $N$-element set $X$ there exists an $n$-element _homogeneous_ set $Y\subseteq X$, i.e., a subset in which all $\ell$-tuples have the same color. The smallest $N$ for which the claim holds is usually denoted by $R_{\ell}(n)$. In our case, we set $X=P$ and color each $(k+1)$-tuple $K\subseteq P$ with the color $\mathop{\rm sgn}\nolimits(K)\in\\{\pm 1\\}$. Then homogeneous subsets are exactly $k$th-order monotone subsets. $\Box$ Let us denote by $\operatorname{ES}_{k}(n)$ the smallest value of $N$ for which the claim in this proposition holds. We have $\operatorname{ES}_{1}(n)\leq(n-1)^{2}+1$ and $\operatorname{ES}_{2}(n)\leq{2n-4\choose n-2}+1$ according to Theorems 1.1 and 1.2, respectively; moreover, these inequalities actually hold with equality [ES35]. Our main goal is to estimate the order of magnitude of $\operatorname{ES}_{k}(n)$ for $k\geq 3$. The above proof gives $\operatorname{ES}_{k}(n)\leq R_{k+1}(n)$. However, for $k=1$, and most likely for all $k$, the order of magnitude of $R_{k+1}(n)$ is much larger than that of $\operatorname{ES}_{k}(n)$. Indeed, considering $k$ fixed and $n$ large, the best known lower and upper bounds of $R_{k+1}(n)$ are of the form333We employ the usual asymptotic notation for comparing functions: $f(n)=O(g(n))$ means that $\|f(n)\|\leq C\|g(n)\|$ for some $C$ and all $n$, where $C$ may depend on parameters declared as constants (in our case on $k$); $f(n)=\Omega(g(n))$ is equivalent to $g(n)=O(f(n))$; and $f(n)=\Theta(g(n))$ means that both $f(n)=O(g(n))$ and $f(n)=\Omega(g(n))$. $R_{2}(n)=2^{\Theta(n)}$ and, for $k\geq 2$, $\mathop{\rm twr}\nolimits_{k}(\Omega(n^{2}))\leq R_{k+1}(n)\leq\mathop{\rm twr}\nolimits_{k+1}(O(n)),$ where the tower function $\mathop{\rm twr}\nolimits_{k}(x)$ is defined by $\mathop{\rm twr}\nolimits_{1}(x)=x$ and $\mathop{\rm twr}\nolimits_{i+1}(x)=2^{\mathop{\rm twr}\nolimits_{i}(x)}$. It is widely believed that the upper bound is essentially the truth. This belief is supported by known bounds for more than two colors, where the lower bound for $(k+1)$-tuples is also a tower of height $k+1$; see Conlon, Fox, and Sudakov [CFS11] for a recent improvement and more detailed overview of the known bounds. The coloring of the $(k+1)$-tuples in the above proof of Proposition 1.3 is not arbitrary. In particular, it has a property we call _transitivity_ (see Lemma 2.5). Transitive colorings were introduced earlier in the recent preprint Fox et al. [FPSS11, Section 6], under the same name. To define a transitive coloring in general, we need to consider a hypergraph whose vertex set is linearly ordered; w.l.o.g. we can identify it with the set $[N]:=\\{1,2,\ldots,N\\}$. A coloring $c\colon{[N]\choose\ell}\to[m]$ is _transitive_ if, for every $i_{1},\ldots,i_{\ell+1}\in[N]$, $i_{1}<\cdots<i_{\ell+1}$, whenever the $\ell$-tuples $\\{i_{1},\ldots,i_{\ell}\\}$ and $\\{i_{2},\ldots,i_{\ell+1}\\}$ have the same color, then _all_ $\ell$-element subsets of $\\{i_{1},\ldots,i_{\ell+1}\\}$ have the same color. Let $R^{\rm trans}_{\ell}(n)$ denote the Ramsey number for transitive colorings, i.e., the smallest $N$ such that any transitive coloring of the complete $\ell$-uniform hypergraph on $[N]$ contains an $n$-element homogeneous subset. We have the following bound.444By inspecting the proof of the next theorem, it is easy to verify that the transitivity condition is not used in full strength—it suffices to assume only that the subsets obtained by omitting one of $i_{2}$, $i_{3}$ have the same color. ###### Theorem 1.4 For $k=1,2$, we have $R^{\rm trans}_{k+1}(n)=\operatorname{ES}_{k}(n)$, and for every fixed $k\geq 3$, $\operatorname{ES}_{k}(n)\leq R^{\rm trans}_{k+1}(n)\leq\mathop{\rm twr}\nolimits_{k}(O(n)).$ We note that Fox et al. [FPSS11] proved the slightly weaker upper bound $R^{\rm trans}_{k+1}(n)\leq\mathop{\rm twr}\nolimits_{k}(O(n\log n))$. The proof of Theorem 1.4 is given in Section 3. The inequality $\operatorname{ES}_{k}(n)\leq R^{\rm trans}_{k+1}(n)$ is clear (since every $N$-point set in $k$-general position provides a transitive coloring of $[N]\choose k+1$). The upper bounds for $R^{\rm trans}_{2}(n)$ and $R^{\rm trans}_{3}(n)$ follow by translating the proofs of Theorem 1.1 and 1.2 to the setting of transitive colorings almost word by word, and they are contained in [FPSS11]. The upper bound on $R^{\rm trans}_{k+1}(n)$ is then obtained by induction on $k$, with $k=3$ as the base case, following one of the usual proofs of Ramsey’s theorem. A set with no large third-order monotone subsets. For $k\leq 2$, the numbers $\operatorname{ES}_{k}(n)$ (and thus $R^{\rm trans}_{k+1}(n)$) are known exactly. Our perhaps most interesting result is an asymptotically matching lower bound for $\operatorname{ES}_{3}(n)$. ###### Theorem 1.5 For all $n\geq 2$ we have $R^{\rm trans}_{4}(2n+1)\geq\operatorname{ES}_{3}(2n+1)\geq 2^{2^{n-1}}+1$. Consequently, $\operatorname{ES}_{3}(n)=2^{2^{\Theta(n)}}$. The proof is given in Section 4. A Ramsey function with known doubly exponential growth seems to be rare in geometric Ramsey-type problems (a notable example is a result of Valtr [Val04]). Order types. Here we change the setting from the plane to ${\mathbb{R}}^{d}$ and we consider an ordered sequence $P=(p_{1},p_{2},\ldots,p_{N})$ in ${\mathbb{R}}^{d}$. This time we do _not_ assume the first coordinates to be increasing. For simplicity, we assume $P$ to be in general position, which now means that no $d+1$ points of $P$ lie on a common hyperplane. We recall that _order type_ of $P$ specifies the orientation of every $(d+1)$-tuple of points of $P$, and it this way, it describes purely combinatorially many of the geometric properties of $P$. More formally, the order type of $P$ is the mapping $\chi\colon{[N]\choose d+1}\to\\{-1,+1\\}$, where for a $(d+1)$-tuple $I=\\{i_{1},\ldots,i_{d+1}\\}$, $i_{1}<i_{2}<\cdots<i_{d+1}$, $\chi(I):=\mathop{\rm sgn}\nolimits\det M(p_{i_{1}},p_{i_{2}},\ldots,p_{i_{d+1}})$, where $M(q_{1},\ldots,q_{d+1})$ is the $(d+1)\times(d+1)$ matrix whose $j$th column is $(1,q_{j})$, i.e., $1$ followed by the vector of the $d$ coordinates of $q_{j}$. See, e.g., Goodman and Pollack [GP93] or [Mat02] for more background about order types. From Ramsey’s theorem for $(d+1)$-tuples, we can immediately derive a Ramsey- type result for order types: for every $d$ and $n$ there exists $N$ such that every $N$-point sequence contains an $n$-point subsequence in which all the $(d+1)$-tuples have the same orientation (we call such a subsequence _order- type homogeneous_). Let us write $\mathop{\rm OT}\nolimits_{d}(n)$ for the smallest such $N$. In Section 5 we first observe that, by simple and probably well known considerations, $\mathop{\rm OT}\nolimits_{1}(n)=(n-1)^{2}+1$ and $\mathop{\rm OT}\nolimits_{2}(n)=2^{\Theta(n)}$. For $d\geq 3$, the best upper bound for $\mathop{\rm OT}\nolimits_{d}(n)$ we are aware of is the one from the Ramsey argument above, i.e., $\mathop{\rm OT}\nolimits_{d}(n)\leq R_{d+1}(n)\leq\mathop{\rm twr}\nolimits_{d+1}(O(n))$. In particular, for $\mathop{\rm OT}\nolimits_{3}(n)$ this upper bound is triply exponential; in Section 5 we prove a doubly exponential lower bound. ###### Proposition 1.6 For all $d$ and $n$, $\mathop{\rm OT}\nolimits_{d}(n)\geq\operatorname{ES}_{d}(n)$. In particular, $\mathop{\rm OT}\nolimits_{3}(n)=2^{2^{\Omega(n)}}$. A Ramsey-type result for hyperplanes. Let us consider a finite set $H$ of hyperplanes in ${\mathbb{R}}^{d}$ in general position (every $d$ intersecting at a single point). Let us say that $H$ is _one-sided_ if $V(H)$, the vertex set of the arrangement of $H$, lies completely on one side of the coordinate hyperplane $x_{d}=0$. Let $\mathop{\rm OSH}\nolimits_{d}(n)$ be the smallest $N$ such that every set $H$ of $N$ hyperplanes in ${\mathbb{R}}^{d}$ in general position contains a one-sided subset of $n$ hyperplanes. Ramsey’s theorem for $d$-tuples immediately gives $\mathop{\rm OSH}\nolimits_{d}(n)\leq R_{d}(n)$ (a $d$-tuple gets color $+1$ if its intersection has a positive last coordinate, and color $-1$ otherwise). Matoušek and Welzl [MW92] observed that, actually, $\mathop{\rm OSH}\nolimits_{2}(n)=\operatorname{ES}_{1}(n)=(n-1)^{2}+1$, and applied this in a range-searching algorithm. Recently Dujmović and Langerman [DL11] used the existence of $\mathop{\rm OSH}\nolimits_{d}(n)$ (essentially Lemma 9 in the arXiv version of their paper) to prove several interesting results, such as a ham-sandwich and centerpoint theorems for hyperplanes. In Section 5 we show that lower bounds for $k$th-order monotone subsets in the plane can be translated into lower bounds for $\mathop{\rm OSH}\nolimits_{d}$. ###### Proposition 1.7 We have $\mathop{\rm OSH}\nolimits_{d}(n)\geq\operatorname{ES}_{d-1}(n)$, and in particular, $\mathop{\rm OSH}\nolimits_{3}(n)=2^{\Omega(n)}$ and555An exponential lower bound for $\mathop{\rm OSH}\nolimits_{3}$ was known to the authors of [MW92], and perhaps to others as well, but as far as we know, it hasn’t appeared in print. $\mathop{\rm OSH}\nolimits_{4}(n)=2^{2^{\Omega(n)}}$. The lower bounds for $\mathop{\rm OSH}\nolimits_{d}(n)$ can also be translated into lower bounds in the theorems of Dujmović and Langerman. For example, in their ham-sandwich theorem, we have $d$ collections $H_{1},\ldots,H_{d}$ of hyperplanes in ${\mathbb{R}}^{d}$, each of size $N$, and we want a hyperplane $g$ such that in each $H_{i}$, we can find disjoint subsets $A_{i},B_{i}$ of $n$ hyperplanes each such $V(A_{i})$ lies on one side of $g$ and $V(B_{i})$ on the other side. To derive a lower bound for the smallest necessary $N$, we fix $d$ affinely independent points $p_{1},\ldots,p_{d}$ in the $x_{d}=0$ hyperplane, and a set $H$ of $N$ hyperplanes in general position with no one-sided subset of size $n$. We let $H_{i}$ be an affinely transformed copy of $H$ such that all of $V(H_{i})$ lies very close to $p_{i}$. Then every potential ham-sandwich hyperplane $g$ for these $H_{i}$ has to be almost parallel to the $x_{d}=0$ hyperplane, and thus there cannot be $A_{i},B_{i}$ of size $n$ for all $i$. The work of Fox et al. While preparing a draft of the present paper, we learned about a recent preprint of Fox, Pach, Sudakov, and Suk [FPSS11]. They investigated various combinatorial and geometric problems inspired by Theorems 1.1 and 1.2, and as was mentioned above, among others, they introduced transitive colorings,666With still another geometric source of such colorings besides the Erdős–Szekeres theorems, namely, noncrossing convex bodies in the plane but mainly they studied a related but different Ramsey-type quantity: let $N_{\ell}(q,n)$ be the smallest integer $N$ such that, for every coloring of ${[N]\choose\ell}$ with $q$ colors, there exists an $n$-element $I=\\{i_{1},\ldots,i_{n}\\}\subseteq[N]$, $i_{1}<\cdots<i_{n}$, inducing a _monochromatic monotone path_ , i.e., such that all the $\ell$-tuples of the form $\\{i_{j},i_{j+1},\ldots,i_{j+\ell-1}\\}$, $j=1,2,\ldots,n-\ell+1$, have the same color. They note that $R^{\rm trans}_{\ell}(n)\leq N_{\ell}(2,n)$, and they obtained the following bounds for $N_{\ell}(2,n)$: $N_{2}(2,n)=\operatorname{ES}_{1}(n)$, $N_{3}(2,n)=\operatorname{ES}_{2}(n)$, and for every fixed $k\geq 3$, $\mathop{\rm twr}\nolimits_{k}(\Omega(n))\leq N_{k+1}(2,n)\leq\mathop{\rm twr}\nolimits_{k}(O(n\log n)).$ As we mentioned after Theorem 1.4, this also yields an upper bound for $R^{\rm trans}_{k+1}(n)$ only slightly weaker than the one in that theorem. Open problems. 1. 1. We have obtained reasonably tight bounds for $\operatorname{ES}_{3}(n)$, but the gaps are much more significant for $\operatorname{ES}_{k}(n)$ with $k\geq 4$. According to the cases $k=1,2,3$, one may guess that $\operatorname{ES}_{k}(n)$ is of order $\mathop{\rm twr}\nolimits_{k}(\Theta(n))$, and thus that stronger lower bounds are needed, but a possibility of a better upper bound shouldn’t also be overlooked. This question looks both interesting and challenging. 2. 2. A perhaps more manageable task might be a better lower bound for $R^{\rm trans}_{k}(n)$, $k\geq 4$. A natural approach would be to imitate the Stepping-Up Lemma used for lower bounds for the Ramsey numbers $R_{k}(n)$ (see, e.g., [CFS11]). But so far we have not succeeded in this, since even if we start with a transitive coloring of $k$-tuples, we could not guarantee transitivity for the coloring of $(k+1)$-tuples. 3. 3. As for order-type homogeneous sequences, for $\mathop{\rm OT}\nolimits_{3}(n)$ we have the lower bound of $2^{2^{\Omega(n)}}$, but upper bound only $\mathop{\rm twr}\nolimits_{4}(O(n))$ directly from Ramsey’s theorem. It seems that the colorings given by the order type are not transitive in any reasonable sense, and we have no good guess of which of the upper and lower bounds should be closer to the truth. Similar comments apply to the problem with one-sided subsets of planes in ${\mathbb{R}}^{3}$ (concerning $\mathop{\rm OSH}\nolimits_{3}(n)$), and the higher-dimensional cases are even more widely open. 4. 4. Another interesting question is whether $n\log n$ can be replaced by $n$ in the upper bound for the quantity $N_{\ell}(2,n)$ considered by Fox et al. [FPSS11]. 5. 5. In our definition of $k$th-order positivity, every $(k+1)$-tuple of points should lie on the graph of a function with a nonnegative $k$th derivative, and different functions can be used for different $(k+1)$-tuples. In an earlier version of this paper, we conjectured that, assuming $k$-general position, a single function should suffice for all $(k+1)$-tuples; in other words, that every $k$th-order monotone finite set finite set in $k$-general position lies on a graph of a $k$-times differentiable function $f\colon{\mathbb{R}}\to{\mathbb{R}}$ whose $k$th derivative is everywhere nonnegative or everywhere nonpositive. However, Rote [Rot12] disproved this for $k=3$ (while the cases $k=1,2$ do hold, as is not hard to check). With his kind permission, we reproduce his example at the end of Section 2. Naturally, this opens up interesting new questions: How can one characterize point sets lying on the graph of a function whose $k$th derivative is positive everywhere? Is there a Ramsey-type theorem for such sets, and if yes, how large is the corresponding Ramsey function? ## 2 On the definition of $k$th-order monotonicity Here we provide several equivalent characterizations of $k$th-order monotonicity of planar point sets and some of their properties. First we recall several known results. Divided differences and Newton’s interpolation. Let $p_{1},p_{2},\ldots,p_{k+1}$ be points in the plane, $p_{i}=(x_{i},y_{i})$, where the $x_{i}$ are all distinct (but not necessarily increasing). We recall that the _$k$ th divided difference_ $\mbox{$\Delta\\!\\!\\!\\!\\!\;\raisebox{2.15277pt}{\mbox{\boldmath$\scriptscriptstyle\mid$}}\,\,$}_{k}(p_{1},p_{2},\ldots,p_{k+1})$ is defined recursively as follows: $\displaystyle\mbox{$\Delta\\!\\!\\!\\!\\!\;\raisebox{2.15277pt}{\mbox{\boldmath$\scriptscriptstyle\mid$}}\,\,$}_{0}(p_{1})$ $\displaystyle:=$ $\displaystyle y_{1}$ $\displaystyle\mbox{$\Delta\\!\\!\\!\\!\\!\;\raisebox{2.15277pt}{\mbox{\boldmath$\scriptscriptstyle\mid$}}\,\,$}_{k}(p_{1},p_{2},\ldots,p_{k+1})$ $\displaystyle:=$ $\displaystyle\frac{\mbox{$\Delta\\!\\!\\!\\!\\!\;\raisebox{2.15277pt}{\mbox{\boldmath$\scriptscriptstyle\mid$}}\,\,$}_{k-1}(p_{2},p_{3},\ldots,p_{k+1})-\mbox{$\Delta\\!\\!\\!\\!\\!\;\raisebox{2.15277pt}{\mbox{\boldmath$\scriptscriptstyle\mid$}}\,\,$}_{k-1}(p_{1},p_{2},\ldots,p_{k})}{x_{k+1}-x_{1}}.$ For example, $\mbox{$\Delta\\!\\!\\!\\!\\!\;\raisebox{2.15277pt}{\mbox{\boldmath$\scriptscriptstyle\mid$}}\,\,$}_{1}(p_{1},p_{2})$ equals the slope of the line $p_{1}p_{2}$. In general, the $k$th divided difference is related to the $k$th derivative as follows (see, e.g., [Phi03, Eq. 1.33]; note that the case $k=1$ is the Mean Value Theorem): ###### Lemma 2.1 (Cauchy) Let the points $p_{1},\ldots,p_{k+1}$, $a:=x_{1}<x_{2}<\cdots<b:=x_{k+1}$, lie on the graph of a function $f$ such that the $k$th derivative $f^{(k)}$ exists everywhere on the interval $(a,b)$. Then there exists $\xi\in(a,b)$ such that $\mbox{$\Delta\\!\\!\\!\\!\\!\;\raisebox{2.15277pt}{\mbox{\boldmath$\scriptscriptstyle\mid$}}\,\,$}_{k}(p_{1},\ldots,p_{k+1})=\frac{f^{(k)}(\xi)}{k!}.$ We will also need the following result (see, e.g., [Phi03, Eq. 1.11–1.19]). ###### Lemma 2.2 (Newton’s interpolation) Let $p_{1},\ldots,p_{k+1}\in{\mathbb{R}}^{2}$ be points with distinct $x$-coordinates (here we need not assume that the $x$-coordinates are increasing). Then the unique polynomial $f$ of degree at most $k$ whose graph contains $p_{1},\ldots,p_{k+1}$ is given by $f(x)=\sum_{i=1}^{k+1}\biggl{(}\mbox{$\Delta\\!\\!\\!\\!\\!\;\raisebox{2.15277pt}{\mbox{\boldmath$\scriptscriptstyle\mid$}}\,\,$}(p_{1},\ldots,p_{i})\prod_{j=1}^{i-1}(x-x_{j})\biggr{)}$ In particular, the coefficient of $x^{k}$ is $\mbox{$\Delta\\!\\!\\!\\!\\!\;\raisebox{2.15277pt}{\mbox{\boldmath$\scriptscriptstyle\mid$}}\,\,$}(p_{1},\ldots,p_{k+1})$, and it equals $f^{(k)}(x)/k!$ (which is a constant function). We recall that a $(k+1)$-tuple $K=\\{p_{1},\ldots,p_{k+1}\\}$ was defined to be positive if it is contained in the graph of a function having a nonnegative $k$th derivative everywhere. We obtain the following equivalent characterization: ###### Corollary 2.3 A $(k+1)$-tuple $K=\\{p_{1},\ldots,p_{k+1}\\}$ is positive iff $\mbox{$\Delta\\!\\!\\!\\!\\!\;\raisebox{2.15277pt}{\mbox{\boldmath$\scriptscriptstyle\mid$}}\,\,$}_{k}(p_{1},\ldots,p_{k+1})\geq 0$ (and similarly for a negative $(k+1)$-tuple). If $K$ is in $k$-general position, we have $\mathop{\rm sgn}\nolimits K=\mathop{\rm sgn}\nolimits\mbox{$\Delta\\!\\!\\!\\!\\!\;\raisebox{2.15277pt}{\mbox{\boldmath$\scriptscriptstyle\mid$}}\,\,$}_{k}(p_{1},\ldots,p_{k+1})$. Proof. If $K$ is contained in the graph of $f$ with $f^{(k)}\geq 0$ everywhere, then $\mbox{$\Delta\\!\\!\\!\\!\\!\;\raisebox{2.15277pt}{\mbox{\boldmath$\scriptscriptstyle\mid$}}\,\,$}_{k}(p_{1},\ldots,p_{k+1})\geq 0$ by Lemma 2.1. Conversely, if $\mbox{$\Delta\\!\\!\\!\\!\\!\;\raisebox{2.15277pt}{\mbox{\boldmath$\scriptscriptstyle\mid$}}\,\,$}_{k}(p_{1},\ldots,p_{k+1})\geq 0$, then by Lemma 2.2, the unique polynomial of degree at most $k$ whose graph contains $K$ is the required function with nonnegative $k$th derivative. If, moreover, $K$ is in $k$-general position, then $\mbox{$\Delta\\!\\!\\!\\!\\!\;\raisebox{2.15277pt}{\mbox{\boldmath$\scriptscriptstyle\mid$}}\,\,$}_{k}(p_{1},\ldots,p_{k+1})\neq 0$, and so $K$ cannot be both $k$th-order positive and $k$th-order negative by Lemma 2.1. $\Box$ We will also need the following criterion for the sign of a $(k+1)$-tuple. ###### Lemma 2.4 Let $K=\\{p_{1},p_{2},\ldots,p_{k+1}\\}$ be a $(k+1)$-tuple of points in $k$-general position, $x_{1}<\cdots<x_{k+1}$, let $i\in[k+1]$, and let $f_{i}$ be the (unique) polynomial of degree at most $k-1$ whose graph passes through the points of $K\setminus\\{p_{i}\\}$. Then $\mathop{\rm sgn}\nolimits K=(-1)^{k-i}$ if $p_{i}$ lies below the graph of $f_{i}$, and $\mathop{\rm sgn}\nolimits K=(-1)^{k+1-i}$ if $p_{i}$ lies above the graph. Let $f$ be the polynomial of degree at most $k$ passing through all of $K$. We use Newton’s interpolation (Lemma 2.2), but with the points reordered so that $p_{i}$ comes last, and we get that $f(x)=f_{i}(x)+\mbox{$\Delta\\!\\!\\!\\!\\!\;\raisebox{2.15277pt}{\mbox{\boldmath$\scriptscriptstyle\mid$}}\,\,$}_{k}(p_{1},\ldots,p_{i-1},p_{i+1},\ldots,p_{k+1},p_{i})\prod_{j\in[k+1]\setminus\\{i\\}}(x-x_{j}).$ Using this with $x=x_{i}$, we get $\displaystyle\mathop{\rm sgn}\nolimits(y_{i}-f_{i}(x_{i}))$ $\displaystyle=$ $\displaystyle\mathop{\rm sgn}\nolimits(f(x_{i})-f_{i}(x_{i}))$ $\displaystyle=$ $\displaystyle\mathop{\rm sgn}\nolimits\mbox{$\Delta\\!\\!\\!\\!\\!\;\raisebox{2.15277pt}{\mbox{\boldmath$\scriptscriptstyle\mid$}}\,\,$}_{k}(p_{1},\ldots,p_{i-1},p_{i+1},\ldots,p_{k+1},p_{i})\cdot\mathop{\rm sgn}\nolimits\prod_{j\in[k+1]\setminus\\{i\\}}(x_{i}-x_{j}).$ Divided differences are invariant under permutations of the points (as can be seen, e.g., from Lemma 2.2, since the interpolating polynomial does not depend on the order of the points), and so $\mathop{\rm sgn}\nolimits\mbox{$\Delta\\!\\!\\!\\!\\!\;\raisebox{2.15277pt}{\mbox{\boldmath$\scriptscriptstyle\mid$}}\,\,$}_{k}(p_{1},\ldots,p_{i-1},p_{i+1},\ldots,p_{k+1},p_{i})=\mathop{\rm sgn}\nolimits K$. Finally, the product $\prod_{j\in[k+1]\setminus\\{i\\}}(x_{i}-x_{j})$ has $k+1-i$ negative factors, thus its sign is $(-1)^{k+1-i}$, and the lemma follows. $\Box$ It remains to prove transitivity. ###### Lemma 2.5 Let $P=\\{p_{1},\ldots,p_{N}\\}$ be a point set in $k$-general position. Then the $2$-coloring of $(k+1)$-tuples $K\in{P\choose k+1}$ by their sign is transitive. Proof. We consider a $(k+2)$-tuple $L=\\{p_{1},\ldots,p_{k+2}\\}$ with $\mathop{\rm sgn}\nolimits\\{p_{1},\ldots,p_{k+1}\\}=\mathop{\rm sgn}\nolimits\\{p_{2},\ldots,p_{k+2}\\}=+1$, and we fix $i\in\\{2,\ldots,{k+1}\\}$. Let $f_{i,k+2}$ be the polynomial of degree at most $k-1$ passing through $L\setminus\\{p_{i},p_{k+2}\\}$, and similarly for $f_{1,k+2}$. Our goal is to show that $f_{i,k+2}(x_{k+2})<y_{k+2}$, since this gives $\mathop{\rm sgn}\nolimits(L\setminus\\{p_{i}\\})=+1$ by Lemma 2.4. Since $\mathop{\rm sgn}\nolimits(L\setminus\\{p_{1}\\})=+1$, we have $f_{1,k+2}(x_{k+2})<y_{k+2}$ (Lemma 2.4 again), and so it suffices to prove $f_{i,k+2}(x_{k+2})<f_{1,k+2}(x_{k+2})$. Let us consider the polynomial $g:=f_{1,k+2}-f_{i,k+2}$; as explained above, our goal is proving $\mathop{\rm sgn}\nolimits g(x_{k+2})=+1$. To this end, we first determine $\mathop{\rm sgn}\nolimits g(x_{1})$: We have $f_{i,k+2}(x_{1})=y_{1}$ and $\mathop{\rm sgn}\nolimits(y_{1}-f_{1,k+2}(x_{1}))=(-1)^{k}$ (using $\mathop{\rm sgn}\nolimits(L\setminus\\{p_{1}\\})=+1$ and Lemma 2.4). Hence $\mathop{\rm sgn}\nolimits g(x_{1})=(-1)^{k-1}$. Next, we observe that $g$ is a polynomial of degree at most $k-1$, and it vanishes at $x_{2},\ldots,x_{i-1},x_{i+1},\ldots,x_{k+1}$. These are $k-1$ distinct values; thus, they include all roots of $g$, and each of them is a simple root. Consequently, $g$ changes sign $(k-1)$-times between $x_{1}$ and $x_{k+2}$. Hence, finally, $\mathop{\rm sgn}\nolimits g(x_{k+2})=(-1)^{k-1}\mathop{\rm sgn}\nolimits g(x_{1})=+1$ as claimed. $\Box$ Rote’s example. Fig. 1 shows a 6-point set $P=\\{p_{1},\ldots,p_{6}\\}$ in 3-general position (no four points on a parabola). It is easy to check 3rd- order positivity using Lemma 2.4: By transitivity, it suffices to look at $4$-tuples of consecutive points. For $p_{1},\ldots,p_{4}$ we use the parabola through $p_{1},p_{2},p_{3}$ (which actually degenerates to the $x$-axis); for $p_{2},\ldots,p_{5}$ we use the dashed parabola through $p_{2},p_{3},p_{4}$ (which is very close to the $x$-axis in the relevant region); and for $p_{3},\ldots,p_{6}$, the parabola through $p_{4},p_{5},p_{6}$ (drawn full). Figure 1: Rote’s example: a 6-point 3rd-order positive set in 3-general position that does not lie on the graph of any function with nonnegative 3rd derivative. It remains to check that $P$ does not lie on the graph of a function $f$ with $f^{(3)}\geq 0$ everywhere. Assuming for contradiction that there is such an $f$, we consider the point $q:=(x_{0},f(x_{0}))$, where $x_{0}$ is such that the full parabola is below the $x$-axis at $x_{0}$. For the $4$-tuple $\\{p_{1},p_{2},p_{3},q\\}$ to be positive, $q$ has to lie above the $x$-axis, but the $4$-tuple $\\{q,p_{4},p_{5},p_{6}\\}$ is positive only if $q$ lies below the parabola through $p_{4},p_{5},p_{6}$—a contradiction. ## 3 Upper bounds on the Ramsey numbers for transitive colorings In this section we prove Theorem 1.4. As we mentioned in the remark following that theorem, it suffices to establish the case $k\geq 3$. Thus, we want to prove that $R^{\rm trans}_{k+1}(n)\leq\mathop{\rm twr}\nolimits_{k}(C_{k}n)$ for all $n$ and for every $k\geq 3$, with suitable constants $C_{k}$ depending on $k$. As the base of the induction we use $R^{\rm trans}_{3}(n)\leq 4^{n}$, which, as was remarked earlier, follows by imitating the proof of Theorem 1.2. Thus, let $k\geq 3$ be fixed, let $n$ be given, and let us set $M:=R^{\rm trans}_{k}(n)$. We will prove that $R^{\rm trans}_{k+1}(n)\leq N:=2^{M^{k}}.$ (1) Theorem 1.4 then follows from this recurrence and from the fact that $2^{\mathop{\rm twr}\nolimits_{k-1}(n)^{k}}\leq\mathop{\rm twr}\nolimits_{k}(kn)$ for $k\geq 3$, which is easy to check. To prove (1), we follow an inductive proofs of Ramsey’s theorem going back to Erdős and Rado [ER52]. Let $\chi\colon{[N]\choose k+1}\to\\{1,2\\}$ be an arbitrary transitive 2-coloring. We set $A_{k-1}:=\\{1,2,\ldots,k-1\\}$ and $X_{k-1}:=[N]\setminus A_{k-1}$. For $i=k,k+1,\ldots,M$ we will inductively construct sets $A_{i},X_{i}\subseteq[N]$ such that 1. (i) $A_{i}<X_{i}$ (i.e., all elements of $A_{i}$ precede all elements of $X_{i}$); 2. (ii) $\|A_{i}\|=i$ and $\|X_{i}\|\geq\|X_{i-1}\|/2^{M^{k-1}}$; and 3. (iii) the color of a $(k+1)$-tuple whose first $k$ elements all belong to $A_{i}$ does not depend on its last element; in other words, for $K\in{A_{i}\choose k}$ and $x,y\in A_{i}\cup X_{i}$ with $K<\\{x,y\\}$, we have $\chi(K\cup\\{x\\})=\chi(K\cup\\{y\\})$. For the inductive step, suppose that $A_{i}$ and $X_{i}$ have already been constructed. We let $x_{i}$ be the smallest element of $X_{i}$, we set $A_{i+1}:=A_{i}\cup\\{x_{i}\\}$, and we write $X^{\prime}_{i}:=X_{i}\setminus\\{x_{i}\\}$. Let us call two elements $x,y\in X^{\prime}_{i}$ _equivalent_ if we have, for every $K\in{A_{i-1}\choose k-1}$, $\chi(K\cup\\{x_{i},x\\})=\chi(K\cup\\{x_{i},y\\})$. There are ${i\choose k-1}$ possible choices of $K$, and hence there are at most $2^{i\choose k-1}<2^{M^{k-1}}$ equivalence classes. We choose $X_{i+1}\subseteq X^{\prime}_{i}$ as the largest equivalence class. Then (i), (iii) obviously hold for $A_{i+1}$ and $X_{i+1}$, and we have $\|X_{i+1}\|\geq(\|X_{i}\|-1)/(2^{M^{k-1}}-1)\geq\|X_{i}\|/2^{M^{k-1}}$ (since $i\leq M$ and thus we have $\|X_{i}\|\geq N/(2^{M^{k}-1})^{i-1}=2^{M^{k}-(i-1)M^{k-1}}\geq 2^{M^{k-1}}$). This finishes the inductive construction of $A_{i}$ and $X_{i}$. In this way, we construct the sets $A:=A_{M}$ and $X_{M}$ (note that $\|X_{M}\|\geq 1$ by (ii)). Let $x$ be the first element of $X_{M}$, and let us define a 2-coloring $\chi^{}\colon{A\choose k}\to\\{1,2\\}$ of the $k$-tuples of $A$ by $\chi^{}(K):=\chi(K\cup\\{x\\})$. We claim that, crucially, $\chi^{}$ is transitive (which is not entirely obvious). So we consider elements $a_{1}<a_{2}<\cdots<a_{k+1}$ of $A$, and we suppose that $\chi^{}(\\{a_{1},\ldots,a_{k}\\})=\chi^{}(\\{a_{2},\ldots,a_{k+1}\\})=:c$. We want to show that $\chi^{}(\\{a_{1},\ldots,a_{k+1}\\}\setminus\\{a_{i}\\})=c$ for every $i=2,3,\ldots,k$. We have $c=\chi^{}(\\{a_{1},\ldots,a_{k}\\})=\chi(\\{a_{1},\ldots,a_{k},x\\})=\chi(\\{a_{1},\ldots,a_{k+1}\\})$ (by definition and by the independence of $\chi$ of the last element), and $c=\chi^{}(\\{a_{2},\ldots,a_{k+1}\\})=\chi(\\{a_{2},\ldots,a_{k+1},x\\})$. Next we use the transitivity of $\chi$ on the $(k+2)$-tuple $(a_{1},\ldots,a_{k+1},x)$, obtaining $\chi(\\{a_{1},\ldots,a_{k+1},x\\}\setminus\\{a_{i}\\})=c=\chi^{}(\\{a_{1},\ldots,a_{k+1}\\}\setminus\\{a_{i}\\})$ as needed. Now we can apply the inductive hypothesis to $A$, which yields an $n$-element subset of $A$ homogeneous w.r.t. $\chi^{}$, and this subset is homogeneous w.r.t. $\chi$ as well, finishing the proof of Theorem 1.4. $\Box$ ## 4 A lower bound for $\operatorname{ES}_{3}$ Here we prove Theorem 1.5, a lower bound for $\operatorname{ES}_{3}(2n+1)$. We proceed by induction on $n$; the goal is to construct a set $P_{n}$ of $N:=2^{2^{n-1}}$ points with no $(2n+1)$-point third-order monotone subset. The induction starts for $n=2$ with an arbitrary $P_{2}$ of size $2^{2^{1}}=4$. In the inductive step, given $P_{n}$, we will construct $P_{n+1}$ so that $\|P_{n+1}\|=\|P_{n}\|^{2}$; then the bound on the size of $P_{n}$ clearly holds. We may assume that $P=P_{n}$ is in $3$-general position (this can always be achieved by a small perturbation). By an affine transformation we also make sure that $P\subset[1,2]\times[0,1]$; or actually, $P\subset[1,1.9]\times[0,1]$ so that there is some room for perturbation. Moreover, there is a small $\delta>0$ such that if $P^{\prime}$ is obtained from $P$ by moving each point arbitrarily by at most $\delta$, then $P^{\prime}$ is still in $3$-general position, the order of the points of $P^{\prime}$ along the $x$-axis is the same as that for $P$, and the sign of every $4$-tuple in $P^{\prime}$ is the same as the sign of the corresponding $4$-tuple in $P$. The construction. The construction of $P_{n+1}$ from $P=P_{n}$ as above proceeds in the following steps. 1. 1. We choose a sufficiently large number $A=A(P)$ (the requirements on it will be specified later), and we set $\varepsilon:=1/A^{2}$. 2. 2. For every point $p\in P$, let $Q_{p}$ be the image of $P$ under the affine map that sends the square $[1,2]\times[0,1]$ to the axis-parallel rectangle of width $\varepsilon$, height $\varepsilon^{2}$, and with the lower left corner at $p$; see Fig. 2. 3. 3. Let $\psi_{p}(x)=Ax^{2}+C_{p}$ be a quadratic function, where $A$ is as above and $C_{p}$ is chosen so that $\psi_{p}(x(p))=0$ (where $x(p)$ is the $x$-coordinate of $p$). Let $\breve{Q}_{p}$ be the set obtained by “adding $\psi_{p}$ to $Q_{p}$”, i.e., by shifting each point $(x,y)\in Q_{p}$ vertically upwards by $\psi_{p}(x)$. We set $P_{n+1}:=\bigcup_{p\in P}\breve{Q}_{p}$. We call the $\breve{Q}_{p}$ the _clusters_ of $P_{n+1}$. Figure 2: A schematic illustration of the construction of $P_{n+1}$. First we check that each cluster $\breve{Q}_{p}$ lies close to $p$. ###### Lemma 4.1 Each $\breve{Q}_{p}$ is contained in an $O(\sqrt{\varepsilon}\,)$-neighborhood of $p$. Proof. Writing $p=(x_{0},y_{0})$, the set $Q_{p}$ obviously lies in the $2\varepsilon$-neighborhood of $p$, and the maximum amount by which a point of $Q_{p}$ was translated upwards is at most $\psi_{p}(x_{0}+\varepsilon)=A\left((x_{0}+\varepsilon)^{2}-x_{0}^{2}\right)=A(2x_{0}\varepsilon+\varepsilon^{2})=O(\sqrt{\varepsilon}\,).$ $\Box$ Here is a key property of the construction. ###### Lemma 4.2 (Slope lemma) Let $\lambda$ be a parabola passing through three points of $P_{n+1}$ that belong to three different clusters, or a line passing through two points of different clusters. Let $\mu$ be a parabola passing through three points of a single cluster $\breve{Q}_{p}$, or a line passing through two such points. Then the maximum slope (first derivative) of $\lambda$ on the interval $[1,2]$ is smaller than the minimum slope of $\mu$ on $[1,2]$, provided that $A$ was chosen sufficiently large. Proof. Clearly, the maximum slope of any such $\lambda$ can be bounded from above by some finite number depending only on $P$ but not on $A$. Thus, it suffices to show that, with $A$ large, for every $\mu$ as in the lemma, the minimum slope is bounded from below by $A$. First let us assume that $\mu$ is a parabola passing through three points of $\breve{Q}_{p}$, where $p=(x_{0},y_{0})$, let $\tilde{\mu}$ be the parabola passing through the corresponding three points of $P$, and let the equation of $\tilde{\mu}$ be $y=ax^{2}+bx+c$. By the construction of $\breve{Q}_{p}$, the affine map transforming $P$ to $Q_{p}$ sends a point with coordinates $(x,y)$ to the point $(\varepsilon(x-1)+x_{0},\varepsilon^{2}y+y_{0})$. Calculation shows that the image of $\tilde{\mu}$ under this affine map has the equation $y=ax^{2}+(2a\varepsilon+b\varepsilon-2ax_{0})x+c^{\prime}$, where the value of the absolute term $c^{\prime}$ need not be calculated since it doesn’t matter. Hence the minimum slope of this curve on $[1,2]$ is bounded from below by $-(8\|a\|+4\|a\|\varepsilon+2\|b\|\varepsilon+8\|a\|)$. Finally, $\mu$ is obtained by adding $\psi_{p}(x)=Ax^{2}+C_{p}$ to this curve, and the minimum slope of $\psi_{p}$ on $[1,2]$ is at least $2A$. Next, let $\mu$ be a line passing through two points $q,r\in\breve{Q}_{p}$. Let us choose another point $s\in\breve{Q}_{p}$ and consider the parabola $\mu^{\prime}$ through $q,r,s$. By the Mean Value Theorem, the slope of $\mu$ equals the slope of $\mu^{\prime}$ at some point between $q$ and $r$, and the latter is at least $A$ by the above. The lemma is proved. $\Box$ Let $K=\\{p_{1},p_{2},p_{3},p_{4}\\}\subseteq P_{n+1}$ be a 4-tuple, $p_{i}=(x_{i},y_{i})$, $x_{1}<\cdots<x_{4}$. We assign a _type_ to $K$, which is an ordered partition of $4$ given by the distribution of $K$ among the clusters; for example, $K$ has type $1+1+2$ if the first point $p_{1}$ lies in some $\breve{Q}_{p}$, $p_{2}$ lies in $\breve{Q}_{p^{\prime}}$ for $p^{\prime}\neq p$, and $p_{3},p_{4}\in\breve{Q}_{p^{\prime\prime}}$, $p^{\prime\prime}\neq p,p^{\prime}$. The next lemma shows that the sign $K$ is determined by its type. We provide a complete classification, although we will not use all of the types in the subsequent proof. ###### Lemma 4.3 Let $K=\\{p_{1},p_{2},p_{3},p_{4}\\}\subseteq P_{n+1}$ be a $4$-tuple. If $K$ is of type $1+1+1+1$ or $4$, then the sign of $K$ is the same as that of the corresponding $4$-tuple in $P$. Otherwise, the sign of $K$ is determined by its type as follows: * • for types $3+1$ and $1+3$ it is $-1$; * • for types $1+1+2$ and $2+1+1$ it is $+1$; * • for type $1+2+1$ it is $-1$; and * • for type $2+2$ it is $+1$. Proof. Since the transformation that converts $P$ into $\breve{Q}_{p}$ preserves the types of $4$-tuples, the statement for type 4 is clear. The statement for type $1+1+1+1$ follows since, by Lemma 4.1, $K$ is obtained by a sufficiently small perturbation of the corresponding $4$-tuple in $P$ (this gives one of the lower bounds on $A$, since we need the bound in Lemma 4.1 to be smaller than the $\delta$ considered at the beginning of our description of the construction). The statements for the remaining types are obtained by simple application of the slope lemma (Lemma 4.2) together with Lemma 2.4. Namely, for type $3+1$, we get that the parabola through $p_{1},p_{2},p_{3}$ lies above $p_{4}$ (by comparing its slope to the slope of the line $p_{3}p_{4}$); see Fig. 3. For type $1+3$ we similarly get that $p_{1}$ lies above the parabola through $p_{2},p_{3},p_{4}$, and so the sign is $-1$ in both of these cases. Figure 3: Determining the signs of $4$-tuples by type. For type $1+1+2$, the segment $p_{3}p_{4}$ is steeper than the parabola through $p_{1}p_{2}p_{3}$, and so the sign is $+1$. Similarly for type $2+1+1$ we get that $p_{1}$ lies below the parabola through $p_{2},p_{3},p_{4}$, which again gives sign $+1$. For type $1+2+1$, $p_{3}$ lies above the parabola through $p_{1},p_{2},p_{4}$, giving sign $-1$. Finally, for type $2+2$, the segment $p_{1}p_{2}$ is steeper than $p_{2}p_{3}$, thus the parabola through $p_{1},p_{2},p_{3}$ is concave, and hence its slope at $p_{3}$ and after it is no larger than the slope of the segment $p_{2}p_{3}$. Thus, $p_{4}$ lies above this parabola and the sign is $+1$ as claimed. $\Box$ Finishing the proof of Theorem 1.5. It remains to show that $P_{n+1}$ contains no $(2n+3)$-point third-order monotone subset. For contradiction, suppose that $M\subseteq P_{n+1}$ is such a $(2n+3)$-point subset. Let $2n+3=n_{1}+n_{2}+\cdots+n_{s}$ be the type of $M$ (i.e., $M$ has $n_{i}\geq 1$ points in the $i$th leftmost cluster it intersects). By the inductive assumption we have $s\leq 2n$ and $n_{i}\leq 2n$ for all $i$. Let $n_{a}=\max_{i}n_{i}$ and $n_{b}=\max_{i\neq a}n_{i}$ be the two largest among the $n_{i}$. For convenience, let us assume $a<b$; the case $a>b$ is handled symmetrically. We distinguish three cases. First, if $n_{a}\geq 3$ and $n_{b}\geq 2$, then we can select 4-tuples of types $3+1$ and $2+2$ from the corresponding two clusters, which have different signs, and so $M$ is not homogeneous. Second, if $n_{a}\geq 3$ and $n_{b}=1$, then we have at least three $n_{i}$ equal to 1 (since $n_{a}\leq 2n$), and at least two of them lie on the same side of the cluster corresponding to $n_{a}$, say to the right of it. Then we can select 4-tuples of types $3+1$ and $2+1+1$, again of opposite signs. Third, if $n_{a}=2$, then there are at least two other clusters of size 2. From these three 2-element clusters, we can select 4-tuples of types 2+2 and 1+2+1, again of opposite signs. This exhausts all possibilities ($n_{a}=1$ cannot happen, because $s\leq 2n$), and Theorem 1.5 is proved. $\Box$ ## 5 Order types and one-sided sets of hyperplanes First we substantiate the two claims made above Proposition 1.6, concerning $\mathop{\rm OT}\nolimits_{1}$ and $\mathop{\rm OT}\nolimits_{2}$. For $d=1$, an order-type homogeneous sequence in ${\mathbb{R}}^{1}$ is just a monotone sequence of real numbers, so $\mathop{\rm OT}\nolimits_{1}(n)=(n-1)^{2}+1$ by Theorem 1.1. In a similar spirit, it is easy to check that a planar order-type homogeneous sequence corresponds to the vertices of a convex $n$-gon, enumerated in a clockwise or counterclockwise order. Thus, $\mathop{\rm OT}\nolimits_{2}(n)\geq\operatorname{ES}_{2}(\lceil n/2\rceil)=2^{\Omega(n)}$. On the other hand, given any $N$-point sequence, we can first select a subsequence of $\lceil\sqrt{N}\,\rceil$ points with increasing or decreasing $x$-coordinates, and then we select a convex or concave configuration from it. Thus, by Theorem 1.2, we have $\mathop{\rm OT}\nolimits_{2}(n)=2^{O(n)}$. Proof of Proposition 1.6. For a point $p=(x,y)\in{\mathbb{R}}^{2}$, we define the point $\tilde{p}:=(x,x^{2},\ldots,x^{d-1},y)\in{\mathbb{R}}^{d}$. To prove that $\operatorname{ES}_{d}(n)\leq\mathop{\rm OT}\nolimits_{d}(n)$, we consider a set $P=\\{p_{1},\ldots,p_{N}\\}\subset{\mathbb{R}}^{2}$ in $d$-general position, $p_{i}=(x_{i},y_{i})$, where $N=\operatorname{ES}_{d}(n)-1$ and $x_{1}<\cdots<x_{N}$, with no $d$th-order monotone subset of $n$ points. It suffices to prove that the sequence $\tilde{P}:=(\tilde{p}_{1},\tilde{p}_{2},\ldots,\tilde{p}_{N})$ has no $n$-point order-type homogeneous subsequence. This follows from the next lemma. ###### Lemma 5.1 For every $(d+1)$-tuple $(p_{1},\ldots,p_{d+1})$ of points in ${\mathbb{R}}^{2}$, $x_{1}<\cdots<x_{d+1}$, we have $\mathop{\rm sgn}\nolimits(\\{p_{1},\ldots,p_{d+1}\\})=\mathop{\rm sgn}\nolimits\det M(\tilde{p}_{1},\tilde{p}_{2},\ldots,\tilde{p}_{d+1})$, where $M(q_{1},\ldots,q_{d+1})$ is the matrix from the definition of order type above Proposition 1.6. Proof. By Lemma 2.2 and Corollary 2.3, the sign of $\\{p_{1},\ldots,p_{d+1}\\}$ equals the sign of the coefficient $a_{d}$ of the unique polynomial $f(x)=\sum_{j=0}^{d}a_{j}x^{j}$ of degree at most $d$ whose graph passes through the points $p_{1},\ldots,p_{d+1}$. The vector $a=(a_{0},\ldots,a_{d})$ can be expressed as the solution of the linear system $Va=y$, where $y=(y_{1},\ldots,y_{d+1})$ and $V$ is the _Vandermonde matrix_ with $v_{ij}=x_{i}^{j-1}$, $i,j=1,2,\ldots,d+1$. By Cramer’s rule, we obtain $a_{d}=\frac{\det W}{\det V},$ where $W$ stands for the matrix $V$ with the last column replaced with the vector $y$. As is well known, $\det V=\prod_{1\leq i<j\leq d+1}(x_{j}-x_{i})$, and since $x_{1}<\cdots<x_{d+1}$, we have $\det V>0$. Thus, $\mathop{\rm sgn}\nolimits a_{d}=\mathop{\rm sgn}\nolimits\det W$. Finally, we have $W=\left(\begin{array}[]{cccccc}1&x_{1}&x_{1}^{2}&\ldots&x_{1}^{d-1}&y_{1}\\\ \vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\\ 1&x_{d+1}&x_{d+1}^{2}&\ldots&x_{d+1}^{d-1}&y_{d+1}\end{array}\right)=M(\tilde{p}_{1},\tilde{p}_{2},\ldots,\tilde{p}_{d+1})^{T}.$ The lemma follows, and Proposition 1.6 is proved. $\Box$ Proof of Proposition 1.7. The proof is very similar to the previous one. This time we start with a set $P=\\{p_{1},\ldots,p_{N}\\}\subset{\mathbb{R}}^{2}$ in $(d-1)$-general position, $p_{i}=(x_{i},y_{i})$, where $N=\operatorname{ES}_{d-1}(n)-1$ and $x_{1}<\cdots<x_{N}$, with no $(d-1)$th- order monotone subset of $n$ points. We define a collection $H=\\{h_{1},\ldots,h_{N}\\}$ of $N$ hyperplanes in ${\mathbb{R}}^{d}$, where $h_{i}$ is given by $h_{i}=\biggl{\\{}(\xi_{1},\ldots,\xi_{d})\in{\mathbb{R}}^{d}:\sum_{j=1}^{d}x_{i}^{j-1}\xi_{j}=y_{i}\biggr{\\}}.$ The intersection point $\xi=(\xi_{1},\ldots,\xi_{d})$ of, say, $h_{1},\ldots,h_{d}$ is the solution of the linear system $V\xi=y$, where $V$ is the $d\times d$ Vandermonde matrix this time, $v_{ij}=x_{i}^{j-1}$. Cramer’s rule then gives that the $d$th coordinate $\xi_{d}$, whose sign we are interested in, equals $(\det W)/(\det V)$, where $W$ is obtained from $V$ by replacing the last column with $y$. As we saw in the proof of Proposition 1.6, $(\det W)/(\det V)$ also expresses the leading coefficient in the polynomial of degree $d-1$ passing through $p_{1},\ldots,p_{d}$, and thus its sign equals $\mathop{\rm sgn}\nolimits\mbox{$\Delta\\!\\!\\!\\!\\!\;\raisebox{2.15277pt}{\mbox{\boldmath$\scriptscriptstyle\mid$}}\,\,$}_{d-1}(p_{1},\ldots,p_{d})$. It follows that one-sided subsets of $H$ precisely correspond to $(d-1)$st- order monotone subsets in $P$, and the proposition is proved. $\Box$ ## Acknowledgment We would like to thank János Pach for kindly discussing some of the results of Fox et al. [FPSS11] with us. We also thank Günter Rote for informing us about about his refutation of our conjecture and for permission to present it in this paper. ## References * [CFS11] D. Conlon, J. Fox, and B. Sudakov. An improved bound for the stepping-up lemma. Discrete Applied Mathematics, 2011. In press. * [DL11] V. Dujmović and S. Langerman. A center transversal theorem for hyperplanes and applications to graph drawing. In Proc. 27th ACM Symposium on Computational Geometry, pages 117–124, 2011. Full version arXiv:1012.0548. * [ER52] P. Erdős and R. Rado. Combinatorial theorems on classifications of subsets of a given set. Proc. London Math. Soc., 3:417–439, 1952. * [ES35] P. Erdős and G. Szekeres. A combinatorial problem in geometry. Compositio Math., 2:463–470, 1935. * [FPSS11] J. Fox, J. Pach, B. Sudakov, and A. Suk. Erdős–Szekeres-type theorem for monotone paths and convex bodies. Arxiv preprint 1105.2097v1, 2011. _Proc. London Math. Soc._ , in press. * [GP93] J. E. Goodman and R. Pollack. Allowable sequences and order types in discrete and computational geometry. In J. Pach, editor, New Trends in Discrete and Computational Geometry, volume 10 of Algorithms and Combinatorics, pages 103–134. Springer, Berlin etc., 1993. * [GRS90] R. L. Graham, B. L. Rothschild, and J. Spencer. Ramsey Theory. J. Wiley & Sons, New York, 1990. * [Mat02] J. Matoušek. Lectures on Discrete Geometry. Springer, New York, 2002. * [MS00] W. Morris and V. Soltan. The Erdős–Szekeres problem on points in convex position—a survey. Bull. Amer. Math. Soc., New Ser., 37(4):437–458, 2000. * [MW92] J. Matoušek and Emo Welzl. Good splitters for counting points in triangles. J. Algorithms, 13:307–319, 1992. * [Phi03] George M. Phillips. Interpolation and approximation by polynomials. Springer, Berlin etc., 2003. * [Rot12] G. Rote. Private communication, February 2012. * [Ste95] M. J. Steele. Variations on the monotone subsequence theme of Erdős and Szekeres. In D. Aldous et al., editors, Discrete Probability and Algorithms, IMA Volumes in Mathematics and its Applications 72, pages 111–131. Springer, Berlin etc., 1995. * [Val04] P. Valtr. Open caps and cups in planar point sets. Discr. Comput. Geom., 37:365–567, 2004.	arxiv-papers	2011-11-16T14:55:15	2024-09-04T02:49:24.387802	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Marek Elias, Jiri Matousek", "submitter": "Marek Elias", "url": "https://arxiv.org/abs/1111.3824" }
1111.3828	Curves on Oeljeklaus-Toma Manifolds Sima Verbitsky Abstract Oeljeklaus-Toma manifolds are complex non-Kähler manifolds constructed by Oeljeklaus and Toma from certain number fields, and generalizing the Inoue surfaces $S_{m}$. We prove that Oeljeklaus-Toma manifolds contain no compact complex curves. ###### Contents 1. 1 Introduction 1. 1.1 Oeljeklaus-Toma manifolds 2. 2 Curves on the Oeljeklaus-Toma manifolds 1. 2.1 The exact semipositive (1,1)-form on the Oeljeklaus-Toma manifold 2. 2.2 The $(1,1)$-form $\omega$ and curves on the Oeljeklaus-Toma manifold 3. 3 Closing remarks ## 1 Introduction Oeljeklaus-Toma manifolds (defined in [O–T]) are compact complex manifolds that are a generalization of Inoue surfaces (defined in [I]). Let us describe them in detail. ### 1.1 Oeljeklaus-Toma manifolds Let $K$ be a number field (i.e. a finite extension of $\mathbb{Q}$), $s>0$ be the number of its real embeddings and $2t>0$ be the number of its complex embeddings. One can easily prove that for each $s$ and $t$ there exists a field $K$ which has these numbers of real and complex embeddings (see e.g. [O–T]). Definition 1.1: The ring of algebraic integers $O_{K}$ is a subring of $K$ that consists of all roots of polynomials with integer coefficients which lie in $K$. Unit group $O^{}_{K}$ is the multiplicative subgroup of invertible elements of $O_{K}$. Let $m$ be $s+t$. Let $\sigma_{1},\ldots,\sigma_{s}$ be real embeddings of the field $K$, $\sigma_{s+1},\ldots,\sigma_{s+2t}$ be complex embeddings such that $\sigma_{s+i}$ and $\sigma_{s+t+i}$ are complex conjugate for each $i$ from $1$ to $t$. Now we can define a map $l:O_{K}^{}\rightarrow\mathbb{R}^{m}$ where $l(u)=(\ln\|\sigma_{1}(u)\|,\ldots,\ln\|\sigma_{s}(u)\|,2\ln\|\sigma_{s+1}(u)\|,\ldots,2\ln\|\sigma_{m}(u)\|)$. Denote $O_{K}^{,+}=\\{a\in O_{K}^{}:\sigma_{i}(a)>0,i=1,\ldots,s\\}$. Let us consider following definitions: Definition 1.2: A lattice $\Lambda$ in $\mathbb{R}^{n}$ is a discrete additive subgroup such that $\Lambda\otimes\mathbb{R}=\mathbb{R}^{n}$. Definition 1.3: [O–T] The group $U\subset O_{K}^{,+}$ of rank $s$ is called admissible for the field $K$ if the projection of $l(U)$ to the first $s$ components is a lattice in $\mathbb{R}^{s}$. Consider a linear space $L=\\{x\in\mathbb{R}^{m}\mid\sum_{i=1}^{m}x_{i}=0\\}$. The projection of $L\subset R^{m}$ to the first $s$ coordinates is surjective, because $s<m$. Using the Dirichlet unit theorem (see e.g. [Mil09]) one can prove that $l(O_{K}^{,+})$ is a full lattice in $L$. Therefore there exists a group $U$ that is admissible. Let $\mathbb{H}=\\{z\in\mathbb{C}\mid\operatorname{im}z>0\\}$. Let $U\subset O_{K}^{,+}$ be a group which is admissible for $K$. The group $U$ acts on $O_{K}$ multiplicatively. This defines a structure of semidirect product $U^{\prime}:=U\ltimes O_{K}$. Define the action of $U^{\prime}$ on $\mathbb{H}^{s}\times\mathbb{C}^{t}$ as follows. The element $u\in U$ acts on $\mathbb{H}^{s}\times\mathbb{C}^{t}$ mapping $(z_{1},\ldots,z_{m})$ to $(\sigma_{1}(u)z_{1},\ldots,\sigma_{m}(u)z_{m})$. Since $U$ lies in $O_{K}^{,+}$, the action $U$ on the first $s$ coordinates preserves $\mathbb{H}$. The additive group $O_{K}$ acts on $\mathbb{H}^{s}\times\mathbb{C}^{t}$ by parallel translations: $a\in O_{K}$ is mapping $(z_{1},\ldots,z_{m})$ to $(\sigma_{1}(a)+z_{1},\ldots,\sigma_{m}(a)+z_{m})$. Since the first $s$ embeddings are real, this action preserves $\mathbb{H}$ in the first $s$ coordinates. One can see that $(u,a)\in U\ltimes O_{K}$ maps $(z_{1},\ldots,z_{m})$ to $(\sigma_{1}(u)z_{1}+\sigma_{1}(a),\ldots,\sigma_{m}(u)z_{m}+\sigma_{m}(a))$. One can easily show that this action is compatible with the group operation in the semidirect product. Definition 1.4: An Oeljeklaus-Toma manifold is the quotient of $\mathbb{H}^{s}\times\mathbb{C}^{t}$ by the action of the group $U\ltimes O_{K}$, which was defined above. This quotient exists because $U\ltimes O_{K}$ acts properly discontiniously on $\mathbb{H}^{s}\times\mathbb{C}^{t}$. Additionally $\mathbb{H}^{s}\times\mathbb{C}^{t}/U\ltimes O_{K}$ is a compact complex manifold. To prove it, let $U$ be admissible for $K$. The quotient $\mathbb{H}^{s}\times\mathbb{C}^{t}/O_{K}$ is obviously diffeomorphic to the trivial toric bundle $(\mathbb{R}_{>0})^{s}\times(S^{1})^{n}$. The group $U$ acts properly discontinuously on the base $(\mathbb{R}_{>0})^{s}$. Therefore it acts properly discontinuously on $\mathbb{H}^{s}\times\mathbb{C}^{t}/O_{K}$. Also, the groups $U$ and $O_{K}$ act holomorphically on $\mathbb{H}^{s}\times\mathbb{C}^{t}$. Therefore the quotient has a holomorphic structure. ## 2 Curves on the Oeljeklaus-Toma manifolds In this section we shall prove that there are no complex curves on the Oeljeklaus-Toma manifolds, just as on Inoue surfaces of type $S_{M}$ (see [I]). ### 2.1 The exact semipositive (1,1)-form on the Oeljeklaus-Toma manifold The $(1,1)$-form we will be using was previously introduced in the paper Subvarieties in Oeljeklaus-Toma manifolds by Ornea and Verbitsky. Authors use this form to prove that Oeljeklaus-Toma manifolds with $t=1$ (that means that the corresponding number field has only two complex embeddings) do not contain any submanifolds. Our result works for all Oeljeklaus-Toma manifolds, but we consider only curves instead of submanifolds of any dimension. Later on we will explain how our method differs from the one in [O–V] and how this implies the difference between our results. Define the notion of $(1,1)$-form. Let $M$ be a smooth complex manifold, $z_{1},\ldots,z_{n}$ — local complex coordinates in the open neighborhood of the point $y\in M$. Definition 2.1: A $(1,1)$-form on a complex manifold $M$ is a 2-form $\omega$, such that $\omega(Iu,v)=-\omega(u,Iv)=\sqrt{-1}\omega(u,v)$ for each $u,v\in T_{y}M$, where $I$ is the almost complex structure on $M$. Definition 2.2: A $(1,1)$-form $\omega$ on a complex manifold $M$ is semipositive if $\omega(u,Iu)\geqslant 0$ for each tangent vector $u\in T_{y}M$. As in [O–V], we consider a certain semipositive $(1,1)$-form on the Oeljeklaus-Toma manifold $M=\mathbb{H}^{s}\times\mathbb{C}^{t}/(U\ltimes O_{K})$. We introduce a $(1,1)$-form $\widetilde{\omega}$ on $\widetilde{M}=\mathbb{H}^{s}\times\mathbb{C}^{t}$ which is preserved by the action of the group $\Gamma=(U\ltimes O_{K})$ and since then it would be a $(1,1)$-form on $M$. Let $(z_{1},\ldots,z_{m})$ be complex coordinates on $\widetilde{M}$. Define $\varphi(z)=\Pi_{i=1}^{s}\operatorname{im}(z_{i})^{-1}$. Since the first $s$ components of $\widetilde{M}$ correspond to upper half-planes $\mathbb{H}\subset\mathbb{C}$, this function is positive on $\widetilde{M}$. Let us now consider the form $\widetilde{\omega}=\sqrt{-1}\partial\bar{\partial}\log\varphi$. Using standard coordinates on $\widetilde{M}$ one can write this form as $\widetilde{\omega}=\sqrt{-1}\sum_{i=1}^{s}\frac{dz_{i}\wedge d\bar{z}_{i}}{4(\operatorname{im}z_{i})^{2}}$. Therefore $\widetilde{\omega}$ is a semipositive $(1,1)$-form on $\widetilde{M}$. Let us show that this form is $\Gamma$-invariant. The group $\Gamma$ is a semidirect product of the additive group $O_{K}$ and the multiplicative group $U$.The additive group acts on the first $s$ components of $\widetilde{M}$ (which correspond to upper half-planes $\mathbb{H}\subset\mathbb{C}$) by translations along the real line. Therefore it does not change $\operatorname{im}z_{i}$ for $i=1\ldots s$. Hence the function $\log\varphi$ is preserved by the action of the additive component. The multiplicative component acts on the first $s$ coordinates of $\widetilde{M}$ by multiplying them by a real number (since the first $s$ embeddings of the number field $K$ are real). Then every $\operatorname{im}z_{i}$ is multiplied by a real number and so there is a real number added to $\log(\operatorname{im}z_{i})$. Since $\log\varphi(z)=-\sum_{i=1}^{s}\log(\operatorname{im}z_{i})$, there is a real number added to $\log\varphi$. The operator $\bar{\partial}$ is zero on the constants, so $\widetilde{\omega}=\sqrt{-1}\partial\bar{\partial}\log\varphi$ is preserved by action of the group $\Gamma$. Since the $(1,1)$-form $\widetilde{\omega}$ is $\Gamma$-invariant it is the pullback of $(1,1)$-form $\omega$ on the Oeljeklaus-Toma manifold $M=\widetilde{M}/\Gamma$. Let us now show that the form $\widetilde{\omega}$ is exact on $\widetilde{M}$. For that we define the operator $d^{c}$. Definition 2.3: Define the twisted differential $d^{c}=I^{-1}dI$ where $d$ is a De Rham differential and $I$ is the almost complex structure. Since $dd^{c}=2\sqrt{-1}\partial\bar{\partial}$ (see [G–H]), one can see that $\widetilde{\omega}=\sqrt{-1}\partial\bar{\partial}\log\varphi=\frac{1}{2}dd^{c}\log\varphi$ and so $\widetilde{\omega}$ is exact as a form on $\widetilde{M}$. Also since the operator $d^{c}$ vanishes on constants the form $d^{c}\log\varphi$ is $\Gamma$-invariant, so $\omega$ is exact on $M$. ### 2.2 The $(1,1)$-form $\omega$ and curves on the Oeljeklaus-Toma manifold Since the form $\omega$ on the manifold $M$ is semipositive, its integral on any complex curve $C\subset M$ is nonnegative. The form $\omega$ is exact. Hence Stokes’ theorem implies that its integral on any complex curve vanishes. Therefore if $C\subset M$ is a closed complex curve, $\omega$ vanishes on it. To find out on which curves $\omega$ vanishes, let us define the zero foliation of the form $\omega$. Definition 2.4: An involutive distribution (or foliation) on $M$ is a subbundle $B\subset TM$ of the tangent bundle that is closed under the Lie bracket: $[B,B]\subset B$. Definition 2.5: A leaf of a foliation $B$ is a connected submanifold of $M$ such that its dimension is equal to $\dim B$ and that is tangent to $B$ at every point. Theorem 2.6: (Frobenius) Let $B\subset TM$ be an involutive distribution. Then for each point of the manifold $M$, there is exactly one leaf of this distribution that contains this point (see e.g. [Boo] Section IV. 8. Frobenius Theorem). Theorem 2.7: Let $N\subset M$ be a connected submanifold such that its tangent space at every point lies in a foliation $F\subset TM$. Then $N$ lies in a leaf of the foliation $F$ (see e.g. [Boo] Section IV. 8. Theorem 8.5). Definition 2.8: The zero foliation of a semipositive $(1,1)$-form $\omega$ on $M$ is the subundle of $TM$ that consists of tangent vectors $u\in T_{y}M$ such that $\omega(u,Iu)=0$, where $I$ is the almost complex structure on $M$. Consider the zero foliation of $\widetilde{\omega}$ on $\widetilde{M}$. The form $\widetilde{\omega}$ is strictly positive on each vector $v=(z_{1},\ldots,z_{m})$ such that at least one of $z_{i}$ for $i=1,\ldots,s$ is nonzero. Such a vector cannot be tangent to a leaf of the zero foliation. Therefore on each leaf of the zero foliation of the form $\widetilde{\omega}$ the first $s$ coordinates are constant. Hence a leaf of the zero foliation of $\widetilde{\omega}$ on $\widetilde{M}$ is isomorphic to $\mathbb{C}^{t}$. Let us now consider the zero foliation of $\omega$ on $M$. We show that the non-trivial image of the action of $\Gamma$ on any leaf $L$ of the zero foliation of the form $\widetilde{\omega}$ does not intersect with $L$. One can see that $L$ is $(z_{1},\ldots,z_{s})\times\mathbb{C}^{t}$ for some fixed $(z_{1},\ldots,z_{s})$. Therefore, for any $\gamma\in\Gamma$ such that $L\cap\gamma(L)\neq\emptyset$, the first $s$ coordinates of the points in $L$ coincide with the first $s$ coordinates of the points in $\gamma(L)$. Then for such $\gamma$ have the following system of equations: $\sigma_{i}(u)z_{i}+\sigma_{i}(a)=z_{i},\quad i=1\ldots s,$ where $\gamma=(u,a)$. These equations imply that $z_{i}=\frac{\sigma_{i}(a)}{1-\sigma_{i}(u)}$. Therefore $z_{i}$ are real but $\mathbb{H}$ does not have real elements. We showed that $L\cap\gamma(L)=\emptyset$ for every $\gamma\neq 1$ in $\Gamma$. Since $\omega$ vanishes on each compact curve $C\subset M$, each curve is contained in some leaf of the zero foliation of $\omega$. Since $\widetilde{\omega}$ is $\Gamma$-invariant, each leaf of the zero foliation of $\omega$ on $M$ is isomorphic to a component of the leaf of the zero foliation of $\widetilde{\omega}$ on $\widetilde{M}$. Therefore, it is isomorphic to $\mathbb{C}^{t}$. And $\mathbb{C}^{t}$ does not contain any compact complex submanifolds. We proved the following theorem: Theorem 2.9: There are no compact complex curves on the Oeljeklaus-Toma manifolds. ## 3 Closing remarks Let us now briefly explain the connection between our work and [O–V]. As in [O–V] we use the zero foliation of a certain $(1,1)$-form. The leaves of this zero foliation are $t$-dimensional complex manifolds. In [O–V] authors consider $t=1$, and submanifolds of any dimension. We consider any $t\geq 1$, and submanifolds of dimension 1. In both works one uses the semipositivity of the $(1,1)$-form to prove that a submanifold and a leaf of the zero foliation, which contains a point in this submanifold, could not intersect transversely. In both cases, one of the manifolds is of dimension one. Non-transversality implies that it is contained in the second one. Authors of [O–V] use the fact that the leaves of the zero foliation are Zariski dense, and therefore could not be contained in any submanifold. In our work we prove that each leaf is isomorphic to $\mathbb{C}^{t}$ and therefore could not contain any submanifolds. The arguments used in our work and [O–V] could not be applied to the case of higher dimensions. It seems obvious that there should be Oeljeklaus-Toma manifols which contain submanifolds — other Oeljeklaus-Toma manifolds, corresponding to smaller number fields, but we do not have a formal proof. In [O–T] it was proved that Oeljeklaus-Toma manifolds do not admit non-trivial meromorphic functions. Therefore all the divisors on these manifolds are fixed in their linear systems (which are 0-dimensional). It is conjectured that Oeljeklaus-Toma manifolds admit no divisors. ## References * [Boo] Boothby W.M. An Introduction to Differentiable Manifolds and Riemannian Geometry. Academic Press, San Diego, California, 2003. * [G–H] Griffiths Ph., Harris J. Principles of Algebraic Geometry. Wiley-Interscience, 1994. * [I] Inoue M. On surfaces of Class VII0, Invent. Math. 24 (1974), 269-310. * [Mil08] Milne J.S. Fields and Galois Theory, September 2008. This paper can be found on http://www.jmilne.org/math/CourseNotes/ft.html, version 4.21 * [Mil09] Milne J.S. Algebraic Number Theory, April 2009. This paper can be found on http://www.jmilne.org/math/CourseNotes/ant.html, version 3.02 * [O–T] Oeljeklaus K., Toma M. Non-Kähler compact complex manifolds associated to number fields. Ann. Inst. Fourier 55 (2005), 1291-1300. * [O–V] Ornea L., Verbitsky M. Subvarieties in Oeljeklaus-Toma manifolds. * [P–V] Parton M., Vuletescu V. Examples of non-trivial rank in locally conformal Kähler geometry. Math. Z. (2010), DOI 10.1007/s00209-010-0791-5, arXiv:1001.4891. * [R] Raghunathan M.S. Discrete subgroups of Lie groups. Springer 1972. * [V] Voisin C. Hodge Theory and Complex Algebraic Geometry Volume 1. Cambridge University Press, 2002. Sima Verbitsky Moscow State University, Faculty of Mathematics and Mechanics GSP-1 1, Leninskie gory, 119991 Moscow, Russia. sverb57@gmail.com	arxiv-papers	2011-11-16T15:12:56	2024-09-04T02:49:24.396903	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Sima Verbitsky", "submitter": "Sima Verbitsky", "url": "https://arxiv.org/abs/1111.3828" }
1111.3919	# Recipe recommendation using ingredient networks Chun-Yuen Teng Yu-Ru Lin Lada A. Adamic School of Information University of Michigan Ann Arbor, MI, USA chunyuen@umich.edu IQSS, Harvard University CCS, Northeastern University Boston, MA yuruliny@gmail.com School of Information University of Michigan Ann Arbor, MI, USA ladamic@umich.edu (30 July 1999) ###### Abstract The recording and sharing of cooking recipes, a human activity dating back thousands of years, naturally became an early and prominent social use of the web. The resulting online recipe collections are repositories of ingredient combinations and cooking methods whose large-scale and variety yield interesting insights about both the fundamentals of cooking and user preferences. At the level of an individual ingredient we measure whether it tends to be essential or can be dropped or added, and whether its quantity can be modified. We also construct two types of networks to capture the relationships between ingredients. The complement network captures which ingredients tend to co-occur frequently, and is composed of two large communities: one savory, the other sweet. The substitute network, derived from user-generated suggestions for modifications, can be decomposed into many communities of functionally equivalent ingredients, and captures users’ preference for healthier variants of a recipe. Our experiments reveal that recipe ratings can be well predicted with features derived from combinations of ingredient networks and nutrition information. ###### category: H.2.8 Database Management Database applications ###### keywords: Data mining ###### keywords: ingredient networks, recipe recommendation ††terms: Measurement; Experimentation ## 1 Introduction The web enables individuals to collaboratively share knowledge and recipe websites are one of the earliest examples of collaborative knowledge sharing on the web. Allrecipes.com, the subject of our present study, was founded in 1997, years ahead of other collaborative websites such as the Wikipedia. Recipe sites thrive because individuals are eager to share their recipes, from family recipes that had been passed down for generations, to new concoctions that they created that afternoon, having been motivated in part by the ability to share the result online. Once shared, the recipes are implemented and evaluated by other users, who supply ratings and comments. The desire to look up recipes online may at first appear odd given that tombs of printed recipes can be found in almost every kitchen. The Joy of Cooking [12] alone contains 4,500 recipes spread over 1,000 pages. There is, however, substantial additional value in online recipes, beyond their accessibility. While the Joy of Cooking contains a single recipe for Swedish meatballs, Allrecipes.com hosts “Swedish Meatballs I”, “II”, and “III”, submitted by different users, along with 4 other variants, including “The Amazing Swedish Meatball”. Each variant has been reviewed, from 329 reviews for “Swedish Meatballs I" to 5 reviews for “Swedish Meatballs III". The reviews not only provide a crowd-sourced ranking of the different recipes, but also many suggestions on how to modify them, e.g. using ground turkey instead of beef, skipping the “cream of wheat” because it is rarely on hand, etc. The wealth of information captured by online collaborative recipe sharing sites is revealing not only of the fundamentals of cooking, but also of user preferences. The co-occurrence of ingredients in tens of thousands of recipes provides information about which ingredients go well together, and when a pairing is unusual. Users’ reviews provide clues as to the flexibility of a recipe, and the ingredients within it. Can the amount of cinnamon be doubled? Can the nutmeg be omitted? If one is lacking a certain ingredient, can a substitute be found among supplies at hand without a trip to the grocery store? Unlike cookbooks, which will contain vetted but perhaps not the best variants for some individuals’ tastes, ratings assigned to user-submitted recipes allow for the evaluation of what works and what does not. In this paper, we seek to distill the collective knowledge and preference about cooking through mining a popular recipe-sharing website. To extract such information, we first parse the unstructured text of the recipes and the accompanying user reviews. We construct two types of networks that reflect different relationships between ingredients, in order to capture users’ knowledge about how to combine ingredients. The complement network captures which ingredients tend to co-occur frequently, and is composed of two large communities: one savory, the other sweet. The substitute network, derived from user-generated suggestions for modifications, can be decomposed into many communities of functionally equivalent ingredients, and captures users’ preference for healthier variants of a recipe. Our experiments reveal that recipe ratings can be well predicted by features derived from combinations of ingredient networks and nutrition information (with accuracy .792), while most of the prediction power comes from the ingredient networks (84%). The rest of the paper is organized as follows. Section 2 reviews the related work. Section 3 describes the dataset. Section 4 discusses the extraction of the ingredient and complement networks and their characteristics. Section 5 presents the extraction of recipe modification information, as well as the construction and characteristics of the ingredient substitute network. Section 6 presents our experiments on recipe recommendation and Section 7 concludes. ## 2 Related work Recipe recommendation has been the subject of much prior work. Typically the goal has been to suggest recipes to users based on their past recipe ratings [15][3] or browsing/cooking history [16]. The algorithms then find similar recipes based on overlapping ingredients, either treating each ingredient equally [4] or by identifying key ingredients [19]. Instead of modeling recipes using ingredients, Wang et al. [17] represent the recipes as graphs which are built on ingredients and cooking directions, and they demonstrate that graph representations can be used to easily aggregate Chinese dishes by the flow of cooking steps and the sequence of added ingredients. However, their approach only models the occurrence of ingredients or cooking methods, and doesn’t take into account the relationships between ingredients. In contrast, in this paper we incorporate the likelihood of ingredients to co- occur, as well as the potential of one ingredient to act as a substitute for another. Another branch of research has focused on recommending recipes based on desired nutritional intake or promoting healthy food choices. Geleijnse et al. [7] designed a prototype of a personalized recipe advice system, which suggests recipes to users based on their past food selections and nutrition intake. In addition to nutrition information, Kamieth et al. [9] built a personalized recipe recommendation system based on availability of ingredients and personal nutritional needs. Shidochi et al. [14] proposed an algorithm to extract replaceable ingredients from recipes in order to satisfy users’ various demands, such as calorie constraints and food availability. Their method identifies substitutable ingredients by matching the cooking actions that correspond to ingredient names. However, their assumption that substitutable ingredients are subject to the same processing methods is less direct and specific than extracting substitutions directly from user- contributed suggestions. Ahn et al. [1] and Kinouchi et al [10] examined networks involving ingredients derived from recipes, with the former modeling ingredients by their flavor bonds, and the latter examining the relationship between ingredients and recipes. In contrast, we derive direct ingredient-ingredient networks of both compliments and substitutes. We also step beyond characterizing these networks to demonstrating that they can be used to predict which recipes will be successful. ## 3 Dataset Allrecipes.com is one of the most popular recipe-sharing websites, where novice and expert cooks alike can upload and rate cooking recipes. It hosts 16 customized international sites for users to share their recipes in their native languages, of which we study only the main, English, version. Recipes uploaded to the site contain specific instructions on how to prepare a dish: the list of ingredients, preparation steps, preparation and cook time, the number of servings produced, nutrition information, serving directions, and photos of the prepared dish. The uploaded recipes are enriched with user ratings and reviews, which comment on the quality of the recipe, and suggest changes and improvements. In addition to rating and commenting on recipes, users are able to save them as favorites or recommend them to others through a forum. We downloaded 46,337 recipes including all information listed from allrecipes.com, including several classifications, such as a region (e.g. the midwest region of US or Europe), the course or meal the dish is appropriate for (e.g.: appetizers or breakfast), and any holidays the dish may be associated with. In order to understand users’ recipe preferences, we crawled 1,976,920 reviews which include reviewers’ ratings, review text, and the number of users who voted the review as useful. ### 3.1 Data preprocessing The first step in processing the recipes is identifying the ingredients and cooking methods from the freeform text of the recipe. Usually, although not always, each ingredient is listed on a separate line. To extract the ingredients, we tried two approaches. In the first, we found the maximal match between a pre-curated list of ingredients and the text of the line. However, this missed too many ingredients, while misidentifying others. In the second approach, we used regular expression matching to remove non-ingredient terms from the line and identified the remainder as the ingredient. We removed quantifiers, such as e.g. “1 lb” or “2 cups”, words referring to consistency or temperature, e.g. chopped or cold, along with a few other heuristics, such as removing content in parentheses. For example “1 (28 ounce) can baked beans (such as Bush’s Original®)" is identified as “baked beans". By limiting the list of potential terms to remove from an ingredient entry, we erred on the side of not conflating potentially identical or highly similar ingredients, e.g. “cheddar cheese”, used in 2450 recipes, was considered different from “sharp cheddar cheese”, occurring in 394 recipes. We then generated an ingredient list sorted by frequency of ingredient occurrence and selected the top 1000 common ingredient names as our finalized ingredient list. Each of the top 1000 ingredients occurred in 23 or more recipes, with plain salt making an appearance in 47.3% of recipes. These ingredients also accounted for 94.9% of ingredient entries in the recipe dataset. The remaining ingredients were missed either because of high specificity (e.g. yolk-free egg noodle), referencing brand names (e.g. Planters almonds), rarity (e.g. serviceberry), misspellings, or not being a food (e.g. “nylon netting"). The remaining processing task was to identify cooking processes from the directions. We first identified all heating methods using a listing in the Wikipedia entry on cooking [18]. For example, baking, boiling, and steaming are all ways of heating the food. We then identified mechanical ways of processing the food such as chopping and grinding, and other chemical techniques such as marinating and brining. ### 3.2 Regional preferences Choosing one cooking method over another appears to be a question of regional taste. 5.8% of recipes were classified into one of five US regions: Mountain, Midwest, Northeast, South, and West Coast (including Alaska and Hawaii). Figure 1 shows significantly ($\chi^{2}$ test p-value < 0.001) varying preferences in the different US regions among 6 of the most popular cooking methods. Boiling and simmering, both involving heating food in hot liquids, are more common in the South and Midwest. Marinating and grilling are relatively more popular in the West and Mountain regions, but in the West more grilling recipes involve seafood (18/42 = 42%) relative to other regions combined (7/106 = 6%). Frying is popular in the South and Northeast. Baking is a universally popular and versatile technique, which is often used for both sweet and savory dishes, and is slightly more popular in the Northeast and Midwest. Examination of individual recipes reflecting these frequencies shows that these differences in preference can be tied to differences in demographics, immigrant culture and availability of local ingredients, e.g. seafood. Figure 1: The percentage of recipes by region that apply a specific heating method. ## 4 Ingredient complement network Can we learn how to combine ingredients from the data? Here we employ the occurrences of ingredients across recipes to distill users’ knowledge about combining ingredients. We constructed an ingredient complement network based on pointwise mutual information (PMI) defined on pairs of ingredients $(a,b)$: $\mathrm{PMI(a,b)}=log\frac{p(a,b)}{p(a)p(b)},$ where $p(a,b)=\frac{\mathrm{\\#\>of\>recipes\>containing}\>a\mathrm{\>and}\>b}{\mathrm{\\#\>of\>recipes}},$ $p(a)=\frac{\mathrm{\\#\>of\>recipes\>containing}\>a}{\mathrm{\\#\>of\>recipes}},$ $p(b)=\frac{\mathrm{\\#\>of\>recipes\>containing}\>b}{\mathrm{\\#\>of\>recipes}}.$ The PMI gives the probability that two ingredients occur together against the probability that they occur separately. Complementary ingredients tend to occur together far more often than would be expected by chance. Figure 2 shows a visualization of ingredient complementarity. Two distinct subcommunities of recipes are immediately apparent: one corresponding to savory dishes, the other to sweet ones. Some central ingredients, e.g. egg and salt, actually are pushed to the periphery of the network. They are so ubiquitous, that although they have many edges, they are all weak, since they don’t show particular complementarity with any single group of ingredients. Figure 2: Ingredient complement network. Two ingredients share an edge if they occur together more than would be expected by chance and if their pointwise mutual information exceeds a threshold. We further probed the structure of the complementarity network by applying a network clustering algorithm [13]. The algorithm confirmed the existence of two main clusters containing the vast majority of the ingredients. An interesting satellite cluster is that of mixed drink ingredients, which is evident as a constellation of small nodes located near the top of the sweet cluster in Figure 2. The cluster includes the following ingredients: lime, rum, ice, orange, pineapple juice, vodka, cranberry juice, lemonade, tequila, etc. For each recipe we recorded the minimum, average, and maximum pairwise pointwise mutual information between ingredients. The intuition is that complementary ingredients would yield higher ratings, while ingredients that don’t go together would lower the average rating. We found that while the average and minimum pointwise mutual information between ingredients is uncorrelated with ratings, the maximum is very slightly positively correlated with the average rating for the recipe ($\rho=0.09$, p-value < $10^{-10}$). This suggests that having at least two complementary ingredients very slightly boosts a recipe’s prospects, but having clashing or unrelated ingredients does not seem to do harm. ## 5 Recipe modifications Co-occurrence of ingredients aggregated over individual recipes reveals the structure of cooking, but tells us little about how flexible the ingredient proportions are, or whether some ingredients could easily be left out or substituted. An experienced cook may know that apple sauce is a low-fat alternative to oil, or may know that nutmeg is often optional, but a novice cook may implement recipes literally, afraid that deviating from the instructions may produce poor results. While a traditional hardcopy cookbook would provide few such hints, they are plentiful in the reviews submitted by users who implemented the recipes, e.g. “This is a great recipe, but using fresh tomatoes only adds a few minutes to the prep time and makes it taste so much better", or another comment about the same salsa recipe “This is by far the best recipe we have ever come across. We did however change it just a little bit by adding extra onion.” As the examples illustrate, modifications are reported even when the user likes the recipe. In fact, we found that 60.1% of recipe reviews contain words signaling modification, such as “add", “omit", “instead”, “extra" and 14 others. Furthermore, it is the reviews that include changes that have a statistically higher average rating (4.49 vs. 4.39, t-test p-value $<10^{-10}$), and lower rating variance (0.82 vs. 1.05, Bartlett test p-value $<10^{-10}$), as is evident in the distribution of ratings, shown in Fig. 3. This suggests that flexibility in recipes is not necessarily a bad thing, and that reviewers who don’t mention modifications are more likely to think of the recipe as perfect, or to dislike it entirely. In the following, we describe the recipe modifications extracted from user reviews, including adjustment, deletion and addition. We then present how we constructed an ingredient substitute network based on the extracted information. Figure 3: The likelihood that a review suggests a modification to the recipe depends on the star rating the review is assigning to the recipe. ### 5.1 Adjustments Some modifications involve increasing or decreasing the amount of an ingredient in the recipe. In this and the following analyses, we split the review on punctuation such as commas and periods. We used simple heuristics to detect when a review suggested a modification: adding/using more/less of an ingredient counted as an increase/decrease. Doubling or increasing counted as an increase, while reducing, cutting, or decreasing counted as a decrease. While it is likely that there are other expressions signaling the adjustment of ingredient quantities, using this set of terms allowed us to compare the relative rate of modification, as well as the frequency of increase vs. decrease between ingredients. The ingredients themselves were extracted by performing a maximal character match within a window following an adjustment term. Figure 4 shows the ratios of the number of reviews suggesting modifications, either increases or decreases, to the number of recipes that contain the ingredient. Two patterns are immediately apparent. Ingredients that may be perceived as being unhealthy, such as fats and sugars, are, with the exception of vegetable oil and margarine, more likely to be modified, and to be decreased. On the other hand, flavor enhancers such as soy sauce, lemon juice, cinnamon, Worcestershire sauce, and toppings such as cheeses, bacon and mushrooms, are also likely to be modified; however, they tend to be added in greater, rather than lesser quantities. Combined, the patterns suggest that good-tasting but “unhealthy" ingredients can be reduced, if desired, while spices, extracts, and toppings can be increased to taste. Figure 4: Suggested modifications of quantity for the 50 most common ingredients, derived from recipe reviews. The line denotes equal numbers of suggested quantity increases and decreases. ### 5.2 Deletions and additions Recipes are also frequently modified such that ingredients are omitted entirely. We looked for words indicating that the reviewer did not have an ingredient (and hence did not use it), e.g. “had no" and “didn’t have". We further used “omit/left out/left off/bother with” as indication that the reviewer had omitted the ingredients, potentially for other reasons. Because reviewers often used simplified terms, e.g. “vanilla" instead of “vanilla extract", we compared words in proximity to the action words by constructing 4-character-grams and calculating the cosine similarity between the n-grams in the review and the list of ingredients for the recipe. To identify additions, we simply looked for the word “add", but omitted possible substitutions. For example, we would use “added cucumber", but not “added cucumber instead of green pepper", the latter of which we analyze in the following section. We then compared the addition to the list of ingredients in the recipes, and considered the addition valid only if the ingredient does not already belong in the recipe. Table 1 shows the correlation between ingredient modifications. As might be expected, the more frequently an ingredient occurs in a recipe, the more times its quantity has the opportunity to be modified, as is evident in the strong correlation between the the number of recipes the ingredient occurs in and both increases and decreases recommended in reviews. However, the more common an ingredient, the more stable it appears to be. Recipe frequency is negatively correlated with deletions/recipe ($\rho=-0.22$), additions/recipe ($\rho=-0.25$), and increases/recipe ($\rho=-0.26$). For example, salt is so essential, appearing in over 21,000 recipes, that we detected only 18 reviews where it was explicitly dropped. In contrast, Worcheshire sauce, appearing in 1,542 recipes, is dropped explicitly in 148 reviews. As might also be expected, additions are positively correlated with increases, and deletions with decreases. However, additions and deletions are very weakly negatively correlated, indicating that an ingredient that is added frequently is not necessarily omitted more frequently as well. Table 1: Correlations between ingredient modifications \| addition \| deletion \| increase \| decrease ---\|---\|---\|---\|--- # recipes \| 0.41 \| 0.22 \| 0.61 \| 0.68 addition \| \| -0.15 \| 0.79 \| 0.11 deletion \| \| \| 0.09 \| 0.58 increase \| \| \| \| 0.39 ### 5.3 Ingredient substitute network Replacement relationships show whether one ingredient is preferable to another. The preference could be based on taste, availability, or price. Some ingredient substitution tables can be found online111e.g., http://allrecipes.com/HowTo/common-ingredient-substitutions/detail.aspx, but are neither extensive nor contain information about relative frequencies of each substitution. Thus, we found an alternative source for extracting replacement relationships – users’ comments, e.g. “I replaced the butter in the frosting by sour cream, just to soothe my conscience about all the fatty calories". To extract such knowledge, we first parsed the reviews as follows: we considered several phrases to signal replacement relationships: “replace $a$ with $b$”, “substitute $b$ for $a$”, “$b$ instead of $a$”, etc, and matched $a$ and $b$ to our list of ingredients. We constructed an ingredient substitute network to capture users’ knowledge about ingredient replacement. This weighted, directed network consists of ingredients as nodes. We thresholded and eliminated any suggested substitutions that occurred fewer than 5 times. We then determined the weight of each edge by $p(b\|a)$, the proportion of substitutions of ingredient $a$ that suggest ingredient $b$. For example, 68% of substitutions for white sugar were to splenda, an artificial sweetener, and hence the assigned weight for the $sugar\rightarrow splenda$ edge is 0.68. Figure 5: Ingredient substitute network. Nodes are sized according to the number of times they have been recommended as a substitute for another ingredient, and colored according to their indegree. Table 2: Clusters of ingredients that can be substituted for one another. A maximum of 5 additional ingredients for each cluster are listed, ordered by PageRank. main \| other ingredients ---\|--- chicken \| turkey, beef, sausage, chicken breast, bacon olive oil \| butter, apple sauce, oil, banana, margarine sweet \| yam, potato, pumpkin, butternut squash, potato \| parsnip baking \| baking soda, cream of tartar powder \| almond \| pecan, walnut, cashew, peanut, sunflower s. apple \| peach, pineapple, pear, mango, pie filling egg \| egg white, egg substitute, egg yolk tilapia \| cod, catfish, flounder, halibut, orange roughy spinach \| mushroom, broccoli, kale, carrot, zucchini italian \| basil, cilantro, oregano, parsley, dill seasoning \| cabbage \| coleslaw mix, sauerkraut, bok choy \| napa cabbage The resulting substitution network, shown in Figure 5, exhibits strong clustering. We examined this structure by applying the map generator tool by Rosvall et al. [13], which uses a random walk approach to identify clusters in weighted, directed networks. The resulting clusters, and their relationships to one another, are shown in Fig. 6. The derived clusters could be used when following a relatively new recipe which may not receive many reviews, and therefore many suggestions for ingredient substitutions. If one does not have all ingredients at hand, one could examine the content of one’s fridge and pantry and match it with other ingredients found in the same cluster as the ingredient called for by the recipe. Table 2 lists the contents of a few such sample ingredient clusters, and Fig. 7 shows two example clusters extracted from the substitute network. Figure 6: Ingredient substitution clusters. Nodes represent clusters and edges indicate the presence of recommended substitutions that span clusters. Each cluster represents a set of related ingredients which are frequently substituted for one another. \| ---\|--- (a) milk substitutes \| (b) cinammon substitutes Figure 7: Relationships between ingredients located within two of the clusters from Fig. 6. Finally, we examine whether the substitution network encodes preferences for one ingredient over another, as evidenced by the relative ratings of similar recipes, one which contains an original ingredient, and another which implements a substitution. To test this hypothesis, we construct a “preference network", where one ingredient is preferred to another in terms of received ratings, and is constructed by creating an edge $(a,b)$ between a pair of ingredients, where $a$ and $b$ are listed in two recipes $X$ and $Y$ respectively, if recipe ratings $R_{X}>R_{Y}$. For example, if recipe $X$ includes beef, ketchup and cheese, and recipe $Y$ contains beef and pickles, then this recipe pair contributes to two edges: one from pickles to ketchup, and the other from pickles to cheese. The aggregate edge weights are defined based on PMI. Because PMI is a symmetric quantity ($\mathrm{PMI}(a;b)=\mathrm{PMI}(b;a)$), we introduce a directed PMI measure to cope with the directionality of the preference network: $\mathrm{PMI}(a\to b)=\mathrm{log}\frac{p(a\to b)}{p(a)p(b)},$ where $p(a\to b)=\frac{\mathrm{\\#\>of\>recipe\>pairs\>from}\>a\mathrm{\>to}\>b}{\mathrm{\\#\>of\>recipe\>pairs}},$ and $p(a)$, $p(b)$ are defined as in the previous section. We find high correlation between this preference network and the substitution network ($\rho=0.72,p<0.001$). This observation suggests that the substitute network encodes users’ ingredient preference, which we use in the recipe prediction task described in the next section. ## 6 Recipe recommendation We use the above insights to uncover novel recommendation algorithms suitable for recipe recommendations. We use ingredients and the relationships encoded between them in ingredient networks as our main feature sets to predict recipe ratings, and compare them against features encoding nutrition information, as well as other baseline features such as cooking methods, and preparation and cook time. Then we apply a discriminative machine learning method, stochastic gradient boosting trees [6], to predict recipe ratings. In the experiments, we seek to answer the following three questions. (1) Can we predict users’ preference for a new recipe given the information present in the recipe? (2) What are the key aspects that determine users’ preference? (3) Does the structure of ingredient networks help in recipe recommendation, and how? ### 6.1 Recipe Pair Prediction The goal of our prediction task is: given a pair of similar recipes, determine which one has higher average rating than the other. This task is designed particularly to help users with a specific dish or meal in mind, and who are trying to decide between several recipe options for that dish. Recipe pair data. The data for this prediction task consists of pairs of similar recipes. The reason for selecting similar recipes, with high ingredient overlap, is that while apples may be quite comparable to oranges in the context of recipes, especially if one is evaluating salads or desserts, lasagna may not be comparable to a mixed drink. To derive pairs of related recipes, we computed similarity with a cosine similarity between the ingredient lists for the two recipes, weighted by the inverse document frequency, $log(\\#\>of\>recipes/\\#\>of\>recipes\>containing\>the\>ingredient)$. We considered only those pairs of recipes whose cosine similarity exceeded 0.2. The weighting is intended to identify higher similarity among recipes sharing more distinguishing ingredients, such as Brussels sprouts, as opposed to recipes sharing very common ones, such as butter. A further challenge to obtaining reliable relative rankings of recipes is variance introduced by having different users choose to rate different recipes. In addition, some users might not have a sufficient number of reviews under their belt to have calibrated their own rating scheme. To control for variation introduced by users, we examined recipe pairs where the same users are rating both recipes and are collectively expressing a preference for one recipe over another. Specifically, we generated 62,031 recipe pairs ($a,b$) where $rating_{i}(a)$ > $rating_{i}(b)$, for at least 10 users $i$, and over 50% of users who rated both recipe $a$ and recipe $b$. Furthermore, each user $i$ should be an active enough reviewer to have rated at least 8 other recipes. Features. In the prediction dataset, each observation consists of a set of predictor variables or features that represent information about two recipes, and the response variable is a binary indicator of which gets the higher rating on average. To study the key aspects of recipe information, we constructed different set of features, including: * • Baseline: This includes cooking methods, such as chopping, marinating, or grilling, and cooking effort descriptors, such as preparation time in minutes, as well as the number of servings produced, etc. These features are considered as primary information about a recipe and will be included in all other feature sets described below. * • Full ingredients: We selected up to 1000 popular ingredients to build a “full ingredient list”. In this feature set, each observed recipe pair contains a vector with entries indicating whether an ingredient from the full list is present in either recipe in the pair. * • Nutrition: This feature set does not include any ingredients but only nutrition information such the total caloric content, as well as quantities of fats, carbohydrates, etc. * • Ingredient networks: In this set, we replaced the full ingredient list by structural information extracted from different ingredient networks, as described in Sections 4 and 5.3. Co-occurrence is treated separately as a raw count, and a complementarity, captured by the PMI. * • Combined set: Finally, a combined feature set is constructed to test the performance of a combination of features, including baseline, nutrition and ingredient networks. To build the ingredient network feature set, we extracted the following two types of structural information from the co-occurrence and substitution networks, as well as the complement network derived from the co-occurrence information: Network positions are calculated to represent how a recipe’s ingredients occupy positions within the networks. Such position measures are likely to inform if a recipe contains any “popular” or “unusual” ingredients. To calculate the position measures, we first calculated various network centrality measures, including degree centrality, betweenness centrality, etc., from the ingredient networks. A centrality measure can be represented as a vector $\vec{g}$ where each entry indicates the centrality of an ingredient. The network position of a recipe, with its full ingredient list represented as a binary vector $\vec{f}$, can be summarized by $\vec{g}^{T}\cdot\vec{f}$, i.e., an aggregated centrality measure based on the centrality of its ingredients. Network communities provide information about which ingredient is more likely to co-occur with a group of other ingredients in the network. A recipe consisting of ingredients that are frequently used with, complemented by or substituted by certain groups may be predictive of the ratings the recipe will receive. To obtain the network community information, we applied latent semantic analysis (LSA) on recipes. We first factorized each ingredient network, represented by matrix $W$, using singular value decomposition (SVD). In the matrix $W$, each entry $W_{ij}$ indicates whether ingredient $i$ co- occurrs, complements or substitues ingredient $j$. Suppose $W_{k}=U_{k}\Sigma_{k}V_{k}^{T}$ is a rank-$k$ approximation of $W$, we can then transform each recipe’s full ingredient list using the low- dimensional representation, $\Sigma_{k}^{-1}V_{k}^{T}\vec{f}$, as community information within a network. These low-dimensional vectors, together with the vectors of network positions, constitute the ingredient network features. Figure 8: Prediction performance. The nutrition information and ingredient networks are more effective features than full ingredients. The ingredient network features lead to impressive performance, close to the best performance. Figure 9: Relative importance of features in the combined set. The individual items from nutrition information are very indicative in differentiating highly rated recipes, while most of the prediction power comes from ingredient networks. Learning method. We applied discriminative machine learning methods such as support vector machines (SVM) [2] and stochastic gradient boosting trees [5] to our prediction problem. Here we report and discuss the detailed results based on the gradient boosting tree model. Like SVM, the gradient boosting tree model seeks a parameterized classifier, but unlike SVM that considers all the features at one time, the boosting tree model considers a set of features at a time and iteratively combines them according to their empirical errors. In practice, it not only has competitive performance comparable to SVM, but can serve as a feature ranking procedure [11]. In this work, we fitted a stochastic gradient boosting tree model with 8 terminal nodes under an exponential loss function. The dataset is roughly balanced in terms of which recipe is the higher-rated one within a pair. We randomly divided the dataset into a training set (2/3) and a testing set (1/3). The prediction performance is evaluated based on accuracy, and the feature performance is evaluated in terms of relative importance [8]. For each single decision tree, one of the input variables, $x^{j}$, is used to partition the region associated with that node into two subregions in order to fit to the response values. The squared relative importance of variable $x^{j}$ is the sum of such squared improvements over all internal nodes for which it was chosen as the splitting variable, as: $imp(j)=\sum_{k}\hat{i}_{k}^{2}I(\mbox{splits on }x^{j})$ where $\hat{i}_{k}^{2}$ is the empirical improvement by the $k$-th node splitting on $x^{j}$ at that point. ### 6.2 Results Figure 10: Relative importance of features representing the network structure. The substitution network has the strongest contribution ($39.8\%$) to the total importance of network features, and it also has more influential features in the top 100 list, which suggests that the substitution network is complementary to other features. Figure 11: Relative importance of features from nutrition information. The carbs item is the most influential feature in predicting higher-rated recipes. The overall prediction performance is shown in Fig. 8. Surprisingly, even with a full list of ingredients, the prediction accuracy is only improved from .712 (baseline) to .746. In contrast, the nutrition information and ingredient networks are more effective (with accuracy .753 and .786, respectively). Both of them have much lower dimensions (from tens to several hundreds), compared with the full ingredients that are represented by more than 2000 dimensions (1000 ingredients per recipe in the pair). The ingredient network features lead to impressive performance, close to the best performance given by the combined set (.792), indicating the power of network structures in recipe recommendation. Figure 9 shows the influence of different features in the combined feature set. Up to 100 features with the highest relative importance are shown. The importance of a feature group is summarized by how much the total importance is contributed by all features in the set. For example, the baseline consisting of cooking effort and cooking methods contribute $8.9\%$ to the overall performance. The individual items from nutrition information are very indicative in differentiating highly-rated recipes, while most of the prediction power comes from ingredient networks ($84\%$). Figure 10 shows the top 100 features from the three networks. In terms of the total importance of ingredient network features, the substitution network has slightly stronger contribution ($39.8\%$) than the other two networks, and it also has more influential features in the top 100 list. This suggests that the structural information extracted from the substitution network is not only important but also complementary to information from other aspects. Looking into the nutrition information (Fig. 11), we found that carbohydrates are the most influential feature in predicting higher-rated recipes. Since carbohydrates comprise around $50\%$ or more of total calories, the high importance of this feature interestingly suggests that a recipe’s rating can be influenced by users’ concerns about nutrition and diet. Another interesting observation is that, while individual nutrition items are powerful predictors, a higher prediction accuracy can be reached by using ingredient networks alone, as shown in Fig. 8. This implies the information about nutrition may have been encoded in the ingredient network structure, e.g. substitutions of less healthful ingredients with “healthier” alternatives. Figure 12: Prediction performance over reduced dimensionality. The best performance is given by reduced dimension $k=50$ when combining all three networks. In addition, using the information about the complement network alone is more effective in prediction than using other two networks. Figure 13: Influential substitution communities. The matrix shows the most influential feature dimensions extracted from the substitution network. For each dimension, the six representative ingredients with the highest intensity values are shown, with colors indicating their intensity. These features suggest that the communities of ingredient substitutes, such as the sweet and oil in the first dimension, are particularly informative in prediction. Constructing the ingredient network feature involves reducing high-dimensional network information through SVD, as described in the previous section. The dimensionality can be determined by cross-validation. As shown in Fig. 12, features with a very large dimension tend to overfit the training data. Hence we chose $k=50$ for the reduced dimension of all three networks. The figure also shows that using the information about the complement network alone is more effective in prediction than using either the co-occurrence and substitute networks, even in the case of low dimensions. Consistently, as shown in terms of relative importance (Fig. 10), the substitution network alone is not the most effective, but it provides more complementary information in the combined feature set. In Figure 13 we show the most representative ingredients in the decomposed matrix derived from the substitution network. We display the top five influential dimensions, evaluated based on the relative importance, from the SVD resultant matrix $V_{k}$, and in each of these dimensions we extracted six representative ingredients based on their intensities in the dimension (the squared entry values). These representative ingredients suggest that the communities of ingredient substitutes, such as the sweet and oil substitutes in the first dimension or the milk substitutes in the second dimesion (which is similar to the cluster shown in Fig. 6), are particularly informative in predicting recipe ratings. To summarize our observations, we find we are able to effectively predict users’ preference for a recipe, but the prediction is not through using a full list of ingredients. Instead, by using the structural information extracted from the relationships among ingredients, we can better uncover users’ preference about recipes. ## 7 Conclusion Recipes are little more than instructions for combining and processing sets of ingredients. Individual cookbooks, even the most expansive ones, contain single recipes for each dish. The web, however, permits collaborative recipe generation and modification, with tens of thousands of recipes contributed in individual websites. We have shown how this data can be used to glean insights about regional preferences and modifiability of individual ingredients, and also how it can be used to construct two kinds of networks, one of ingredient complements, the other of ingredient substitutes. These networks encode which ingredients go well together, and which can be substituted to obtain superior results, and permit one to predict, given a pair of related recipes, which one will be more highly rated by users. In future work, we plan to extend ingredient networks to incorporate the cooking methods as well. It would also be of interest to generate region- specific and diet-specific ratings, depending on the users’ background and preferences. A whole host of user-interface features could be added for users who are interacting with recipes, whether the recipe is newly submitted, and hence unrated, or whether they are browsing a cookbook. In addition to automatically predicting a rating for the recipe, one could flag ingredients that can be omitted, ones whose quantity could be tweaked, as well as suggested additions and substitutions. ## 8 Acknowledgments This work was supported by MURI award FA9550-08-1-0265 from the Air Force Office of Scientific Research. The methodology used in this paper was developed with support from funding from the Army Research Office, Multi- University Research Initiative on Measuring, Understanding, and Responding to Covert Social Networks: Passive and Active Tomography. The authors gratefully acknowledge D. Lazer for support. ## References * [1] Ahn, Y., Ahnert, S., Bagrow, J., and Barabasi, A. Flavor network and the principles of food pairing. Bulletin of the American Physical Society 56 (2011). * [2] Cortes, C., and Vapnik, V. Support-vector networks. Machine learning 20, 3 (1995), 273–297. * [3] Forbes, P., and Zhu, M. Content-boosted matrix factorization for recommender systems: Experiments with recipe recommendation. Proceedings of Recommender Systems (2011). * [4] Freyne, J., and Berkovsky, S. Intelligent food planning: personalized recipe recommendation. In IUI, ACM (2010), 321–324. * [5] Friedman, J. Stochastic gradient boosting. Computational Statistics & Data Analysis 38, 4 (2002), 367–378. * [6] Friedman, J., Hastie, T., and Tibshirani, R. Additive logistic regression: a statistical view of boosting. Annals of Statistics 28 (1998), 2000. * [7] Geleijnse, G., Nachtigall, P., van Kaam, P., and Wijgergangs, L. A personalized recipe advice system to promote healthful choices. In IUI, ACM (2011), 437–438. * [8] Hastie, T., Tibshirani, R., Friedman, J., and Franklin, J. The elements of statistical learning: data mining, inference and prediction. The Mathematical Intelligencer 27, 2 (2005). * [9] Kamieth, F., Braun, A., and Schlehuber, C. Adaptive implicit interaction for healthy nutrition and food intake supervision. Human-Computer Interaction. Towards Mobile and Intelligent Interaction Environments (2011), 205–212. * [10] Kinouchi, O., Diez-Garcia, R., Holanda, A., Zambianchi, P., and Roque, A. The non-equilibrium nature of culinary evolution. New Journal of Physics 10 (2008), 073020. * [11] Lu, Y., Peng, F., Li, X., and Ahmed, N. Coupling feature selection and machine learning methods for navigational query identification. In CIKM, ACM (2006), 682–689. * [12] Rombauer, I., Becker, M., Becker, E., and Maestro, L. Joy of cooking. Scribner Book Company, 1997. * [13] Rosvall, M., and Bergstrom, C. Maps of random walks on complex networks reveal community structure. PNAS 105, 4 (2008), 1118. * [14] Shidochi, Y., Takahashi, T., Ide, I., and Murase, H. Finding replaceable materials in cooking recipe texts considering characteristic cooking actions. In Proc. of the ACM multimedia 2009 workshop on Multimedia for cooking and eating activities, ACM (2009), 9–14. * [15] Svensson, M., Höök, K., and Cöster, R. Designing and evaluating kalas: A social navigation system for food recipes. ACM Transactions on Computer-Human Interaction (TOCHI) 12, 3 (2005), 374–400. * [16] Ueda, M., Takahata, M., and Nakajima, S. User’s food preference extraction for personalized cooking recipe recommendation. Proc. of the Second Workshop on Semantic Personalized Information Management: Retrieval and Recommendation (2011). * [17] Wang, L., Li, Q., Li, N., Dong, G., and Yang, Y. Substructure similarity measurement in chinese recipes. In WWW, ACM (2008), 979–988. * [18] Wikipedia. Outline of food preparation, 2011. [Online; accessed 22-Oct-2011]. * [19] Zhang, Q., Hu, R., Mac Namee, B., and Delany, S. Back to the future: Knowledge light case base cookery. In Proc. of The 9th European Conference on Case-Based Reasoning Workshop (2008), 15.	arxiv-papers	2011-11-16T19:32:57	2024-09-04T02:49:24.404707	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Chun-Yuen Teng, Yu-Ru Lin, and Lada A. Adamic", "submitter": "Lada A. Adamic", "url": "https://arxiv.org/abs/1111.3919" }
1111.3925	# A Low-Delay Low-Complexity EKF Design for Joint Channel and CFO Estimation in Multi-User Cognitive Communications Pengkai Zhao Electrical Engineering, UCLA, CA, USA Cong Shen Qualcomm Inc., San Diego, CA, USA ###### Abstract Parameter estimation in cognitive communications can be formulated as a multi- user estimation problem, which is solvable under maximum likelihood solution but involves high computational complexity. This paper presents a time-sharing and interference mitigation based EKF (Extended Kalman Filter) design for joint CFO (carrier frequency offset) and channel estimation at multiple cognitive users. The key objective is to realize low implementation complexity by decomposing high-dimensional parameters into multiple separate low- dimensional estimation problems, which can be solved in a time-shared manner via pipelining operation. We first present a basic EKF design that estimates the parameters from one TX user to one RX antenna. Then such basic design is time-shared and reused to estimate parameters from multiple TX users to multiple RX antennas. Meanwhile, we use interference mitigation module to cancel the co-channel interference at each RX sample. In addition, we further propose adaptive noise variance tracking module to improve the estimation performance. The proposed design enjoys low delay and low buffer size (because of its online real-time processing), as well as low implementation complexity (because of time-sharing and pipeling design). Its estimation performance is verified to be close to Cramer-Rao bound. ## I Introduction Cognitive communication system is widely accepted as a perspective way in increasing the spectrum efficiency of wireless networks, where primary links and secondary links can usually co-exist in the network, resulting in an interference limited environment. Parameter estimation in cognitive communications is a challenging problem because of (i) the existence of co- channel interference, and (ii) the high-dimensional parameters from multiple TX users to multiple RX antennas. In particular, note that different TX users often have independent carrier frequency offset (CFO) values (including both oscillator offsets and doppler offsets), which usually introduce serious nonlinear components within the observed signal, complicating the estimation problem. Meanwhile, channel responses from multiple TX users to multiple RX antennas can result in a set of high-dimensional parameters, which are also difficult to estimate. Finally, due to the existence of multi-user interference, CFO and channel parameters usually have to be treated together and be estimated in a joint way so as to approach the optimal performance, which further increases the estimation complexity. Without loss of generality, this paper assumes Orthogonal Frequency Division Multiplexing (OFDM) system, which is an overwhelming choice for modern wireless systems. The classical CFO and channel estimation method in a single- user OFDM system is based on two repeated training symbols [1]. It has low implementation complexity and near-optimal performance, but only applies to a single-user scenario111It is also applicable for multi-user scenario with non- overlapping training symbols, but this is not the case considered in this paper.. In multi-user OFDMA systems with unique subcarrier set per TX user, CFO and channel parameters can be recovered by exploiting distinct subcarrier structures among TX users ([2, 3]). But this method requires separate subcarrier allocation for different users. Consequently, in a general multi- user cognitive system without specific subcarrier allocation per user and with overlapped training symbols, ML and EM related methods seem to be the only applicable choice, where all TX users’ parameters have to be formulated into a maximum likelihood (ML) estimation problem [4], which is solvable under Expectation Maximization (EM) method [5] in an iterative way. However, since the entire OFDM block is stored offline and is iteratively processed multiple times, these ML and EM approaches often require high computational complexity and high processing delay. Based on above considerations, this paper will focus on using Kalman filter structure to estimate the CFO and channel parameters in multi-user cognitive communications. Our major objective is to achieve low-complexity and low-delay estimation performance in cognitive systems. In general, Kalman filter is a good candidate for low delay and low complexity parameter estimation primarily due to its real-timing processing property. It has been conventionally used for CFO and channel estimation in multi-user OFDM systems, e.g., the FFT-Block EKF design in [6], the parallel EKF design in [7], and the particle filter design in [7]. However, these existing designs inherently suffer from multiple issues related to complexity, delay and buffer size as follows: 1. 1. Block EKF design in [6] operates on an FFT-block basis, which grows increasingly complex as FFT size becomes large (e.g., 2048 FFT size). Also, parameters estimated in this method are handled in a high dimension manner. 2. 2. Parallel EKF design in [7] also operates on an FFT-block basis, marking it complex under large FFT size. Parameters in this design are jointly estimated by calculating the covariance information between different users, leading to a high matrix dimension. 3. 3. Beyond FFT size and parameter dimension issues similar to item 1 and 2, particle filter design in [7] needs to repeat the Kalman operation at multiple particle samples, yielding a multiplicative effect on complexity. To summarize, the major challenge in implementing a low-complexity EKF design lies in the factors of: (i) multiple TX users; (ii) multiple RX antennas; (iii) high parameter dimension; and (iv) large FFT size (e.g., 2048 size). With low-complexity and low-delay requirement in mind, this paper will present a time-sharing and interference mitigation based Extended Kalman Filter (EKF) design for multi-user cognitive communications, which can estimate the CFO and channel parameters from multiple TX users to multiple RX antennas in a time- sharing manner. Here low delay property is achieved by using Kalman filter estimation at each RX sample in a real-time manner, and low complexity property is achieved by reusing a single user EKF design in a time- sharing222Time-sharing in this paper indicates that the same hardware module can be reused by different processes at separate time slots. and pipeling way. We first present a fundamental EKF design that estimates the CFO and channel parameters from one TX user to one RX antenna. Then such basic EKF design is reused in a time-shared way to estimate the parameters from multiple TX users to multiple RX antennas. Meanwhile, at each RX sample, an interference mitigation strategy is developed to estimate and remove the expected multi- user interference. In addition, we provide an adaptive noise variance tracking module to further enhance the estimation performance. Because of the usage of EKF structure, our design is essentially different from the particle filters in [7] and the EM method in [5]. Our design is also different from the Parallel-EKF design in [7] and the FFT-Block EKF design in [6] at the following perspectives: (i) our design runs at each time domain RX sample, not at an FFT-block basis; (ii) our design treats each user separately, not jointly; (iii) our design can be implemented in a time-sharing way, which is less considered in [6] and [7]; (iv) system model in our design is different from the ones in [6, 7] by integrating CFO parameter into channel response (see Eqn. (5) in section II). Analysis and simulations results validate that our proposed design can closely approach the Cramer-Rao bound, and has lower computational complexity than the ones in [6, 7]. Finally, although cognitive communication is a typical application scenario for our proposed design, it is also applicable in many other multi-user systems that satisfy the conditions presented in section II. ## II System Model ### II-A Problem Formulation We consider a total of $Q$ TX users in the cognitive network. One of them is the primary TX user (i.e., base station), and the rest are all secondary TX users. Primary TX user’s transmission is based on a time division MAC protocol, where time is divided into different time frames with equal duration. Secondary users can maintain time synchronization with the primary TX user by learning and synchronizing with its time frames. Each secondary RX user is equipped with $N_{A}$ multiple antennas to decode the packets. Without loss of generality, we assume that every TX user has only one spatial stream, and there exists $Q\leq N_{A}$. Also, every TX user has a distinct training symbol333This unique training symbol can be determined according to either the unique user ID in the network, or the access order in the current time frame. $s_{q}(n)$ with $1\leq q\leq Q$ and $0\leq n\leq N_{F}-1$. Here $N_{F}$ is the FFT size of OFDM system. Each TX user has an independent carrier frequency offset (CFO) that is caused by both the oscillator offset and the doppler offset. Denote TX user $q$’s CFO value as $\varepsilon_{q}$. For a given secondary RX user, the channel from TX user $q$ to the $m$th RX antenna of this secondary user is denoted as $h_{q,m}(p_{l}^{q,m})$, $\forall\ 1\leq l\leq L_{\rm max}$. Here $L_{\rm max}$ is the number of time domain paths in the channel response, and $p_{l}^{q,m}$ is an integer value representing the relative delay of the $l$th path. We assume that all $p_{l}^{q,m}$ values have already been determined at an early stage (e.g., using PN sequences at coarse synchronization). The received signal at the $m$th RX antenna is derived as: $\displaystyle y_{m}(n)$ $\displaystyle=$ $\displaystyle\sum_{q=1}^{Q}\exp\left(j\frac{2\pi\varepsilon_{q}n}{N_{F}}\right)\sum_{l=1}^{L_{\rm max}}h_{q,m}(p_{l}^{q,m})s_{q}\left[(n-p_{l}^{q,m})_{N_{F}}\right]$ (1) $\displaystyle+z_{m}(n),0\leq n\leq N_{F}-1$ where $(n-p_{l}^{q,m})_{N_{F}}=\left\\{(n-{p_{l}^{q,m}}){\ \rm mod\ }N_{F}\right\\}$ is circular shift, and $z_{m}(n)$ is the background noise at the $m$th RX antenna. The task in this work is to estimate CFO parameter $\varepsilon_{q}$ and channel parameter $h_{q,m}(p_{l}^{q,m})$ for all users ($1\leq q\leq Q$) and all antennas ($1\leq m\leq N_{A}$). Obviously, the optimal estimation is the solution to this maximum likelihood (ML) problem: $\displaystyle\min\sum_{m=1}^{N_{A}}{\big{\|}}y_{m}(n)-$ (2) $\displaystyle\sum_{q=1}^{Q}\exp\left(j\frac{2\pi{\widehat{\varepsilon}}_{q}n}{N_{F}}\right)\sum_{l=1}^{L_{\rm max}}{\widehat{h}}_{q,m}({p_{l}^{q,m}})s_{q}\left[(n-p_{l}^{q,m})_{N_{F}}\right]{\big{\|}}^{2}$ where ${\widehat{\varepsilon}}_{q}$ and ${\widehat{h}}_{q,m}({p_{l}^{q,m}})$ represent the estimated values. There are a total of $(L_{\rm max}N_{A}+1)Q$ parameters in Eqn. (2), which constitutes a high-dimensional parameter estimation problem. ### II-B State-Space Formulation ML solution can generally approach the optimal performance but it requires huge computations, which are highly undesirable in most systems. Instead, this paper proposes an EKF design for the estimation of the CFO and channel parameters, which can sequentially update the estimation results at each RX sample, resulting in low buffer size and low estimation delay. Initially, it is straightforward to directly apply an EKF design at Eqn. (2) by building all CFO and channel parameters into one state vector, whose dimension is as high as $(L_{\rm max}N_{A}+1)Q$. This method will significantly increase the complexity of the derived Kalman filter. With such complexity consideration in mind, we first propose a low-dimensional EKF design that can estimate the parameters from one TX user to one RX antenna, which has only $(L_{\rm max}+1)$ parameters. Then we reuse this fundamental EKF design in a time- shared manner to estimate the parameters from multiple TX users to multiple RX antennas. In this way, high-dimensional parameters are estimated by sequentially reusing a low-dimensional estimator, which reduces the complexity of the proposed EKF design. We first present an RX signal formulation from the perspective of TX user $q$ and the $m$th RX antenna as: $\displaystyle y_{q,m}(n)$ $\displaystyle=$ $\displaystyle\exp\left(j\frac{2\pi\varepsilon_{q}n}{N_{F}}\right)\sum_{l=1}^{L_{\rm max}}h_{q,m}(p_{l}^{q,m})s_{q}\left[(n-p_{l}^{q,m})_{N_{F}}\right]$ (3) $\displaystyle+z_{q,m}(n),$ here $y_{q,m}(n)$ is extracted from $y_{m}(n)$ with the aid of interference mitigation module, and $z_{q,m}(n)$ represents the residual noise at TX user $q$ and the $m$th RX antenna, which includes both the residual co-channel interference from other TX users and the background noise at the $m$th RX antenna. Additionally, the initial value of $y_{q,m}(n)$ without any interference mitigation is set to $y_{q,m}(n)=y_{m}(n)$. Details about the interference mitigation module will be given in section III. Now we define the associated state vector as: $\displaystyle{\rm{\bf X}}_{q,m}(n)=\left[\varepsilon_{q},{\rm{\bf H}}_{q,m}(n)\right]^{T},$ (4) $\displaystyle{\rm{\bf H}}_{q,m}(n)=\exp\left(j\frac{2\pi n\varepsilon_{q}}{N_{F}}\right)\left[h_{q,m}(p^{q,m}_{1}),...,h_{q,m}(p^{q,m}_{L_{\rm max}})\right].$ (5) The state-space model for TX user $q$ and the $m$th RX antenna can be derived from (3) as: $\displaystyle{\rm{\bf X}}_{q,m}(n)=f\left\\{{\rm{\bf X}}_{q,m}(n-1)\right\\}={\rm{\bf D}}_{q,m}(\varepsilon_{q}){\rm{\bf X}}_{q,m}(n-1),$ (6) $\displaystyle{\rm{\bf D}}_{q,m}(\varepsilon_{q})=\left[\begin{array}[]{cc}1&{\rm{\bf 0}}_{1\times L_{\rm max}}\\\ {\rm{\bf 0}}_{L_{\rm max}\times 1}&\exp\left(j2\pi\varepsilon_{q}/N_{F}\right){\rm{\bf I}}_{L_{\rm max}\times L_{\rm max}}\end{array}\right]$ (9) $\displaystyle y_{q,m}(n)={\rm{\bf G}}_{q,m}(n){\rm{\bf X}}_{q,m}(n)+z_{q,m}(n),$ (10) $\displaystyle{\rm{\bf G}}_{q,m}(n)=\big{[}0,s_{q}\left[(n-p^{q,m}_{1})_{N_{F}}\right],s_{q}\left[(n-p^{q,m}_{2})_{N_{F}}\right],...,$ $\displaystyle s_{q}\left[(n-p^{q,m}_{L_{\rm max}})_{N_{F}}\right]\big{]}.$ (11) Finally, it is worth mentioning that the derived state-space formulation is a nonlinear model, since there is a nonlinear component $\exp\left(j2\pi\varepsilon_{q}/N_{F}\right)$ in the matrix ${\rm{\bf D}}_{q,m}(\varepsilon_{q})$. ## III Interference Mitigation based EKF Design This section sequentially describes (i) the basic EKF design that aims at only one TX user and one RX antenna, (ii) the interference mitigation module that cancels co-channel interference at each RX sample, (iii) the proposed adaptive noise variance tracking module, and (iv) the overall paradigm of the proposed design. ### III-A Fundamental EKF Design The key idea behind the EKF design is using Jacobian derivation to linearize the nonlinear matrix ${\rm{\bf D}}_{q,m}(\varepsilon_{q})$ at local estimates: $\displaystyle{\rm{\bf F}}_{q,m}(n-1)=\left.\frac{\partial f\left({\rm{\bf X}}_{q,m}(n-1)\right)}{\partial{\rm{\bf X}}_{q,m}(n-1)}\right\|_{\widehat{{\rm{\bf X}}}_{q,m}(n-1\|n-1)}$ $\displaystyle=\left[\begin{array}[]{cc}1&{\rm{\bf 0}}_{1\times L_{\rm max}}\\\ \alpha(n-1){\widehat{\rm{\bf H}}}_{q,m}^{T}(n-1\|n-1)&\exp\left(\alpha(n-1)\right){\rm{\bf I}}_{L_{\rm max}\times L_{\rm max}}\end{array}\right]$ (14) (15) $\displaystyle\alpha(n-1)=j2\pi\widehat{\varepsilon}_{q,m}(n-1\|n-1)/{N_{F}}$ (16) where ${\widehat{{\rm{\bf X}}}_{q,m}(n-1\|n-1)}$ represents the estimated state vector after processing the $(n-1)$th RX sample. Based on (III-A), the prediction steps in our fundamental EKF design are: $\displaystyle{\widehat{{\rm{\bf X}}}_{q,m}(n\|n-1)}=$ $\displaystyle{\rm{\bf D}}_{q,m}(\widehat{\varepsilon}_{q,m}(n-1\|n-1)){\widehat{{\rm{\bf X}}}_{q,m}(n-1\|n-1)},$ (17) $\displaystyle{\rm{\bf P}}_{q,m}(n\|n-1)=$ $\displaystyle{\rm{\bf F}}_{q,m}(n-1){\rm{\bf P}}_{q,m}(n-1\|n-1){\rm{\bf F}}_{q,m}^{H}(n-1).$ (18) And the updating steps are as follows: $\displaystyle{\rm{\bf K}}_{q,m}(n)={\rm{\bf P}}_{q,m}(n\|n-1){\rm{\bf G}}_{q,m}^{H}(n)\times$ $\displaystyle\left[{\rm{\bf G}}_{q,m}(n){\rm{\bf P}}_{q,m}(n\|n-1){\rm{\bf G}}_{q,m}^{H}(n)+\sigma_{q,m}^{2}(n)\right]^{-1},$ (19) $\displaystyle{\rm{\bf P}}_{q,m}(n\|n)=$ $\displaystyle\left[{\rm{\bf I}}-{\rm{\bf K}}_{q,m}(n){\rm{\bf G}}_{q,m}(n)\right]{\rm{\bf P}}_{q,m}(n\|n-1),$ (20) $\displaystyle{\widehat{{\rm{\bf X}}}_{q,m}(n\|n)}={\widehat{{\rm{\bf X}}}_{q,m}(n\|n-1)}+$ $\displaystyle{\rm{\bf K}}_{q,m}(n)\left[y_{q,m}(n)-{\rm{\bf G}}_{q,m}(n){\widehat{{\rm{\bf X}}}_{q,m}(n\|n-1)}\right].$ (21) Here $\sigma_{q,m}^{2}(n)$ represents the variance of the observation noise $z_{q,m}(n)$ in Eqn. (3). Also, CFO estimate $\widehat{\varepsilon}_{q,m}(n\|n)$ in the state vector ${\widehat{{\rm{\bf X}}}_{q,m}(n\|n)}$ should only use its real part as $\widehat{\varepsilon}_{q,m}(n\|n)={\rm Real}\left[\widehat{\varepsilon}_{q,m}(n\|n)\right]$. Finally, the EKF design presented above is only used for the parameter estimation of one TX user and one RX antenna. This basic EKF design is then iterated in a time-shared manner to estimate the parameters of all TX users ($1\leq q\leq Q$) and all RX antennas ($1\leq m\leq N_{A}$). ### III-B Interference Mitigation and Refined CFO Estimation Before describing the interference mitigation module, we first look at the refinement of the CFO estimates. Although TX user $q$ has only one CFO parameter, our proposed EKF design can result in $N_{A}$ different estimates that are derived from $N_{A}$ RX antennas, which are denoted as $\widehat{\varepsilon}_{q,m}(n\|n-1)$, $1\leq m\leq N_{A}$. As a result, we can use these $N_{A}$ different estimates to get a refined result $\widehat{\varepsilon}_{q}(n\|n-1)$, which is calculated as: $\displaystyle\widehat{\varepsilon}_{q}(n\|n-1)=$ $\displaystyle\sum_{m=1}^{N_{A}}\frac{1/{\rm{\bf P}}_{q,m}(n\|n-1)_{(1,1)}}{\sum_{r=1}^{N_{A}}1/{\rm{\bf P}}_{q,r}(n\|n-1)_{(1,1)}}\widehat{\varepsilon}_{q,m}(n\|n-1),$ (22) where ${\rm{\bf P}}_{q,m}(n\|n-1)_{(1,1)}$ denotes ${\rm{\bf P}}_{q,m}(n\|n-1)$’s element located at the $1^{\rm st}$ row and $1^{\rm st}$ column. Recall that $y_{q,m}(n)$ involved in (3) and (III-A) is extracted from the original RX signal $y_{m}(n)$ with the help of an interference mitigation strategy. Having derived the refined CFO estimate $\widehat{\varepsilon}_{q}(n\|n-1)$, now the interference mitigation process can be applied at $y_{q,m}(n)$ as follows: $\displaystyle y_{q,m}(n)=y_{m}(n)-$ $\displaystyle\sum_{u=1,u\neq q}^{Q}\exp\left(j\frac{2\pi\widehat{\varepsilon}_{u}(n\|n-1)}{N_{F}}\right)\cdot{\rm{\bf G}}_{u,m}(n){\widehat{{\rm{\bf X}}}_{u,m}(n\|n-1)}.$ ### III-C Adaptive Noise Variance Tracking Since $y_{q,m}(n)$ is extracted from $y_{m}(n)$ via interference mitigation module, the variance of $z_{q,m}(n)$, (i.e., $\sigma_{q,m}^{2}(n)$ used in Eqn. (III-A)), is varying during the convergence process of the interference mitigation module, which has to be adaptively tracked. Such variance tracking is based on the following observation: $\displaystyle\mathbb{E}\left\|y_{q,m}(n)-{\rm{\bf G}}_{q,m}(n){\widehat{\rm{\bf X}}}_{q,m}(n\|n-1)\right\|^{2}\approx$ $\displaystyle{\rm{\bf G}}_{q,m}(n){\rm{\bf P}}_{q,m}(n\|n-1){\rm{\bf G}}^{H}_{q,m}(n)+\sigma_{q,m}^{2}(n).$ (24) Using (19), noise variance $\sigma_{q,m}^{2}(n)$ can be tracked as: $\displaystyle\sigma_{q,m}^{2}(n)$ $\displaystyle=$ $\displaystyle\left[1-\frac{1-b}{1-b^{n+1}}\right]\cdot\sigma_{q,m}^{2}(n-1)+$ (25) $\displaystyle\frac{1-b}{1-b^{n+1}}\cdot\left\\{\max\left[e_{q,m}(n),0\right]\right\\},$ $\displaystyle e_{q,m}(n)$ $\displaystyle=$ $\displaystyle\left\|y_{q,m}(n)-{\rm{\bf G}}_{q,m}(n){\widehat{\rm{\bf X}}}_{q,m}(n\|n-1)\right\|^{2}-$ (26) $\displaystyle{\rm{\bf G}}_{q,m}(n){\rm{\bf P}}_{q,m}(n\|n-1){\rm{\bf G}}^{H}_{q,m}(n),$ where $b=0.99$ is the decay factor used to exponentially weight the history values. ### III-D Block Diagram The complete functional diagram of our proposed design is shown in Fig. 1. In this paradigm, received samples at all RX antennas are first processed in the interference mitigation module. Then the resultant samples are sequentially processed in the basic EKF module and noise variance tracking module. For the ease of description, all components in Fig. 1 are depicted in a parallel manner. However, in practice these design components can be implemented in a time-shared manner, and only one single EKF module is physically required. Figure 1: Block diagram of our proposed design. ## IV Simulations and Discussions ### IV-A Parameters Setup The OFDM system built in the simulation has a bandwidth of 20MHz and a FFT size $N_{F}=2048$. Each TX user’s CFO value is independently and randomly generated within the range of [-2, 2]444Since integer frequency offsets can generally be estimated during the coarse synchronization stage, CFO value at fine synchronization stage is usually between -0.5 and 0.5. But here we use range 2 to demonstrate our design’s estimation performance.. Wireless channels are generated using the SUI-3 channel model [8], which has $L_{\rm max}=3$ non-zero paths at the time domain. SNR in this paper is defined as the ratio of the signal power to the noise power at one RX antenna, i.e., ${\rm SNR}=\sigma_{R}^{2}/\sigma_{Z}^{2}$ where $\sigma_{R}^{2}$ is the total received signal power at one RX antenna that is coming from all TX users, and $\sigma_{Z}^{2}$ is the power of the background noise. In the simulation, CFO and channel parameters are estimated using one OFDM training symbol with $N_{F}=2048$ samples. Estimation results are validated via the mean square error (MSE) performance. Specifically, MSE for channel estimation is defined as a normalized version: $\displaystyle{\rm MSE}(h_{q,m})=\frac{\sum_{l=1}^{L_{\rm max}}\left\|{\widehat{h}}_{q,m}(p_{l}^{q,m})-h_{q,m}(p_{l}^{q,m})\right\|^{2}}{\sum_{l=1}^{L_{\rm max}}\left\|h_{q,m}(p_{l}^{q,m})\right\|^{2}}.$ (27) Cramer-Rao bounds for the MSE results of CFO and channel estimation can be derived according to [9] as: $\displaystyle{\rm CRB}_{\rm CFO}({\rm SNR})=\frac{3Q}{2\pi^{2}\cdot N_{F}\cdot{\rm SNR}\cdot N_{A}},$ (28) $\displaystyle{\rm CRB}_{\rm Chan}({\rm SNR})=\left(\frac{L_{\rm max}}{N_{F}}+\frac{3}{2N_{F}}\right)\frac{Q}{{\rm SNR}}.$ (29) TABLE I: Complexity Comparison (Number of Complex Multiplications) Design Name \| Number of Complex Multiplications ---\|--- Proposed Design \| $\approx L_{\rm max}^{3}+10L_{\rm max}^{2}+14L_{\rm max}+2$ Full-State EKF \| $\mathcal{O}\left\\{Q^{3}(L_{\rm max}N_{A}+1)^{3}\right\\}$ FFT-Block EKF [6] \| \| $\mathcal{O}\\{N_{A}N_{F}(QN_{A})^{2}(L_{\rm max}+1)^{2}$ --- $+QN_{A}(N_{A}N_{F})^{2}(L_{\rm max}+1)^{2}$ $+(QN_{A})^{3}(L_{\rm max}+1)^{3}\\}$ Parallel EKF [7] \| \| $\mathcal{O}\\{Q^{2}N_{F}(L_{\rm max}+1)^{2}+QN_{F}^{2}(L_{\rm max}+1)^{2}$ --- $+QN_{F}(L_{\rm max}+1)^{3}\\}$ Particle Filter [7] \| \| $\mathcal{O}\\{N_{P}Q^{2}N_{F}(L_{\rm max}+1)^{2}+N_{P}QN_{F}^{2}(L_{\rm max}+1)^{2}$ --- $+N_{P}QN_{F}(L_{\rm max}+1)^{3}\\}$ $N_{P}$ is the number of particle samples. EM method [5] \| \| $\mathcal{O}\\{N_{L}QN_{F}^{2}L_{\rm max}+N_{L}QN_{F}L_{\rm max}^{2}\\}$ --- $N_{L}$ is the number of iterations. ### IV-B Simulation Results We consider a cognitive network with one primary link and three secondary links (a total of 4 links). We investigate the CFO and channel estimations at one secondary RX user with $N_{A}=4$ RX antennas. Without loss of generality, we assume that this secondary RX user has the same received power from all TX users. We plot the MSE results of CFO and channel estimation in Fig. 2 and Fig. 3, respectively. The results show that our proposed design can closely approach the Cramer-Rao bounds. In addition, we repeat our simulation by disabling interference mitigation module, or noise variance tracking module. The corresponding results (Fig. 2 and Fig. 3) indicate that without interference mitigation, the estimation performance can be dramatically degraded. And without noise variance tracking module, there could be an error floor at high SNR values because of the inaccurate tracking of noise variance information. Using the values in Fig. 2 and Fig. 3, it is feasible to further investigate the BER/PER performance. But such discussions heavily depend on the designed receiver structure, which is omitted here for page limitation. ### IV-C Delay, Buffer Size and Complexity Analysis This subsection evaluates the issues of complexity, delay and buffer size in the considered designs. We first look at the complexity issue. In particular, we count the number of complex multiplications involved in our proposed EKF design, which is listed in Table I. And for comparison, in that table, we also list the complexity results of Full-State EKF, FFT-Block EKF [6], Parallel EKF [7], Particle filter [7], and EM method [5]. Here Full-State EKF represents the EKF that builds all $(L_{\rm max}N_{A}+1)Q$ states in (2) into one state vector, yielding high state dimension. We can see that our proposed design enjoys the lowest computation complexity, which is only at the order of $L_{\rm max}^{3}$. But Full-State EKF’s complexity is around $Q^{3}N_{A}^{3}$ higher than our design. Moreover, FFT-Block EKF, Parallel EKF, Particle Filter, and EM method’s complexities555Particle filters in [7] and EM designs in [5] have even higher complexity because of either the number of particle samples, or the number of iterations. all rely on FFT size $N_{F}$, which is significantly large in our case ($N_{F}=2048$). Now we further look at the delay and buffer size in the proposed design. Since our EKF scheme updates the Kalman estimate at each RX sample (not at each FFT block) in an online and real-time manner, it has low estimation delay and requires low buffer size. However, Particle Filter [7], Parallel EKF [7], and EM approach [5] all operate at an FFT-block basis with buffer size $N_{F}=2048$ samples, resulting in both a large delay and a large buffer size. Even worse, particle filter and EM method both need to process the FFT-block multiple times (e.g., particle samples in particle filter, and iterations in EM method), leading to additional estimation delay. ## V Conclusion This paper has presented a low-delay and low-complexity EKF design that can estimate the CFO and channel parameters in multi-user cognitive communications. We first present a fundamental EKF design that works at one TX user and one RX antenna. Then this basic EKF design is reused in a time-shared way to estimate the parameters for multiple TX users at multiple RX antennas. Besides, an interference mitigation strategy is proposed to estimate and cancel the multi-user interference at each RX sample. Moreover, adaptive noise variance tracking module is further employed to further enhance the estimation performance. Compared with existing related designs (FFT-Block EKF [6], Parallel EKF [7], Particle filter [7], and EM method [5]), our proposed design enjoys low computation complexity (because of pipelining and time-sharing design), low delay and low buffer size (due to its online and run-time estimation). Besides, its estimation performance can closely approach the Cramer-Rao bound. ## References * [1] R. van Nee and R. Prasad, _OFDM for Wireless Multimedia Communications_. Artech House, Inc., 2000\. * [2] Z. Cao, U. Tureli, and Y.-D. Yao, “Deterministic multiuser carrier-frequency offset estimation for interleaved OFDMA uplink,” _IEEE Transactions on Communications_ , vol. 52, no. 9, pp. 1585–1594, Sept. 2004. * [3] Y. Ma and R. Tafazolli, “Channel estimation for OFDMA uplink: a hybrid of linear and BEM interpolation approach,” _IEEE Transactions on Signal Processing_ , vol. 55, no. 4, pp. 1568–1573, Apr. 2007. * [4] M.-O. Pun, M. Morelli, and C.-C. J. Kuo, “Maximum-likelihood synchronization and channel estimation for OFDMA uplink transmissions,” _IEEE Transactions on Communications_ , vol. 54, no. 4, pp. 726–736, Apr. 2006. * [5] M.-O. Pun, S.-H. Tsai, and C.-C. J. Kuo, “An EM-based joint maximum likelihood estimation of carrier frequency offset and channel for uplink OFDMA systems,” in _IEEE 60th VTC_ , vol. 1, 26-29 2004, pp. 598–602. * [6] T. Roman, M. Enescu, and V. Koivunen, “Joint time-domain tracking of channel and frequency offsets for MIMO OFDM systems,” _Wireless Personal Communications_ , vol. 31, pp. 181–200, 2004. * [7] K. J. Kim, M.-O. Pun, and R. A. Iltis, “Joint carrier frequency offset and channel estimation for uplink MIMO-OFDMA systems using Parallel Schmidt Rao-Blackwellized particle filters,” _Communications, IEEE Transactions on_ , vol. 58, no. 9, pp. 2697 –2708, sep. 2010. * [8] _Channel models for fixed wireless applications_ , IEEE 802.16 Broadband Wireless Access Working Group IEEE 802.16a-03/01, 2003. * [9] P. Stoica and O. Besson, “Training sequence design for frequency offset and frequency-selective channel estimation,” _IEEE Transactions on Communications_ , vol. 51, no. 11, pp. 1910–1917, Nov. 2003. Figure 2: CFO estimation’s MSE results under different SNR values. Figure 3: Channel estimation’s MSE results under different SNR values.	arxiv-papers	2011-11-16T19:54:22	2024-09-04T02:49:24.414652	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Pengkai Zhao, Cong Shen", "submitter": "Pengkai Zhao", "url": "https://arxiv.org/abs/1111.3925" }
1111.4103	# Quantum correlations by four–wave mixing in an atomic vapor in a non- amplifying regime: a quantum beam splitter for photons. Quentin Glorieux quentin.glorieux@univ-paris-diderot.fr Luca Guidoni Samuel Guibal Jean-Pierre Likforman Thomas Coudreau Univ Paris Diderot, Sorbonne Paris Cité, Laboratoire Matériaux et Phénomènes Quantiques, UMR 7162 CNRS, F-75205 Paris, France ###### Abstract We study the generation of intensity quantum correlations using four–wave mixing in a rubidium vapor. The absence of cavity in these experiments allows to deal with several spatial modes simultaneously. In the standard, amplifying, configuration, we measure relative intensity squeezing up to 9.2 dB below the standard quantum limit. We also theoretically identify and experimentally demonstrate an original regime where, despite no overall amplification, quantum correlations are generated. In this regime a four–wave mixing set–up can therefore play the role of a photonic beam splitter with non–classical properties, i.e. a device that splits a coherent state input into two quantum correlated beams. ###### pacs: 42.50.Dv, 42.50.Lc, 42.50.Nn Non–classical ”intense” beam have been widely studied in a large variety of contexts including potential applications to quantum information protocols Braunstein05 ; Sangouard10 , fundamental issues in quantum mechanics such as entanglement and non–locality Reid09 , quantum imaging QuantumImaging or enhancement of the sensitivity of gravitational wave interferometers Caves81 . Quantum correlated beams are usually obtained through optical non–linear effects described as $\chi^{(2)}$ or $\chi^{(3)}$ non linearities present in a variety of media (see BachorBook for a review). In this paper, we study the generation of quantum correlation by using four–wave mixing (4WM) in a hot atomic vapor. Based on $\chi^{(3)}$ non linearity, 4WM is well known to generate intense non classical beams Yuen79 ; Slusher1985 ; Shelby86 ; Maeda87 . However, over the last 20 years, attention has been focused mainly on $\chi^{(2)}$ media Wu87 ; Heidmann87 ; Laurat05 ; Mehmet10 mainly because of their low losses (availability of high quality optical crystals). On the contrary, in hot vapors the presence of an atomic resonance enhances the non-linearity but also usually increases the losses. Yet, recently, it was shown that non–degenerate 4WM in atomic vapors can produce very large amounts of quantum correlations between intense beams McCormick2007 ; McCormick08 ; GlorieuxSPIE10 . Such a set–up has a significant advantage over $\chi^{(2)}$ media in that it does not require an optical cavity to enhance the nonlinearity and the related quantum effects. This is particularly important in the case of quantum imaging where spatially multimode quantum effects are involved QuantumImaging ; Boyer08 . Furthermore, the generated beams directly match the atomic resonance frequency of an atom–based quantum memory, a key requirement for quantum communications Sangouard10 . As noted above, the large nonlinear and quantum effects observed in 4WM originate from the presence of an atomic resonance. This resonance also induces incoherent effects, most notably absorption and spontaneous emission, which, in general, decrease the degree of quantum correlations. These possible drawbacks are often reduced by increasing the detuning from resonance, resulting in an overall amplification of the probe and conjugate beams. However, as we show, there exists a regime where quantum correlations can be observed despite the fact that the probe beam is de-amplified by propagation through the atomic vapor. In this regime, a 4WM setup then behaves as a beam–splitter separating an incoming beam in two different beams, without overall amplification. However, when the input beam is in a coherent state, the two output states are quantum correlated : we thus call this new device a quantum beam–splitter for photons. On the two input ports of the device are respectively sent the vacuum state and a coherent state and through the two output ports are emitted the quantum correlated states. This denomination omits the role of the pump which is crucial in this scheme as naturally no classical beam splitter can generate non classical states starting from coherent states. The simplest way to model theoretically such a device is to chain an ideal linear phase–insensitive amplifier with a partially transmitting medium. Despite the introduction of large losses, up to a level that cancels the gain, we show that quantum correlated beams can be generated in such a configuration. We then introduce the gemellity Treps05 , a criterion well adapted to describe experiments with non balanced beams. Figure 1: (Color Online) a) 4WM in hot atomic vapor schematic setup. b) Relevant levels of the Rb D1 line described as a double-$\Lambda$ system. $\Delta$ is the so called one–photon detuning and $\delta$ the two–photon detuning. We demonstrate, using a microscopic modelGlorieuxPRA10 , that 4WM in a hot atomic vapor can efficiently implement a quantum beam–splitter and we show that the limit for the maximum gemellity predicted in the linear amplifier model can be theoretically exceeded in this new regime. Finally we test these predictions experimentally. ## I 4WM in the amplifying regime The experiment is based on McCormick2007 and is described in detail in GlorieuxSPIE10 so that we only recall here its main features. A linearly polarized intense pump beam, frequency locked near the 85Rb $D_{1}$ line, is mixed with an orthogonally polarized weak probe beam inside an isotopically pure cell of length $L$. The relevant levels are shown in Fig. 1, a. At the output of the cell, due to 4WM, the probe beam is amplified and a conjugate beam is generated (see Fig. 1, b). After filtering out the pump beam with a polarizing beam splitter, intensity correlations between the probe and conjugate beams are measured by a pair of high quantum-efficiency photodiodes coupled to a spectrum analyzer. A high gain can be observed for a relatively large set of experimental parameters. The use of a heated cell yields a large number of atoms: for a temperature $T$ ranging from 100∘C to 150∘C, the atomic density $\mathcal{N}$ calculated from the Clausius–Clapeyron formula Alcock84 varies from 6$\times 10^{12}~{}$cm-3 to $10^{14}~{}$cm-3. Thus, the equivalent optical depth, $\mathcal{N}\sigma L$ varies between $5\times 10^{3}$ and $10^{5}$ where $\sigma$ is the atomic cross section for the 5$S^{1/2}$ $\to$ 5$P^{1/2}$ transition in 85Rb. These atoms interact with beams close to resonance : the single photon detuning $\Delta$ is typically 1 GHz (on the order of the Doppler broadening) while the two–photon detuning $\delta$ is smaller than 10 MHz. Within these domains of parameters, explored systematically GlorieuxSPIE10 , we have identified an optimal noise reduction regime. For $\Delta=+750$ MHz, $\delta=+6$ MHz, $T=118^{\circ}$C, $P_{pump}=1200$ mW (corresponding to a Rabi frequency $\Omega=1$ GHz), gain on the incoming probe beam up to 20 can be observed. In these conditions, Fig. 2 shows the noise power of the intensity difference of the probe and conjugate as a function of the analysis frequency after correcting for the electronic noise:significant noise reduction is observed in the range 500 kHz to 5 MHz and with a maximal noise reduction of 9.2 dB$\pm$0.5 dB below the SQL is observed between 1 and 2 MHz. This value is slightly larger than the best results obtained to date with 4WM McCormick08 and very close to those obtained with OPOs Laurat05 . The matching of the atomic resonance of Rb turns this setup into an ideal source of non-classical light to interact with Rb vapor quantum memory PL1 ; PL2 . Figure 2: (Color Online) Noise power of the intensity difference between the probe and conjugate beams as a function of the frequency after correcting for the electronic noise. A reduction of 9.2 dB$\pm$0.5 dB below the SQL is reached at 1 MHz analysis frequency. ## II Quantum beam splitter regime In the previously described regime,the larger was the gain, the larger were the quantum correlations. However, we will show here that this not a necessary condition and that one can, somewhat counter-intuitively, observe significant quantum correlations in the absence of overall gain. ### II.1 Ideal linear amplifier model In an ideal phase–insensitive amplifier, an input probe beam is amplified while a conjugate beam is generated. At the output of the amplifier, neglecting the contribution of the noise to the average number of photons, the probe beam has an intensity $GI_{0}$ and the conjugate beam has an intensity $(G-1)I_{0}$ where $G$ denotes the gain and $I_{0}$ the input probe beam intensity. Taking into account the ideal character of the amplifier, no noise is added and the intensity difference at the output has a noise ratio $1/(2G-1)$ with respect to the input. For probe and conjugate at the input respectively coherent and vacuum states, this noise ratio is equal to the quantum correlations at the output of the amplifier. If we now extend this model by including losses at the output of the medium on the probe and/or conjugate beams, one would expect a reduction in these correlations as it is well known that losses are detrimental to squeezing. Let us recall that this is not always the case as the beams intensity are not balanced: a small amount of extra losses on the probe beam will tend to make the two beams more balanced and thus improve the noise reduction on the intensity difference as noted e.g. in Jasperse11 . Contrary to the case of Optical Parametric Oscillators above threshold Laurat05 , 4WM naturally generates unbalanced beams. Unbalanced beams may exhibit strong quantum correlations but the measurement of the noise on the intensity difference is not an ideal criterion in this case. It is useful to introduce the gemellity $\mathcal{G}$ Laurat03 ; Laurat04 ; Treps05 ; Laurat05b defined by : $\mathcal{G}=\frac{F_{a}+F_{b}}{2}-\sqrt{C_{ab}^{2}F_{a}F_{b}+\left(\frac{F_{a}-Fb}{2}\right)^{2}},$ (1) where $F_{i}=\langle\hat{X}_{i}\hat{X}_{i}\rangle$ with $i$ used for $a$ (probe) and $b$ (conjugate), $C_{ab}=\frac{\langle\hat{X}_{a}\hat{X}_{b}\rangle}{\sqrt{F_{a}F_{b}}}$ and $\hat{X}_{i}$ is the amplitude quadrature of the related field as defined in Treps05 . In case of balanced beams the gemellity is equal to the normalized noise on the difference between the fluctuations of the two measurements : $\mathcal{G}=\frac{\langle(\hat{X}_{i}-\hat{X}_{j})^{2}\rangle}{2}$, which is the quantitative measure of the maximal “non–classicality” that can be extracted from the correlated beams Treps05 . For balanced beams, such as the ones produced in the limit of infinite gain, this value is equal to the standard criterion, namely the intensity noise difference. In the conditions above where the intensity difference noise is -9.2 dB, the noise on the individual beams are $F_{a}=F_{b}$ = +12 dB at 1 MHz yielding a gemellity $\mathcal{G}$ = -9.8 dB $\pm$ 0.5 dB. This value is comparable with record values measured with an OPO above threshold Laurat05 and moreover a large number of spatial modes (estimated to 100 in this particular configuration) are squeezed simultaneously Boyer08 . Using this criteria and introducing losses on the probe ($T_{a}$) and conjugate ($T_{b}$) beams so that the overall transmission is equal to one ($T_{a}G+T_{b}(G-1)=1$), it is straightforward to show that there always exists a region in the parameter space where a gemellity lower that one is expected. To our knowledge, this phenomenon, albeit simple, has neither been discussed nor observed. The larger quantum correlations reachable with no overall amplification corresponds to the situation of a gain $G=1.23$, a transmission of 0.62 on the probe beam and perfect transmission on the conjugate beam. This configuration gives the limit for the gemellity reachable by this simple model : $\mathcal{G}=-2.8$ dB. ### II.2 Microscopic model To investigate further this effect, we have studied the 4WM process using a microscopic model based on the cold–atom model described extensively in GlorieuxPRA10 . This model assumes the simplified double–$\Lambda$ level structure of Fig. 1, right. The Heisenberg–Langevin approach is used to obtain the relevant classical quantities (probe gain $G_{a}$, conjugate gain $G_{b}$ defined with respect to $I_{0}$) as well as the quantum properties of the output beams. In particular, it is possible to calculate noise spectra that allow for quantifying quantum correlations both in terms of intensity–difference noise $S_{N_{-}}$ and for the unbalanced case in terms of gemellity $\mathcal{G}$. In the regime of high amplification previously described, this model is in good quantitative agreement with the measured correlations GlorieuxPhD . Exploring the parameters space in this model, we have found a new region where the 4WM process generates quantum correlations in the absence of overall amplification. This regime is therefore very similar to the linear amplifier model followed by a lossy medium described above. Nevertheless, the microscopic model predicts that in this regime, the gemellity can be significantly enhanced in contrast to the linear model and exceeds the -2.8 dB limit discussed previously. Let us start by presenting the classical behavior of the probe and conjugate beams in the region of interest of parameter space (theoretical data are compared to the experimental results). We plot in Fig. 3, the gain for the two fields as a function the two–photon detuning $\delta$. The main difference with respect to the high gain parameter region is the choice of the atomic density (experimentally driven by the temperature). The large gain results of Fig. 2 were obtained for a temperature of 118 ∘C while the curves in Fig. 3 are obtained for $T=95^{\circ}$C. The approximately one order of magnitude lower optical density, together with the different choice of $\delta$ and $\Delta$, explain the drastic reduction of $G_{a}$ and $G_{b}$. ”Beam–splitter” regime is obtained near the two-photon resonance, where $G_{a}$ goes to zero due to a Raman process involving a probe and a pump photonGlorieuxPRA10 . Due to the pump–induced AC-Stark shift, this two–photon resonance is shifted to negative values of $\delta$ and its exact position depends on the one-photon detuning $\Delta$ and on the pump Rabi frequency $\Omega$. Within a very narrow region of parameter space, the sum of the two beams output intensities becomes slightly smaller or almost equal to the input probe intensity. It is interesting to note that for potential applications this very narrow feature could be considered as a limitation. Notwithstanding we have verified numerically, by changing simultaneously $\Delta$, $\Omega$ and the optical depth $\mathcal{N}\alpha L$, that the detuning for which this system exhibits the behavior of a quantum beam splitter can be tuned over more than 100 MHz. As already remarked in GlorieuxPRA10 , we note that, despite the fact that the model is based on a cold atom sample, it yields without any adjustable parameter a qualitative agreement with the experimental data obtained in a hot vapor. Figure 3: (Color Online) Theoretically predicted (left) and experimentally measured (right) gain for the probe beam ($G_{a}$, ) and conjugate beam ($G_{b}$, black) as a function of the two–photon detuning $\delta$. The parameters used in the simulations are : , optical depth $\mathcal{N}\alpha L=500$, Pump Rabi frequency $\Omega$ = 0.42 GHz, single photon detuning $\Delta/2\pi$=0.8 GHz. Measured parameters are : pump power $P$=0.6 W ($\Omega/2\pi=0.4$ GHz), T=95∘C, single photon detuning $\Delta/2\pi$=0.8 GHz. Figure 4: (Color Online) Quantum intensity correlations between the probe and conjugate beams as a function of the analysis frequency with the same parameters as above and $\delta/2\pi$=-52 MHz. ### II.3 Demonstration of the quantum beam splitter Motivated by these theoretical predictions, we have experimentally investigated this original regime. We plot in Fig. 4 the experimentally measured intensity difference noise as a function of the analysis frequency $\omega$. We observe significant quantum correlations, down to 1.0$\pm 0.2$ dB below the SQL around an analysis frequency of 1 MHz. At the same time, the power of the two beams normalized to the probe input power is measured to be 0.65 and 0.35 for the probe and conjugate respectively. This demonstrates clearly the behavior of a quantum beam splitter for photons where one laser beam is split into two beams without gain but generating quantum correlations. We note that the measured noise reduction is slightly smaller than the one predicted theoretically (Fig. 4): this discrepancy can be attributed to the fact that the model is based on a cold atomic sample, far from the experimental regime. In this situation, $\mathcal{G}$ can be calculated to compare it to the theoretical limit of the linear amplifier model. By measuring the noise on the two individuals beams respectively equal to + 3 and + 2 dB for probe and conjugate, we obtain a value of the gemellity equal to $\mathcal{G}=-1.8\pm 0.5$ dB. This value does not exceed the maximum limit of -2.8 dB predicted by the linear amplifier model. As previously noted, the theoretical model does not take into account the velocity distribution of the atoms, thus time transit effects and Doppler broadening which are expected to play a detrimental role. This can explain why the linear amplifier model limit cannot be reached in this configuration whereas the microscopic model predicts that gemellities better than $\mathcal{G}$=-3.2 dB can be obtained with the above parameters and an optical depth, $\mathcal{N}\sigma L=1500$. We have thus shown firstly that generating quantum correlations does not require overall amplification and, secondly, that the ideal linear amplifier is not the ideal device to perform this operation but that 4WM in atomic vapours presents an interesting avenue in this context. This setup could also be used as the input beam splitter introducing quantum correlations for an original version of the Mach-Zender interferometer as well as in a so-called SU(1,1) interferometer Yurke86 . ## III Conclusion We have studied the production of quantum correlated beams in four–wave mixing in a 85Rb cell. First, we have identified and experimentally realized an optimal regime in the high gain region where intensity–difference noise down to -9.2 dB below the standard quantum limit (gemellity $\mathcal{G}=-9.8~{}dB$) have been measured. This result is important in the domain of quantum communications where both large non–classical effects and the availability of an atom–based storage media form strong requirements Sangouard10 ; PL1 ; PL2 . We have also predicted and observed an original regime where quantum correlations are present despite significant losses on the probe beam. This regime is of particular interest, because it can occur in a situation in which the sum of the two output beam intensities is smaller or equal to the input probe intensity. Therefore the atomic medium controlled by the pump laser acts like a beam splitter device that creates quantum correlations (quantum beam splitter). Although this effect could in principle be observed with an ideal amplifier, it is to our knowledge the first demonstration of it. In this context, we have discussed the use of the gemellity criterion more appropriate in the case of unbalanced beams produced by four–wave mixing. Finally, a microscopic model allowed us to demonstrate that 4WM in the quantum beam splitter regime can beat theoretically the limit of quantum correlations predicted by the model of a linear amplifier followed by a lossy medium. In our experiment, with a hot atomic vapor, a value of $\mathcal{G}=-1.8\pm 0.5$ dB has been reported. Although the parameter values required to beat the linear amplifier model limit are presently beyond reach of experiments performed with cold atoms, our model provides an interesting avenue to surpass this limit using hot or cold atoms. ## Acknowledgements We thank E. Arimondo, R. Dubessy and P. Lett for fruitful discussions. We thank N. Treps for the loan of high–efficiency photodiodes. ## References * (1) V. Boyer et al., Phys. Rev. Lett. 100, 143601 (2008) * (2) S. Braunstein and P. Van Loock, Rev. Mod. Phys. 77, 513 (2005) * (3) N. Sangouardet al., J. Opt. Soc. Am. B 27, 137 (2010) * (4) M. D Reidet al., H. A Bachor, U. L Andersen, G Leuchs, Rev. Mod. Phys, 81, 1727 (2009) * (5) M. I. Kolobov, ed., Quantum imaging (Springer, New York, 2007); A. Gatti, E. Brambilla, and L. Lugiato, in Progress in Optics ed. E. Wolf, 51 p. 251 (Elsevier, 2008). * (6) C. M. Caves , Phys. Rev. D 23, 1693 (1981) * (7) H.A. Bachor and T.C. Raplh, A Guide to Experiments in Quantum Optics, (Wiley-VCH, 2004). * (8) H. P. Yuen and J. H. Shapiro,Opt. Let. 4, 334 (1979) * (9) R.E. Slusher et al., Phys. Rev. Lett. 55, 2409 (1985). * (10) R.M. Shelby et al., Phys. Rev. Lett. 57, 691 (1986) * (11) M. W. Maeda et al., Opt. Lett. 12, 161 (1987) * (12) L.A. Wu et al., Phys. Rev. Lett. 57, 2520 (1986) * (13) A. Heidmann et al., Phys. Rev. Lett. 59, 2555 (1987) * (14) J. Laurat et al., Opt. Lett. 30, 1177 (2005) * (15) J. Laurat et al., arXiv:quant-ph/0510063v1. * (16) J. Laurat et al., Phys. Rev. Lett. 91, 213601 (2003) * (17) J. Laurat et al., Phys. Rev. A 69, 033808 (2004) * (18) M. Mehmet et al., Phys. Rev. A 81, 013814 (2010) * (19) C. F. McCormick et al., Opt. Lett.32, 178 (2007). * (20) C.F. McCormick et al., Phys. Rev. A 78, 43816 (2008) * (21) Q. Glorieux et al., Proc. SPIE, 7727, 772703 (2010) * (22) M. Hosseini et al., Nat. Comm. 2,174 (2011) * (23) M. Hosseini et al., Nat. Phys. (2011) - doi:10.1038/nphys2021 * (24) N. Treps and C. Fabre, Laser Physics 15, 187 (2005) * (25) C. B. Alcock et al., Can. Metal. Quart. 23, 309 (1984). * (26) Q. Glorieux et al., Phys. Rev. A 82, 033819(2010) * (27) C.M. Caves, Phys. Rev. D 26, 1817 (1982) * (28) M. Jasperse et al., Opt. Exp., 19, 3765 (2011) * (29) Q. Glorieux, Thèse de l’Université Paris Diderot (2010), available at http://tel.archives-ouvertes.fr/tel-00558505/fr/ * (30) B. Yurke et al., Phys. Rev. A 33, 4033 (1986)	arxiv-papers	2011-11-17T14:08:45	2024-09-04T02:49:24.424992	{ "license": "Public Domain", "authors": "Quentin Glorieux, Luca Guidoni, Samuel Guibal, Jean-Pierre Likforman,\n and Thomas Coudreau", "submitter": "Quentin Glorieux", "url": "https://arxiv.org/abs/1111.4103" }
1111.4108	Jordan product determined points in matrix algebras Wenlei Yang 111E-mail address: yang121045760@yahoo.cn, Jun Zhu222E-mail address: zhu$\\_$gjun@yahoo.com.cn Institute of Mathematics, Hangzhou Dianzi University, Hangzhou 310018, People’s Republic of China Abstract Let $M_{n}(R)$ be the algebra of all $n\times n$ matrices over a unital commutative ring $R$ with 6 invertible. We say that $A\in M_{n}(R)$ is a Jordan product determined point if for every $R$-module $X$ and every symmetric $R$-bilinear map $\\{\cdot,\cdot\\}$ : $M_{n}(R)\times M_{n}(R)\to X$ the following two conditions are equivalent: (i) there exists a fixed element $w\in X$ such that $\\{x,y\\}=w$ whenever $x\circ y=A$, $x,y\in M_{n}(R)$; (ii) there exists an $R$-linear map $T:M_{n}(R)^{2}\to X$ such that $\\{x,y\\}=T(x\circ y)$ for all $x,y\in M_{n}(R)$. In this paper, we mainly prove that all the matrix units are the Jordan product determined points in $M_{n}(R)$ when $n\geq 3$. In addition, we get some corollaries by applying the main results. AMS Classification: 15A04 15A27 Keywords : Jordan product determined point; matrix algebra; Jordan all- multiplicative point; Jordan all-derivable point ## 1\. Introduction In this paper, we will mainly discuss Jordan product determined points on matrix algebras. Before proceeding let us fix some symbols and notations in this paper. Let $M_{n}(R)$ be the algebra of all $n\times n$ matrices over a unital commutative ring $R$ with 6 invertible. Matrix units are denoted by $e_{ij}$ and the Jordan product “$\circ$” is defined as $x\circ y=xy+yx$. The identity matrix is denoted by $I$. By $M_{n}(R)^{2}$ we denote the $R$-linear span of all elements of the form $xy$ where $x,y\in M_{n}(R)$. The concept of zero product (resp. Jordan product, Lie product) determined algebras was introduced by Brešar et al. [5]. According to [5], $M_{n}(B)$ ($n\geq 2$) is zero product determined where $B$ is a unital algebra. If $B$ is a unital algebra with 2 invertible, then $M_{n}(B)$ ($n\geq 3$) is zero Jordan product determined. From the results above, we can study the linear maps preserving zero product (resp. Jordan product) and derivable (resp. Jordan derivable) at zero point respectively. Wang et al. [1, 2] showed that (1) if a symmetric bilinear map $\\{\cdot,\cdot\\}$ : $M_{n}(R)\times M_{n}(R)\to X$ satisfies the condition that $\\{u,u\\}=\\{e,u\\}$ whenever $u^{2}=u$, then there exists a linear map $f$ from $M_{n}(R)$ to $X$ such that $\\{x,y\\}=f(x\circ y)$ for all $x,y\in M_{n}(R)$; and (2) if an invertible linear map $\delta$ on $M_{n}(R)$ preserves identity-product, then it is a Jordan automorphism; and a linear map $\sigma$ on $M_{n}(R)$ is derivable at the identity matrix if and only if it is an inner derivation. Zhu et al. [3] showed that for every $G\in M_{n}$, det$G=0$, is an all-multiplicative point in $M_{n}$. Gong and Zhu [6] considered the case of Jordan all-multiplicative point in $M_{n}$. Zhu et al. [4] showed that a matrix $G$ is an all-derivable point in $M_{n}$ if and only if $G\neq 0$. Zhao et al. [7] showed that every element of the algebra of all upper triangular matrices is a Jordan all- derivable point. Motivated by the concepts and results above, we will consider Jordan product determined points in matrix algebras. For $A\in M_{n}(R)$, we say that $A$ is a Jordan product determined point if for every $R$-module $X$ and every symmetric $R$-bilinear map $\\{\cdot,\cdot\\}$ : $M_{n}(R)\times M_{n}(R)\to X$ the following two conditions are equivalent: (i) there exists a fixed element $w\in X$ such that $\\{x,y\\}=w$ whenever $x\circ y=A$, $x,y\in M_{n}(R)$; (ii) there exists an $R$-linear map $T:M_{n}(R)^{2}\to X$ such that $\\{x,y\\}=T(x\circ y)$ for all $x,y\in M_{n}(R)$. We say that $G\in M_{n}(R)$ is a Jordan all-multiplicative point in $M_{n}(R)$ if for every $M_{n}(R)$-module $X$ and every Jordan multiplicative $R$-linear map $\varphi$ : $M_{n}(R)\to X$ at $G$ (i.e. $\varphi(S\circ T)=\varphi(S)\circ\varphi(T)$ for any $S,T\in M_{n}(R)$, $S\circ T=G$) with $\varphi(I)=I$ is a multiplicative mapping in $M_{n}(R)$. We say that $H\in M_{n}(R)$ is a Jordan all-derivable point in $M_{n}(R)$ if for every $M_{n}(R)$-module $X$ and every Jordan derivable $R$-linear map $\varphi$ : $M_{n}(R)\to X$ at $H$ (i.e. $\varphi(S\circ T)=\varphi(S)\circ T+S\circ\varphi(T)$ for any $S,T\in M_{n}(R)$, $S\circ T=H$) with $\varphi(I)=0$ is a Jordan derivation in $M_{n}(R)$. The above two definitions are somewhat different from [6] and [7]. In this paper, we will prove that every matrix unit $e_{ij}$ is a Jordan product determined point in $M_{n}(R)$ when $n\geq 3$. As an application of the result above, we will show that every matrix unit $e_{ij}$ is a Jordan all-multiplicative point and a Jordan all-derivable point respectively. This paper is organized as follows. Section 2 concerns Jordan product determined points in $M_{n}(R)$, and we obtain the major results Theorem 2.2 and Theorem 2.3 in this paper. In Section 3, we get some corollaries by applying the main results in section 2. ## 2\. Jordan product determined points in $M_{n}(R)$ According to [5], we give the lemma below. Lemma 2.1. For $A\in M_{n}(R)$, $A$ is a Jordan product determined point if and only if for every $R$-module $X$ and every symmetric $R$-bilinear map $\\{\cdot,\cdot\\}$ satisfy the condition (i), the following condition (iii) holds true. (iii) for every $x_{t},y_{t}\in M_{n}(R)$ with $\sum_{t=1}^{l}x_{t}\circ y_{t}=0$, $t=1,2,\dots,l$, $\sum_{t=1}^{l}\\{x_{t},y_{t}\\}=0$ holds true. Proof. Obviously, the “only if” part holds true. Conversely, if the condition (iii) holds true, we can define $R$-linear map $T$ : $M_{n}(R)^{2}\to X$ as $T(\sum_{t}x_{t}\circ y_{t})=\sum_{t}\\{x_{t},y_{t}\\}$ according to [5]. Then $T$ satisfies condition (ii). We only need to prove that $T$ is well defined. Indeed, if condition (iii) is fulfilled, $T$ is well defined obviously. Hence $A$ is a Jordan product determined point. $\Box$ Theorem 2.2. $e_{ss}$, $s\in\\{1,2,\dots,n\\}$, is a Jordan product determined point in $M_{n}(R)$ when $n\geq 3$. Proof. Let $s$ be a fixed number, $X$ be an $R$-module, $\\{\cdot,\cdot\\}$ : $M_{n}(R)\times M_{n}(R)\to X$ be a symmetric (i.e. $\\{x,y\\}=\\{y,x\\}$) $R$-bilinear map. Now we assume that there exists a fixed element $w\in X$ such that $\\{x,y\\}=w$ whenever $x\circ y=e_{ss}$, $x,y\in M_{n}(R)$. Throughout the proof, $i,j,k,m$ will denote arbitrary indices. We begin by noticing that $(\frac{1}{2}e_{ss})\circ e_{ss}=e_{ss}$ and as $\\{\cdot,\cdot\\}$ is symmetric, then $\\{\frac{1}{2}e_{ss},e_{ss}\\}=w=\\{e_{ss},\frac{1}{2}e_{ss}\\}.$ (1) Next we suppose $n\geq 3$ and divide the proof into three steps. Step 1. In this step, we assume $i\neq s,j\neq s,k\neq s\;\mathrm{and}\;m\neq s$. Case 1.1. $i\neq m\;\mathrm{and}\;j\neq k$. Since $(\frac{1}{2}e_{ss}+e_{ij})\circ e_{ss}=e_{ss}$ and as $\\{\cdot,\cdot\\}$ is symmetric, we have that $\\{e_{ij},e_{ss}\\}=0=\\{e_{ss},e_{ij}\\}.$ (2) Noting $(\frac{1}{2}e_{ss}+e_{ij})\circ(e_{ss}+e_{km})=e_{ss}$, it follows $\\{\frac{1}{2}e_{ss}+e_{ij},e_{ss}+e_{km}\\}=w$. As $\\{\cdot,\cdot\\}$ is symmetric, applying (1) and (2), this yields $\\{e_{ij},e_{km}\\}=0.$ (3) Case 1.2. $i,j,k$ are distinct. From $(\frac{1}{2}e_{ss}+e_{ik})\circ(e_{ss}+e_{kk}-e_{ii})=e_{ss}$, we obtain that $\\{\frac{1}{2}e_{ss}+e_{ik},e_{ss}+e_{kk}-e_{ii}\\}=w$. Because $\\{\cdot,\cdot\\}$ is symmetric, it follows from (1) and (2) that $\\{e_{ik},e_{kk}\\}=\\{e_{ik},e_{ii}\\}=\\{e_{ii},e_{ik}\\}=\\{e_{kk},e_{ik}\\}.$ (4) Now we assume $n>3$. As $(\frac{1}{2}e_{ss}+e_{ij}+e_{ik})\circ(e_{ss}+e_{jk}-e_{kk})=e_{ss}$, we derive $\\{\frac{1}{2}e_{ss}+e_{ij}+e_{ik},e_{ss}+e_{jk}-e_{kk}\\}=w$. Using (1), (2) and (3), this can be reduced to $\\{e_{ij},e_{jk}\\}=\\{e_{ik},e_{kk}\\}.$ (5) (4) together with (5) yield $\\{e_{ij},e_{jk}\\}=\\{e_{ik},e_{kk}\\}=\\{e_{ik},e_{ii}\\}=\\{e_{ii},e_{ik}\\}.$ (6) Since $\\{\cdot,\cdot\\}$ is symmetric, it follows that $\\{e_{ii},e_{ik}\\}=\\{e_{ik},e_{ii}\\}=\\{e_{kk},e_{ik}\\}=\\{e_{jk},e_{ij}\\}.$ (7) Case 1.3. $i\neq j$. By $(\frac{1}{2}e_{ss}+\frac{1}{2}e_{ii}+e_{ij}-\frac{1}{2}e_{jj})\circ(e_{ss}+e_{ji}-e_{ii}+e_{jj})=e_{ss}$, it is clear that $\\{\frac{1}{2}e_{ss}+\frac{1}{2}e_{ii}+e_{ij}-\frac{1}{2}e_{jj},e_{ss}+e_{ji}-e_{ii}+e_{jj}\\}=w$. Then we can get from (1), (2), (3) and (4) that $\\{e_{ij},e_{ji}\\}=\frac{1}{2}\\{e_{ii},e_{ii}\\}+\frac{1}{2}\\{e_{jj},e_{jj}\\}.$ (8) Step 2. In this step, we consider some of the indices of the matrix units equal to $s$. Case 2.1. $i\neq s,j\neq s,k\neq s\;\mathrm{and}\;m\neq s$. For $(\frac{1}{2}e_{ss}-e_{si})\circ(e_{ss}-e_{ii})=e_{ss}$, we have that $\\{\frac{1}{2}e_{ss}-e_{si},e_{ss}-e_{ii}\\}=w$. Applying (1) and (2), it can be reduced to $\\{e_{si},e_{ss}\\}=\\{e_{si},e_{ii}\\}$. As $\\{\cdot,\cdot\\}$ is symmetric, then it follows that $\\{e_{si},e_{ss}\\}=\\{e_{si},e_{ii}\\}=\\{e_{ss},e_{si}\\}=\\{e_{ii},e_{si}\\}.$ (9) Since $(\frac{1}{2}e_{ss}-e_{is})\circ(e_{ss}-e_{ii})=e_{ss}$, a similar discussion as above shows that $\\{e_{is},e_{ss}\\}=\\{e_{is},e_{ii}\\}=\\{e_{ss},e_{is}\\}=\\{e_{ii},e_{is}\\}.$ (10) Let $j\neq k$, then we have $(\frac{1}{2}e_{ss}+e_{sj})\circ(e_{ss}+e_{km}-e_{jj})=e_{ss}$ and it implies that $\\{\frac{1}{2}e_{ss}+e_{sj},e_{ss}+e_{km}-e_{jj}\\}=w$. As $\\{\cdot,\cdot\\}$ is symmetric and by (1), (2) and (9), this yields $\\{e_{sj},e_{km}\\}=0=\\{e_{km},e_{sj}\\}\;\mathrm{if}\;j\neq k.$ (11) From $(\frac{1}{2}e_{ss}+e_{si})\circ(e_{ss}-2e_{si})=e_{ss}$, we derive $\\{\frac{1}{2}e_{ss}+e_{si},e_{ss}-2e_{si}\\}=w$. Then it follows from (1) and (9) that $\\{e_{si},e_{si}\\}=0.$ (12) Since $(\frac{1}{2}e_{ss}+e_{sj}-\frac{1}{2}e_{si})\circ(e_{ss}+e_{si}-e_{jj})=e_{ss}$ when $i\neq j$, then we have $\\{\frac{1}{2}e_{ss}+e_{sj}-\frac{1}{2}e_{si},e_{ss}+e_{si}-e_{jj}\\}=w$. Using (1), (2), (9), (11) and (12), it is clear that $\\{e_{sj},e_{si}\\}=0\;\mathrm{if}\;i\neq j.$ (13) If $j\neq m$, from $(\frac{1}{2}e_{ss}+e_{js})\circ(e_{ss}+e_{km}-e_{jj})=e_{ss}$ we can get that $\\{\frac{1}{2}e_{ss}+e_{js},e_{ss}+e_{km}-e_{jj}\\}=w$. As $\\{\cdot,\cdot\\}$ is symmetric and by (1), (2) and (10), it follows $\\{e_{js},e_{km}\\}=0=\\{e_{km},e_{js}\\}\;\mathrm{if}\;j\neq m.$ (14) For $(\frac{1}{2}e_{ss}+e_{is})\circ(e_{ss}-2e_{is})=e_{ss}$, we have that $\\{\frac{1}{2}e_{ss}+e_{is},e_{ss}-2e_{is}\\}=w$. Applying (1) and (10), it can be reduced to $\\{e_{is},e_{is}\\}=0.$ (15) Assume $i\neq j$, it follows from $(\frac{1}{2}e_{ss}+e_{js}-\frac{1}{2}e_{is})\circ(e_{ss}+e_{is}-e_{jj})=e_{ss}$ that $\\{\frac{1}{2}e_{ss}+e_{js}-\frac{1}{2}e_{is},e_{ss}+e_{is}-e_{jj}\\}=w$. Using (1), (2), (10), (14) and (15), this yields $\\{e_{js},e_{is}\\}=0\;\mathrm{if}\;i\neq j.$ (16) Case 2.2. $i\neq s,k\neq s\;\mathrm{and}\;j\neq s$. Since $(\frac{1}{2}e_{ss}+e_{ki}-e_{ks}-\frac{1}{2}e_{ii})\circ(e_{ss}+e_{is})=e_{ss}$ if $k\neq i$, then we have that $\\{\frac{1}{2}e_{ss}+e_{ki}-e_{ks}-\frac{1}{2}e_{ii},e_{ss}+e_{is}\\}=w$. By (1), (2), (10) and (16), this yields $\\{e_{ki},e_{is}\\}=\\{e_{ks},e_{ss}\\}$ if $k\neq i$. For our purpose, it is more convenient to rewrite this equation as $\\{e_{ik},e_{ks}\\}=\\{e_{is},e_{ss}\\}$ if $i\neq k$. As $\\{\cdot,\cdot\\}$ is symmetric, then we can conclude from the above equation and (10) that $\\{e_{ik},e_{ks}\\}=\\{e_{is},e_{ss}\\}=\\{e_{ii},e_{is}\\}=\\{e_{is},e_{ii}\\}=\\{e_{ss},e_{is}\\}=\\{e_{ks},e_{ik}\\}\;\mathrm{if}\;i\neq k.$ (17) If $i\neq k$, then we have $(\frac{1}{2}e_{ss}+e_{ik}-e_{sk}-\frac{1}{2}e_{ii})\circ(e_{ss}+e_{si})=e_{ss}$. By a similar discussion as above, this yields $\\{e_{sk},e_{ki}\\}=\\{e_{ss},e_{si}\\}=\\{e_{si},e_{ii}\\}=\\{e_{ii},e_{si}\\}=\\{e_{si},e_{ss}\\}=\\{e_{ki},e_{sk}\\}\;\mathrm{if}\;i\neq k.$ (18) If $j\neq k$, from $(\frac{1}{2}e_{ss}+e_{sk}+e_{jk}-\frac{1}{2}e_{js})\circ(e_{ss}+e_{js}-e_{kk})=e_{ss}$ we have that $\\{\frac{1}{2}e_{ss}+e_{sk}+e_{jk}-\frac{1}{2}e_{js},e_{ss}+e_{js}-e_{kk}\\}=w$. For $\\{\cdot,\cdot\\}$ is symmetric, applying (1), (2), (14), (15), (17) and (18), it follows $\\{e_{sk},e_{js}\\}=\\{e_{jk},e_{kk}\\}=\\{e_{js},e_{sk}\\}\;\mathrm{if}\;j\neq k.$ (19) Case 2.3. $i\neq s$. Since $(2e_{si}+\frac{1}{2}e_{is}-e_{ii})\circ(e_{si}+\frac{1}{4}e_{is}+\frac{1}{2}e_{ii})=e_{ss}$, we obtain $\\{2e_{si}+\frac{1}{2}e_{is}-e_{ii},e_{si}+\frac{1}{4}e_{is}+\frac{1}{2}e_{ii}\\}=w$. Using (1), (12), (15), (17) and (18), this can be reduced to $\\{e_{si},e_{is}\\}+\\{e_{is},e_{si}\\}=\\{e_{ss},e_{ss}\\}+\\{e_{ii},e_{ii}\\}$. As $\\{e_{si},e_{is}\\}=\\{e_{is},e_{si}\\}$, it leads to $\\{e_{si},e_{is}\\}=\\{e_{is},e_{si}\\}=\frac{1}{2}\\{e_{ss},e_{ss}\\}+\frac{1}{2}\\{e_{ii},e_{ii}\\}.$ (20) Step 3. Now concluding from case 1.1 and case 2.1, we can obtain $\\{e_{ij},e_{km}\\}=0\;\mathrm{for\;every}\;i,j,k,m,\;\mathrm{if}\;i\neq m\;\mathrm{and}\;j\neq k.$ (21) If $i,j,k$ are distinct and $n=3$, it follows from (4), (17), (18) and (19) that $\\{e_{ij},e_{jk}\\}=\\{e_{ik},e_{kk}\\}=\\{e_{ii},e_{ik}\\}=\\{e_{ik},e_{ii}\\}=\\{e_{kk},e_{ik}\\}=\\{e_{jk},e_{ij}\\}.$ (22) If $i,j,k$ are distinct and $n>3$, from (6), (7), (17), (18) and (19) we have the equations above as well. So (22) holds true whenever $n\geq 3$. From case 1.3 and case 2.3, we have $\\{e_{ij},e_{ji}\\}=\frac{1}{2}\\{e_{ii},e_{ii}\\}+\frac{1}{2}\\{e_{jj},e_{jj}\\}\;\mathrm{if}\;i\neq j.$ (23) Let $\sum_{t=1}^{l}x_{t}\circ y_{t}=0$ where $x_{t},y_{t}\in M_{n}(R)$, $t=1,2,\ldots,l$. We write $x_{t}$ and $y_{t}$ for $x_{t}=\sum_{i=1}^{n}\sum_{j=1}^{n}a_{ij}^{t}e_{ij},\;y_{t}=\sum_{k=1}^{n}\sum_{m=1}^{n}b_{km}^{t}e_{km}.$ Then for all $i$ and $m$ we have that $\sum_{t=1}^{l}\sum_{j=1}^{n}(a_{ij}^{t}b_{jm}^{t}+b_{ij}^{t}a_{jm}^{t})=0.$ (24) Now we will show $\sum_{t=1}^{l}\\{x_{t},y_{t}\\}=0$. Note that $\displaystyle\sum_{t=1}^{l}\\{x_{t},y_{t}\\}$ $\displaystyle=$ $\displaystyle\sum_{t=1}^{l}\sum_{i=1}^{n}\sum_{j=1}^{n}\sum_{k=1}^{n}\sum_{m=1}^{n}\\{a_{ij}^{t}e_{ij},b_{km}^{t}e_{km}\\}$ $\displaystyle=$ $\displaystyle\sum_{t=1}^{l}\sum_{i=1}^{n}\sum_{j=1}^{n}\sum_{k=1}^{n}\sum_{m=1}^{n}a_{ij}^{t}b_{km}^{t}\\{e_{ij},e_{km}\\}.$ According to our assumptions, it follows from (21), (22), (23) and (24) that $\displaystyle\sum_{t=1}^{l}\\{x_{t},y_{t}\\}$ $\displaystyle=$ $\displaystyle\sum_{t=1}^{l}\sum_{i=1}^{n}\sum_{j=1}^{n}\sum_{k=1\atop k\neq j}^{n}\sum_{m=1\atop m\neq i}^{n}a_{ij}^{t}b_{km}^{t}\\{e_{ij},e_{km}\\}+\sum_{t=1}^{l}\sum_{i=1}^{n}\sum_{j=1}^{n}\sum_{m=1\atop m\neq i}^{n}a_{ij}^{t}b_{jm}^{t}\\{e_{ij},e_{jm}\\}$ $\displaystyle+$ $\displaystyle\sum_{t=1}^{l}\sum_{i=1}^{n}\sum_{j=1}^{n}\sum_{k=1\atop k\neq j}^{n}a_{ij}^{t}b_{ki}^{t}\\{e_{ij},e_{ki}\\}+\sum_{t=1}^{l}\sum_{i=1}^{n}\sum_{j=1}^{n}a_{ij}^{t}b_{ji}^{t}\\{e_{ij},e_{ji}\\}$ $\displaystyle=$ $\displaystyle\sum_{t=1}^{l}\sum_{i=1}^{n}\sum_{j=1}^{n}\sum_{m=1\atop m\neq i}^{n}(a_{ij}^{t}b_{jm}^{t}+b_{ij}^{t}a_{jm}^{t})\\{e_{ij},e_{jm}\\}$ $\displaystyle+$ $\displaystyle\frac{1}{2}\sum_{t=1}^{l}\sum_{i=1}^{n}\sum_{j=1}^{n}a_{ij}^{t}b_{ji}^{t}\left(\\{e_{ii},e_{ii}\\}+\\{e_{jj},e_{jj}\\}\right)$ $\displaystyle=$ $\displaystyle\sum_{i=1}^{n}\sum_{m=1\atop m\neq i}^{n}\left[\sum_{t=1}^{l}\sum_{j=1}^{n}(a_{ij}^{t}b_{jm}^{t}+b_{ij}^{t}a_{jm}^{t})\\{e_{im},e_{mm}\\}\right]$ $\displaystyle+$ $\displaystyle\frac{1}{2}\sum_{i=1}^{n}\left[\sum_{t=1}^{l}\sum_{j=1}^{n}(a_{ij}^{t}b_{ji}^{t}+b_{ij}^{t}a_{ji}^{t})\\{e_{ii},e_{ii}\\}\right]$ $\displaystyle=$ $\displaystyle 0.$ By Lemma 2.1, $e_{ss}$ is a Jordan product determined point. $\Box$ Theorem 2.3. $e_{pq}$, $p\neq q$, is a Jordan product determined point in $M_{n}(R)$ when $n\geq 3$. Proof. Let $p,q$ be the distinct fixed indices, $\\{\cdot,\cdot\\}$ : $M_{n}(R)\times M_{n}(R)\to X$ be a symmetric (i.e. $\\{x,y\\}=\\{y,x\\}$) $R$-bilinear map where $X$ is an $R$-module. And we assume that there exists a fixed element $w\in X$ such that $\\{x,y\\}=w$ whenever $x\circ y=e_{pq}$, $x,y\in M_{n}(R)$. Throughout the proof, $i,j,k,m$ will denote arbitrary indices. According to Lemma 2.1 and Step 3 in the proof of Theorem 2.2, we only need to verify (21), (22) and (23) hold true when $n\geq 3$. Now we suppose $n\geq 3$ and divide the proof into several steps. Step 1. In this step, we assume $i\neq q\;\mathrm{and}\;m\neq p$. Since $e_{ps}\circ e_{sq}=e_{pq}\>(s=1,2,\ldots,n)$ and as $\\{\cdot,\cdot\\}$ is symmetric, hence we have $\\{e_{ps},e_{sq}\\}=w=\\{e_{sq},e_{ps}\\},\;s=1,2,\ldots,n.$ (25) Choosing $s\neq j\;\mathrm{and}\;s\neq k$, then we obtain $e_{ps}\circ(e_{sq}+e_{km})=e_{pq}$ and $(e_{ps}+e_{ij})\circ e_{sq}=e_{pq}$ respectively. For $\\{\cdot,\cdot\\}$ is symmetric, applying (25), it follows $\\{e_{ps},e_{km}\\}=\\{e_{km},e_{ps}\\}=\\{e_{ij},e_{sq}\\}=\\{e_{sq},e_{ij}\\}=0\;\mathrm{if}\>s\neq j\;\mathrm{and}\;s\neq k.$ (26) Given $s\neq j\;\mathrm{and}\;s\neq k$, then if $i\neq m\;\mathrm{and}\;j\neq k$ we have $(e_{ps}+e_{ij})\circ(e_{sq}+e_{km})=e_{pq}$. It is clear that $\\{e_{ps}+e_{ij},e_{sq}+e_{km}\\}=w$. As $\\{\cdot,\cdot\\}$ is symmetric, using (25) and (26), this yields $\\{e_{ij},e_{km}\\}=0=\\{e_{km},e_{ij}\\}\>\mathrm{if}\>i\neq m\;\mathrm{and}\;j\neq k.$ (27) Step 2. In this step, we suppose $i\neq q\;\mathrm{and}\;m\neq p$. Assuming $s\neq i\;\mathrm{and}\;s\neq m$, then if $i\neq m\;\mathrm{and}\;i\neq p$ we can verify $(e_{ps}+e_{im})\circ(e_{sq}+e_{mm}-e_{ii})=e_{pq}$. It follows that $\\{e_{ps}+e_{im},e_{sq}+e_{mm}-e_{ii}\\}=w$. Since $\\{\cdot,\cdot\\}$ is symmetric, by (25) and (27), this yields $\\{e_{im},e_{mm}\\}=\\{e_{im},e_{ii}\\}=\\{e_{ii},e_{im}\\}\>\mathrm{if}\>i\neq m\;\mathrm{and}\;i\neq p.$ (28) From $e_{pm}\circ(e_{mq}+e_{mm}-e_{pp})=e_{pq}$, we have that $\\{e_{pm},e_{mq}+e_{mm}-e_{pp}\\}=w$. As $\\{\cdot,\cdot\\}$ is symmetric, it follows from (25) that $\\{e_{pm},e_{mm}\\}=\\{e_{pm},e_{pp}\\}=\\{e_{pp},e_{pm}\\}.$ (29) Choosing $s\neq j\;\mathrm{and}\;s\neq m$, if $i,j,k$ are distinct we obtain $(e_{ps}+e_{ij}+e_{im})\circ(e_{sq}+e_{jm}-e_{mm})=e_{pq}$. Then it leads to $\\{e_{ps}+e_{ij}+e_{im},e_{sq}+e_{jm}-e_{mm}\\}=w$. Applying (25) and (27), this yields $\\{e_{ij},e_{jm}\\}=\\{e_{im},e_{mm}\\}\;\mathrm{if}\;i,j,k\;\mathrm{aredistinct}.$ (30) Since $\\{\cdot,\cdot\\}$ is symmetric, if $i,j,m$ are distinct, we can conclude from (28), (29) and (30) that $\\{e_{ij},e_{jm}\\}=\\{e_{im},e_{mm}\\}=\\{e_{ii},e_{im}\\}=\\{e_{im},e_{ii}\\}=\\{e_{mm},e_{im}\\}=\\{e_{jm},e_{ij}\\}.$ (31) Step 3. In this step, we assume $i\neq q\;\mathrm{and}\;m\neq p$. Choosing $s\neq i\;\mathrm{and}\;s\neq m$, we suppose $n>3,\>i\neq p,\>i\neq m\;\mathrm{and}\;m\neq q$. As $(e_{ps}+\frac{1}{2}e_{ii}+e_{im}-\frac{1}{2}e_{mm})\circ(e_{sq}+e_{mi}-e_{ii}+e_{mm})=e_{pq}$, it follows that $\\{e_{ps}+\frac{1}{2}e_{ii}+e_{im}-\frac{1}{2}e_{mm},e_{sq}+e_{mi}-e_{ii}+e_{mm}\\}=w$. Using (25), (27) and (31), if $n>3,\>i\neq p,\>i\neq m\;\mathrm{and}\;m\neq q$ we have that $\\{e_{im},e_{mi}\\}=\frac{1}{2}\\{e_{ii},e_{ii}\\}+\frac{1}{2}\\{e_{mm},e_{mm}\\}.$ (32) Step 4. In this step, we assume $i=q\;\mathrm{and}\;m\neq p$. Case 4.1. $j\neq p\;\mathrm{and}\;k\neq q$. By (27), we have $\\{e_{km},e_{qj}\\}=0$ if $m\neq q\;\mathrm{and}\;j\neq k$. As $\\{\cdot,\cdot\\}$ is symmetric, this yields $\\{e_{qj},e_{km}\\}=\\{e_{km},e_{qj}\\}=0\;\mathrm{if}\;m\neq q\;\mathrm{and}\;j\neq k.$ (33) Case 4.2. $j\neq p\;\mathrm{and}\;k=q$. Since $(e_{pq}+e_{qj})\circ(e_{qq}-e_{jj})=e_{pq}$ if $j\neq q$, then we have $\\{e_{pq}+e_{qj},e_{qq}-e_{jj}\\}=w$. For $\\{\cdot,\cdot\\}$ is symmetric, it follows from (25) and (27) that $\\{e_{qj},e_{qq}\\}=\\{e_{qj},e_{jj}\\}=\\{e_{jj},e_{qj}\\}=\\{e_{qq},e_{qj}\\}\;\mathrm{if}\;j\neq q.$ (34) Noting $(e_{pq}+e_{qj}+e_{pp})\circ(e_{qq}+e_{qm}-e_{jj}-e_{pm})=e_{pq}$ if $j\neq q\;\mathrm{and}\;m\neq q$, then we obtain $\\{e_{pq}+e_{qj}+e_{pp},e_{qq}+e_{qm}-e_{jj}-e_{pm}\\}=w$. Applying (25), (27), (31), (33) and (34), it can be reduced to $\\{e_{qj},e_{qm}\\}=0\;\mathrm{if}\;j\neq q\;\mathrm{and}\;m\neq q.$ (35) Case 4.3. $j=p\;\mathrm{and}\;k\neq q$. From $(\frac{1}{2}e_{pp}+\frac{1}{2}e_{pq}+e_{qp})\circ(e_{pq}-2e_{qp}+e_{qq})=e_{pq}$, we have that $\\{\frac{1}{2}e_{pp}+\frac{1}{2}e_{pq}+e_{qp},e_{pq}-2e_{qp}+e_{qq}\\}=w$. As $\\{\cdot,\cdot\\}$ is symmetric, it follows from (25) and (27) that $\\{e_{qp},e_{qq}\\}-\\{e_{pp},e_{qp}\\}-2\\{e_{qp},e_{qp}\\}=0.$ (36) Since $(\frac{1}{2}e_{pp}+e_{pq}+2e_{qp})\circ(e_{pq}-2e_{qp}+\frac{1}{2}e_{qq})=e_{pq}$, a similar discussion shows that $\\{e_{qp},e_{qq}\\}-\\{e_{pp},e_{qp}\\}-4\\{e_{qp},e_{qp}\\}=0.$ (37) Because $\\{\cdot,\cdot\\}$ is symmetric, comparing (36) with (37), we get that $\\{e_{qp},e_{qp}\\}=0,$ (38) $\\{e_{qp},e_{qq}\\}=\\{e_{pp},e_{qp}\\}=\\{e_{qp},e_{pp}\\}=\\{e_{qq},e_{qp}\\}.$ (39) Noting $(\frac{1}{2}e_{pp}+\frac{1}{2}e_{pq}+e_{qp})\circ(e_{pq}-2e_{qp}+e_{qq}+e_{km})=e_{pq}$ if $k\neq p\;\mathrm{and}\;m\neq q$, we have that $\\{\frac{1}{2}e_{pp}+\frac{1}{2}e_{pq}+e_{qp},e_{pq}-2e_{qp}+e_{qq}+e_{km}\\}=w$. As $\\{\cdot,\cdot\\}$ is symmetric, applying (25), (27) and (36), if $k\neq p\;\mathrm{and}\;m\neq q$ it leads to $\\{e_{qp},e_{km}\\}=0=\\{e_{km},e_{qp}\\}.$ (40) Case 4.4. $j=p\;\mathrm{and}\;k=q$. If $m\neq q$, then we can verify that $(\frac{1}{2}e_{pp}+\frac{1}{2}e_{pq}+e_{qp}-e_{mm})\circ(e_{pq}-2e_{qp}+e_{qq}+e_{qm}+e_{pm})=e_{pq},$ $(\frac{1}{2}e_{pp}+\frac{1}{2}e_{pq}+e_{qp}+\frac{1}{2}e_{mm})\circ(e_{pq}-2e_{qp}+e_{qq}-2e_{qm}+e_{pm})=e_{pq}.$ Hence we have that $\\{\frac{1}{2}e_{pp}+\frac{1}{2}e_{pq}+e_{qp}-e_{mm},e_{pq}-2e_{qp}+e_{qq}+e_{qm}+e_{pm}\\}=w,$ $\\{\frac{1}{2}e_{pp}+\frac{1}{2}e_{pq}+e_{qp}+\frac{1}{2}e_{mm},e_{pq}-2e_{qp}+e_{qq}-2e_{qm}+e_{pm}\\}=w.$ For $\\{\cdot,\cdot\\}$ is symmetric, it follows from (25), (27), (31), (38), (39) and (40) that $\\{e_{qp},e_{pm}\\}+\\{e_{qp},e_{qm}\\}-\\{e_{mm},e_{qm}\\}=0,$ $\\{e_{qp},e_{pm}\\}-2\\{e_{qp},e_{qm}\\}-\\{e_{mm},e_{qm}\\}=0.$ Thus if $m\neq q$ we have that $\\{e_{qp},e_{qm}\\}=0=\\{e_{qm},e_{qp}\\},$ (41) $\\{e_{qp},e_{pm}\\}=\\{e_{mm},e_{qm}\\}=\\{e_{qm},e_{mm}\\}=\\{e_{pm},e_{qp}\\}.$ (42) Step 5. In this step, we assume $i\neq q\;\mathrm{and}\;m=p$. Case 5.1. $j\neq p\;\mathrm{and}\;k\neq q$. From (27), if $i\neq p\;\mathrm{and}\;j\neq k$ it follows that $\\{e_{ij},e_{kp}\\}=0=\\{e_{kp},e_{ij}\\}.$ (43) Case 5.2. $j=p\;\mathrm{and}\;k\neq q$. Since $(e_{pq}+e_{ip}-e_{kq})\circ(e_{qq}+e_{kp})=e_{pq}$ if $i\neq p\;\mathrm{and}\;k\neq p$, it is clear that $\\{e_{pq}+e_{ip}-e_{kq},e_{qq}+e_{kp}\\}=w$. Applying (25), (27) and (31), it can be reduced to $\\{e_{ip},e_{kp}\\}=0.$ (44) Case 5.3. $j\neq p\;\mathrm{and}\;k=q$. By (40), if $i\neq p\;\mathrm{and}\;j\neq q$ we have that $\\{e_{ij},e_{qp}\\}=0.$ (45) Case 5.4. $j=p\;\mathrm{and}\;k=q$. If $i\neq p\;\mathrm{and}\;i\neq q$, then we can verify that $\displaystyle(\frac{1}{2}e_{pp}+\frac{1}{2}e_{pq}+e_{qp}-e_{ii})\circ(e_{pq}-2e_{qp}+e_{qq}+2e_{ip}+e_{iq})=e_{pq},$ $\displaystyle(\frac{1}{2}e_{pp}+\frac{1}{2}e_{pq}+e_{qp}+\frac{1}{2}e_{ii})\circ(e_{pq}-2e_{qp}+e_{qq}-e_{ip}+e_{iq})=e_{pq}.$ According to our assumptions, using (25), (27), (31), (38), (39) and (40), we have $\\{e_{pp},e_{ip}\\}+\\{e_{qp},e_{iq}\\}-2\\{e_{ii},e_{ip}\\}+2\\{e_{qp},e_{ip}\\}=0,$ (46) $\\{e_{pp},e_{ip}\\}-2\\{e_{qp},e_{iq}\\}+\\{e_{ii},e_{ip}\\}+2\\{e_{qp},e_{ip}\\}=0.$ (47) Comparing the two equations above, as $\\{\cdot,\cdot\\}$ is symmetric, it follows that $\\{e_{qp},e_{iq}\\}=\\{e_{ii},e_{ip}\\}=\\{e_{iq},e_{qp}\\}=\\{e_{ip},e_{ii}\\}\;\mathrm{if}\;i\neq p\;\mathrm{and}\;i\neq q.$ (48) Substituting (48) into (46), we obtain $\\{e_{pp},e_{ip}\\}-\\{e_{ii},e_{ip}\\}+2\\{e_{qp},e_{ip}\\}=0\;\mathrm{if}\;i\neq p\;\mathrm{and}\;i\neq q.$ (49) Noting $(e_{pq}+e_{ip})\circ(e_{ii}-e_{pp}+2e_{qq})=e_{pq}$ if $i\neq p\;\mathrm{and}\;i\neq q$, then we have that $\\{e_{pq}+e_{ip},e_{ii}-e_{pp}+2e_{qq}\\}=w$. Since $\\{\cdot,\cdot\\}$ is symmetric, it follows from (25) and (27) that $\\{e_{ip},e_{ii}\\}=\\{e_{ip},e_{pp}\\}=\\{e_{pp},e_{ip}\\}=\\{e_{ii},e_{ip}\\}\;\mathrm{if}\;i\neq p\;\mathrm{and}\;i\neq q.$ (50) Substituting (50) into (49), we derive $\\{e_{qp},e_{ip}\\}=0=\\{e_{ip},e_{qp}\\}\;\mathrm{if}\;i\neq p\;\mathrm{and}\;i\neq q.$ (51) Step 6. In this step, we assume $i=q\;\mathrm{and}\;m=p$. Case 6.1. $j\neq p\;\mathrm{and}\;k\neq q$. From (27), if $j\neq k$ it is clear that $\\{e_{qj},e_{kp}\\}=\\{e_{kp},e_{qj}\\}=0.$ (52) Case 6.2. $j\neq p\;\mathrm{and}\;k=q$. From (41), we have $\\{e_{qj},e_{qp}\\}=\\{e_{qp},e_{qj}\\}=0\;\mathrm{if}\;j\neq q.$ (53) Case 6.3. $j=p\;\mathrm{and}\;k\neq q$. From (51), it follows $\\{e_{qp},e_{kp}\\}=\\{e_{kp},e_{qp}\\}=0\;\mathrm{if}\;k\neq p.$ (54) Case 6.4. $j=p\;\mathrm{and}\;k=q$. From (38), we obtain that $\\{e_{qp},e_{qp}\\}=0.$ (55) Step 7. In this step, we assume $i=q\;\mathrm{and}\;m\neq p$. If $n>3,\>j\neq p,\>j\neq q,\>j\neq m\;\mathrm{and}\;m\neq q$, choosing $s\neq q\;\mathrm{and}\;s\neq m$, then we can verify $(e_{ps}+e_{mm}-e_{jm})\circ(e_{sq}+e_{qm}+e_{qj})=e_{pq}$. It follows that $\\{e_{ps}+e_{mm}-e_{jm},e_{sq}+e_{qm}+e_{qj}\\}=w$. As $\\{\cdot,\cdot\\}$ is symmetric, by (25) and (27), if $n>3,\>j\neq p,\>j\neq q,\>j\neq m\;\mathrm{and}\;m\neq q$ we have that $\\{e_{mm},e_{qm}\\}=\\{e_{jm},e_{qj}\\}=\\{e_{qj},e_{jm}\\}=\\{e_{qm},e_{mm}\\}.$ (56) In the case of $n=3$, if $q,j,m$ are distinct we can conclude from (34) and (42) that $\\{e_{qj},e_{jm}\\}=\\{e_{qm},e_{mm}\\}=\\{e_{qq},e_{qm}\\}=\\{e_{qm},e_{qq}\\}=\\{e_{mm},e_{qm}\\}=\\{e_{jm},e_{qj}\\}.$ (57) In the case of $n>3$, if $q,j,m$ are distinct, from (34), (42) and (56) we have the same equations as well. So (57) holds true whenever $n\geq 3$. Step 8. In this step, we assume that $i\neq q\;\mathrm{and}\;m=p$. If $n>3,\>i\neq p,\>i\neq j,\>j\neq p\;\mathrm{and}\;j\neq q$, choosing $s\neq p\;\mathrm{and}\;s\neq i$, then we have that $(e_{ps}+e_{jp}-e_{ip})\circ(e_{sq}+e_{ij}+e_{ii})=e_{pq}$. From the assumptions this yields $\\{e_{ps}+e_{jp}-e_{ip},e_{sq}+e_{ij}+e_{ii}\\}=w$. Because $\\{\cdot,\cdot\\}$ is symmetric, if $n>3,\>i\neq p,\>i\neq j,\>j\neq p\;\mathrm{and}\;j\neq q$ it follows from (25) and (27) that $\\{e_{jp},e_{ij}\\}=\\{e_{ip},e_{ii}\\}=\\{e_{ij},e_{jp}\\}=\\{e_{ii},e_{ip}\\}.$ (58) If $i,j,p$ are distinct and $n=3$, then we can conclude from (48) and (50) that $\\{e_{ij},e_{jp}\\}=\\{e_{ip},e_{pp}\\}=\\{e_{ii},e_{ip}\\}=\\{e_{ip},e_{ii}\\}=\\{e_{pp},e_{ip}\\}=\\{e_{jp},e_{ij}\\}.$ (59) If $i,j,p$ are distinct and $n>3$, from (48), (50) and (58) we obtain the equations above as well. Hence, (59) holds true whenever $n\geq 3$. Step 9. Selecting $s\neq p\;\mathrm{and}\;s\neq m$, then if $m\neq p\;\mathrm{and}\;m\neq q$ we can verify $(e_{ps}+\frac{1}{2}e_{pp}+e_{mp}-\frac{1}{2}e_{mm})\circ(e_{sq}+e_{ss}+e_{pm}-e_{pp}+e_{mm})=e_{pq}$. It implies that $\\{e_{ps}+\frac{1}{2}e_{pp}+e_{mp}-\frac{1}{2}e_{mm},e_{sq}+e_{ss}+e_{pm}-e_{pp}+e_{mm}\\}=w$. As $\\{\cdot,\cdot\\}$ is symmetric, by (25), (27), (31) and (59), if $m\neq p\;\mathrm{and}\;m\neq q$ this yields $\\{e_{mp},e_{pm}\\}=\frac{1}{2}\\{e_{pp},e_{pp}\\}+\frac{1}{2}\\{e_{mm},e_{mm}\\}=\\{e_{pm},e_{mp}\\}.$ (60) Choosing $s\neq q\;\mathrm{and}\;s\neq m$, then if $m\neq p\;\mathrm{and}\;m\neq q$ we have $(e_{ps}+e_{mq}+\frac{1}{2}e_{qq}-\frac{1}{2}e_{ss}-\frac{1}{2}e_{mm})\circ(e_{sq}+e_{qm}-e_{qq}+e_{mm})=e_{pq}$. It is clear that $\\{e_{ps}+e_{mq}+\frac{1}{2}e_{qq}-\frac{1}{2}e_{ss}-\frac{1}{2}e_{mm},e_{sq}+e_{qm}-e_{qq}+e_{mm}\\}=w$. Because $\\{\cdot,\cdot\\}$ is symmetric, if $m\neq p\;\mathrm{and}\;m\neq q$ it follows from (25), (27), (31) and (57) that $\\{e_{mq},e_{qm}\\}=\frac{1}{2}\\{e_{mm},e_{mm}\\}+\frac{1}{2}\\{e_{qq},e_{qq}\\}=\\{e_{qm},e_{mq}\\}.$ (61) Since $(e_{pp}+\frac{5}{4}e_{pq}+e_{qp}+e_{qq})\circ(e_{pp}-\frac{3}{4}e_{pq}-e_{qp}+e_{qq})=e_{pq}$, then we have that $\\{e_{pp}+\frac{5}{4}e_{pq}+e_{qp}+e_{qq},e_{pp}-\frac{3}{4}e_{pq}-e_{qp}+e_{qq}\\}=w$. As $\\{\cdot,\cdot\\}$ is symmetric, using (25), (27) and (38), it can be reduced to $\\{e_{pq},e_{qp}\\}=\frac{1}{2}\\{e_{pp},e_{pp}\\}+\frac{1}{2}\\{e_{qq},e_{qq}\\}=\\{e_{qp},e_{pq}\\}.$ (62) Step 10. If $j\neq p\;\mathrm{and}\;j\neq q$, we can verify that $(e_{pp}+e_{pq}-4e_{qp}-3e_{qq}+e_{qj}+e_{jj})\circ(2e_{pp}-4e_{qp}-e_{qq}-4e_{jp}-2e_{jq}+e_{jj})=e_{pq}$. It follows that $\\{e_{pp}+e_{pq}-4e_{qp}-3e_{qq}+e_{qj}+e_{jj},2e_{pp}-4e_{qp}-e_{qq}-4e_{jp}-2e_{jq}+e_{jj}\\}=w$. Since $\\{\cdot,\cdot\\}$ is symmetric, applying (25), (27), (31), (38), (39), (40), (53), (54), (57), (59), (61) and (62), if $j\neq p\;\mathrm{and}\;j\neq q$ we have that $\\{e_{qj},e_{jp}\\}=\\{e_{qp},e_{qq}\\}=\\{e_{qp},e_{pp}\\}=\\{e_{qq},e_{qp}\\}=\\{e_{pp},e_{qp}\\}=\\{e_{jp},e_{qj}\\}.$ (63) Step 11. Now, if $i\neq m\;\mathrm{and}\;j\neq k$ we can conclude from step 1, step 4, step 5 and step 6 that $\\{e_{ij},e_{km}\\}=0.$ (64) If $i,j,m$ are distinct, it follows from step 2, step 7, step 8 and step 10 that $\\{e_{ij},e_{jm}\\}=\\{e_{im},e_{mm}\\}=\\{e_{ii},e_{im}\\}=\\{e_{im},e_{ii}\\}=\\{e_{mm},e_{im}\\}=\\{e_{jm},e_{ij}\\}.$ (65) If $i\neq m$ and $n=3$, from (60), (61) and (62) we have that $\\{e_{im},e_{mi}\\}=\frac{1}{2}\\{e_{ii},e_{ii}\\}+\frac{1}{2}\\{e_{mm},e_{mm}\\}.$ (66) If $i\neq m$ and $n>3$, from step 3 and step 9 we have the same equations as well. So we complete the proof. $\Box$ ## 3\. Several applications Now, we will give two applications of the theorems above. Definition 3.1. We say that $G\in M_{n}(R)$ is a Jordan all-multiplicative point in $M_{n}(R)$ if for every $M_{n}(R)$-module $X$ and every Jordan multiplicative $R$-linear map $\varphi$ : $M_{n}(R)\to X$ at $G$ (i.e. $\varphi(S\circ T)=\varphi(S)\circ\varphi(T)$ for any $S,T\in M_{n}(R),\>S\circ T=G$) with $\varphi(I)=I$ is a multiplicative mapping in $M_{n}(R)$. Corollary 3.2. Every matrix units $e_{ij}$ in $M_{n}(R)$, $n\geq 3$, is a Jordan all-multiplicative point. Proof. Let $X$ be an $M_{n}(R)$-module, $\varphi$ be a Jordan multiplicative $R$-linear map at $e_{ij}$. Then it follows $\varphi(I)=I,\;\varphi(S\circ T)=\varphi(S)\circ\varphi(T)\ \mathrm{for\;all}\>S,T\in M_{n}(R),\>S\circ T=e_{ij}.$ Set $\\{S,T\\}=\varphi(S)\circ\varphi(T)$ for any $S,T\in M_{n}(R)$, thus $\\{\cdot,\cdot\\}$ is a symmetric $R$-bilinear map. Since $\varphi$ is a multiplicative map at $e_{ij}$, we have $\\{S,T\\}=\varphi(S)\circ\varphi(T)=\varphi(e_{ij})\ \mathrm{for\;all}\>S,T\in M_{n}(R)\;\mathrm{with}\;S\circ T=e_{ij}.$ As $e_{ij}$ is a Jordan product determined point in $M_{n}(R)$, then there exists an $R$-linear map $\phi$ : $M_{n}(R)^{2}\to X$ such that $\\{S,T\\}=\phi(S\circ T)$ for all $S,T\in M_{n}(R)$. So $\varphi(S)\circ\varphi(T)=\\{S,T\\}=\phi(S\circ T),\;\forall\>S,T\in M_{n}(R).$ Set $S=I$ in the equation above, then we have $\varphi(T)=\phi(T)$ for every $T\in M_{n}(R)$. It follows that $\varphi(S\circ T)=\varphi(S)\circ\varphi(T),\;\forall\;S,T\in M_{n}(R).$ Hence $\varphi$ is a multiplicative mapping in $M_{n}(R)$. The proof is completed. $\Box$ Definition 3.3. We say that $H\in M_{n}(R)$ is a Jordan all-derivable point in $M_{n}(R)$ if for every $M_{n}(R)$-module $X$ and every Jordan derivable $R$-linear map $\varphi$ : $M_{n}(R)\to X$ at $H$ (i.e. $\varphi(S\circ T)=\varphi(S)\circ T+S\circ\varphi(T)$ for any $S,T\in M_{n}(R),\>S\circ T=H$) with $\varphi(I)=0$ is a Jordan derivation in $M_{n}(R)$. Corollary 3.4. Every matrix units $e_{ij}$ in $M_{n}(R)$, $n\geq 3$, is a Jordan all-derivable point. Proof. Let $X$ be an $M_{n}(R)$-module, $\tau$ be a Jordan derivable $R$-linear map at $e_{ij}$. Then we have $\tau(I)=0,\;\tau(S\circ T)=\tau(S)\circ T+S\circ\tau(T)\;\mathrm{for\;all}\;S,T\in M_{n}(R),\;S\circ T=e_{ij}.$ Set $\\{S,T\\}=\tau(S)\circ T+S\circ\tau(T)$ for any $S,T\in M_{n}(R)$, so $\\{\cdot,\cdot\\}$ is a symmetric $R$-bilinear map. Since $\tau$ is a derivable map at $e_{ij}$, it follows $\\{S,T\\}=\tau(S)\circ T+S\circ\tau(T)=\tau(e_{ij})\;\mathrm{for\;all}\;S,T\in M_{n}(R)\;\mathrm{with}\;S\circ T=e_{ij}.$ As $e_{ij}$ is a Jordan product determined point in $M_{n}(R)$, then there exists an $R$-linear map $\psi$ : $M_{n}(R)^{2}\to X$ such that $\\{S,T\\}=\psi(S\circ T)$ for all $S,T\in M_{n}(R)$. Hence $\tau(S)\circ T+S\circ\tau(T)=\\{S,T\\}=\psi(S\circ T),\;\forall\>S,T\in M_{n}(R).$ Set $S=I$ in the equation above, then we have $\tau(T)=\psi(T)$ for every $T\in M_{n}(R)$. It follows that $\tau(S\circ T)=\tau(S)\circ T+S\circ\tau(T),\;\forall\>S,T\in M_{n}(R).$ Then $\tau$ is a Jordan derivation in $M_{n}(R)$. The proof is completed. $\Box$ ## References * [1] D. Wang, X. Li, H. Ge, Idempotent elements determined matrix algebras, Linear Algebra Appl. 435 (2011) 2889-2895. * [2] D. Wang, X. Li, H. Ge, Maps determined by action on identity-product elements, Linear Algebra Appl. 436 (2012) 112-119. * [3] J. Zhu, C. Xiong, H. Zhu, Multiplicative mappings at some points on matrix algebras, Linear Algebra Appl. 433 (2010) 914-927. * [4] J. Zhu, C. Xiong, L. Zhang, All-derivable points in matrix algebras, Linear Algebra Appl. 430 (2009) 2070-2079. * [5] M. Brešar, M. Grašič, J. Ortega, Zero product determined matrix algebras, Linear Algebra Appl. 430 (2009) 1486-1498. * [6] M. Gong, J. Zhu, Jordan multiplicative mappings at some points on matrix algebras, Journal of Advanced Research in Pure Mathematics. 4 (2010) 84-93. * [7] S. Zhao, J. Zhu, Jordan all-derivable points in the algebra of all upper triangular matrices, Linear Algebra Appl. 433 (2010) 1922-1938.	arxiv-papers	2011-11-17T14:23:22	2024-09-04T02:49:24.431615	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Yang Wenlei and Zhu Jun", "submitter": "Jun Zhu Professor", "url": "https://arxiv.org/abs/1111.4108" }
1111.4122	# Reversible electron beam heating for suppression of microbunching instabilities at free-electron lasers Christopher Behrens1, Zhirong Huang2, and Dao Xiang2 1 Deutsches Elektronen- Synchrotron DESY, Notkestr. 85, 22607 Hamburg, Germany 2 SLAC National Accelerator Laboratory, Menlo Park, CA 94025, USA ###### Abstract The presence of microbunching instabilities due to the compression of high- brightness electron beams at existing and future X-ray free-electron lasers (FELs) results in restrictions on the attainable lasing performance and renders beam imaging with optical transition radiation impossible. The instability can be suppressed by introducing additional energy spread, i.e., “heating” the electron beam, as demonstrated by the successful operation of the laser heater system at the Linac Coherent Light Source. The increased energy spread is typically tolerable for self-amplified spontaneous emission FELs but limits the effectiveness of advanced FEL schemes such as seeding. In this paper, we present a reversible electron beam heating system based on two transverse deflecting radio-frequency structures (TDSs) up and downstream of a magnetic bunch compressor chicane. The additional energy spread is introduced in the first TDS, which suppresses the microbunching instability, and then is eliminated in the second TDS. We show the feasibility of the microbunching gain suppression based on calculations and simulations including the effects of coherent synchrotron radiation. Acceptable electron beam and radio- frequency jitter are identified, and inherent options for diagnostics and on- line monitoring of the electron beam’s longitudinal phase space are discussed. ###### pacs: 29.27.-a, 41.60.Cr, 41.85.Ct ## I Introduction X-ray free-electron lasers (FELs) provide an outstanding tool for studying matter at ultrafast time and atomic length scales LCLSnature1 , and have become a reality with the operation of the Free-Electron Laser in Hamburg (FLASH) FLASHAnature , the Linac Coherent Light Source (LCLS) LCLSnature2 , and the SPring-8 Angstrom Compact Free Electron Laser (SACLA) SACLAnature . The required high transverse and longitudinal brightness of the X-ray FEL driving electron bunches may encounter various degradation effects due to collective effects like coherent synchrotron radiation (CSR) or microbunching instabilities (e.g., Refs. CSR ; CSR-ub ; lsc-ub ), and need to be preserved and controlled. In order to suppress a microbunching instability associated with longitudinal bunch compression that deteriorates the FEL performance, the LCLS uses a laser heater system to irreversibly increase the uncorrelated energy spread within the electron bunches, i.e., the slice energy spread, to a level tolerable for operation of a self-amplified spontaneous emission FEL LH ; LCLSheater . For future X-ray FELs that plan to use external quantum lasers (seed lasers) to seed the FEL process in order to achieve better temporal coherence and synchronization for pump-probe experiments, a smaller slice energy spread is required to leave room for the additional energy modulation imprinted by the seed laser. Thus, the amount of tolerable beam heating is more restrictive and the longitudinal phase space control becomes more critical (e.g., Refs. NLS ; flatflat ). The same strict requirement on small slice energy spreads is valid for optical klystron enhanced self-amplified spontaneous emission free-electron lasers okly . Originally designed for high-energy particle separation by radio-frequency (rf) fields LOLA , transverse deflecting rf structures (TDSs) are routinely used for high-resolution temporal electron beam diagnostics at present X-ray FELs (e.g., Refs. Kick ; Roehrs ; Filippetto ; xtcav ) and are proposed to use for novel beam manipulation methods (e.g., Refs. psxray ; exchange1 ; exchange2 ; mapping ; ramp ; psex ). Recently, a TDS was used to increase the slice energy spread in an echo-enabled harmonic generation FEL experiment Xiang ; Xiang2 . In this paper, we present a reversible electron beam heating system that uses two TDSs located up and downstream of a magnetic bunch compressor chicane. The additional slice energy spread is introduced in the first TDS, which suppresses the microbunching instability, and then is eliminated in the second TDS. The method of reversible beam heating is shown in Sec. II by means of linear beam optics and a corresponding matrix formalism. In Sec. III, we show the feasibility of this scheme to preserve both the transverse and longitudinal brightness of the electron beam, and discuss the impact of coherent synchrotron radiation. Section IV covers the gain suppression of microbunching instabilities by analytical calculations and numerical simulations, and in Sec. V we discuss the impact of beam and rf jitter, and show inherent options for diagnosis and on-line monitoring of the electron beam’s longitudinal phase space. The results and conclusions are summarized in Sec. VI. ## II Method Figure 1: Layout of a reversible electron beam heater system including two transverse deflecting rf structures located up and downstream of a magnetic bunch compressor (BC) chicane, and longitudinal phase space diagnostics using screens and synchrotron radiation monitors (SRM). Parameters related to the reversible beam heater system are denoted in curly brackets. In this and the following sections, we consider a linear accelerator (linac) employing a single bunch compressor for a soft X-ray FEL, such as the proposed linac configuration for the Next Generation Light Source (NGLS) at LBNL Corlett . The choice of a single magnetic bunch compressor simplifies our consideration and analysis, although the concept is also applicable for typical bunch compressor arrangements with multiple stages. We note that a single bunch compressor arrangement has also been considered for the FERMI@Elettra FEL in order to minimize the impact of microbunching instabilities Venturini . The generic layout of the reversible electron beam heater system is depicted in Fig. 1. It consists of linac sections providing and accelerating high- brightness electron beams, a magnetic bunch compressor chicane in order to achieve sufficient peak currents to drive the FEL process, and two transverse deflecting rf structures located up and downstream of the bunch compressor. An additional higher-harmonic rf linearizer system (Linearizer), like at the LCLS or FLASH Lin , can be used to achieve uniform bunch compression by means of longitudinal phase space linearization upstream of the bunch compressor. The whole system can be supplemented by dedicated longitudinal phase space diagnostics (see Sec. V), and except for the two TDSs, the layout is commonly used for bunch compression at present and future X-ray FELs. The principle of the reversible electron beam heater relies on the physics of TDSs arising from the Panofsky-Wenzel theorem Panowsky ; Bro , which states that the transverse momentum gain $\Delta\vec{p}_{\perp}$ of a relativistic electron imprinted by a TDS is related to the transverse gradient of the longitudinal electric field $\nabla_{\perp}\mathcal{E}_{z}$ inside the TDS, and yields $\Delta\vec{p}_{\perp}=-i\frac{e}{\omega}\int_{0}^{L}\nabla_{\perp}\mathcal{E}_{z}d\tilde{z}\,,$ (1) where $\omega/(2\pi)$ is the operating rf frequency, $e$ is the elementary charge, $L$ is the structure length, and $\tilde{z}$ is the longitudinal position inside the structure (not to be confused with the beamline coordinate, which is given by $s$ in the following). Operating a TDS with vertical deflection, i.e., in y-direction, near the zero-crossing rf phase $\psi=\omega/c\,z$, electrons experience transverse kicks Kick $\Delta y^{\prime}=\frac{e\omega V_{y}}{cE}z=K_{y}z$ (2) and relative energy deviations ($\delta=\Delta E/E$) IES1 ; IES2 $\Delta\delta=K_{y}\frac{1}{L}\int_{0}^{L}{y(s)ds}=K_{y}\overline{y}\,,$ (3) where $K_{y}=e\omega V_{y}/(cE)$ is the vertical kick strength, $V_{y}$ is the peak deflection voltage in the TDS, $c$ is the speed of light in vacuum, $E$ is the electron energy, and $\overline{y}$ is the mean vertical position over the structure length $L$ along the beamline relative to the central axis inside the finite TDS. Here, $z$ is the internal bunch length coordinate of the electron relative to the zero-crossing rf phase. Both the additional transverse kicks and relative energy deviations are induced by the TDS operation itself and generate correlations within an electron bunch. In fact, near the zero-crossing rf phase (see Eq. (2)), the induced transverse kick correlates linearly with the internal bunch length coordinate ($z=ct$) and enables high-resolution temporal diagnostics (e.g., Refs. Kick ; Roehrs ; Filippetto ), whereas the induced relative energy deviation correlates with the vertical offset inside the TDS and results in an induced relative energy spread $\Delta\sigma_{\delta}=K_{y}\sigma_{y}$. Here, the symbol $\sigma$ denotes the root mean square (r.m.s.) value, and $\sigma_{y}$ is the vertical r.m.s. beam size. This additional energy spread (cf. laser heater LH ; LCLSheater ), in combination with the momentum compaction $R_{56}$ of a bunch compressor chicane, is able to smear microbunch structures, and correspondingly suppresses the associated instability as is shown in Sec. IV. The effect of induced energy spread (“beam heating”) is generated by off-axis longitudinal electric fields, related to the principle of a TDS by the Panofsky-Wenzel theorem, and has been observed experimentally at FLASH IES3 and the LCLS emma . The induced energy spread is uncorrelated in the longitudinal phase space $(z,\delta)$, but shows correlations in the phase space $(y,\delta)$, which is the reason that it can be eliminated (“beam cooling”) with a second TDS in a reversible mode as is shown in the following by two different approaches. ### II.1 Linear beam optics The transverse betatron motion of an electron passing through a TDS with vertical deflection (in $y$) is given by $y(s)=y_{0}(s)+S_{y}(s,s_{0})z$ (4) with the vertical shear function (e.g., Refs. Kick ; Roehrs ; IES2 ) $S_{y}(s,s_{0})=R_{34}K_{y}=\sqrt{\beta_{y}(s)\beta_{y}(s_{0})}\mathrm{sin}(\Delta\phi_{y}(s,s_{0}))\frac{e\omega V_{y}}{cE}\,,$ (5) where $R_{34}$ is the angular-to-spatial element of the vertical beam transfer matrix from the TDS at $s_{0}$ to any position $s$, $\beta_{y}$ is the vertical beta function, $\Delta\phi_{y}$ is the vertical phase advance between $s_{0}$ and $s$, and $y_{0}$ describes the vertical beam offset independent of any TDS shearing effect. Referring to the layout depicted in Fig. 1 and taking bunch compression into account, the induced vertical beam offset ($\Delta y=y-y_{0}$) downstream of the second TDS becomes (omitting the subscript $y$ in $S_{y}$) $\displaystyle\Delta y(s)=$ $\displaystyle S_{1}(s,s_{1})z_{1}+S_{2}(s,s_{2})z_{2}$ $\displaystyle=$ $\displaystyle(CS_{1}(s,s_{1})+S_{2}(s,s_{2}))z_{2}$ (6) with the bunch compression factor $C=z_{1}/z_{2}$ and the shear functions $S_{1,2}(s,s_{1,2})$ of the corresponding TDSs. Here, $S_{1}(s,s_{1})z_{1}$ describes the vertical beam offset induced by the first TDS located at $s_{1}$ that is independent of the second TDS. In order to cancel the spatial chirp induced by the combined TDS operation, the beam offset $\Delta y$ in Eq. (6) must vanish for any $z_{2}$. Hence, using Eq. (5) for $S_{1,2}$ in Eq. (6) and taking acceleration from $E_{1}$ to $E_{2}$ in Linac2 into account by making the replacement $\beta_{y}(s)\beta_{y}(s_{1})\rightarrow\beta_{y}(s)\beta_{y}(s_{1})E_{1}/E_{2}$ Kick , we get $\displaystyle C\sqrt{\beta_{y}(s_{1})}\mathrm{sin}(\Delta\phi_{y}(s,s_{1}))\sqrt{E_{1}}K_{1}$ $\displaystyle+\sqrt{\beta_{y}(s_{2})}\mathrm{sin}(\Delta\phi_{y}(s,s_{2}))\sqrt{E_{2}}K_{2}=0\,,$ (7) where $K_{1,2}$ are the vertical kick strengths of the corresponding TDSs, and $\Delta\phi_{y}(s,s_{1,2})$ describes the vertical phase advances between $s_{1,2}$ and $s$, respectively. As a consequence, the phase advance between both TDSs is $\Delta\phi_{y}(s_{2},s_{1})=\Delta\phi_{y}(s,s_{1})-\Delta\phi_{y}(s,s_{2})$. A general solution, valid for any position $s$ downstream of the second TDS, is only possible for a phase advance difference of $\Delta\phi_{y}(s_{2},s_{1})=n\cdot\pi$ (8) with $n$ being integer, and the kick strength $K_{2}=\pm C\sqrt{\frac{\beta_{y}(s_{1})}{\beta_{y}(s_{2})}}\sqrt{\frac{E_{1}}{E_{2}}}K_{1}\,,$ (9) where the sign depends on the actual phase advance, i.e., $\Delta\phi_{y}(s_{2},s_{1})=\pi+n\cdot 2\pi$ for ($+$) and $\Delta\phi_{y}(s_{2},s_{1})=n\cdot 2\pi$ for ($-$). The different sign of $K$ can technically be achieved by changing the rf phase in the TDS by $180^{\circ}$ which results in a zero-crossing rf phase with opposite slope and deflection. Besides cancelation of the induced spatial chirps, the induced energy spread of the first TDS needs to be eliminated in the second structure in order to have a fully reversible electron beam heater. Applying Eq. (3) similar to Eq. (6), the relative energy deviation downstream of the second TDS for finite structure lengths become (omitting the argument in $y(s)$ and $S(s)$) $\Delta\delta=K_{1}\overline{y_{1}}C\frac{E_{1}}{E_{2}}+K_{2}\overline{(y_{2}+S_{1}z_{1})}$ (10) with the mean vertical offsets $\overline{y_{1}}$ and $\overline{(y_{2}+S_{1}z_{1})}$ inside the TDSs. For constant vertical offsets inside the TDSs or short structure lengths, the mean vertical offsets can be replaced by the actual offsets, i.e., $\overline{y_{1}}\rightarrow y_{1}$ and $\overline{(y_{2}+S_{1}z_{1})}\rightarrow(y_{2}+S_{1}z_{1})$. The latter describes the offset in the second TDS and involves the spatial chirp induced by the first TDS with $S_{1}\sim\mathrm{sin}(\Delta\phi_{y}(s_{2},s_{1}))$, which vanishes in the case of spatial chirp cancelation given by Eq. (8). In order to cancel the relative energy spread induced by the combined TDS operation, it follows $K_{1}\overline{y_{1}}C\frac{E_{1}}{E_{2}}+K_{2}\overline{y_{2}}=0\,.$ (11) The general transverse beam transport optics with the vertical phase advance condition in Eq. (8) gives $\overline{y_{2}}=\pm\overline{y_{1}}\sqrt{\beta_{y}(s_{2})/\beta_{y}(s_{1})}$, and taking $\beta_{y}(s_{2})\rightarrow\beta_{y}(s_{2})E_{1}/E_{2}$ (see prior Eq. (7)) into account yields exactly the same condition as in Eq. (9). Simultaneous spatial chirp and energy spread cancelation in the second TDS is the basic principle for reversible electron beam heating and enables local increase of slice energy spread. The additional energy spread in the bunch compressor, which is added in quadrature by the first TDS, can be controlled by the kick strength $K_{1}$ and the vertical beam size $\sigma_{y}(s_{1})$. In the following, a complementary approach to discuss the reversible beam heating system is shown. It uses the matrix formalism for beam transport and provides an analytical way to show microbunching gain suppression and to discuss the impact of beam and rf jitter. ### II.2 Matrix formalism We adopt the beam transport matrix notation of a 6x6 matrix for $(x,x^{\prime},y,y^{\prime},z,\delta)$ but leaves $(x,x^{\prime})$ out for simplicity, i.e., $(y,y^{\prime},z,\delta)$ is used in the following. The 4x4 beam transport matrix for a vertical deflecting TDS in thin-lens approximation reads (e.g., Refs. exchange1 ; IES1 ; IES3 ) ${\mathbf{R}}_{T}^{thin}=\begin{pmatrix}1&0&0&0\\\ 0&1&K&0\\\ 0&0&1&0\\\ K&0&0&1\end{pmatrix}.$ (12) As discussed above, the main components of the given reversible heater system shown in Fig. 1 consist of TDS1 with the kick strength $K_{1}$, a bunch compressor with the momentum compaction factor $R_{56}$, and TDS2 with the kick strength $K_{2}$. Including the momentum compaction factor $R_{56}$ and acceleration in Linac2 ($E_{1}\rightarrow E_{2}$), the 4x4 beam matrix between the two TDSs is given by ${\mathbf{R}}_{C}=\begin{pmatrix}R_{33}&R_{34}&0&0\\\ R_{43}&R_{44}&0&0\\\ 0&0&1&R_{56}\\\ 0&0&0&\frac{E_{1}}{E_{2}}\end{pmatrix}\,.$ (13) In order to allow the energy change in the first TDS to be compensated for in the second TDS, we require the point-to-point imaging from TDS1 to TDS2 (i.e., $R_{34}=0$), which corresponds to an equivalent vertical phase advance of $\Delta\phi_{y}(s_{2},s_{1})=n\cdot\pi$ with $n$ being integer (see Eq. (8)). Then we get the magnification factor $R_{33}=\pm\sqrt{\beta_{y}(s_{2})/\beta_{y}(s_{1})}$ and $R_{44}=1/R_{33}$. The linear accelerator section with higher-harmonic rf linearizer (Linac1 and Linearizer) upstream of the first TDS introduces an appropriate energy chirp $h$ for uniform bunch compression. Without loss of generality, we neglect acceleration between the two TDSs, i.e., we do not consider Linac2 anymore. Including Linac2 would simply result in a correction term $\sqrt{E_{1}/E_{2}}$ (cf. Eqs. (9) and (15) below) but would leave the physics unchanged. Then the entire 4x4 beam transport matrix from the beginning of TDS1 to the end of TDS2 becomes $\begin{pmatrix}R_{33}&0&0&0\\\ R_{43}+K_{1}K_{2}R_{56}&\frac{1}{R_{33}}&\frac{K_{1}}{R_{33}}+K_{2}(1+hR_{56})&K_{2}R_{56}\\\ K_{1}R_{56}&0&1+hR_{56}&R_{56}\\\ {K_{1}}+R_{33}K_{2}&0&0&\frac{1}{1+hR_{56}}\end{pmatrix}\,.$ (14) Cancelation of the induced spatial chirp ($\Delta y^{\prime}\sim z_{0}$, cf. Eq. (2)) requires $R_{45}=0$ (6x6-matrix notation), i.e., $K_{1}/R_{33}+K_{2}(1+hR_{56})=0\,,$ (15) where $R_{45}$ describes the coupling between $y^{\prime}$ and $z_{0}$. We note that the coupling between $\delta$ and $y_{0}$ (i.e., $R_{63}$ element) is nonzero in Eq. (14) because the bunch is energy-chirped after compression ($\delta\sim z\sim y_{0}$), which can be removed by Linac3 downstream of TDS2. For uniform bunch compression with $C^{-1}=(1+hR_{56})$, no acceleration in Linac2, i.e., $E_{2}=E_{1}$, and taking into account that $R_{33}=\pm\sqrt{\beta_{y}(s_{2})/\beta_{y}(s_{1})}$, Eq. (15) is identical to Eq. (9). Thus, both formalisms yield the same result. Since the kick strength of the first TDS is very weak, it can be implemented by means of a short rf structure and the thin-lens approximation is still valid. However, the kick strength of the second TDS is usually stronger, and the effect of the finite structure length should be taken into account. The symplectic beam transport matrix of a finite TDS with the length $L_{2}$ is given in Ref. exchange1 by ${\mathbf{R}}_{T}^{thick}=\begin{pmatrix}1&L_{2}&K_{2}L_{2}/2&0\\\ 0&1&K_{2}&0\\\ 0&0&1&0\\\ K_{2}&K_{2}L_{2}/2&K_{2}^{2}L_{2}/6&1\end{pmatrix}.$ (16) In this case, we require the point-to-point imaging is from the first TDS to the middle of the second TDS in order to have a complete cancellation. The overall matrix from TDS1 to the end of TDS2, when Eq. (15) is fulfilled, becomes more complicated. A few correction terms containing the length $L_{2}$ of TDS2 appear, which however does not change the working principle of the reversible beam heater system. It should be pointed out that downstream of the reversible heater system, the beam is slightly coupled in $y^{\prime}-\delta_{0}$ and $y-z_{0}$, which results in a small growth of the projected emittance given by $\epsilon_{y,z}^{2}=\epsilon_{y0,z0}^{2}+\epsilon_{y0}\epsilon_{z0}\frac{\beta_{y0}\gamma_{z0}K_{1}^{2}R_{56}^{2}}{(1+hR_{56})^{2}}\,,$ (17) where $\epsilon_{y0,z0}$ is the initial vertical (longitudinal) emittance, and $\beta_{y0}$ and $\gamma_{z0}$ are the initial Twiss parameters. As is shown in the following section, this projected emittance growth is typically negligible. ## III Reversible heating and emittance preservation We demonstrate the feasibility of the reversible beam heater system by numerical simulations using the particle tracking code elegant Elegant , and the simulations in the following include $5\,\times\,10^{5}$ particles. Table 1 summarizes the main parameters used in the simulations, and the adopted accelerator optics model, including the positions of the TDSs, is shown in Fig. 2. The magnetic bunch compressor chicane is assumed to bend in the horizontal plane, and the TDSs are oriented perpendicularly with vertical deflection. In the previous section, Table 1: Parameters of the electron beam, of the bunch compressor system, and of the transverse deflecting rf structures. Parameter \| Symbol \| Value \| Unit ---\|---\|---\|--- Beam energy at TDS1/2 \| $E$ \| 350 \| MeV Lorentz factor at TDS1/2 \| $\gamma$ \| 685 \| Initial transverse emittance \| $\gamma\epsilon_{x,y}$ \| 0.6 \| $\mu$m Initial slice energy spread \| $\sigma_{E}$ \| $\sim$ 1 \| keV Momentum compaction factor \| $R_{56}$ \| $-138$ \| mm Compression factor \| $C$ \| $\sim$ 13 \| Final bunch current \| $I_{f}$ \| $\sim$ 520 \| A TDS1/2 rf frequency \| $\omega/2\pi$ \| 3.9 \| GHz Voltage of TDS1 \| $V_{1}$ \| 0.415 \| MV Voltage of TDS2 (without CSR) \| $V_{2}$ \| 5.440 \| MV Length of TDS1 \| $L_{1}$ \| 0.1 \| m Length of TDS2 \| $L_{2}$ \| 0.5 \| m Figure 2: Relevant accelerator optics (Twiss parameters) and positions of the transverse deflecting rf structures used to numerically demonstrate the reversible beam heater system. we included Linac2 for a general derivation of the method, but in practice, due to wakefield concerns, we recommend putting TDS2 right after the bunch compressor. In order to show numerical examples based on this approach, Linac2 is not considered anymore throughout the rest of this paper. Except for the TDSs, the parameters are similar to the magnetic bunch compressor system discussed for the Next Generation Light Source at LBNL Corlett ; Venturini2 . The initial longitudinal electron bunch profile is assumed to be flat-top with a peak current of $\sim$ 40 A and a slice energy spread of $\sim$ 1 keV (r.m.s.). The initial linear and quadratic chirp is set for a uniform compression factor $C$ of about 13 across the entire bunch length. This is possible even with bunch compressor nonlinearities by using a higher-harmonic rf linearizer upstream of the bunch compressor Lin and needed to achieve uniform cancelation of the induced energy spread downstream of TDS2. Figure 3 shows the principle of the reversible beam heater system by means of simulation of the longitudinal phase space at different positions along the beamline. (a) Upstream of TDS1. (b) Downstream of TDS1. (c) Upstream of TDS2. (d) Downstream of TDS2. Figure 3: Simulation of the longitudinal phase space after removing the correlated energy chirp: (a) upstream of the first TDS, (b) directly downstream of the first TDS, (c) directly downstream of the bunch compressor and upstream of the second TDS, and (d) downstream of the second TDS. The axes scales change from (b) to (c) when bunch compression takes place. The bunch head is on the left, i.e., where $z/c<0$. The impact of CSR is not taken into account (cf. next subsection for CSR effects). The initial slice energy spread is heated up to $\sim$ 10 keV (r.m.s.) in the first TDS, increased by the compression factor in the bunch compressor to $\sim$ 130 keV (r.m.s.), and finally cooled down to $\sim$ 13 keV (r.m.s.) by the second TDS (see Figs. 3(a)-3(d)). The plot in Fig. 4(b) shows that the heating induced by the first TDS is perfectly reversible, and the final slice energy spread is simply the initial slice energy spread scaled with the compression factor, which would be exactly the same like in the case without using the reversible beam heater system. Figure 4(a) shows the heater system impact on both the projected emittance (horizontal and vertical) and the core energy spread, i.e., the slice energy spread in the center of the bunch, for different voltages in the second TDS. The minimum of the vertical emittance is related to the cancelation of the spatial chirp and energy spread induced by the first TDS. The horizontal emittance is not affected at all, and the small projected emittance growth (6 %) in the vertical plane at the minimum is due to residual coupling generated by the system that is described by Eq. (17). Nevertheless, as shown in Sec. III.1, even in the case with CSR effects, the horizontal slice emittance stays unaffected at all and the vertical slice emittance exhibits only deviations in the bunch head ($z/c<0$) and tail ($z/c>0$). (a) Projected emittances and core energy spread. (b) Slice energy spread. Figure 4: Simulations without CSR effects on the impact of the reversible heater system on projected emittances, core energy spread, and slice energy spread : (a) Projected emittances (normalized) and core energy spread, and (b) slice energy spread for $V_{\mathrm{2}}$ at minimum emittance (see Fig. 4(a)). The longitudinal coordinate is normalized to the bunch length. ### III.1 Impact of coherent synchrotron radiation The previous results undergo small modifications when including CSR effects, which is shown in Fig. 5. The voltage of the second TDS for minimum projected emittance in the vertical is shifted by about 0.2 MV to lower values which is due the additional energy chirp induced by CSR. In comparison to the case without any CSR effects (cf. Fig. 4), the projected emittance in the vertical plane is slightly increased and the slice energy spread is not perfectly canceled in the head and tail. The slice energy spread in the core part of the bunch is also slightly increased to 17.5 keV (r.m.s.) (instead of 13.5 keV (r.m.s.) in the absence of CSR). The projected emittance in the horizontal is about 1.7 larger which is independent of the reversible beam heater operation. This horizontal emittance growth can further be reduced by minimizing the horizontal beta function in the last dipole of the chicane where the bunch length becomes the shortest. This optimization is independent of the relevant motion in the vertical and does not affect the results of the reversible heater system. (a) Projected emittances and core energy spread. (b) Slice energy spread. Figure 5: Simulation on the impact of the reversible beam heater system on projected emittances, core energy spread, and slice energy spread: (a) Projected emittances (normalized) and core energy spread, and (b) slice energy spread for $V_{\mathrm{2}}$ at minimum emittance (see Fig. 5(a)). CSR effects are included by means of the 1-dimensional model in elegant Elegant . Albeit the fact that the projected emittances are increased, the horizontal slice emittance stays unaffected and the vertical slice emittance exhibits only deviations in the head and tail due to CSR effects as is shown in Fig. 6. Thus, the core emittances are well preserved. We note that vertically streaked bunches in the bunch compressor chicane may change the impact of CSR effects but require a 3-dimensional “point-to-point” tracking which is not available neither in elegant nor in CSRtrack CSRtrack , and is beyond the scope of this paper. Figure 6: Simulation of the normalized slice emittance for both the vertical and horizontal upstream of the first and downstream of the second TDS. CSR effects are included. ## IV Microbunching gain suppression The principle of the microbunching gain suppression with the reversible beam heater system is shown by an analytical treatment following Ref. LCLSheater and by using the beam transport matrix in Eq. (14). Then we show the feasibility of the reversible heater system to suppress microbunching instabilities by means of particle tracking simulations with initial density and energy modulations. ### IV.1 Analytical calculations Using the vector notation $(y_{0},y_{0}^{\prime},z_{0},\delta_{0})$ for particles in the first linac upstream of the first TDS, the longitudinal position downstream of the second TDS is given by $z=K_{1}R_{56}y_{0}+(1+hR_{56})z_{0}+R_{56}\delta_{0}\,.$ (18) Suppose that $\delta_{0}=\delta_{u}+\delta_{m}$, where $\delta_{u}$ is the uncorrelated relative energy deviation, and $\delta_{m}(z_{0})$ is the relative energy modulation accumulated before and in the first linac (Linac1). Following Ref. LCLSheater , the initial energy modulation at the wavenumber $k_{0}$ is converted into additional density modulation at a compressed wavenumber $k$. For a 4-dimensional (4-D) distribution function $F(y,y^{\prime},z,\delta)$, the bunching factor $b(k)$ is given by $\displaystyle b(k)=$ $\displaystyle\int dydy^{\prime}dzd\delta e^{-ikz}F(y,y^{\prime},z,\delta)$ $\displaystyle=$ $\displaystyle\int dy_{0}dy_{0}^{\prime}dz_{0}d\delta_{u}e^{-ikK_{1}R_{56}y_{0}-ik(1+hR_{56})z_{0}}$ $\displaystyle e^{-ikR_{56}(\delta_{u}+\delta_{m}(z_{0}))}F_{0}(y_{0},y_{0}^{\prime},z_{0},\delta_{u})\,,$ (19) where $F_{0}(y_{0},y_{0}^{\prime},z_{0},\delta_{u})$ is the initial 4-D distribution. If the induced energy modulation is small such that $\|kR_{56}\delta_{m}\|\ll 1$, we can expand the exponent of Eq. (19) up to the first order in $\delta_{m}$ to obtain $\displaystyle b(k)\approx$ $\displaystyle~{}b_{0}(k_{0})-ikR_{56}\int dz_{0}\delta_{m}(z_{0})e^{-ik_{0}z_{0}}$ $\displaystyle\times$ $\displaystyle\int dy_{0}d\delta_{u}e^{-ikK_{1}R_{56}y_{0}-ikR_{56}\delta_{u}}U(y_{0})V(\delta_{u})\,,$ (20) where $k=Ck_{0}$, $C=1/(1+hR_{56})$, $U(y_{0})$ describes the transverse profile, and $V(\delta_{u})$ is the initial energy distribution. For both Gaussian profiles ($U$ and $V$), we have $\displaystyle b(k)=b_{0}(k_{0})-$ $\displaystyle ikR_{56}\delta_{m}(k_{0})\exp\left[-(k^{2}R_{56}^{2}K_{1}^{2}\sigma_{y1}^{2}/2)\right]$ $\displaystyle\times$ $\displaystyle\exp\left[-(k^{2}R_{56}^{2}\sigma_{\delta u}^{2}/2)\right]\,.$ (21) Here, we denote the Fourier transform of $\delta_{m}(z_{0})$ as $\delta_{m}(k_{0})$, which is the accumulated energy modulation at the wavenumber $k_{0}$ in the first linac due to longitudinal space charge and other collective effects. The initial energy spread is given by $\sigma_{\delta u}$, and $\sigma_{y1}$ is the vertical beam size in the first TDS. We see that $K_{1}\sigma_{y1}$ acts like effective energy spread for microbunching gain suppression. ### IV.2 Numerical simulations (a) Downstream of TDS2: Heater system off. (b) Downstream of TDS2: Heater system on. Figure 7: Simulation on suppression of microbunching instabilities due to an initial density modulation, i.e., simulating CSR-driven microbunching. The entire longitudinal phase space, after removing the correlated energy chirp, is shown. Suppression of microbunching instabilities is demonstrated by using both a pure initial density modulation with 5 $\%$ peak amplitude and $100\,\mathrm{\mu m}$ modulation wavelength ($\lambda_{m}$), and a pure initial energy modulation with 3 keV peak amplitude and $\lambda_{m}=50\,\mathrm{\mu m}$. Whereas the case with initial energy modulation is immediately consistent with the previous analytical treatment and describes the longitudinal space charge driven microbunching instability lsc-ub , the initial density modulations need to be converted into energy modulations by longitudinal CSR-impedance which expresses the consistency and describes the CSR-driven microbunching instability CSR-ub . The simulations were performed using the code elegant with $1\,\times\,10^{6}$ particles. Figure 7 shows the longitudinal phase space downstream of the second TDS, after removing the correlated energy chirp (linear and quadratic chirp), for both the reversible beam heater system switched off (Fig. 7(a)) and on (Fig. 7(b)). In the case without reversible beam heater, energy and density modulations at the compressed modulation wavelength $\lambda_{m}/C$ appear, i.e., CSR-driven microbunching becomes visible. When switching the reversible beam heater on, the microbunching instability disappears and the resulting longitudinal phase space remains smooth. The reason is that the microbunches at the compressed wavelength are smeared due to $R_{56}K_{1}\sigma_{y1}$ (cf. Eq. (21)), and accordingly, the modulations appear as correlations in the phase spaces $(y,z)$ and $(y^{\prime},\delta)$. The same effect of microbunching suppression is given for initial energy modulations as shown in Fig. 8. (a) Downstream of TDS2: Heater system off. (b) Downstream of TDS2: Heater system on. Figure 8: Simulation on suppression of microbunching instabilities due to an initial energy modulation, i.e., simulating longitudinal space charge driven microbunching. For the sake of clarity, only the core of the longitudinal phase space, after removing the correlated energy chirp, is shown. The effect of the microbunching instability appears even stronger compared to the simulations case with initial density modulations, but the performance of the reversible heater system is the same with a smooth residual longitudinal phase space when the reversible beam heater is switched on (see Fig. 8(b)). Figures 7 and 8 are obtained for a magnetic bunch compressor system as shown in Fig. 1. The electron bunch will be further accelerated and transported throughout the rest of the accelerator to reach the final beam energy and peak current in order to drive an X-ray FEL (not studied in this paper). A microbunched electron beam as illustrated in Figs. 7(a) and 8(a), i.e., when the reversible beam heater system is switched off, will accumulate additional energy and density modulations, which would lead to unacceptable longitudinal phase space properties for an X-ray FEL such as a large slice energy spread. ## V Practical considerations The previous sections covered the principle of reversible electron beam heating and microbunching gain suppression by means of analytical calculations and numerical simulations. In real accelerators, we also have to deal with imperfections, jitter and drifts of various parameters, and accordingly supplementary studies with respect to sensitivity on jitter sources and tolerances have to be performed. In the following, we discuss the impact of beam and rf jitter on the reversible beam heater system, and also point out the inherent possibility of longitudinal phase space diagnostics and on-line monitoring. ### V.1 Jitter and tolerances The impact of beam and rf jitter on the reversible beam heater method can effectively be discussed using the Eqs. (2) and (14) with the condition in Eq. (15). Deviations from the conditions in Eq. (15) can appear from jitter of the individual peak deflection voltages $V_{1}$ and $V_{2}$ of the TDSs, and lead to growth of the projected vertical emittance as is shown in Fig. 5(a), where the voltage of the second TDS is varied. Even in the case of a large TDS voltage jitter of 1 %, the vertical projected emittance growth is less than 2 $\%$ (see Fig. 5(a)). In the case of acceleration between the first and second TDS, also energy jitter, which is similar or smaller than TDS voltage jitter, due to this intermediate acceleration leads to deviation of the condition in Eq. (15). The choice of superconducting accelerator technology even provide much better rf stability HS ; CS . Pure arrival time jitter upstream of the first TDS has no impact as long as the condition in Eq. (15), which describes the coupling between $y^{\prime}$ and $t=z/c$, is fulfilled. In the case that Eq. (15) is not exactly fulfilled, e.g., due to TDS voltage jitter which is on the percent-level, the impact of typical arrival time jitter well below 100 fs, like at the LCLS LCLSnature2 or FLASH CS , is negligible. The most critical jitter sources arise from energy jitter upstream of the bunch compressor chicane and from rf phase jitter in the TDSs. The momentum compaction factor translates energy jitter into arrival time jitter, which leads to vertical kicks in the second TDS. The same effect of additional vertical kicks is generated by rf phase jitter in the TDSs. In order to have small impact of vertical kicks on the remaining beam transport, we demand $\Delta\sigma_{y^{\prime}}\ll\sigma_{y^{\prime}}$ directly downstream of the second TDS with the induced vertical r.m.s. kick $\Delta\sigma_{y^{\prime}}$ and the intrinsic beam divergence $\sigma_{y^{\prime}}$. The relevant total vertical r.m.s. kick is given by $\displaystyle\Delta\sigma_{y^{\prime}}=$ $\displaystyle\sqrt{\left(K_{2}R_{56}\frac{\sigma_{E}}{E}\right)^{2}+\left(K_{2}\frac{c}{\omega}\sigma_{\varphi_{2}}\right)^{2}+\left(\frac{K_{1}}{R_{33}}\frac{c}{\omega}\sigma_{\varphi_{1}}\right)^{2}}$ $\displaystyle=$ $\displaystyle\sqrt{\left(K_{2}R_{56}\frac{\sigma_{E}}{E}\right)^{2}+\left(K_{2}\frac{c}{\omega}\right)^{2}\left(\sigma_{\varphi_{2}}^{2}+\frac{1}{C^{2}}\sigma_{\varphi_{1}}^{2}\right)}$ $\displaystyle\approx$ $\displaystyle\sqrt{\left(K_{2}R_{56}\frac{\sigma_{E}}{E}\right)^{2}+\left(K_{2}\frac{c}{\omega}\right)^{2}\sigma_{\varphi_{2}}^{2}}$ (22) with the energy jitter $\sigma_{E}/E$ upstream of the bunch compressor, the rf phase jitter $\sigma_{\varphi_{1,2}}$ of the TDSs, the magnification factor $R_{33}$ from the first to the second TDS (see Eq. (13)), and using Eq. (15) with the compression factor $C=(1+hR_{56})^{-1}$. We see that the vertical r.m.s. kick due to rf phase jitter in the first TDS scales with $C^{-2}$ and can be neglected compared to the vertical r.m.s. kick induced by the second TDS when we assume the same amount of rf phase jitter in both TDSs. The condition for trajectory stability $\Delta\sigma_{y^{\prime}}\ll\sigma_{y^{\prime}_{2}}=\sqrt{\epsilon_{y_{2}}/\beta_{y_{2}}}$ with the intrinsic (uncorrelated) r.m.s. beam divergence $\sigma_{y^{\prime}_{2}}$ downstream of the bunch compressor at TDS2, where $\epsilon_{y_{2}}$ is the geometrical emittance, can be restated as $\sqrt{\left(R_{56}\frac{\sigma_{E}}{E}\right)^{2}+\left(\frac{c}{\omega}\sigma_{\varphi_{2}}\right)^{2}}\ll\frac{\epsilon_{y_{2}}}{K_{2}\sqrt{\beta_{y_{2}}\epsilon_{y_{2}}}}=\frac{\sqrt{\epsilon_{y_{2}}\epsilon_{y_{1}}}}{C\Delta\sigma_{\delta_{1}}}\,.$ (23) Here, $\Delta\sigma_{\delta_{1}}$ is the additional relative energy spread induced by the first TDS for suppression of microbunching instabilities, and $\epsilon_{y_{1}}$ denotes the geometrical emittance upstream of the bunch compressor at TDS1. For the example parameters discussed throughout this paper (see also Table 1), i.e., $C=13$, $\gamma\epsilon_{y_{1}}=0.6\,\mathrm{\mu m}$, $\gamma\epsilon_{y_{2}}=0.72\,\mathrm{\mu m}$ (see Fig. 5(a)), and $\Delta\sigma_{\delta_{1}}E\approx 10\,\mathrm{keV}$ with $E=350\,\mathrm{MeV}$ ($\gamma=685$), the stability condition in Eq. (23) yields pure relative energy jitter (neglecting rf phase jitter) of $\sigma_{E}/E\ll 1.9\cdot 10^{-5}$ or pure rf phase jitter (neglecting energy jitter) of $\sigma_{\varphi_{2}}\ll 0.012^{\circ}$. A combination of both will obviously tighten the acceptable jitter. This level of rf stability is difficult to achieve in normal conducting linacs with single bunch operation, but might be achieved with superconducting accelerator technology like at FLASH or as planned for NGLS, where many bunches can be accelerated in a long rf pulse, i.e., in a bunch train. Currently, several rf feedforward and feedback controls are able to stabilize the bunches at FLASH to $\sigma_{E}/E=3.0\cdot 10^{-5}$ and $\sigma_{\varphi}=0.007^{\circ}$ at 150 MeV HS ; SP , and further improvements towards $\sigma_{E}/E\leq 1.0\cdot 10^{-5}$ are planned using a fast normal conducting cavity upstream of the bunch compressors CS ; HS . With perfect scaling of rf jitter from several independent rf power stations that adds uncorrelated, we would expect an improvement of $\sqrt{150\,\mathrm{MeV}/350\,\mathrm{MeV}}\approx 0.66$ compared to the results at FLASH with $150\,\mathrm{MeV}$ and assuming the beam energy of $350\,\mathrm{MeV}$ in the bunch compressor of the NGLS design. Continuous-wave rf operation, as planned for the NGLS design Corlett , and a proper choice of rf working points for FEL operation might improve the stability further. ### V.2 Integrated longitudinal phase space diagnostics A practical spin-off of the reversible beam heater system is the availability of longitudinal phase space diagnostics. The vertical betatron motion of electrons passing through a TDS is described by Eq. (4), which enables a mapping from time (longitudinal coordinate) to the vertical Kick ; Roehrs ; Filippetto , and finally a possibility to obtain temporal bunch information by means of transverse beam diagnostics. In a similar manner, the relative energy deviation is mapped to the horizontal in the presence of horizontal momentum dispersion, like in a magnetic bunch compressor chicane (see, e.g., Refs. Roehrs ; Filippetto ). The combined operation makes single-shot measurements of the longitudinal phase space possible, and in the case of the generic layout of a reversible electron beam heater system as depicted in Fig. 1, longitudinal phase space measurements become feasible using the first TDS and observation screens in the dispersive section of the bunch compressor chicane. In order to get information of the bunch length after the bunch compression, the second TDS can be used with downstream observation screens (not shown in Fig. 1). In addition to invasive longitudinal phase space measurements of a single bunch using observation screens, even fully noninvasive measurements utilizing incoherent synchrotron radiation, emitted in the bunch compressor bending magnets, are possible (see, e.g., Ref. Gerth ). When using a fast gated camera, the implication will be the possibility of on-line monitoring the longitudinal phase space of individual bunches in multi-bunch accelerators. ## VI Summary and conclusions Our studies show that the reversible beam heater system proposed here can suppress microbunching instabilities and preserve the high beam brightness at the same time. Due to CSR effects, some vertical emittance degradation in the head and tail region of the bunch occurs, but the core emittances are well preserved. In the numerical demonstrations using the code elegant, the first TDS generates about 10 keV (r.m.s.) slice energy spread, which is similar to the laser heater but with a more Gaussian energy distribution (cf. laser heater). The bunch compression process increases the slice energy spread to $\sim$ 130 keV (r.m.s.), which is then reversed to $\sim$ 17 keV (r.m.s.) after the second TDS in the presence of CSR effects. Without CSR effects, the slice energy spread is reversed to $\sim$ 13 keV (r.m.s.), which demonstrates perfect cancelation. The simulations also show that initial bunching in energy and density in the beam can be smeared out during the process in the reversible beam heater system, i.e., microbunching instabilities can be suppressed. The resulting smooth beam can then propagate through the remaining accelerator without further generation of much additional energy spread and is advantageous for any kind of laser seeding manipulations and experiments. For example, this scheme significantly loosen the required laser power for short- wavelength HHG seeding NLS and may strongly impact the design of future seeded FELs. In addition, the reversible beam heater system exhibits integrated options for diagnosis and on-line monitoring of the longitudinal phase space applicable for multi-bunch machines, which is also the preferred type of accelerator for the reversible heater system due to large sensitivities on energy and rf jitter. Linear accelerators based on superconducting rf technology might be able to match the strict tolerances in order to keep vertical kicks small and to achieve a sufficient trajectory stability in the downstream undulators. ###### Acknowledgements. We would like to thank P. Emma, Ch. Gerth, A. Lumpkin, H. Schlarb, and J. Thangaraj for useful discussions and suggestions. This work was supported by Department of Energy Contract No. DE-AC02-76SF00515. ## References * (1) S. Jamison, Nature Photonics 4, 589 - 591 (2010). * (2) W. Ackermann et al., Nature Photonics 1, 336 - 342 (2007). * (3) P. Emma et al., Nature Photonics 4, 641 - 647 (2010). * (4) D. Pile, Nature Photonics 5, 456 - 457 (2011). * (5) E.L. Saldin, E.A. Schneidmiller, and M.V. Yurkov, Nucl. Instrum. Methods Phys. Res., Sect. A 398, 373 (1997). * (6) M. Borland et al., Nucl. Instrum. Methods Phys. Res., Sect. A 483, 268 (2002). * (7) E.L. Saldin, E.A. Schneidmiller, and M.V. Yurkov, Nucl. Instrum. Methods Phys. Res., Sect. A 528, 355 (2004). * (8) Z. Huang, M. Borland, P. Emma, J. Wu, C. Limborg, G. Stupakov, and J. Welch, Phys. Rev. ST Accel. Beams 7, 074401 (2004). * (9) Z. Huang et al., Phys. Rev. ST Accel. Beams 13, 020703 (2010). * (10) D.J. Dunning et al., Proceedings of the 1st International Particle Accelerator Conference, Kyoto, Japan, 2010, TUPE049. * (11) M. Cornacchia, S. Di Mitri, G. Penco, and A. Zholents, Phys. Rev. ST Accel. Beams 9, 120701 (2006). * (12) Y. Ding, P. Emma, Z. Huang, and V. Kumar, Phys. Rev. ST Accel. Beams 9, 070702 (2006). * (13) O. Altenmueller, R. Larsen, and G. Loew, Rev. Sci. Instrum. 35, 438 (1964). * (14) P. Emma, J. Frisch, and P. Krejcik, Technical Report No. LCLS-TN-00-12, 2000. * (15) M. Röhrs, Ch. Gerth, H. Schlarb, B. Schmidt, and P. Schmüser, Phys. Rev. ST Accel. Beams 12, 050704 (2009). * (16) D. Filippetto et al., Phys. Rev. ST Accel. Beams 14, 092804 (2011). * (17) Y. Ding, C. Behrens, P. Emma, J. Frisch, Z. Huang, H. Loos, P. Krejcik, and M-H. Wang, Phys. Rev. ST Accel. Beams 14, 120701 (2011). * (18) A. Zholents, P. Heimann, M. Zolotorev, and J. Byrd, Nucl. Instrum. Methods Phys. Res., Sect. A 425, 385 (1999). * (19) M. Cornacchia and P. Emma, Phys. Rev. ST Accel. Beams 5, 084001 (2002). * (20) P. Emma, Z. Huang, K.-J. Kim, and P. Piot, Phys. Rev. ST Accel. Beams 9, 100702 (2006). * (21) D. Xiang and Y. Ding, Phys. Rev. ST Accel. Beams 13, 094001 (2010). * (22) P. Piot, Y.-E Sun, J.G. Power, and M. Rihaoui1, Phys. Rev. ST Accel. Beams 14, 022801 (2011). * (23) D. Xiang and A. Chao, Phys. Rev. ST Accel. Beams 14, 114001 (2011). * (24) D. Xiang et al., Phys. Rev. Lett. 105, 114801 (2010). * (25) D. Xiang et al., Phys. Rev. Lett. 108, 024802 (2012). * (26) J. Corlett et al., Proceedings of the 24th Particle Accelerator Conference, New York, USA, 2011, TUOCS5. * (27) M. Venturini and A. Zholents, Nucl. Instrum. Methods Phys. Res., Sect. A 593, 53 (2008). * (28) H. Edwards, C. Behrens, and E. Harms, Proceedings of the 25th International Linear Accelerator Conference, Tsukuba, Japan, 2010, MO304. * (29) W. Panofsky and W. Wenzel, Rev. Sci. Instrum. 27, 967 (1956). * (30) M.J. Browman, Proceedings of the 15th Particle Accelerator Conference, Washington, D.C., USA, 1993, p. 800. * (31) S. Korepanov, M. Krasilnikov, F. Stephan, D. Alesini, and L. Ficcadenti, Proceedings of the 8th European Workshop on Beam Diagnostics and Instrumentation for Particle Accelerators, Venice, Italy, 2007, TUPB32. * (32) C. Behrens and Ch. Gerth, Proceedings of the 9th European Workshop on Beam Diagnostics and Instrumentation for Particle Accelerators, Basel, Switzerland, 2009, TUPB44. * (33) C. Behrens and Ch. Gerth, Proceedings of the 10th European Workshop on Beam Diagnostics and Instrumentation for Particle Accelerators, Hamburg, Germany, 2011, TUPD31. * (34) P. Emma (private communication). * (35) M. Borland, ANL/APS Report No. LS-287, 2000. * (36) M. Venturini et al., Proceedings of the 24th Particle Accelerator Conference, New York, USA, 2011, THP180. * (37) M. Dohlus and T. Limberg, Proceedings of the 26th International Free Electron Laser Conference, Trieste, Italy, 2004, MOCOS05. * (38) H. Schlarb (private communication). * (39) C. Schmidt et al., Proceedings of the 33rd International Free Electron Laser Conference, Shanghai, China, 2011, THPA26. * (40) S. Pfeiffer et al., LLRF-2011 workshop, Hamburg, Germany, 2011, “Feedback Strategies for longitudinal Beam Stabilization”. * (41) Ch. Gerth, Proceedings of the 8th European Workshop on Beam Diagnostics and Instrumentation for Particle Accelerators, Venice, Italy, 2007, TUPC03.	arxiv-papers	2011-11-17T15:03:00	2024-09-04T02:49:24.439895	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Christopher Behrens, Zhirong Huang, Dao Xiang", "submitter": "Christopher Behrens", "url": "https://arxiv.org/abs/1111.4122" }
1111.4424	# Initial conditions for star formation in clusters: physical and kinematical structure of the starless core OphA-N6 Tyler L. Bourke11affiliation: Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138; email tbourke@cfa.harvard.edu , Philip C. Myers11affiliation: Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138; email tbourke@cfa.harvard.edu , Paola Caselli22affiliation: School of Physics & Astronomy, E.C. Stoner Building, The University of Leeds, Leeds, LS2 9JT, UK , James Di Francesco33affiliation: National Research Council Canada, Herzberg Institute of Astrophysics, Victoria, BC, Canada , Arnaud Belloche44affiliation: Department of Physics and Astronomy, University of Calgary, Calgary, AB, Canada , René Plume55affiliation: Max-Planck-Institut für Radioastronomie, Auf dem Hügel 69, D-53121 Bonn, Germany , David J. Wilner11affiliation: Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138; email tbourke@cfa.harvard.edu ###### Abstract We present high spatial ($<$300 AU) and spectral (0.07 km s-1) resolution Submillimeter Array observations of the dense starless cluster core Oph A N6, in the 1 mm dust continuum and the 3-2 line of N2H+ and N2D+. The dust continuum observations reveal a compact source not seen in single-dish observations, of size $\sim$1000 AU and mass 0.005-0.01 ${\rm M}_{\sun}$. The combined line and single-dish observations reveal a core of size 3000 $\times$ 1400 AU elongated in a NW-SE direction, with almost no variation in either line width or line center velocity across the map, and very small non-thermal motions. The deuterium fraction has a peak value of $\sim$0.15 and is $>$ 0.05 over much of the core. The N2H+ column density profile across the major axis of Oph A-N6 is well represented by an isothermal cylinder, with temperature 20 K, peak density $7.1\times 10^{6}$ cm-3, and N2H+ abundance $2.7\times 10^{-10}$. The mass of Oph A-N6 is estimated to be 0.29 ${\rm M}_{\sun}$, compared to a value of 0.18 ${\rm M}_{\sun}$ from the isothermal cylinder analysis, and 0.63 ${\rm M}_{\sun}$ for the critical mass for fragmentation of an isothermal cylinder. Compared to isolated low-mass cores, Oph A-N6 shows similar narrow line widths and small velocity variation, with a deuterium fraction similar to “evolved” dense cores. It is significantly smaller than isolated cores, with larger peak column and volume density. The available evidence suggests Oph A-N6 has formed through the fragmentation of the Oph A filament and is the precursor to a low-mass star. The dust continuum emission suggests it may already have begun to form a star. ISM: individual (Oph-A N6) – stars: formation – stars: low-mass ## 1 Introduction It is now well established that low-mass (i.e., solar-like) stars form from the collapse of dense cores within molecular clouds (e.g., Larson 2003). The initial conditions of isolated star formation are known from continuum and line observations of many tens of dense cores in molecular cloud complexes such as Taurus, with relatively sparse concentrations of young stars (Di Francesco et al. 2007). The best commonly observed molecular line tracers of the central few thousand AU of centrally condensed cores on the verge of star formation are NH3 (ammonia), N2H+, and N2D+ (Caselli et al. 2002c; Tafalla et al. 2002; Crapsi et al. 2005, 2007). Observations of these lines and the dust continuum have enabled the properties of dense isolated starless cores thought to be near to or at the point of star formation to be well determined in recent years (Di Francesco et al. 2007; Bergin & Tafalla 2007). These properties include (i) a high degree of deuterium fractionation, (i.e., large N(N2D+)/N(N2H+)) (ii) a large central density ($\sim 10^{6}$ cm-3), (iii) depletion of CO and other C-bearing species, (iv) cold central regions ($<$10 K), and (v) line asymmetries indicative of infall motions. In this paper, we present the first detailed observational study of the internal structure of a starless cluster core, to explore how star formation in clusters compares to isolated star formation. Most stars are believed to form in close proximity to other stars within embedded clusters (Lada & Lada 2003; Allen et al. 2007; Bressert et al. 2010). As a result, a fundamental problem in astrophysics is whether star formation in cluster environments is similar to better-understood isolated star formation, or whether cluster star formation is more turbulent and dynamic. An important question is to what degree do stars in clusters form in the same way as in isolation, i.e., from cores whose properties strongly influence the stellar properties (Shu et al. 2004; Larson 2005; Tan et al. 2006; see reviews by Shu et al. 1987; Larson 2003), and to what degree do they form in a different, more dynamic way, where external forces and interactions matter more than initial conditions (e.g., Bonnell et al. 2001; Bonnell & Bate 2006). A number of observational studies made over recent years have provided information on the global properties of dense cores within nearby cluster- forming regions (Ward-Thompson et al. 2007). Dust continuum observations tracing high column densities have been more numerous, due to their faster mapping speeds compared to molecular line observations. As a result, large maps have been made of molecular clouds containing embedded clusters in Ophiuchus (L1688; Motte et al. 1998; Johnstone et al. 2000; Stanke et al. 2006; Young et al. 2006), Perseus (NGC 1333, IC348-SW; Sandell & Knee 2001; Hatchell et al. 2005; Enoch et al. 2006; H. Kirk et al. 2006), Corona Australis (Chini et al. 2003; Nutter et al. 2005), Serpens (Davis et al. 1999; Enoch et al. 2007), and the more distant and massive Orion cluster (Chini et al. 1997; Lis et al. 1998; Johnstone & Bally 1999; Li et al. 2007). Some progress has been made in large area mapping of molecular line dense gas tracers with resolution approaching that of dust continuum observations of $\leq 30\arcsec$, in nearby low-mass regions (Williams & Myers 2000; André et al. 2007; Walsh et al. 2007; Friesen et al. 2009, 2010a,b), and Orion (Ikeda et al. 2007, 2009; Tatematsu et al. 2008). Although some studies have been able to “resolve” starless cores in clusters in molecular spectral lines (Williams & Myers 2000; Walsh et al 2007), in the sense that the measured core size is larger than the beam, no study has examined the internal structure of any cluster core in detail. Observations of cores in cluster-forming regions that resolve individual cores are thus needed to understand how these regions can make stars so much more efficiently than in isolation. Progress toward this goal has been slower than for isolated regions. Most young clusters are more crowded so their cores suffer more confusion; their cores are smaller and more distant so they are harder to resolve, and they form stars more frequently so that starless cores are less common (Jijina et al. 1999). ### 1.1 Ophiuchus A N6 The Ophiuchus molecular cloud, at a distance of only 125 pc (de Geus et al. 1989; Knude & Hog 1998; Loinard et al. 2008; Lombardi et al. 2008), is the nearest example of cluster formation embedded within dense gas and dust, and is an ideal region in which to study the initial conditions for cluster formation (see review by Wilking, Gagné and Allen 2008; Motte et al. 1998; Johnstone et al. 2000, 2004; André et al. 2007; Enoch et al. 2008; Jørgensen et al. 2008; Simpson et al. 2008; Padgett et al. 2008; Friesen et al. 2009, 2010a,b; van Kempen et al. 2009; Gutermuth et al. 2009; Maruta et al. 2010). Within Ophiuchus, the Oph A ridge is the brightest of the clumps in both dust emission and N2H+ 1-0, containing 8 dust cores identified throught 1.3 mm continuum emission (Motte et al. 1998) and 6 local maxima of integrated N2H+ 1-0 emission (Di Francesco et al. 2004, hereafter DAM04; André et al. 2007). The N2H+ cores in the main part of the ridge, where the dust emission is strongest, show line widths that are not significantly different from those observed toward isolated starless cores (Di Francesco et al. 2004; Crapsi et al. 2005). In this region, N6 is the best core for studies of internal structure as it is isolated from the other Oph A cores (thereby suffering less from confusion), is bright in molecular lines, and is larger and thus better resolved than the other Oph A cores (Di Francesco et al. 2004; Pon et al. 2009). For these reasons, we have undertaken high angular resolution submillimeter observations of N6 using the Submillimeter Array in the high density gas tracers N2H+ and N2D+ 3-2, and combined those data with single dish observations using the James Clerk Maxwell Telescope (JCMT) and Institut de Radioastronomie Millimétrique (IRAM) 30-m telescope. This paper is divided into the following sections. In §2 we present our observations using the Submillimeter Array (SMA), JCMT, and IRAM 30-m, in §3 we present the line and continuum results, §4 presents the analysis, including the column densities, deuterium fraction, and structure of N6, §5 presents a discussion on the structure and evolution of N6, and compares it to both isolated cores and cores within cluster-forming regions, while a summary is presented in §6. ## 2 Observations ### 2.1 Submillimeter Array Observations in the N2D+ 3-2 line toward Oph A-N6 were undertaken with the SMA on 2006 July 22. The array was in its compact configuration and zenith opacities at 225 GHz were typically 0.1-0.13. The SMA correlator was configured with 2048 channels over 104 MHz for the N2D+ 3-2 line at 231.321 GHz, providing a channel spacing of 0.066 km s-1. This high resolution mode decreases the available bandwidth for continuum observations, resulting in 1 GHz of continuum bandwidth in the upper sideband and 1.15 GHz in the lower. The effective continuum frequency is 226 GHz (1.3 mm). The observations of Oph A N6 were interleaved with the quasar J1626-298 for complex gain calibration. Uranus and 3c454.3 were used for bandpass calibration, and Uranus was also used for flux calibration. The data were calibrated and edited in the SMA’s MIR software package. Observations in the N2H+ 3-2 line toward Oph A-N6 were undertaken with the SMA on 2007 March 30. The array was in its compact configuration and zenith opacities at 225 GHz were typically $\sim 0.06$. The SMA correlator was configured with 2048 channels over 104 MHz for the N2H+ 3-2 line at 279.512 GHz, providing a channel spacing of 0.055 km s-1. This high resolution mode decreases the available bandwidth for continuum observations, resulting in 1.3 GHz of continuum bandwidth in both the upper and lower sidebands. The effective continuum frequency is 276 GHz (1.1 mm). The observations of Oph A-N6 were interleaved with the quasars J1626-298 and J1517-243 for complex gain calibration. The quasar 3c279 was used for bandpass calibration, and Titan and 3c279 were used for flux calibration. The data were calibrated and edited in the SMA’s MIR software package. Further observations in the N2H+ 3-2 line toward Oph A-N6 were undertaken with the SMA on 2007 May 2, with the array in its subcompact configuration. Zenith opacities at 225 GHz were typically 0.06-0.08. The correlator setup, resolution, available continuum bandwidth, and complex gain calibrators were the same as for the 2007 March 30 observations. Neptune and 3c273 were used for bandpass calibration, and Neptune was also used for flux calibration. The data were calibrated and edited in the SMA’s MIR software package. Based upon independent observations of the gain and passband quasars at similar frequencies ($\pm 10$ GHz) and times (within one month), and upon independent calibration of the observations presented here, as part of the SMA’s ongoing monitoring of quasar fluxes111see http://sma1.sma.hawaii.edu/callist/callist.html, we estimate the flux calibration to be good to 20% for all our observations. ### 2.2 JCMT Observations in the N2D+ 3-2 line toward Oph A-N6 were undertaken with the James Clerk Maxwell Telescope (JCMT) between February and July 2005. All observations were made on a $5\times 5$ grid with 10″ spacing, with angular resolution of 22″, and a spectral resolution of 0.1 km s-1. Details of the observations have been presented in Pon et al. (2009), and the reader is referred to that paper for further information. ### 2.3 IRAM 30-m Observations of Oph A-N6 with the IRAM 30-m telescope were undertaken in May 2007 as part of a larger mapping program of the Oph A ridge. Although four SIS heterodyne receivers were used simultaneously in the 3, 2, and 1.2 mm atmospheric windows, here we only focus on the observations of N2H+ 3-2 at 279.511 GHz that are used in this paper. The autocorrelation spectrometer VESPA was used as backend with a channel spacings of 40 kHz and bandwidth of 80 MHz. The system temperature ranged from 650 to 3620 K and the pointing was checked every 1–2 hours on bright quasars and found to be good to 2–3″ (rms). The telescope focus was optimized on Saturn and Jupiter every 3–4 hours. At the frequency of N2H+ 3-2 the telescope beam size (full-width at half-power) is 9″. The observations were performed in position-switching mode with the OFF position offset by ($\Delta\alpha$,$\Delta\delta$) = (-900″,0″) from the nominal map center of 16h26m26$\fs$46, -24°24′30.8″, located at VLA 1623. No emission was found at the OFF position down to an rms noise level of 0.42 K in $T_{a}^{}$ scale. Mapping was done in on-the-fly scanning mode with a step of 4′′, providing fully sampled maps. We scanned alternately in right ascension and declination to avoid striping artefacts. The data were reduced using the CLASS software in its Fortran 90 version222see http://www.iram.fr/IRAMFR/GILDAS ### 2.4 Combined Interferometric and Single-Dish Observations The procedure used to combine the interferometric and single dish data sets is similar to that described by Zhang et al. (2000) and Takakuwa et al. (2007), and is based on the methods described in Vogel et al. (1984) and Wilner & Welch (1994). The MIRIAD software package (Sault et al. 1995) was used for the combination and subsequent imaging. #### 2.4.1 N2D+ 3-2: SMA + JCMT For the N2D+ 3-2 observations, the data sets were resampled along the velocity axis to a channel spacing of 0.07 km s-1. The JCMT data were converted to Jy using a conversion factor of $S(Jy)=27.4\times T_{A}^{}(K)$, and deconvolved with a 22″ FWHM Gaussian used to represent the JCMT beam at 231 GHz. Next, the JCMT data were convolved by a 55″ Gaussian representing the SMA primary beam (full width at half power). Side lobe effects are not well known and are thus ignored. Then the JCMT image cube was fourier transformed into a visibility data set, with a sampling density in the (u,v) plane chosen to closely match that of the SMA in their overlap region. Finally, the JCMT and SMA visibility data sets were fourier transformed together back into the image plane. Because of the extended nature of the emission, a correction for the SMA primary beam attenuation away from the phase center was applied. With a robust weighting of 0 applied during the transform, the resultant image cube has a resolution of $5\farcs 1\times 3\farcs 4$ (synthesised beam full-width at half power) with a 1$\sigma$ rms sensitivity of 0.43 Jy beam-1 channel-1. #### 2.4.2 N2H+ 3-2: SMA + 30-m The procedure for combining the SMA and 30-m N2H+ 3-2 data sets is the same as that used for the SMA and JCMT data sets. The data sets were first resampled along the velocity axis to a channel spacing of 0.07 km s-1, for direct comparison with the N2D+ 3-2 data. A conversion factor of $S(Jy)=9.3\times T_{A}^{}(K)$ was used for the 30-m data, and the assumed Gaussian beams sizes used were 8$\farcs$8 and 45″ for the 30-m and SMA respectively at 279.5 GHz. The 30-m and SMA visibility data sets were fourier transformed together using a robust weighting of 0 with a Gaussian taper of 2″. Because of the extended nature of the emission, a correction for the SMA primary beam attenuation away from the phase center was applied. The resultant image cube has a resolution of $5\farcs 6\times 3\farcs 7$, similar to that of the N2D+ 3-2 cube, with a 1$\sigma$ rms sensitivity of 0.72 Jy beam-1 channel-1. A comparison of the central spectra of the combined SMA + 30-m dataset with the 30-m only dataset, after smoothing the SMA + 30-m dataset to the angular resolution of the 30-m dataset, shows that essentially all of the single-dish flux is recovered, and the line-shapes are very similar. ### 2.5 Continuum imaging The MIRIAD software package was used to fourier transform and produce images from the interferometric-only continuum data. At 226 GHz (1.3 mm) a robust weighting of 2 was used with a 3″ taper, to improve the sensitivity, resulting in a resolution of $5\farcs 2\times 4\farcs 3$ with a 1$\sigma$ rms sensitivity of 2.6 mJy beam-1. At 276 GHz (1.1 mm) a robust weighting of 0 was used with a 3″ taper (note that with both compact and subcompact array data, a robust weighting of 2 would excessively down-weight the longer baselines), resulting in a resolution of $4\farcs 6\times 3\farcs 5$ with a 1$\sigma$ rms sensitivity of 3.6 mJy beam-1. ## 3 Results ### 3.1 Molecular line maps Figure 1 shows the integrated line maps of N2D+ 3-2 and N2H+ 3-2, compared to the distribution of integrated N2H+ 1-0 emission within the entire Oph A ridge (DAM04). The integration is performed over the hyperfine structure, corresponding to velocity ranges of 3.23–4.56 km s-1 (N2D+) and 0.22–6.18 km s-1 (N2H+). The general agreement between the N2D+ and N2H+ emission is good, but their peaks are offset by 9″ (Figure 2). Similarly, the N2H+ 1-0 map has its peak offset from the N2D+ 3-2 map, and shows good positional coincidence with N2H+ 3-2. The offsets between N2H+ and N2D+ could be due to optical depth effects (see §4.1), or chemical differentiation (Pon et al. 2009). Figure 2 also shows that the 3-2 maps have slightly different position angles. The N2D+ 3-2 map is closely aligned with the N2H+ 1-0 map, although their peaks do not coincide. The N2H+ 3-2 spectrum is composed of three groups of hyperfine (hf) features. Integrated maps of these hyperfine groups are shown in Figure 3, labelled according to their velocity offsets relative to the line frequency as “low-V”, “main-V”, and “high-V”. The low-V and high-V groups are sometimes referred to as the satellite hyperfine groups. Figure 3(a) shows the spectrum at the position of peak integrated emission in the main-V map, with the positions and relative intensities of the hyperfine components in the optically thin case indicated. The main-V group shows considerable saturation for the components in the range 3.5–4.0 km s-1, indicating large optical depths (confirmed through fitting of the hyperfine structure, see §4.1). As a result, the integrated map of the main-V group is larger than that of the other hyperfine groups and the region of peak emission is more extended. The peaks of the satellite hyperfine groups, low-V and high-V, are not coincident, as might be expected as they both account for the same relative line strength. Instead, the high-V group peaks at the same position as the main-V group, while the low-V group peaks at the position of peak N2H+ 1-0 emission (DAM04). The reason for this is not clear; perhaps it suggests that non-LTE excitation anomalies, as seen in the 1-0 line (Caselli, Myers & Thaddeus 1995; Daniel, Cernicharo, & Dubernet 2006) are present in the 3-2 line. Modelling of the current observations with a non-LTE line code that does not assume the hyperfines are in statistical equilibrium (Keto & Caselli 2010) could potentially address this question, but is beyond the scope of this paper. The satellite hyperfines are particularly strong in N6, both in absolute intensity, and in their relative intensity compared to the main-V group. Their relative strength compared to the main-V group is mostly due to the high total optical depth, as noted. However, their absolute intensity is about an order- of-magnitude brighter than has been seen toward any other low-mass starless core, for example by comparison to L1544, using just the 30-m data for each core (Caselli et al. 2002a,b; Daniel et al. 2007). The map sizes as traced by the molecular line emission have been estimated through two dimensional Gaussian fitting to the integrated maps, and through approximate measurements by eye using the contour level tracing 50% of the peak emission in the same maps. The results are similar for both methods for each line, suggesting the integrated emission can be approximated by a Gaussian. For N2H+ 3-2, we only used the maps of the low-V and high-V emission to estimate the size, so including N2D+ we used three maps in all. All maps and methods give consistent results, with the half-maximum diameter measured to be $\sim 3100\times 1600$ AU, with uncertainties of a few hundred AU for each axis. The geometric mean diameter is $\sim 2200$ AU, which is smaller than the geometric mean diameter of $\sim 3400$ AU determined by DAM04 through N2H+ 1-0 observations, likely due to the finer resolution of the observations presented here (5″ cf. 10″). The ratio of major-to-minor axes is about 2:1. The core is well-resolved, as each axis is significantly greater than the beam diameter ($640\times 425$ AU for N2D+, $700\times 460$ AU for N2H+). ### 3.2 Dust continuum emission Weak dust continuum emission is detected at both 1.3 mm and 1.1 mm (Fig. 4). Unlike the emission in the single-dish map (Fig. 4a; Motte et al. 1998), N6 is a local peak of emission at these frequencies. The interferometer has effectively filtered out the larger scale bright emission to reveal the weak emission associated with N6. More extended emission is missing from the 1.3 mm map due to the array configurations used, and this may be enough to cause the emission in this map to look smaller than the 1.1 mm emission. The dust emission is similar in size and orientation at both wavelengths, with a small offset (3$\farcs$5) between their peaks. However, this offset is at the 1$\sigma$ flux level and so is probably not significant. The orientation of the dust emission is very similar to that of the molecular line emission (Figure 5), and to that of the large scale dust emission. The mass of the region traced by the dust emission can be determined in the standard manner (Hildebrand 1983) using the flux density of the region. For these calculations we use the peak dust temperature of 20 K (Pon et al. 2009), a gas-to-dust ratio of 100, and assume the dust opacity is given by the commonly used “OH5” opacities that are believed to best represent the dust properties within cold dense regions (Ossenkopf & Henning 1994; Evans et al. 2001). Using the flux density above the 2 sigma level leads to a total mass estimate of 0.011 ${\rm M}_{\sun}$ at both 1.3 mm and 1.1 mm. Using the 3 sigma level as the cutoff, the mass estimates are 0.006 ${\rm M}_{\sun}$ at 1.3 mm and 0.005 ${\rm M}_{\sun}$ at 1.1 mm. Thus the values at 1.3 mm and 1.1 mm are in agreement. At the 3 sigma intensity level the core size is of order 1000 AU at both wavelengths, much smaller than the size of the line emitting regions. This small size may be due to large scale structure being resolved out by the interferometer. ### 3.3 Kinematic Structure Figures 6 and 7 show the velocity and linewidth maps of N2D+ and N2H+ 3-2. The line velocity was determined through fits of the hyperfine structure of each line using the hfs method in CLASS333http://www.iram.fr/IRAMFR/GILDAS, which performs a simultaneous fitting of all hyperfine components using CERN’s “Function Minimization and Error Analysis” package MINUIT444http://wwwasdoc.web.cern.ch/wwwasdoc/minuit/minmain.html. MINUIT has been shown to produce accurate values for line velocities and widths, even in the case of severe line overlap (Pon et al. 2009). Typical uncertainties reported by CLASS for the fits reported here are $\lesssim$ 0.01 km s-1 in velocity and $\lesssim$ 0.02 km s-1 in line-width, for N2H+, and $\lesssim$ 0.01 km s-1 in velocity and $\lesssim$ 0.03 km s-1 in line-width, for N2D+. There is very little variation in line center velocity across the N2D+ 3-2 map (Fig. 6a), with a possible hint of a gradient across the long axis in the southern part of the core. The variation in line center velocity is mostly $<0.1$ km s-1, larger than the typical uncertainty from hyperfine fitting, and of the same order as the channel separation. The two integrated line peaks have velocities that are separated by about a line width ($\sim 0.25$ km s-1). The results are similar for N2H+ 3-2, in that there is very little variation in line center velocity across the map, with the largest variation occurring along the western edge. DAM04 found that the linewidth of N2H+ 1-0 over N6 is generally $\leq$ 0.3 km s-1 with a mean value of 0.25 km s-1, with typical uncertainties of 0.005-0.01 km s-1. Our observations with higher angular resolution confirm this result, and suggest that the linewidth varies by less than the channel width of the observations over most of the map where significant emission is present (Fig. 7). There is a suggestion of large line widths on the western edge, which may be due to the nearby dust continuum source SM2, or simply due to lower S/N in the edges of the map. There appears to be an increase in N2H+ line width of order $\sim 0.03-0.04$ km s-1 near the N2D+ SE peak, which is also near the position of the continuum source (Fig. 7). This increase is seen in the N2D+ data (Figure 8), as the line width of the SE peak of N2D+ ($0.279\pm 0.018$ km s-1) is significantly larger than that of the NW peak ($0.217\pm 0.015$ km s-1) As in the case of N2H+ 1-0 (DAM04), the observed line width across N6 in the 3-2 lines of N2H+ and N2D+ is $\sim 0.25\pm 0.02$ km s-1. Observations of the (1,1) and (2,2) lines of NH3 indicate a gas temperature of $20\pm 2$ K (Pon et al. 2009). Using this gas temperature, the thermal velocity dispersion $\sigma_{\rm T}$ is 0.26 km s-1, implying that the non-thermal velocity dispersion $\sigma_{\rm NT}$ is $\sim 0.08$ km s-1. Thus the non-thermal motions within N6 are highly subsonic, with $\sigma_{\rm NT}/\sigma_{\rm T}\sim 0.3$. ## 4 Analysis ### 4.1 N2H+ and N2D+ Column Density Line optical depths ($\tau$) and excitation temperatures ($T_{\rm ex}$) were determined from fits to the hyperfine components of each transition, using the fitting routines in CLASS (DAM04). CLASS provides an estimate of the optical depth, and the product [$J_{\nu}(T_{\rm ex})-J_{\nu}(T_{\rm bg})$]$\tau$, where $J_{\nu}(T_{\rm ex})$ and $J_{\nu}(T_{\rm bg})$ are the equivalent Rayleigh-Jeans excitation and background temperatures. The method of Caselli et al. (2002b; their appendix, in particular equation (A4)) was used to determine the column density of N2H+ and N2D+ by integrating over the hyperfine features. This method assumes that the line emission is optically thin. Equation (A4) from Caselli et al. (2002b) is repeated here, as it is important for the following discussion, $N_{\rm tot}=\frac{8\pi W}{\lambda^{3}A}\frac{g_{l}}{g_{u}}\frac{1}{J_{\nu}(T_{\rm ex})-J_{\nu}(T_{\rm bg})}\frac{1}{1-\exp(-h\nu/kT_{\rm ex})}\frac{Q_{\rm rot}}{g_{l}\exp(-E_{l}/kT_{\rm ex})}\,\,,$ (1) where $N_{\rm tot}$ is the total column density, $W$ is the integrated line emission, $\lambda$ and $\nu$ are the wavelength and frequency of the observations, $A$ is the Einstein coefficient, $g_{l}$ and $g_{u}$ are the statistical weights of the lower and upper levels, and $Q_{\rm rot}$ is the partition function. Fits to the hyperfine structure of N2D+ 3-2 show that this line is optically thin at most positions (but with CLASS usually reporting values $>$ 0.1), with the largest total opacities measured near to the map center at $\tau\sim 2-3$. Positions where the total optical depth was $\geq$ 1 were used to estimate a single excitation temperature for the whole map, for which we find $\mbox{$T_{\rm ex}$}=10.0\pm 3.3$ K, so we assume a constant $T_{\rm ex}$ of 10 K for N2D+. For N2D+, the total integrated line emission and $T_{\rm ex}$ were used to calculate the column density $N$ at each map position. The results are shown in Figure 9(a). The column density of N2D+ ranges from $9.8\times 10^{11}$ cm-2 to $4.7\times 10^{12}$ cm-2, with values greater than $2\times 10^{12}$ cm-2 over much of the map. The N2H+ 3-2 emission was found to be very optically thick over much of N6, making it difficult to estimate $\tau$, and thus determine $T_{\rm ex}$. In addition, the saturated lines means that the observed integrated line emission is only a lower limit of the true emission, and equation (1) is not valid for optically thick emission. To overcome this problem, a multistep approach was used to obtain an estimate of $T_{\rm ex}$and measure the integrated line emission, so that the column density could be determined. Most of the optical depth of N2H+ 3-2 is due to the main-V hyperfine group. The low-V and high-V groups only account for 0.0742 of the total line strength (normalized to 1.0; Daniel et al. 2006; Pagani, Daniel, & Dubernet 2009). The total integrated line emission was thus found by integrating only over the low-V ($0.22-1.58$ km s-1) and high-V ($5.07-6.18$ km s-1) hf groups, and scaling by the inverse of their relative line strength. In the outer part of the N2H+ map, the total optical depth drops to reasonable values ($<15$), allowing the total line emission to be measured, and compared to the value obtained using only the low-V and high-V hf groups scaled by 1/0.0742. At these positions, the results were found to be in general agreement (better than 20%). Although the total optical depth is high, the individual hyperfine features are optically thin (39 hyperfine features in total, 17 in the low-V and high-V hyperfine groups), and the total optical depth of the low-V and high-V hf groups together are also thin in these data, or at most $\tau\sim 2$ with and uncertainty of similar size. While it was possible to obtain good fits to essentially every map position using only the low-V and high-V hf groups, in most cases this resulted in an optically thin fit ($\tau$ = 0.1 in CLASS), so that $T_{\rm ex}$ is unconstrained. In order to obtain an estimate of $T_{\rm ex}$, full hf fitting is needed. Using only those positions away from the map center where the full hf fit gives $\tau$ $<$ 20, we obtain $\mbox{$T_{\rm ex}$}=10.0\pm 2.2$ K. A full hf fit to a spectrum generated from the inner $8\times 10$ positions, gives a similar result. As we were unable to obtain a reliable estimate for each individual map position, we assume that $\mbox{$T_{\rm ex}$}=10\pm 2$ K across the whole map. This value is significantly lower than the value of 17 K determined by DAM04 for N2H+ 1-0, and the value of the kinetic temperature of 20 K. This difference could suggest that while the 1-0 line is thermalized, the 3-2 line is not. Alternatively, the denser interior of the core, better traced by the 3-2 line, could be colder. However, $T_{\rm ex}$ is fairly constant over the region mapped in N2H+ 3-2, and the temperature derived from dust observations is closer to 20 K, so this alternative is the less likely of the two possibilities. To determine the N2H+ 3-2 column density, we assumed that the total column density $N_{\rm tot}=N_{\rm hf}/0.0742$, where $N_{\rm hf}$ is the column density of the outer hyperfines, calculated using the integrated line emission of the low-V and high-V hyperfine groups, and assuming a constant $T_{\rm ex}$ of 10 K. As shown in equation (1), the column density $N$ (whether $N_{\rm tot}$ or $N_{\rm hf}$) is simply a function of $T_{\rm ex}$, $f(\mbox{$T_{\rm ex}$})$, times the integrated line intensity, $W$, so that $N=W\times f(\mbox{$T_{\rm ex}$})$. The column density of N2H+ determined in this manner ranges from $3.5\times 10^{12}$ cm-2 to $4.6\times 10^{13}$ cm-2, with most values being greater than $10^{13}$ cm-2, and with a significant fraction of the inner map region having values $>2.5\times 10^{13}$ cm-2 (Figure 9(b)). The typical uncertainty in a particular measurement of the column density is $N^{+100\%}_{-50\%}$. Similar values for the N2H+ column density were found by DAM04. We have checked our results, using the outer hyperfine satellite groups and assuming optically thin emission, against N2H+ column densities determined from hyperfine fits to the full hyperfine spectra (Caselli et al. 2002b; Di Francesco et al. 2004; Friesen et al. 2010a). We find that the results are consistent, in that the values from the full fit are within the uncertainties of the method we have used. However, $N$ determined from the full fit case are typically, but not systematically, higher (but are sometimes lower) by up to 50%. Because the total optical depth is so high its actual value is not well constrained by the full fit at any particular position, so we prefer the method we have used for estimating $N$. ### 4.2 Deuterium Fraction The ratio of N2H+ and N2D+ column densities can be used to estimate the deuteration fraction within N6. This is shown in Figure 10, where the ratio $N$(N2D+)/$N$(N2H+) is shown, compared to the integrated intensity maps of each molecule. From this Figure it can be seen that the D/H ratio is of order 0.05 over a large fraction of the map, reaching higher values toward the western side, of order 0.15. These values are larger than those determined by Pon et al. (2009), from lower resolution observations. Figure 10 also shows that the NW N2D+ peak has a higher D/H ratio than the SE peak, as might be expected from Pon et al. (2009), where only the NW peak is clearly detected in the JCMT data. This result shows that Oph A-N6 has a high central degree of deuteration, and is similar to values found for isolated low-mass starless cores (Crapsi et al. 2005). In some map locations the D/H value is close to the dividing line of 0.1 used to characterize the isolated cores as prestellar or starless, with the idea that prestellar cores are those closest to star formation (Crapsi et al. 2005). Of the prestellar cores identified by Crapsi et al. (2005), all but one, like OphA-N6, have $N$(N2H+) $>10^{13}$ cm-2. It is notable that even though the kinetic temperatures are near to 20 K, where the D/H ratio should decrease dramatically (Caselli et al. 2008), and significantly higher than in isolated cores, the D/H ratio is as high as in most starless cores, if not higher. ### 4.3 Structure & Mass N6 is elongated and may represent a fragment of a filament. The simplest model of a filament is a self-gravitating isothermal cylinder, whose radial density profile is (Ostriker 1964; Johnstone et al. 2003), $n(r)=\frac{n_{0}}{\left[1+\left(\frac{r^{2}}{8H^{2}}\right)\right]^{2}}\,,$ (2) where $n_{0}$ is the peak number density, $r$ is the radial offset, and the scale length $H$ is $H^{2}\equiv\frac{c^{2}}{4\pi G\rho_{0}}\,,$ (3) where $c$ is the sound speed, $\rho_{0}$ the peak density, and $G$ is the gravitational constant. If N6 is viewed perpendicular to its axis, then the column density along the line-of-sight is $\displaystyle N(r)$ $\displaystyle=$ $\displaystyle\frac{\pi}{2}\frac{n_{0}H}{\left[1+\left(\frac{r^{2}}{8H^{2}}\right)^{2}\right]^{3/2}}$ (4) $\displaystyle=$ $\displaystyle N_{0}\frac{\pi}{4R}\frac{H}{\left[1+\left(\frac{r^{2}}{8H^{2}}\right)^{2}\right]^{3/2}}$ (5) where $N_{0}$ is the peak column density and $R$ is the radius. Figure 12 shows the radial column density profile across the minor axis of N6 derived from N2H+ 3-2 compared to the profile of an isothermal cylinder (dark continuous curve). This profile was constructed from N2H+ 3-2 data imaged with a 2$\farcs$4 beam and 1$\farcs$2 pixels (Nyquist sampling), using the method of “super-resolution” (Briggs 1994; Chandler et al. 2005), in order to better sample the radial profile. Eighteen independent, consecutive profiles were extracted across the major axis at 1$\farcs$2 intervals along the major axis. The region over which the profiles were extracted is shown in Figure 11. Each profile was normalized to its peak values, and the normalized profiles averaged together to form the composite profile shown in Figure 12. This figure shows that the column density profile of N6 is very well represented by an isothermal cylinder, as the model matches the data within its 1$\sigma$ uncertainties at eight consecutive positions across the peal of the profile. The model allows the peak density and hence abundance of N2H+ to be estimated, keeping other parameters fixed at their previously determined values; radius $R$ = 800 AU, temperature of 20 K (Pon et al. 2009), and peak N2H+ column density of 4.6 $\times 10^{13}$ cm-2. Using these values, we find a good match to the data, as shown in Figure 12, assuming a constant N2H+ abundance $X_{N_{2}H^{+}}=2.7\pm 0.2\times 10^{-10}$, resulting in values of peak density $n_{0}=7.1^{+0.6}_{-0.5}\times 10^{6}$ cm-3, and scale length $H=362^{+12}_{-14}$ AU. Allowing for a 5 pc uncertainty in the distance does not change these values. Even though N6 is not a local dust emission peak, DAM04 estimated the column density, $N$(H2), to be $3\times 10^{23}$ cm-2 using the dust continuum emission, assuming isothermal dust at a temperature of 20 K. From this and their value for $N$(N2H+) they infer an abundance $X$(N2H+) of $3\times 10^{-10}$, in very close agreement with the value used here that provides an excellent match between the isothermal cylinder model and the data. This abundance is in good agreement with values inferred for isolated low-mass cores, including the evolved prestellar cores discussed above (Benson, Caselli & Myers 1998; Caselli et al. 2002c; Crapsi et al. 2005). The mass per unit length of an isothermal cylinder is $M(r)=2\pi\rho_{0}\int_{0}^{R}rdr\left[1+\left(\frac{r^{2}}{8H^{2}}\right)\right]^{-2}$ (6) After integrating, the mass of a cylinder of length $L$ can be written: $M=L\,\frac{2c^{2}}{G}\left[1+\left(\frac{2c^{2}}{\pi G\rho_{0}R^{2}}\right)\right]^{-1}\,.$ (7) For $T=20$ K, $n_{0}=7.1\times 10^{6}$ cm-2, $R$ = 800 AU, and $L$ = 3100 AU, the mass is $M=0.18\pm 0.02$ ${\rm M}_{\sun}$, where the uncertainty is due to the uncertainties in $n_{0}$ given above and the distance uncertainty. We can determine the total mass traced by N2H+, using the N2H+ column density map (Fig. 9(b)), with the result for the N2H+ abundance. The map gives the column density per pixel, from which the mass per pixel ($M_{p}$) can be determined, and hence the total mass, using $M_{p}=X\,\mu_{m}\,A_{p}\,N_{X}\,\,,$ (8) where $\mu_{m}$ is the mean particle mass (2.37 amu; Stahler & Palla 2005; Kauffmann et al. 2008), $A_{p}$ is the area per pixel, $X$ is the abundance of the molecule used, and $N_{X}$ is its column density. In a Nyquist sampled map, the total mass is then just the sum over all pixels. For $T_{\rm ex}$ = 10 K and $X_{N_{2}H^{+}}$ of $2.7\times 10^{-10}$, we measure $M=0.29^{+0.05}_{-0.04}$ ${\rm M}_{\sun}$ for positions within the half-power level of the column density map. The uncertainties come from the uncertainties in $X$ and the distance. The change in mass by assuming $T_{\rm ex}$ = 9 or 11 K is much smaller than either of these. The critical mass is the mass of a condensation whose radius is equal to the shortest wavelength of a periodic perturbation that will grow. Larson (1985) has studied the critical mass for fragmentation of a number of geometries, and for an isothermal filament (i.e., a cylinder) finds (Larson 1985, equation 21) $\displaystyle M_{c}$ $\displaystyle=$ $\displaystyle\frac{7.88c^{4}}{G^{2}\mu_{m}N}$ (9) $\displaystyle=$ $\displaystyle 1.1\left(\frac{T}{20\,{\rm K}}\right)^{2}\left(\frac{10^{23}\,\,\mbox{cm${}^{-2}$}}{N}\right)[\mbox{${\rm M}_{\sun}$}].$ (10) With $T=20$ K and $N=1.7\times 10^{23}\mbox{cm${}^{-2}$}$ (from the peak N2H+ column density and $X$), $M_{c}=0.63^{+0.05}_{-0.04}$ ${\rm M}_{\sun}$. This value is within about a factor of 2 of the mass computed for N6 of 0.29 ${\rm M}_{\sun}$ from eqn. (8). Given the uncertainties in computing masses, such as determining the “size” of a core, and our method of measuring $N$ at each position, this result suggests that N6 is consistent with having formed from the fragmentation of an isothermal filament, in this case Oph A. ## 5 Discussion ### 5.1 Kinematics Internally, Oph A-N6 is rather quiescent. It shows very narrow N2H+ and N2D+ line-widths of about 0.25 km s-1 across its extent, barely more than the thermal line width for the measured gas temperature of 20 K, of 0.18 km s-1. Its non-thermal motions are very sub-sonic, but the surrounding gas shows significantly larger line-widths (DAM04; André et al. 2007; Pon et al. 2009) suggesting that N6 has lost any turbulent motions it may have had. The lack of significant variation in line centroid velocity and line-width over the core indicate that N6 is an example of a coherent core, as has been seen in more isolated cores (Barranco & Goodman 1998; Goodman et al. 1998; Caselli et al. 2002a; Tafalla et al. 2004; Pineda et al. 2010). This result suggests that small non-thermal motions typical of isolated cores are found in some starless cores within turbulent molecular clouds. Observations of HCO+ and DCO+ 3-2 show the expected signature of inward motions (Evans 1999; Pon et al. 2009), but the complex hyperfine structure of N2H+ and N2D+ 3-2 makes identifying any similar signature in these lines impossible. In addition, the very narrow line widths of N2H+ 1-0 together with the spectral resolution and signal-to-noise of the data make it difficult to identify any signature of inward motions (DAM04), regardless of the hyperfine structure. Data with finer spectral resolution and improved signal-to-noise are required to search for inward motions in N2H+. However, the very narrow line-widths already suggest that any inward motions on the size scales probed by N2H+ ($\sim 300$ AU) must be small. The ratio of non-thermal-to-thermal line-width in N6 is about 0.3, which is lower than observed in most starless dense cores in Perseus (Walsh et al. 2007; H. Kirk et al. 2007; Rosolowsky et al. 2008), for dense cores elsewhere in Ophiuchus (André et al. 2007; Friesen et al. 2009), or for most isolated low-mass dense cores (Myers 1983). Further, the absence of line-broadening toward the center of N6 suggests a lack of a central source. The motions observed in HCO+ may be infall onto the core, rather than core collapse (Pon et al. 2009). ### 5.2 Dust Emission Starless cores are usually defined through observations of the dust continuum or molecular lines in single-dish observations at millimeter wavelengths, with angular resolution 10-20″ (typically the line observations are of lower resolution than the continuum). As a result, by definition they generally only show a single peak of emission, and fairly simple structures, being round or elongated with small aspect ratios (less than 2). When observed with an interferometer, which acts as a spatial filter, many such cores are not detected, or still only appear as single peaks of emission, due to their smooth large-scale structure and lack of significant sub-structure (Williams & Myers 1999; Williams et al. 1999, 2006; Harvey et al. 2003a; Olmi et al. 2005; Schnee et al. 2010). Combining the single-dish and interferometer line data, as we have done here, allows the small scale structure to be studied, without concerns about missing flux. These studies usually show that starless cores do not break up into sub-cores on small scales. One exception is L183, which is composed of 3 sub-cores in N2H+ 1-0 (J. Kirk et al. 2009), but it shows a very elongated structure in single dish maps, so perhaps this is not too surprising. The nature of the compact dust emission detected toward the peak of integrated N2H+ emission is unclear, given that N6 is not a local maximum in single-dish continuum observations between 1300 and 450 µm, with 10-15″ resolution (Motte et al. 1998; Wilson et al. 1999; Johnstone et al. 2000). However, a dust temperature map derived from the ratio of 450-to-850 µm flux, assuming a constant dust emissivity, shows a similar structure to the N2H+ maps, although with lower resolution (Pon et al. 2009). The dust temperature map shows a peak of 20 K at the N2H+ peak, and is elongated in the NW-SE direction. It is not yet known if the gas temperature varies in a similar manner on similar scales, as the NH3 observations only have a resolution of about 30″. However, NH3 may not probe the highest densities toward the center of N6, and so determining the gas temperature there with confidence will be difficult. Supporting evidence for a relatively constant gas temperature within N6 comes from the comparison of its column density profile with that of an isothermal cylinder (Fig. 12), and from the almost constant line-widths. N6 is embedded within the Oph A ridge, and the large column of dust due to the ridge may make it difficult to distinguish a compact core within it as a separate entity. The dust temperature map suggests that it is a local temperature maximum, at about 20 K (Pon et al. 2009). This result is unlike those in detailed studies of isolated starless cores, which show flat temperature profiles in low-resolution observations (Jijina et al. 1999; Tafalla et al. 2004), but a drop in temperature toward the core center in observations with finer resolution (Crapsi et al. 2007). The mass of dust seen in N6 is very low, only of order 0.005-0.01 ${\rm M}_{\sun}$, and the inferred peak column density of $\sim 1.3\times 10^{22}$ cm-2 is an order of magnitude below that found from N2H+ observations, and from single-dish continuum observations. Recently, interferometers have detected compact millimeter dust emission toward three “starless” cores (Chen et al. 2010; Enoch et al. 2010; Pineda et al. 2011; Dunham et al. 2011). Supporting evidence, in the form of CO outflows or faint, compact 70 µm emission, and SED modeling, suggests that in all cases the emission is due to an internal heating source of very low temperature ($>$100 K), and the inferred luminosities are very low ($<$0.1 $L_{\sun}$). These cores are all candidates to be the long theorized first hydrostatic core (FHSC; Larson 1969; Boss & Yorke 1995; Omukai 2007; Tomida et al. 2010), although none are in complete agreement with theoretical predictions. They have outflows that are too fast (L1448-IRS2E – Chen et al. 2010), too collimated ((L1448-IRS2E – Chen et al. 2010; Per-Bolo 58 – Dunham et al. 2011), or are detected at too short a wavelength (Per-Bolo 58 – Enoch et al. 2011), to be consistent with current models. For one source, L1451-mm, the observations are in better agreement with theoretical models, but a model of a protostar plus a disk provides an equally good fit to its SED and continuum interferometric visibilities (Pineda et al. 2011). Theoretically, a FHSC is expected to be of low mass, with a maximum value of order 0.05 ${\rm M}_{\sun}$, essentially undetectable at $<100$ µm, and with an observed SED resembling a blackbody at 30 K (Omukai 2007; Saigo & Tomisaka 2011). A low- velocity, compact outflow may also be present. Due to crowding, it is difficult to determine if 70 µm Spitzer emission is present toward N6, and as stated it does not show a local maximum in 450-1300 µm emission in single-dish observations. Observations to search for the presence of an outflow in CO 2-1 are compromised by the bright outflow from VLA 1623 (André et al. 1990), that passes close to N6 and is detected in the SMA observations presented here. The compact 1 mm dust emission detected with the SMA has the right mass to be considered a candidate FHSC, but further evidence is needed, particularly detections at other wavelengths. At present there is insufficient information to suggest whether N6 is a better FHSC candidate than the other candidates. ### 5.3 Comparison to other Starless Cores Studies of starless cores with resolution similar to the one presented here are rare, particularly in molecular lines. This is a greater problem for cores in clusters, which are typically smaller than more isolated cores, such as those in Taurus (Ward-Thompson et al. 2007, and references therein). Studies with 10-15″ resolution have been made for the cluster-forming regions in Ophiuchus (125 pc; Motte et al. 1998; Johnstone et al. 2000; DAM04; Simpson et al. 2008; Friesen et al. 2009), Perseus ($\sim 235$ pc; Hatchell et al. 2005; Walsh et al. 2007) and Serpens ($>$300 pc; Testi & Sargent 1998; Williams & Myers 1999), and for isolated cores in Taurus (140 pc; Ward-Thompson et al. 1994; Caselli et al. 2002a,b; Tafalla et al. 2002; J. Kirk et al. 2005). A comparison of the properties of N6 with the cores in these studies shows that N6 is denser than most starless cores, by about an order-of-magnitude ($10^{7}$ cm-3 cf. $10^{6}$ cm-3). These observations typically do not have the resolution of our observations of N6, and derived average densities of $10^{6}$ cm-3 over the central 1000 AU could be consistent with peak densities of $\sim 10^{7}$ cm-3 (Keto & Caselli 2010). N6 is smaller than most cores in all three cluster-forming regions listed above, (Walsh et al. 2007; Friesen et al. 2009), but this could be partly an effect of resolution, as this study has finer resolution by at least a factor of two over previous work. The small size could also be partly due to the molecular transition used, as we have used higher J transitions that preferentially trace higher column density material. Walsh et al. (2007) list a few cores with sizes comparable to N6, but these are at the limit of their resolution. N6 is significantly smaller than all isolated cores that have been well resolved, whether studied in line-emission, dust continuum, or extinction (Ward-Thompson et al. 1999; Bacmann et al. 2000; Crapsi et al. 2005; Kandori et al. 2005; Kauffmann et al. 2008). As noted earlier, the N2H+ linewidths in N6 are narrower than almost all other cluster cores (André et al. 2007; Walsh et al. 2007; H. Kirk et al. 2007; Friesen et al. 2010a), and are almost totally due to thermal motions (as discussed in detail in DAM04). The linewidth barely varies across N6, and it is an excellent example of a coherent core, a core where the non-thermal motions are subsonic, and constant, so that it appears to be cut-off from the surrounding turbulent gas (Mouschovias 1991; Myers 1998; Barranco & Goodman 1998; Goodman et al. 1998; Caselli et al. 2002c; Pineda et al. 2010). The N2H+ column density in N6 is larger than most cores elsewhere in Ophiuchus (Friesen et al. 2010a) and Perseus (H. Kirk et al. 2007), with a peak value of $\sim 5\times 10^{13}$ cm-2 compared to values of $\sim 10^{13}$ cm-2. This peak value is about three times greater than the peak value observed in a sample of 28 isolated starless cores, and about eight times greater than the sample mean ($\sim 8\times 10^{12}$ cm-2; Crapsi et al. 2005; see also Daniel et al. 2007). Similarly, the N2D+ column density of N6, $\sim 5\times 10^{12}$ cm-2, is greater than that of cores in Oph B, where the peak value is $\sim 7\times 10^{11}$ cm-2, and greater than the mean value of 25 isolated starless cores, of $<10^{12}$ cm-2 (Friesen et al. 2010b; Crapsi et al. 2005; see also Daniel et al. 2007). However, the peak value of N(N2D+) for isolated starless cores, observed toward L1544, L429 and L694-2, is similar to N6 (Crapsi et al. 2005). The deuterium fraction, ranging from a mean near 0.05 to a maximum value of about 0.15, is similar to that seen in 28 isolated cores (range 0.03 – 0.44), where 22 of the cores have D/H $<$ 0.1 (Crapsi et al. 2005). The range of values in N6 is also similar to that of the cluster-core Oph B2, which has a peak of 0.16 but with most of the core showing values around 0.03 (Friesen et al. 2010b). The mean temperature of the isolated cores is about 10 K, while the mean in Oph B2 is higher at around 13-14 K. The deuterium fraction is expected to be significantly higher in the cold ($<$20 K) dense interiors of starless cores than the cosmic D/H ratio. This is due to two main factors. First, the main pathway for the formation of H2D+, the parent molecule of deuterated molecules, is exothermic by 230 K, and the backward reaction is not available. Second, CO will freeze out at temperatures below 20 K, removing the main destroyer of H2D+. However, N2 should also freeze out at essentially the same rate at CO at $<$20 K, so the picture is not so simple. A number of more subtle factors that affect the D/H ratio, such as grain size (surface chemistry), ionization rate, ortho-to-para H2 value, and the CO depletion factor are examined in detail by Caselli et al. (2008). They show that the D/H ratio can still be relatively high near 20 K, but drops sharply at $<$15 K. One obvious explanation for the relatively high values of D/H in N6 may be temperatures a few degrees lower than 20 K, but the complete reason for the high D/H values is likely to be more complicated. The D/H ratio is largest away from the dust temperature peak, to the NW, and Pon et al. (2009) do infer a radial temperature drop. Thus, N6 appears to be denser and smaller than starless cores in both cluster- forming and isolated environments. While a very small number of cluster-cores have a similar size, no isolated core does, and no starless core has a mean density as high. ### 5.4 Structure and Evolution Starless cores have typically been modeled with spherically symmetric geometries, with a radial density profile that is almost constant at small radii but decrease as a power law at larger radii (Ward-Thompson et al. 1994, 1999; Bacmann et al. 2000; Evans et al. 2001; Kandori et al. 2005; J. Kirk et al. 2005). However, N6 is clearly elongated, with an aspect ratio of at least 2:1, as observed in many cores (Myers et al. 1991; Ryden 1996), suggesting that it is prolate or filamentary in nature. Over its half-maximum size its dust temperature is fairly uniform at about 20 K, to within 1-2 K, decreasing at larger distances (Pon et al. 2009). We have thus compared the column density profile of N6 to that of an isothermal cylinder (Ostriker 1964; Curry 2000), finding an extremely good match between the data and the model (Fig. 12). The mass of the isothermal model cylinder is $\sim 0.2$ ${\rm M}_{\sun}$, similar to the observationally derived mass of 0.3 ${\rm M}_{\sun}$, given the uncertainties in the input values (cylinder size, column density per pixel, temperature). Similarly, the critical mass for fragmentation of an isothermal filament with properties similar to those of N6 is $\sim 0.6$ ${\rm M}_{\sun}$ (Larson 1985), also close to the observational value given the uncertainties (in particular as the observed value of mass depends on column density, whereas the critical cylinder mass depends on its inverse). The excellent match between the column density profile of N6 and the isothermal cylinder model, and the similarity of the observed mass to the mass of an isothermal filament with the properties of N6, strongly suggest that N6 has formed via fragmentation of the Oph A filament, and is in a critical state at the beginning of star formation, or has already started the star-formation process (as evidenced by the low-mass compact dust continuum emission). N6 is aligned with its parent core, Oph A, supporting the view that it has formed as the result of fragmentation of Oph A along its axis. Although non-spherically symmetric models are rarely used, they have been quite successful in explaining the observed elongated structure of dense cores and filamentary nature of molecular clouds (Harvey et al. 2003b; Johnstone et al. 2003). It has also been shown that the density profiles of collapsing centrally condensed (“Bonner-Ebert”) spheres and cylinders are remarkably similar, and distinguishing between these cases observationally, using only column density maps, may not be possible (Myers 2005). Resorting to the ease of using spherical models without considering alternatives should be avoided, a message that is often given without heed (Hartmann 2004). Harvey et al. (2003b) performed a detailed study of the isolated starless core L694-2, through near-infrared extinction mapping. They found that spherical models where the radial density profile is described by a power-law, or a Bonnor-Ebert sphere, did not provide accurate matches to the data. Instead they found that a cylindrical model, like that described here, provided an excellent fit to their data. In the case of L694-2, the cylinder is inclined to the line-of-sight, so initially it was not obvious that a cylindrical model was needed. The steep power-law index that best matched the data, significantly steeper than predicted by inside-out collapse, motivated the cylindrical model. Many distant infrared dark clouds and nearby star-forming complexes have filamentary or elongated structures (Schneider & Elmegreen 1979; Myers 2009), and in at least one case, that of the infrared dark cloud G11.11-0.12, the radial profile of 850 µm emission closely matches that of an isothermal filament of radius $\approx$ 0.1 pc over a length of more than 10 pc (Johnstone et al. 2003). While many molecular clouds have been compared to models of infinite or finite sheets, models of individual clouds as cylinders are almost absent, although theoretical studies suggest that such models could provide good matches to available data (Curry 2000). New results from the Herschel Space Observatory show filamentary structures in nearby molecular clouds (André et al. 2010; Arzoumanian et al. 2011), and modelling suggests radial profiles much shallower than isothermal cylinders, at radii $>$ 0.1 pc. At such large radii the assumptions of constant temperature and/or hydrostatic equilibrium are unlikely, so this result is not surprising. The observations do not have the resolution to probe scales similar to that of N6. The profiles could be consistent with collapsing polytropic cylinders (Arzoumanian et al. 2011), or magnetized filaments in virial equilibrium (Fiege & Pudritz 2000). It will be interesting to observe cores within these filaments with finer resolution to study their structure. The physical, kinematic, and chemical properties of dense cores have been used to assess their evolutionary state (Ward-Thompson et al. 1999; Crapsi et al. 2005; Di Francesco et al. 2007). Crapsi et al. (2005) searched for evolutionary indicators in a sample of 31 isolated cores, using observations of N2H+, N2D+, CO and the dust continuum. They proposed eight chemical and kinematic evolutionary indicators, and identified as “evolved” those cores that met at least four of the conditions. These conditions include large column densities of N2H+ and N2D+, a large deuterium fraction, large CO depletion and central density, broad linewidths, infall asymmetry in the line profiles, and compact central regions. All of these conditions, with the exception of CO depletion, can be tested in N6 with our data. The peak column densities of N2H+ ($4.6\times 10^{13}$ cm-2) and N2D+ ($4.7\times 10^{12}$ cm-2) are greater than the dividing values given by Crapsi et al. (2005), of $8.5\times 10^{12}$ cm-2 and $1.0\times 10^{12}$ cm-2, respectively. Similarly, the peak of the deuterium fraction, 0.15, is greater than the value of 0.1 used by Crapsi et al. to separate candidate prestellar cores from starless cores. The peak density we determine, $7.0\times 10^{6}$ cm-3, is far greater than the separator of $5.1\times 10^{5}$ cm-3 used by Crapsi et al., as is the degree of central concentration (this is not a function of resolution, as the single dish data have been included in our results). The narrow line widths, and hyperfine structure, makes the search for line skewness difficult. As mentioned, Pon et al. (2009) observe infall asymmetry in HCO+, so there is some evidence of contraction in N6. However, the N2H+ and N2D+ lines widths are of similar size to the separating value of 0.25 km s-1 used by Crapsi et al. (2005). So of the seven evolutionary indicators we can measure, five clearly indicate that N6 is an evolved prestellar core, according to the analysis of Crapsi et al. (2005), while the other two indicators are borderline (line widths) or unclear (infall asymmetry or line skewness). These results strongly suggest that N6 is at an advanced stage of prestellar evolution. The compact dust continuum emission further supports this idea. ## 6 Summary & Conclusions We have presented high spatial ($\sim$500 AU) and spectral (0.07 km s-1) resolution observations in N2H+ 3-2, N2D+ 3-2, and the dust continuum at 1-mm, of the starless core Oph A-N6, embedded within the Oph A molecular ridge in the Ophuichus cluster-forming molecular cloud. These are the highest resolution observations of a starless dense core presented to date. Such a small condensation as N6 would be hard to detect with coarser angular resolution observations, especially in clouds much more distant than Ophiuchus. The major results of this study are summarized below: 1. 1. The observations reveal a compact dust continuum peak, of size $\sim$1000 AU and mass 0.005-0.01 ${\rm M}_{\sun}$, not seen in single dish observations, except in a dust temperature map (Pon et al. 2009). The small size and mass suggests it might be the first indication of collapse, either representing a temperature increase, or possibly a first hydrostatic core. 2. 2. The size of N6 from the line observations is larger, with a projected half- power diameter of $3100\times 1600$ AU, and thus an aspect ratio of 2:1. The N2H+ and N2D+ integrated line maps show slightly different position angles, probably representing chemical variations, or the very high optical depth in N2H+. 3. 3. Very little variation is seen in either linewidth or line center velocity in either line across their maps. The variations are so small that N6 appears to be a coherent core, with very small non-thermal motions. 4. 4. The peak column densities are $4.6\times 10^{13}$ cm-2 for N2H+, and $4.7\times 10^{12}$ cm-2 for N2D+. The positions of peak column density are offset, with the N2D+ peak located to the NW of the N2H+ peak. 5. 5. The deuterium fraction has a peak value of 0.15, and is greater than or about equal to 0.05 over much of the mapped area. The maximum value of deuteration lies in the NW, and not at the position of the dust continuum peak, nor the N2H+ peak. 6. 6. The column density profile of N6 across its minor axis, as determined from the N2H+ observations, is very well represented by an isothermal cylinder (at 20 K), of peak density $7.1\times 10^{6}$ cm-3, and N2H+ abundance $2.7\times 10^{-10}$. 7. 7. The mass of N6, determined from the mapped positions, lies in the range 0.25-0.34 ${\rm M}_{\sun}$, depending strongly on the assumed N2H+ abundance ($2.7\pm 0.2\times 10^{-10}$). The low value compares favourably to the mass determined from the cylindrical analysis, of $\sim 0.2$ ${\rm M}_{\sun}$, while the high value compares well to the critical mass for fragmentation of an isothermal filament with similar properties, of $\sim 0.6$ ${\rm M}_{\sun}$. 8. 8. Compared to isolated low-mass cores, Oph A-N6 shows similar narrow line widths and small velocity variation, with a deuterium fraction that is similar to “evolved” dense cores. It is significantly smaller than isolated cores, with larger peak column density and volume density, while the previously measured kinetic temperature is significantly higher than isolated starless cores. These results strongly suggest Oph A-N6 has formed from the fragmentation of the Oph A filament, and is a precursor to a low-mass star in a cluster-forming region. The results also suggest Oph A-N6 has completed almost all of its prestellar evolution, and may even have begun to form a star. This research is supported in part by the National Science Foundation under grant number 0708158 (T.L.B.). We thank Andy Pon and Rachel Friesen for sharing results in advance of publication, and Rachel Friesen and Chris de Vries for checking column density estimates for optically thick N2H+. We thank Mark Gurwell for his diligent maintenance of the “Submillimeter Calibrator List”. The Submillimeter Array is a joint project between the Smithsonian Astrophysical Observatory and the Academia Sinica Institute of Astronomy and Astrophysics and is funded by the Smithsonian Institution and the Academia Sinica. The James Clerk Maxwell Telescope is operated by The Joint Astronomy Centre on behalf of the Science and Technology Facilities Council of the United Kingdom, the Netherlands Organisation for Scientific Research, and the National Research Council of Canada. Based on observations carried out with the IRAM 30 m telescope. IRAM is supported by INSU/CNRS (France), MPG (Germany) and IGN (Spain). This research has made use of NASA’s Astrophysics Data System Bibliographic Services ## References Allen et al. (2007) Allen, L., Megeath, S. T., Gutermuth, R., et al. 2007, Protostars and Planets V, ed. B. Reipurth, D. Jewitt, & K. Keil, (Tuscon: University of Arizona Press), 361 * Andre et al. (1990) Andre, P., Martin-Pintado, J., Despois, D., & Montmerle, T. 1990, A&A, 236, 180 * André et al. (2007) André, P., Belloche, A., Motte, F., & Peretto, N. 2007, A&A, 472, 519 * Arzoumanian et al. (2011) Arzoumanian, D., André, P., Didelon, P., et al. 2011, A&A, 529, L6 * Bacmann et al. (2000) Bacmann, A., André, P., Puget, J.-L., et al., 2000, A&A, 361, 555 * Barranco & Goodman (1998) Barranco, J. A., & Goodman, A. A. 1998, ApJ, 504, 207 * Benson et al. (1998) Benson, P. J., Caselli, P., & Myers, P. C. 1998, ApJ, 506, 743 * Bonnell et al. (2001) Bonnell, I. A., Bate, M. R., Clarke, C. J., & Pringle, J. E. 2001, MNRAS, 323, 785 * Bonnell & Bate (2006) Bonnell, I. A., & Bate, M. R. 2006, MNRAS, 370, 488 * Boss & Yorke (1995) Boss, A. P., & Yorke, H. W. 1995, ApJ, 439, L55 * Bressert et al. (2010) Bressert, E., Bastian, N., Gutermuth, R., et al. 2010, MNRAS, 409, L54 * Briggs (1994) Briggs, D. S. 1994, in The Restoration of HST Images and Spectra - II, ed. R.J. Hanisch, & R.L. White (Baltimore: Space Telescope Science Institute), 250 * Caselli et al. (1995) Caselli, P., Myers, P. C., & Thaddeus, P. 1995, ApJ, 455, L77 * Caselli et al. (2002) Caselli, P., Walmsley, C. M., Zucconi, A., et al., 2002a, ApJ, 565, 331 * Caselli et al. (2002) Caselli, P., Walmsley, C. M., Zucconi, A., et al., 2002b, ApJ, 565, 344 * Caselli et al. (2002) Caselli, P., Benson, P. J., Myers, P. C., & Tafalla, M. 2002c, ApJ, 572, 238 * Caselli et al. (2008) Caselli, P., Vastel, C., Ceccarelli, C., et al., 2008, A&A, 492, 703 * Chandler et al. (2005) Chandler, C. J., Brogan, C. L., Shirley, Y. L., & Loinard, L. 2005, ApJ, 632, 371 * Chen et al. (2010) Chen, X., Arce, H. G., Zhang, Q., et al., 2010, ApJ, 715, 1344 * Chini et al. (1997) Chini, R., Reipurth, B., Ward-Thompson, D., et al., 1997, ApJ, 474, L135 * Chini et al. (2003) Chini, R., Kämpgen, K., Reipurth, B., et al. 2003, A&A, 409, 235 * Crapsi et al. (2005) Crapsi, A., Caselli, P., Walmsley, C. M., et al., 2005, ApJ, 619, 379 * Crapsi et al. (2007) Crapsi, A., Caselli, P., Walmsley, M. C., & Tafalla, M. 2007, A&A, 470, 221 * Curry (2000) Curry, C. L. 2000, ApJ, 541, 831 * Daniel et al. (2006) Daniel, F., Cernicharo, J., & Dubernet, M.-L. 2006, ApJ, 648, 461 * Daniel et al. (2007) Daniel, F., Cernicharo, J., Roueff, E., Gerin, M., & Dubernet, M. L. 2007, ApJ, 667, 980 * Davis et al. (1999) Davis, C. J., Matthews, H. E., Ray, T. P., Dent, W. R. F., & Richer, J. S. 1999, MNRAS, 309, 141 * de Geus et al. (1989) de Geus, E. J., de Zeeuw, P. T., & Lub, J. 1989, A&A, 216, 44 * Di Francesco et al. (2004) di Francesco, J., André, P., & Myers, P. C. 2004, ApJ, 617, 425 [DAM04] * di Francesco et al. (2007) di Francesco, J., Evans, N. J., II, Caselli, P., et al., 2007, in Protostars and Planets V, ed. B. Reipurth, D. Jewitt, & K. Keil, (Tuscon: University of Arizona Press), 17 * Dunham et al. (2011) Dunham, M. M., Chen, X., Arce, H. G., et al. 2011, arXiv:1108.1342 * Enoch et al. (2006) Enoch, M. L., Young, K. E., Glenn, J., et al. 2006, ApJ, 638, 293 * Enoch et al. (2007) Enoch, M. L., Glenn, J., Evans, N. J., II, et al., 2007, ApJ, 666, 982 * Enoch et al. (2008) Enoch, M. L., Evans, N. J., II, Sargent, A. I., et al., 2008, ApJ, 684, 1240 * Enoch et al. (2010) Enoch, M. L., Lee, J.-E., Harvey, P., Dunham, M. M., & Schnee, S. 2010, ApJ, 722, L33 * Evans (1999) Evans, N. J., II 1999, ARA&A, 37, 311 * Evans et al. (2001) Evans, N. J., II, Rawlings, J. M. C., Shirley, Y. L., & Mundy, L. G. 2001, ApJ, 557, 193 * Fiege & Pudritz (2000) Fiege, J. D., & Pudritz, R. E. 2000, MNRAS, 311, 85 * Friesen et al. (2009) Friesen, R. K., Di Francesco, J., Shirley, Y. L., & Myers, P. C. 2009, ApJ, 697, 1457 * Friesen et al. (2010) Friesen, R. K., Di Francesco, J., Shimajiri, Y., & Takakuwa, S. 2010a, ApJ, 708, 1002 * Friesen et al. (2010) Friesen, R. K., Di Francesco, J., Myers, P. C., et al., 2010b, ApJ, 718, 666 * Goodman et al. (1998) Goodman, A. A., Barranco, J. A., Wilner, D. J., & Heyer, M. H. 1998, ApJ, 504, 223 * Gutermuth et al. (2009) Gutermuth, R. A., Megeath, S. T., Myers, P. C., et al., 2009, ApJS, 184, 18 * Hartmann (2004) Hartmann, L. 2004, in IAU Symp. 221, Star Formation at High Angular Resolution, ed. M. Burton, R. Jayawardhana, & T. Bourke, (San Francisco, CA: ASP), 201 * Harvey et al. (2003) Harvey, D. W. A., Wilner, D. J., Myers, P. C., & Tafalla, M. 2003a, ApJ, 597, 424 * Harvey et al. (2003) Harvey, D. W. A., Wilner, D. J., Lada, C. J., Myers, P. C., & Alves, J. F. 2003b, ApJ, 598, 1112 * Hatchell et al. (2005) Hatchell, J., Richer, J. S., Fuller, G. A., et al., 2005, A&A, 440, 151 * Hildebrand (1983) Hildebrand, R. H. 1983, QJRAS, 24, 267 * Ikeda et al. (2007) Ikeda, N., Sunada, K., & Kitamura, Y. 2007, ApJ, 665, 1194 * Ikeda et al. (2009) Ikeda, N., Kitamura, Y., & Sunada, K. 2009, ApJ, 691, 1560 * Jijina et al. (1999) Jijina, J., Myers, P. C., & Adams, F. C. 1999, ApJS, 125, 161 * Johnstone & Bally (1999) Johnstone, D., & Bally, J. 1999, ApJ, 510, L49 * Johnstone et al. (2000) Johnstone, D., Wilson, C. D., Moriarty-Schieven, G., et al., 2000, ApJ, 545, 327 * Johnstone et al. (2003) Johnstone, D., Fiege, J. D., Redman, R. O., Feldman, P. A., & Carey, S. J. 2003, ApJ, 588, L37 * Johnstone et al. (2004) Johnstone, D., Di Francesco, J., & Kirk, H. 2004, ApJ, 611, L45 * Jørgensen et al. (2008) Jørgensen, J. K., Johnstone, D., Kirk, H., et al., ApJ, 683, 822 * Kandori et al. (2005) Kandori, R., Nakajima, Y., Tamura, M., et al. 2005, AJ, 130, 2166 * Kauffmann et al. (2008) Kauffmann, J., Bertoldi, F., Bourke, T. L., Evans, N. J., II, & Lee, C. W. 2008, A&A, 487, 993 * Keto & Caselli (2010) Keto, E., & Caselli, P. 2010, MNRAS, 402, 1625 * Kirk et al. (2006) Kirk, H., Johnstone, D., & Di Francesco, J. 2006, ApJ, 646, 1009 * Kirk et al. (2007) Kirk, H., Johnstone, D., & Tafalla, M. 2007, ApJ, 668, 1042 * Kirk et al. (2005) Kirk, J. M., Ward-Thompson, D., & André, P. 2005, MNRAS, 360, 1506 * Kirk et al. (2009) Kirk, J. M., Crutcher, R. M., & Ward-Thompson, D. 2009, ApJ, 701, 1044 * Knude & Hog (1998) Knude, J., & Hog, E. 1998, A&A, 338, 897 * Lada & Lada (2003) Lada, C. J., & Lada, E. A. 2003, ARA&A, 41, 57 * Larson (1969) Larson, R. B. 1969, MNRAS, 145, 271 * Larson (1985) Larson, R. B. 1985, MNRAS, 214, 379 * Larson (2003) Larson, R. B. 2003, Reports on Progress in Physics, 66, 1651 * Larson (2005) Larson, R. B. 2005, MNRAS, 359, 211 * Li et al. (2007) Li, D., Velusamy, T., Goldsmith, P. F., & Langer, W. D. 2007, ApJ, 655, 351 * Lis et al. (1998) Lis, D. C., Serabyn, E., Keene, J., et al., 1998, ApJ, 509, 299 * Loinard et al. (2008) Loinard, L., Torres, R. M., Mioduszewski, A. J., & Rodríguez, L. F. 2008, ApJ, 675, L29 * Lombardi et al. (2008) Lombardi, M., Lada, C. J., & Alves, J. 2008, A&A, 489, 143 * Maruta et al. (2010) Maruta, H., Nakamura, F., Nishi, R., Ikeda, N., & Kitamura, Y. 2010, ApJ, 714, 680 * Motte et al. (1998) Motte, F., Andre, P., & Neri, R. 1998, A&A, 336, 150 * Mouschovias (1991) Mouschovias, T. C. 1991, ApJ, 373, 169 * Myers (1983) Myers, P. C. 1983, ApJ, 270, 105 * Myers et al. (1991) Myers, P. C., Fuller, G. A., Goodman, A. A., & Benson, P. J. 1991, ApJ, 376, 561 * Myers (1998) Myers, P. C. 1998, ApJ, 496, L109 * Myers (2005) Myers, P. C. 2005, ApJ, 623, 280 * Myers (2009) Myers, P. C. 2009, ApJ, 700, 1609 * Nutter et al. (2005) Nutter, D. J., Ward-Thompson, D., & André, P. 2005, MNRAS, 357, 975 * Olmi et al. (2005) Olmi, L., Testi, L., & Sargent, A. I. 2005, A&A, 431, 253 * Omukai (2007) Omukai, K. 2007, PASJ, 59, 589 * Ossenkopf & Henning (1994) Ossenkopf, V., & Henning, T. 1994, A&A, 291, 943 * Ostriker (1964) Ostriker, J. 1964, ApJ, 140, 1056 * Padgett et al. (2008) Padgett, D. L., et al. 2008, ApJ, 672, 1013 * Pagani et al. (2009) Pagani, L., Daniel, F., & Dubernet, M.-L. 2009, A&A, 494, 719 * Pineda et al. (2010) Pineda, J. E., Goodman, A. A., Arce, H. G., et al., 2010, ApJ, 712, L116 * Pineda et al. (2011) Pineda, J. E., Arce, H. G., Schnee, S., et al. 2011, arXiv:1109.1207 * Pon et al. (2009) Pon, A., Plume, R., Friesen, R. K., et al., 2009, ApJ, 698, 1914 * Rosolowsky et al. (2008) Rosolowsky, E. W., Pineda, J. E., Foster, J. B., et al., 2008, ApJS, 175, 509 * Ryden (1996) Ryden, B. S. 1996, ApJ, 471, 822 * Saigo & Tomisaka (2011) Saigo, K., & Tomisaka, K. 2011, ApJ, 728, 78 * Sandell & Knee (2001) Sandell, G., & Knee, L. B. G. 2001, ApJ, 546, L49 * Schnee et al. (2010) Schnee, S., Enoch, M., Johnstone, D., et al., 2010, ApJ, 718, 306 * Schneider & Elmegreen (1979) Schneider, S., & Elmegreen, B. G. 1979, ApJS, 41, 87 * Shu et al. (1987) Shu, F. H., Adams, F. C., & Lizano, S. 1987, ARA&A, 25, 23 * Shu et al. (2004) Shu, F. H., Li, Z.-Y., & Allen, A. 2004, in ASP Conf. Ser. 323, Star Formation in the Interstellar Medium: In Honor of David Hollenbach, Chris McKee and Frank Shu, ed. D. Johnstone, F.C. Adams, D.N.C. Lin, D.A. Neufeld, & E.C. Ostriker (San Francisco, CA: ASP), 37 * Simpson et al. (2008) Simpson, R. J., Nutter, D., & Ward-Thompson, D. 2008, MNRAS, 391, 205 * Stahler & Palla (2005) Stahler, S. W., & Palla, F. 2005, The Formation of Stars, (Wiley-VCH) * Stanke et al. (2006) Stanke, T., Smith, M. D., Gredel, R., & Khanzadyan, T. 2006, A&A, 447, 609 * Tafalla et al. (2002) Tafalla, M., Myers, P. C., Caselli, P., Walmsley, C. M., & Comito, C. 2002, ApJ, 569, 815 * Tafalla et al. (2004) Tafalla, M., Myers, P. C., Caselli, P., & Walmsley, C. M. 2004, A&A, 416, 191 * Takakuwa et al. (2007) Takakuwa, S., et al. 2007, ApJ, 662, 431 * Tan et al. (2006) Tan, J. C., Krumholz, M. R., & McKee, C. F. 2006, ApJ, 641, L121 * Tatematsu et al. (2008) Tatematsu, K., Kandori, R., Umemoto, T., & Sekimoto, Y. 2008, PASJ, 60, 407 * Testi & Sargent (1998) Testi, L., & Sargent, A. I. 1998, ApJ, 508, L91 * Tomida et al. (2010) Tomida, K., Machida, M. N., Saigo, K., Tomisaka, K., & Matsumoto, T. 2010, ApJ, 725, L239 * van Kempen et al. (2009) van Kempen, T. A., van Dishoeck, E. F., Salter, D. M., et al., 2009, A&A, 498, 167 * Vogel et al. (1984) Vogel, S. N., Wright, M. C. H., Plambeck, R. L., & Welch, W. J. 1984, ApJ, 283, 655 * Walsh et al. (2007) Walsh, A. J., Myers, P. C., Di Francesco, J., et al., 2007, ApJ, 655, 958 * Ward-Thompson et al. (1994) Ward-Thompson, D., Scott, P. F., Hills, R. E., & Andre, P. 1994, MNRAS, 268, 276 * Ward-Thompson et al. (1999) Ward-Thompson, D., Motte, F., & Andre, P. 1999, MNRAS, 305, 143 * Ward-Thompson et al. (2007) Ward-Thompson, D., André, P., Crutcher, R., et al., 2007, in Protostars and Planets V, ed. B. Reipurth, D. Jewitt, & K. Keil, (Tuscon: University of Arizona Press), 33 * Wilking et al. (2008) Wilking, B. A., Gagné, M., & Allen, L. E. 2008, in Handbook of Star Forming Regions, Volume II, ed. B. Reipurth (San Francisco, CA: ASP), 351 * Williams & Myers (1999) Williams, J. P., & Myers, P. C. 1999, ApJ, 518, L37 * Williams et al. (1999) Williams, J. P., Myers, P. C., Wilner, D. J., & di Francesco, J. 1999, ApJ, 513, L61 * Williams & Myers (2000) Williams, J. P., & Myers, P. C. 2000, ApJ, 537, 891 * Williams et al. (2006) Williams, J. P., Lee, C. W., & Myers, P. C. 2006, ApJ, 636, 952 * Wilner & Welch (1994) Wilner, D. J., & Welch, W. J. 1994, ApJ, 427, 898 * Wilson et al. (1999) Wilson, C. D., Avery, L. W., Fich, M., et al. 1999, ApJ, 513, L139 * Young et al. (2006) Young, K. E., Enoch, M. L., Evans, N. J., II, et al. 2006, ApJ, 644, 326 * Zhang et al. (2000) Zhang, Q., Ho, P. T. P., & Wright, M. C. H. 2000, AJ, 119, 1345 Figure 1: Integrated combined single-dish + interferometer line maps of (a) N2H+ 1-0 (from DAM04), (b) N2D+ 3-2, and (c) N2H+ 3-2. The contour levels are (a) 6, 9, 12 … times the $1\sigma$ sensitivity of 0.6 Jy/beam km s-1 for N2H+ 1-0, (b) 9, 12, 15 … times the $1\sigma$ sensitivity of 0.12 Jy/beam km s-1 for N2D+ 3-2, and (c) 12, 15, 18 … times the $1\sigma$ sensitivity of 0.47 Jy/beam km s-1 for N2H+ 3-2. The white cross is the position of the peak integrated N2H+ 1-0 emission from DAM04. The large dashed circles in (b) and (c) indicate the size of the SMA primary beam (full-width at half-maximum sensitivity). The small grey ovals at lower right in each panel indicate the synthesised beam sizes (full-width at half-maximum sensitivity). Figure 2: Comparison of the integrated line maps of N2D+ 3-2 (black contours) and N2H+ 3-2 (grey contours). The black cross is the position of the peak integrated N2H+ 1-0 from DAM04. The synthesised beam size for each observation is shown in the corresponding colours at lower right, and the primary beam size for the N2D+ observations is shown as the dashed circle. Contour levels are 10, 14, 18 … times the $1\sigma$ sensitivity of 0.12 Jy/beam km s-1 for N2D+ 3-2, and 12, 16, 20 … times the $1\sigma$ sensitivity of 0.47 Jy/beam km s-1 for N2H+ 3-2. Figure 3: Spectrum of N2H+ 3-2, panel (a), and maps of integrated N2H+ 3-2 emission over the 3 hyperfine groups, panels (b)-(d). Data is combined 30-m + SMA (compact+subcompact). In (a), the location of the hyperfine components, and their relative weights, are indicated by the light grey lines. The limits of integration are indicated by “low-V”, “main-V”, and “high-V”, for panels (b)-(d). The rms noise in the spectrum is 0.5 Jy/beam. Contour levels are (b) 3, 4, 5, … times the $1\sigma$ sensitivity of 0.26 Jy/beam km s-1, (c) 15, 18, 21, … times the $1\sigma$ sensitivity of 0.30 Jy/beam km s-1, and (c) 3, 4, 5, … times the $1\sigma$ sensitivity of 0.21 Jy/beam. The white cross is the position of the peak integrated N2H+ 1-0 emission (DAM04), while the white square marks the position of the spectrum shown in (a). The large dashed circle indicates the size of the SMA primary beam. The small grey ovals at lower right in panels (b)-(d) indicate the synthesised beam size. Figure 4: Maps of (a) 1.3 mm emission (30-m), (b) 1.3 mm emission (SMA) and (c) 1.1 mm emission (SMA). Contours for (a) the 30-m observations are 10, 20, 30, … times the $1\sigma$ sensitivity of 10 mJy/beam (Motte et al. 1998; DAM04). Contours for the SMA observations are 2, 3, 4, … times the $1\sigma$ sensitivity of (b) 2.6 mJy/beam (1.3 mm) and (c) 3.6 mJy/beam (1.1 mm). Dotted contours indicate negative levels. The cross is the position of the peak integrated N2H+ 1-0 emission (DAM04). The large dashed circles indicates the size of the SMA primary beam. The small grey ovals at lower right each panel indicate the synthesised beam size. Figure 5: Comparison of continuum and integrated line maps. (a) Contours of 1.3 mm continuum emission (white; SMA) over contours of integrated N2D+ 3-2 emission (SMA+JCMT). The greyscale is integrated N2D+ 3-2 emission. (b) Contours of 1.1 mm emission (SMA) over contours of integrated N2H+ 3-2 emission (SMA+30-m). Contour levels for N2D+ are 10, 14, 18, … times the $1\sigma$ sensitivity of 0.12 Jy/beam km s-1, while contour levels for N2H+ are 12, 16, 20, … times $1\sigma$ sensitivity of 0.47 0.12 Jy/beam km s-1. Continuum contour levels are the same as shown in Figures 4(a)–(b). The white cross is the position of the peak integrated N2H+ 1-0 emission (DAM04). The primary beam size of the SMA for each observation is shown as the dashed circle. Figure 6: (a) Velocity map from hyperfine fits to the N2D+ 3-2 data. The data have been resampled onto a 2$\farcs$5 grid with a 5″ circular beam (shown at lower right). The positions where a fit was performed are indicated by crosses. Typical uncertainties in the fitted velocities are 0.01-0.02 km s-1. The contours represent the integrated intensity map of N2D+ and are the same as shown in Figure 1. (b) Velocity map from hyperfine fits to the N2H+ 3-2 data. Typical uncertainties in the fitted velocities are 0.005-0.02 km s-1. Other details are the same as in (a), except that the contours represent the integrated intensity map of N2H+ and are the same as shown in Figure 1. The origin of the maps is the position of the peak integrated N2H+ 1-0 emission (DAM04). The open circle indicates the beam size. [COLOUR FIGURE] Figure 7: (a) Linewidth map from hyperfine fits to the N2D+ 3-2 data. The data have been resampled onto a 2$\farcs$5 grid with a 5″ circular beam (shown at lower right). Typical uncertainties in the fitted linewidths are 0.02-0.04 km s-1. The positions where a fit was performed are indicated by crosses. The contours represent the integrated intensity map of N2D+ and are the same as shown in Figure 1. (b) Linewidth map from hyperfine fits to the N2D+ 3-2 data. Typical uncertainties in the fitted linewidths are 0.01-0.03 km s-1. Other details are the same as in (a), except that the contours represent the integrated intensity map of N2H+ and are the same as shown in Figure 1. The origin of the maps is the position of the peak integrated N2H+ 1-0 emission (DAM04). [COLOUR FIGURE] Figure 8: Spectra at the integrated line map peaks of N2D+ 3-2. (a) The integrated map as shown in Figure 1, with the locations of the two spectra indicated. The large dashed circle indicates the SMA primary beam size, while the small grey oval indicates the synthesised beam size. The white cross is the position of the peak integrated N2H+ 1-0 emission (DAM04). (b) The spectrum at position NW (histogram), with a model fit of the N2D+ 3-2 hyperfine structure (continuous line in grey). (c) The spectrum at position SE (histogram), with a model fit of the N2D+ 3-2 hyperfine structure (continuous line in grey). Figure 9: (a) Column density of N2D+, assuming a constant excitation temperature of 10 K. The data have been resampled onto a 2$\farcs$5 grid with a 5″ circular beam (shown at lower right). The positions where the column density was determined are indicated by crosses. The contours represent the integrated intensity map of N2D+ and are the same as shown in Figure 1. (b) Column density of N2H+, assuming a constant excitation temperature of 10 K. Other details are the same as in (a), except that the contours represent the integrated intensity map of N2H+ and are the same as shown in Figure 1. The origin of the maps is the position of the peak integrated N2H+ 1-0 emission (DAM04). [COLOUR FIGURE] Figure 10: (a) Ratio of N2D+ to N2H+ column densities, using the results presented in Figure 9. The contours are integrated intensity of N2D+ and are the same as shown in Figure 1. (b) Ratio of N2D+ to N2H+ column densities, using the results presented in Figure 9. The contours are integrated intensity of N2H+and are the same as shown in Figure 1. The resolution of the observations is indicated by the circular beam (shown at lower right). The origin of the maps is the position of the peak integrated N2H+ 1-0 emission (DAM04). [COLOUR FIGURE] Figure 11: Location of the N2H+ 3-2 radial column density cuts used to construct the radial profile shown in Figure 12. The box indicates the area from which the cuts were extracted, with the vector indicating the direction of the cuts. The image is the low resolution column density map, with contour levels of 1.5, 2.5, 3.6, and 4.6 $\times 10^{13}$ cm-2. Figure 12: Normalized mean N2H+ 3-2 radial column density cut parallel to the minor axis of N6 (histogram), constructed from a series of cuts across N6 that were averaged and then normalized to the peak value. The dark continuous line represents a model of an isothermal cylinder whose parameters are described in §4.3. The model shown here has the values of radius (800 AU), kinetic temperature (20 K) and peak N2H+ column density ($4.6\times 10^{13}$ cm-2) fixed prior to comparison with the data, an N2H+ abundance ($2.75\times 10^{-10}$), determined by matching the model to the data (by eye), with a resultant peak density of $7.0\times 10^{6}$ cm-3 and scale length of 365 AU. The shaded grey areas indicate where the data and model differ significantly and the core begins to merge into the background emission.	arxiv-papers	2011-11-18T16:55:58	2024-09-04T02:49:24.457193	{ "license": "Public Domain", "authors": "Tyler L. Bourke, Philip C. Myers, Paola Caselli, James Di Francesco,\n Arnaud Belloche, Ren\\'e Plume, David J. Wilner", "submitter": "Tyler Bourke", "url": "https://arxiv.org/abs/1111.4424" }
1111.4526	# Signal Propagation in Feedforward Neuronal Networks with Unreliable Synapses Daqing Guo1, Chunguang Li2 Email: dqguo07@gmail.comAuthor for correspondence, Email: cgli@zju.edu.cn (1 School of Electronic Engineering, University of Electronic Science and Technology of China, Chengdu 610054, People’s Republic of China 2 Department of Information Science and Electronic Engineering, Zhejiang University, Hangzhou 310027, People’s Republic of China ) ###### Abstract In this paper, we systematically investigate both the synfire propagation and firing rate propagation in feedforward neuronal network coupled in an all-to- all fashion. In contrast to most earlier work, where only reliable synaptic connections are considered, we mainly examine the effects of unreliable synapses on both types of neural activity propagation in this work. We first study networks composed of purely excitatory neurons. Our results show that both the successful transmission probability and excitatory synaptic strength largely influence the propagation of these two types of neural activities, and better tuning of these synaptic parameters makes the considered network support stable signal propagation. It is also found that noise has significant but different impacts on these two types of propagation. The additive Gaussian white noise has the tendency to reduce the precision of the synfire activity, whereas noise with appropriate intensity can enhance the performance of firing rate propagation. Further simulations indicate that the propagation dynamics of the considered neuronal network is not simply determined by the average amount of received neurotransmitter for each neuron in a time instant, but also largely influenced by the stochastic effect of neurotransmitter release. Second, we compare our results with those obtained in corresponding feedforward neuronal networks connected with reliable synapses but in a random coupling fashion. We confirm that some differences can be observed in these two different feedforward neuronal network models. Finally, we study the signal propagation in feedforward neuronal networks consisting of both excitatory and inhibitory neurons, and demonstrate that inhibition also plays an important role in signal propagation in the considered networks. Keywords: Feedforward neuronal network, unreliable synapse, signal propagation, synfire chain, firing rate ## 1 Introduction A major challenge in neuroscience is to understand how the neural activities are propagated through different brain regions, since many cognitive tasks are believed to involve this process (Vogels and Abbott, 2005). The feedforward neuronal network is the most used model in investigating this issue, because it is simple enough yet can explain propagation activities observed in experiments. In recent years, two different modes of neural activity propagation have been intensively studied. It has been found that both the synchronous spike packet (synfire), and the firing rate, can be transmitted across deeply layered networks (Abeles 1991; Aertsen et al. 1996; Diesmann et al. 1999; Diesmann et al. 2001; C$\hat{\text{a}}$teau and Fukai 2001; Gewaltig et al. 2001; Tetzlaff et al. 2002; Tetzlaff et al. 2003; van Rossum et al. 2002; Vogels and Abbott 2005; Wang et al. 2006; Aviel et al. 2003; Kumar et al. 2008; Kumar et al. 2010; Shinozaki et al. 2007; Shinozaki et al. 2010). Although these two propagation modes are quite different, the previous results demonstrated that a single network with different system parameters can support stable and robust signal propagation in both of the two modes, for example, they can be bridged by the background noise and synaptic strength (van Rossum et al. 2002; Masuda and Aihara 2002; Masuda and Aihara 2003). Neurons and synapses are fundamental components of the brain. By sensing outside signals, neurons continually fire discrete electrical signals known as action potentials or so-called spikes, and then transmit them to postsynaptic neurons through synapses (Dayan and Abbott 2001). The spike generating mechanism of cortical neurons is generally highly reliable. However, many studies have shown that the communication between neurons is, by contrast, more or less unreliable (Abeles 1991; Raastad et al. 1992; Smetters and Zador 1996). Theoretically, the synaptic unreliability can be explained by the phenomenon of probabilistic transmitter release (Branco and Staras 2009; Katz 1966; Katz 1969; Trommershäuser et al. 1999), i.e., synapses release neurotransmitter in a stochastic fashion, which has been confirmed by well- designed biological experiments (Allen and Stevens 1994). In most cases, the transmission failure rate at a given synapse tends to exceed the fraction of successful transmission (Rosenmund et al. 1993; Stevens and Wang 1995). In some special cases, the synaptic transmission failure rate can be as high as 0.9 or even higher (Allen and Stevens 1994). Further computational studies have revealed that the unreliability of synaptic transmission might be a part of information processing of the brain and possibly has functional roles in neural computation. For instance, it has been reported that the unreliable synapses provide a useful mechanism for reliable analog computation in space- rate coding (Maass and Natschl$\ddot{\text{a}}$ger 2000); and it has been found that suitable synaptic successful transmission probability can improve the information transmission efficiency of synapses (Goldman 2004) and can filter the redundancy information by removing autocorrelations in spike trains (Goldman et al. 2002). Furthermore, it has also been demonstrated that unreliable synapses largely influence both the emergence and dynamical behaviors of clusters in an all-to-all pulse-coupled neuronal network, and can make the whole network relax to clusters of identical size (Friedrich and Kinzel 2009). Although the signal propagation in multilayered feedforward neuronal networks has been extensively studied, to the best of our knowledge the effects of unreliable synapses on the propagation of neural activity have not been widely discussed and the relevant questions still remain unclear (but see the footnote111An anonymous reviewer kindly reminded us that there might be a relevant abstract (Trommershäuser and Diesmann 2001) discussing the effect of synaptic variability on the synchronization dynamics in feedforward cortical neural networks, but the abstract itself does not contain the results presumably presented on the poster and also the follow-up publications do not exist.). In this paper, we address these questions and provide insights by computational modeling. For this purpose, we examine both the synfire propagation and firing rate propagation in feedforward neuronal networks. We mainly investigate the signal propagation in feedforward neuronal networks composed of purely excitatory neurons connected with unreliable synapses in an all-to-all coupling fashion (abbr. URE feedforward neuronal network) in this work. We also compare our results with the corresponding feedforward neuronal networks (we will clarify the meaning of “corresponding” later) composed of purely excitatory neurons connected with reliable synapses in a random coupling fashion (abbr. RRE feedforward neuronal network). Moreover, we study feedforward neuronal networks consisting of both excitatory and inhibitory neurons connected with unreliable synapses in an all-to-all coupling fashion (abbr. UREI feedforward neuronal network). The rest of this paper is organized as follows. The network architecture, neuron model, and synapse model used in this paper are described in Sec. 2. Besides these, the measures to evaluate the performance of synfire propagation and firing rate propagation, as well as the numerical simulation method are also introduced in this section. The main results of the present work are presented in Sec. 3. Finally, a detailed conclusion and discussion of our work are given in Sec. 4. ## 2 Model and method ### 2.1 Network architecture In this subsection, we introduce the network topology used in this paper. Here we only describe how to construct the URE feedforward neuronal network. The methods about how to build the corresponding RRE feedforward neuronal network and the UREI feedforward neuronal network will be briefly given in Secs. 3.4 and 3.5, respectively. The architecture of the URE feedforward neuronal network is schematically shown in Figure 1. The network totally contains $L=10$ layers, and each layer is composed of $N_{s}=100$ excitatory neurons. Since neurons in the first layer are responsible for receiving and encoding the external input signal, we therefore call this layer sensory layer and neurons in this layer are called sensory neurons. In contrast, the function of neurons in the other layers is to propagate neural activities. Based on this reason, we call these layers transmission layers and the corresponding neurons cortical neurons. Because the considered neuronal network is purely feedforward, there is no feedback connection from neurons in downstream layers to neurons in upstream layers, and there is also no connection among neurons within the same layer. For simplicity, we call the $i$-th neuron in the $j$-th layer neuron $(i,j)$ in the following. Figure 1: Network architecture of the URE feedforward neuronal network. The network totally contains 10 layers. The first layer is the sensory layer and the others are the transmission layers. Each layer consists of 100 excitatory neurons. For clarity, only 6 neurons are shown in each layer. ### 2.2 Neuron model We now introduce the neuron model used in the present work. Each cortical neuron is modeled by using the integrate-and-fire (IF) model (Nordlie et al. 2009), which is a minimal spiking neuron model to mimic the action potential firing dynamics of biological neurons. The subthreshold dynamics of a single IF neuron obeys the following differential equation: $\begin{split}\tau_{m}\frac{dV_{ij}}{dt}=V_{\text{rest}}-V_{ij}+RI_{ij},\end{split}$ (2.1) with the total input current $\begin{split}I_{ij}=I_{ij}^{\text{syn}}+I_{ij}^{\text{noise}}.\end{split}$ (2.2) Here $i=1,2,\ldots,N_{s}$ and $j=2,3,\ldots,L$, $V_{ij}$ represents the membrane potential of neuron $(i,j)$, $\tau_{m}=20$ ms is the membrane time constant, $V_{\text{rest}}=-60$ mV is the resting membrane potential, $R=20$ M$\Omega$ denotes the membrane resistance, and $I_{ij}^{\text{syn}}$ is the total synaptic current. The noise current $I_{ij}^{\text{noise}}=\sqrt{2D_{t}}\xi_{ij}(t)$ represents the external or intrinsic fluctuations of the neuron, where $\xi_{ij}(t)$ is a Gaussian white noise with zero mean and unit variance, and $D_{t}$ is referred to as the noise intensity of the cortical neurons. In this work, a deterministic threshold-reset mechanism is implemented for spike generation. Whenever the membrane potential of a neuron reaches a fixed threshold at $V_{\text{th}}=-50$ mV, the neuron fires a spike, and then the membrane potential is reset according to the resting potential, where it remains clamped for a 5-ms refractory period. On the other hand, we use different models to simulate the sensory neurons depending on different tasks. To study the synfire propagation, we assume that each sensory neuron is a simple spike generator, and control their firing behaviors by ourselves. While studying the firing rate propagation, the sensory neuron is modeled by using the IF neuron model with the same expression (see Eq. (2.1)) and the same parameter settings as those for cortical neurons. For each sensory neuron, the total input current is given by $\begin{split}I_{i1}=I(t)+I_{i1}^{\text{noise}},\end{split}$ (2.3) where $i=1,2,...,N_{s}$ index neurons. The noise current $I_{i1}^{\text{noise}}$ has the same form as that for cortical neurons but with the noise intensity $D_{s}$. $I(t)$ is a time-varying external input current which is injected to all sensory neurons. For each run of the simulation, the external input current is constructed by the following process. Let $\eta(t)$ denote an Ornstein-Uhlenbeck process, which is described by $\begin{split}\tau_{c}\frac{d\eta(t)}{dt}=-\eta(t)+\sqrt{2A}\xi(t),\end{split}$ (2.4) where $\xi(t)$ is a Gaussian white noise with zero mean and unit variance, $\tau_{c}$ is a correlation time constant, and $A$ is a diffusion coefficient. The external input current $I(t)$ is defined as $\begin{split}I(t)=\begin{cases}\eta(t)&\text{if $\eta(t)\geq 0$},\\\ 0&\text{if $\eta(t)<0$}.\end{cases}\end{split}$ (2.5) Parameter $A$ can be used to denote the intensity of the external input signal $I(t)$. In this work, we choose $A=200$ $\text{nA}^{2}$ and $\tau_{c}=80$ ms. By its definition, the external input current $I(t)$ corresponds to a Gaussian-distributed white noise low-pass filtered at 80 ms and half-wave rectified. It should be noted that this type of external input current is widely used in the literature, in particular in the research papers which study the firing rate propagation (van Rossum et al. 2002; Vogels and Abbott 2005; Wang and Zhou 2009). ### 2.3 Synapse model The synaptic interactions between neurons are implemented by using the modified conductance-based model. Our modeling methodology is inspired by the phenomenon of probabilistic transmitter release of the real biological synapses. Here we only introduce the model of unreliable excitatory synapses, because the propagation of neural activity is mainly examined in URE feedforward neuronal networks in this work. The methods about how to model reliable excitatory synapses and unreliable inhibitory synapses will be briefly introduced in Secs. 3.4 and 3.5, respectively. The total synaptic current onto neuron $(i,j)$ is the linear sum of the currents from all incoming synapses, $\begin{split}I_{ij}^{\text{syn}}=\sum_{k=1}^{N_{s}}G(i,j;k,j-1)\cdot(E_{{\text{syn}}}-V_{ij}).\end{split}$ (2.6) In this equation, the outer sum runs over all synapses onto this particular neuron, $G(i,j;k,j-1)$ is the conductance from neuron $(k,j-1)$ to neuron $(i,j)$, and $E_{\text{syn}}=0$ mV is the reversal potential of the excitatory synapse. Whenever the neuron $(k,j-1)$ emits a spike, an increment is assigned to the corresponding synaptic conductances according to the synaptic reliability parameter, which process is given by $\begin{split}G(i,j;k,j-1)\leftarrow G(i,j;k,j-1)+J(i,j;k,j-1)\cdot h(i,j;k,j-1),\end{split}$ (2.7) where $h(i,j;k,j-1)$ denotes the synaptic reliability parameter of the synapse from neuron $(k,j-1)$ to neuron $(i,j)$, and $J(i,j;k,j-1)$ stands for the relative peak conductance of this particular excitatory synapse which is used to determine its strength. For simplicity, we assume that $J(i,j;k,j-1)=g$, that is, the synaptic strength is identical for all excitatory connections. Parameter $p$ is defined as the successful transmission probability of spikes. When a presynaptic neuron $(k,j-1)$ fires a spike, we let the corresponding synaptic reliability variables $h(i,j;k,j-1)=1$ with probability $p$ and $h(i,j;k,j-1)=0$ with probability $1-p$. That is to say, whether the neurotransmitter is successfully released or not is in essence controlled by a Bernoulli on-off process in the present work. In other time, the synaptic conductance decays by an exponential law: $\begin{split}\frac{d}{dt}G(i,j;k,j-1)=\frac{1}{\tau_{s}}G(i,j;k,j-1),\end{split}$ (2.8) with a fixed synaptic time constant $\tau_{s}$. In the case of synfire propagation, we choose $\tau_{s}=2$ ms, and in the case of firing rate propagation, we choose $\tau_{s}=5$ ms. ### 2.4 Measures of the synfire and firing rate propagation We now introduce several useful measures used to quantitatively evaluate the performance of the two different propagation modes: the synfire mode and firing rate mode. The propagation of synfire activity is measured by the survival rate and the standard deviation of the spiking times of the synfire packet (Gewaltig et al. 2001). Let us first introduce how to calculate the survival rate for the synfire propagation. In our simulations, we find that the synfire propagation can be divided into three types: the failed synfire propagation, the stable synfire propagation, as well as the synfire instability propagation (for detail, see Sec. 3.1). For neurons in each layer, a threshold method is developed to detect the local highest “energy” region. To this end, we use a 5 ms moving time window with 0.1 ms sliding step to count the number of spikes within each window. Here a high energy region means that the number of spikes within the window is larger than a threshold $\theta=50$. Since we use a moving time window with small sliding step, there might be a continuous series of windows contain more than 50 spikes around a group of synchronous spikes. In this work, we only select the first window which covers the largest number of spikes around a group of synchronous spikes as the local highest energy region. We use the number of local highest energy region to determine which type of synfire propagation occurs. If there is no local highest energy region detected in the final layer of the network, we consider it as the failed synfire propagation. When two or more separated local highest energy regions are detected in one layer, we consider it as the synfire instability propagation. Otherwise, it means the occurrence of the stable synfire propagation. For each experimental setting, we carry out the simulation many times. The survival rate of the synfire propagation is defined as the ratio of the number of occurrence of the stable synfire propagation to the total number of simulations. In additional simulations, it turns out that the threshold value $\theta$ can vary in a wide range without altering the results. Under certain conditions, noise can help the feedforward neuronal network produce the spontaneous spike packets, which promotes the occurrence of synfire instability propagation and therefore decreases the survival rate. For stable synfire propagation, there exists only one highest energy region for neurons in each layer. Spikes within this region are considered as the candidate synfire packet, which might also contain a few spontaneous spikes caused by noise and other factors. In this work, an adaptive algorithm is introduced to eliminate spontaneous spikes from the candidate synfire packet. Suppose now that there is a candidate synfire packet in the $i$-th layer with the number of spikes it contains $\alpha_{i}$ and the corresponding spiking times $\\{t_{1},t_{2},\ldots,t_{\alpha_{i}}\\}$. The average spiking time of the candidate synfire packet is therefore given by $\begin{split}\bar{t}_{i}=\frac{1}{\alpha_{i}}\sum_{k=1}^{\alpha_{i}}t_{k}.\end{split}$ (2.9) Thus the standard deviation of the spiking times in the $i$-th layer can be calculated as follows: $\begin{split}\sigma_{i}=\sqrt{\frac{1}{\alpha_{i}}\sum_{k=1}^{\alpha_{i}}[t_{k}-\bar{t}_{i}]^{2}}.\end{split}$ (2.10) We remove the $j$-th spike from the candidate synfire packet if it satisfies: $\|t_{j}-\bar{t}_{i}\|>\mu\sigma_{i}$, where $\mu$ is a parameter of our algorithm. We recompute the average spiking time as well as the standard deviation of the spiking times for the new candidate synfire packet, and repeat the above eliminating process, until no spike is removed from the new candidate synfire packet anymore. We define the remaining spikes as the synfire packet, which is characterized by the final values of $\alpha_{i}$ and $\sigma_{i}$. Parameter $\mu$ determines the performance of the proposed algorithm. If $\mu$ is too large, the synfire packet will lose several useful spikes at its borders, and if $\mu$ is too small, the synfire packet will contain some noise data. In our simulations, we found that $\mu=4$ can result in a good compromise between these two extremes. It should be emphasized that our algorithm is inspired by the method given in (Gewaltig et al. 2001). Next, we introduce how to measure the performance of the firing rate propagation. The performance of firing rate propagation is evaluated by combining it with a population code. Specifically, we compute how similar the population firing rates in different layers to the external input current $I(t)$ (van Rossum et al. 2002; Vogels and Abbott 2005). To do this, a 5 ms moving time window with 1 ms sliding step is also used to estimate the population firing rates ${r_{i}(t)}$ for different layers as well as the smooth version of the external input current $I_{s}(t)$. The correlation coefficient between the population firing rate of the $i$-th layer and external input current is calculated by $\begin{split}C_{i}(\tau)=\frac{\left\langle\left[I_{s}(k+\tau)-\overline{I}_{s}\right]\left[r_{i}(k)-\overline{r}_{i}\right]\right\rangle_{t}}{\sqrt{\left\langle\left[I_{s}(k+\tau)-\overline{I}_{s}\right]^{2}\right\rangle_{t}\left\langle\left[r_{i}(k)-\overline{r}_{i}\right]^{2}\right\rangle_{t}}},\end{split}$ (2.11) where $\langle\cdot\rangle_{t}$ denotes the average over time. Here we use the maximum cross-correlation coefficient $Q_{i}=\max\\{C_{i}(\tau)\\}$ to quantify the performance of the firing rate propagation in the $i$-th layer. Note that $Q_{i}$ is a normalization measure and a larger value corresponds to a better performance. ### 2.5 Numerical simulation method In all numerical simulations, we use the standard Euler-Maruyama integration scheme to numerically calculate the aforementioned stochastic differential Eqs. (2.1)-(2.8) (Kloeden et al. 1994). The temporal resolution of integration is fixed at 0.02 ms for calculating the measures of the synfire mode and at 0.05 ms for calculating the measures of the firing rate mode, as the measurement of the synfire needs higher precise. In additional simulations, we have found that further reducing the integration time step does not change our numerical results in a significant way. For the synfire mode, all simulations are executed at least 100 ms to ensure that the synfire packet can be successfully propagated to the final layer of the considered network. While studying the firing rate mode, we perform all simulations up to 5000 ms to collect enough spikes for statistical analysis. It should be noted that, to obtain convincing results, we carry out several times of simulations (at least 200 times for the synfire mode and 50 times for the firing rate mode) for each experimental setting to compute the corresponding measures. ## 3 Simulation results In this section, we report the main results obtained in the simulation. We first systematically investigate the signal propagation in the URE feedforward neuronal networks. Then, we compare these results with those for the corresponding RRE feedforward neuronal networks. Finally, we further study the signal propagation in the UREI feedforward neuronal networks. ### 3.1 Synfire propagation in URE feedforward neuronal networks Here we study the role of unreliable synapses on the propagation of synfire packet in the URE feedforward neuronal networks. In the absence of noise, we artificially let each sensory neuron fire and only fire an action potential at the same time ($\alpha_{1}=100$ and $\sigma_{1}=0$ ms). Without loss of generality, we let all sensory neurons fire spikes at $t=10$ ms. Figure 2 shows four typical spike raster diagrams of propagating synfire activity. Note that the time scales in Figs. 2-2 are different. The URE feedforward neuronal network with both small successful transmission probability and small excitatory synaptic strength badly supports the synfire propagation. In this case, due to high synaptic unreliability and weak excitatory synaptic interaction between neurons, the propagation of synfire packet cannot reach the final layer of the whole network (see Fig. 2). For suitable values of $p$ and $g$, we find that the synfire packet can be stably transmitted in the URE feedforward neuronal network. Moreover, it is obvious that the width of the synfire packet at any layer for $p=0.8$ is much narrower than that of the corresponding synfire packet for $p=0.25$ (see Figs. 2 and 2). At the same time, the transmission speed is also enhanced with the increasing of $p$. These results indicate that the neuronal response of the considered network is much more precise and faster for suitable large successful transmission probability. However, our simulation results also reveal that a strong excitatory synaptic strength with large value of $p$ might destroy the propagation of synfire activity. As we see from Fig. 2, the initial tight synfire packet splits into several different synfire packets during the transmission process. Such phenomenon is called the “synfire instability” (Tetzlaff et al. 2002; Tetzlaff et al. 2003), which mainly results from the burst firings of several neurons caused by the strong excitatory synaptic interaction as well as the stochastic fluctuation of the synaptic connections. Figure 2: (Color online) Several typical spike raster diagrams for different values of successful transmission probability $p$ and excitatory synaptic strength $g$. Shown are samples of (a) failed synfire propagation, (b) and (c) stable synfire propagation, and (d) synfire instability. System parameters are $p=0.19$ and $g=1$ nS (a), $p=0.25$ and $g=1$ nS (b), $p=0.8$ and $g=1$ nS (c), and $p=0.7$ and $g=2$ nS (d), respectively. As we see, the synfire packet can reach the final layer of the network successfully only for appropriate values of $p$ and $g$. It should be noted that the time scales in these four subfigures are different. Figure 3: (Color online) Effects of the successful transmission probability and excitatory synaptic strength on the synfire propagation in URE feedforward neuronal networks. (a) The survival rate of the synfire propagation versus $p$ for different values of $g$. (b) Schematic of three different synfire propagation regimes in the $(g,p)$ panel ($41\times 41=1681$ points). Regime I: the failed synfire propagation region; regime II: the stable synfire propagation region; and regime III: the synfire instability propagation region. (c) The value of $\sigma_{10}$ as a function of $p$ for different values of $g$. (d) The value of $\sigma_{10}$ as a function of $g$ for different values of $p$. In all cases, the noise intensity $D_{t}=0$. Each data point shown here is computed based on 200 independent simulations with different random seeds. In Fig. 3, we depict the survival rate of synfire propagation as a function of the successful transmission probability $p$ for different values of excitatory synaptic strength $g$, with the noise intensity $D_{t}=0$. We find that each survival rate curve can be at least characterized by one corresponding critical probability $p_{\text{on}}$. For small $p$, due to low synaptic reliability, any synfire packet cannot reach the final layer of the URE feedforward neuronal network. Once the successful transmission probability $p$ exceeds the critical probability $p_{\text{on}}$, the survival rate rapidly transits from 0 to 1, suggesting that the propagation of synfire activity becomes stable for a suitable high synaptic reliability. On the other hand, besides the critical probability $p_{\text{on}}$, we find that the survival rate curve should be also characterized by another critical probability $p_{\text{off}}$ if the excitatory synaptic strength is sufficiently strong (for example, $g=3.5$ nS in Fig. 3). In this case, when $p\geq p_{\text{off}}$, our simulation results show that the survival rate rapidly decays from 1 to 0, indicating that the network fails to propagate the stable synfire packet again. However, it should be noted that this does not mean that the synfire packet cannot reach the final layer of the network in this situation, but because the excitatory synapses with both high reliability and strong strength lead to the occurrence of the redundant synfire instability in transmission layers. To systematically establish the limits for the appearance of stable synfire propagation as well as to check that whether our previous results can be generalized within a certain range of parameters, we further calculate the survival rate of synfire propagation in the $(g,p)$ panel, which is shown in Fig. 3. As we see, the whole $(g,p)$ panel can be clearly divided into three regimes. These regimes include the failed synfire propagation regime (regime I), the stable synfire propagation regime (regime II), and the synfire instability propagation regime (regime III). Our simulation results reveal that transitions between these regimes are normally very fast and therefore can be described as a sharp transition. The data shown in Fig. 3 further demonstrate that synfire propagation is determined by the combination of both the successful transmission probability and excitatory synaptic strength. For a lower synaptic reliability, the URE feedforward neuronal network might need a larger $g$ to support the stable propagation of synfire packet. In reality, not only the survival rate of the synfire propagation but also its performance is largely influenced by the successful transmission probability and the strength of the excitatory synapses. In Figs. 3 and 3, we present the standard deviation of the spiking times of the output synfire packet $\sigma_{10}$ for different values of $p$ and $g$, respectively. Note that here we only consider parameters $p$ and $g$ within the stable synfire propagation regime. The results illustrated in Fig. 3 clearly demonstrate that the propagation of synfire packet shows a better performance for a suitable higher synaptic reliability. For the ideal case $p=1$, the URE feedforward neuronal network even has the capability to propagate the perfect synfire packet ($\alpha_{i}=100$ and $\sigma_{i}=0$ ms) in the absence of noise. On the other hand, it is also found that for a fixed $p$ the performance of synfire propagation becomes better and better as the value of $g$ is increased (see Fig. 3). The above results indicate that both high synaptic reliability and strong excitatory synaptic strength are able to help the URE feedforward neuronal network maintain the precision of neuronal response in the stable synfire propagation regime. Figure 4: (Color online) Dependence of the synfire propagation on the parameters of the initial spike packet. Here we display the survival rate of the synfire propagation versus $\alpha_{1}$ (a) and $\sigma_{1}$ (b) for different successful transmission probabilities, and the values of $n_{i}$ (c) and $\sigma_{i}$ (d) as a function of the layer number for different initial spike packets and different intrinsic parameters of the network, respectively. Note that the vertical axis in (d) is a log scale. In all cases, the noise intensity $D_{t}=0$. Other parameters are $g=1$ nS and $\sigma_{1}=3$ ms (a), $g=1$ nS and $\alpha_{1}=70$ (b). Each data shown in (a) and (b) is computed based on 600 independent simulations, whereas each data shown in (c) and (d) is calculated based on 500 independent successful synfire propagation simulations. Up to now, we only use the perfect initial spike packet ($\alpha_{1}=100$ and $\sigma_{1}=0$ ms) to evoke the synfire propagation. This is a special case which is simplified for analysis, but it is not necessary to restrict this condition. To understand how a generalized spike packet is propagated through the URE feedforward neuronal network, we randomly choose $\alpha_{1}$ neurons from the sensory layer, and let each of these neurons fire and only fire a spike at any moment according to a Gaussian distribution with the standard deviation $\sigma_{1}$. In Figs. 4 and 4, we plot the survival rate of the synfire propagation as a function of $\alpha_{1}$ and $\sigma_{1}$ for four different values of successful transmission probability, respectively. When the successful transmission probability is not too large (for example, $p=0.24$, 0.27, and 0.3 in Figs. 4 and 4), the synfire activity is well build up after several initial layers for sufficiently strong initial spike packet (large $\alpha_{1}$ and small $\sigma_{1}$), and then this activity can be successfully transmitted along the entire network with high survival rate. In this case, too weak initial spike packet (small $\alpha_{1}$ and large $\sigma_{1}$) leads to the propagation of the neural activities becoming weaker and weaker with the increasing of layer number. Finally, the neural activities are stopped before they reach the final layer of the network. Moreover, with the increasing of the successful transmission probability, neurons in the downstream layers will share more common synaptic currents from neurons in the corresponding upstream layers. This means that neurons in the considered network have the tendency to fire more synchronously for suitable larger $p$ (not too large). On the other hand, for sufficiently high synaptic reliability (for instance, $p=0.6$ in Figs. 4 and 4), a large $\alpha_{1}$ or a suitable large $\sigma_{1}$ may result in the occurrence of synfire instability, which also reduces the survival rate of the synfire propagation. Therefore, for a fixed $g$, the URE feedforward neuronal network with suitable higher synaptic reliability has the ability to build up stable synfire propagation from a slightly weaker initial spike packet (see Figs. 4 and 4). Figures 4 and 4 illustrate the values of $\alpha_{i}$ and $\sigma_{i}$ versus the layer number for different initial spike packets and several different intrinsic system parameters of the network (the successful transmission probability $p$ and excitatory synaptic strength $g$). For each case shown in Figs. 4 and 4, once the synfire propagation is successfully established, $n_{i}$ converges fast to the saturated value 100 and $\sigma_{i}$ approaches to an asymptotic value. Although the initial spike packet indeed determines whether the synfire propagation can be established or not as well as influences the performance of synfire propagation in the first several layers, but it does not determine the value of $\sigma_{i}$ in deep layers provided that the synfire propagation is successfully evoked. For the same intrinsic system parameters, if we use different initial spike packets to evoke the synfire propagation, the value of $\sigma_{i}$ in deep layers is almost the same for different initial spike packets (see Fig. 4). The above results indicate that the performance of synfire propagation in deep layers of the URE feedforward neuronal network is quite stubborn, which is mainly determined by the intrinsic parameters of the network but not the parameters of the initial spike packet. In fact, many studies have revealed that the synfire activity is governed by a stable attractor in the $(\alpha,\sigma)$ space (Diesmann et al. 1999; Diesmann et al. 2001; Diesmann 2002; Gewaltig et al. 2001). Our above finding is a signature that the stable attractor of synfire propagation does also exist for the feedforward neuronal networks with unreliable synapses. Figure 5: (Color online) Effects of the noise intensity $D$ on the synfire propagation. (a) The survival rate versus successful transmission probability $p$ for different excitatory coupling strength and noise intensities. (b) The value of $\sigma_{10}$ versus the noise intensity $D$ for different successful transmission probabilities, with the excitatory synaptic strength $g=1.5$ nS. In all cases, the parameters of the initial spike packet are $\alpha_{1}=100$ and $\sigma_{1}=0$ ms. Each data shown here is computed based on 200 independent simulations with different random seeds. Next, we study the dependence of synfire propagation on neuronal noise. It is found that both the survival rate of synfire propagation and its performance are largely influenced by the noise intensity. There is no significant qualitative difference between the corresponding survival rate curves in low synaptic reliable regime. However, we find important differences between these curves for small $g$ in high synaptic reliable regime, as well as for large $g$ in intermediate synaptic reliable regime, that is, during the transition from the successful synfire regime to the synfire instability propagation regime (see from Fig. 5). For each case, it is obvious that the top region of the survival rate becomes smaller with the increasing of noise intensity. This is at least due to the following two reasons: (i) noise makes neurons desynchronize, thus leading to a more dispersed synfire packet in each layer. For relatively high synaptic reliability, a dispersed synfire packet has the tendency to increase the occurrence rate of the synfire instability. (ii) Noise with large enough intensity results in several spontaneous neural firing activities at random moments, which also promote the occurrence of the synfire instability. Figure 5 presents the value of $\sigma_{10}$ as a function of the noise intensity $D_{t}$ for different values of successful transmission probability $p$. As we see, the value of $\sigma_{10}$ becomes larger and larger as the noise intensity is increased from 0 to 0.1 (weak noise regime). This is also due to the fact that the existence of noise makes neurons desynchronize in each layer. However, although noise tends to reduce the synchrony of synfire packet, the variability of $\sigma_{i}$ in deep layers is quite low (data not shown). The results suggest that, in weak noise regime, the synfire packet can be stably transmitted through the feedforward neuronal network with small fluctuation in deep layers, but displays slightly worse performance compared to the case of $D_{t}=0$. Further increase of noise will cause many spontaneous neural firing activities which might significantly deteriorate the performance of synfire propagation. However, it should be emphasized that, although the temporal spread of synfire packet tends to increase as the noise intensity grows, several studies have suggested that under certain conditions the basin of attraction of synfire activity reaches a maximum extent (Diesmann 2002; Postma et al. 1996; Boven and Aertsen 1990). Such positive effect of noise can be compared to a well known phenomenon called aperiodic stochastic resonance (Collins et al. 1995b; Collins et al. 1996; Diesmann 2002). ### 3.2 Firing rate propagation in URE feedforward neuronal networks In this subsection, we examine the firing rate propagation in URE feedforward neuronal networks. To this end, we assume that all sensory neurons are injected to a same time-varying external current $I(t)$ (see Sec. 2.2 for detail). Note now that the sensory neurons are modeled by using the integrate- and-fire neuron model in the study of the firing rate propagation. Figure 6: The maximum cross-correlation coefficient between the smooth version of external input current $I_{s}(t)$ and the population firing rate of sensory neurons $r_{1}(t)$ for different noise intensities. Figure 7: (Color online) Impacts of noise on the encoding performance (the firing rate mode) of sensory neurons. For each case, the smooth version of the external input current $I_{s}(t)$ (top panel), spike raster diagram of sensory neurons (middle panel), and population firing rate of sensory neurons (bottom panel) are shown. Noise intensities are $D_{s}=0$ (a), $D_{s}=0.1$ (b), $D_{s}=0.6$ (c), $D_{s}=0.8$ (d), $D_{s}=2$ (e), and $D_{s}=5$ (f), respectively. Before we present the results of the firing rate propagation, let us first investigate how noise influences the encoding capability of sensory neurons by the population firing rate. This is an important preliminary step, because how much input information represented by sensory neurons will directly influence the performance of firing rate propagation. The corresponding results are plotted in Figs. 6 and 7, respectively. When the noise is too weak, the dynamics of sensory neurons is mainly controlled by the same external input current, which causes neurons to fire spikes almost at the same time (see Figs. 7 and 7). In this case, the information of the external input current is poorly encoded by the population firing rate since the synchronous neurons have the tendency to redundantly encode the same aspect of the external input signal. When the noise intensity falls within a special intermediate range (about 0.5-10), neuronal firing is driven by both the external input current and noise. With the help of noise, the firing rate is able to reflect the temporal structural information (i.e., temporal waveform) of the external input current to a certain degree (see Figs. 7 to 7), and therefore $Q_{1}$ has large value in this situation. For too large noise intensity, the external input current is almost drowned in noise, thus resulting that the input information cannot be well read from the population firing rate of sensory neurons again. On the other hand, sensory neurons can fire “false” spikes provided that they are driven by sufficiently strong noise (as for example at $t\approx 2800$ ms in Fig. 7). Although the encoding performance of the sensory neurons might be good enough in this case, our numerical simulations reveal that such false spikes will seriously reduce the performance of the firing rate propagation in deep layers, which will be discussed in detail in the later part of this section. By taking these factors into account, we consider the noise intensity of sensory neurons to be within the range of 0.5 to 1 in the present work. Figure 8: An example of the firing rate propagation in the URE feedforward network. Here we show the smooth version of the external input current $I_{s}(t)$, as and the population firing rates of layers 1, 2, 4, 6, 8, and 10, respectively. System parameters are $g=0.4$ nS, $p=0.2$, and $D_{s}=D_{t}=0.7$. Figure 8 shows a typical example of the firing rate propagation. In view of the overall situation, the firing rate can be propagated rapidly and basically linearly in the URE feedforward neuronal network. However, it should be noted that, although the firing rates of neurons from the downstream layers tend to track those from the upstream layers, there are still several differences between the firing rates for neurons in two adjacent layers. For example, it is obvious that some low firing rates may disappear or be slightly amplified in the first several layers, as well as some high firing rates are weakened to a certain degree during the whole transmission process. Therefore, as the neural activities are propagated across the network, the firing rate has the tendency to lose a part of local detailed neural information but can maintain a certain amount of global neural information. As a result, the maximum cross- correlation coefficient between $I_{s}(t)$ and $r_{i}(t)$ basically drops with the increasing of the layer number. Figure 9: (Color online) Effects of the unreliable synapses on the performance of firing rate propagation. (a) The value of $Q_{10}$ as a function of $p$ for different values of excitatory synaptic strength. (b) The value of $Q_{10}$ as a function of $g$ for different values of successful transmission probability. Noise intensities are $D_{s}=D_{t}=0.5$ in all cases. Here each data point is computed based on 50 different independent simulations with different random seeds. Let us now assess the impacts of the unreliable synapses on the performance of firing rate propagation in the URE feedforward neuronal network. Figure 9 presents the value of $Q_{10}$ versus the success transmission probability $p$ for various excitatory synaptic strengths. For a fixed value of $g$, a bell- shaped $Q_{10}$ curve is clearly seen by changing the value of successful transmission probability, indicating that the firing rate propagation shows the best performance at an optimal synaptic reliability level. This is because, for each value of $g$, a very small $p$ will result in the insufficient firing rate propagation due to low synaptic reliability, whereas a sufficiently large $p$ can lead to the excessive propagation of firing rate caused by burst firings. Based on above reasons, the firing rate can be well transmitted to the final layer of the URE feedforward neuronal network only for suitable intermediate successful transmission probabilities. Moreover, with the increasing of $g$, the considered network needs a relatively small $p$ to support the optimal firing rate propagation. In Fig. 9, we plot the value of $Q_{10}$ as a function of the excitatory synaptic strength $g$ for different values of $p$. Here the similar results as those shown in Fig. 9 can be observed. This is due to the fact that increasing $g$ and fixing the value of $p$ is equivalent to increasing $p$ and fixing the value of $g$ to a certain degree. According to the aforementioned results, we conclude that both the successful transmission probability and excitatory synaptic strength are critical for firing rate propagation in URE feedforward networks, and better choosing of these two unreliable synaptic parameters can help the cortical neurons encode neural information more accurately. Figure 10: (Color online) Impacts of noise on the performance of firing rate propagation. (a) The value of $Q_{10}$ as a function of the noise intensity of cortical neurons $D_{t}$ for different values of $D_{s}$. (b) The performance of firing rate propagation in each layer at $D_{t}=0.6$ for two different noise intensities of sensory neurons. In all cases, $p=0.2$ and $g=0.4$ nS. Here each data point is computed based on 50 different independent simulations with different random seeds. Next, we examine the dependence of the firing rate propagation on neuronal noise. The corresponding results are plotted in Figs. 10 and 11, respectively. Figure 10 demonstrates that the noise of cortical neurons plays an important role in firing rate propagation. Noise of cortical neurons with appropriate intensity is able to enhance their encoding accuracy. It is because appropriate intermediate noise, on the one hand, prohibits synchronous firings of cortical neurons in deep layers, and on the other hand, ensures that the useful neural information does not drown in noise. However, the level of enhancement is largely influenced by the noise intensity of sensory neurons. As we see, for a large value of $D_{s}$, such enhancement is weakened to a great extent. This is because slightly strong noise intensity of sensory neurons will cause these neurons to fire several false spikes and a part of these spikes can be propagated to the transmission layers. If enough false spikes appear around the weak components of the external input current, these spikes will help the network abnormally amplify these weak components during the whole transmission process. The aforementioned process can be seen clearly from an example shown in Fig. 11. As a result, the performance of the firing rate propagation might be seriously deteriorated in deep layers. However, it should be noted that this kind of influence typically needs the accumulation of several layers. Our simulation results show that the performance of firing rate propagation can be well maintained or even becomes slightly better (depending on the noise intensity of sensory neurons, see Fig. 6) in the first several layers for large $D_{s}$ (see Fig. 10). In fact, the above results are based on the assumption that each cortical neuron is driven by independent noise current with the same intensity. Our results can be generalized from the sensory layer to the transmission layers if we suppose that noise intensities for neurons in different transmission layers are different. All these results imply that better tuning of the noise intensities of both the sensory and cortical neurons can enhance the performance of firing rate propagation in the URE feedforward neuronal network. Figure 11: (Color online) An example of weak external input signal amplification. System parameters are the successful transmission probability $p=0.2$, excitatory synaptic strength $g=0.4$ nS, and noise intensities $D_{s}=2.5$ and $D_{t}=0.6$, respectively. ### 3.3 Stochastic effect of neurotransmitter release From the numerical results depicted in Secs. 3.1 and 3.2, we find that increasing $g$ with $p$ fixed has similar effects as increasing $p$ while keeping $g$ fixed for both the synfire mode and firing rate mode. Some persons might therefore postulate that the signal propagation dynamics in feedforward neuronal networks with unreliable synapses can be simply determined by the average amount of received neurotransmitter for each neuron in a time instant, which can be reflected by the product of $g\cdot p$. To check whether this is true, we calculate the measures of these two signal propagation modes as a function of $g\cdot p$ for different successful transmission probabilities. If this postulate is true, the URE feedforward neuronal network will show the same propagation performance for different values of $p$ at a fixed $g\cdot p$. Our results shown in Figs. 12-12 clearly demonstrate that the signal propagation dynamics in the considered network can not be simply determined by the product $g\cdot p$ or, equivalently, by the average amount of received neurotransmitter for each neuron in a time instant. For both the synfire propagation and firing rate propagation, although the propagation performance exhibits the similar trend with the increasing of $g\cdot p$, the corresponding measure curves do not superpose in most parameter region for each case, and in some parameter region the differences are somewhat significant (see Figs. 12 and 12). This is because of the stochastic effect of neurotransmitter release, that is, the unreliability of neurotransmitter release will add randomness to the system. Different successful transmission probabilities may introduce different levels of randomness, which will further affect the nonlinear spiking dynamics of neurons. Therefore, the URE feedforward neuronal network might display different propagation performance for different values of $p$ even at a fixed $g\cdot p$. If we set the value of $g\cdot p$ constant, a low synaptic reliability will introduce large fluctuations in the synaptic inputs. For small $p$, according to the above reason, some neurons will fire spikes more than once in the large $g\cdot p$ regime. This mechanism increases the occurrence rate of the synfire instability. Thus, the URE feedforward neuronal network has the tendency to stop the stable synfire propagation for a small synaptic transmission probability (see Fig. 12). On the other hand, a high synaptic reliability will introduce small fluctuations in the synaptic inputs for a fixed $g\cdot p$. This makes neurons in the considered network fire spikes almost synchronously for a large $p$, thus resulting the worse performance for the firing rate propagation in large $g\cdot p$ regime (see Fig. 12). Our above results suggest that the performance of the signal propagation in feedforward neuronal networks with unreliable synapses is not only purely determined by the change of synaptic parameters, but also largely influenced by the stochastic effect of neurotransmitter release. Figure 12: (Color online) Dependence of signal propagation dynamics on the product of $g\cdot p$ in the URE feedforward neuronal network. _Synfire_ mode: survival rate (a) and $\sigma_{10}$ versus $g\cdot p$ with $D_{t}=0$. The parameters of the initial spike packet are $\alpha_{1}=100$ and $\sigma_{1}=0$ ms. _Firing rate_ mode: $Q_{10}$ as a function of $g\cdot p$ with $D_{s}=D_{t}=0.5$. Here each data point shown in (a) and (b) is calculated based on 200 different independent simulations, whereas each data point shown in (c) are based on 50 different independent simulations. ### 3.4 Comparison with corresponding RRE feedforward neuronal networks In this subsection, we make comparisons on the propagation dynamics between the URE and the RRE feedforward networks. We first introduce how to generate a corresponding RRE feedforward neuronal network for a given URE feedforward neuronal network. Suppose now that there is a URE feedforward neuronal network with successful transmission probability $p$. A corresponding RRE feedforward neuronal network is constructed by using the connection density $p$ (on the whole), that is, a synapse from one neuron in the upstream layer to one neuron in the corresponding downstream layer exists with probability $p$. As in the URE feedforward neuronal network given in Sec. 2.1, there is no feedback connection from downstream neurons to upstream neurons and also no connection among neurons within the same layer in the RRE feedforward neuronal network. It is obvious that parameter $p$ has different meanings in these two different feedforward neuronal network models. The synaptic interactions between neurons in the RRE feedforward neuronal network are also implemented by using the conductance-based model (see Eqs. (2.6) and (2.7) for detail). However, here we remove the constraint of the synaptic reliability parameter for the RRE feedforward neuronal network, e.g., $h(i,j;k,j-1)=1$ in all cases. A naturally arising question is what are the differences, if have, between the synfire propagation and firing propagation in URE feedforward neuronal networks and those in RRE feedforward neuronal networks, although the numbers of active synaptic connections that taking part in transmitting spikes in a time instant are the same from the viewpoint of mathematical expectation. Figure 13: (Color online) The difference between the synfire propagation in the URE feedforward neuronal network and the RRE feedforward neuronal network. Here we show the value of survival rate as a function of $\sigma_{1}$ for different network models. In all cases, $D_{t}=0$ and $\alpha_{1}=70$. Other system parameters are $g=4$ nS and $p=0.15$ (dot: “$\bullet$”, and circle: “$\circ$” ), and $g=3.5$ nS and $p=0.12$ (square: “$\square$”, and asterisk: “$\ast$”). Each data point is calculated based on 500 different independent simulations with different random seeds. For the synfire propagation, our simulation results indicate that, compared to the RRE feedforward neuronal network, the URE feedforward neuronal network is able to suppress the occurrence of synfire instability to a certain degree, which can be seen clearly in Fig. 13. Typically, this phenomenon can be observed in strong excitatory synaptic strength regime. Due to the heterogeneity of connectivity, some neurons in the RRE feedforward neuronal network will have more input synaptic connections than the other neurons in the same network. For large value of $g$, these neurons tend to fire spikes very rapidly after they received synaptic currents. If the width of the initial spike packet is large enough, these neurons might fire spikes again after their refractory periods, which are induced by a few spikes from the posterior part of the dispersed initial spike packet. These spikes may increase the occurrence rate of the synfire instability. While in the case of URE feedforward neuronal network, the averaging effect of unreliable synapses tends to prohibit neurons fire spikes too quickly. Therefore, under the equivalent parameter conditions, less neurons can fire two or more spikes in the URE feedforeard neuronal network. As a result, the survival rate of the synfire propagation for the URE feedforeard neuronal network is larger than that for the RRE feedforward neuronal network (see Fig. 13), though not so significant. In further simulations, we find interesting results in small $p$ regime for the firing rate propagation. Compared to the case of the URE feedforward neuronal network, the RRE feedforward neuronal network can better support the firing rate propagation in this small $p$ regime for strong excitatory synaptic strength (see Fig. 14). It is because the long-time averaging effect of unreliable synapses at small $p$ tends to make neurons fire more synchronous spikes in the URE feedforward neuronal network through the homogenization process of synaptic currents. However, with the increasing of $p$, neurons in the downstream layers have the tendency to share more common synaptic currents from neurons in the corresponding upstream layers for both types of feedforward neuronal networks. The aforementioned factor makes the difference of the performance of firing rate propagation between these two types of feedforward neuronal networks become small so that the $Q_{10}$ curves almost coincide with each other for the case of $p=0.6$ (see Fig. 14). Although from the above results we can not conclude that unreliable synapses have advantages and play specific functional roles in signal propagation, not like those results shown in the previous studies (Goldman et al. 2002; Goldman 2004), at least it is shown that the signal propagation activities are different in URE and RRE to certain degrees. We should be cautioned when using random connections to replace unreliable synapses in modelling research. However, it should be noted that the RRE feedforward neuronal network considered here is just one type of diluted feedforward neuronal networks. There exists several other possibilities to construct the corresponding diluted feedforward neuronal networks (Hehl et al. 2001). The similar treatments for these types of diluted feedforward neuronal networks require further investigation. Figure 14: (Color online) Firing rate propagation in URE feedforward neuronal network and RRE feedforward neuronal network. The value of $Q_{10}$ as a function of excitatory synaptic strength $g$ for $p=0.2$ (a) and $p=0.6$ (b), respectively. Noise intensities are $D_{t}=D_{s}=0.5$. Each data point is calculated based on 50 different independent simulations with different random seeds. ### 3.5 Signal propagation in UREI feedforward neuronal networks In this subsection, we further study the signal propagation in the feedforward neuronal networks composed of both excitatory and inhibitory neurons connected in an all-to-all coupling fashion (i.e., the UREI feedforward neuronal networks). This study is necessary because real biological neuronal networks, especially mammalian neocortex, consist not only of excitatory neurons but also of inhibitory neurons. The UREI feedforward neuronal network studied in this subsection has the same topology as that shown in Fig. 1. In simulations, we randomly choose 80 neurons in each layer as excitatory and the rest of them as inhibitory, as the ratio of excitatory to inhibitory neurons is about $4:1$ in mammalian neocortex. The dynamics of the unreliable inhibitory synapse is also modeled by using Eqs. (2.6) and (2.7). The reversal potential of the inhibitory synapse is fixed at -75 mV, and its strength is set as $J=K\cdot g$, where $K$ is a scale factor used to control the relative strength of inhibitory and excitatory synapses. Since the results of the signal propagation in UREI feedforward neuronal networks are quite similar to those in URE feedforward neuronal networks, we omit most of them and only discuss the effects of inhibition in detail. Figure 15: (Color online) Partition of three different synfire propagation regimes in the $(K,p)$ panel ($41\times 41=1681$ points). Regime I: the failed synfire propagation region; regime II: the stable synfire propagation region; and regime III: the synfire instability propagation region. we set $g=2.5$ nS (a), $g=3$ nS (b), and $g=3.5$ nS (c). In all cases, the parameters of the initial spike packet are $\alpha_{1}=100$ and $\sigma_{1}=0$ ms, and the noise intensity is $D_{t}=0$. Each data point shown here is calculated based on 200 different independent simulations with different random seeds. Figure 15 shows the survival rate of synfire propagation in the $(K,p)$ panel for three different excitatory synaptic strengths. Depending on whether the synfire packet can be successfully and stably transmitted to the final layer of the UREI feedforward neuronal network, the whole $(K,p)$ panel can also be divided into three regimes. For each considered case, the network with both small successful transmission probability and strong relative strength of inhibitory and excitatory synapses (failed synfire regime) prohibits the stable propagation of the synfire activity. While in the case of high synaptic reliability and small $K$ (synfire instability propagation regime), the synfire packet also cannot be stably transmitted across the whole network due to the occurrence of synfire instability. Therefore, the UREI feedforward neuronal network is able to propagate the synfire activity successfully in a stable way only for suitable combination of parameters $p$ and $K$. Moreover, due to the competition between excitation and inhibition, the transitions between these different regimes cannot be described as a sharp transition anymore, in particular, for large scale factor $K$. Our results suggest that such non-sharp character is strengthen with the increasing of $g$. On the other hand, the partition of these different propagation regimes depends not only on parameters $p$ and $K$ but also on the excitatory synaptic strength $g$. As the value of $g$ is decreased, both the synfire instability propagation regime and stable synfire propagation regime are shifted to the upper left of the $(K,p)$ panel at first, and then disappear one by one (data not shown). In contrast, a strong excitatory synaptic strength has the tendency to extend the areas of the synfire instability propagation regime, and meanwhile makes the stable synfire propagation regime move to the lower right of the $(K,p)$ panel. Figure 16: (Color online) Effect of inhibition on firing rate propagation. Here we show the value of $Q_{10}$ as a function of scale factor $K$ for different excitatory synaptic strengths. System parameters are $p=0.2$, and $D_{s}=D_{t}=0.6$ in all cases. Each data point is calculated based on 50 different independent simulations with different random seeds. For the case of firing rate propagation, we plot the value of $Q_{10}$ versus the scale factor $K$ for different excitatory synaptic strengths in Fig. 16, with a fixed successful transmission probability $p=0.2$. When the excitatory synaptic strength is small (for instance $g=0.4$ nS), due to weak excitatory synaptic interaction between neurons the UREI feedforward neuronal network cannot transmit the firing rate sufficiently even for $K=0$. In this case, less and less neural information can be propagated to the final layer of the considered network with the increasing of $K$. Therefore, $Q_{10}$ monotonically decreases with the scale factor $K$ at first and finally approaches to a low steady state value. Note that here the low steady state value is purely induced by the spontaneous neural firing activities, which are caused by the additive Gaussian white noise. As the excitatory synaptic strength grows, more neural information can be successfully transmitted for small value of $K$. When $g$ is increased to a rather large value, such as $g=0.6$ nS, the coupling is so strong that too small scale factor will lead to the excessive propagation of firing rate. However, in this case, the propagation of firing rate can still be suppressed provided that the relative strength of inhibitory and excitatory synapses is strong enough. As a result, there always exists an optimal scale factor to best support the firing rate propagation for each large excitatory synaptic strength (see Fig. 16). If we fix the value of $g$ (not too small), then the similar results can also be observed by changing the scale factor for a large successful transmission probability (data not shown). Once again, this is due to the fact that increasing $g$ and fixing $p$ is equivalent to increasing $p$ and fixing $g$ to a certain degree. ## 4 Conclusion and discussion The feedforward neuronal network provides us an effective way to examine the neural activity propagation through multiple brain regions. Although biological experiments suggested that the communication between neurons is more or less unreliable (Abeles 1991; Raastad et al. 1992; Smetters and Zador 1996), so far most relevant computational studies only considered that neurons transmit spikes based on reliable synaptic models. In contrast to these previous work, we took a different approach in this work. Here we first built a 10-layer feedforward neuronal network by using purely excitatory neurons, which are connected with unreliable synapses in an all-to-all coupling fashion, that is, the so-called URE feedforward neuronal network in this paper. The goal of this work was to explore the dependence of both the synfire propagation and firing rate propagation on unreliable synapses in the URE neuronal network, but was not limited this type of feedforward neuronal network. Our modelling methodology was motivated by experimental results showing the probabilistic transmitter release of biological synapses (Branco and Staras 2009; Katz 1966; Katz 1969; Trommershäuser et al. 1999). In the study of synfire mode, it was observed that the synfire propagation can be divided into three types (i.e., the failed synfire propagation, the stable synfire propagation, and the synfire instability propagation) depending on whether the synfire packet can be successfully and stably transmitted to the final layer of the considered network. We found that the stable synfire propagation only occurs in the suitable region of system parameters (such as the successful transmission probability and excitatory synaptic strength). For system parameters within the stable synfire propagation regime, it was found that both high synaptic reliability and strong excitatory synaptic strength are able to support the synfire propagation in feedforward neuronal networks with better performance and faster transmission speed. Further simulation results indicated that the performance of synfire packet in deep layers is mainly influenced by the intrinsic parameters of the considered network but not the parameters of the initial spike packet, although the initial spike packet determines whether the synfire propagation can be evoked to a great degree. In fact, this is a signature that the synfire activity is governed by a stable attractor, which is in agreement with the results given in (Diesmann et al. 1999; Diesmann et al. 2001; Diesmann 2002; Gewaltig et al. 2001). In the study of firing rate propagation, our simulation results demonstrated that both the successful transmission probability and the excitatory synaptic strength are critical for firing rate propagation. Too small successful transmission probability or too weak excitatory synaptic strength results in the insufficient firing rate propagation, whereas too large successful transmission probability or too strong excitatory synaptic strength has the tendency to lead to the excessive propagation of firing rate. Theoretically speaking, better tuning of these two synaptic parameters can help neurons encode the neural information more accurately. On the other hand, neuronal noise is ubiquitous in the brain. There are many examples confirmed that noise is able to induce many counterintuitive phenomena, such as stochastic resonance (Collins et al. 1995a; Collins et al. 1995b; Collins et al. 1996; Chialvo et al. 1997; Guo and Li 2009) and coherence resonance (Pikovsky and Kurths 1996; Lindner and Schimansky-Geier 2002; Guo and Li 2009). Here we also investigated how the noise influences the performance of signal propagation in URE feedforward neuronal networks. The numerical simulations revealed that noise tends to reduce the performance of synfire propagation because it makes neurons desynchronized and causes some spontaneous neural firing activities. Further studies demonstrated that the survival rate of synfire propagation is also largely influenced by the noise. In contrast to the synfire propagation, our simulation results showed that noise with appropriate intensity is able to enhance the performance of firing rate propagation in URE feedforward neuronal networks. In essence, it is because suitable noise can help neurons in each layer maintain more accurate temporal structural information of the the external input signal. Note that the relevant mechanisms about noise have also been discussed in several previous work (van Rossum et al. 2002; Masuda and Aihara 2002; Masuda and Aihara 2003), and our results are consistent with the findings given in these work. Furthermore, we have also investigated the stochastic effect of neurotransmitter release on the performance of signal propagation in the URE feedforward neuronal networks. For both the synfire propagation and firing rate propagation, we found that the URE feedforward neuronal networks might display different propagation performance, even when their average amount of received neurotransmitter for each neuron in a time instant remains unchanged. This is because the unreliability of neurotransmitter release will add randomness to the system. Different synaptic transmission probabilities will introduce different levels of stochastic effect, and thus might lead to different spiking dynamics and propagation performance. These findings revealed that the signal propagation dynamics in feedforward neuronal networks with unreliable synapses is also largely influenced by the stochastic effect of neurotransmitter release. Finally, two supplemental work has been also performed in this paper. In the first work, we compared both the synfire propagation and firing rate propagation in URE feedforward neuronal networks with the results in corresponding feedforward neuronal networks composed of purely excitatory neurons but connected with reliable synapses in an random coupling fashion (RRE feedforward neuronal network). Our simulations showed that several different results exist for both the synfire propagation and firing rate propagation in these two different feedforward neuronal network models. These results tell us that we should be cautioned when using random connections to replace unreliable synapses in modelling research. In the second work, we extended our results to more generalized feedforward neuronal networks, which consist not only of the excitatory neurons but also of inhibitory neurons (UREI feedforward neuronal network). The simulation results demonstrated that inhibition also plays an important role in both types of neural activity propagation, and better choosing of the relative strength of inhibitory and excitatory synapses can enhance the transmission capability of the considered network. The results presented in this work might be more realistic than those obtained based on reliable synaptic models. This is because the communication between biological neurons indeed displays the unreliable properties. In real neural systems, neurons may make full use of the characteristics of unreliable synapses to transmit neural information in an adaptive way, that is, switching between different signal propagation modes freely as required. Further work on this topic includes at least the following two aspects: (i) since all our results are derived from numerical simulations, an analytic description of the synfire propagation and firing rate propagation in our considered feedforward neuronal networks requires investigation. (ii) Intensive studies on signal propagation in the feedforward neuronal network with other types of connectivity, such as the Mexican-hat-type connectivity (Hamaguchi et al. 2004; Hamaguchi and Aihara 2005) and the Gaussian-type connectivity (van Rossum et al. 2002), as well as in the feedforward neuronal network imbedded into a recurrent network (Aviel et al. 2003; Vogels and Abbott 2005; Kumar et al. 2008), from the unreliable synapses point of view are needed as well. ## Acknowledgement We thank Feng Chen, Yuke Li, Qiuyuan Miao, Xin Wei and Qunxian Zheng for valuable discussions on an early version of this manuscript. We gratefully acknowledge the anonymous reviewers for providing useful comments and suggestion, which greatly improved our paper. We also sincerely thank one reviewer for reminding us of a critical citation (Trommershäuser and Diesmann 2001). This work is supposed by the National Natural Science Foundation of China (Grant No. 60871094), the Foundation for the Author of National Excellent Doctoral Dissertation of PR China, and the Fundamental Research Funds for the Central Universities (Grant No. 1A5000-172210126). Daqing Guo would also like to thank the award of the ongoing best PhD thesis support from the University of Electronic Science and Technology of China. ## References * [1] Abeles, M. (1991). _Corticonics: Neural circuits of the cerebral cortex_. New York: Cambridge Uinversity Press. * [2] Aertsen, A., Diesmann, M., $\&$ Gewaltig, M. O. (1996). Propagation of synchronous spiking activity in feedforward neural networks. _J Physiology_ , 90, 243-247. * [3] Allen, C., $\&$ Stevens, C. F. (1994). An evaluation of causes for unreliability of synaptic transmission. _Proc. Natl. Acad. Sci. USA_ , 91, 10380-10383. * [4] Aviel, Y., Mehring, C., Abeles, M., $\&$ Horn, D. (2003). On embedding synfire chains in a balanced network. _Neural Computation_ , 15, 1321-1340. * [5] Boven, K., H., $\&$ Aertsen, A., M., H., J. (1990). Dynamics of activity in neuronal networks give rise to fast modulations of functional connectivity. In _Parallel processing and neural systems and computers_ (pp. 53-56). Amsterdam: Elsevier. * [6] Branco, T., $\&$ and Staras, K. (2009). The probability of neurotransmitter release: variability and feedback control at single synapses. Nature Reviews Neuroscience, 10, 373-383. * [7] C$\hat{\text{a}}$teau, H., $\&$ Fukai, T. (2001). Fokker-Planck approach to the pulse packet propagation in synfire chain. _Neural Networks_ , 14, 675-685. * [8] Chialvo, D. R., Longtin, A., $\&$ Muller-Gerking, J. (1997). Stochastic resonance in models of neuronal ensembles. _Phys. Rev. E_ , 55, 1798-1808. * [9] Collins, J. J., Chow, C. C., $\&$ Imhoff, T. T. (1995a). Stochastic resonance without tuning. _Nature_ , 376, 236-238. * [10] Collins, J. J., Chow, C. C., $\&$ Imhoff, T. T. (1995b). Aperiodic stochastic resonance in excitable systems. _Phys. Rev. E_ 52, R3321-R3324. * [11] Collins, J. J., Chow, C. C., Capela, A. C., $\&$ Imhoff, T. T. (1996). Aperiodic stochastic resonance. _Phys. Rev. E_ , 54, 5575-5584. * [12] Dayan, P., $\&$ Abbott, L. F. (2001). _Theoretical neuroscience: Computaional and mathematical modeling of neural systems_. Cambridge: MIT Press. * [13] Diesmann, M., Gewaltig, M. O., $\&$ Aertsen, A. (1999). Stable propagation of synchronous spiking in cortical neural networks. _Nature_ , 402, 529-533. * [14] Diesmann, M., Gewaltig, M., O., Rotter, S., $\&$ Aertsen, A. (2001). State Space Analysis of Synchronous Spiking in Cortical Neural Networks. _Neurocomputing_ , 38-40, 565-571. * [15] Diesmann, M. (2002). Conditions for stable propagation of synchronous spiking in cortical neural networks: single neuron dynamics and network properties. PhD thesis, University of Bochum. * [16] Friedrich, J., $\&$ Kinzel, W. (2009). Dynamics of recurrent neural networks with delayed unreliable synapses: metastable clustering. _J. Comput. Neurosci._ , 27, 65-80. * [17] Gewaltig, M. O., Diesmann, M., $\&$ Aertsen, A. (2001). Propagation of cortical synfire activity: survival probability in single trials and stability in the mean. _Neural Networks_ , 14, 657-673. * [18] Goldman, M. S., Maldonado, P., $\&$ Abbott., L. F. (2002). Redundancy reduction and sustained firing with stochastic depressing synapses. _J. Neurosci._ , 22, 584-591. * [19] Goldman, M. S. (2004). Enhancement of information transmission efficiency by synaptic failures. _Neural Computation_ , 16, 1137-1162. * [20] Guo, D., Q., $\&$, Li, C., G. (2009). Stochastic and coherence resonance in feed-forward-loop neuronal network motifs. _Phys. Rev. E_ , 79, 051921. * [21] Hamaguchi, K., $\&$ Aihara, K. (2005). Quantitative information transfer through layers of spiking neurons connected by Mexican-Hat-type connectivity. _Neurocomputing_ , 58-60, 85-90. * [22] Hamaguchi, K., Okada, M., Yamana, M., $\&$ Aihara, K. (2005). Correlated firing in a feedforward network with mexican-hat-type connectivity. _Neural Compuation_ , 17, 2034-2059. * [23] Hehl, U., Hellwig, B., Rotter, S., Diesmann, M., Aertsen, A. (2001). Localization of synchronous spiking as a result of anatomical connectivity. _Soc. of Neuroscience_ , 64, 1. * [24] Katz, B. (1966). _Nerve, muscle and synapse_. New York: McGraw-Hill. * [25] Katz, B. (1969). _The release of neural transmitter substances_. Liverpool, Liverpol University Press. * [26] Kloeden P. E., Platen E., $\&$ Schurz H. (1994). _Numerical solution of SDE through computer experiments_. Berlin, Springer-Verlag Press. * [27] Kumar A., Rotter, S., $\&$ Aertsen, A., (2008). Conditions for propagating synchronous spiking and asynchronous firing rates in a cortical network model. _J. Neurosci._ , 28, 5268-5280. * [28] Kumar A., Rotter, S., $\&$ Aertsen, A., (2010). Spiking activity propagation in neuronal networks: reconciling different perspectives on neural coding. _Nat. Revs. Neurosci._ , 11, 615-627. * [29] Lindner, B., $\&$ Schimansky-Geier, L. (2002). Maximizing spike train coherence or incoherence in the leaky integrate-and-fire model. _Phys. Rev. E_ , 66, 031916. * [30] Maass W., $\&$ Natschl$\ddot{\text{a}}$ger T. (2000). A model for fast analog computation based on unreliable synapses. _Neural Computation_ , 12, 1679-1704. * [31] Masuda, N., $\&$ Aihara, K., (2002). Bridging rate coding and temporal spike coding by effect of noise. _Phys. Rev. Lett._ , 88, 248101. * [32] Masuda, N., $\&$ Aihara, K., (2003). Duality of rate coding and temporal coding in multilayered feedforward networks. _Neural Computation_ , 15, 103-125. * [33] Nordlie, E., Gewaltig, M., O., $\&$ Plesser, H., E. (2009). Towards reproducible descriptions of neuronal network models. _PLoS Computational Biological_ , 5, e1000456. * [34] Pikovsky, A. S., $\&$ Kurths, J. (1996). Coherence resonance in a noise-driven excitable system, _Phys. Rev. Lett._ , 78, 775-778. * [35] Postma, E., O., van den Herik, H., J., $\&$ Hudson, P., T., W. (1996). Robust feedforward processing in synfire chains. _Int. J. Neural. Sys._ , 7, 537-542. * [36] Raastad, M., Storm, J. F., $\&$ Andersen P. (1992). Putative single quantum and single fibre excitatory postsynaptic currents show similar amplitude range and variability in rat hippocampal slices. _Eur. J. Neurosci._ , 4, 113-117. * [37] Rosenmund, C., Clements, J. D., $\&$ Westbrook, G. L. (1993). Nonuniform probability of glutamate release at a hippocampal synapse. _Science_ , 262, 754-757. * [38] Shinozaki, T., C$\hat{\text{a}}$teau, H., Urakubo, H., $\&$ Okada, M. (2007). Controlling synfire chain by inhibitory synaptic input. _Journal of the Physical Society of Japan_ , 76, 044806. * [39] Shinozaki, T., Okada, M., Reyes, A. D., $\&$ C$\hat{\text{a}}$teau, H. (2010). Flexible traffic control of the synfire-mode transmission by inhibitory modulation: Nonlinear noise reduction. _Phys. Rev. E_ , 81, 011913. * [40] Smetters, D. K., $\&$ Zador, A. (1996). Synaptic transmission: Noisy synapses and noisy neurons. _Current Biology_ , 6, 1217-1218. * [41] Stevens, C. F., $\&$ Wang, Y. (1995). Facilitation and depression at single central synapses. _Neuron_ , 14, 795-802. * [42] Tetzlaff, T., Geisel, T., $\&$ Diesmann, M. (2002), The ground Sstate of cortical feed-forward networks. _Neurocomputing_ 44-46, 673-678. * [43] Tetzlaff, T., Buschermoehle, M., Geisel, T., $\&$ Diesmann, M. (2003), The spread of rate and correlation in stationary cortical networks, _Neurocomputing_ , 52-54, 949-954. * [44] Trommershäuser J., Marienhagen J., $\&$ Zippelius A. (1999), Stochastic model of central synapses: slow diffusion of transmitter interacting with spatially distributed receptors and transporters. _J Theor. Biol._ , 198, 101-120. * [45] Trommershäuser, J., $\&$ Diesmann, M. (2001), The effect of synaptic variability on the synchronization dynamics in feed-forward cortical networks. In _Soc. of Neuroscience_ , 64, 4. * [46] van Rossum, M. C. W., Turrigiano, G. G., $\&$ Nelson, S. B. (2002). Fast propagation of firing rates through layered networks of noisy neurons. _J. Neurosci._ , 22, 1956-1966. * [47] Vogels, P. T., $\&$ Abbott, L. F. (2005). Signal propagation and logic gating in networks of integrate-and-fire neurons. _J. Neurosci._ , 25, 10786-10795. * [48] Wang, S. T., Wang, W., $\&$ Liu, F. (2006). Propagation of firing rate in a feed-forward neuronal network. _Phys. Rev. Lett._ , 96, 018103, 2006. * [49] Wang, S. T., $\&$ Zhou, C. S. (2009). Information encoding in an oscillatory network. _Phys. Rev. E_ , 79, 061910, 2009.	arxiv-papers	2011-11-19T02:36:39	2024-09-04T02:49:24.472642	{ "license": "Public Domain", "authors": "Daqing Guo and Chunguang Li", "submitter": "Daqing Guo", "url": "https://arxiv.org/abs/1111.4526" }
1111.4537	# Some common fixed points results on metric spaces over topological modules Ion Marian Olaru ###### Abstract In this paper, we replace the real numbers by a topological R-module and define R-metric spaces $(X,d)$. Also, we prove some common fixed point theorems on R-module metric spaces. We obtain, as a particular case the Perov theorem (see [3]) 2010 Mathematical Subject Classification: 47H10 Keywords: R-metric spaces, fixed point theory, topological rings, topological modules ## 1 R-metric spaces In this section we shall define $R$-metric spaces and prove some properties. All axioms for an ordinary metric space can be meaningfully formulated for an abstract metric space, where the abstract metric takes values in a partially ordered topological module of a certain type which will be defined below. Such a space will be called $R$-metric space. We begin this section by recalling a few facts concerning topological rings, topological modules and partially ordered rings. Unless explicitly stated otherwise all rings will be assumed to possess an identity element, denoted by $1$. ###### Definition 1.1. (see [5]) A topology $\tau$ on a ring $(R,+,\cdot)$ is a ring topology and $R$, furnished with $\tau$, is a topological ring if the following conditions hold: * (TR 1) $(x,y)\rightarrow x+y$ is continuous from $R\times R$ to $R$; * (TR 2) $x\rightarrow-x$ is continuous from $R$ to $R$; * (TR 3) $(x,y)\rightarrow x\cdot y$ is continuous from $R\times R$ to $R$, where $R$ is given topology $\tau$ and $R\times R$ the cartesian product determined by topology $\tau$. ###### Definition 1.2. (see [5]) Let $R$ be a topological ring, $E$ an R-module. A topology $\mathcal{T}$ on $E$ is a R-module topology and E, furnished with $\mathcal{T}$, is a topological R-module if the following conditions hold: * (TM 1) $(x,y)\rightarrow x+y$ is continuous from $E\times E$ to $E$; * (TM 2) $x\rightarrow-x$ is continuous from $E$ to $E$; * (TM 3) $(a,x)\rightarrow a\cdot x$ is continuous from $R\times E$ to $E$, where $E$ is given topology $\mathcal{T}$, $E\times E$ the cartesian product determined by topology $\mathcal{T}$ and $A\times E$ the cartesian product determined by topology of R and E. By a partially ordered ring is meant a pair consisting of a ring and a compatible partial order, denoted by $\preceq$(see [4]). In the following we always suppose that $R$ is an ordered topological ring such that $0\preceq 1$ and $E$ is a topological $R$-module. ###### Definition 1.3. A subset $P$ of $E$ is called a cone if: * (i) $P$ is closed, nonempty and $P\neq\\{0_{E}\\}$; * (ii) $a,b\in R$, $0\preceq a$, $0\preceq b$ and $x,y\in P$ implies $a\cdot x+b\cdot y\in P$; * (iii) $P\cap-P=\\{0\\}.$ Given a cone $P\subset E$, we define on E the partial ordering $\leq_{P}$ with respect to $P$ by (1.1) $x\leq_{P}y\ if\ and\ only\ if\ y-x\in P.$ We shall write $x<_{P}y$ to indicate that $x\leq_{P}y$ but $x\neq y$, while $x\ll y$ will stand for $y-x\in intP$(interior of $P$). ###### Example 1.1. Let $R=\mathcal{M}_{n\times n}(\mathbb{R})$ be the ring of all matrices with n rows and n columns with entries in $\mathbb{R}$ and $E=\mathbb{R}^{n}$. We define the partial order $\preceq$ on $M_{n\times n}(\mathbb{R})$ as follows $A\preceq B\ if\ and\ only\ if\ for\ each\ i,j=\overline{1,n}\ we\ have\ a_{ij}\leq b_{ij}.$ Then * (a) the topology $\tau$, generated by matrix norm $N:M_{n\times n}(\mathbb{R})\rightarrow\mathbb{R},$ $N(A)=\max\limits_{i=\overline{1,n}}\sum\limits_{j=1}^{n}\|a_{ij}\|,$ is a ring topology; * (b) the standard topology $\mathcal{D}$ is a R-module topology on $\mathbb{R}^{n}$; * (c) $P=\\{(x_{1},x_{2},\cdots,x_{n})\in\mathbb{R}^{n}\mid x_{i}\geq 0,(\forall)i=\overline{1,n}\\}$ is a cone in E. Indeed, Theorem 1.3 pp 2 of [5] leads us to $(a)$. It is obvious that $(TM\ 1)$ and $(TM\ 2)$ are satisfied. Now we consider $A_{n}\stackrel{{\scriptstyle\tau}}{{\rightarrow}}A$ and $x_{n}\stackrel{{\scriptstyle\mathcal{D}}}{{\rightarrow}}x$ as $n\rightarrow\infty$. Then $\\|A_{n}\cdot x_{n}-A\cdot x\\|_{\mathbb{R}^{n}}=\\|A_{n}\cdot x_{n}-A_{n}\cdot x+A_{n}\cdot x-A\cdot x\\|_{\mathbb{R}^{n}}\leq$ $\\|A_{n}\cdot(x_{n}-x)\\|_{\mathbb{R}^{n}}+\\|(A_{n}-A)\cdot x\\|_{\mathbb{R}^{n}}\leq N(A_{n})\\|x_{n}-x\\|_{\mathbb{R}^{n}}+N(A_{n}-A)\\|x\\|_{\mathbb{R}^{n}}\stackrel{{\scriptstyle n\rightarrow\infty}}{{\rightarrow}}0.$ Hence, $(TM\ 3)$ holds. Thus, we have obtained $(b)$. Finally, it easy to see that $P$ is a cone in $E$. In the following we always suppose that $E$ is a topological $R$-module, $P$ is a cone in $E$ with $intP\neq\emptyset$ and $\leq_{P}$ is a partial ordering with respect to $P$. ###### Definition 1.4. Let $X$ be a nonempty set. Suppose that a mapping $d:X\times X\rightarrow E$ satisfies: * $(d_{1})$ $0_{E}\leq_{P}d(x,y)$ for all $x,y\in X$ and $d(x,y)=0_{E}$ if and only if $x=y$; * $(d_{2})$ $d(x,y)=d(y,x)$, for all $x,y\in X$ ; * $(d_{3})$ $d(x,y)\leq_{P}d(x,z)+d(z,y)$, for all $x,y,z\in X$. Then $d$ is called a R-metric on $X$ and $(X,d)$ is called a R-metric space. ###### Example 1.2. Any cone metric space is a R-metric space. ###### Example 1.3. Let $R=M_{n\times n}(\mathbb{R})$ be the ring of all matrices with n rows and n columns with entries in $\mathbb{R}$, $E=\mathbb{R}^{n}$, $X=\mathbb{R}^{n}$ and $P=\\{(x_{1},x_{2},\cdots,x_{n})\in\mathbb{R}^{n}\mid x_{i}\geq 0,(\forall)i=\overline{1,n}\\}$ a cone in E. We define the partial order $\preceq$ on $M_{n\times n}(\mathbb{R})$ as follows $A\preceq B\ if\ and\ only\ if\ for\ each\ i,j=\overline{1,n}\ we\ have\ a_{ij}\leq b_{ij}.$ Then for all $A=(a_{ij})$, $a_{ij}>0$ we have that $d:\mathbb{R}^{n}\times\mathbb{R}^{n}\rightarrow\mathbb{R}^{n},$ $d(x,y)=(\sum\limits_{j=1}^{n}a_{1j}\|x_{j}-y_{j}\|,\cdots,\sum\limits_{j=1}^{n}a_{ij}\|x_{j}-y_{j}\|,\cdots,\sum\limits_{j=1}^{n}a_{nj}\|x_{j}-y_{j}\|)$ is a $R$-metric on $X$. Indeed, * $(d_{1})$ Since $\sum\limits_{j=1}^{n}a_{ij}\|x_{j}-y_{j}\|\geq 0$ for all $i=\overline{1,n}$, we have that $0\leq_{P}d(x,y)$ for all $x,y\in\mathbb{R}^{n}$. Also, $d(x,y)=0$ involve that $\sum\limits_{j=1}^{n}a_{ij}\|x_{j}-y_{j}\|=0$ which means that $x_{j}=y_{j}$ for all $j=\overline{1,n}$. It follows that $x=y$. * $(d_{2})$ It is obvious that $d(x,y)=d(y,x)$, for all $x,y\in\mathbb{R}^{n}$. * $(d_{3})$ Let be $x,y,z\in\mathbb{R}^{n}$. Since $\sum\limits_{j=1}^{n}a_{ij}\|x_{j}-y_{j}\|\leq\sum\limits_{j=1}^{n}a_{ij}\|x_{j}-z_{j}\|+\sum\limits_{j=1}^{n}a_{ij}\|z_{j}-y_{j}\|$, we have that $d(x,y)\leq_{P}d(x,z)+d(z,y)$, for all $x,y,z\in\mathbb{R}^{n}$. In the following, we shall write $x\prec y$ to indicate that $x\preceq y$ but $x\neq y$. ###### Remark 1.1. We have that: * (i) $intP+intP\subseteq intP$; * (ii) $\lambda\cdot intP\subseteq intP$, where $\lambda$ is a invertible element of the ring $R$ such that $0\prec\lambda$; * (iii) if $x\leq_{P}y$ and $0\preceq\alpha$, then $\alpha\cdot x\preceq\alpha\cdot y$. Proof: $(i)$ Let be $x\in intP+intP$. Then there exists $x_{1},x_{2}\in intP$ such that $x=x_{1}+x_{2}$. It follows that there exists $V_{1}$ neighborhood of $x_{1}$ and $V_{2}$ neighborhood of $x_{2}$ such that $x_{1}\in V_{1}\subset P\ and\ x_{2}\in V_{2}\subset P.$ Since for each $x_{0}\in E$, the mapping $x\rightarrow x+x_{0}$ is a homeomorphism of $E$ onto itself,we have that $V_{1}+V_{2}$ is a neighborhood of $x$ with respect to topology $\mathcal{T}$. Thus, $x\in intP$. $(ii)$ Let $0\prec\lambda$ be an invertible element of the ring $R$ and $x=\lambda\cdot c$, $c\in intP$. It follows that there exists a neighborhood $V$ of $c$ such that $c\in V\subset P$. Since the mapping $x\rightarrow\lambda x$ is a homeomorphism of $E$ onto itself, we have that $\lambda\cdot V$ is a neighborhood of $x$ with respect to the topology $\mathcal{T}$. Thus, $x\in intP$. $(iii)$ If $x\leq_{P}y$, then $y-x\in P$. It follows that for all $0\preceq\alpha$ we have that $\alpha\cdot(y-x)\in P$ i.e. $\alpha\cdot x\preceq\alpha\cdot y$. In the following, unless otherwise specified, we always suppose that there exists at least one a sequence $\\{\alpha_{n}\\}\subset R$ of invertible elements such that $0\prec\alpha_{n}$ and $\alpha_{n}\rightarrow 0$ as $n\rightarrow\infty$. ###### Remark 1.2. Let $E$ be a $R$-topological module and $P\subset E$ a cone. We have that: * (i) If $u\leq_{P}v$ and $v\ll w$, then $u\ll w$; * (ii) If $u\ll v$ and $v\leq_{P}w$, then $u\ll w$; * (iii) If $u\ll v$ and $v\ll w$, then $u\ll w$; * (iv) If $0\leq_{P}u\ll c$ for each $c\in IntP$, then $u=0$; * (v) If $a\leq_{P}b+c$ for each $c\in IntP$, then $a\leq_{P}b$; * (vi) If $0\ll c$, $0\leq_{P}a_{n}$ and $a_{n}\rightarrow 0$ then there exists $n_{0}\in\mathbb{N}$ such that $a_{n}\ll c$ for all $n\geq n_{0}$. Proof: * (i) We have to prove that $w-u\in intP$ if $v-u\in P$ and $w-v\in intP$. The condition $(TM_{1})$ implies that there exists a neighborhood $V$ of $0$ such that $w-v+V\subset P$. It follows that $w-u+V=(w-v)+V+(v-u)\subset P+P\subset P$. Since for each $x_{0}\in E$ the mapping $x\rightarrow x+x_{0}$ is a homeomorphism of $E$ onto itself we have that $w-u+V$ is a neighborhood of $w-u$ with respect to the topology $\mathcal{T}$. Thus, $w-u\in int\ P$. * (ii) Analogous with $(i)$. * (iii) We have to prove that $w-u\in intP$ if $v-u\in intP$ and $w-v\in intP$. Since we have $intP+intP\subset intP$, it easy to see that the above assertion is satisfied. * (iv) Let $\\{\alpha_{n}\\}_{n\in\mathbb{N}}\subset R$ be a sequence of invertible elements such that for each $n\in\mathbb{N}$ we have $0\prec\alpha_{n}$ and $\alpha_{n}\rightarrow 0$ as $n\rightarrow\infty$. Since for each invertible element $\lambda_{0}\in R$ the mapping $x\rightarrow\lambda_{0}\cdot x$ is a homeomorphism of $E$ onto itself, we have that if $V$ is a neighborhood of zero then $\lambda_{0}\cdot V$ is a neighborhood of zero. Hence, $\alpha_{n}\cdot c\in intP$ for each $n\in\mathbb{N}$. Then $\alpha_{n}\cdot c-u\in intP$. It follows that $\lim\limits_{n\rightarrow\infty}\alpha_{n}\cdot c-u=-u\in\overline{P}=P$. Thus, $u\in P\cap-P=\\{0\\}$. * (v) Analogous with $(iv)$. * (vi) Let be $0\ll c$, $0\leq_{P}a_{n}$ and $a_{n}\rightarrow 0$. Since for each invertible element $\lambda_{0}\in R$ the mapping $x\rightarrow\lambda_{0}\cdot x$ is a homeomorphism of $E$ onto itself, it follows that for all neighborhood $V$ of zero we have that $-V$ is a neighborhood of zero. Now, $0\ll c$ implies that there exists a neighborhood $U$ of zero such that $c+U\subset P$. Let $V=U\cap-U$ be a neighborhood of zero. Since $a_{n}$ converges to zero, it follows that there exists $n_{0}\in\mathbb{N}$ such that $a_{n}\in V$ for all $n\geq n_{0}$. Then we have that $c-a_{n}\in c+V\subset c+U\subset P$ for all $n\geq n_{0}$. Thus, $a_{n}\ll c$ for all $n\geq n_{0}$. ###### Definition 1.5. Let $\\{x_{n}\\}$ be a sequence in a R-metric space $(X,d)$ and $x\in X$. We say that: * (i) the sequence $\\{x_{n}\\}$ converges to $x$ and is denoted by $\lim\limits_{n\rightarrow\infty}x_{n}=x$ if for every $0\ll c$ there exists $N\in\mathbb{N}$ such that $d(x_{n},x)\ll c$, for all $n>N$; * (ii) the sequence $\\{x_{n}\\}$ is a Cauchy sequence if for every $c\in E$, $0\ll c$ there exists $N\in\mathbb{N}$ such that $d(x_{m},x_{n})\ll c$ for all $m,n>N$; The R-metric space $(X,d)$ is called complete if every Cauchy sequence is convergent. From the above remark we obtain ###### Remark 1.3. Let $(X,d)$ be a R-metric space and $\\{x_{n}\\}$ be a sequence in $X$. If $\\{x_{n}\\}$ converges to $x$ and $\\{x_{n}\\}$ converges to $y$, then $x=y$. Indeed, for all $0\ll c$ $d(x,y)\leq_{P}d(x,x_{n})+d(x_{n},y)\ll 2\cdot c.$ Hence, $d(x,y)=0$ i.e. $x=y$. ## 2 Common fixed points theorems In this section we obtain several coincidence and common fixed point theorems for mappings defined on a $R$-metric space. ###### Definition 2.1. (see [1]) Let $f$ and $g$ be self maps of a set $X$. If $w=fx=gx$ for some $x\in X$, then $x$ is called a coincidence point of $f$ and $g$, and $w$ is called a point of coincidence of $f$ and $g$. Jungck [2], defined a pair of self mappings to be weakly compatible if they commute at their coincidence points. ###### Proposition 2.1. (see [1]) Let $f$ and $g$ be weakly compatible self maps of a set $X$. If $f$ and $g$ have a unique point of coincidence $w=fx=gx$, then $w$ is the unique common fixed point of $f$ and $g$. Let $\mathcal{K}$ be the set of all $k\in R$, $0\preceq k$ which have the property that there exists a unique $S\in R$ such that $S=\lim\limits_{n\rightarrow\infty}(1+k+\cdots+k^{n})$. ###### Example 2.1. Let be $A\in\mathcal{M}_{n\times n}(\mathbb{R}_{+})$ such that $\rho(A)<1$. Then $A\in\mathcal{K}$. ###### Theorem 2.1. Let $(X,d)$ be a R-metric space and suppose that the mappings $f,g:X\rightarrow X$ satisfy: * (i) the range of $g$ contains the range of $f$ and $g(X)$ is a complete subspace of $X$; * (ii) there exists $k\in\mathcal{K}$ such that $d(fx,fy)\leq_{P}k\cdot d(gx,gy)$ for all $x,y\in X$. Then $f$ and $g$ have a unique point of coincidence in $X$. Moreover, if $f$ and $g$ are weakly compatible then, $f$ and $g$ have a unique common fixed point. Proof: Let $x_{0}$ be an arbitrary point in $X$. We choose a point $x_{1}\in X$ such that $f(x_{0})=g(x_{1}).$ Continuing this process, having chosen $x_{n}\in X$, we obtain $x_{n+1}\in X$ such that $f(x_{n})=g(x_{n+1})$. Then $d(gx_{n+1},gx_{n})=d(fx_{n},fx_{n-1})\leq_{P}k\cdot d(gx_{n},gx_{n-1})\leq_{P}$ $\leq_{P}k^{2}\cdot d(gx_{n-1},gx_{n-2})\leq_{P}\cdots\leq_{P}k^{n}\cdot d(gx_{1},gx_{0}).$ We denote $S_{n}=1+k+\cdots+k^{n}$and we get that $d(gx_{n},gx_{n+p})\leq_{P}d(gx_{n},gx_{n+1})+d(gx_{n+1},gx_{n+2})+\cdots+d(gx_{n+p-1},gx_{n+p})\leq_{P}$ $\leq_{P}(k^{n}+k^{n+1}+\cdots+k^{n+p-1})\cdot d(gx_{1},gx_{0})=(S_{n+p-1}-S_{n-1})\cdot d(gx_{1},gx_{0}),$ for all $p\geq 1$. Thus, via Remark 1.2 $(vi)$, we obtain that $gx_{n}$ is a Cauchy sequence. Since $g(X)$ is complete, there exists $q\in g(X)$ such that $gx_{n}\rightarrow q$ as $n\rightarrow\infty$. Consequently, we can find $p\in X$ such that $g(p)=q$. Further, for each $0\ll c$ there exists $n_{0}\in\mathbb{N}$ such that for all $n\geq n_{0}$ $d(gx_{n},fp)=d(fx_{n-1},f(p))\leq_{P}k\cdot d(gx_{n-1},gp)\ll c.$ It follows that $gx_{n}\rightarrow fp$ as $n\rightarrow\infty$. The uniqueness of the limit implies that $fp=gp=q$. Next we show that $f$ and $g$ have a unique point of coincidence. For this, we assume that there exists another point $p_{1}\in X$ such that $fp_{1}=gp_{1}$. We have $d(gp_{1},gp)=d(fp_{1},fp)\leq_{P}k\cdot d(gp_{1},gp)=k\cdot d(fp_{1},fp)\leq_{P}k^{2}\cdot d(gp_{1},gp)\leq_{P}\cdots\leq_{P}k^{n}\cdot d(gp_{1},gp).$ Let be $0\ll c$. Since $k^{n}\rightarrow 0$ as $n\rightarrow\infty$ it follows that there exists $n_{0}\in\mathbb{N}$ such that $k^{n}\cdot d(gp_{1},gp)\ll c$ for all $n\geq n_{0}$. Then $d(gp_{1},gp)\ll c$ for each $0\ll c$. Thus $d(gp_{1},gp)=0$ i.e. $gp_{1}=gp$. From Proposition 2.1 $f$ and $g$ have a unique common fixed point. ###### Remark 2.1. The above theorem generalizes Theorem 2.1 of Abbas and Jungck [1], which itself is a generalization of Banach fixed point theorem. ###### Corollary 2.1. Let $(X,d)$ be a complete $R$-metric space and we suppose that the mapping $f:X\rightarrow X$ satisfies: * (i) there exists $k\in\mathcal{K}$ such that $d(fx,fy)\leq_{P}k\cdot d(x,y)$ for all $x,y\in X$. Then $f$ has in $X$ a unique fixed point point. Proof: The proof uses Theorem 3.1 by replacing $g$ with the identity mapping. From the above corollary using Example 1.1, we obtain the Perov fixed point theorem (see [3]) ###### Corollary 2.2. Let $(X,d)$ be a complete $\mathcal{M}_{n\times n}(\mathbb{R}_{+})-$ metric space and $E=\mathbb{R}^{n}$ and we suppose the mapping $f:X\rightarrow X$ satisfies: * (i) there exists $A\in\mathcal{M}_{n\times n}(\mathbb{R}_{+})$ with $\rho(A)<1$ such that $d(fx,fy)\leq_{P}A\cdot d(x,y),$ for all $x,y\in X$. Then $f$ has in $X$ a unique fixed point point. ## 3 Comparison function ###### Definition 3.1. Let P be a cone in a topological R-module E. A function $\varphi:P\rightarrow P$ is called a comparison function if * (i) $\varphi(0)=0$ and $\varphi(t)<_{P}t$ for all $t\in P-\\{0\\}$; * (ii) $t_{1}\leq_{P}t_{2}$ implies $\varphi(t_{1})\leq_{P}\varphi(t_{2})$; * (iii) $t\in intP$ implies $t-\varphi(t)\in intP$; * (iv) if $t\in P-\\{0\\}$ and $0\ll c$, then there exists $n_{0}\in\mathbb{N}$ such that $\varphi^{n}(t)\ll c$ for each $n\geq n_{0}$. ###### Example 3.1. Let P be a cone in a topological R-module E and $\lambda\in\mathcal{K}$ such that $0\prec 1-\lambda$. Then $\varphi:P\rightarrow P$, defined by $\varphi(t)=\lambda\cdot t$ is a comparison function. Indeed, $(i)$ It is obvious that $\varphi(0)=0$ and $\varphi(t)<_{P}t$ for all $t\in P-\\{0\\}$. $(ii)$ if $t_{1}\leq_{P}t_{2}$ and $\lambda\in\mathcal{K}$ then $\lambda\cdot(t_{2}-t_{1})\in P$. Thus $\varphi(t_{1})\leq_{P}\varphi(t_{2})$. $(iii)$ We remark that if $\lambda\in\mathcal{K}$, then $1-\lambda$ is an invertible element of the ring $R$. Now, let be $t\in intP$. Then $(1-\lambda)\cdot t\in(1-\lambda)intP\subset intP$. $(iv)$ Let be $t\in P-\\{0\\}$ and $0\ll c$. Then $\varphi^{n}(t)=\lambda^{n}\cdot t\stackrel{{\scriptstyle n\rightarrow\infty}}{{\rightarrow}}0.$ We obtain, via Remark 1.2, that there exists $n_{0}\in\mathbb{N}$ such that $\varphi^{n}(t)\ll c$ for each $n\geq n_{0}$. Let $(X,d)$ be a $R$-metric space and let $\varphi:K\rightarrow K$ be a comparison function. For a pair $(f,g)$ of self-mappings on $X$ consider the following condition * (C) for arbitrary $x,y\in X$ there exists $u\in\\{d(gx,gy),d(gx,fx),d(gy,fy)\\}$ such that $d(fx,fy)\leq_{P}\varphi(u)$. ###### Theorem 3.1. Let $(X,d)$ be a R-metric space and let $f,g:X\rightarrow X$ such that * (i) the pair $(f,g)$ satisfies the condition (C) for some comparison function $\varphi$; * (ii) $f(X)\subset g(X)$ and f(X) or g(X) is a complete subspace of X. Then f and g have a unique point of coincidence in X. Moreover if $f$ and $g$ are weakly compatible, then $f$ and $g$ have a unique common fixed point. Proof: Let $x_{0}$ be an arbitrary point in $X$. We choose a point $x_{1}\in X$ such that $fx_{0}=gx_{1}.$ Continuing this process, having chosen $x_{n}\in X$, we obtain $x_{n+1}\in X$ such that $fx_{n}=gx_{n+1}$. $\ulcorner$ We shall prove that the sequence $\\{y_{n}\\}$, where $y_{n}=fx_{n}=gx_{n+1}$( the so-called Jungck sequence ) is a Cauchy sequence in $R$-metric space $(X,d)$. If $y_{N}=y_{N+1}$ for some $N\in\mathbb{N}$ then $y_{m}=y_{N}$ for each $m>N$ and the conclusion follows. Indeed, we prove by induction arguments that (3.1) $y_{N+k}=y_{N+k+1},(\forall)k\in\mathbb{N}.$ For $k=0$ we have $y_{N}=y_{N+1}$. Let (3.1) hold for all $k=\overline{0,i}$. Then $d(y_{N+i+1},y_{N+i+2})=d(fx_{N+i},fx_{N+i+1})\leq_{P}\varphi(u),$ where $u\in\\{d(gx_{N+i},gx_{N+i+1}),d(gx_{N+i},fx_{N+i}),d(gx_{N+i+1},fx_{N+i+1})\\}=$ $\\{d(y_{N+i-1},y_{N+i}),d(y_{N+i-1},y_{N+i}),d(y_{N+i},y_{N+i+1})\\}=\\{0\\}.$ Hence, $d(y_{N+i+1},y_{N+i+2})\leq_{P}\varphi(u)=0$ i.e. $y_{N+i+1}=y_{N+i+2}$. Q.E.D Suppose that $y_{n}\neq y_{n+1}$ for each $n\in\mathbb{N}$. The condition (C) implies that $d(y_{n},y_{n+1})=d(fx_{n},fx_{n+1})\leq_{P}\varphi(u),$ where $u\in\\{d(gx_{n},gx_{n+1}),d(fx_{n},gx_{n}),d(fx_{n+1},gx_{n+1})\\}=\\{d(y_{n-1},y_{n}),d(y_{n},y_{n-1}),d(y_{n+1},y_{n})\\}.$ The case $u=d(y_{n+1},y_{n})$ is impossible, since this would imply $d(y_{n+1},y_{n})\leq_{P}\varphi(d(y_{n+1},y_{n}))<_{P}d(y_{n+1},y_{n}).$ Thus, $u=d(y_{n},y_{n-1})$ and $d(y_{n+1},y_{n})\leq_{P}\varphi(d(y_{n},y_{n-1}))\leq_{P}\cdots\leq_{P}\varphi^{n}(d(y_{1},y_{0})).$ Using Remark 1.2 (i) and property $(iv)$ of the comparison function we obtain that for all $0\ll\varepsilon$ there exists $n_{0}\in\mathbb{N}$ such that (3.2) $d(y_{n},y_{n+1})\ll\varepsilon,\ (\forall)n\geq n_{0}.$ Now, let be $0\ll c$. Then, using property $(iii)$ of the comparison function, we get that (3.3) $d(y_{n},y_{n+1})\ll c-\varphi(c),\ (\forall)n\geq n_{0}.$ Let us fix now $n\geq n_{0}$ and let us prove that (3.4) $d(y_{n},y_{k+1})\ll c,\ (\forall)k\geq n.$ Indeed, for $k=n$ we have $d(y_{n},y_{n+1})\ll c-\varphi(c)\leq_{P}c.$ Hence, $d(y_{n},y_{n+1})\ll c.$ Let (3.4 ) hold for some $k\geq n$. Then we have $d(y_{n},y_{k+2})\leq_{P}d(y_{n},y_{n+1})+d(y_{n+1},y_{k+2})\ll$ $c-\varphi(c)+d(fx_{n+1},fx_{k+2})\leq_{P}c-\varphi(c)+\varphi(u),$ where $u\in\\{d(gx_{n+1},gx_{k+2}),d(gx_{n+1},fx_{n+1}),d(gx_{k+2},fx_{k+2})\\}.$ Consider now the following three possible cases: * Case 1: $u=d(gx_{n+1},gx_{k+2})$. Then $\varphi(u)=\varphi(d(gx_{n+1},gx_{k+2}))=\varphi(d(y_{n},y_{k+1}))\leq_{P}\varphi(c).$ From the above relation it follows that, $d(y_{n},y_{k+2})\ll c-\varphi(c)+\varphi(u)\leq_{P}c-\varphi(c)+\varphi(c)=c.$ Hence, $d(y_{n},y_{k+2})\ll c.$ * Case 2: $u=d(gx_{n+1},fx_{n+1})=d(y_{n},y_{n+1})$. Then $\varphi(u)\leq_{P}\varphi(d(y_{n},y_{n+1}))\leq_{P}\varphi(c-\varphi(c))\leq_{P}\varphi(c).$ From the above relation we get that, $d(y_{n},y_{k+2})\ll c-\varphi(c)+\varphi(u)\leq_{P}c-\varphi(c)+\varphi(c)=c.$ Hence, $d(y_{n},y_{k+2})\ll c.$ * Case 3: $u=d(gx_{k+2},fx_{k+2})$. Then $\varphi(u)=\varphi(d(gx_{k+2},fx_{k+2}))=\varphi(d(y_{k+1},y_{k+2}))\leq_{P}\varphi(c-\varphi(c))\leq_{P}\varphi(c).$ From the above relation we get that, $d(y_{n},y_{k+2})\ll c-\varphi(c)+\varphi(u)\leq_{P}c-\varphi(c)+\varphi(c)=c.$ Hence, $d(y_{n},y_{k+2})\ll c.$ So, it has been proved by induction that $\\{y_{n}\\}$ is a Cauchy sequence.$\lrcorner$ Since, by assumption, $f(X)$ or $g(X)$ is a complete subspace of $X$, we conclude that there exists $q\in g(X)$ such that $y_{n}=fx_{n}=gx_{n+1}\rightarrow q$ as $n\rightarrow\infty$ and there exists $p\in X$ such that $q=gp$. $\ulcorner$ We claim that $q=fp$. Indeed, if we suppose that $d(fp,q)\neq 0$, then we have $d(fp,q)\leq_{P}d(fp,fx_{n})+d(fx_{n},q)\leq_{P}\varphi(u)+d(y_{n},q),$ where $u\in\\{d(gp,gx_{n}),d(gp,fp),d(gx_{n},fx_{n})\\}$ Let $0\ll c$. At least one of the following three cases holds for infinitely many $n\in\mathbb{N}$: * Case 1: $u=d(gp,gx_{n})$. Then, there exists $n_{0}(c)\in\mathbb{N}$ such that for all $n\geq n_{0}(c)$ $d(fp,q)\leq_{P}\varphi(d(gp,gx_{n}))+d(y_{n},q)<_{P}d(q,y_{n-1})+d(y_{n},q)\ll 2\cdot c.$ It follows that $d(fp,q)=0$, which is a contradiction. * Case 2: $u=d(gp,fp)=d(fp,q)$ . Then we have $d(fp,q)\leq_{P}\varphi(d(q,fp))+d(y_{n},q)\ll\varphi(d(q,fp))+c.$ Thus, $d(fp,q)\leq_{P}\varphi(d(q,fp))$. So, by using of the properties $(ii)$ and $(iv)$ of the comparison function, we obtain that there exists $n_{0}\in\mathbb{N}$ such that $d(fp,q)\leq_{P}\varphi^{n}(d(fp,q))\ll c$ i.e. $d(fp,q)=0$, which is a contradiction. * Case 3: $u=d(gx_{n},fx_{n})=d(y_{n-1},y_{n})$. Then, there exists $n_{0}(c)\in\mathbb{N}$ such that for all $n\geq n_{0}(c)$ we have $d(fp,q)\leq_{P}\varphi(d(y_{n-1},y_{n}))+d(y_{n},q)<_{P}d(y_{n-1},y_{n})+d(y_{n},q)\ll 2\cdot c,$ i.e. $d(fp,q)=0$, which is a contradiction. It follows that $fp=gp=q$ i.e. $p$ is a coincidence point of the pair $(f,g)$ and $q$ is a point of coincidence.$\lrcorner$ $\ulcorner$ Next we show that $f$ and $g$ have a unique point of coincidence. For this we assume that there exists another point $p_{1}\in X$ such that $fp_{1}=gp_{1}$. If we suppose that $d(fp_{1},fp)\neq 0$ we get that $d(fp_{1},fp)\leq_{P}\varphi(u)$, where $u\in\\{d(gp_{1},gp),d(gp_{1},fp_{1}),d(gp,fp)\\}=\\{d(gp_{1},gp),0\\}.$ In both possible cases a contradiction easily follows : $d(fp_{1},fp)\leq_{P}\varphi(d(fp_{1},fp))<_{P}d(fp_{1},fp)$ or $d(fp_{1},fp)\leq_{P}\varphi(0)=0$. We conclude that the mappings $f$ and $g$ have a unique point of coincidence. From Proposition 2.1 $f$ and $g$ have a unique common fixed point.$\lrcorner$ ## References * [1] M. Abbas, G. Jungck,Common fixed point results for noncommuting mappings without continuity in cone metric spaces, J. Math.Anal.Appl. 341 (2008) * [2] G. Jungck, Common fixed points for noncontinuous nonself maps on non-metric spaces, Far East J. Math. Sci.(FJMS) 4(1996) 199-215. * [3] A. I. Perov, On the Cauchy problem for a system of ordinary differential equations, Pviblizhen. Met. Reshen. Differ. Uvavn., vol. 2, pp. 115 134, 1964. * [4] Stuart A. Steinberg, Lattice-ordered rings and modules, Springer Science + Business Media, LLC 2010, DOI 10.1007/978-1-4419-1721-8-1. * [5] Seth Warner, Topological rings, North-Holland Math. Studies 178, 1993. Department of Mathematics, Faculty of Science, University ”Lucian Blaga” of Sibiu, Dr. Ion Ratiu 5-7, Sibiu, 550012, Romania E-mail: marian.olaru@ulbsibiu.ro	arxiv-papers	2011-11-19T07:46:08	2024-09-04T02:49:24.485558	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Ion Olaru", "submitter": "Ion Olaru", "url": "https://arxiv.org/abs/1111.4537" }
1111.4579	# Canalization of the evolutionary trajectory of the human influenza virus Trevor Bedford Department of Ecology and Evolutionary Biology, University of Michigan, Ann Arbor, MI, USA. Howard Hughes Medical Institute, University of Michigan, Ann Arbor, MI, USA. Andrew Rambaut Institute of Evolutionary Biology, University of Edinburgh, Edinburgh, UK. Fogarty International Center, National Institutes of Health, Bethesda, MD, USA. Mercedes Pascual Department of Ecology and Evolutionary Biology, University of Michigan, Ann Arbor, MI, USA. Howard Hughes Medical Institute, University of Michigan, Ann Arbor, MI, USA. Abstract Since its emergence in 1968, influenza A (H3N2) has evolved extensively in genotype and antigenic phenotype. Antigenic evolution occurs in the context of a two-dimensional ‘antigenic map’, while genetic evolution shows a characteristic ladder-like genealogical tree. Here, we use a large-scale individual-based model to show that evolution in a Euclidean antigenic space provides a remarkable correspondence between model behavior and the epidemiological, antigenic, genealogical and geographic patterns observed in influenza virus. We find that evolution away from existing human immunity results in rapid population turnover in the influenza virus and that this population turnover occurs primarily along a single antigenic axis. Thus, selective dynamics induce a canalized evolutionary trajectory, in which the evolutionary fate of the influenza population is surprisingly repeatable and hence, in theory, predictable. ## Author Summary Each year, hundreds of millions become sick with the influenza virus and hundreds of thousands die from the infection. After recovery from the flu, a person gains permanent immunity specific to the infecting strain. However, owing to its RNA makeup, mutations occur rapidly to the virus genome. Some of these mutations change the shape of proteins visible to the human immune system, and thus alter the virus’s antigenic phenotype. These mutations allow the virus to re-infect those who have previously recovered from an earlier strain, and thus will quickly spread through the virus population. It is because of influenza’s rapid antigenic evolution that the flu vaccine needs frequent updating. However, despite strong pressure to evolve away from human immunity and to diversify in antigenic phenotype, influenza, and especially influenza A (H3N2), shows paradoxically limited genetic and antigenic diversity present at any one time. Here, we propose a simple model of influenza that displays rapid evolution but low standing diversity and simultaneously accounts for the epidemiological, genetic, antigenic and geographical patterns displayed by the virus. We find that in evolving directly away from existing human immunity, the virus severely limits its future evolutionary potential. ## Introduction Epidemic influenza is responsible for between 250,000 and 500,000 global deaths annually, with influenza A (and in particular, A/H3N2) having caused the bulk of human mortality and morbidity [1]. Influenza A (H3N2) has continually circulated within the human population since its introduction in 1968, exhibiting recurrent seasonal epidemics in temperate regions and less periodic transmission in the tropics. During this time, H3N2 influenza has continually evolved both genetically and antigenically. Most antigenic drift is thought to be driven by changes to epitopes in the hemagglutinin (HA) protein [2]. Phylogenetic analysis of the genetic relationships among HA sequences has revealed a distinctive genealogical tree showing a single predominant trunk lineage and side branches that persist for only 1–5 years before going extinct [3]. This tree shape is indicative of serial replacement of strains over time; H3N2 influenza shows rapid evolution, but low standing genetic diversity. This observation has remained puzzling from an epidemiological standpoint. Antigenic evolution occurs rapidly and strong diversifying selection exists to escape from human immunity; why then do we see serial replacement of strains rather than continual genetic and antigenic diversification? Indeed, simple epidemiological models show explosive diversity of genotype and phenotype over time [4, 5]. Previous work has sought model-based explanations of the limited diversity of influenza, relying on short-lived strain-transcending immunity [4, 5], complex genotype-to-phenotype maps [6] or a limited repertoire of antigenic phenotypes [7]. Experimental characterization of antigenic phenotype is possible through the hemagglutination inhibition (HI) assay, which measures the cross-reactivity of HA from one virus strain to serum raised against another strain [8]. The results of many HI assays can be combined to yield a two-dimensional map, representing antigenic similarity and distance between strains as an easily visualized and quantified measure [9]. The path traced across this map by influenza A (H3N2) from 1968 until present is largely linear, showing serial replacement of one strain by another; there are no major bifurcations of antigenic phenotype [9]. Herein, we seek to simultaneously model the genetic and antigenic evolution of the influenza virus. We represent antigenic phenotypes as points in a $N$-dimensional Euclidean space. Based on the finding that a two-dimensional map adequately explains observed antigenic distance between strains [9], we begin with antigenic phenotypes as points on a plane, but relax this assumption later on in the analysis. After exposure to a virus, a host’s risk of infection is proportional to the Euclidean distance between the infecting phenotype and the closest phenotype in the host’s immune history. Mutations perturb antigenic phenotype, moving phenotype in a random radial direction and for a randomly distributed distance. We implemented this geometrical model in a large-scale individual-based simulation intended to directly model the antigenic map and genealogical tree of the global influenza population. The simulation includes multiple host populations with different seasonal forcing, hosts with complete immune histories of infection, and viruses with antigenic phenotypes. As the simulation proceeds, infections are tracked and a complete genealogy connecting virus samples is constructed. Results shown here are for a single representative simulation of 40 years of virus evolution in a population of 90 million hosts. ## Results The virus persists over the course of the 40-year simulation, infecting a significant fraction of the host population through annual winter epidemics in temperate regions and through less periodic epidemics in the tropics (Figure 1A). Across replicate simulations, we observe average yearly attack rates of 6.8% in temperate regions and rates of 7.1% in the tropics, comparable with estimated attack rates of influenza A (H3N2) of 3–8% per year [10, 11]. Over the course of the simulation, the virus population evolves in antigenic phenotype exhibiting, at any point, a handful of highly abundant phenotypes sampled repeatedly and a large number of phenotypes appearing at low abundance (Figure 1B). By including measurement noise on antigenic locations, we approximate an experimental antigenic map of H3N2 influenza (Figure 1D). The appearance of clusters in the antigenic map comes from the regular spacing of high abundance phenotypes combined with measurement noise. Over time, clusters of antigenically similar strains are replaced by novel clusters of more advanced strains (Figure S1A). Across replicate simulations, clusters persist for an average of 5.0 years measured as the time it takes for a new cluster to reach 10% frequency, peak and decline to 10% frequency. The transition between clusters occurs quickly, taking an average of 1.8 years. Remarkably, although antigenic phenotype is free to mutate in any direction in the two-dimensional space, selection pressures force the virus population to move in nearly a straight line in antigenic space (Figure 1B). Across replicate simulations, 94% of the variance of antigenic phenotype can be explained by a single dimension of variation. This mirrors the empirical results showing a largely linear antigenic map for H3N2 influenza isolates from 1968 to 2003 [9]. Because of the primarily one-dimensional movement, antigenic distance from the original phenotype increases nearly linearly with time (Figure S1B). Antigenic evolution occurs in a punctuated fashion; periods of relative stasis are interspersed with more rapid antigenic change (Figure S1B). Antigenic and epidemiological dynamics show a fundamental linkage, so that large jumps of antigenic phenotype result in increased rates of infection (Figure 1, Figure S2). In general, evolution via many smaller mutations results in a smoother antigenic map and less variation in yearly epidemics (Figure S3), while evolution via rare mutations of large effect exhibits a more clustered antigenic map and wider variation in seasonal incidence (Figure S4). The genealogical tree connecting the evolving virus population appears characteristically sparse with pronounced trunk and short-lived side branches (Figure 1C). This tree shape is reflected in low levels of standing diversity; across replicates, an average of 5.68 years of evolution separate two randomly sampled viruses in the population. This level of diversity matches what is observed in phylogenies of influenza A (H3N2) [12]. A spindly genealogical tree is indicative of population turnover, wherein novel antigenic phenotypes continually replace more primitive ‘spent’ phenotypes, purging their genealogical diversity. In general, natural selection reduces effective population size and genealogical diversity [13]. Here, by comparing mutations occurring on trunk branches vs. mutations occurring on side branches, we find evidence for pervasive positive selection for antigenic change (Table 1). Trunk mutations tend to push antigenic phenotype forward along the line of primary antigenic variation (Figure S5). We find a roughly linear relationship between the antigenic effect of a mutation and the likelihood that this mutation becomes incorporated into the trunk (Figure S6). Additionally, we find that trunk mutations occur at strikingly regular intervals, with less variation of waiting times than expected under a simple random process (Figure S7). There is a relative scarcity of mutation events occurring in intervals under 1 year and a relative excess of a mutation events occurring in 2–3 year intervals (Figure S7). The genealogical tree also contains detailed information on the history of migration between regions. We find that, consistent with empirical estimates [14, 15], the trunk resides primarily within the tropics, where seasonal dynamics are less prevalent (Figure 2A). Across replicate simulations, we observe 72% of the trunk’s history within the tropics and 28% within temperate regions. With symmetric host contact rates and equal host population sizes, and without seasonal forcing, we would expect trunk proportions of one third for each region. We calculated rates of migration based on observed event counts across replicate simulations, separating region-specific rates on side branches from region-specific rates on trunk branches. We find that migration patterns on side branches are close to symmetric, with similar rates between all regions, while migration patterns on trunk branches are highly asymmetric, with high rates of movement between temperate regions and from temperate regions into the tropics (Figure 2B). Extrapolating from these rates, we arrive at an expected stationary distribution of 76% tropics and 24% temperate regions, in line with the observed residency patterns of the trunk. It may at first seem counter-intuitive to see higher rates of movement from the temperate regions into the tropics along trunk branches, but it makes sense when thought of in terms of conditional probability. Only those lineages that migrate into the tropics or those lineages which rapidly migrate between the north and south have a chance at becoming the trunk lineage, while lineages that remain within the temperate regions are doomed to extinction. These findings suggest that persistence and migration are fundamentally connected and have important implications for future phylogeographic analyses. Russell et al. [14] emphasize a source-sink model of movement of the HA protein of influenza A (H3N2) based on their finding of a trunk lineage residing within China and the Southeast Asian tropics. Whereas, Bedford et al. [15] emphasize a global metapopulation model based on phylogenetic inference of migration rates across the entire tree. Our results suggest that both scenarios are simultaneously possible; side branches may be highly volatile moving rapidly and symmetrically between regions, while the trunk lineage may be more stable remaining within a region (or within a highly connected network of regions) that has more persistent transmission. In light of these results, we suggest that future work on the phylogeography of influenza take into account trunk vs. side branch differences in migration patterns. ## Discussion ### Correspondence between model and data Although multiple epidemiological/evolutionary mechanisms have been proposed to explain the restricted genetic diversity and rapid population turnover of influenza A (H3N2) [4, 5, 6, 7], our results show that a simple model coupling antigenic and genealogical evolution exhibits broad explanatory power. We find a strong correspondence between the antigenic and genealogical patterns generated by our model (Figure 1) and patterns of genetic and antigenic evolution exhibited by influenza A (H3N2) [3, 9]. Our model suggests that punctuated antigenic evolution need only be explained by a lack of mutational opportunity and predicts that more detailed classification of influenza strains will support a relatively small number of predominant phenotypes (Figure 1B). We suggest that a large proportion of intra-cluster variation in the observed antigenic map is due to experimental noise, rather than each strain possessing a unique antigenic location. Additionally, our model accurately predicts the contrasting dynamics of other types/subtypes of influenza. We find that lowering mutation size/effect or lowering intrinsic $R_{0}$ results in decreased incidence, slower antigenic movement and greater genealogical diversity, all distinguishing characteristics of H1N1 influenza and influenza B (Figure S8, Figure S9). In our model, when antigenic phenotype remains static, there may be multiple consecutive seasons without appreciable incidence (Figure 1A), a pattern apparently absent from H3N2 influenza [16]. We suggest that any model exhibiting punctuated evolution broadly consistent with the punctuated change seen in the experimental antigenic map will show similar patterns of incidence. We can ‘fix’ the incidence patterns, but at the cost of too smooth an antigenic map (Figure S3). Evolutionary patterns of the neuraminidase (NA) protein may provide an explanation. Epitopes in the HA and NA proteins are jointly responsible for determining antigenicity [2], and it is now clear that levels of adaptive evolution are similar between HA and NA [17]. Thus, changes in NA may be driving incidence patterns as well, resulting in an observed timeseries of incidence partially divorced from the antigenic map of HA. It remains a central question as to the extent that short-lived strain- transcending immunity is responsible for influenza’s limited diversity and spindly genealogical tree [4, 5]. Our findings suggest a possible resolution. Although lacking short-lived immunity, our model shows a detailed correspondence to both the antigenic map and genealogical tree of H3N2 influenza. If an antigenic map were to show a deep bifurcation, where two viral lineages move in different antigenic directions, then we would expect the same bifurcation to be evident in the genealogical tree. Short-lived strain-transcending immunity provides a mechanism by which lineages may diverge in antigenic phenotype, but still show epidemiological interference. This mechanism would explain a situation where bifurcations emerge in the antigenic map, but competition results in the extinction of divergent antigenic lineages. The empirical antigenic map [9] suggests that this is not the case; one cluster leads to another cluster in orderly succession and there is never competition between antigenically distant clusters. This supports the hypothesis that antigenic evolution is primarily limited by a lack of mutational availability. This is not to say that short-lived strain- transcending immunity is not present; observed interference between subtypes [4, 18] and evolution at CTL epitopes [19] provides substantial evidence for its existence. Instead, we suggest that short-lived strain-transcending immunity does not automatically generate antigenic maps and genealogical trees consistent with empirical evidence. ### Linear antigenic movement It would seem possible for one viral lineage to move in one antigenic direction, while another lineage moves tangentially, eventually resulting in two non-interacting viral lineages. Instead, we find that only movement in a single antigenic direction is favored. The origins of this pattern can be seen in the interaction between virus evolution and host immunity (Figure 3). As the virus population evolves forward it leaves a wake of immunity in the host population, and evolution away from this immunity results in the canalization of the antigenic phenotype; mutations that continue along the line of primary antigenic variation will show a transmission advantage compared to more tangential mutations. Following the work of Smith et al. [9], it remained an open question of why a two-dimensional map should explain the antigenic variation of H3N2 influenza. Although the authors astutely speculated that “there is a selective advantage for clusters that move away linearly from previous clusters as they most effectively escape existing population-level immunity, and this is a plausible explanation for the somewhat linear antigenic evolution in regions of the antigenic map.” This hypothesis remained to be tested. Here, we show from a simple model of epidemiology and evolution that a linear trajectory of antigenic evolution dynamically emerges due to basic selective pressures. This result simultaneously explains the linear pattern of antigenic drift [9] and the characteristically spindly genealogical tree [3] exhibited by influenza A (H3N2). To consider to what extent these results were contingent on the dimensionality of the underlying antigenic model, we further implemented our model in a 10-dimensional antigenic space. Here, mutations occur as 10-spheres, but the distance moved by a mutation is the same as in the previous two-dimensional formulation. We arrive at nearly the same results with this model; principal components analysis shows that the first and second dimensions of variation account for 87% and 7%, respectively, of the total variance (Figure S10). Thus, our model predicts that future work probing mutational effects will support an underlying high-dimensional antigenic space, even though a two- dimensional map is sufficient to explain observed antigenic relationships among strains. ### Winding back the tape It seems clear that, in our model, selection reduces the degrees of freedom of antigenic evolution. In light of this, we wanted to examine the degree of stochasticity in replicated evolutionary trajectories, and thereby test what happens when we “wind back the tape” [20] on the evolution of the virus. We ran 100 replicate simulations, each starting from the endpoint of the original 40-year simulation (Figure 4, Figure 5). Initially, we find a great detail of repeatability; during the first year of evolution, every replicate virus population undergoes a similar antigenic transition (Figure 4), resulting in a repeatable peak in northern hemisphere incidence (Figure 5). After three years, repeatability has mostly disappeared, with antigenic phenotype and incidence appearing highly variable across replicates (Figure 4, Figure 5). The 1–2 year timescale of repeatability can be explained by the presence of standing antigenic variation. In the initial virus population, there are several novel antigenic variants present at low frequency (Figure 3), one of which, without fail, comes to predominate the virus population. We see that the initial evolutionary trajectory, during which time standing variation plays out, is highly repeatable, and thus predictable given enough information and the right methods of analysis. However, prediction of longer- term evolutionary scenarios will necessarily be difficult or impossible except in a vague sense. Through careful surveillance efforts and genetic and antigenic characterization of influenza strains, the World Health Organization makes twice-yearly vaccine strain recommendations [21]. It should be possible to combine these sorts of modeling approaches with surveillance data to gauge the likelihood that a sampled variant will spread through the population. Recent work on empirical fitness landscapes has shown that natural selection follows few mutational paths [22]. The spindly genealogical tree and the almost linear serial replacement of influenza strains has remained a puzzling phenomenon. We suggest that the evolutionary and epidemiological dynamics displayed by the influenza virus may simply be explained as an outgrowth of selection to avoid host immunity. Natural selection can only ‘see’ one step ahead, and so sacrifices long-term gains for short-term advantages. The result is a canalized evolutionary trajectory lacking antigenic diversification. ## Materials and Methods ### Transmission model To characterize the joint epidemiological, genealogical, antigenic and spatial patterns of influenza, we implemented a large-scale individual-based model. This model consists of daily time steps, in which the states of hosts and viruses are updated. Hosts may be born, may die, may contact other hosts allowing viral transmission, or may recover from infection. Viruses may mutate in antigenic phenotype. Each simulation ran for 40 years of model time. Hosts in this model are divided between three regions: North, South and Tropics. There are 30 million hosts within each of the three regions, giving $N=9\times 10^{7}$ hosts. Host population size remains fixed at this number, but vital dynamics cause births and deaths of hosts at a rate of $1/30$ years $=9.1\times 10^{-5}$ per host per day. Within each region, transmission proceeds through mass-action with contacts between hosts occurring at a rate of $\beta=0.36$ per host per day. Regional transmission rates in temperate regions vary according to sinusoidal seasonal forcing with amplitude $\epsilon=0.15$ and opposite phase in the North and in the South. Transmission rate does not vary over time in the Tropics. Transmission between region $i$ and region $j$ occurs at rate $m\,\beta_{i}$, where $m=0.001$ and is the same between each pair of regions and $\beta_{i}$ is the within-region contact rate. Hosts recover from infection at rate $\nu=0.2$ per host per day, so that $R_{0}$ in a naive host population is 1.8. There is no super-infection in the model. Each virus possesses an antigenic phenotype, represented as a location in Euclidean space. Here, we primarily use a two-dimensional antigenic location. After recovery, a host ‘remembers’ the phenotype of its infecting virus as part of its immune history. When a contact event occurs and a virus attempts to infect a host, the Euclidean distance from infecting phenotype $\phi_{v}$ is calculated to each of the phenotypes in the host immune history $\phi_{h_{1}},\dots,\phi_{h_{n}}$. Here, one unit of antigenic distance is designed to correspond to a twofold dilution of antiserum in a hemagglutination inhibition (HI) assay [9]. The probability that infection occurs after exposure is proportional to the distance $d$ to the closest phenotype in the host immune history. Risk of infection follows the form $\rho=\textrm{min}\\{d\,s,1\\}$, where $s=0.07$. Cross-immunity $\sigma$ equals $1-\rho$. The initial host population begins with enough immunity to slow down the initial virus upswing and place the dynamics closer to their equilibrium state; initial $R$ was 1.28. Our model follows Gog and Grenfell [23] in representing antigenic distance as distance between points in a geometric space. By forcing one-dimension to directly modulate $\beta$, Gog and Grenfell find an orderly replacement of strains. Kryazhimskiy et al. [24] use a two-dimensional strain-space, but enforce a cross-immunity kernel that directly favors moving along a diagonal line away from previous strains. Our model does not ‘build in’ the one- dimensional direction of antigenic drift, which instead emerges dynamically from competition among strains. The initial virus population consisted of 10 infections each with the identical antigenic phenotype of $\\{0,0\\}$. Over time viruses evolve in antigenic phenotype. Each day there is a chance $\mu=10^{-4}$ that an infection mutates to a new phenotype. This mutation rate represents a phenotypic rate, rather than genetic mutation rate, and can be thought of as arising from multiple genetic sources. When a mutation occurs, the virus’s phenotype is moved in a completely random direction $\sim\textrm{Uniform}(0,360)$ degrees. Mutation size is sampled from the distribution $\sim\textrm{Gamma}(\alpha,\beta)$, where $\alpha$ and $\beta$ are chosen to give a mean mutation size of 0.6 units and a standard deviation of 0.4 units. This distribution is parameterized so that mutation usually has little effect on antigenic phenotype, but occasionally has a large effect. This is similar to the neutral networks implemented by Koelle et al. [6], wherein most amino acid changes result in little decrease to cross-immunity between strains, but some changes result in large jumps in cross-immunity. ### Model output Daily incidence and prevalence are recorded for each region. During the course of the simulation, samples of current infections are taken from the evolving virus population at a rate proportional to prevalence. Each viral infection is assigned a unique ID, and in addition, infections have their phenotypes, locations and dates of infection recorded. In this model, viruses lack sequences and so standard phylogenetic inference of the evolutionary relationships among strains is impossible. Instead, the viral genealogy is directly recorded. This is made possible by tracking transmission events connecting infections during the simulation; infections record the ID of their ‘parent’ infection. Proceeding from a sample of infections, their genealogical history can be reconstructed by following consecutive links to parental infections. During this procedure, lineages coalesce to the ancestral lineages shared by the sampled infections, eventually arriving at the initial infection introduced at the beginning of the simulation. Commonly, phylodynamic simulations generate sequences that are subsequently analyzed with phylogenetic software to produce an estimated genealogy [4, 6, 25]. This step of phylogenetic inference is imperfect and computationally intensive, and by side-stepping phylogenetic reconstruction we arrive at genealogies quickly and accurately. Other authors have implemented similar tracking of infection trees [26, 27]. This genealogy-centric approach makes many otherwise difficult calculations transparent, such as calculating lineage-specific region-specific migration rates (Figure 2) and lineage-specific mutation effects (Figure S5). Infections are sampled at a rate designed to give approximately 6000 samples over the course of the simulation, with genealogies constructed from a subsample of approximately 300 samples. The results presented in Figure 1 represent a single representative model output; one hundred replicate simulations were conducted to arrive at statistical estimates. ### Parameter selection and sensitivity analysis Estimating what the basic reproductive number $R_{0}$ for seasonal influenza would be in a naive population is notoriously difficult. Season-to-season estimates of effective reproductive number $R$ for the USA and France gathered from mortality timeseries display an interquartile range of 0.9–1.8 [28]. Geographic spread within the USA suggests an average seasonal $R$ of 1.35 [29]. These estimates of $R$ will be lower than the $R_{0}$ of influenza due to the effects of human immunity. We assumed $R_{0}$ of 1.8, consistent with the upper range of seasonal estimates. Duration of infection was chosen based on patterns of viral shedding shown during challenge studies [30]. The linear form of the risk of infection and its increase as a function of antigenic distance $s$ was chosen as 0.07 based on experimental work on equine influenza [31] and from studies of vaccine effectiveness [32]. Between-region contact rate $m$ was chosen to yield a trunk lineage that resides predominantly in the tropics. With much higher rates of mixing, the trunk lineage ceases to show a preference the tropics, and with much lower rates of mixing, particular seasons in the north and the south will often be skipped. The amplitude of seasonal forcing $\epsilon$ was chosen to be just large enough to get consistent fade-outs in the summer months and is consistent with empirical estimates [33]. Mutational parameters were based, in part, on model behavior. We assumed 10 amino acid sites involved in antigenicity, each mutating at a rate of $10^{-5}$ [12] to give a phenotypic mutation rate $\mu=10^{-4}$ per infection per day. We chose mutational effect parameters ($\textrm{mean}=0.6$, $\textrm{sd}=0.4$) that would give suitably fast rates of antigenic evolution corresponding to approximately 1.2 units of antigenic change per year, while simultaneously giving clustered patterns of antigenic evolution [9]. Similar outcomes are possible under a variety of parameterizations. If mutations are more common ($\mu=3\times 10^{-4}$) and show less variation in effect size ($\textrm{mean}=0.6$, $\textrm{sd}=0.2$), then antigenic drift occurs in a more continuous fashion, resulting in less variation in seasonal incidence and a smoother distribution of antigenic phenotypes (Figure S3). If mutations are less common ($\mu=5\times 10^{-5}$) and show more variance in effect ($\textrm{mean}=0.7$, $\textrm{sd}=0.5$), then antigenic change occurs in a more punctuated fashion (Figure S4). Basic reproductive number $R_{0}$ can be traded off with mutational parameters to some extent. Less mutational input and higher $R_{0}$ will give similar patterns of antigenic drift and seasonal incidence. Similarly, Kucharski and Gog [34] find that increasing $R_{0}$ results in increased rates of emergence of antigenically novel strains. In 20 out of the 100 replicate simulations, we observed a major bifurcation of antigenic phenotype and a consequent increase in incidence and genealogical diversity. These simulations were removed from the analysis. Similar to Koelle et al. [35], we assume that although the historical evolution of H3N2 influenza followed the path of a single lineage, it could have included a major bifurcation. Further work in these directions will help to determine the likelihoods of single lineage vs. bifurcating scenarios. ### Antigenic map Antigenic phenotypes are modeled as discrete entities on the Euclidean plane; multiple samples have the same antigenic location. However, in the empirical antigenic map of influenza A (H3N2), each strain appears in a unique location [9]. We would argue that some of this pattern comes from experimental noise. Indeed, Smith et al. [9] find that observed measurements and measurements predicted from the map differ by an average of 0.83 antigenic units with a standard deviation of 0.67 antigenic units. We take this as a proxy for experimental noise and add jitter to each sampled antigenic phenotype by moving it in a random direction for an exponentially distributed distance with mean of 0.53 antigenic units. If two samples with the same underlying antigenic phenotype are jittered in this fashion, the distance between them averages 0.83 antigenic units with a standard deviation of 0.64 units. We added noise to each of the 5943 sampled viruses in this fashion resulting in an approximated antigenic map (Figure 1D). Virus samples were then clustering following standard clustering algorithms. We tried clustering by the $k$-means algorithm and also agglomerative hierarchical clustering with a variety of linkage criterion. We found that clustering by Ward’s criterion consistently outperformed other methods, when measured in terms of within- cluster and between-cluster variances. However, the exact clustering algorithm had little effect on our overall results. ### Acknowledgments We would like to thank Sarah Cobey, Aaron King, Pejman Rohani and the attendees of the 2011 RAPIDD Workshop on Phylodynamics for helpful discussion. We would also like to thank Ed Baskerville and Daniel Zinder for programming advice. The term ‘canalization’ was originally suggested by Micaela Martinez- Bakker. ### Funding TB is supported by the Howard Hughes Medical Institute and by the European Molecular Biology Organization. AR works as a part of the Interdisciplinary Centre for Human and Avian Influenza Research (ICHAIR) and the University of Edinburgh’s Centre for Immunity, Infection and Evolution (CIIE). MP is an investigator of the Howard Hughes Medical Institute. ## References * 1. WHO (2009) Influenza Fact sheet. Available at http://www.who.int/mediacentre/factsheets/fs211/en/. * 2. Nelson MI, Holmes EC (2007) The evolution of epidemic influenza. Nat Rev Genet 8: 196–205. * 3. Fitch WM, Bush RM, Bender CA, Cox NJ (1997) Long term trends in the evolution of H(3) HA1 human influenza type A. Proc Natl Acad Sci USA 94: 7712-8. * 4. Ferguson NM, Galvani AP, Bush RM (2003) Ecological and immunological determinants of influenza evolution. Nature 422: 428–433. * 5. Tria F, Lässig M, Peliti L, S F (2005) A minimal stochastic model for influenza evolution. J Stat Mech . * 6. Koelle K, Cobey S, Grenfell B, Pascual M (2006) Epochal evolution shapes the phylodynamics of interpandemic influenza A (H3N2) in humans. Science 314: 1898–1903. * 7. Recker M, Pybus O, Nee S, Gupta S (2007) The generation of influenza outbreaks by a network of host immune responses against a limited set of antigenic types. Proc Natl Acad Sci USA 104: 7711–7716. * 8. Hirst G (1943) Studies of antigenic differences among strains of influenza A by means of red cell agglutination. J Exp Med 78: 407–423. * 9. Smith DJ, Lapedes AS, de Jong JC, Bestebroer TM, Rimmelzwaan GF, et al. (2004) Mapping the antigenic and genetic evolution of influenza virus. Science 305: 371–376. * 10. Monto A, Sullivan K (1993) Acute respiratory illness in the community. Frequency of illness and the agents involved. Epidemiol Infect 110: 145–160. * 11. Koelle K, Kamradt M, Pascual M (2009) Understanding the dynamics of rapidly evolving pathogens through modeling the tempo of antigenic change: influenza as a case study. Epidemics 1: 129–137. * 12. Rambaut A, Pybus OG, Nelson MI, Viboud C, Taubenberger JK, et al. (2008) The genomic and epidemiological dynamics of human influenza A virus. Nature 453: 615–619. * 13. Bedford T, Cobey S, Pascual M (2011) Strength and temp of selection revealed in viral gene genealogies. BMC Evol Biol 11: 220. * 14. Russell CA, Jones TC, Barr IG, Cox NJ, Garten RJ, et al. (2008) The global circulation of seasonal influenza A (H3N2) viruses. Science 320: 340–346. * 15. Bedford T, Cobey S, Beerli P, Pascual M (2010) Global migration dynamics underlie evolution and persistence in human influenza A (H3N2). PLoS Pathog 6: e1000918. * 16. Finkelman B, Viboud C, Koelle K, Ferrari M, Bharti N, et al. (2007) Global patterns in seasonal activity of influenza A/H3N2, A/H1N1, and B from 1997 to 2005: viral coexistence and latitudinal gradients. PLoS One 2: e1296. * 17. Bhatt S, Holmes E, Pybus O (2011) The genomic rate of molecular adaptation of the human influenza A virus. Mol Biol Evol Advance access. * 18. Goldstein E, Cobey S, Takahashi S, Miller J, Lipsitch M (2011) Predicting the epidemic sizes of influenza A/H1N1, A/H3N2, and B: a statistical method. PLoS Med In press. * 19. Voeten J, Bestebroer T, Nieuwkoop N, Fouchier R, Osterhaus A, et al. (2000) Antigenic drift in the influenza A virus (H3N2) nucleoprotein and escape from recognition by cytotoxic T lymphocytes. J Virol 74: 6800–6807. * 20. Gould S (1989) Wonderful life. Norton New York. * 21. Barr I, McCauley J, Cox N, Daniels R, Engelhardt O, et al. (2010) Epidemiological, antigenic and genetic characteristics of seasonal influenza A (H1N1), A (H3N2) and B influenza viruses: basis for the WHO recommendation on the composition of influenza vaccines for use in the 2009-2010 Northern Hemisphere season. Vaccine 28: 1156–1167. * 22. Weinreich D, Delaney N, DePristo M, Hartl D (2006) Darwinian evolution can follow only very few mutational paths to fitter proteins. Science 312: 111–114. * 23. Gog J, Grenfell B (2002) Dynamics and selection of many-strain pathogens. Proc Natl Acad Sci USA 99: 17209–17214. * 24. Kryazhimskiy S, Dieckmann U, Levin S, Dushoff J (2007) On state-space reduction in multi-strain pathogen models, with an application to antigenic drift in influenza A. PLoS Comp Biol 3: e159. * 25. Koelle K, Khatri P, Kamradt M, Kepler T (2010) A two-tiered model for simulating the ecological and evolutionary dynamics of rapidly evolving viruses, with an application to influenza. J R Soc Interface 7: 1257–1274. * 26. Volz E, Kosakovsky Pond S, Ward M, Leigh Brown A, Frost S (2009) Phylodynamics of infectious disease epidemics. Genetics 183: 1421–1430. * 27. O’Dea E, Wilke C (2011) Contact heterogeneity and phylodynamics: how contact networks shape parasite evolutionary trees. Interdiscip Perspect Infect Dis 2011: 238743. * 28. Chowell G, Miller M, Viboud C (2008) Seasonal influenza in the United States, France, and Australia: transmission and prospects for control. Epidemiol Infect 136: 852–864. * 29. Viboud C, Bjørnstad O, Smith D, Simonsen L, Miller M, et al. (2006) Synchrony, waves, and spatial hierarchies in the spread of influenza. Science 312: 447–451. * 30. Carrat F, Vergu E, Ferguson NM, Lemaitre M, Cauchemez S, et al. (2008) Time lines of infection and disease in human influenza: a review of volunteer challenge studies. Am J Epidemiol 167: 775–785. * 31. Park A, Daly J, Lewis N, Smith D, Wood J, et al. (2009) Quantifying the impact of immune escape on transmission dynamics of influenza. Science 326: 726–728. * 32. Gupta V, Earl D, Deem M (2006) Quantifying influenza vaccine efficacy and antigenic distance. Vaccine 24: 3881–3888. * 33. Truscott J, Fraser C, Cauchemez S, Meeyai A, Hinsley W, et al. (2011) Essential epidemiological mechanisms underpinning the transmission dynamics of seasonal influenza. J R Soc Interface In press. * 34. Kucharski A, Gog JR (2011) Influenza emergence in the face of evolutionary constraints. Proc R Soc Lond B Biol Sci Advance access. * 35. Koelle K, Ratmann O, Rasmussen D, Pasour V, Mattingly J (2011) A dimensionless number for understanding the evolutionary dynamics of antigenically variable rna viruses. Proc R Soc Lond B Biol Sci Advance access. ## Tables Table Table 1: Rates of mutation and phenotypic change on trunk and side branches and mutational expectation. \| Baseline \| Side branch \| Trunk \| Trunk / side branch ---\|---\|---\|---\|--- Mutation size (AG units) \| 0.60 \| 0.79 \| 1.58 \| 1.99$\times$ Mutation rate (mut per year) \| 0.04 \| 0.06 \| 0.81 \| 13.23$\times$ Antigenic flux (AG units per year) \| 0.02 \| 0.05 \| 1.27 \| 26.25$\times$ ## Figures Figure Figure 1: Simulation results showing epidemiological, antigenic and genealogical dynamics. (A) Weekly timeseries of incidence of viral infection in north and tropics regions. (B) Two-dimensional antigenic phenotypes of 5943 viruses sampled over the course of the simulation. Each discrete virus phenotype is shown as a bubble, with bubble area proportional to the number of times this phenotype was sampled. (C) Genealogical tree depicting the infection history of 376 samples from the virus population. Parent/offspring relationships were tracked over the course of the simulation, giving a direct observation of the genealogy rather than a phylogenetic inference. (D) Antigenic map depicting phenotypes of 5943 viruses sampled over the course of the simulation. To construct the map, noise was added to each sample and the resulting observations grouped into 11 clusters and colored accordingly. Grid lines show single units of antigenic distance. Cluster assignments were used to color all panels in a consistent fashion. Figure Figure 2: Patterns of geographic movement of virus lineages. (A) Evolutionary relationships among 376 viruses sampled evenly through time colored by geographic location. Lineages residing in the north (N), south (S) and tropics (T) are colored yellow, red and blue respectively. (B) Observed migration rates between regions on side branch lineages (left) and on trunk lineages (right). Arrows denote movement of lineages and arrow width is proportional to migration rate. Circle area is proportional to the expected stationary frequency of a region given the observed migration rates. In both cases, migration rates are calculated across 80 replicate simulations. Figure Figure 3: Host immunity and antigenic history of the virus population. Contour lines represent the state of host immunity at the end of the 40-year simulation. They show the mean risk of infection (as a percentage) after a random host in the population encounters a virus bearing a particular antigenic phenotype. Contour lines are spaced in intervals of 2.5%. Bubbles represent a sample of antigenic phenotypes present at the end of the 40-year simulation. The area of each bubble is proportional to the number of samples with this phenotype. Lines leading into these bubbles show past antigenic history. The current phenotypes rapidly coalesce to a trunk phenotype. The movement of the virus population from the left to the center of the figure can be seen from the antigenic history. At the end of the simulation several virus phenotypes exist with similar antigenic locations; all of these phenotypes lie significantly ahead of the peak of host immunity. Figure Figure 4: Antigenic phenotypes over the course of 4 years of evolution across 100 replicate simulations starting from identical initial conditions. Replicate simulations were initialized with the end state of the 40-year simulation shown in Figure 1. Each panel shows an additional year of evolution, with black points representing the mean antigenic phenotypes of the 100 replicate simulations and gray lines representing the history of each mean antigenic phenotype. Figure Figure 5: Timeseries of incidence across 100 replicate simulations with identical initial conditions. Panels show incidence in the North, Tropics and South regions over the course of 6 years. Solid black lines represent the median weekly incidence across the 100 replicate simulations, while gray intervals represent the interquartile range across simulations. There is little variability for the first year of replicate simulations. Replicate simulations were initialized with the end state of the 40-year simulation shown in Figure 1. ## Supporting Information Figure Figure S1: Antigenic evolution over the course of the 40-year simulation. (A) Proportion of virus population comprised of each antigenic cluster through time. (B) Antigenic distance from initial phenotype ($x=0$, $y=0$) for each of 5943 virus samples relative to time of virus sampling. Viruses were sampled at a constant rate proportional to prevalence and coloring was determined from the antigenic map in Figure 1D. Figure Figure S2: Correlation between antigenic drift and attack rate. Antigenic drift is measured as the distance between the centroid of phenotypes of one year and the centroid of phenotypes of the following year. Measurements were taken across 80 replicate simulations. Individual pairs of measurements are shown as gray points and a locally-linear regression (LOESS) is shown as a black dashed line. Figure Figure S3: Simulation results showing epidemiological, antigenic and genealogical dynamics for ‘smoother’ mutation model. (A) Weekly timeseries of incidence of viral infection in north and tropics regions. (B) Antigenic map depicting phenotypes of viruses sampled over the course of the simulation. Grid lines show single units of antigenic distance. (C) Genealogical tree depicting the infection history of samples from the virus population. Cluster assignments were used to color panels (A), (B) and (C) in a consistent fashion. Alternative mutational parameters are $\mu=3\times 10^{-4}$, mean mutation size of 0.6 units and standard deviation of mutation size of 0.2 units. Figure Figure S4: Simulation results showing epidemiological, antigenic and genealogical dynamics for ‘rougher’ mutation model. (A) Weekly timeseries of incidence of viral infection in north and tropics regions. (B) Antigenic map depicting phenotypes of viruses sampled over the course of the simulation. Grid lines show single units of antigenic distance. (C) Genealogical tree depicting the infection history of samples from the virus population. Cluster assignments were used to color panels (A), (B) and (C) in a consistent fashion. Alternative mutational parameters are $\mu=5\times 10^{-5}$, mean mutation size of 0.7 units and standard deviation of mutation size of 0.5 units. Figure Figure S5: Mutation spectrum in two-dimensional antigenic space of side branch mutations and trunk mutations. (A) Histogram of mutation effects along the axis of primary antigenic variation across 80 replicate simulations. The left panel shows the distribution of effects of side branch mutations and the right panel shows the distribution of effects of trunk mutations. (B) Smoothed two-dimensional histogram of mutation effects along the primary and secondary axes of antigenic variation across 80 replicate simulations. Histograms were constructed from 21,405 side branch mutations and 1584 trunk mutations. Figure Figure S6: Relationship between a mutation’s phenotypic effect and its likelihood of being part of the trunk. The $x$-axis represents the effect of a mutation along the line of primary antigenic variation, and the $y$-axis represents the probability that the mutation is part of the trunk. Mutations of large effect are increasingly rare, but when they do occur are increasingly likely to be part of the trunk. Figure Figure S7: Observed vs. expected distributions of waiting times between phenotypic mutations along genealogy trunk. (A) Histogram bins show the observed distribution of waiting times in years across 80 replicate simulations representing 1584 mutations. The mean of this distribution is 1.76 years. The dashed line shows the Poisson process expectation of exponentially distributed waiting times. (B) The density distribution of waiting times is transformed into a hazard function, representing the rate of trunk mutation after a specific waiting time. The dashed line shows the memoryless hazard function of the Poisson process expectation. Figure Figure S8: Simulation results showing epidemiological, antigenic and genealogical dynamics with weaker mutation. (A) Weekly timeseries of incidence of viral infection in north and tropics regions. (B) Antigenic map depicting phenotypes of viruses sampled over the course of the simulation. Grid lines show single units of antigenic distance. (C) Genealogical tree depicting the infection history of samples from the virus population. Cluster assignments were used to color panels (A), (B) and (C) in a consistent fashion. Here, $\mu=5\times 10^{-5}$, mean mutation size is 0.42 units and standard deviation of mutation size is 0.28 units. Figure Figure S9: Simulation results showing epidemiological, antigenic and genealogical dynamics with lower intrinsic $R_{0}$. (A) Weekly timeseries of incidence of viral infection in north and tropics regions. (B) Antigenic map depicting phenotypes of viruses sampled over the course of the simulation. Grid lines show single units of antigenic distance. (C) Genealogical tree depicting the infection history of samples from the virus population. Cluster assignments were used to color panels (A), (B) and (C) in a consistent fashion. Here, $\beta=0.3$, giving $R_{0}=1.5$. Figure Figure S10: Principal components of antigenic variation under a 10-sphere mutation model. Each panel shows 5991 samples of antigenic phenotype over the course of a 40-year simulation. Each phenotype is represented as a bubble, with bubble area proportional to the number of samples with this phenotype. Bubbles are colored based on clustering the 10-dimensional antigenic phenotypes. The original 10-dimensional space was rotated using principal components analysis to give orthogonal axes in the order of their contribution to antigenic variation. Each panel shows a two-dimensional slice of the this rotated space. Principal components 7–10 were left out of the figure for clarity.	arxiv-papers	2011-11-19T19:41:26	2024-09-04T02:49:24.493026	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Trevor Bedford, Andrew Rambaut and Mercedes Pascual", "submitter": "Trevor Bedford", "url": "https://arxiv.org/abs/1111.4579" }
1111.4593	footnote # A graph counterexample to davies’ conjecture Gady Kozma ###### Abstract. There exists a graph with two vertices $x$ and $y$ such that the ratio of the heat kernels $p(x,x;t)/p(y,y;t)$ does not converge as $t\to\infty$. This paper is concerned with a conjecture of Brian Davies from 1997 on the heat kernel of Riemannian manifolds [D97, §5]. We will not disprove the conjecture as stated, but rather transform it to the realm of graphs using a well-known (though informal) “dictionary” between these two categories, and build a graph that will serve as a counterexample. We will make some remarks on how the construction might be carried over back to the category of manifolds in the end, but we will not give all details. The bulk of this paper is about graphs. We start by describing the conjecture in its original setting. Let $M$ be a connected Riemannian manifold, and let $p$ be the _heat kernel_ associated with the _Laplace-Beltrami operator_ on $M$. Then Davies conjecture states that for any $M$ and any 3 points $x,y,z\in M$ the limit $\lim_{t\to\infty}\frac{p(x,y;t)}{p(z,z;t)}$ (1) exists and is positive. Here $p(x,y;t)$ is the value of the heat kernel at time $t$ and at points $x$ and $y$. This property is known as the “strong ratio limit property” (where the “weak” version is an averaged result due to Döblin, [D38]) or SRLP for short. So Davies’ conjecture is that in these settings SRLP always holds. SRLP holds for manifolds with one end [D82] and for strongly Liouville manifolds (i.e. manifolds where any positive harmonic function is constant), see [ABJ02, corollary 2.7] who also make interesting connections between these properties and the infinite Brownian loop. Ratio limit properties were considered for Markov chains even earlier. If $M$ is any Markov chain on a countable state space, then we say that $M$ satisfies SRLP if (1) holds for any three states $x$, $y$ and $z$, where $p(x,y;t)$ is the probability that the Markov chain started at $x$ will be at $y$ at time $t$. For general Markov chains there are a few examples where SRLP does not hold. Clearly it does not hold when the Markov chain has some kind of periodicity. F. J. Dyson constructed an example of an aperiodic recurrent Markov chain which does not satisfy SRLP [C60, part I, §10]. That example utilizes long chains of states with only one outgoing edge, which the walker must traverse sequentially. In particular it is not _reversible_. Now, the Laplace-Beltrami operator is self-adjoint so a proper analog of Davies conjecture needs to assume that the Markov chain is reversible. Reversible Markov chains are also known as random walks on weighted graphs. The issue of periodicity can be dealt with by looking at random walk in continuous time or at lazy random walk. Lazy random walk is a walk where the walker, at every step, chooses with probability $\frac{1}{2}$ to stay where it is, and with probability $\frac{1}{2}$ moves to one of the neighbours (with probability proportional to the weights). The result is this paper is ###### Theorem. There exists a connected graph $G$ with bounded weights and two vertices $x,y\in G$ such that the heat kernel of the lazy random walk satisfies $\frac{p(x,x;t)}{p(y,y;t)}\nrightarrow$ (2) as $t\to\infty$. Let us remark on the “bounded weights” clause. When doing analogies between manifolds and graphs, it is often assumed that the manifold has bounded geometry and the graph has bounded weights. Davies, however, explicitly does not make the assumption of bounded geometry. Thus one might wonder what is exactly the graph analog. All this is mute, of course, since the counterexample does has bounded weights (and hence a manifold example constructed along the same line should have bounded geometry). It is easy to see that in these setting (reversible, irreducible, lazy) this ratio must be bounded between two constants independent of $n$. Hence if it does not converge then it must fluctuate between two values. The proof constructs the graph with two halves, denoted $H^{e}$ and $H^{o}$ ($e$ and $o$ standing for even and odd, $H$ standing for half), which are connected by one edge, $(x,y)$. On the “odd scales”, $H^{e}$ will “look like $\mathbb{Z}^{22}$” while $H^{o}$ will “look like $\mathbb{Z}^{3}$”. This means that to get from $x$ to $x$ (where $x$ is on the $H^{e}$ side), the most beneficial strategy is to move to $y$ as fast as possible, spend most of your time on the $H^{o}$ side and return to the $H^{e}$ side only in the last minute. Clearly this would mean that $p(x,x;t)$ is smaller than $p(y,y;t)$ as the random walk starting from $y$ can stay on its side at all time, not losing the constant that $x$ needs for the maneuver. On the “even scales” the picture is reversed and $y$ is at a disadvantage. See figure 1 — drawing in 22 dimensions might have distracted the reader, so the figure demonstrates the construction in 1 and 2 dimensions. The smallest two braces in the figure are the first scale, in which $H^{o}$ is really one dimensional and $H^{e}$ is really two- dimensional. The larger two braces indicate the second scale. This time $H^{o}$ is a network of lines so it should be thought about as two dimensional, while $H^{e}$ is a thick column, so it should be thought about as one dimensional. The third scale is only hinted at in this figure, but one can imagine that $H^{o}$ now becomes a thick band, so it is again one-dimensional, while $H^{e}$ becomes a network of these thick columns and bands, so it is back to being two dimensional. $x$$y$$H^{e}$$H^{o}$ Figure 1. The graphs $H^{o}$ and $H^{e}$ As one might expect, the numbers $3$ and $22$ have no particular significance. They both have to be $>2$, since otherwise our graph would be recurrent and recurrent graphs always satisfy SRLP [O61, theorem 3]. And of course they have to be different. We took here the large value $22$ in order to be able to be wasteful in various points (sum over times and such stuff), but the proof could proceed with any value larger than $3$. This paper was first written in 2006. I wish to take this opportunity to apologize to all those who has to wait so long for it to appear, with no real reason. My intentions were good but my time management was abysmal. I wish to thank Yehuda Pinchover for telling me about the problem and for reading early drafts. Partially supported by the Israel Science Foundation. ## 1\. Proof The construction uses $\mathbb{Z}^{d}$-like graphs as building blocks, so we start with quoting a few results on these. We first recall the notion of rough isometry [K85]. ###### Definition. Let $X$ and $Y$ be two metric spaces. We say that $X$ and $Y$ are roughly isometric if there is a constant $C$ and a map $\varphi:X\to Y$ with the following properties: 1. (i) For all $x$ and $y$ in $X$, $\frac{1}{C}d(x,y)-C\leq d(\varphi(x),\varphi(y))\leq Cd(x,y)+C$ 2. (ii) The image of $\varphi$ is roughly dense, i.e. for all $y\in Y$ there is an $x\in X$ such that $d(y,\varphi(x))\leq C$ If $G$ and $H$ are graphs we say that they are roughly isometric if they are roughly isometric when considered with the metric $d$ being the graph distance, namely $d(x,y)$ is the length of the shortest path between $x$ and $y$, or $\infty$ if no such path exists. With this definition we can state the following standard result, essentially due to Delmotte. ###### Lemma 1. Let $G$ be a graph roughly isometric to $\mathbb{Z}^{d}$. Then the heat kernel $p$ for the lazy walk on $G$ satisfies, for all $t\geq 1$, $ct^{-d/2}\leq p(x,x;t)\leq Ct^{-d/2}.$ Here $G$ is a simple graph — we do not allow weights or multiple edges. $C$ and $c$ are constant which do not depend on $t$. In general we will use $c$ for constants which are small enough and $C$ for constants which are large enough, and different appearance of $c$ and $C$ might relate to different constants. ###### Proof. By Delmotte’s theorem [D99] any $G$ which satisfies volume-doubling and the Poincaré inequality, satisfies $p(x,x;t)\approx\frac{1}{\|B(x,\sqrt{t})\|}$ where $B(x,r)$ is the ball around $x$ with radius $r$ (again with the graph distance), and $\|B(x,r)\|$ is the sum of the degrees of the vertices in $B$. The notation $X\approx Y$ is short for $cY\leq X\leq CY$. The fact that $G$ is roughly-isometric to $\mathbb{Z}^{d}$ gives $\|B(x,r)\|\approx r^{d}$ (3) so we would get $p(x,x;t)\approx t^{-d/2}$, as needed. So we need only show that $G$ satisfies volume doubling and Poincaré inequality. Now, the definition of volume doubling is that for every $x$ a vertex of $G$ and every $r\geq 1$ $\|B(x,2r)\|\leq C\|B(x,r)\|$ and it follows immediately from (3). The Poincaré inequality is not much more complicated. The definition is: for every vertex $x$, for every $r$ and for every function $f:B(x,2r)\to\mathbb{R}$, $\sum_{y\in B(x,r)}\deg(y)\|f(y)-\overline{f}\|^{2}\leq Cr^{2}\sum_{(y,z)\in E(B(x,2r))}(f(y)-f(z))^{2}$ (4) where $\overline{f}=\frac{1}{\|B(x,r)\|}\sum_{y\in B(x,r)}\deg(y)f(y).$ and where $\deg(y)$ is the degree of $y$, and $E(B)$ is the set of edges both whose vertices are in $B$. Now, $\mathbb{Z}^{d}$ satisfies the Poincaré inequality (see e.g. [PSC, §4.1.1]). It is well-known and not difficult to see that the Poincaré inequality is preserved by rough isometries (it uses the fact that $\sum\deg(y)\|f(y)-a\|^{2}$ is minimized when $a=\overline{f}$). This finishes the proof.∎ ###### Lemma 2. Let $G$ be a graph roughly-isometric to $\mathbb{Z}^{d}$, $d\geq 3$ and let $x$ be some vertex. Let $p$ be the probability that lazy random walk starting from $x$ returns to $x$ for the first time at time $t$. Then $p\geq ct^{-d/2}$. ###### Proof. Let $p_{1}$ be the same probability but without the restriction that this is the first return to $x$. This is exactly $p(x,x;t)$ and by theorem 1 we have $p_{1}\geq ct^{-d/2}$. Fix some $K$ and examine the event that the random walk returns to $x$ at $t$ and also at some time $s\in[K,t-K]$. Let $p_{2}$ be its probability. Using the other direction in theorem 1 we can write $p_{2}\leq\sum_{s=K}^{t-K}p(x,x;s)p(x,x;t-s)\leq C\sum_{s=K}^{t-K}s^{-d/2}(t-s)^{-d/2}\leq CK^{1-d/2}t^{-d/2}.$ Since $d\geq 3$ we can choose $K$ sufficiently large such that $p_{2}\leq\frac{1}{2}p_{1}$. So we know that with probability $p_{1}-p_{2}\geq ct^{-d/2}$ the walk does not return to $x$ between $K$ and $t-K$. If it does reach $x$ before time $K$, do some local modification so that it does not. For example, if the original walker reached $x$ at some time $s<K$ and on the next step went to some neighbour $y$ of $x$, modify it to walk to $y$ in the first step and stay there for $s$ steps (remember that our walk is lazy) and then continue like the original walker. Clearly this “costs” only a constant and ensures our walker does not visit $x$ in the interval $[1,K]$. Do the same for the interval $[t-K,t-1]$, losing another constant. The finishes the lemma.∎ ###### Lemma 3. Let $G$ be a graph and let $p$ be the heat kernel for the lazy walk on $G$. Let $t$ and $s$ satisfy that $\|t-s\|\leq\sqrt{t}$. Then $\|p(x,x;t)-p(x,x;s)\|\leq C\frac{\|t-s\|\log^{3}t}{\sqrt{t}}p(x,x;t)+Ce^{-c\log^{2}t}.$ ###### Proof. Denote by $q(x,y;t)$ the heat kernel for the _simple_ random walk on $G$. Then by definition $p(x,x;t)=\sum_{i=0}^{t}q(x,x;i)\binom{t}{i}2^{-t}.$ Writing the same formula for $p(x,x;s)$ and subtracting we get $\|p(x,x;t)-p(x,x;s)\|\leq\sum_{i}q(x,x;i)\left(\binom{t}{i}2^{-t}-\binom{s}{i}2^{-s}\right)=\Sigma_{1}+\Sigma_{2}$ where $\Sigma_{1}$ is the sum over all $\|t-2i\|\leq\sqrt{t}\log t$ and $\Sigma_{2}$ is the rest. A simple calculation with Stirling’s formula shows that $2^{-t}\binom{t}{i}=\sqrt{\frac{2}{\pi t}}\exp\left(-\frac{(t-2i)^{2}}{2t}\left(1+O\left(\frac{\|t-2i\|+1}{t}\right)\right)\right)$ And with some more calculations $\Sigma_{1}\leq C\sum_{\|t-2i\|\leq\sqrt{t}\log t}q(x,x;i)\sqrt{\frac{2}{\pi t}}e^{-(t-2i)^{2}/2t}\frac{\|t-s\|\log^{3}t}{\sqrt{t}}\leq C\frac{\|t-s\|\log^{3}t}{\sqrt{t}}p(x,x;t)$ while $\Sigma_{2}\leq C\sum_{\|t-2i\|>\sqrt{t}\log t}e^{-c(t-2i)^{2}/t}\leq Ce^{-c\log^{2}t}$ proving the lemma. ∎ ###### Proof of the theorem. Abusing notations, for subsets $H\subset\mathbb{Z}^{d}$ we will not distinguish between $H$ as a set and as an induced subgraph of $\mathbb{Z}^{d}$ ($d$ will be $22$). For the construction we need a sufficiently fast increasing sequence $a_{1}<a_{2}<\dotsb$. We further assume that $a_{k}$ are all even and that $a_{k-1}$ divides $\frac{1}{2}a_{k}$. It would have probably been enough to choose $a_{k}=2^{a_{k-1}}$, but it turns out simpler to choose the $a_{k}$ inductively, and we perform this as follows. Let $a_{1}=2$. Assume now $a_{1},\dotsc,a_{k-1}$ have been defined. Define, for integers $m<\frac{1}{2}l$ and $i\in\\{1,\dotsc,22\\}$, $\displaystyle Q_{l,m,i}$ $\displaystyle:=\left\\{\vec{n}\in\mathbb{Z}^{22}:\left\|n_{i}\textrm{ mod }l\right\|\leq m\right\\}$ $\displaystyle Q_{l,m}$ $\displaystyle:=\bigcup_{\begin{subarray}{c}I\subset\\{1,\dotsc,22\\}\\\ \|I\|=19\end{subarray}}\bigcap_{i\in I}Q_{l,m,i}.$ Here $n\textrm{ mod }l\in\\{-\lfloor\frac{l-1}{2}\rfloor,\dotsc,\lfloor\frac{l}{2}\rfloor\\}$. In words, $Q_{l,m,i}$ is a $21$-dimensional subspace of $\mathbb{Z}^{22}$ orthogonal to one of the axes, fattened up by $2m+1$ (a “slab”) and repeated periodically with period $l$. $Q_{l,m}$ is the collection of all $3$-dimensional subspaces, fattened and repeated similarly. The particular point $\vec{0}$ is in fact contained in all $\binom{22}{3}$ of these $3$-dimensional slabs which will be a little inconvenient, so let us shift $Q_{l,m}$ by $v(m)=\Big{(}\underbrace{{\textstyle\frac{1}{2}}m,\dotsc,{\textstyle\frac{1}{2}}m}_{3\textrm{ times}},\underbrace{\vphantom{{\textstyle\frac{1}{2}}m}0,\dotsc,0}_{19\textrm{ times}}\Big{)}.$ In the shifted set $Q_{l,m}+v(m)$ the geometry of the neighbourhood of $\vec{0}$ is simpler, it is contained in just one slab. Compare to the figure on page 1. The point $x$ is in the _middle_ of a fat column and not at the intersection of a column and a band. We want to use these graphs with $l=a_{j}$ and $m$ a little larger than $a_{j-1}$. Precisely, define $b_{j}=\sum_{k=1}^{j}a_{k}.$ With this choice of $b_{j}$, $Q(a_{j},b_{j-1},i)$ contains only complete components of $Q(a_{l},\linebreak[4]b_{l-1},i)$ for each $l<j$. Each such component is either contained in $Q(a_{j},b_{j-1},i)$ or disjoint from it. The same holds for the translations $Q(a_{l},b_{l-1},i)+v(a_{l})$ (we need here that $a_{l}>4a_{l-1}$ so let us assume this from now on). For brevity, define $v_{j}=v(a_{j})$. We may now define two graphs, denoted by $H_{k-1}^{\textrm{e}}$ and $H_{k-1}^{o}$ (“e” and “o” standing for even and odd) by $H_{k-1}^{\textrm{e/o}}:=\bigcap_{\begin{subarray}{c}2\leq j\leq k-1\\\ j\textrm{ even/odd}\end{subarray}}(Q_{a_{j},b_{j-1}}+v_{j}).$ We shall usually suppress the $k-1$ from the notation. It is not difficult to check that $H^{\textrm{e/o}}$ are both roughly isometric to $\mathbb{Z}^{22}$ (the rough isometry constant depends on the “past” $a_{1},\dotsc,a_{k-1})$. Therefore by lemma 1 we see that there exists an $\alpha$ (again, depending on the past) such that $p_{H^{\textrm{e/o}}}(x,x;t)\leq\alpha t^{-11}.$ (5) Examine now the graphs $F_{k-1}^{\textrm{e/o}}:=H_{k-1}^{\textrm{e/o}}\cap\left\\{\vec{n}\in\mathbb{Z}^{22}:\left\|n_{i}\right\|\leq b_{k-1}\>\forall i=4,\dotsc,22\right\\}.$ $F^{\textrm{e/o}}$ are both roughly isometric to $\mathbb{Z}^{3}$ so by lemma 2 there exists some $\beta$ such that $\mathbb{P}_{F^{\textrm{e/o}}}(\mbox{the walk returns to }\vec{0}\mbox{ for the first time at }t)\geq\frac{1}{\beta}t^{-3/2}.$ (6) Define $\gamma_{k}:=\left\lceil\max\\{\alpha,\beta\\}\right\rceil$ (as usual, $\left\lceil\cdot\right\rceil$ stands for the upper integer value). With these we can define $a_{k}$ to be any even number satisfying $a_{k}>2\gamma_{k}^{4}+4a_{k-1}$ and such that $a_{k-1}$ divides $\frac{1}{2}a_{k}$. This completes the description of the induction, and we define $H_{\infty}^{\textrm{e/o}}:=\bigcap_{\begin{subarray}{c}2\leq j\\\ j\textrm{ even/odd}\end{subarray}}(Q_{a_{j},a_{j-1}}+v_{j}).$ These graphs will be the two halves of our target graph $G$. Before continuing, let us collect some simple facts about $H_{\infty}^{\textrm{e/o}}$: 1. (i) $H_{\infty}^{\textrm{e/o}}$ is connected — in fact we used this indirectly when we claimed $H_{k}^{\textrm{e/o}}$ are roughly isometric to $\mathbb{Z}^{22}$. 2. (ii) $H_{\infty}^{\textrm{e/o}}$ are transient — this follows because each contains a copy of $\mathbb{Z}^{3}$ (namely $\\{n_{4}=\dotsb=n_{22}=0\\}$) and transience is preserved on adding edges. This last fact follows from conductance arguments, see e.g. [DS84]. Define therefore the escape probabilities $\varepsilon^{\textrm{e/o}}:=\mathbb{P}_{H_{\infty}^{\textrm{e/o}}}^{\vec{0}}(R(t)\neq\vec{0}\,\forall t>0)$ ($R$ being the random walk on the graph) and let $\delta:=\frac{1}{2}\min\\{\varepsilon^{\textrm{e}},\varepsilon^{\textrm{o}}\\}$. Define the graph $G$ by connecting $H_{\infty}^{\textrm{e}}$ to $H_{\infty}^{\textrm{o}}$ with a single edge between the two $\vec{0}$ with weight $\delta$. Define $x:=\vec{0}^{\textrm{e}}$ and $y=\vec{0}^{\textrm{o}}$. This is our construction and we need to show (2), which will follow if we show that, for $k$ sufficiently large, $\left.\begin{aligned} p(x,x;t_{2k})&\geq 3p(y,y;t_{2k})\\\ p(x,x;t_{2k+1})&\leq{\textstyle\frac{1}{3}}p(y,y;t_{2k+1})\end{aligned}\right\\}\quad t_{k}:=\gamma_{k}^{4}.$ (7) We will only prove the even case, the odd will follow similarly. Examine therefore $p(x,x;t_{2k})$. Since $a_{2k}>t_{2k}$ we get that $H_{\infty}^{\textrm{e/o}}\cap[-t_{2k},t_{2k}]^{22}=H_{2k}^{\textrm{e/o}}\cap[-t_{2k},t_{2k}]^{22}$ or in other words, the steps after $2k$ do not effect us at all. Similarly it is possible to simplify the last stage namely $\displaystyle H_{2k}^{\textrm{e}}\cap[-t_{2k},t_{2k}]^{22}$ $\displaystyle=H_{2k-1}^{\textrm{e}}\cap(Q_{a_{2k},b_{2k-1}}+v_{2k})\cap[-t_{2k},t_{2k}]^{22}=$ $\displaystyle=H_{2k-1}^{\textrm{e}}\cap\left\\{\vec{n}\in\mathbb{Z}^{22}:\left\|n_{i}\right\|\leq b_{2k-1}\>\forall i=4,\dotsc,22\right\\}=F_{2k-1}^{\textrm{e}}$ (here is where these translations by $v_{j}$ are used). By (6), $\displaystyle p_{G}(x,x;t_{2k})$ $\displaystyle\geq\frac{1}{2}\mathbb{P}_{H_{2k}^{e}}(R\mbox{ returns to }x\mbox{ for the first time at }t)\geq$ $\displaystyle\stackrel{{\scriptstyle\textrm{(\ref{eq:beta})}}}{{\geq}}\frac{1}{2\gamma_{2k}}t_{2k}^{-3/2}=\frac{1}{2}t_{2k}^{-7/4}$ (8) (the $\frac{1}{2}$ comes from the first step). To estimate $p(y,y;t_{2k})$ we divide the event $\\{R(t_{2k})=y\\}$ according to whether $R$ “essentially goes through $x$” or not. Formally, denote by $T_{1}$ and $T_{2}$ the first and last time before $t_{2k}$ when $R(T)=x$ (if this does not happen, denote $T_{1}=\infty$ and $T_{2}=-\infty$). Then we define $\displaystyle p_{1}:=\mathbb{P}(T_{1}>\gamma_{2k},\,R(t_{2k})=y)\qquad p_{2}:=\mathbb{P}(T_{2}<t_{2k}-\gamma_{2k},\,R(t_{2k})=y)$ $\displaystyle p_{3}:=\mathbb{P}(T_{1}\leq\gamma_{2k},\,T_{2}\geq t_{2k}-\gamma_{2k},\,R(t_{2k})=y)$ so that $p(y,y;t_{2k})\leq p_{1}+p_{2}+p_{3}$. Now, $p_{1}$ and $p_{2}$ are easy to estimate. As above we have $H_{2k}^{\textrm{o}}\cap[-t_{2k},t_{2k}]^{22}=H_{2k-1}^{\textrm{o}}\cap[-t_{2k},t_{2k}]^{22}$ so (5) applies and we get $\mathbb{P}_{H_{\infty}^{\textrm{o}}}^{y}(R(t)=y)\leq\gamma_{2k}t^{-11}\quad\forall t\leq t_{2k}.$ (9) Therefore $\displaystyle p_{1}$ $\displaystyle\leq\sum_{t=\gamma_{2k}}^{t_{2k}-1}\mathbb{P}(T_{1}=t,\,R(t_{2k})=y)+\mathbb{P}(T_{1}=\infty,\,R(t_{2k})=y)\leq$ $\displaystyle\leq\sum_{t=\gamma_{2k}-1}^{t_{2k}-2}\mathbb{P}_{H_{\infty}^{\textrm{o}}}(R(t)=y)+\mathbb{P}_{H_{\infty}^{\textrm{o}}}(R(t_{2k})=y)\leq$ $\displaystyle\stackrel{{\scriptstyle\textrm{(\ref{eq:yy})}}}{{\leq}}\sum_{t=\gamma_{2k}-1}^{t_{2k}-2}\gamma_{2k}\cdot t^{-11}+\gamma_{2k}\cdot t_{2k}^{-11}\leq C\gamma_{2k}^{-9}=Ct_{2k}^{-9/4}\stackrel{{\scriptstyle\textrm{(\ref{eq:pxxle})}}}{{=}}o(p(x,x;t))$ (10) and similarly for $p_{2}$. As for $p_{3}$, we have $\mathbb{P}(T_{1}\leq\gamma_{2k})\leq\Big{(}\sum_{i=0}^{\infty}\mathbb{P}_{H_{\infty}^{\textrm{o}}}(r\textrm{ visits }y\textrm{ }i\textrm{ times before }\gamma_{2k})\Big{)}\cdot\delta\leq\frac{\delta}{\epsilon^{\textrm{o}}}\leq\frac{1}{2}$ and similarly (using time reversal) for $\mathbb{P}(T_{2}\geq t_{2k}-\gamma_{2k})$. Hence we get $p_{3}\leq\frac{1}{4}\max_{t_{2k}-2\gamma_{2k}\leq s\leq t_{2k}}p(x,x;s)$ and by lemma 3, $\displaystyle p_{3}$ $\displaystyle\leq\frac{1}{4}p(x,x;t_{2k})\left(1+O\left(\frac{\gamma_{2k}\log^{3}t_{2k}}{\sqrt{t_{2k}}}\right)\right)+O(e^{-c\log^{2}t_{2k}})\leq$ $\displaystyle\stackrel{{\scriptstyle\textrm{(\ref{eq:pxxle})}}}{{\leq}}\frac{1}{4}p(x,x;t_{2k})(1+o(1)).$ With the estimate (10) for $p_{1}$ and the corresponding estimate for $p_{2}$ we get $p(y,y;t_{2k})\leq p(x,x;t_{2k})\left(\frac{1}{4}+o(1)\right).$ A completely symmetric argument shows that at $t_{2k+1}$ the opposite occurs: $p(x,x;t_{2k+1})\leq p(y,y;t_{2k+1})\left(\frac{1}{4}+o(1)\right)$ proving the theorem.∎ ###### Remark. If you want an example with unweighted graphs, this is not a problem. $H^{e}$ and $H^{o}$ are already unweighted, so the only thing needed is to connect them, instead of with an edge of weight $\delta$, with a segment sufficiently long such that the probability to traverse it is $\leq\delta$. The proof remains essentially the same. ## 2\. Manifolds We would like to show a Manifold $M$ and two points $x,y\in M$ such that the heat kernel on $M$ satisfies $\frac{p(x,x;t)}{p(y,y;t)}\nrightarrow$ as $t\to\infty$. Here is how one might translate the construction of our theorem to the settings of manifolds. The dimension of the manifold plays little role, so we might as well construct a surface. For a subset $H\subset\mathbb{Z}^{22}$ one can associate a manifold $H^{}$ by replacing each vertex $v\in H$ with a sphere $v^{}$ and every edge with a empty, baseless cylinder. Since the degree of every vertex in $H$ is $\leq 44$ we may simply designate 44 disjoint circles on $\mathbb{S}^{2}$ and attach the cylinders to the spheres at these circles. This is reminiscent of the well- known “infinite jungle gym” construction, see some lovely pictures in [ON]. The exact method of doing so is unimportant since anyway the manifold that we get is roughly isometric to $H$, considered as an induced subgraph of $\mathbb{Z}^{22}$ (one of the nice features of rough isometry is that continuous and discrete objects may be roughly isometric, rough isometry inspects only the large scale geometry). Clearly $H^{}$ can be made $C^{\infty}$. One can then construct a (possibly different) sequence $a_{k}$ and two manifolds $\big{(}H_{\infty}^{\textrm{e/o}}\big{)}^{}$ with the only difference is that the $\alpha$ and $\beta$ must satisfy (5) and (6) for our choice of the ∗ operation. This should be possible since $\big{(}H_{k}^{\textrm{e/o}}\big{)}^{}$ and $\big{(}F_{k}^{\textrm{e/o}}\big{)}^{}$ are roughly isometric to $\mathbb{Z}^{22}$ and $\mathbb{Z}^{3}$ respectively. Instead of Delmotte one can use the manifold version [SC95] (or rather, [D99] is the graph version of earlier results for manifolds, see [SC95] for historical remarks). The argument for the transience of $\big{(}H_{\infty}^{\textrm{e/o}}\big{)}^{}$ should also be direct translation. Each contains a submanifold (with boundary) which is roughly isometric to $\mathbb{Z}^{3}$ and therefore is transient. Since transience is equivalent to the fact that for some $c>0$ every function which is $1$ at $x$ and $0$ at infinity satisfies that the Dirichlet form $\langle\nabla f,\nabla f\rangle>c$, and since restricting to a submanifold only decreases the Dirichlet form, we see that $\big{(}H_{\infty}^{\textrm{e/o}}\big{)}^{}$ are transient. Denote by $\epsilon^{\textrm{e/o}}=\inf_{x\in v^{},v\sim\vec{0}^{\textrm{e/o}}}\mathbb{P}^{x}\left(W[0,\infty)\cap\big{(}\vec{0}^{\textrm{e/o}}\big{)}^{}=\emptyset\right)$ where $W$ here is the Brownian motion on the manifold $\big{(}H_{\infty}^{\textrm{e/o}}\big{)}^{}$; and where the infimum is taken over all $x$ belonging to a sphere $v^{}$ where $v$ is some neighbor of $\vec{0}^{\textrm{e/o}}$ in $H_{\infty}^{\textrm{e/o}}$. One can now define $\delta=\frac{1}{2}\min(\varepsilon^{\textrm{e}},\varepsilon^{\textrm{o}})$ and connect $\vec{0}^{\textrm{e}}$ to $\vec{0}^{\textrm{o}}$ by a cylinder sufficiently thin (or sufficiently long) such that the probability to traverse it in either direction before reaching a neighboring sphere is $\leq\delta$. This concludes a possible construction of a manifold $M$, and one may take $x$ to be an arbitrary point in $\big{(}\vec{0}^{\textrm{e}}\big{)}^{}$ and $y$ and arbitrary point in $\big{(}\vec{0}^{\textrm{o}}\big{)}^{}$. The proof that $p(x,x;t)/p(y,y;t)$ does not converge should not require significant changes. We note that in our case it is possible for a Brownian motion at time $t$ to exit the box $[-t,t]^{22}$, but it is exponentially difficult to do so. Hence, for example, instead of (8) we get $p(x,x;t_{2K})\geq\frac{1}{2\gamma_{2k}^{2}}t_{2k}^{-3/2}-Ce^{-ct_{2k}}\geq\frac{1}{4}t_{2k}^{-7/4}$ for $k$ sufficiently large. Another point to note is that lemma 3 needs to be replaced with an appropriate analog. ## References * [ABJ02] Jean-Philippe Anker, Philippe Bougerol and Thierry Jeulin, _The infinite Brownian loop on a symmetric space_. Rev. Mat. Iberoamericana 18:1 (2002), 41–97. Available at: projecteuclid.org * [C60] Kai Lai Chung, _Markov chains with stationary transition probabilities_. Die Grundlehren der mathematischen Wissenschaften 104, Springer-Verlag, 1960. * [D82] Brian E. Davies, _Metastable states of symmetric Markov semigroups. II_. J. London Math. Soc. 26:3 (1982), 541–556. Available at: oxfordjournals.org * [D97] Brian E. Davies, _Non-Gaussian aspects of heat kernel behaviour_. J. London Math. Soc. 55:1 (1997), 105–125. Available at: oxfordjournals.org * [D38] Wolfgang Doeblin, _Sur deux problèmes de M. Kolmogoroff concernant les chaînes dénombrables_. Bull. de la Soc. Math. de France, 66 (1938), 210–220. Available at: numdam.org * [DS84] Peter G. Doyle and Laurie J. Snell, _Random walks and electric networks_. Carus Mathematical Monographs, 22. Mathematical Association of America, Washington DC, 1984. Available at: dartmouth.edu/~doyle * [D99] Thierry Delmotte, _Parabolic Harnack inequality and estimates of Markov chains on graphs_ , Rev. Mat. Iberoamericana 15:1 (1999), 181–232. Available at: rsme.es * [K85] Masahiko Kanai, _Rough isometries, and combinatorial approximations of geometries of non-compact Riemannian manifolds_. J. Math. Soc. Japan 37:3 (1985), 391–413. projecteuclid.org * [ON] Barrett O’Neill, _An infinite jungle gym_. Available at: ucla.edu/~bon * [O61] Steven Orey, _Strong ratio limit property_. Bull. Amer. Math. Soc. 67:5 (1961), 571–574. Available at: ams.org * [PSC] Christophe Pittet and Laurent Saloff-Coste, _A survey on the relationships between volume growth, isoperimetry, and the behavior of simple random walk on Cayley graphs, with examples_. Available from: cornell.edu/~lsc * [SC95] Laurent Saloff-Coste, _Parabolic Harnack inequality for divergence-form second-order differential operators_. Potential Anal. 4:4 (special issue, 1995), 429–467. Available at: springerlink.com	arxiv-papers	2011-11-19T23:28:07	2024-09-04T02:49:24.502353	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Gady Kozma", "submitter": "Gady Kozma", "url": "https://arxiv.org/abs/1111.4593" }
1111.4712	# An $L_{p}$-theory of stochastic parabolic equations with the random fractional Laplacian driven by Lévy processes Kyeong-Hun Kim111Department of Mathematics, Korea University, 1 Anam-dong, Sungbuk-gu, Seoul 136-701, Republic of Korea. E-mail: kyeonghun@korea.ac.kr. The research of this author was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education, Science and Technology (20090087117). and Panki Kim222 Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, San56-1 Shinrim-dong Kwanak-gu, Seoul 151-747, Republic of Korea. E-mail: pkim@snu.ac.kr. This work was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) grant funded by the Korea government(MEST)(2010-0001984). ###### Abstract In this paper we give an $L_{p}$-theory for stochastic parabolic equations with random fractional Laplacian operator. The driving noises are general Lévy processes. Keywords: Fractional Laplacian, Stochastic partial differential equations, Lévy processes, $L_{p}$-theory. AMS 2000 subject classifications: 60H15, 35R60. ## 1 Introduction Let $d,m\geq 1$ be positive integers, $p\in[2,\infty)$ and $\alpha\in(0,2)$. As usual $\mathbb{R}^{d}$ stands for the Euclidean space of points $x=(x^{1},...,x^{d})$. We will use $dx$ to denote the Lebesgue measure in either $\mathbb{R}^{d}$ or $\mathbb{R}^{m}$, which is clear in each context. In this article we are dealing with $L_{p}$-theory of the stochastic partial differential equations of the type $du=\left(a(\omega,t)\Delta^{\alpha/2}u+f(u)\right)dt\,+\sum_{k=1}^{\infty}g^{k}(u)\cdot dZ^{k}_{t},\quad u(0)=u_{0}$ (1.1) given for $\omega\in\Omega,t\geq 0$ and $x\in\mathbb{R}^{d}$. Here $\Omega$ is a probability space, $\Delta^{\alpha/2}$ is the fractional Laplacian defined in (2.2), $Z^{k}_{t}$ are independent $m$-dimensional Lévy processes, and the functions $f$ and vector-valued function $g^{k}=(g^{k,1},\dots g^{k,m})$ depend on $(\omega,t,x,u)$ satisfying certain continuity conditions. Our result will cover the case $\displaystyle f(u)$ $\displaystyle=b(\omega,t,x)\Delta^{\beta_{1}/2}u+c^{i}(\omega,t,x)u_{x^{i}}I_{\alpha>1}+d(\omega,t,x)u+f_{0},$ $\displaystyle g^{k,j}(u)$ $\displaystyle=\sigma^{k,j}(\omega,t,x)\Delta^{\beta^{j}_{2}/2}u+\nu^{k,j}(\omega,t,x)u+g^{k,j}_{0},\quad j=1,\dots m.$ where $\beta_{1}<\alpha$ and $\beta^{j}_{2}<\alpha/2$ (see Assumptions 2.13 and 3.7). An $L_{p}$-theory of (1.1) is introduced in [7] for the case that $a(\omega,t)=1$ and are only finitely many Winer processes appear in the equation. The approach in [7] cannot cover the case when there are infinitely many Wiener processes, and the assumptions on $g$ in [7] are stronger than conditions in our paper (See Remark 2.12 below). Moreover equations driven by jump processes are not considered in [7]. A Hölder space theory for more general (but non-random) integro-differential equations driven by Hilbert space-valued Wiener process is given in [19] (also see [18] for a deterministic equation). Even though the main result in [19] provide a nice Hölder regularity of the solution to such problem, due to the Hölder-type function spaces defined there, assumptions on $f$ and $g$ are quite strong. Furthermore in [19] the equations with discontinuous Lévy processes are not considered. We emphasize that the approach of this paper, based on $L_{p}$ theory in [14], is different from [19]. Our results include the case when $f$ and $g$ are only distributions and the number of derivatives of $f$ and $g$ are negative and fractional. On the other hand if $f$ and $g$ are sufficiently smooth in $x$ then Sobolev embedding theorem combined with our $L_{p}$-theory gives pointwise Hölder continuity of the solution even when $Z^{k}$ are general Lévy processes. $L_{p}$-theory for second-order stochastic parabolic equations driven by Wiener processes was first established by Krylov [14]. Recently in [6] $L_{p}$ regularity theory for second-order stochastic parabolic equations driven by Lévy processes is discussed. In this paper, we establish an $L_{p}$-theory for stochastic parabolic equations with the random fractional Laplacian driven by arbitrary Lévy processes. Our result includes the case when the equation is driven by Lévy space-time white noise (see Theorems 4.3 and 4.4). Among main tools used in the article to study $L_{p}$-regularity theory are Burkholder-Davis-Gundy inequality and a parabolic version of Littlewood-Paley inequality for the fractional Laplacian operator introduced [11]. The organization of this article is as follows. First, in section 2, we prove uniqueness and existence results of equation (1.1) driven by Wiener processes in the space $L_{p}(\Omega\times[0,T],H^{\gamma+\alpha/2}_{p})$ (Theorem 2.15). Here $p\in[2,\infty)$ and $\gamma\in\mathbb{R}$. In section 3 we extend Theorem 2.15 for the case when $Z^{k}_{t}$ are Lévy processes and $Z^{k}_{t}$ have finite $p$-th moments (see condition (3.2)). In section 4, the condition (3.2) is weakened, and the uniqueness and existence results are proved in the space $L_{p,\text{loc}}(\Omega\times[0,T],H^{\gamma+\alpha/2}_{p})$. The condition (3.2) can be completely dropped if only finitely many Lévy processes appear in the equation. If we write $c=c(...)$, this means that the constant $c$ depends only on what are in parenthesis. The constant $c$ stands for constants whose values are unimportant and which may change from one appearance to another. The dependence of the lower case constants on the dimensions $d,m$ may not be mentioned explicitly. We will use “$:=$” to denote a definition, which is read as “is defined to be”. For $a,b\in\mathbb{R}$, $a\wedge b:=\min\\{a,b\\}$ and $a\vee b:=\max\\{a,b\\}$. Let $C^{\infty}_{0}(\mathbb{R}^{d})$ be the collection of smooth functions with compact supports in $\mathbb{R}^{d}$. Most of functions we discuss in this paper are random (depend on $\omega\in\Omega$). For notational convenience, we suppress the dependency on $\omega$ in most of expressions ## 2 Stochastic Parabolic equations with the random fractional Laplacian driven by Wiener processes Let $(\Omega,\mathcal{F},P)$ be a complete probability space, $\\{\mathcal{F}_{t},t\geq 0\\}$ be an increasing filtration of $\sigma$-fields $\mathcal{F}_{t}\subset\mathcal{F}$, each of which contains all $(\mathcal{F},P)$-null sets. We assume that on $\Omega$ we are given independent one-dimensional Wiener processes $W^{1}_{t},W^{2}_{t},...$ relative to $\\{\mathcal{F}_{t},t\geq 0\\}$. Let $\mathcal{P}$ be the predictable $\sigma$-field generated by $\\{\mathcal{F}_{t},t\geq 0\\}$. Let $p(t,x)$, where $t>0$, denote the inverse Fourier transform of $e^{-\|\xi\|^{\alpha}t}$ in $\mathbb{R}^{d}$, that is, $p(t,x):=\frac{1}{(2\pi)^{d/2}}\int_{\mathbb{R}^{d}}e^{i\xi\cdot x}e^{-\|\xi\|^{\alpha}t}d\xi.$ For a suitable function $g$ and $t>0$, define the corresponding convolution operator $T_{t}g(x):=(p(t,\cdot)g(\cdot))(x):=\int_{\mathbb{R}^{d}}p(t,x-y)g(y)dy,$ (2.1) and define $\partial^{\alpha}_{x}g(x)={\Delta}^{\frac{\alpha}{2}}g(x)=-(-{\Delta})^{\frac{\alpha}{2}}g(x):=\mathcal{F}^{-1}(-\|\xi\|^{\alpha}\mathcal{F}(g)(\xi))(x),$ (2.2) where $\mathcal{F}(g)(\xi)=\hat{g}(\xi):=\frac{1}{(2\pi)^{d/2}}\int_{\mathbb{R}^{d}}e^{-i\xi\cdot x}g(x)dx$ is the Fourier transform of $g$ in $\mathbb{R}^{d}$. In this section we study the nonlinear equations of the type $du=\left(a(\omega,t)\Delta^{\alpha/2}u+f(u)\right)\,dt+\sum_{k=1}^{\infty}g^{k}(u)dW^{k}_{t},\quad u(0)=u_{0},$ (2.3) where $a(\omega,t)\in(\delta,\delta^{-1})$ for some $\delta>0$, and $f(u)=f(\omega,t,x,u)$ and $g^{k}(u)=g^{k}(\omega,t,x,u)$ satisfy certain continuity conditions, which we will put below. First we introduce some stochastic Banach spaces. Let $(\phi,\psi):=\int_{\mathbb{R}^{d}}\phi(x)\psi(x)dx$ and for $p\geq 1$, $L_{p}=L_{p}(\mathbb{R}^{d}):=\\{\phi:\mathbb{R}^{d}\to\mathbb{R},\\|\phi\\|^{p}_{p}:=\int_{\mathbb{R}^{d}}\|\phi(x)\|^{p}dx<\infty\\}.$ For $n=0,1,2,...$, define $H^{n}_{p}=H^{n}_{p}(\mathbb{R}^{d}):=\left\\{u:u,Du,...,D^{n}u\in L_{p}(\mathbb{R}^{d})\right\\}.$ In general, for $\gamma\in\mathbb{R}$ define the space $H^{\gamma}_{p}=H^{\gamma}_{p}(\mathbb{R}^{d})=(1-\Delta)^{-\gamma/2}L_{p}$ (called the space of Bessel potentials or the Sobolev space with fractional derivatives) as the set of all distributions $u$ on $\mathbb{R}^{d}$ such that $(1-\Delta)^{\gamma/2}u\in L_{p}$. For $u\in H^{\gamma}_{p}$, we define $\\|u\\|_{H^{\gamma}_{p}}:=\\|(1-\Delta)^{\gamma/2}u\\|_{p}:=\\|\mathcal{F}^{-1}[(1+\|\xi\|^{2})^{\gamma/2}\mathcal{F}(u)(\xi)]\\|_{p},$ (2.4) where $\mathcal{F}$ is the Fourier transform in $\mathbb{R}^{d}$. For $\ell_{2}$-valued $g=(g^{1},g^{2},\dots)$, we define $\\|g\\|_{H^{\gamma}_{p}(\ell_{2})}:=\\|(1-\Delta)^{\gamma/2}g\|_{\ell_{2}}\\|_{p}:=\\|\mathcal{F}^{-1}[(1+\|\xi\|^{2})^{\gamma/2}\mathcal{F}(g)(\xi)]\|_{\ell_{2}}\\|_{p}.$ Let $\overline{\mathcal{P}}$ be the completion of $\mathcal{P}$ with respect to $dP\times dt$, and $\mathbb{H}^{\gamma}_{p}(T):=L_{p}(\Omega\times[0,T],\overline{\mathcal{P}},H^{\gamma}_{p})$, that is, $\mathbb{H}^{\gamma}_{p}(T)$ is the set of all $\overline{\mathcal{P}}$-measurable processes $u:\Omega\times[0,T]\to H^{\gamma}_{p}$ so that $\\|u\\|_{\mathbb{H}^{\gamma}_{p}(T)}:=\left({\mathbb{E}}\left[\int^{T}_{0}\,\\|u(\omega,t)\\|^{p}_{H^{\gamma}_{p}}\,dt\right]\right)^{1/p}<\infty.$ ###### Lemma 2.1 For any $\beta>0$, $\eta^{1}_{\beta}(\xi):=\frac{(1+\|\xi\|^{2})^{\beta/2}}{1+\|\xi\|^{\beta}}$, $\eta^{2}_{\beta}=(\eta^{1}_{\beta})^{-1}$, $\eta^{3}:=\frac{\|\xi\|^{\beta}}{1+\|\xi\|^{\beta}}$ and $\eta^{4}:=\frac{\|\xi\|^{\beta}}{(1+\|\xi\|^{2})^{\beta/2}}$ are $L^{p}(\mathbb{R}^{d})$-multipliers, that is, $\\|\mathcal{F}^{-1}\left(\eta^{i}_{\beta}(\xi)(\mathcal{F}u)(\xi)\right)\\|_{L_{p}}\leq c(p,\beta)\\|u\\|_{L_{p}},\quad\quad i=1,2,3,4.$ Proof. See Theorem 0.2.6 of [24] (also see the remark below the theorem). $\Box$ Lemma 2.36 easily yields the following results. ###### Corollary 2.2 (i) Let $\gamma\geq 0$. There exists a constant $c=c(\gamma)>0$ so that $c\\|u\\|_{H^{\gamma}_{p}}\leq(\\|u\\|_{L_{p}}+\\|\partial^{\gamma/2}_{x}u\\|_{L_{p}})\leq c^{-1}\\|u\\|_{H^{\gamma}_{p}}.$ (ii) For any $\beta\in\mathbb{R}$, $\\|\Delta^{\alpha/2}u\\|_{H^{\beta}_{p}}\leq c(\alpha,\beta)\\|u\\|_{H^{\beta+\alpha}_{p}}.$ ###### Remark 2.3 Let $\gamma,\beta\geq 0$. Then due to the well-known inequality $\\|u\\|_{L_{p}}\leq\varepsilon\\|u\\|_{H^{\gamma}_{p}}+c(\varepsilon,\gamma,\beta)\\|u\\|_{H^{-\beta}_{p}},$ it also follows $\\|u\\|_{H^{\gamma}_{p}}\leq c(\gamma,\beta,p)(\\|u\\|_{H^{-\beta}_{p}}+\\|\partial^{\gamma/2}_{x}u\\|_{L_{p}}).$ For $\ell_{2}$-valued $\overline{\mathcal{P}}$-measurable processes $g=(g^{1},g^{2},\dots)$, we write $g\in\mathbb{H}^{\gamma}_{p}(T,\ell_{2})$ if $\\|g\\|_{\mathbb{H}^{\gamma}_{p}(T,\ell_{2})}:=\left({\mathbb{E}}\int^{T}_{0}\\|\,\|(1-\Delta)^{\gamma/2}g(\omega,t)\|_{\ell_{2}}\,\\|^{p}_{p}\,dt\right)^{1/p}<\infty.$ (2.5) Denote $\mathbb{L}_{p}(T):=\mathbb{H}^{0}_{p}(T)$ and $\mathbb{L}_{p}(T,\ell_{2})=\mathbb{H}^{0}_{p}(T,\ell_{2})$. Finally, we say $u_{0}\in U^{\gamma}_{p}$ if $u_{0}$ is $\mathcal{F}_{0}$-measurable function $\Omega\to H^{\gamma}_{p}$ and $\\|u_{0}\\|_{U^{\gamma}_{p}}:=\left({\mathbb{E}}\left[\\|u_{0}\\|^{p}_{H^{\gamma}_{p}}\right]\right)^{1/p}<\infty.$ ###### Remark 2.4 It is easy to check (see Remark 3.2 in [14] for detailed proof) that for any $\gamma\in(-\infty,\infty)$, $g\in\mathbb{H}^{\gamma}_{p}(T,\ell_{2})$ and $\phi\in C^{\infty}_{0}(\mathbb{R}^{d})$ we have $\sum_{k=1}^{\infty}\int^{T}_{0}(g^{k}(\omega,t),\phi)^{2}dt<\infty$ a.s., and consequently the series of stochastic integral $\sum_{k=1}^{\infty}\int^{t}_{0}(g^{k}(\omega,s),\phi)dW^{k}_{s}$ converges uniformly in $t$ in probability on $[0,T]$. ###### Definition 2.5 Write $u\in\mathcal{H}^{\gamma+\alpha}_{p}(T)$ if $u\in\mathbb{H}^{\gamma+\alpha}_{p}(T),u(0)\in U^{\gamma+\alpha-\alpha/p}_{p}$, and for some $f\in\mathbb{H}^{\gamma}_{p}(T)$ and $g\in\mathbb{H}^{\gamma+\alpha/2}_{p}(T,\ell_{2})$ $du=fdt+\sum_{k=1}^{\infty}g^{k}dW^{k}_{t},\quad\hbox{for }t\in[0,T]$ in the sense of distributions, that is, for any $\phi\in C^{\infty}_{0}(\mathbb{R}^{d})$, $(u(t),\phi)=(u(0),\phi)+\int^{t}_{0}(f(s),\phi)ds+\sum_{k=1}^{\infty}\int^{t}_{0}(g^{k}(s),\phi)dW^{k}_{s}$ (2.6) holds for all $t\leq T$ $a.s.$. In this case we write $\mathbb{D}u:=f,\quad\mathbb{S}^{k}u:=g^{k},\quad\mathbb{S}u:=(\mathbb{S}^{1}u,\dots,\mathbb{S}^{k}u,\dots)$ and define the norm $\\|u\\|_{\mathcal{H}^{\gamma+\alpha}_{p}(T)}:=\\|u\\|_{\mathbb{H}^{\gamma+\alpha}_{p}(T)}+\\|\mathbb{D}u\\|_{\mathbb{H}^{\gamma}_{p}(T)}+\\|\mathbb{S}u\\|_{\mathbb{H}^{\gamma+\alpha/2}_{p}(T,\ell_{2})}+\\|u(0)\\|_{U^{\gamma+\alpha-\alpha/p}_{p}}.$ (2.7) ###### Theorem 2.6 The space $\mathcal{H}^{\gamma+\alpha}_{p}(T)$ is a Banach space, and for every $0<t\leq T$ ${\mathbb{E}}\Big{[}\sup_{s\leq t}\\|u(s,\cdot)\\|^{p}_{H^{\gamma}_{p}}\Big{]}\leq c(p,T,\alpha)\left(\\|\mathbb{D}u\\|^{p}_{\mathbb{H}^{\gamma}_{p}(t)}+\\|\mathbb{S}u\\|^{p}_{\mathbb{H}^{\gamma}_{p}(t,\ell_{2})}+\\|u(0)\\|^{p}_{U^{\gamma}_{p}}\right).$ (2.8) In particular, for any $t\leq T$, $\\|u\\|^{p}_{\mathbb{H}^{\gamma}_{p}(t)}\leq c(p,T,\alpha)\int^{t}_{0}\\|u\\|^{p}_{\mathcal{H}^{\gamma+\alpha}_{p}(s)}\,ds.$ (2.9) ###### Remark 2.7 Note that $\alpha$ is not involved in (2.8). Proof. See Theorem 3.7 in [14]. Actually in [14] the theorem is proved only for $\alpha=2$, but the proof works for any $\alpha\in(0,2)$. We will give the detailed proof of Theorem 3.4 below, which is the counterpart of Theorem 2.6 for pure-jump Lévy processes. $\Box$ ###### Remark 2.8 It follows from (2.4) that for any $\mu,\gamma\in\mathbb{R}$, the operator $(1-\Delta)^{\mu/2}:H^{\gamma}_{p}\to H^{\gamma-\mu}_{p}$ is an isometry. Indeed, $\\|(1-\Delta)^{\mu/2}u\\|_{H^{\gamma-\mu}_{p}}=\\|(1-\Delta)^{(\gamma-\mu)/2}(1-\Delta)^{\mu/2}u\\|_{p}=\\|(1-\Delta)^{\gamma/2}u\\|_{p}=\\|u\\|_{H^{\gamma}_{p}}.$ The same reason shows that $(1-\Delta)^{\mu/2}:\mathcal{H}^{\gamma}_{p}(T)\to\mathcal{H}^{\gamma-\mu}_{p}(T)$ is an isometry. ###### Theorem 2.9 (i) For any deterministic functions $f=f(t,x)$ and $u_{0}=u_{0}(x)$ with $\int^{T}_{0}\,\\|f(t,\cdot)\\|^{p}_{H^{\gamma}_{p}}\,dt<\infty,\quad\\|u_{0}\\|_{H^{\gamma+\alpha-\alpha/p}_{p}}<\infty,$ the (deterministic) equation $u_{t}=\Delta^{\alpha/2}u+f,\quad u(0)=u_{0}$ has a unique solution $u$ with $\int^{T}_{0}\,\\|u(t,\cdot)\\|^{p}_{H^{\gamma+\alpha}_{p}}\,dt<\infty$, and for every $0<t\leq T$ $\int^{t}_{0}\,\\|u(s,\cdot)\\|^{p}_{H^{\gamma+\alpha}_{p}}ds\leq c(p,T)\left(\int^{t}_{0}\,\\|f(s,\cdot)\\|^{p}_{H^{\gamma}_{p}}\,ds+\\|u_{0}\\|^{p}_{H^{\gamma+\alpha-\alpha/p}_{p}}\right).$ (2.10) (ii) For any $f\in\mathbb{H}^{\gamma}_{p}(T)$ and $u_{0}\in U^{\gamma+\alpha-\alpha/p}_{p}$, the equation $u_{t}=\Delta^{\alpha/2}u+f,\quad u(0)=u_{0}$ (2.11) has a unique solution $u\in\mathbb{H}^{\gamma+\alpha}_{p}(T)$ and for every $0<t\leq T$ $\\|u\\|_{\mathcal{H}^{\gamma+\alpha}_{p}(t)}\leq c(p,T)\left(\\|f\\|_{\mathbb{H}^{\gamma}_{p}(t)}+\\|u_{0}\\|_{U^{\gamma+\alpha-\alpha/p}_{p}}\right).$ (2.12) Proof. (i). See, for instance, Theorem 2.1 in [18]. (ii). This result is also known. See, for instance, Lemma 3.2 and Lemma 3.4 of [7]. Actually since equation (2.11) is deterministic for each fixed $\omega$, the claim of (ii) can be obtained from (i). Indeed, the uniqueness and estimate (2.12) are obvious by (i). For the existence of solution, assume that $u_{0}$ and $f$ are sufficiently smooth in $x$, then using Fourier transform one can easily check that $u(t):=T_{t}u_{0}+\int^{t}_{0}T_{t-s}fds$ solves (2.11) and is in $\mathcal{H}^{\gamma+\alpha}_{p}(T)$. For general $u_{0}$ and $f$ it is enough to use a standard approximation argument (see, for instance, the proof Theorem 2.11). $\Box$ Now we give our assumption on $a(\omega,t)$. ###### Assumption 2.10 The process $a(\omega,t)$ is predictable and there is a constant $\delta>0$ so that $\delta<a(\omega,t)<\delta^{-1},\quad\quad\forall\omega,t.$ Now we present an $L_{p}$-theory for linear stochastic parabolic equations with random fractional Laplacian. ###### Theorem 2.11 Let $p\in[2,\infty)$ and $\gamma\in\mathbb{R}$. For any $f\in\mathbb{H}^{\gamma}_{p}(T)$, $g\in\mathbb{H}^{\gamma+\alpha/2}_{p}(T,\ell_{2})$ and $u_{0}\in U^{\gamma+\alpha-\alpha/p}_{p}$, the linear equation $du=\left(a(\omega,t)\Delta^{\alpha/2}u+f\right)\,dt+\sum_{k=1}^{\infty}g^{k}dW^{k}_{t},\quad u(0)=u_{0},$ (2.13) admits a unique solution $u$ in $\mathcal{H}^{\gamma+\alpha}_{p}(T)$, and for this solution $\\|u\\|_{\mathcal{H}^{\gamma+\alpha}_{p}(T)}\leq c(p,T,\delta)\left(\\|f\\|_{\mathbb{H}^{\gamma}_{p}(T)}+\\|g\\|_{\mathbb{H}^{\gamma+\alpha/2}_{p}(T,\ell_{2})}+\\|u_{0}\\|_{U^{\gamma+\alpha-\alpha/p}_{p}}\right).$ (2.14) ###### Remark 2.12 (i) Recall that the unique solution $u\in\mathcal{H}^{\gamma+\alpha}_{p}(T)$ is understood in the sense of distributions as in Definition 2.5, that is, for any $\phi\in C^{\infty}_{0}(\mathbb{R}^{d})$, $(u(t),\phi)=(u(0),\phi)+\int^{t}_{0}\left(a(\omega,s)(u,\Delta^{\alpha/2}\phi)+(f(s),\phi)\right)ds+\sum_{k=1}^{\infty}\int^{t}_{0}(g^{k}(s),\phi)dW^{k}_{s}$ holds for all $t\leq T$ $a.s.$. (ii) A version of Theorem 2.11 is proved in [7] under stronger conditions on $g$ and the processes. Precisely in [7] it is assumed that $a(\omega,t)=1$, $g\in\mathbb{H}^{\gamma+\alpha/2+\varepsilon}_{p}(T)$, $\varepsilon>0$, and there are only finitely many Wiener processes in equation (2.13). Proof. Step 1. Owing to Remark 2.8, we only need to show that the theorem holds for a particular $\gamma=\gamma_{0}$. Indeed, suppose that the theorem holds when $\gamma=\gamma_{0}$. Then it is enough to notice that $u\in\mathcal{H}^{\gamma+\alpha}_{p}(T)$ is a solution of the equation if and only if $\bar{u}:=(1-\Delta)^{(\gamma-\gamma_{0})/2}u\in\mathcal{H}^{\gamma_{0}+\alpha}_{p}$ is a solution of the equation with $\bar{f}:=(1-\Delta)^{\frac{(\gamma-\gamma_{0})}{2}}f,\quad\bar{g}:=(1-\Delta)^{\frac{(\gamma-\gamma_{0})}{2}}g,\quad\bar{u}_{0}:=(1-\Delta)^{\frac{(\gamma-\gamma_{0})}{2}}u_{0},$ in place of $f,g$ and $u_{0}$, respectively. Furthermore, $\displaystyle\\|u\\|_{\mathbb{H}^{\gamma+\alpha}_{p}(T)}=\\|\bar{u}\\|_{\mathbb{H}^{\gamma_{0}+\alpha}_{p}(T)}$ $\displaystyle\leq$ $\displaystyle c\left(\\|\bar{f}\\|_{\mathbb{H}^{\gamma_{0}}_{p}(T)}+\\|\bar{g}\\|_{\mathbb{H}^{\gamma_{0}+\alpha/2+\varepsilon_{0}}_{p}(T,\ell_{2})}+\\|\bar{u}_{0}\\|_{U^{\gamma_{0}+\alpha-\alpha/p}_{p}}\right)$ $\displaystyle=$ $\displaystyle c\left(\\|f\\|_{\mathbb{H}^{\gamma}_{p}(T)}+\\|g\\|_{\mathbb{H}^{\gamma+\alpha/2+\varepsilon_{0}}_{p}(T,\ell_{2})}+\\|u_{0}\\|_{U^{\gamma+\alpha-\alpha/p}_{p}}\right).$ Step 2. Next we assume $a(\omega,t)=1$ and prove the theorem for the equation: $du=\Delta^{\alpha/2}udt+\sum_{k=1}^{\infty}g^{k}dW^{k}_{t},\quad u(0)=0.$ (2.15) Remember that we may assume $\gamma=-\alpha/2$. Since the uniqueness of (2.15) follows from results for the deterministic equations (Theorem 2.9), we only need to show that there exists a solution $u\in\mathbb{H}^{\alpha/2}_{p}(T)$ of (2.15) and $u$ satisfies estimate (2.14) with $f=u_{0}=0$ and $\gamma=-\alpha/2$. For a moment, assume $N_{0}>0$ is a fixed non-random constant, $g^{k}=0$ for all $k>N_{0}$ and $g^{k}(t,x)=\sum_{i=0}^{m_{k}}I_{(\tau^{k}_{i},\tau^{k}_{i+1}]}(t)g^{k_{i}}(x)\qquad\text{for }k\leq N_{0},$ (2.16) where $\tau^{k}_{i}$ are bounded stopping times and $g^{k_{i}}(x)\in C^{\infty}_{0}(\mathbb{R}^{d})$. Define $v(t,x):=\sum_{k=1}^{N_{0}}\int^{t}_{0}g^{k}(s,x)dW^{k}_{s}=\sum_{k=1}^{N_{0}}\sum_{i=1}^{m_{k}}g^{k_{i}}(x)(W^{k}_{t\wedge\tau^{k}_{i+1}}-W^{k}_{t\wedge\tau^{k}_{i}})$ and $u(t,x):=v(t,x)+\int^{t}_{0}\Delta^{\alpha/2}T_{t-s}v(s,x)\,ds=v(t,x)+\int^{t}_{0}T_{t-s}\Delta^{\alpha/2}v(s,x)\,ds.$ (2.17) Using Fourier transform one can easily show (See, for instance, [7]) that if functions $h_{1}=h_{1}(t,x)$ and $h_{2}=h_{2}(x)$ are sufficiently smooth in $x$ then $w_{1}(t,x):=\int^{t}_{0}T_{t-s}h_{1}(s,x)ds,\quad w_{2}(t,x)=T_{t}h_{2}(x)$ solve $dw_{1}=(\Delta^{\alpha/2}w_{1}+h_{1})\,dt,\quad w_{1}(0)=0,$ $dw_{2}=\Delta^{\alpha/2}w_{2}\,dt,\quad w_{2}(0)=h_{2}.$ Therefore we have $d(u-v)=(\Delta^{\alpha/2}(u-v)+\Delta^{\alpha/2}v)dt=\Delta^{\alpha/2}udt$, and $du=\Delta^{\alpha/2}udt+dv=\Delta^{\alpha/2}udt+\sum_{k=1}^{N_{0}}g^{k}dW^{k}_{t}.$ Also by (2.17) and stochastic Fubini theorem ([22, Theorem 64]), almost surely, $\displaystyle u(t,x)$ $\displaystyle=$ $\displaystyle v(t,x)+\sum_{k=1}^{N_{0}}\int^{t}_{0}\int^{s}_{0}\Delta^{\alpha/2}T_{t-s}g^{k}(r,x)dW^{k}_{r}ds$ (2.18) $\displaystyle=$ $\displaystyle v(t,x)-\sum_{k=1}^{N_{0}}\int^{t}_{0}\int^{t}_{r}\frac{\partial}{\partial s}T_{t-s}g^{k}(r,x)dsdW^{k}_{r}$ $\displaystyle=$ $\displaystyle\sum_{k=1}^{N_{0}}\int^{t}_{0}T_{t-s}g^{k}(s,x)dW^{k}_{s}.$ Hence, $\partial^{\alpha/2}_{x}u(t,x)=\sum_{k=1}^{N_{0}}\int^{t}_{0}\partial^{\alpha/2}_{x}T_{t-s}g^{k}(s,\cdot)(x)dW^{k}_{s},$ and by Burkholder-Davis-Gundy’s inequality, we have ${\mathbb{E}}\left[\big{\|}\partial^{\alpha/2}_{x}u(t,x)\big{\|}^{p}\right]\leq c(p){\mathbb{E}}\left[\left(\int^{t}_{0}\sum_{k=1}^{N_{0}}\|\partial^{\alpha/2}_{x}T_{t-s}g^{k}(s,\cdot)(x)\|^{2}ds\right)^{p/2}\right].$ Now we use a parabolic version of Littlewood-Paley inequality for fractional Laplacian (Theorem 2.3 in [11]) $\int_{\mathbb{R}^{d}}\int^{T}_{0}\left[\int^{t}_{0}\|\partial^{\alpha/2}_{x}T_{t-s}g(s,\cdot)(x)\|^{2}_{\ell_{2}}ds\right]^{p/2}dtdx\leq c(\alpha,p)\int_{\mathbb{R}^{d}}\int^{T}_{0}\|g(t,x)\|^{p}_{\ell_{2}}\,dtdx$ (2.19) and get ${\mathbb{E}}\left[\int^{T}_{0}\\|\partial^{\alpha/2}_{x}u(t,\cdot)\\|^{p}_{p}\,dt\right]\leq c(p){\mathbb{E}}\left[\int^{T}_{0}\\|\|g(t,\cdot)\|_{\ell_{2}}\\|^{p}_{p}\,dt\right].$ (2.20) Similarly, (2.18) and Burkholder-Davis-Gundy’s inequality yield ${\mathbb{E}}\left[\|u(t,x)\|^{p}\right]\leq c(p)\,{\mathbb{E}}\left[\left(\int^{t}_{0}\sum_{k=1}^{N_{0}}\|T_{t-s}g^{k}(s,x)\|^{2}ds\right)^{p/2}\right].$ (2.21) Since $(\sum_{k=1}^{N_{0}}\|a_{n}\|^{2})^{p/2}\leq c(N_{0},p)\sum_{k=1}^{N_{0}}\|a_{n}\|^{p}$ and $\\|T_{t}f\\|_{p}\leq c\\|f\\|_{p}$, we see that for every $t>0$ $\displaystyle\int_{\mathbb{R}^{d}}\left(\int^{t}_{0}\sum_{k=1}^{N_{0}}\|T_{t-s}g^{k}(s,x)\|^{2}ds\right)^{p/2}dx$ $\displaystyle\leq t^{p/2-1}\int_{\mathbb{R}^{d}}\int^{t}_{0}\left(\sum_{k=1}^{N_{0}}\|T_{t-s}g^{k}(s,x)\|^{2}\right)^{p/2}dtdx$ $\displaystyle\leq c(T,N_{0},p)\int^{t}_{0}\int_{\mathbb{R}^{d}}\sum_{k=1}^{N_{0}}\|g^{k}(t,x)\|^{p}dxdt.$ Consequently, $\displaystyle{\mathbb{E}}\int_{0}^{T}\int_{\mathbb{R}^{d}}\|u(t,x)\|^{p}dxdt\leq c(T,N_{0},p){\mathbb{E}}\int^{T}_{0}\int_{\mathbb{R}^{d}}\|g(t,x)\|^{p}_{\ell_{2}}dxds.$ (2.22) Thus we proved $\partial^{\alpha/2}_{x}u,u\in\mathbb{L}_{p}(T)$, and hence by Corollary 2.2 we have $u\in\mathcal{H}^{\alpha/2}_{p}(T)$. Note that by Corollary 2.2(ii) $\\|\Delta^{\alpha/2}u\\|_{H^{-\alpha/2}_{p}}=\\|\Delta^{\alpha/4}(\partial^{\alpha/2}_{x}u)\\|_{H^{-\alpha/2}_{p}}\leq c\\|\partial^{\alpha/2}_{x}u\\|_{L_{p}}.$ By definition (2.7) and Remark 2.3, for any $t\leq T$, $\displaystyle\\|u\\|^{p}_{\mathcal{H}^{\alpha/2}_{p}(t)}$ $\displaystyle\leq$ $\displaystyle c(p)\left(\\|u\\|^{p}_{\mathbb{H}^{\alpha/2}_{p}(t)}+\\|\Delta^{\alpha/2}u\\|^{p}_{\mathbb{H}^{-\alpha/2}_{p}(t)}+\\|g\\|^{p}_{\mathbb{L}_{p}(t,\ell_{2})}\right)$ (2.23) $\displaystyle\leq$ $\displaystyle c\left(\\|u\\|^{p}_{\mathbb{H}^{-\alpha/2}_{p}(t)}+\\|\partial^{\alpha/2}_{x}u\\|^{p}_{\mathbb{L}_{p}(t)}+\\|g\\|^{p}_{\mathbb{L}_{p}(t,\ell_{2})}\right).$ Combining this with (2.20) and (2.9) we have that for every $0<t\leq T$ $\displaystyle\\|u\\|^{p}_{\mathcal{H}^{\alpha/2}_{p}(t)}$ $\displaystyle\leq$ $\displaystyle c(p,T,\alpha)\left(\\|u\\|^{p}_{\mathbb{H}^{-\alpha/2}_{p}(t)}+\\|g\\|^{p}_{\mathbb{L}_{p}(t,\ell_{2})}\right)$ (2.24) $\displaystyle\leq$ $\displaystyle c(p,T,\alpha)\int_{0}^{t}\\|u\\|^{p}_{\mathcal{H}^{\alpha/2}_{p}(s)}ds+c(p,T,\alpha)\\|g\\|^{p}_{\mathbb{L}_{p}(T,\ell_{2})}.$ Finally, Gronwall leads to (2.14). Now we drop the additional assumptions on $g$ by using the following standard approximation argument: By Theorem 3.10 in [14], for $g\in\mathbb{L}_{p}(T,\ell_{2})$ we can take a sequence $g_{n}\in\mathbb{L}_{p}(T,\ell_{2})$ so that $g_{n}\to g$ in $\mathbb{L}_{p}(T,\ell_{2})$ and each $g_{n}=(g^{1}_{n},g^{2}_{n},\cdots)$ satisfies above assumed assumptions, that is, $g^{k}_{n}=0$ for all large $k$ and each $g^{k}_{n}$ is of type (2.16). By the above result, the equation $du_{n}=\Delta^{\alpha/2}u_{n}dt+\sum_{k=1}^{\infty}g^{k}_{n}dW^{k}_{t},\quad u_{n}(0)=0$ has a unique solution $u_{n}$. It also follows that $u_{n}-u_{m}$ is the unique solution of $d(u_{n}-u_{m})=\Delta^{\alpha/2}(u_{n}-u_{m})dt+(g^{k}_{n}-g^{k}_{m})dW^{k}_{t},\quad(u_{n}-u_{m})(0)=0$ and, by the previous argument $\\|u_{n}-u_{m}\\|_{\mathcal{H}^{\alpha/2}_{p}(T)}\leq c(p,T)\\|g_{n}-g_{m}\\|_{\mathbb{L}_{p}(T,\ell_{2})}.$ Consequently, there is $u\in\mathcal{H}^{\alpha/2}_{p}(T)$ so that $u_{n}\to u$ in $\mathcal{H}^{\alpha/2}_{p}(T)$. We only need to prove $u$ is a solution of (2.15). Equivalently, we need to prove that for any $\phi\in C^{\infty}_{0}(\mathbb{R}^{d})$, the equality $(u(t,\cdot),\phi)=\int^{t}_{0}(\Delta^{\alpha/2}u(s,\cdot),\phi)ds+\sum_{k=1}^{\infty}\int^{t}_{0}(g^{k}(s,\cdot),\phi)dW^{k}_{s}.$ (2.25) holds for for all $t\leq T$ (a.s.), or equivalently $((1-\Delta)^{-\alpha/2}u(t,\cdot),(1-\Delta)^{\alpha/2}\phi)=\int^{t}_{0}(\Delta^{\alpha/4}u(s,\cdot),\Delta^{\alpha/4}\phi)ds+\sum_{k=1}^{\infty}\int^{t}_{0}(g^{k}(s,\cdot),\phi)dW^{k}_{s}.$ (2.26) By (2.8), $\lim_{n\to\infty}{\mathbb{E}}\left[\sup_{t\leq T}\\|(1-\Delta)^{-\alpha/2}(u_{n}(t,\cdot)-u(t,\cdot))\\|^{p}_{L_{p}}\right]=0,\quad\text{a.s.}$ which implies that one can take a subsequence $n_{j}$ so that $(1-\Delta)^{-\alpha/2}u_{n_{j}}\to(1-\Delta)^{-\alpha/2}u$ in $L_{p}(\mathbb{R}^{d})$ uniformly on $[0,T]$ (a.s) and consequently $t\to((1-\Delta)^{-\alpha/2}u(t,\cdot),(1-\Delta)^{\alpha/2}\phi)$ is continuous on $[0,T]$. By taking the limit from $((1-\Delta)^{-\alpha/2}u_{n_{j}}(t,\cdot),(1-\Delta)^{\alpha/2}\phi)=\int^{t}_{0}(\Delta^{\alpha/4}u_{n_{j}}(s,\cdot),\Delta^{\alpha/4}\phi)ds+\sum_{k=1}^{\infty}\int^{t}_{0}(g^{k}_{n_{j}}(s,\cdot),\phi)dW^{k}_{s}$ and remembering that both sides of (3.7) are continuous in $t$, one easily get that equality (3.7) holds for all $t\leq T$ (a.s.). Step 3. Next we prove the theorem for the equation $du=(\Delta^{\alpha/2}u+f)dt+g^{k}dW^{k}_{t},\quad u(0)=u_{0}.$ (2.27) Again we may assume $\gamma=-\alpha/2$, and due to Theorem 2.9 we only need to show that there exists a solution $u$ and it satisfies estimate (2.14). By Theorem 2.9, the equation $dv=(\Delta^{\alpha/2}v+f)dt,\quad v(0)=u_{0}$ has a solution $v\in\mathcal{H}^{\alpha/2}_{p}(T)$ and $\\|v\\|_{\mathcal{H}^{\alpha/2}_{p}(T)}\leq c(p,T)\left(\\|f\\|_{\mathbb{H}^{-\alpha/2}_{p}(T)}+\\|u_{0}\\|_{U^{\alpha/2-\alpha/p}_{p}}\right).$ Also by the result of Step 2, the equation $dw=\Delta^{\alpha/2}wdt+\sum_{k=1}^{\infty}g^{k}dW^{k}_{t},\quad w(0)=0$ has a unique solution and $\\|w\\|_{\mathcal{H}^{\alpha/2}_{p}(T)}\leq c(p,T)\\|g\\|_{\mathbb{L}_{p}(T,\ell_{2})}.$ Now it is enough to take $u=v+w$. Step 4 (A priori estimate). We prove the a priori estimate (2.14) holds given that a solution $u\in\mathcal{H}^{\gamma+\alpha}_{p}(T)$ of the following equation already exists : $du=\left(a(\omega,t)\Delta^{\alpha/2}u+f\right)\,dt+\sum_{k=1}^{\infty}g^{k}dW^{k}_{t},\quad u(0)=u_{0}.$ This time we prove (2.14) only for $\gamma=0$. This is enough due to the reason given in Step 1. By Step 3, the equation $dv=(\Delta^{\alpha/2}v+f)dt+\sum_{k=1}^{\infty}g^{k}dW^{k}_{t},\quad v(0)=u_{0}.$ has a solution $v\in\mathcal{H}^{\alpha}_{p}(T)$ and $\\|v\\|_{\mathcal{H}^{\alpha}_{p}(T)}\leq c(p,T)\left(\\|f\\|_{\mathbb{L}_{p}(T)}+\\|g\\|_{H^{\alpha/2}_{p}(T,\ell_{2})}+\\|u_{0}\\|_{U^{\alpha-\alpha/p}_{p}}\right).$ Note that $\bar{u}:=u-v$ satisfies $d\bar{u}=(a(\omega,t)\Delta^{\alpha/2}\bar{u}+\bar{f})dt,\quad\bar{u}(0)=0,$ where $\bar{f}:=(a(\omega,t)-1)\Delta^{\alpha/2}v$, and $\\|\bar{f}\\|_{\mathbb{L}_{p}(T)}\leq c\\|\Delta^{\alpha/2}v\\|_{\mathbb{L}_{p}(T)}\leq c\left(\\|f\\|_{\mathbb{L}_{p}(T)}+\\|g\\|_{H^{\alpha/2}_{p}(T,\ell_{2})}+\\|u_{0}\\|_{U^{\alpha-\alpha/p}_{p}}\right).$ Since $u=v+\bar{u}$, $\\|u\\|_{\mathcal{H}^{\alpha}_{p}(T)}\leq\\|v\\|_{\mathcal{H}^{\alpha}_{p}(T)}+\\|\bar{u}\\|_{\mathcal{H}^{\alpha}_{p}(T)}$ and $\\|\bar{u}\\|_{\mathcal{H}^{\alpha}_{p}(T)}\leq c\\|\bar{u}\\|_{\mathbb{H}^{\alpha}_{p}(T)}+c\\|\bar{f}\\|_{\mathbb{L}_{p}(T)}$, to prove (2.14) we only need to show that for each $\omega\in\Omega$, $\int^{T}_{0}\\|\bar{u}(t,\cdot)\\|^{p}_{H^{\alpha}_{p}}dt\leq c(p,T,\alpha,\delta)\int^{T}_{0}\\|\bar{f}(t,\cdot)\\|^{p}_{L_{p}}dt.$ (2.28) For fixed $\omega$, define a non-random functions $\tilde{u}(t,x)=\bar{u}(\omega,\xi(\omega,t),x)\quad\tilde{f}(t,x)=a(\omega,t)^{-1}\bar{f}(\omega,\xi(\omega,t),x).$ (2.29) where $\xi(\omega,t):=\int^{t}_{0}\frac{ds}{a(\omega,s)}$. Then clearly $\tilde{u}$ satisfies $\tilde{u}_{t}=\Delta^{\alpha/2}\tilde{u}+\tilde{f},\quad\tilde{u}(0)=0.$ Let $\tilde{T}(\omega,T)$ be such that $T=\int^{\tilde{T}(\omega,T)}_{0}\frac{ds}{a(\omega,s)}$. Since $\delta T<\tilde{T}(\omega,T)<\delta T$, applying (2.10), we get $\int^{\tilde{T}(\omega,T)}_{0}\\|\tilde{u}(t,\cdot)\\|^{p}_{H^{\alpha}_{p}}dt\leq c(p,T,\delta,\alpha)\int^{\tilde{T}(\omega,T)}_{0}\\|\tilde{f}(t,\cdot)\\|^{p}_{L_{p}}dt.$ This and relations in (2.29) easily lead to (2.28). Step 5 (Method of continuity). The solvability of equation (2.27), the a priori estimate (2.14) and the method of continuity obviously finish the proof of the theorem. But below we show how the method of continuity works only for reader’s convenience. For $\lambda\in[0,1]$, denote $a_{\lambda}(\omega,t)=(1-\lambda)+\lambda a(\omega,t)$. Then obviously $a_{\lambda}$ is predictable and $a_{\lambda}\in(\delta,\delta^{-1})$. It follows from Step 4 that if $u\in\mathcal{H}^{\gamma+\alpha}_{p}(T)$ is a solution of the equation $du=\left(a_{\lambda}(\omega,t)\Delta^{\alpha/2}u+f\right)\,dt+\sum_{k=1}^{\infty}g^{k}dW^{k}_{t},\quad u(0)=u_{0},$ (2.30) then the estimate (2.14) holds with the same constant $c=c(p,T,\delta)$. Now let $J$ be the collection of $\lambda\in[0,1]$ so that for any $f\in\mathbb{H}^{\gamma}_{p}(T),g\in\mathbb{H}^{\gamma+\alpha/2}_{p}(T,\ell_{2})$ and $u_{0}\in U^{\gamma+\alpha/2-\alpha/p}_{p}$, equation (2.30) has a solution. By Step 3, $0\in J$. Note that to finish the proof of the theorem we only need to show $1\in J$. Now let $\lambda_{0}\in J$. Obviously $u\in\mathcal{H}^{\gamma+\alpha}_{p}(T)$ is a solution of (2.30) if and only if $du=\left(a_{\lambda_{0}}(\omega,t)\Delta^{\alpha/2}u+[(a_{\lambda}(\omega,t)-a_{\lambda_{0}}(\omega,t))\Delta^{\alpha/2}u+f]\right)\,dt+\sum_{k=1}^{\infty}g^{k}dW^{k}_{t},\quad u(0)=u_{0}.$ (2.31) Now fix $u^{1}\in\mathcal{H}^{\gamma+\alpha}_{p}(T)$ with initial date $u_{0}$ (for instance take the solution of (2.27)), and define $u^{2},u^{3},\cdots$ so that $u^{n+1}\in\mathcal{H}^{\gamma+\alpha}_{p}(T)$ is the solution of $du^{n+1}=\left(a_{\lambda_{0}}(\omega,t)\Delta^{\alpha/2}u^{n+1}+[(a_{\lambda}(\omega,t)-a_{\lambda_{0}}(\omega,t))\Delta^{\alpha/2}u^{n}+f]\right)dt\,dt+g^{k}dW^{k}_{t},\quad u(0)=u_{0}.$ (2.32) Then $v^{n+1}:=u^{n+1}-u^{n}$ satisfies $dv^{n+1}=\left(a_{\lambda_{0}}(\omega,t)\Delta^{\alpha/2}v^{n+1}+(a_{\lambda}(\omega,t)-a_{\lambda_{0}}(\omega,t))\Delta^{\alpha/2}v^{n}\right)\,dt$ By the a priori estimate (2.14), $\displaystyle\\|v^{n+1}\\|_{\mathcal{H}^{\gamma+\alpha}_{p}(T)}$ $\displaystyle\leq$ $\displaystyle c\\|(a_{\lambda}-a_{\lambda_{0}})\Delta^{\alpha/2}v^{n}\\|_{\mathbb{H}^{\gamma}_{p}(T)}$ $\displaystyle\leq$ $\displaystyle N(p,T,\delta)\|\lambda-\lambda_{0}\|\\|v^{n}\\|_{\mathcal{H}^{\gamma+\alpha}_{p}(T)}.$ Thus if $\|\lambda-\lambda_{0}\|<1/(2N(p,T,\delta))$, the map which send $u^{n}$ to $u^{n+1}$ is a contraction in $\mathcal{H}^{\gamma+\alpha}_{p}(T)$, and has a unique fixed point $u$. Thus $u$ satisfies (2.30)–(2.31). Since the above constant $N$ is independent of $\lambda$, it follows that $J=[0,1]$ and the theorem is proved. $\Box$ Finally we consider the nonlinear equation $du=\left(a(\omega,t)\Delta^{\alpha/2}u+f(u)\right)\,dt+\sum_{k=1}^{\infty}g^{k}(u)dW^{k}_{t},\quad u(0)=u_{0},$ (2.33) where $f(u)=f(\omega,t,x,u)$ and $g^{k}(u)=g^{k}(\omega,t,x,u)$. ###### Assumption 2.13 Assume $f(0)\in\mathbb{H}^{\gamma}_{p}(T)$ and $g(0)\in\mathbb{H}^{\gamma+\alpha/2}_{p}(T,\ell_{2})$. Moreover, for any $\varepsilon>0$, there exists a constant $K_{\varepsilon}$ so that for any $u=u(x),v=v(x)\in H^{\gamma+\alpha}_{p}$ and $\omega,t$, we have $\displaystyle\\|f(t,\cdot,u(\cdot))-f(t,\cdot,v(\cdot))\\|_{H^{\gamma}_{p}}+\\|g(t,\cdot,u(\cdot))-g(t,\cdot,v(\cdot))\\|_{H^{\gamma+\alpha/2}_{p}(\ell_{2})}$ $\displaystyle\leq$ $\displaystyle\varepsilon\\|u-v\\|_{H^{\gamma+\alpha}_{p}}+K(\varepsilon)\\|u-v\\|_{H^{\gamma}_{p}}.$ (2.34) To give an example of $f(u)$ and $g(u)$ satisfying Assumption 2.13, we introduce the space of point-wise multipliers in $H^{\gamma}_{p}$. For each $r\geq 0$, define $B^{r}=\begin{cases}B(\mathbb{R}^{d})\qquad&\hbox{if }r=0,\\\ C^{r-1,1}(\mathbb{R}^{d})&\hbox{if }r=1,2,\cdots,\\\ C^{r}(\mathbb{R}^{d})&\hbox{otherwise},\end{cases}$ (2.35) where $B(\mathbb{R}^{d})$ is the space of bounded Borel measurable functions on $\mathbb{R}^{d}$, $C^{r-1,1}(\mathbb{R}^{d})$ is the space of $r-1$ times continuously differentiable functions whose $(r-1)$st order derivatives are Lipschitz continuous, and $C^{r}(\mathbb{R}^{d})$ is the usual Hölder space. Also we use the space $B^{r}$ for $\ell_{2}$-valued functions. For instance, if $g=(g^{1},g^{2},...)$, then $\|g\|_{B^{0}}=\sup_{x}\|g(x)\|_{\ell_{2}}$ and $\|g\|_{C^{n-1,1}}=\sum_{\|\alpha\|\leq n-1}\|D^{\alpha}g\|_{B^{0}}+\sum_{\|\alpha\|=n-1}\sup_{x\neq y}\frac{\|D^{\alpha}g(x)-D^{\alpha}g(y)\|_{\ell_{2}}}{\|x-y\|}.$ Fix $\kappa_{0}=\kappa_{0}(\gamma)\geq 0$ so that $\kappa_{0}>0$ if $\gamma$ is not integer. It is known (see, for instance, Lemma 5.2 in [14]) that for any $a\in B^{\|\gamma\|+\kappa_{0}}$ and $h\in H^{\gamma}_{p}$, $\\|ah\\|_{H^{\gamma}_{p}}\leq c(\gamma,\kappa_{0})\|a\|_{B^{\|\gamma\|+\kappa_{0}}}\|h\|_{H^{\gamma}_{p}}$ (2.36) and the same inequality holds for $\ell_{2}$-valued functions $a$. ###### Example 2.14 Fix $\kappa_{0}=\kappa_{0}(\gamma)\geq 0$ so that $\kappa_{0}>0$ if $\gamma$ is not integer. Consider $f(u)=b(\omega,t,x)\Delta^{\beta_{1}/2}u+\sum_{i=1}^{d}c^{i}(\omega,t,x)u_{x^{i}}I_{\alpha>1}+d(\omega,t,x)u+f_{0},$ $g^{k}(u)=\sigma^{k}(\omega,t,x)\Delta^{\beta_{2}/2}u+v^{k}(\omega,t,x)u+g^{k}_{0},$ where $\beta_{1}<\alpha$, $\beta_{2}<\alpha/2$, $f_{0}\in\mathbb{H}^{\gamma}_{p}(T)$ and $g_{0}\in\mathbb{H}^{\gamma+\alpha/2}_{p}(T,\ell_{2})$. Assume for each $\omega,t$, $\|b\|_{B^{\|\gamma\|+\kappa_{0}}}+\sum_{i=1}^{d}\|c^{i}\|_{B^{\|\gamma\|+\kappa_{0}}}+\|d\|_{B^{\|\gamma\|+\kappa_{0}}}+\|\sigma\|_{B^{\|\gamma\|+\alpha/2+\kappa_{0}}}+\|\nu\|_{B^{\|\gamma\|+\alpha/2+\kappa_{0}}}\leq K.$ Then by (2.36), for each $t$ $\displaystyle\\|f(t,\cdot,u(\cdot))-f(t,\cdot,v(\cdot))\\|_{H^{\gamma}_{p}}+\\|g(t,\cdot,u(\cdot))-g(t,\cdot,v(\cdot))\\|_{H^{\gamma+\alpha/2}_{p}(\ell_{2})}$ $\displaystyle\leq$ $\displaystyle c\left(\\|\Delta^{\beta_{1}/2}(u-v)\\|_{H^{\gamma}_{p}}+I_{\alpha>1}\\|D(u-v)\\|_{H^{\gamma}_{p}}+\\|u-v\\|_{H^{\gamma+\alpha/2}_{p}}+\\|\Delta^{\beta_{2}/2}(u-v)\\|_{H^{\gamma+\alpha/2}_{p}}\right).$ Since for any $\alpha_{1}<\alpha$ and $\varepsilon>0$, by interpolation theory, $\\|u\\|_{H^{\gamma+\alpha_{1}}_{p}}\leq c(\alpha,\alpha_{1})\\|u\\|_{H^{\gamma+\alpha}_{p}}^{\alpha_{1}/\alpha}\\|u\\|_{H^{\gamma}_{p}}^{1-\alpha_{1}/\alpha}\leq\varepsilon\\|u\\|_{H^{\gamma+\alpha}_{p}}+c(\varepsilon,\alpha_{1},\alpha)\\|u\\|_{H^{\gamma}_{p}},$ one easily gets (2.13). Here is the main result of this section. ###### Theorem 2.15 Suppose Assumptions 2.10 and 2.13 hold. Then equation (2.33) has a unique solution $u\in\mathcal{H}^{\gamma+\alpha}_{p}(T)$, and for this solution $\\|u\\|_{\mathcal{H}^{\gamma+\alpha}_{p}(T)}\leq c\left(\\|f(0)\\|_{\mathbb{H}^{\gamma}_{p}(T)}+\\|g(0)\\|_{\mathbb{H}^{\gamma+\alpha/2}_{p}(T,\ell_{2})}+\\|u_{0}\\|_{U^{\gamma+\alpha/2-\alpha/p}_{p}}\right),$ where $c=c(p,T,\delta)$. Proof. Our proof is virtually identical to the that of Theorem 6.4 in [14], where the theorem is proved when $\alpha=2$. The only difference is that one has to use Theorem 2.11 in this article, in place the corresponding result in [14]. We skip the proof here since we will give the proof for more general case in next section. $\Box$ By Sobolev embedding theorem, we immediately get the following ###### Corollary 2.16 Suppose Assumptions 2.10 and2.13 hold. If $\gamma+\alpha>d/p$, then the unique solution $u\in\mathcal{H}^{\gamma+\alpha}_{p}(T)$ of equation (2.33) is $C^{\gamma+\alpha-d/p}$-valued process on $[0,T]\times\Omega$ a.s.. ## 3 General case Let $Z^{1}_{t},Z^{2}_{t},...$ be independent $m$-dimensional Lévy processes relative to $\\{\mathcal{F}_{t},t\geq 0\\}$. For $t\geq 0$ and Borel set $A\in\mathcal{B}(\mathbb{R}^{m}\setminus\\{0\\})$, define $N_{k}(t,A):=\\#\left\\{0\leq s\leq t;\,Z^{k}_{s}-Z^{k}_{s-}\in A\right\\},\quad\widetilde{N}_{k}(t,A):=N_{k}(t,A)-t\nu_{k}(A)$ where $\nu_{k}(A):={\mathbb{E}}[N_{k}(1,A)]$ is the Lévy measure of $Z^{k}$. By Lévy-Itô decomposition, there exist a vector $\alpha^{k}$, a non-negative definite matrix $\beta^{k}$ and $m$-dimensional Wiener process $B^{k}$ so that $Z^{k}(t)=\alpha^{k}t+\beta^{k}B^{k}_{t}+\int_{\|z\|<1}z\widetilde{N}_{k}(t,dz)+\int_{\|z\|\geq 1}zN_{k}(t,dz).$ (3.1) For any $q,k=1,2,\cdots$, denote $\widehat{c}_{k,q}:=\left(\int_{\mathbb{R}^{m}}\|z\|^{q}\nu_{k}(dz)\right)^{1/q}.$ Now we fix $p\in[2,\infty)$ and denote $\widehat{c}_{k}:=\left(\widehat{c}_{k,2}\vee\widehat{c}_{k,p}\right)$. In this section we assume $\widehat{c}:=\sup_{k\geq 1}\widehat{c}_{k}<\infty.$ (3.2) ((3.2) will be weaken in section 4). Then for any $2<q<p$, by Hölder’s inequality, $\widehat{c}_{k,q}\leq\left(\int_{\mathbb{R}^{m}}\|z\|^{2}\nu_{k}(dz)\right)^{(p-q)/(q(p-2))}\left(\int_{\mathbb{R}^{m}}\|z\|^{p}\nu_{k}(dz)\right)^{(q-2)/(q(p-2))}\leq\widehat{c}_{k}.$ By (3.2), $\int_{\|z\|\geq 1}\|z\|\nu_{k}(dz)\leq\int_{\|z\|\geq 1}\|z\|^{2}\nu_{k}(dz)<\infty$, and $\int_{\|z\|\geq 1}zN_{k}(t,dz)=\int_{\|z\|\geq 1}z\widetilde{N}_{k}(t,dz)+t\int_{\|z\|\geq 1}z\nu_{k}(dz).$ Thus by absorbing $\widetilde{\alpha}_{k}:=\int_{\|z\|\geq 1}z\nu_{k}(dz)$ into $\alpha_{k}$ we can rewrite (3.1) as $Z^{k}_{t}=\tilde{\alpha}_{k}t+\beta_{k}B^{k}_{t}+\int_{\mathbb{R}^{m}}z\widetilde{N}_{k}(t,dz).$ We first consider the following linear equation: $du=\left(a(\omega,t)\Delta^{\alpha/2}u+f\right)dt\,+\sum_{k=1}^{\infty}g^{k}\cdot dZ^{k}_{t},\quad u(0)=u_{0}.$ (3.3) Relocation of the term $\sum_{k=1}^{\infty}g^{k}\cdot\tilde{\alpha}_{k}dt$ into the deterministic part of (3.3) allow us to assume $\tilde{\alpha}_{k}=(0,\dots,0)$. Moreover, since $B^{k,j}$’s are independent 1-dimensional Wiener processes where $B^{k}=(B^{k,1},\dots B^{k,m})$, (3.3) can be written as $du=\left(a(\omega,t)\Delta^{\alpha/2}u+f\right)\,dt+\sum_{i=1}^{\infty}h^{k}dW^{k}_{t}+\sum_{k=1}^{\infty}\sum_{j=1}^{m}g^{k,j}dY^{k,j}_{t},\quad u(0)=u_{0},$ (3.4) for some $h=(h^{1},h^{2},\cdots)$ and independent one-dimensional Wiener processes $W^{k}_{t}$ and $Y^{k}_{t}:=\int_{\mathbb{R}^{m}}z\widetilde{N}_{k}(t,dz).$ Note that $Y^{k}_{t}$ are independent $m$-dimensional pure jump Lévy processes with Lévy measure of $\nu^{k}$. Furthermore by considering $u-v$, where $v$ is the solution of $dv=a(\omega,t)\Delta^{\alpha/2}v\,dt\,+\sum_{i=k}^{\infty}h^{k}dW^{k}_{t},\quad u(0)=0$ from Theorem 2.11, we find that without loss of generality we may also assume $h^{k}$’s are all zero. By $<M,N>$ we denote the bracket of real-valued square integrable martingales $M$ and $N$. Also let $[M]$ denote the quadratic variation of $M$. ###### Remark 3.1 (i) Note that, if $\widehat{c}_{k,2}<\infty$, then $Y^{k,i}=\int_{\mathbb{R}^{m}}z^{i}\widetilde{N}_{k}(t,dz)$ is a square integrable martingale for each $k\geq 1$ and $i=1,\cdots,m$. Also for any $\overline{\mathcal{P}}$-measurable process $H=(H^{1},\dots,H^{m})\in L_{2}(\Omega\times[0,T],\mathbb{R}^{m})$ which has a predictable version $\bar{H}=(\bar{H}^{1},\dots,\bar{H}^{m})$, $M^{k}_{t}:=\int_{0}^{t}H_{s}\cdot dY^{k}_{s}=\sum_{i=1}^{m}\int_{0}^{t}\int_{\mathbb{R}^{m}}H^{i}_{s}z^{i}\widetilde{N}_{k}(ds,dz)=\sum_{i=1}^{m}\int_{0}^{t}\int_{\mathbb{R}^{m}}\bar{H}^{i}_{s}z^{i}\widetilde{N}_{k}(ds,dz)$ is a square integrable martingale with $[M^{k}]_{t}=\sum_{i,j=1}^{m}\int^{t}_{0}\int_{\mathbb{R}^{m}}H^{i}H^{j}z^{i}z^{j}N(ds,dz),$ $E[M^{k}]_{t}=\sum_{i,j}(\int_{\mathbb{R}^{m}}z^{i}z^{j}\nu^{k}(dz)){\mathbb{E}}\int^{t}_{0}H^{i}(s)H^{j}(s)ds\leq\widehat{c}^{2}m^{2}{\mathbb{E}}\int_{0}^{t}\|H_{s}\|^{2}ds.$ (ii) Suppose that (3.2) holds. Then, for any $1\leq j\leq m$, $g^{\cdot,j}\in\mathbb{H}^{\gamma}_{p}(T,\ell_{2})$ and $\phi\in C^{\infty}_{0}(\mathbb{R}^{d})$, the series of stochastic integral $\sum_{k=1}^{\infty}\sum_{j=1}^{m}\int^{t}_{0}(g^{k,j}(s,\cdot),\phi)dY^{k,j}_{s}$ defines a square integrable martingale on $[0,T]$, which is right continuous with left limits. Indeed, denote $M_{n}:=\sum_{k=1}^{n}\sum_{j=1}^{m}\int^{t}_{0}(g^{k,j}(s,\cdot),\phi)dY^{k,j}_{s}$, then the quadratic variation of $M_{n}$ is $[M_{n}]_{t}=\sum_{k=1}^{n}\sum_{i,j=1}^{m}\int^{t}_{0}\int_{\mathbb{R}^{m}}(g^{k,i},\phi)(g^{k,j}(s,\cdot),\phi)z^{i}z^{k}N^{k}(ds,dz),$ and $\displaystyle{\mathbb{E}}[M_{n}]_{t}=\sum_{k=1}^{n}\sum_{i,j=1}^{m}{\mathbb{E}}\int^{t}_{0}(g^{k,i},\phi)(g^{k,j}(s,\cdot),\phi)\int_{\mathbb{R}^{m}}z^{i}z^{j}\nu_{k}(dz)ds\leq c(m,\widehat{c})\sum_{k=1}^{n}\sum_{j=1}^{m}{\mathbb{E}}\int^{t}_{0}(g^{k,j}(s,\cdot),\phi)^{2}ds.$ Also, with $q:=p/(p-2)$, for every $1\leq j\leq m$, $\displaystyle\sum_{k=1}^{\infty}\,{\mathbb{E}}\left[\int^{T}_{0}(g^{k,j}(s,\cdot),\phi)^{2}ds\right]=\sum_{k=1}^{\infty}{\mathbb{E}}\left[\int_{0}^{T}((1-\Delta)^{\gamma/2}g^{k,j}(s,\cdot),(1-\Delta)^{-\gamma/2}\phi)^{2}\,ds\right]$ $\displaystyle\leq$ $\displaystyle\\|(1-\Delta)^{-\gamma/2}\phi\\|_{1}\,{\mathbb{E}}\left[\int^{T}_{0}\Big{(}\sum_{k=1}^{\infty}\|(1-\Delta)^{\gamma/2}g^{k,j}(s,\cdot)\|^{2},\,\|(1-\Delta)^{-\gamma/2}\phi\|\Big{)}\,ds\right]$ $\displaystyle\leq$ $\displaystyle\\|(1-\Delta)^{-\gamma/2}\phi\\|_{1}\,\\|(1-\Delta)^{-\gamma/2}\phi\\|_{q}\,{\mathbb{E}}\left[\int^{T}_{0}\Big{\\|}\,\sum_{k=1}^{\infty}\|(1-\Delta)^{\gamma/2}g^{k,j}(s,\cdot)\|^{2}\Big{\\|}_{p/2}\,ds\right]$ $\displaystyle\leq$ $\displaystyle\\|(1-\Delta)^{-\gamma/2}\phi\\|_{1}\,\\|(1-\Delta)^{-\gamma/2}\phi\\|_{q}\,T^{1-\frac{2}{p}}\,\\|g^{\cdot,j}\\|_{\mathbb{H}^{\gamma}_{p}(T,l^{2})}^{2}<\infty.$ It follows that $[M_{n}]_{t}$ converges in probability uniformly on $[0,T]$ and this certainly proves the claim. ###### Definition 3.2 Write $u\in\mathcal{H}^{\gamma+\alpha}_{p}(T)$ if $u\in\mathbb{H}^{\gamma+\alpha}_{p}(T),u(0)\in U^{\gamma+\alpha-\alpha/p}_{p}$, and for some $f\in\mathbb{H}^{\gamma}_{p}(T)$ $h\in\mathbb{H}^{\gamma+\alpha/2}_{p}(T,\ell_{2})$ and $g^{\cdot,j}\in\mathbb{H}^{\gamma+\alpha/2}_{p}(T,\ell_{2}),1\leq j\leq m$ $du=f\,dt+\sum_{k=1}^{\infty}h^{k}dW^{k}_{t}+\sum_{k=1}^{\infty}\sum_{j=1}^{m}g^{k,j}dY^{k,j}_{t},\quad u(0)=u_{0},\quad\hbox{for }t\in[0,T]$ in the sense of distributions, that is, for any $\phi\in C^{\infty}_{0}(\mathbb{R}^{d})$, $(u(t,\cdot),\phi)=(u(0,\cdot),\phi)+\int^{t}_{0}(f(s,\cdot),\phi)ds+\sum_{k=1}^{\infty}\int^{t}_{0}(h^{k}(s,\cdot),\phi)ds+\sum_{k=1}^{\infty}\sum_{j=1}^{m}\int^{t}_{0}(g^{k,j}(s,\cdot),\phi)dY^{k,j}_{s}$ (3.5) holds for all $t\leq T$ $a.s.$. In this case we write $\mathbb{D}u:=f,\quad\mathbb{S}_{c}u:=(h^{1},\dots h^{k},\dots),\quad\mathbb{S}^{k,j}_{d}u:=g^{k,j},\quad\mathbb{S}^{\cdot,j}_{d}u:=(g^{1,j},\dots g^{k,j},\dots)$ and define $\\|u\\|_{\mathcal{H}^{\gamma+\alpha}_{p}(T)}:=\\|u\\|_{\mathbb{H}^{\gamma+\alpha}_{p}(T)}+\\|\mathbb{D}u\\|_{\mathbb{H}^{\gamma}_{p}(T)}+\\|\mathbb{S}_{c}u\\|_{\mathbb{H}^{\gamma+\alpha/2}_{p}(T,\ell_{2})}+\sum_{j=1}^{m}\\|\mathbb{S}^{\cdot,j}_{d}u\\|_{\mathbb{H}^{\gamma+\alpha/2}_{p}(T,\ell_{2})}+\\|u(0)\\|_{U^{\gamma+\alpha-\alpha/p}_{p}}.$ To prove that $\mathcal{H}^{\gamma+\alpha}_{p}(T)$ is a Banach space we need the following result, which is an infinite dimensional extension of Kunita’s inequality (for example, see [2, Theorem 4.4.23]). In fact, if $m=1$ then the proof is given in [5]. ###### Lemma 3.3 Suppose $1\leq j\leq m$, $g^{\cdot,j}(\omega,t)=(g^{1,j},g^{2,j},\cdots)$’s are $\ell_{2}$-valued predictable processes such that each $g^{k}=(g^{k,1},\dots g^{k,m})$ is bounded. Then, under the assumption (3.2), $\displaystyle{\mathbb{E}}\left[\left(\sum_{k=1}^{\infty}\int^{t}_{0}\int_{\mathbb{R}^{m}}\|g^{k}(s)\|^{2}\,\|z\|^{2}N_{k}(s,dz)ds\right)^{p/2}\right]$ (3.6) $\displaystyle\leq$ $\displaystyle c(p)\,{\mathbb{E}}\left[\left(\int^{t}_{0}\sum_{k=1}^{\infty}\|g^{k}(s)\|^{2}ds\right)^{p/2}+\int^{t}_{0}\sum_{k=1}^{\infty}\|g^{k}(s)\|^{p}\,ds\right].$ Proof. Due to monotone convergence theorem we may assume $g^{k,j}=0$ for all $i>M$ and $1\leq j\leq m$. By monotone convergence theorem, $\displaystyle A$ $\displaystyle:=$ $\displaystyle{\mathbb{E}}\left[\left(\sum_{k=1}^{M}\int^{t}_{0}\int_{\mathbb{R}^{m}}\|g^{k}(s)\|^{2}\|z\|^{2}N_{k}(s,dz)ds\right)^{p/2}\right]$ $\displaystyle=$ $\displaystyle\lim_{N\to\infty}{\mathbb{E}}\left[\left(\sum_{k=1}^{M}\int^{t}_{0}\int_{\|z\|\leq N}\|g^{k}(s)\|^{2}\|z\|^{2}N_{k}(s,dz)ds\right)^{p/2}\right].$ Since $(a+b)^{p/2}\leq c(p)(\|a\|^{p/2}+\|b\|^{p/2})$ and $\widetilde{N}_{k}(s,dz):=N_{k}(s,dz)-s\nu_{k}(dz)$, $\displaystyle A\leq c(p)\lim_{N\to\infty}{\mathbb{E}}\left[(J_{2,t})^{p/2}\right]+c(p){\mathbb{E}}\left[\left(\int^{t}_{0}\int_{\mathbb{R}^{m}}\sum_{k=1}^{M}\|g^{k}(s)\|^{2}\,\|z\|^{2}\nu_{k}(dz)ds\right)^{p/2}\right]$ where $J_{n,t}:=\sum_{k=1}^{M}\int^{t}_{0}\int_{\|z\|\leq N}\|g^{k}(s)\|^{n}\|z\|^{n}\widetilde{N}_{k}(s,dz)ds$, which is a square integrable martingale becuase $g^{k}$ are bounded predictable processes. By Burkholder-Davis-Gundy inequality (For example, see [22, Theorem 48].) $\displaystyle{\mathbb{E}}\left[(J_{2,t})^{p/2}\right]\leq c(p){\mathbb{E}}\left[[J_{2}]_{t}^{p/4}\right]=c(p){\mathbb{E}}\left[\left(\sum_{k=1}^{M}\int^{t}_{0}\int_{\|z\|\leq N}\|g^{k}(s)\|^{4}\|z\|^{4}N_{k}(s,dz)ds\right)^{p/4}\right]$ (3.7) $\displaystyle\leq c(p){\mathbb{E}}\left[\left(\sum_{k=1}^{M}\,\,\sum_{0\leq s\leq t}\|g^{k}(s)\|^{4}\,\|\Delta Y^{k}_{s}\|^{4}\right)^{p/4}\right].$ Recall that for any $q>1$, $(\sum\|a_{n}\|^{q})^{1/q}\leq\sum\|a_{n}\|$. Thus if $2<p\leq 4$, then $\displaystyle{\mathbb{E}}\left[(J_{2,t})^{p/2}\right]\leq c(p){\mathbb{E}}\left[\sum_{k=1}^{M}\,\,\sum_{0\leq s\leq t}\|g^{k}(s)\|^{p}\,\|\Delta Y^{k}_{s}\|^{p}\right]\leq c(p,\widehat{c}){\mathbb{E}}\left[\int^{t}_{0}\sum_{k=1}^{M}\|g^{k}(s)\|^{p}\,ds\right].$ If $4<p\leq 8$ then, by the relation $\widetilde{N}_{k}(s,dz)=N_{k}(s,dz)-s\nu_{k}(dz)$ and Burkholder-Davis-Gundy inequality, $\displaystyle{\mathbb{E}}\left[(J_{2,t})^{p/2}\right]\leq c(p){\mathbb{E}}\left[\left(\sum_{k=1}^{M}\int^{t}_{0}\int_{\|z\|\leq N}\|g^{k}(s)\|^{4}\,\|z\|^{4}N_{k}(s,dz)ds\right)^{p/4}\right]$ $\displaystyle\leq$ $\displaystyle c(p){\mathbb{E}}\left[(J_{8,t})^{p/8}\right]+c(p){\mathbb{E}}\left[\left(\sum_{k=1}^{M}\int^{t}_{0}\int_{\|z\|\leq N}\|g(s)\|^{4}\,\|z\|^{4}\nu_{k}(dz)ds\right)^{p/4}\right]$ $\displaystyle\leq$ $\displaystyle c(p,\widehat{c}){\mathbb{E}}\left[\left(\sum_{k=1}^{M}\int^{t}_{0}\int_{\|z\|\leq N}\|g^{k}(s)\|^{8}\|z\|^{8}N_{k}(s,dz)ds\right)^{p/8}+\left(\int^{t}_{0}\sum_{k=1}^{M}\|g^{k}(s)\|^{4}ds\right)^{p/4}\right]$ $\displaystyle\leq$ $\displaystyle c(p,\widehat{c}){\mathbb{E}}\left[\left(\sum_{k=1}^{M}\int^{t}_{0}\int_{\|z\|\leq N}\|g^{k}(s)\|^{p}\|z\|^{p}N_{k}(s,dz)ds\right)+\left(\int^{t}_{0}\sum_{k=1}^{M}\|g^{k}(s)\|^{4}ds\right)^{p/4}\right]$ $\displaystyle\leq$ $\displaystyle c(p,\widehat{c}){\mathbb{E}}\left[\int^{t}_{0}\sum_{k=1}^{M}\|g^{k}(s)\|^{p}\,ds+\left(\int^{t}_{0}\sum_{k=1}^{M}\|g^{k}(s)\|^{4}ds\right)^{p/4}\right].$ Similarly, in general, for $p\in(2^{n-1},2^{n}]$, $\displaystyle A\leq c(p,\widehat{c})\,\sum_{j=1}^{n}{\mathbb{E}}\left[\left(\int^{t}_{0}\sum_{k=1}^{M}\|g^{k}(s)\|^{2^{j}}ds\right)^{p2^{-j}}\right]+c(p,\widehat{c})\,{\mathbb{E}}\left[\int^{t}_{0}\sum_{k=1}^{M}\|g^{k}(s,x)\|^{p}ds\right].$ Also since for each $2\leq q\leq p$, $\left(\int^{t}_{0}\sum_{k=1}^{M}\|g^{k}(s)\|^{q}ds\right)^{1/q}\leq\left(\left(\int^{t}_{0}\sum_{k=1}^{M}\|g^{k}(s)\|^{2}ds\right)^{1/2}+\left(\int^{t}_{0}\sum_{k=1}^{M}\|g^{k}(s)\|^{p}ds\right)^{1/p}\right),$ we get $\displaystyle A\leq c(p,\widehat{c})\,{\mathbb{E}}\left[\left(\int^{t}_{0}\sum_{k=1}^{\infty}\|g^{k}(s)\|^{2}ds\right)^{p/2}+\int^{t}_{0}\sum_{k=1}^{\infty}\|g^{k}(s)\|^{p}\,ds\right].$ (3.8) Thus the lemma is proved. $\Box$ ###### Theorem 3.4 Suppose that (3.2) holds. For any $p\in[2,\infty)$ and $\gamma\in\mathbb{R}$, $\mathcal{H}^{\gamma+\alpha}_{p}(T)$ is a Banach space with norm $\\|\cdot\\|_{\mathcal{H}^{\gamma+\alpha}_{p}(T)}$. Moreover, there is a constant $c=c(d,p,T)>0$ such that for every $u\in\mathcal{H}^{\gamma+2}_{p}(T)$ and $0<t\leq T$, ${\mathbb{E}}\left[\sup_{s\leq t}\\|u(s,\cdot)\\|^{p}_{H^{\gamma}_{p}}\right]\leq c(p,d,T)\left(\\|\mathbb{D}u\\|^{p}_{\mathbb{H}^{\gamma}_{p}(t)}+\\|\mathbb{S}_{c}u\\|_{\mathbb{H}^{\gamma}_{p}(t,\ell_{2})}+\sum_{j=1}^{m}\\|\mathbb{S}^{\cdot,j}_{d}u\\|^{p}_{\mathbb{H}^{\gamma}_{p}(t,\ell_{2})}+\\|u_{0}\\|^{p}_{U^{\gamma}_{p}}\right).$ (3.9) Proof. By Theorem 2.6 and the reasons explained just before Remark 3.1, without loss of generality we assume that $Y^{k}_{t}=\int_{\mathbb{R}^{m}}z\widetilde{N}_{k}(t,dz)$. Moreover, due to Remark 2.8 it suffices to prove the theorem only for $\gamma=0$. First we prove (3.9). Let $du=fdt+\sum_{k=1}^{\infty}g^{k}\cdot dY^{k}_{t}$ with $u(0)=u_{0}$. For a moment, we assume that $g^{k,j}=0$ for all $k\geq N_{0},1\leq j\leq m$ and $g^{k,j}$ is of the type $g^{k,j}(t,x)=\sum_{i=0}^{m_{k}}I_{(\tau^{k,j}_{i},\tau^{k,j}_{i+1}]}(t)g^{k_{i},j}(x),$ (3.10) where $\tau^{k,j}_{i}$ are bounded stopping times and $g^{k_{i},j}\in C^{\infty}_{0}(\mathbb{R}^{d})$. Define $v(t,x)=\sum_{k=1}^{N_{0}}\int^{t}_{0}g^{k}(s,x)\cdot dY^{k}_{s}.$ Then by Burkholder-Davis-Gundy inequality and Lemma 3.3, $\displaystyle{\mathbb{E}}\left[\sup_{s\leq t}\|v(s,x)\|^{p}\right]={\mathbb{E}}\left[\sup_{s\leq t}\left\|\sum_{k=1}^{N_{0}}\sum_{j=1}^{m}\int^{t}_{0}\int_{\mathbb{R}^{m}}g^{k,j}(s,x)z^{j}\widetilde{N}_{k}(s,dz)ds\right\|^{p}\right]$ $\displaystyle\leq$ $\displaystyle c(p){\mathbb{E}}\left[\left(\sum_{k=1}^{\infty}\sum_{i,j=1}^{m}\int^{t}_{0}\int_{\mathbb{R}^{m}}g^{k,i}(s,x)g^{k,j}(s,x)z^{i}z^{j}N_{k}(s,dz)ds\right)^{p/2}\right]$ $\displaystyle\leq$ $\displaystyle c(p,\widehat{c}){\mathbb{E}}\left[\left(\sum_{k=1}^{\infty}\int^{t}_{0}\int_{\mathbb{R}^{m}}\|g^{k}(s,x)\|^{2}\|z\|^{2}N_{k}(s,dz)ds\right)^{p/2}\right]$ $\displaystyle\leq$ $\displaystyle c(p,\widehat{c})\,{\mathbb{E}}\left[\left(\int^{t}_{0}\sum_{k=1}^{\infty}\|g^{k}(s,x)\|^{2}ds\right)^{p/2}+\int^{t}_{0}\sum_{k=1}^{\infty}\|g^{k}(s,x)\|^{p}ds\right].$ Since $\sum_{n}\|a_{n}\|^{p}\leq(\sum_{n}\|a_{n}\|^{2})^{p/2}$ and $(\int^{t}_{0}\|f\|ds)^{p}\leq t^{p-1}\int^{t}_{0}\|f\|^{p}ds$, by integrating over $\mathbb{R}^{d}$ we get that for every $t\leq T$ ${\mathbb{E}}\left[\sup_{s\leq t}\\|v\\|^{p}_{p}\right]\leq c(T,p)\sum_{j=1}^{m}\\|g^{\cdot,j}\\|^{p}_{\mathbb{L}_{p}(t,\ell_{2})}:=c(T,p)\,\sum_{j=1}^{m}{\mathbb{E}}\int^{t}_{0}\int_{\mathbb{R}^{d}}\|g^{\cdot,j}\|^{p}_{\ell_{2}}\,dxds.$ (3.11) Next we prove (3.11) for general $g^{\cdot,j}\in\mathbb{L}_{p}(T,\ell_{2})$. By Theorem 3.10 in [14], we can take a sequence $g^{\cdot,j}_{n}\in\mathbb{L}_{p}(T,\ell_{2})$ so that for each fixed $n$, $g^{k,j}_{n}=0$ for all large $k$ and each $g^{k,j}_{n}$ is of of the type (3.10), and $g^{\cdot,j}_{n}\to g^{\cdot,j}$ in $\mathbb{L}_{p}(T,\ell_{2})$ as $n\to\infty$. Define $v_{n}(t,x)=\sum_{k=1}^{\infty}\sum_{j=1}^{m}\int^{t}_{0}g^{k,j}_{n}dY^{k,j}_{t}$, then for every $t\leq T$ ${\mathbb{E}}\left[\sup_{s\leq t}\\|v_{n}\\|^{p}_{p}\right]\leq c(T,p)\sum_{j=1}^{m}\\|g^{\cdot,j}_{n}\\|^{p}_{\mathbb{L}_{p}(t,\ell_{2})},\quad{\mathbb{E}}\left[\sup_{s\leq t}\\|v_{n_{1}}-v_{n_{2}}\\|^{p}_{p}\right]\leq c(T,p)\sum_{j=1}^{m}\\|g^{\cdot,j}_{n_{1}}-g^{\cdot,j}_{n_{2}}\\|^{p}_{\mathbb{L}_{p}(t,\ell_{2})}.$ Thus (3.11) follows by taking $n\to\infty$. Now note that $d(u-v)=fdt\quad\hbox{ with}\quad(u-v)(0)=u_{0}.$ Thus it is easy to check that ${\mathbb{E}}\left[\sup_{s\leq t}\\|u-v\\|^{p}_{p}\right]\leq N{\mathbb{E}}\left[\\|u_{0}\\|^{p}_{p}\right]+N{\mathbb{E}}\left[\int^{t}_{0}\\|f(s,\cdot)\\|^{p}_{p}\,ds\right].$ Consequently, ${\mathbb{E}}\left[\sup_{s\leq t}\\|u\\|^{p}_{p}\right]\leq N\\|f\\|^{p}_{\mathbb{H}^{0}_{p}(t)}+c\sum_{j=1}^{m}\\|g^{\cdot,j}\\|^{p}_{\mathbb{L}_{p}(t,\ell_{2})}+N{\mathbb{E}}\\|u_{0}\\|^{p}_{p}.$ The completeness of the space $\mathcal{H}^{\alpha}_{p}(T)$ easily follows from (3.9). Indeed, let $\\{u_{n}:n=1,2,\cdots\\}$ be a Cauchy sequence in $\mathcal{H}^{\alpha}_{p}(T)$. Then $\\{u_{n}\\}$, $\\{\mathbb{D}u_{n}\\}$, $\\{\mathbb{S}^{\cdot,j}_{d}u_{n}\\}$ and $\\{u_{n}(0)\\}$ are Cauchy sequences in $\mathbb{H}^{\alpha}_{p},\mathbb{L}_{p}(T),\mathbb{H}^{\alpha/2}_{p}(T,\ell_{2})$ and $U^{\alpha/2-\alpha/p}_{p}$ respectively. Thus there exist $u\in\mathbb{H}^{\alpha}_{p}(T)$, $f\in\mathbb{L}_{p}(T),g^{\cdot,j}\in\mathbb{H}^{\alpha/2}_{p}(T,\ell_{2})$ and $u_{0}\in U^{\alpha/2-\alpha/p}_{p}$ so that $u_{n},\mathbb{D}u_{n},\mathbb{S}^{\cdot,j}_{d}u_{n},u_{n}(0)$ converge to $u,f,g^{\cdot,j},u_{0}$ respectively, that is, $\\|u_{n}-u\\|_{\mathbb{H}^{\alpha}_{p}(T)}+\\|\mathbb{D}u_{n}-f\\|_{\mathbb{L}_{p}(T)}+\\|\mathbb{S}^{\cdot,j}u_{n}-g^{j}\\|_{\mathbb{H}^{\alpha/2}_{p}(T,\ell_{2})}+\\|u_{n}(0)-u_{0}\\|_{U^{\alpha/2-\alpha/p}_{p}}\to 0$ as $n\to\infty$. Thus to prove $u\in\mathcal{H}^{\alpha}_{p}(T)$ and $u_{n}\to u$ in $\mathcal{H}^{\alpha}_{p}(T)$, we only need to show that for any $\phi\in C^{\infty}_{0}(\mathbb{R}^{d})$, the equality $(u(t,\cdot),\phi)=(u_{0},\phi)+\int^{t}_{0}(f(s,\cdot),\phi)ds+\sum_{k=1}^{\infty}\sum_{j=1}^{m}\int^{t}_{0}(g^{k,j}(s,\cdot),\phi)dY^{k,j}_{s}$ (3.12) holds for all $t\leq T$ (a.s.). Taking the limit from $(u_{n}(t,\cdot),\phi)=(u_{n}(0),\phi)+\int^{t}_{0}(\mathbb{D}u_{n}(s,\cdot),\phi)ds+\sum_{k=1}^{\infty}\sum_{j=1}^{m}\int^{t}_{0}(\mathbb{S}^{k,j}_{d}u_{n}(s,\cdot),\phi)dY^{k,j}_{s}$ and using the argument used in Remark 3.1(ii) one can show that (3.12) holds in $\Omega\times[0,T]$ (a.e.). Also using the inequality (see (3.9)) ${\mathbb{E}}\left[\sup_{t\leq T}\\|u_{n}(\cdot,t)-u_{m}(\cdot,t)\\|^{p}_{L_{p}}\right]\leq N\\|u_{n}-u_{m}\\|_{\mathcal{H}^{\alpha}_{p}(T)}$ and taking $m\to\infty$, one finds that $(u(t,\cdot),\phi)$ is right continuous with left limits, and consequently (3.12) holds for all $t\leq T$ (a.s.). The theorem is proved. $\Box$ ###### Lemma 3.5 Let $p\in(2,\infty)$, $t>0$ and $f\in L_{p}([0,t\,]\times\mathbb{R}^{d})$. Then for any $\varepsilon>\alpha(1/2-1/p)$, $\int_{\mathbb{R}^{d}}\int^{t}_{0}\int^{s}_{0}\|\partial^{\alpha/2}_{x}T_{s-r}f(r,x)\|^{p}\,drds\,dx\leq c\int^{t}_{0}\\|f(s,\cdot)\\|^{p}_{H^{\varepsilon}_{p}}\,ds,$ (3.13) where $c=c(d,p,\alpha,\varepsilon)$ is independent of $t$. Proof. Note that we may assume $\alpha(1/2-1/p)<\varepsilon<\alpha/2$. Let $q>p$ be chosen so that $\frac{1}{p}=(1-\frac{2\varepsilon}{\alpha})\times\frac{1}{2}+\frac{2\varepsilon}{\alpha}\times\frac{1}{q}.$ Such choice of $q$ is possible since $1/p>(1-\frac{2\varepsilon}{\alpha})\times\frac{1}{2}$. We will use an interpolation theorem. First, note that $\varepsilon=(1-\frac{2\varepsilon}{\alpha})\times 0+\frac{2\varepsilon}{\alpha}\times\frac{\alpha}{2}.$ Define an operator $\mathcal{A}$ by $\mathcal{A}f(s,r,x)=\begin{cases}\partial^{\alpha/2}T_{s-r}f\quad&\hbox{if }r<s,\\\ 0&\hbox{otherwise}.\end{cases}$ Then, due to (2.19) and the inequality $\\|T_{s-r}\partial^{\alpha/2}f\\|_{q}\leq\\|\partial^{\alpha/2}f\\|_{q}\leq\\|f\\|_{H^{\alpha/2}_{q}}$, the linear mappings $\mathcal{A}:L_{2}([0,t],L_{2}(\mathbb{R}^{d}))\to L_{2}([0,t]\times[0,t]\times\mathbb{R}^{d})$ and $\mathcal{A}:L_{q}([0,t],H^{\alpha/2}_{q})\to L_{q}([0,t]\times[0,t]\times\mathbb{R}^{d})$ are bounded and their norms are independent of $t$. It follows from the interpolation theory (see, for instance, [3, Theorem 5.1.2]) that the operator $\mathcal{A}:L_{p}([0,t],H^{\varepsilon}_{p}(\mathbb{R}^{d}))\to L_{p}([0,t]\times[0,t]\times\mathbb{R}^{d})$ is bounded and its norm is independent of $t$. The lemma is proved. $\Box$ ###### Theorem 3.6 Fix a constant $\varepsilon_{1}$ so that $\varepsilon_{1}=0$ if $p=2$, and $\varepsilon_{1}>\alpha(1/2-1/p)$ if $p>2$. suppose (3.2) holds. Then for any $f\in\mathcal{H}^{\gamma}_{p}(T),h\in\mathbb{H}^{\gamma+\alpha/2}_{p}(T,\ell_{2}),g^{\cdot,j}\in\mathcal{H}^{\gamma+\alpha/2+\varepsilon_{1}}_{p}(T,\ell_{2}),1\leq j\leq m$ and $u_{0}\in U^{\gamma+\alpha/2-\alpha/p}_{p}$, equation (3.4) has a unique solution $u$ in $\mathcal{H}^{\gamma+\alpha}_{p}(T)$, and for this solution $\\|u\\|_{\mathcal{H}^{\gamma+\alpha}_{p}(t)}\leq c(p,T,\delta)\left(\\|f\\|_{\mathbb{H}^{\gamma}_{p}(t)}+\\|h\\|_{\mathbb{H}^{\gamma+\alpha/2}_{p}(t,\ell_{2})}+\sum_{j=1}^{m}\\|g^{\cdot,j}\\|_{\mathbb{H}^{\gamma+\alpha/2+\varepsilon_{1}}_{p}(t,\ell_{2})}+\\|u_{0}\\|_{U^{\gamma+\alpha-\alpha/p}_{p}}\right)$ (3.14) for every $t\leq T$. Proof. As explained before, without loss of generality we assume that $h^{i}$’s are all zeros. Step 1. As in the proof of Theorem 2.11, we only need to prove the theorem for a particular $\gamma=\gamma_{0}$. Step 2. We assume $a(\omega)=1$ and prove the theorem for the equation: $du=\Delta^{\alpha/2}udt+\sum_{k=1}^{\infty}g^{k}dY^{k}_{t},\quad u(0)=0.$ (3.15) By the result of Step 1, we may assume that $\gamma=-\alpha/2$. The uniqueness is obvious and we only prove the existence and the estimate (3.14). Considering approximation arguments, for a moment, we assume that $g^{k,j}=0$ for all $k>N_{0}$ and $1\leq j\leq m$ and that $g^{k,j}(t,x)=\sum_{i=0}^{m_{k}}I_{(\tau^{k,j}_{i},\tau^{k,j}_{i+1}]}(t)g^{k_{i},j}(x),$ where $\tau^{k,j}_{i}$ are bounded stopping times and $g^{k_{i},j}(x)\in C^{\infty}_{0}(\mathbb{R}^{d})$. Define $v(t,x):=\sum_{k=1}^{N_{0}}\int^{t}_{0}g^{k}(s,x)\cdot dY^{k}_{s}=\sum_{k=1}^{N_{0}}\sum_{i=1}^{m_{k}}\sum_{j=1}^{m}g^{k_{i},j}(x)(Y^{k,j}_{t\wedge\tau^{k}_{i+1}}-Y^{k,j}_{t\wedge\tau^{k}_{i}})$ and $u(t,x):=v(t,x)+\int^{t}_{0}\Delta^{\alpha/2}T_{t-s}v\,ds=v(t,x)+\int^{t}_{0}T_{t-s}\Delta^{\alpha/2}v\,ds.$ (3.16) Now we remember from the proof of Theorem 2.11 that if functions $h_{1}=h_{1}(t,x)$ and $h_{2}=h_{2}(x)$ are sufficiently smooth, then $w_{1}(t,x):=\int^{t}_{0}T_{t-s}h_{1}(s)ds,\quad w_{2}(t,x)=T_{t}h_{2}$ solve $dw_{1}=(\Delta^{\alpha/2}w+h_{1})\,dt,\quad w_{1}(0)=0,$ $dw_{2}=\Delta^{\alpha/2}w_{2}\,dt,\quad w_{2}(0)=h_{2}.$ Therefore we have $d(u-v)=(\Delta^{\alpha/2}(u-v)+\Delta^{\alpha/2}v)dt=\Delta^{\alpha/2}udt$, and $du=\Delta^{\alpha/2}udt+dv=\Delta^{\alpha/2}udt+\sum_{k=1}^{N_{0}}g^{k}\cdot dY^{k}_{t}.$ Let $T_{t-s}g^{k}(r,x)=(T_{t-s}g^{k,1}(r,x),\dots T_{t-s}g^{k,m}(r,x))$. By (3.16) and stochastic Fubini theorem ([22, Theorem 64]), almost surely, $\displaystyle u(t,x)$ $\displaystyle=$ $\displaystyle v(t,x)+\sum_{k=1}^{N_{0}}\int^{t}_{0}\int^{s}_{0}\Delta^{\alpha/2}T_{t-s}g^{k}(r,x)\cdot dY^{k}_{r}ds$ (3.17) $\displaystyle=$ $\displaystyle v(t,x)-\sum_{k=1}^{N_{0}}\sum_{j=1}^{m}\int^{t}_{0}\int^{t}_{r}\frac{\partial}{\partial s}T_{t-s}g^{k,j}(r,x)dsdY^{k,j}_{r}$ $\displaystyle=$ $\displaystyle\sum_{k=1}^{N_{0}}\int^{t}_{0}T_{t-s}g^{k}(s,x)\cdot dY^{k}_{s}.$ Hence, $\partial^{\alpha/2}_{x}u(t,x)=\sum_{k=1}^{N_{0}}\int^{t}_{0}\partial^{\alpha/2}_{x}T_{t-s}g^{k}(s,x)\cdot dY^{k}_{s}=\sum_{k=1}^{N_{0}}\sum_{j=1}^{m}\int^{t}_{0}\partial^{\alpha/2}_{x}T_{t-s}g^{k,j}(s,x)dY^{k,j}_{s}.$ By Burkholder-Davis-Gundy’s inequality and Lemma 3.3, we have for every $0<t\leq T$ $\displaystyle{\mathbb{E}}\left[\|\partial^{\alpha/2}_{x}u(t,x)\|^{p}\right]\leq c(p){\mathbb{E}}\left[\left(\sum_{k=1}^{N_{0}}\sum_{i,j=1}^{m}\int^{t}_{0}\int_{\mathbb{R}^{m}}\partial^{\alpha/2}_{x}T_{t-s}g^{k,i}(s,x)\partial^{\alpha/2}_{x}T_{t-s}g^{k,j}(s,x)z^{i}z^{j}N^{k}(dz,ds)\right)^{p/2}\right]$ $\displaystyle\leq\,c(p){\mathbb{E}}\left[\left(\sum_{k=1}^{N_{0}}\int^{t}_{0}\int_{\mathbb{R}^{m}}\|\partial^{\alpha/2}_{x}T_{t-s}g^{k}(s,x)\|^{2}\|z\|^{2}N^{k}(dz,ds)\right)^{p/2}\right]$ $\displaystyle\leq c(p)\,{\mathbb{E}}\left[\left(\int^{t}_{0}\sum_{k=1}^{\infty}\|\partial^{\alpha/2}_{x}T_{t-s}g^{k}(s,x)\|^{2}ds\right)^{p/2}+\int^{t}_{0}\sum_{k=1}^{\infty}\|\partial^{\alpha/2}_{x}T_{t-s}g^{k}(s,x)\|^{p}ds\right].$ By (2.19), Lemma 3.5 and the inequality $\sum_{k=1}^{\infty}\|a_{k}\|^{p}\leq(\sum_{k=1}^{\infty}\|a_{n}\|^{2})^{p/2}$, ${\mathbb{E}}\left[\int^{t}_{0}\\|\partial^{\alpha/2}_{x}u(s,\cdot)\\|^{p}_{p}\,ds\right]\leq c(p,\alpha)\sum_{j=1}^{m}{\mathbb{E}}\left[\int^{t}_{0}\\|g^{\cdot,j}(s,\cdot)\\|^{p}_{H^{\varepsilon_{1}}_{p}(\ell_{2})}\,dt\right].$ (3.18) Similarly from (3.17) we also get, for every $0<t\leq T$, ${\mathbb{E}}\left[\|u(t,x)\|^{p}\right]\leq c(p)\,{\mathbb{E}}\left[\left(\int^{t}_{0}\sum_{k=1}^{N_{0}}\|T_{t-s}g^{k}(s,x)\|^{2}ds\right)^{p/2}+\int^{t}_{0}\sum_{k=1}^{N_{0}}\|T_{t-s}g^{k}(s,x)\|^{p}\,ds\right].$ (3.19) By the same argument which leads to (2.22), we see that the right side of (3.19) is finite. Thus we proved $\partial^{\alpha/2}_{x}u,u\in\mathbb{L}_{p}(T)$, and hence $u\in\mathcal{H}^{\alpha/2}_{p}(T)$. As in (2.23) and (2.24), $\displaystyle\\|u\\|^{p}_{\mathcal{H}^{\alpha/2}_{p}(t)}$ $\displaystyle\leq$ $\displaystyle c(p)\left(\\|u\\|^{p}_{\mathbb{H}^{\alpha/2}_{p}(t)}+\\|\Delta^{\alpha/2}u\\|^{p}_{\mathbb{H}^{-\alpha/2}_{p}(t)}+\\|g\\|^{p}_{\mathbb{L}_{p}(t,\ell_{2})}\right)$ $\displaystyle\leq$ $\displaystyle c\left(\\|u\\|^{p}_{\mathbb{H}^{-\alpha/2}_{p}(t)}+\\|\partial^{\alpha/2}_{x}u\\|^{p}_{\mathbb{L}_{p}(t)}+\\|g\\|^{p}_{\mathbb{L}_{p}(t,\ell_{2})}\right)$ $\displaystyle\leq$ $\displaystyle c(p,T,\alpha)\left(\\|u\\|^{p}_{\mathbb{H}^{-\alpha/2}_{p}(t)}+\\|g\\|^{p}_{\mathbb{H}^{\varepsilon_{1}}_{p}(t,\ell_{2})}\right)$ $\displaystyle\leq$ $\displaystyle c(p,T,\alpha)\int_{0}^{t}\\|u\\|^{p}_{\mathcal{H}^{\alpha/2}_{p}(s)}ds+c(p,T,\alpha)\\|g\\|^{p}_{\mathbb{H}^{\varepsilon_{1}}_{p}(T,\ell_{2})}.$ Finally, Gronwall leads to (3.14). Once one has a unique solvability of equation (3.15) and estimate (3.14) for sufficiently smooth $g$, we repeat the same approximation argument used in the Step 2 of the proof of Theorem 2.11. Step 3. Now, we follow Step 3–Step 5 of the proof of Theorem 2.11 word for word except obvious changes from $W^{k}_{t}$ to $Y^{k}_{t}$. The theorem is proved. $\Box$ Finally we consider the nonlinear equation $du=\left(a(\omega,t)\Delta^{\alpha/2}u+f(u)\right)\,dt+\sum_{k=1}^{\infty}h^{k}(u)dW^{i}_{t}+\sum_{k=1}^{\infty}\sum_{j=1}^{m}g^{k,j}(u)dY^{k,j}_{t},\quad u(0)=u_{0},$ (3.20) where $f(u)=f(\omega,t,x,u)$, $h^{k}(u)=h^{k}(\omega,t,x,u)$, $g^{k}(u)=(g^{k,1}(\omega,t,x,u),\dots,g^{k,m}(\omega,t,x,u))$, $W_{t}$ are independent $1$-dimensional Wiener processes and $Y^{k}_{t}:=\int_{\mathbb{R}^{m}}z\widetilde{N}_{k}(t,dz)$ are independent $m$-dimensional pure jump Lévy processes with Lévy measure $\nu_{k}$. ###### Assumption 3.7 Fix a constant $\varepsilon_{1}$ so that $\varepsilon_{1}=0$ if $p=2$, and $\varepsilon_{1}>\alpha(1/2-1/p)$ if $p>2$. Assume that $f(0)\in\mathbb{H}^{\gamma}_{p}(T)$, $g^{\cdot,j}(0)\in\mathbb{H}^{\gamma+\alpha/2+\varepsilon_{1}}_{p}(T,\ell_{2})$ and $h(0)\in\mathbb{H}^{\gamma+\alpha/2}_{p}(T,\ell_{2}).$ Moreover, for any $\varepsilon>0$, there exists a constant $K_{\varepsilon}$ so that for any $u=u(x),v=v(x)\in H^{\gamma+\alpha}_{p}$ and $t,\omega$ we have $\displaystyle\\|f(t,\cdot,u(\cdot))-f(t,\cdot,v(\cdot))\\|_{H^{\gamma}_{p}}+\sum_{j=1}^{m}\\|g^{\cdot,j}(t,\cdot,u(\cdot))-g^{\cdot,j}(t,\cdot,v(\cdot))\\|_{H^{\gamma+\alpha/2+\varepsilon_{1}}_{p}(\ell_{2})}$ $\displaystyle+\\|h(t,\cdot,u(\cdot))-h(t,\cdot,v(\cdot))\\|_{H^{\gamma+\alpha/2}_{p}(\ell_{2})}\leq\varepsilon\\|u-v\\|_{H^{\gamma+\alpha}_{p}}+K(\varepsilon)\\|u-v\\|_{H^{\gamma}_{p}}.$ (3.21) ###### Example 3.8 Recall that the space $B^{r}$ is defined in (2.35). Fix $\kappa_{0}=\kappa_{0}(\gamma)\geq 0$ so that $\kappa_{0}>0$ if $\gamma$ is not integer. Consider $\displaystyle f(u)$ $\displaystyle=b(\omega,t,x)\Delta^{\beta_{1}/2}u+\sum_{i=1}^{d}c^{i}(\omega,t,x)u_{x^{i}}I_{\alpha>1}+d(\omega,t,x)u+f_{0},$ $\displaystyle h^{k}(u)$ $\displaystyle=\eta^{k}(\omega,t,x)\Delta^{\beta_{2}/2}u+l^{k}(\omega,t,x)u+h^{k}_{0},$ $\displaystyle g^{k,j}(u)$ $\displaystyle=\sigma^{k,j}(\omega,t,x)\Delta^{\beta^{j}_{3}/2}u+v^{k,j}(\omega,t,x)u+g^{k,j}_{0},\quad j=1,\dots m.$ Here $\beta_{1}<\alpha$, $\beta_{2}<\alpha/2$, $\beta^{j}_{3}<\alpha/2-\varepsilon_{1}$ and $f_{0}\in\mathbb{H}^{\gamma}_{p}(T)$, $h_{0}\in\mathbb{H}^{\gamma+\alpha/2}_{p}(T,\ell_{2})$, $g^{\cdot,j}_{0}\in\mathbb{H}^{\gamma+\alpha/2+\varepsilon_{1}}_{p}(T,\ell_{2})$. Assume for each $\omega,t,i,j$, $\displaystyle\|b\|_{B^{\|\gamma\|+\kappa_{0}}}+\|c^{i}\|_{B^{\|\gamma\|+\kappa_{0}}}+\|d\|_{B^{\|\gamma\|+\kappa_{0}}}+\|\eta\|_{B^{\|\gamma\|+\alpha/2+\kappa_{0}}}+\|l\|_{B^{\|\gamma\|+\alpha/2+\kappa_{0}}}$ $\displaystyle+\|\sigma^{\cdot,j}\|_{B^{\|\gamma\|+\alpha/2+\varepsilon_{1}+\kappa_{0}}}+\|v^{\cdot,j}\|_{B^{\|\gamma\|+\alpha/2+\varepsilon_{1}+\kappa_{0}}}\leq K<\infty.$ Then the calculus in Example 2.14 shows that (3.7) holds. Here is the main result of this section. ###### Theorem 3.9 Suppose (3.2) and Assumptions 2.10 and 3.7 hold. Then the equation (3.20) has a unique solution $u\in\mathcal{H}^{\gamma+\alpha}_{p}(T)$, and for this solution we have $\\|u\\|_{\mathcal{H}^{\gamma+\alpha}_{p}(t)}\leq c\left(\\|f(0)\\|_{\mathbb{H}^{\gamma}_{p}(t)}+\\|h(0)\\|_{\mathbb{H}^{\gamma+\alpha/2}_{p}(t,\ell_{2})}+\sum_{j=1}^{m}\\|g^{\cdot,j}(0)\\|_{\mathbb{H}^{\gamma+\alpha/2+\varepsilon_{1}}_{p}(t,\ell_{2})}+\\|u_{0}\\|_{U^{\gamma+\alpha/2-\alpha/p}_{p}}\right),$ (3.22) for every $t\leq T$, where $c=c(p,T,\delta)$. Proof. As we mentioned in the previous section, our proof is a repetition of that of Theorem 6.4 in [14]. By Theorem 3.6, for any $u\in\mathcal{H}^{\gamma+\alpha}_{p}(T)$ with initial data $u_{0}$ we can define $v=\mathcal{R}u$ as the solution of $\displaystyle dv=\left(a(\omega,t)\Delta^{\alpha/2}v(t,x)+f(t,x,u)\right)dt+\sum_{i=1}^{\infty}h^{i}(t,x,u)dW^{i}_{t}+\sum_{k=1}^{\infty}\sum_{j=1}^{m}g^{k,j}(t,x,u)dY^{k,j}_{t},\quad v(0)=u_{0}.$ Then for any $u,v$ initial data $u_{0}$, we have $(\mathcal{R}u-\mathcal{R}v)(0,x)=0$ and $\displaystyle d(\mathcal{R}u-\mathcal{R}v)=$ $\displaystyle\left(a(\omega,t)\Delta^{\alpha/2}(\mathcal{R}u-\mathcal{R}v)+(f(t,x,u)-f(t,x,v))\right)dt$ $\displaystyle+\sum_{i=1}^{\infty}\int_{0}^{t}(h^{i}(t,x,u)-h^{i}(t,x,u))dW^{i}_{t}+\sum_{k=1}^{\infty}\sum_{j=1}^{m}\int_{0}^{t}(g^{k,j}(t,x,u)-g^{k,j}(t,x,v))dY^{k,j}_{t}.$ By Theorems 2.11 and Assumption 3.7, for every $t\in(0,T]$, $\displaystyle\\|\mathcal{R}u-\mathcal{R}v\\|^{p}_{\mathcal{H}^{\gamma+\alpha}_{p}(t)}$ $\displaystyle\leq c(p,T,\delta)\Big{(}\\|f(u)-f(v)\\|^{p}_{\mathbb{H}^{\gamma}_{p}(t)}+\\|h(u)-h(v)\\|^{p}_{\mathbb{H}^{\gamma+\alpha/2}_{p}(t,\ell_{2})}+\sum_{j=1}^{m}\\|g^{\cdot,j}(u)-g^{\cdot,j}(v)\\|^{p}_{\mathbb{H}^{\gamma+\alpha/2+\varepsilon_{1}}_{p}(t,\ell_{2})}\Big{)}$ $\displaystyle\leq\varepsilon^{p}c(p,T,\delta)\\|u-v\\|^{p}_{\mathcal{H}^{\gamma+\alpha}_{p}(t)}+K(\varepsilon)c(p,T,\delta)\int_{0}^{t}{\mathbb{E}}\\|u(s,\cdot)-v(s,\cdot)\\|^{p}_{H^{\gamma}_{p}}ds$ $\displaystyle\leq\theta\\|u-v\\|^{p}_{\mathcal{H}^{\gamma+\alpha}_{p}(t)}+N\int_{0}^{t}\\|u-v\\|^{p}_{\mathcal{H}^{\gamma+\alpha}_{p}(s)}ds$ where $\theta:=\varepsilon^{p}c(p,T,\delta)$ and $N=c(p,T,\delta,\varepsilon)$. Denote $\mathcal{R}^{n+1}u:=\mathcal{R}(\mathcal{R}^{n}u)$. Then by induction, for every $t\in(0,T]$ $\displaystyle\\|\mathcal{R}^{n}u-\mathcal{R}^{n}v\\|^{p}_{\mathcal{H}^{\gamma+\alpha}_{p}(t)}$ $\displaystyle\leq$ $\displaystyle\theta^{n}\\|u-v\\|^{p}_{\mathcal{H}^{\gamma+\alpha}_{p}(t)}+\sum_{k=1}^{n}{{n}\choose{k}}\theta^{n-k}N^{k}\int_{0}^{t}\frac{(t-s)^{k-1}}{(k-1)!}\\|u-v\\|^{p}_{\mathcal{H}^{\gamma+\alpha}_{p}(s)}ds$ Therefore, $\displaystyle\\|\mathcal{R}^{n}u-\mathcal{R}^{n}v\\|^{p}_{\mathcal{H}^{\gamma+\alpha}_{p}(T)}$ $\displaystyle\leq$ $\displaystyle\theta^{n}\sum_{k=0}^{n}{{n}\choose{k}}\frac{(NT/\theta)^{k}}{k!}\\|u-v\\|^{p}_{\mathcal{H}^{\gamma+\alpha}_{p}(T)}$ $\displaystyle\leq$ $\displaystyle(2\theta)^{n}\left(\sup_{k\geq 0}\frac{(NT/\theta)^{k}}{k!}\right)\\|u-v\\|^{p}_{\mathcal{H}^{\gamma+\alpha}_{p}(T)}.$ Choose $\varepsilon>0$ so that $2\theta<1/2$, and then fix $n$ large enough so that $(2\theta)^{n}\left(\sup_{k\geq 0}\frac{(NT/\theta)^{k}}{k!}\right)<1/2$. Then $\bar{\mathcal{R}}:=\mathcal{R}^{n}$ is a contraction in $\mathcal{H}^{\gamma+\alpha}_{p}(T)$ and obviously the unique fixed point $u$ under this map becomes the unique solution of (3.20). Moreover, the estimate (3.22) also easily from Assumption 3.7, Theorems 3.6 and 3.4. We leave the details to the readers as an exercise. $\Box$ ## 4 Application and Extension First, we consider equations with the random fractional Laplacian driven by (Lévy) space-time white noise; Let $d=1$ and consider the equation $du=(a(\omega,t)\Delta^{\alpha/2}u(t,x)+f(\omega,t,x,u(t,x)))dt+\xi(\omega,t,x)h(\omega,t,x,u(t,x))d\mathcal{Z}_{t}$ (4.23) where $\mathcal{Z}_{t}$ is a cylindrical Lévy process on $L_{2}(\mathbb{R})$, that is $\mathcal{Z}_{t}$ has an expansion of the form $\mathcal{Z}_{t}=\sum_{k=1}^{\infty}\eta^{k}(x)Z^{k}_{t}$ where $\\{\eta^{k}:k=1,2,\dots\\}$ is an orthonormal basis in $L_{2}$ and $Z^{k}_{t}$ are i.i.d. one-dimensional $\mathcal{F}_{t}$-adapted Lévy processes (see [10] for the details). Using this expansion we can rewrite (4.23) as follows : $du=(a(\omega,t)\Delta^{\alpha/2}u+f(u))dt+\sum_{k=1}^{\infty}g^{k}(u)dZ^{k}_{t},$ (4.24) where $g^{k}(u):=\xi(\omega,t,x)h(\omega,t,x,u(t,x))\eta^{k}(x)$. Let $\gamma,p,s,r$ be constants satisfying $0>\gamma+\alpha/2>-1,\quad p\geq 2r\geq 2,\quad 1\leq r<(2\gamma+\alpha+2)^{-1},\quad s^{-1}+r^{-1}=1\,\,(1\leq s\leq\infty).$ (4.25) Define $R_{\gamma}(x):=\|x\|^{-(\gamma+\alpha/2+1)}\int^{\infty}_{0}t^{-(\gamma+\alpha/2+3)/2}e^{-tx^{2}-1/(4t)}dt.$ It is known that there exists a constant $c>0$ so that $cR_{\gamma}(x)$ is the kernel of the operator $(1-\Delta)^{(\gamma+\alpha/2)/2}$, that is $(1-\Delta)^{(\gamma+\alpha/2)/2}f=(cR_{\gamma}f)(x)$. ###### Assumption 4.1 (i) For each $x$, $\xi=\xi(\omega,t,x)$ is predictable, and $\\|\xi(\omega,t,\cdot)\\|_{L_{2s}}\leq K$ for each $\omega,t$. (ii) For each $x,u$, the processes $f(\omega,t,x,u),h(\omega,t,x,u)$ are predictable, and $\|f(\omega,t,x,u)-f(\omega,t,x,v)\|\leq K\|u-v\|,\quad\|h(\omega,t,x,u)-h(\omega,t,x,v)\|\leq K\|u-v\|.$ By following the arguments in the proof of [14, Lemma 8.4], we get the following ###### Lemma 4.2 Let (4.25) hold. Take some functions $h_{0}=h_{0}(x)\in L_{p}(\mathbb{R})$, $\xi_{0}=\xi_{0}(x)\in L_{2s}(\mathbb{R})$, and set $g^{k}_{0}=\xi_{0}h_{0}\eta^{k}$. Then $g_{0}=\\{g^{k}_{0}\\}\in H^{\gamma+\alpha/2}_{p}(\ell_{2})$ and $\\|g_{0}\\|_{H^{\gamma+\alpha/2}_{p}(\ell_{2})}=\\|\overline{h}_{0,\gamma}\\|_{p}\leq N\\|\xi_{0}\\|_{2s}\\|h_{0}\\|_{p},$ where $N=\\|R_{\gamma}\\|_{2r}<\infty$ and $\bar{h}_{0,\gamma}(x):=\left(\int_{\mathbb{R}}R_{\gamma}^{2}(x-y)\xi^{2}_{0}(y)h^{2}_{0}(y)dy\right)^{1/2}.$ We first discuss the case when $Z^{k}_{t}$ are independent one-dimensional Wiener processes. ###### Theorem 4.3 Let $Z^{k}_{t}$ be independent one-dimensional Wiener processes. Suppose (4.25) and Assumption 4.1 hold. Also assume $\gamma\in(-\alpha,\frac{-1-\alpha}{2})$, $u_{0}\in U^{\gamma+\alpha-\alpha/p}_{p}$ and $I(p,T):=\left({\mathbb{E}}\int^{T}_{0}\left(\\|f(t,\cdot,0)\\|^{p}_{H^{\gamma}_{p}}+\\|\bar{h}(t,\cdot,0)\\|^{p}_{p}\right)ds\right)^{1/p}<\infty,$ (4.26) where $\bar{h}(t,x,0):=\left(\int_{\mathbb{R}}R_{\gamma}^{2}(x-y)\xi^{2}(y)h^{2}(t,y,0)dy\right)^{1/2}.$ Then equation (4.23) with initial data $u_{0}$ has a unique solution $u\in\mathcal{H}^{\gamma+\alpha}_{p}(T)$ and for this solution, $\\|u\\|_{\mathcal{H}^{\gamma+\alpha}_{p}(T)}\leq c\left(I(p,T)+\\|u_{0}\\|_{U^{\gamma+\alpha-\alpha/p}_{p}}\right).$ Proof. We check whether $f(u)$ and $g(u)$ satisfy condition (2.13). Since $\gamma<0$ and $\gamma+\alpha>0$, $\\|f(u)-f(v)\\|_{H^{\gamma}_{p}}\leq\\|f(u)-f(v)\\|_{L_{p}}\leq K\\|u-v\\|_{L_{p}}\leq\varepsilon\\|u-v\\|_{H^{\gamma+\alpha}_{p}}+K(\varepsilon)\\|u-v\\|_{H^{\gamma}_{p}}.$ Also for $g(u)=\\{g^{k}(u)\\}$, by Lemma 4.2, $\\|g(0)\\|_{H^{\gamma+\alpha/2}_{p}(\ell_{2})}\leq\\|R_{\gamma}\\|_{2r}\\|\xi\\|_{2s}\\|h(0)\\|_{p}\leq c\\|h(0)\\|_{p},$ $\\|g(u)-g(v)\\|_{H^{\gamma+\alpha/2}_{p}(\ell_{2})}\leq\\|R_{\gamma}\\|_{2r}\xi\\|_{2s}\\|h(u)-h(v)\\|_{p}\leq c\\|u-v\\|_{L_{p}}\leq\varepsilon\\|u-v\\|_{H^{\gamma+\alpha}_{p}}+K(\varepsilon)\\|u-v\\|_{H^{\gamma}_{p}}.$ Therefore condition (2.13) is satisfied and the theorem is proved. $\Box$ Now we consider space-time white noise with jump Lévy processes. Unlike Theorem 4.3, in the case space-time white noise with jump Lévy processes, $L_{p}$-theory is not satisfactory due to the condition $\varepsilon_{1}>\alpha(1/2-1/p)$ if $p>2$. Thus we only give an $L_{2}$-theory. ###### Theorem 4.4 Suppose $Z^{k}_{t}$ are independent one-dimensional jump Lévy processes with Lévy measure $\nu$. Suppose (3.2), (4.25) and Assumption 4.1 hold with $p=2$. Also assume $\gamma\in(-\alpha,\frac{-1-\alpha}{2})$, $u_{0}\in U^{\gamma+\alpha-\alpha/2}_{2}$ and $I(2,T)<\infty$, where $I(2,T)$ is taken from (4.26). Then equation (4.23) with initial data $u_{0}$ has a unique solution $u\in\mathcal{H}^{\gamma+\alpha}_{2}(T)$ and for this solution, $\\|u\\|_{\mathcal{H}^{\gamma+\alpha}_{2}(T)}\leq c\left(I(2,T)+\\|u_{0}\\|_{U^{\gamma+\alpha-\alpha/2}_{2}}\right).$ Proof. There is nothing to prove since conditions on $f$ and $g$ were already checked in the proof of Theorem 4.3. $\Box$ For a stopping time $\tau$ relative to $\\{\mathcal{F}_{t}\\}$, denote $(\\![0,\tau]\\!]:=\\{(\omega,t):0<t\leq\tau(\omega)\\}.$ Then obviously the process ${\bf 1}_{(\\![0,\tau]\\!]}(\omega,t)$ is left- continuous and predictable. For an $H^{\gamma}_{p}$-valued $\mathcal{P}^{dP\times dt}$-measurable process $u$, write $u\in\mathbb{H}^{\gamma}_{p}(\tau)$ if $\\|u\\|^{2}_{\mathbb{H}^{\gamma}_{p}(\tau)}:={\mathbb{E}}\left[\int^{\tau}_{0}\\|u\\|^{2}_{H^{\gamma}_{p}}ds\right]<\infty.$ We define the Banach spaces $\mathbb{L}_{p}(\tau)$, $\mathbb{L}_{p}(\tau,\ell_{2})$ and $\mathcal{H}^{\gamma}_{p}(\tau)$ similarly. The following theorem plays the key role when we weaken condition (3.2) later in the next section. ###### Theorem 4.5 Let $\tau\leq T$ be a stopping time. Fix a constant $\varepsilon_{1}$ so that $\varepsilon_{1}=0$ if $p=2$, and $\varepsilon_{1}>\alpha(1/2-1/p)$ if $p>2$. Then, under Assumption 2.10 and (3.2), for any $f\in\mathbb{H}^{\gamma}_{p}(\tau)$, $h\in\mathbb{H}^{\gamma+\alpha/2}_{p}(\tau,\ell_{2})$, $g^{\cdot,j}\in\mathbb{H}^{\gamma+\alpha/2+\varepsilon_{1}}_{p}(\tau,\ell_{2}),1\leq j\leq m$ and $u_{0}\in U^{\gamma+\alpha/2-\alpha/p}_{p}$, equation (3.4) has a unique solution $u$ in $\mathcal{H}^{\gamma+\alpha}_{p}(\tau)$, and for this solution $\\|u\\|_{\mathcal{H}^{\gamma+\alpha}_{p}(\tau)}\leq c\left(\\|f\\|_{\mathbb{H}^{\gamma}_{p}(\tau)}+\\|h\\|_{\mathbb{H}^{\gamma+\alpha/2}_{p}(\tau,\ell_{2})}+\sum_{j=1}^{m}\\|g^{\cdot,j}\\|_{\mathbb{H}^{\gamma+\alpha/2+\varepsilon_{1}}_{p}(\tau,\ell_{2})}+\\|u_{0}\\|_{U^{\gamma+\alpha-\alpha/p}_{p}}\right),$ (4.27) where $c=c(p,T,\delta)$ independent of $\tau$. Proof. First we prove the existence and (4.27). Obviously we have $\bar{f}:={\bf 1}_{(\\![0,\tau]\\!]}\,f\in\mathbb{H}^{\gamma}_{p}(T),\quad\bar{h}:={\bf 1}_{(\\![0,\tau]\\!]}\,h\in\mathbb{H}^{\gamma+\alpha/2}_{p}(T,\ell_{2}),\quad\bar{g^{\cdot,j}}:={\bf 1}_{(\\![0,\tau]\\!]}\,g^{\cdot,j}\in\mathbb{H}^{\gamma+\alpha/2+\varepsilon_{1}}_{p}(T,\ell_{2}).$ Let $u\in\mathcal{H}^{\gamma+\alpha}_{p}(T)$ be the solution of (2.14) with $\bar{f},\bar{h}$ and $\bar{g}$ instead of $f,h$ and $g$ respectively. Then, since $\tau\leq T$, we have $\\|u\\|_{\mathcal{H}^{\gamma+\alpha}_{p}(\tau)}\leq\\|u\\|_{\mathcal{H}^{\gamma+\alpha}_{p}(T)}$, and by Theorem 3.6, $\displaystyle\\|u\\|_{\mathcal{H}^{\gamma+\alpha}_{p}(\tau)}$ $\displaystyle\leq$ $\displaystyle c\left(\\|\bar{f}\\|_{\mathbb{H}^{\gamma}_{p}(T)}+\\|\bar{h}\\|_{\mathbb{H}^{\gamma+\alpha/2}_{p}(T,\ell_{2})}+\sum_{j=1}^{m}\\|\bar{g}^{\cdot,j}\\|_{\mathbb{H}^{\gamma+\alpha/2+\varepsilon_{1}}_{p}(T,\ell_{2})}+\\|u_{0}\\|_{U^{\gamma+\alpha/2-\alpha/p}_{p}}\right)$ $\displaystyle=$ $\displaystyle c\left(\\|f\\|_{\mathbb{H}^{\gamma}_{p}(\tau)}+\\|h\\|_{\mathbb{H}^{\gamma+\alpha/2}_{p}(\tau,\ell_{2})}+\sum_{j=1}^{m}\\|g^{\cdot,j}\\|_{\mathbb{H}^{\gamma+\alpha/2+\varepsilon_{1}}_{p}(\tau,\ell_{2})}+\\|u_{0}\\|_{U^{\gamma+\alpha/2-\alpha/p}_{p}}\right).$ Now we prove the uniqueness. Let $u\in\mathcal{H}^{\gamma+\alpha}_{p}(\tau)$ be a solution of equation (2.14). Then obviously, ${\bf 1}_{(\\![0,\tau]\\!]}\cdot(\mathbb{D}u-a(\omega,t)\Delta^{\alpha/2}u)\in\mathbb{H}^{\gamma}_{p}(T),\quad{\bf 1}_{(\\![0,\tau]\\!]}\cdot\mathbb{S}_{c}u\in\mathbb{H}^{\gamma+\alpha/2}_{p}(T,\ell_{2}),\quad{\bf 1}_{(\\![0,\tau]\\!]}\cdot\mathbb{S}^{\cdot,j}_{d}u\in\mathbb{H}^{\gamma+\alpha/2+\varepsilon_{1}}_{p}(T,\ell_{2}).$ According to Theorem 3.6 we can define $v\in\mathcal{H}^{\gamma+\alpha}_{p}(T)$ as the solution of $\displaystyle dv$ $\displaystyle=$ $\displaystyle(a(\omega,t)\Delta^{\alpha/2}v+{\bf 1}_{(\\![0,\tau]\\!]}\,(\mathbb{D}u-a(\omega,t)\Delta^{\alpha/2}u))dt+\sum_{k=1}^{\infty}1_{(\\![0,\tau]\\!]}\,\mathbb{S}^{k}_{c}u\,dW^{k}_{t}$ (4.28) $\displaystyle\quad+\sum_{k=1}^{\infty}\sum_{j=1}^{m}1_{(\\![0,\tau]\\!]}\,\mathbb{S}^{k,j}_{d}u\,dZ^{k}_{t},\qquad v(0)=u(0).$ Then for $t\leq\tau$, $d(u-v)=\Delta^{\alpha/2}(u-v)dt$ and $(u-v)(0)=0$. Therefore by Theorem 2.9, we conclude that $u(t)=v(t)$ for all $t\leq\tau$ a.s.. By replacing $u$ by $v$ for $t\leq\tau$, from (4.28) we find that $v$ satisfies $dv=\left(a\Delta^{\alpha/2}v+f{\bf 1}_{(\\![0,\tau]\\!]}\right)dt+\sum_{k=1}^{\infty}1_{(\\![0,\tau]\\!]}\,h^{k}\,dW^{k}_{t}+\sum_{k=1}^{\infty}\sum_{j=1}^{m}1_{(\\![0,\tau]\\!]}\,g^{k,j}\,dZ^{k}_{t},\quad v(0)=u_{0}.$ (4.29) We proved that if $u\in\mathcal{H}^{\gamma+\alpha}_{p}(\tau)$ is a solution of equation (2.14) then $u(t)=v(t)$ for all $t\leq\tau$ a.s.. This proves the uniqueness of solution of equation (2.14) in the class $\mathcal{H}^{\gamma+\alpha}_{p}(\tau)$ because by Theorem 3.6 $v\in\mathcal{H}^{\gamma+\alpha}_{p}(T)$ is the unique solution of equation (4.29). The theorem is proved. $\Box$ For a stopping time $\tau\leq T$ and $\gamma\in\mathbb{R}$, write $u\in\mathbb{H}^{\gamma}_{p,{\rm loc}}(\tau)$ if there exists a sequence of stopping times $\tau_{n}\uparrow\infty$ so that $u\in\mathbb{H}^{\gamma}_{p}(\tau\wedge\tau_{n})$ for each $n$. The following is a weakened version of (3.2). ###### Assumption 4.6 There exists an integer $N_{0}\geq 1$ so that $\widehat{c}_{k}<\infty$ for all integer $k>N_{0}$. ###### Definition 4.7 Let $u_{0}\in U^{\gamma+\alpha-\alpha/p}_{p}$, $f(0)\in\mathbb{H}^{\gamma}_{p}(\tau)$, $h(0)\in\mathbb{H}^{\gamma+\alpha/2}_{p}(\tau,\ell_{2})$ and $g^{\cdot,j}(0)\in\mathbb{H}^{\gamma+\alpha/2+\varepsilon_{1}}_{p}(\tau,\ell_{2})$, $1\leq j\leq m$. We say that $u\in\mathcal{H}^{\gamma+\alpha}_{p,{\rm loc}}(\tau)$ is a path-wise solution to (3.4) if the followings hold; (i) $u\in\mathbb{H}^{\gamma+\alpha}_{p,{\rm loc}}(\tau)$ and $u(t)$ is right continuous with left limits in $H^{\gamma}_{p}$ for $t<\tau$ ($a.s.$), (ii) for any $\phi\in C^{\infty}_{0}(\mathbb{R}^{d})$, the equality $\displaystyle(u(t,\cdot),\phi)=$ $\displaystyle(u_{0},\phi)+\int^{t}_{0}a(\omega,s)(u(s,\cdot),\Delta^{\alpha/2}\phi)ds+\int^{t}_{0}(f(s,\cdot),\phi)ds$ $\displaystyle+\sum_{k=1}^{\infty}\int^{t}_{0}(h^{k}(s,\cdot),\phi)dW^{k}_{s}+\sum_{k=1}^{\infty}\sum_{j=1}^{m}\int^{t}_{0}(g^{k,j}(s,\cdot),\phi)dY^{k,j}_{s}$ (4.30) holds for all $t<\tau$ $a.s.$. ###### Theorem 4.8 Let $\tau\leq T$. Suppose that Assumptions 2.10 and 4.6 hold. Then for any $u_{0}\in U^{\gamma+\alpha-\alpha/p}_{p}$, $f\in\mathbb{H}^{\gamma}_{p}(\tau)$, $h\in\mathbb{H}^{\gamma+\alpha/2}_{p}(\tau,\ell_{2})$, $g^{\cdot,j}\in\mathbb{H}^{\gamma+\alpha/2+\varepsilon_{1}}_{p}(\tau,\ell_{2}),1\leq j\leq m$, there exists a unique path-wise solution $u\in\mathcal{H}^{\gamma+\alpha}_{p,{\rm loc}}(\tau)$ to (3.4). In particular, if $\gamma+\alpha>d/p$, then the unique path-wise solution $u$ is $C^{\gamma+\alpha-d/p}$-valued process (for $t\leq\tau$) a.s.. Proof. Step 1. First, additionally assume that (3.2) holds. Then the existence of path-wise solution under (3.2) in $\mathcal{H}_{p}^{\gamma+\alpha}(\tau)$ (hence in $\mathcal{H}^{\gamma+\alpha}_{p,{\rm loc}}(\tau))$) follows from Theorem 4.5. Now we show that the pathwise solution is unique in $\mathcal{H}^{\gamma+\alpha}_{p,{\rm loc}}(\tau)$. Let $u\in\mathcal{H}^{\gamma+\alpha}_{p,{\rm loc}}(\tau)$ be a path-wise solution. Define $\tau_{n}=\tau\wedge\inf\\{t:\int^{t}_{0}\\|u\\|^{2}_{H^{\gamma+\alpha}_{p}}ds>n\\}$. Then $u\in\mathbb{H}^{\gamma+\alpha}_{p}(\tau_{n})$ and $\tau_{n}\uparrow\tau$ since $\int^{t}_{0}\\|u\\|^{2}_{H^{\gamma+\alpha}_{p}}ds<\infty$ for all $t<\tau$, a.s. By Theorem 4.5, $\\|u\\|_{\mathcal{H}^{\gamma+\alpha}_{p}(\tau_{n})}\leq c(T,d,\alpha)\left(\\|f\\|_{\mathbb{H}^{\gamma}_{p}(\tau_{n})}+\\|h\\|_{\mathbb{H}^{\gamma+\alpha/2}_{p}(\tau_{n},\ell_{2})}+\sum_{j=1}^{m}\\|g^{\cdot,j}\\|_{\mathbb{H}^{\gamma+\alpha/2+\varepsilon_{1}}_{p}(\tau_{n},\ell_{2})}+\\|u_{0}\\|_{U^{\gamma+\alpha-\alpha/p}_{p}}\right).$ By letting $n\to\infty$ we find that $u\in\mathcal{H}^{\gamma+\alpha}_{p}(\tau)$, and the uniqueness of the pathwise solution under (3.2) follows from the uniqueness result of Theorem 4.5. Step 2. For the general case, note that for each $n>0$ and $k\leq N_{0}$, $\widehat{c}_{k,n}:=\left(\int_{\\{z\in\mathbb{R}^{m}:\|z\|\leq n\\}}\|z\|^{2}\nu_{k}(dz)\right)^{1/2}\vee\left(\int_{\\{z\in\mathbb{R}^{m}:\|z\|\leq n\\}}\|z\|^{p}\nu_{k}(dz)\right)^{1/p}<\infty.$ Consider Lévy processes $(Z^{1}_{n},\cdots,Z^{N_{0}}_{n},Z^{N_{0}+1},\cdots)$ in place of $(Z^{1},Z^{2}\cdots)$, where $Z^{k}_{n}(k\leq N_{0})$ is obtained from $Z^{k}$ by removing all the jumps that has absolute size strictly large than $n$. Note that condition (3.2) is valid with $\widehat{c}_{k}$ replaced by $\widehat{c}_{k,n}$. By Step 1, there is a unique path-wise solution $v_{n}\in\mathcal{H}^{\gamma+\alpha}_{p}(\tau)$ with $Z^{k}_{n}$ in place of $Z^{k}$ for $k=1,2,\cdots,N_{0}$. Let $T_{n}$ be the first time that one of the Lévy processes $\\{Z^{k},1\leq k\leq N_{0}\\}$ has a jump of (absolute) size in $(n,\infty)$. Define $u(t)=v_{n}(t)$ for $t<T_{n}\wedge\tau$. Note that for $n<m$, by Step 1, we have $v_{n}(t)=v_{m}(t)$ for $t<T_{n}\wedge\tau$. This is because, for $t<T_{n}\wedge\tau$, both $v_{n}$ and $v_{m}$ satisfy (4.30) with each term inside the stochastic integral multiplied by $1_{s<T_{n}}$ (and with $Z^{k}_{n}$, $k\leq N_{0}$, in place of $Z^{k}$). Thus $u$ is well defined. By letting $n\to\infty$, one constructs a unique pathwise solution $u$ in $\mathcal{H}^{\gamma+\alpha}_{p,\text{loc}}(\tau)$. The last claim follows from Sobolev embedding theorem. The theorem is proved. $\Box$ ## References * [1] S. Albeverio, J.L. Wu and T.S. Zhang, Parabolic SPDEs driven by Poisson white noise, Stochastic Process Appl. 74 (1998), 21-36. * [2] D. Applebaum, Lévy processes and stochastic calculus. Second edition. Cambridge University Press, Cambridge, 2009. * [3] J. Bergh and J. Löfström, Interpolation spaces, Grundlerhren der Mathematischen Wissenschafter, No. 223. Springer-Verlag, Berlin-New York, 1976. * [4] K. Bogdan, A. Stós and P. Sztonyk, Harnack inequality for stable processes on $d$-sets, Studia Math. 158 (2003), no. 2, 163-198. * [5] Z.-Q. Chen and K.H. Kim, An $L^{2}$-theory of stochastic PDEs driven by Lévy processes. Preprint, 2009. * [6] Z.-Q. Chen and K.H. Kim, An $L^{p}$-theory of non-divergence form SPDE driven by Lévy processes. Preprint, 2011. * [7] T. Chang and K. Lee, On a stochastic partial differential equation with a fractional Laplacian operator, preprint. * [8] L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Communications in partial differential equations, 32 (2007), no. 7-9, 1245-1260. * [9] N. Fournier, Malliavin calculus for parabolic SPDEs with jumps, Probab.Theory Relat.Fields 87 (2000), 115-147. * [10] H. Holden, B. Øksendal, J. Ubøe and T. Zhang, Stochastic partial differential equations. A modeling, white noise functional approach. Second edition. Universitext. Springer, New York, 2010. * [11] I. Kim and K. Kim, A generalization of the Littlewood-paley inequality for the fractional Laplacian $(-\Delta)^{\alpha/2}$, submitted. * [12] K. Kim, On stochastic partial differential equations with variable coefficients in $C^{1}$ domains, Stochastic processes and their applications 112 (2004), no.2, 261-283. * [13] K. Kim and N.V. Krylov, On SPDEs with variable coefficients in one space dimension, Potential Anal. 21 (2004), no.3, 203-239. * [14] N.V. Krylov, An analytic approach to SPDEs, Stochastic Partial Differential Equations: Six Perspectives, Mathematical Surveys and Monographs 64 (1999), AMS, Providence, RI. * [15] N.V. Krylov, Introduction to the Theory of Random Processes, GSM 43, AMS (2002). * [16] N.V. Krylov and S.V. Lototsky, A Sobolev space theory of SPDEs with constant coefficients on a half line, SIAM J. Math. Anal. 30 (1999), no. 2, 298-325. * [17] N.V. Krylov and S.V. Lototsky, A Sobolev space theory of SPDEs with constant coefficients in a half space, SIAM J. on Math. Anal. 31 (1999), no 1, 19-33. * [18] R. Mikulevicius and H. Pragarauskas, On the Cauchy problem for certain integro-differential operators in Sobolev and Hölder spaces, Lithuanian Mathematical Journal, 32 (1992), no.2, 238-264 * [19] R. Mikulevicius and H. Pragarauskas, On Hölder solutions of the integro-differential Zakai equation, Stochastic processes and their applications, 119 (2009), 3319-3355. * [20] S.V. Lototsky, Dirichlet problem for stochastic parabolic equations in smooth domains, Stochastics and Stochastics Reports 68 (1999), no. 1-2, 145-175. * [21] C. Mueller, The heat equation with Levy noise, Stochastic Process Appl. 74 (1998), 67,82. * [22] P. E. Protter, Stochastic Integration and Differential Equations. Second edition. Version 2.1. Corrected third printing. Springer-Verlag, Berlin, 2005 * [23] M. Röckner and T.S. Zhang, Stochastic evolution equations of jumps type: existence, uniqueness and large deviation principle, Potential Anal. 26 (2007), 255-279. * [24] C.D. Sogge, Fourier Integrals in Classical Analysis, Cambridge, 1993.	arxiv-papers	2011-11-21T01:50:37	2024-09-04T02:49:24.514708	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Kyeong-Hun Kim and Panki Kim", "submitter": "Kyeong-Hun Kim", "url": "https://arxiv.org/abs/1111.4712" }
1111.4719	# The stellar metallicity distribution in intermediate latitude fields with BATC and SDSS data Xiyan Peng1, Cuihua Du1 , Zhenyu Wu2 1College of Physical Sciences, Graduate university of the Chinese Academy of Sciences, Beijing 100049, P. R. China 2National Astronomical Observatories, Chinese Academy of Sciences, Beijing 100012, P. R. China E-mail:ducuihua@gucas.ac.cn (Received) ###### Abstract Based on the Beijing-Arizona-Taiwan-Connecticut (BATC) and Sloan Digital Sky Survey (SDSS) photometric data, we adopt SEDs fitting method to evaluate the metallicity distribution for $\sim$40, 000 main-sequence stars in the Galaxy. According to the derived photometric metallicities of these sample stars, we find that the metallicity distribution shift from metal-rich to metal-poor with the increase of distance from the Galactic center. The mean metallicity is about of $-1.5\pm 0.2$ dex in the outer halo and $-1.3\pm 0.1$ dex in the inner halo. The mean metallicity smoothly decreases from $-0.4$ to $-0.8$ in interval $0<r\leqslant 5$ kpc. The fluctuation in the mean metallicity with Galactic longitude can be found in interval $4<r\leqslant 8$ kpc. There is a vertical abundance gradients d[Fe/H]/dz $\sim-0.21\pm 0.05$ dex kpc-1 for the thin disk ($z\leqslant 2$ kpc). At distance $2<z\leqslant 5$ kpc, where the thick disk stars are dominated, the gradients are about of $-0.16\pm 0.06$ dex kpc-1, it can be interpreted as a mixture of stellar population with different mean metallicities at all $z$ levels. The vertical metallicity gradient is $-0.05\pm 0.04$ dex kpc-1 for the halo ($z$ $>$ 5 kpc). So there is little or no metallicity gradient in the halo. ###### keywords: Galaxy: structure-Galaxy: metallicity-Galaxy: formation. ††pagerange: The stellar metallicity distribution in intermediate latitude fields with BATC and SDSS data–References††pubyear: 2002 ## 1 Introduction The structure, formation and evolution of the Galaxy are very important issues in contemporary astrophysics. The basic components of Galaxy are the thin disk, the thick disk, the halo and central bulge, albeit that the inter- relationships and distinction among different components remain subject to some debate (Gilmore 1983; 1984; Lemon et al. 2004). Recent studies, based on accurate large-area surveys, have revealed that the Galaxy is marked by numerous irregular substructure such as the Sagittarius dwarf tidal stream in the halo (Ivezic et al. 2000; Yanny et al. 2000; Vivas et al. 2001; Majewski et al. 2003) and the Monoceros stream closer to the Galactic plane (Newberg et al. 2002; Rocha-Pinto et al. 2003). Carollo et al. (2007) shown that the the halo is clearly divisible into two broadly overlapping structural components - an inner and outer halo from a local kinematic analysis. It is now apparent that our Galaxy is much more complex system than we thought before. The formation of galaxies was long thought to be a steady process resulting in a smooth distribution of stars (Bahcall & Soneira 1981; Gilmore et al. 1989; Majewski 1993). But the view of the formation of the Galaxy has changed dramatically since the discoveries of complex substructures (Newberg et al. 2002; Belokurov et al. 2007). The presence of these lumpy and complex substructure are in qualitatively agreement with models for the formation of the stellar halo through the accretion and merging of nearby dwarf galaxies. Numerical simulations also suggest that this merger process plays a crucial role in setting the structure and motions of stars within galaxies (Bullock & Johnston 2005). The abundance distribution is particular importance to understanding the formation and chemical evolution of the Galaxy (Freeman & Bland-Hawthorn 2002). Researchers have long sought to constrain models for the Galactic formation and evolution on the basis of observation of the stellar and clusters populations that it contains. Specific models of galaxy formation make specific predictions about the stellar abundance distribution. For example, stars on more radial orbits are more metal-poor than stars on planar orbits. This may indicate that the Milky Way formation began with a relatively rapid collapse of the initial proto-galactic cloud, which means the halo stars formed during the initial collapse, the disk stars formed after the gas had settled into the galactic plane. But the global collapse theory was unable to account for the lack of an abundance gradient in the Galactic halo (Searle & Zinn 1978). The current view is that the Galactic halo formed at least partly through the accretion of small satellite galaxies or merger of larger systems (Freeman et al., 2002), which is well-supported by studies of stellar kinematics and spatial distribution (Yannny et al. 2003; Juric et al. 2008). For the thick disk, it may be one of the most significant components for studying signatures of galaxy formation because it presents a snap frozen relic of the state of the early disk (Freeman 2002). An intrinsic abundance gradient in the thick disk would favor a scenarios which the thick disk was formed either in the slow late stages of the early Galactic collapse or the gradual kinematical diffusion of disk stars. On the contrary, a irregular metallicity distribution or absence of gradient would favor the thick disk having formed via the kinematical heating of thin disk or from merger debris (Siegel et al. 2009). The metallicity distribution of the Galaxy is best probed directly through spectroscopic surveys (Yoss et al. 1987; Allende-Prieto et al. 2006). However, it has the advantage of using the photometric metallicity of many more stars out to limiting magnitude of photometric survey. Accurate determination of the properties of the Galactic components requires surveys with sufficient sky coverage to assess the overall geometry, sufficient depth for mapping stars to larger distance and sufficient information to obtain reasonable distance estimates for these stars (De Jong et al. 2010). Over the past few years, numerous surveys have been used to investigate the existence and size of the Galactic abundance gradient in the disk and halo. The existence of a radial gradient in the Galaxy is now well established. An average gradient of about $-0.06$ dex kpc-1 is observed in the Galactic disk for most of the elements (Chen et al, 2003). De Jong et al. (2010) provided evidence for a radical metallicity gradients in the Galactic stellar halo. However, there is considerable disagreement about whether there is a vertical metallicity gradient among field and/or open cluster stars of the Galaxy. The BATC multicolor photometric survey accumulated a large data base which is very useful for studying the Galactic structure and formation. Du et al. (2003) provided some information on the density distribution of the main components of the Galaxy, which can present constraints on the parameters of models of the Galactic structure. Later, they use F and G dwarfs from the BATC data to study the metal-abundance information (Du et al., 2004). With the new improved observation and improved knowledge regarding galaxy formation, it becomes possible to further discuss the metallicity gradient from different observation direction of the Galaxy. In this paper, we attempt to study the metallicity gradient of the Milky Way galaxy using the 21 BATC photometric survey fields combined with the SDSS photometric data. The outlines of this paper is as follows: The BATC photometric system and data reduction are introduced briefly in Section 2. In Sect. 3 we describe the theoretical model atmospheric spectra and synthetic photometry. The metallicity distribution is discussed in Sect. 4. Finally in Sect. 5 we summarize our main conclusions in this study. ## 2 Observations and data ### 2.1 BATC photometric system and SDSS photometric system The BATC survey performs photometric observations with a large field multi- colour system. There are 15 intermediate-band filters in the BATC filter system, which covers an optical wavelength range from 3000 to 10000 Å (Fan et al. 1996; Zhou et al. 2001). The 60/90 cm f/3 Schmidt Telescope of National Astronomical Observatories (NAOC) was used in the BATC program, with a Ford Aerospace 2048 $\times$ 2048 CCD camera at its main focus. The field of view of the CCD is $58^{\prime}$ $\times$ $58^{\prime}$ with a pixel scale of $1\arcsec{\mbox{}.}7$. The BATC magnitudes adopt the monochromatic AB magnitudes as defined by Oke & Gunn (1983). The PIPELINE II reduction procedure was performed on each single CCD frame to get the point spread function (PSF) magnitude of each point source (Zhou et al. 2003). The detailed description of the BATC photometric system and flux calibration of the standard stars can be found in Fan et al. (1996) and Zhou et al. (2001, 2003). In order to apply more color information to accurately estimate the photometric stellar metallicity, we combine the BATC colors with the SDSS colors message for the sample stars. The SDSS used a dedicated 2.5 m telescope which has an imaging camera and a pair of spectrographs. The imaging camera (Gunn et al. 1998) contained 30 2048 $\times$ 2048 CCDs in the focal plane of the telescope. The flux densities of observed objects were measured almost simultaneously in five broad bands [ $u$, $g$, $r$, $i$, $z$] (Fukugita et al. 1996; Gunn et al. 1998; Hongg et al. 2001). For distinguishing explicitly between BATC and SDSS filter names, we refer to the SDSS filters and magnitudes as $u^{\prime}$, $g^{\prime}$, $r^{\prime}$, $i^{\prime}$, $z^{\prime}$. The photometric pipeline (Luption et al. 2001) detected the objects, matched the data from the five filters and measured instrumental fluxes, positions and shape parameters. The shape parameters allowed the classification of objects as point source or extended. The magnitudes derived from fitting a PSF are currently accurate to about 2 percents in $g$, $r$ and $i$, and 3-5 percents in $u$ and $z$ for bright ($\leq$ 20 mag) point source. In Table 1, we list the parameters of the BATC and SDSS filters. Col. (1) and Col. (2) represent the ID of the BATC and SDSS filters, Col. (3) and Col. (4) the central wavelengths and FWHM of the 20 filters, respectively. The reddening extinction for each star are determined from the SDSS catalog. Table 1: Parameters of the BATC and SDSS filters No. \| Filter \| Wavelength \| FWHM ---\|---\|---\|--- \| \| (Å) \| (Å) 1 \| $a$ \| 3371.5 \| 359 2 \| $b$ \| 3906.9 \| 291 3 \| $c$ \| 4193.5 \| 309 4 \| $d$ \| 4540.0 \| 332 5 \| $e$ \| 4925.0 \| 374 6 \| $f$ \| 5266.8 \| 344 7 \| $g$ \| 5789.9 \| 289 8 \| $h$ \| 6073.9 \| 308 9 \| $i$ \| 6655.9 \| 491 10 \| $j$ \| 7057.4 \| 238 11 \| $k$ \| 7546.3 \| 192 12 \| $m$ \| 8023.2 \| 255 13 \| $n$ \| 8484.3 \| 167 14 \| $o$ \| 9182.2 \| 247 15 \| $p$ \| 9738.5 \| 275 1 \| $u^{\prime}$ \| 3543 \| 569 2 \| $g^{\prime}$ \| 4770 \| 1387 3 \| $r^{\prime}$ \| 6231 \| 1373 4 \| $i^{\prime}$ \| 7625 \| 1526 5 \| $z^{\prime}$ \| 9134 \| 9500 ### 2.2 Direction and data reduction Accurate determination of the properties of components of the Milky Way requires surveys with sufficient sky coverage to assess the overall geometry (De Jong et al. 2010). Most previous investigations about the Galactic metallicity distribution use only one or a few selected lines-of-sight directions (Du et al. 2004; Siegel et al. 2009; Karatas et al. 2009). In this paper, the BATC photometry survey presented 21 intermediate-latitude fields in the multiple directions. These fields used in this paper are towards the Galactic center, the anticentre, the antirotation direction at median and high latitudes, $\|b\|>35^{\circ}$. Since metallicity distribution at high Galactic latitudes are not strongly related to the radial distribution, they are well suited to study the vertical distribution of the Galaxy. Table 2 lists the locations of the observed fields and their general characteristics. In Table 2, column 1 represents the BATC field name, columns 2 and 3 represent the Galactic longitude and latitude, columns 4 and 5 represent the limit magnitude and the number of sample star used in this study. As shown in the Table 2, the most photometric depth of our data is 21.0 mag in $i$ band. In total, there are about 40, 000 sample stars in our study. Table 2: Relative information for the BATC observation fields Observed field \| $l$ (deg) \| $b$ (deg) \| $i$ (Comp) \| star number ---\|---\|---\|---\|--- T485 \| 175.7 \| 37.8 \| 21.0 \| 1550 T518 \| 238.9 \| 39.8 \| 19.5 \| 1584 T288 \| 189.0 \| 37.5 \| 20.0 \| 2115 T477 \| 175.7 \| 39.2 \| 20.0 \| 2001 T328 \| 160.3 \| 41.9 \| 19.5 \| 1666 T349 \| 224.1 \| 35.3 \| 20.5 \| 2436 TA26 \| 191.1 \| 44.4 \| 20.0 \| 1285 T291 \| 167.8 \| 46.4 \| 20.0 \| 1670 T362 \| 245.7 \| 53.4 \| 20.0 \| 1237 T330 \| 147.2 \| 68.3 \| 20.5 \| 1020 U085 \| 121.6 \| 60.2 \| 21.0 \| 1679 T521 \| 56.1 \| -36.8 \| 20.5 \| 4430 T491 \| 62.9 \| -44.0 \| 20.0 \| 2824 T359 \| 79.7 \| -37.8 \| 20.5 \| 2932 T350 \| 251.3 \| 67.3 \| 19.5 \| 1353 T534 \| 91.6 \| 51.1 \| 21.0 \| 2044 T193 \| 59.8 \| -39.7 \| 20.0 \| 2830 T516 \| 125.0 \| -62.0 \| 20.0 \| 1113 T329 \| 169.9 \| 50.4 \| 21.0 \| 1704 TA01 \| 135.7 \| -62.1 \| 20.5 \| 1342 T517 \| 188.6 \| -38.2 \| 20.0 \| 1296 Figure 1: Two-color diagrams for the BATC T329 field stars. The panel (a) is the distribution of sample stars in the ($j-m$)/($g-j$) two-colour diagram, and panel (b) is the ($j-m$)/($g-j$) two-colour diagram after removing those objects that lay significantly off the stellar locus. The line denotes the stellar locus from equation (1). Here, because our fields in this work have also been observed by the Sloan Digital Space Survey (SDSS-DR7) and each object type (stars-galaxies-QSO) has been given. Thus, we can obtain a relative reliable star catalogue. In these star sample, it is complete to 20.5 mag with an error of less than 0.1 mag in the BATC $i$ band. Owing to the BATC observing strategy, data of stars brighter than $m$ = 14 mag, are saturated, and star counts are not completed for visual magnitudes fainter than $m$ = 21.0 mag. So our work is restricted to the magnitudes range 15 $\leqslant$ $i$ $<$ 21\. From two-color diagrams for all objects in any field as an example, we can find that most stars are plotted on a diagonal band. In Fig. 1 show the BATC ($g-j$) versus ($j-m$) two-color diagram for T329 field stars. The line which can be described by equation (1) donates the stellar locus of the sample in the T329 field . $\displaystyle(j-m)=-0.10245+0.81172(g-j)$ (1) The sample shows a main-sequence (MS) stellar locus but has significant contamination from giants, manifested in the broader distribution overlying the narrow stellar locus. In order to determine the MS star sample, we use multi-color selection criteria outlined in Karaali et al. (2003) and Juric et al. (2008), which remove objects based on their location relative to the dominant stellar locus. For example, Juric et al. (2008) applied an extra procedure which consists of rejecting objects at distance larger than 0.3 mag from the stellar locus in order to remove hot dwarfs, low-red shift quasars, and white /red dwarf unresolved binaries from their sample. The procedure does well for high latitude field data from SDSS (Karaali et al. 2003; Yaz et al. 2010). Fig. 1b gives the cleaning sample stars after rejecting those objects which is lay significantly off the stellar locus. ## 3 THEORETICAL MODEL AND CALIBRATION FOR METALLICITY ### 3.1 theoretical stellar library and synthetic photometric A homogeneous and complete stellar library can match any ambitious goals imposed on a standard library. Lejeune et al. (1997) presented a hybrid library of synthetic stellar spectra. The library covers a wide range of stellar parameters $T_{\rm eff}$= 50, 000 K to 2, 000 K in intervals of 250 K, log $g$= $-1.02$ to 5.50 in main increments of 0.5, and [M/H]= $-5.0$ to +1.0. For each model in the library, a flux spectrum is given for the same set of 1221 wavelength points covering the range 9.1 to 160, 000 nm, with a mean resolution of 20Å in the visible. The spectra are thus in a format which has proved to be adequate for synthetic photometry of wide- and intermediate-band systems (Du et al. 2004). On the basis of the theoretical library, we calculate synthetic colors of the BATC and SDSS photometric system. Here, we synthesize colors for simulated stellar spectra with $T_{\rm eff}$ and log $g$ characteristic of main sequences (log $g$ = 3.5, 4.0, 4.5 for dwarfs) and 19 values of metallicity ([M/H]= $-5.0,-4.5,-4.0,-3.5$, $-3.0,-2.5,-2.0,-1.5,-1.0,-0.5$,$-0.3,-0.2,-0.1$, 0.0, +0.1, +0.2, +0.3, +0.5 and +1.0), where [M/H] denotes metallicity relative to hydrogen. The synthetic $i$th filter magnitude can be calculated with equation (2). $m=-2.5~{}{\rm log}\frac{\int{F_{\lambda}\phi_{i}({\lambda}){\rm d}\lambda}}{\int{\phi_{i}({\lambda}){\rm d}\lambda}}-48.60,$ (2) Where ${F_{\lambda}}$ is the flux per unit wavelength, $\phi_{i}$ is the transmission curve of the $i$th filter of the BATC or SDSS filter system (Du et al. 2004) . The bluer colors are sensitive to metallicity down to the lowest observed metallicities because most of the line-blanketing from heavy elements occurs in the shorter wavelength regions. In contrast, the redder colors are primarily sensitive to temperature index. The BATC $a$, $b$ bands contain the Balmier jump, a stellar spectral feature which is sensitive to surface gravity. Since our sample includes only main sequences, it conveys little gravity information. It should be mentioned that, although the metallicity or temperature derived from synthetic photometry is not very accurate for a single star, perhaps which can be distorted by a poor point, it is meaningful for the statistic analysis of sample stars. ### 3.2 METALLICITY AND PHOTOMETRIC PARALLAX The most accurate measurements of stellar metallicity are based on spectroscopic observation. Despite the recent progress in the availability of stellar spectra (e.g. SDSS-III and RAVE), the stellar number detected in photometric surveys is much more than spectroscopic observation. So photometric methods have also often been used to give the stellar metallicity. For example, Sandage (1969) detailed a technique using UBV photometry indices to measure approximate abundance. Karaali (2003) evaluated the metal abundance by ultraviolet-excess photometric parameter using CCD UBVI data. Karaali (2005) extended this method to the SDSS photometry. Ivezic et al. (2008) obtained the mean metallicity of stars as a function of $u-g$ and $g-r$ colors of SDSS data. For the BATC multicolor photometric system, there are 15 intermediate-band filters covering an optical wavelength range from 3000 to 10000 Å. There are 5 filters for the SDSS photometric system. So the SEDs of 20 filters for every object are equivalent to a low resolution spectrum. The sample SEDs simulation with template SEDs can be used to derive the parameter of sample stars (Du et al. 2004). The standard minimization, computing and minimizing the deviations between the photometric SEDs of the star and the templates SEDs obtained with the same photometric system, is used in the fitting process. The minimum indicated the best fit to the observed SED by the set of spectra (Du et al. 2004). ${\chi^{2}=\sum\limits_{l=1}^{N_{filt}=20}\left[\frac{m_{obs,l}-m_{temp,l}}{\sigma_{l}}\right]^{2}},$ (3) where ${m_{obs,l}}$, $m_{temp,l}$ and $\sigma_{l}$ are the observed magnitude, template magnitude and their uncertainty in filter $l$, respectively, and $N_{filt}$ is the total number of filters in the photometry. According to the results of SEDs fitting, the metallicity and temperature of about 40, 000 sample stars are obtained in 21 fields. In addition, we extract the spectroscopic metallicities from sdss DR7 database for our studied 21 fields, and there are about 870 stars for which they also have photometric metallicities from our method. Using these stars, we present a calibration of our SED fitting method. After applying calibration, it is reliable for the derived photometric metallicities from SEDs fitting method. In Figure 2., we present the difference of photometric and spectroscopic metallicity as a function of $(g-r)$ color. The uncertainties of metallicity obtained from comparing SEDs between photometry and theoretical models are due to the observational error and the finite grid of the models. For the metal-poor stars ([Fe/H]$<-1.0$), the metallicity uncertainty is about 0.5 dex, and 0.2 dex for the stars [Fe/H]$>-0.5$ (Du et al. 2004). The metallicity distribution diagram for all sample stars was given in Fig. 3. One local maximum appear at [Fe/H] from -0.5 to 0 dex, and a tail down to -3.0 dex. Figure 2: The difference of photometric and spectroscopic metallicity for 870 stars as a function of $(g-r)$ color is shown. Figure 3: The metallicity distribution for all the sample stars selected in our study is shown. The stellar type can be derived from effective temperature of dwarfs, then the stellar distance relative to the sun can be obtained by equation (4). $\displaystyle m_{v}-M_{v}=5lgr-5+A_{v}$ (4) where $m_{v}$ is the visual magnitude, absolute magnitude $M_{v}$ can be obtained according to the stellar type. The reddening extinction $A_{v}$ is small for most fields. We adopted the absolute magnitude versus stellar type relation for main-sequence stars from Lang (1992). $r$ is the stellar distances. The vertical distance of the star to the galactic plane can be evaluated by equation (5): $\displaystyle z=rsinb$ (5) A variety of errors affect the determination of stellar distances. The first source of errors is from photometric uncertainty less than 0.1 mag in the BATC $i$ band; the second from the misclassification, which should be small due to the multicolor photometry. For luminosity class V, types F/G, the absolute magnitude uncertainty is about 0.3 mag. In addition, there may exist an error from the contamination of binary stars in our sample. We neglect the effect of binary contamination on distance derivation due to the unknown but small influence from mass dsitribution in binary components (Kroupa et al. 1993; Ojha et al. 1996). Figure 4: The metallicity distribution for the T291 field in different distance range is shown. In the short distance, r $<$ 1kpc, most stars are in the range [0, $-0.5$], in the larger distance, 15 $<$ r $<$ 20 kpc, most stars are poorer than $-1$. ## 4 METALLICITY DISTRIBUTION It is well known that the chemical abundance of stellar population contains much information about the population,s early evolution. The stellar metallicity distribution in the Galaxy has been the subject of photometric and spectroscopic surveys (Gilmore et al. 1985; Ratnatunga et al. 1989; Friel 1988). In this study, we want to explore possible stellar metallicity distribution variation with the observation direction. A method of SED combination for the SDSS and BATC photometries has been adopted to give the steller metallicity distribution. At first, the metallicity for the sample stars can be derived by comparing SEDs between photometry and the theoretical models. The SED fitting method is described in Section 3\. 2. The mean metallicity distribution for each field is determined in the following distance intervals (in kpc): $15<r\leqslant 20$, $10<r\leqslant 15$, $8<r\leqslant 10$, $6<r\leqslant 8$, $5<r\leqslant 6$, $4<r\leqslant 5$, $3<r\leqslant 4$, $2.5<r\leqslant 3$, $2<r\leqslant 2.5$, $1.5<r\leqslant 2$, $1<r\leqslant 1.5$, $0.25<r\leqslant 1$. As an example the metallicity distributions as a function of vertical distance for the field T291 is presented in Fig. 4. From the figures (Fig. 4), it is clear that there is a number-shift from metal-rich stars to metal-poor ones with the increasing of distance. In star counts the younger metal rich stars are confined to regions close to the Galactic mid-plane, while the older, metal-poorer stars with a larger scale height dominated at larger vertical distances from the Galactic plane. ### 4.1 Metallicity variation with Galactic longitude Mean metallicity distribution as a function of Galactic longitude for different distance intervals are presented in Fig. 5. The mean metallicity shift from metal-rich to metal-poor with the increase of distance from the Galactic center can be found in Fig. 5. The solid points represent the south galactic latitude fields, The open square points represent the north galactic latitude fields. As shown in Fig. 5, the mean metallicity in interval $10<r\leqslant 20$ kpc is around of $-1.5$ dex. In intervals $8<r\leqslant 10$ kpc our result indicates that the mean metallicity is $\sim$ $-1.3$. The mean metallicity in intervals $5<r\leqslant 8$ kpc is about $-1.0$ and the mean metallicity smoothly decreases from $-0.6$ to $-0.8$ in intervals $2.5<r\leqslant 5$ kpc. Our results are consistent with the results of Siegel et al. (2009) and Karatas et al. (2009). Siegel et al. find a monometallic thick disk and halo with abundances of [Fe/H] = $-0.8$ and $-1.4$ respectively. Karatas et al. derive mean abundance values of [Fe/H]= $-0.77\pm 0.36$ dex for the thick disk, and [Fe/H] = $-1.42\pm 0.98$. The mean metallicity decreases from $-0.4$ to $-0.6$ in intervals $0<r\leqslant 2.5$ kpc. The mean metallicity in interval $0.25<r\leqslant 1$ kpc is [Fe/H] $\sim$ $-0.3$, which is consistent with the result of Yaz et al. (2010). As shown in Fig. 5, at larger distance, r $>$ 10 kpc, compared to the typical error bars, the mean metallicity distributions variation with Galactic longitude is almost flat. For $4<r\leqslant 8$ kpc, there is a fluctuation in the mean metallicity with Galactic longitude. The overall distribution of mean metallicity has a maxminum at $l$ $\sim$ 200∘. For $2<r\leqslant 8$ kpc, the T517 filed (Galactic coordinates : $l=188.6^{\circ},b=-38.2^{\circ}$; Equatorial coordinates : $\alpha=58.59^{\circ}$, $\delta=-0.35^{\circ}$) and the TA26 field (Galactic coordinates : $l=191.1^{\circ},b=44.4^{\circ}$; Equatorial coordinates : $\alpha=139.956^{\circ}$, $\delta=33.745^{\circ}$) show metal rich character related to other fields. This feature may reflect a fluctuation from streams (such as Monocers stream) which are accreted from nearby galaxies. Juric et al.(2008) detect two overdensities in the thick disk region. Klement et al.(2009) also find individual stream from the SSPP in the direction with central coordinates (Equatorial coordinates): $\alpha$ = $58.58^{\circ}$ and $\delta$ = $-4.99^{\circ}$. Maybe the deviant behaviors of the two fields result from systematical error in the observation. In the work of AK et al. (2007), they find that the metallicity distributions for both (relatively) short and large vertical distances show systematic fluctuations. The scaleheight of thick disk varies with the observed direction were found in the works of Du et al. (2006) and Bilir et al. (2008). Figure 5: The mean metallicity distribution as a function of galactic longitude in different distance intervals are shown. ### 4.2 THE VERTICAL METALLICITY GRADIENT Detailed information about the vertical metallicity gradient can provide important clue about the formation scenario of stellar population. Here, we used the mean metallicity to described the metallicity distribution function. As an example, The distribution trend of mean metallicity [Fe/H] with height above the galactic plane [$z$] for the T291 field is shown in Fig. 6. The metallicity gradients for all the fields in different $z$ intervals $z<2$ kpc, $2<z\leqslant 5$ kpc and $5<z\leqslant 15$ kpc are given in Fig. 7 and detailed in Table 3. In Table 3, Column 1 represents the BATC field name, Columns 2 - 7 represent gradient and error of gradient in different $z$ distance: $z\leqslant 2$ kpc, $2<z\leqslant 5$ kpc and $5<z\leqslant 15$ kpc, respectively. From the Fig. 7 we can find that the variation of the gradient for the halo with galactic longitude is flat and the mean gradient of halo is about $-0.05\pm 0.04$ dex kpc-1 ($5<z\leqslant 15$ kpc), which is essentially in agreement with the conclusion of Yaz et al. (2010) and Du et al. (2004). Du et al.(2004) find the small or zero gradient d[Fe/H]/dz = $-0.06\pm 0.09$ in the halo. Yaz et al. (2010) find $d[M/H]/dz=-0.01$ dex kpc-1 for the inner spheroid. The result of Karaali et al. (2003) is slightly steeper than the value of our result. Karaali et al. (2003) find that there is a metallicity gradient d[Fe/H]/dz $\sim-0.1$ dex kpc-1 in the inner halo ($5<z\leqslant 8$ kpc ) and zero in the outer part ($8<z\leqslant 10$ kpc). From Fig. 6 we can find that the incompleteness of the star sample causes significant statistical uncertainties at large distance. Probably, there is little or no metallicity gradient in the halo. It is consistent with the merger or accretion origin of the outer halo. As shown in Fig. 7, at distance $0<z<2$ kpc, the mean vertical abundance gradient is about d[Fe/H]/dz $\sim-0.21\pm 0.05$ dex kpc-1. The value for the vertical metallicity at distance $0<z<2$ kpc is in agreement with the canonical metallicity gradients with the same $z$ distances. For example, Yaz et al. (2009) find the metallicity gradient is d[Fe/H]/dz $\sim-0.3$ dex kpc-1 for short distance. The metallicity gradient is found to be d[Fe/H]/dz $\sim-0.37$ dex kpc-1 for $z$ $<$ 4 kpc in the work of Du et al. (2004). The result of Karaali et al. (2003) can be described as d[Fe/H]/dz $\sim-0.2$ dex kpc-1 for the thin and thick disk. At distance $2<z\leqslant 5$ kpc, where the thick disk stars dominated, the gradient is about $-0.16\pm 0.06$ dex kpc-1 in our work which is consistent with the work of karaali et al. (2003) and less than the value of Du et al. (2004). Du et al. (2004) point out that the metallicity gradient is d[Fe/H]/dz $\sim$ $-0.37$ dex kpc-1. In our study, the thick disk gradient is interpreted as different contribution from three components of the Galaxy at different $z$ distance. The existence of a clear vertical metallicity of the thick disk would be an important clue about the origin of the thick disk. However, it is an open question for the formation of the thick disk component. A number of models have been put forward since the confirmation if its existence. Chen et al. (2001) support that the thick disk formed through the heating of a preexisting thin disk, with the heating mechanism being the merging of a satellite galaxy. Here, we also favor the thick disk having formed via the kinematical heating of thin disk and from merger debris. Thus, there is a irregular metallicity distribution or absence of intrinsic gradient. Figure 6: The mean metallicity as a function of vertical distance $z$ for the T291 field. The metallicity gradients of the thin disk, thick disk and halo are $-0.23\pm 0.03$, $-0.18\pm 0.07$, $-0.05\pm 0.01$ dex kpc-1, respectively. Figure 7: The metallicity gradients distribution for all fields in this study are shown for the intervals $z$ $<$ 2 kpc , 2 kpc $<$ $z$ $<$ 5kpc and 5 kpc $<$ $z$ $<$ 15kpc. Table 3: The gradient distribution in different distance interval for the selected fields \| 0-2 (kpc) \| \| 2-5 (kpc) \| \| 5-15 (kpc) \| ---\|---\|---\|---\|---\|---\|--- Observed field \| gradient \| error \| gradient \| error \| gradient \| error T193 \| -0.136 \| 0.017 \| -0.080 \| 0.020 \| -0.020 \| 0.022 T288 \| -0.258 \| 0.027 \| -0.267 \| 0.032 \| -0.007 \| 0.037 T291 \| -0.228 \| 0.028 \| -0.185 \| 0.071 \| -0.045 \| 0.014 T328 \| -0.110 \| 0.027 \| -0.126 \| 0.032 \| -0.137 \| 0.043 T329 \| -0.170 \| 0.028 \| -0.100 \| 0.042 \| -0.037 \| 0.015 T330 \| -0.171 \| 0.042 \| -0.217 \| 0.055 \| -0.080 \| 0.010 T349 \| -0.180 \| 0.017 \| -0.225 \| 0.050 \| -0.029 \| 0.031 T350 \| -0.267 \| 0.033 \| -0.178 \| 0.074 \| -0.069 \| 0.035 T359 \| -0.203 \| 0.017 \| -0.126 \| 0.023 \| -0.054 \| 0.023 T362 \| -0.237 \| 0.037 \| -0.126 \| 0.060 \| -0.037 \| 0.022 T477 \| -0.181 \| 0.025 \| -0.275 \| 0.030 \| -0.053 \| 0.022 T485 \| -0.247 \| 0.033 \| -0.119 \| 0.044 \| 0.028 \| 0.040 T491 \| -0.194 \| 0.017 \| -0.179 \| 0.020 \| -0.070 \| 0.027 T516 \| -0.291 \| 0.039 \| -0.204 \| 0.061 \| -0.005 \| 0.026 T517 \| 0.054 \| 0.041 \| -0.173 \| 0.059 \| -0.101 \| 0.067 T518 \| -0.163 \| 0.035 \| -0.026 \| 0.045 \| -0.051 \| 0.052 T521 \| -0.197 \| 0.012 \| -0.175 \| 0.013 \| -0.045 \| 0.011 T534 \| -0.254 \| 0.028 \| -0.110 \| 0.035 \| -0.035 \| 0.012 TA01 \| -0.199 \| 0.037 \| -0.192 \| 0.046 \| -0.047 \| 0.015 TA26 \| 0.307 \| 0.030 \| -0.187 \| 0.038 \| -0.085 \| 0.047 U085 \| -0.146 \| 0.024 \| -0.148 \| 0.037 \| -0.035 \| 0.013 ## 5 CONCLUSIONS AND SUMMARY In this work, based on the BATC and SDSS photometric data, we evaluated the stellar metallicity distribution for 40, 000 main-sequence stars in the Galaxy by adopting SEDs fitting method. These selected fields are towards the Galactic center, the anticentre, the antirotation direction at median and high latitudes . The metallicity distribution could be obtained up to distances $r=20$ kpc, which covers the thin disk, thick disk and halo. We determined the mean stellar metallicity as a function of vertical distance in different direction. It can be clearly seen that the metallicity distribution shift from metal-rich to metal-poor with the increase of distance from the Galactic center. The mean metallicity is about $-1.5\pm 0.2$ dex in intervals $10<r\leqslant 20$ kpc and $-1.3\pm 0.1$ dex in interval $8<r\leqslant 10$ kpc. The mean metallicity smoothly decreases from $-0.6$ to $-0.8$ in interval $2.5<r\leqslant 5$ kpc, while the mean metallicity decreases from $-0.4$ to $-0.6$ in interval $0<r\leqslant 2.5$ kpc. In addition, a fluctuation in the mean metallicity with Galactic longitude can be found and the overall distribution has a maximum at about $l$ $\sim$ 200∘ in interval $4<r\leqslant 8$ kpc. Maybe this feature can be related with the substructure or streams (such as Monoceros stream) which are accreted from nearby galaxies. At the same time, we find the vertical abundance gradients for the thin disk ($0<z<2$ kpc) is d[Fe/H]/dz $\sim-0.21\pm 0.05$ dex kpc-1, and a vertical gradient $-0.16\pm 0.06$ dex kpc-1 at distances $2<z\leqslant 5$ kpc where the thick disk stars are dominated. Here, we consider the thick disk gradient may be the result from the different contributions from three components of the Galaxy at different $z$ distance. The vertical gradient d[Fe/H]/dz $\sim-0.05\pm 0.04$ dex kpc-1 is found in distance $5<z\leqslant 15$ kpc. So, there is little or no gradient in the halo. These results are in agreement with the values in the literature (Yoss et al. 1987; Trefzger et al. 1995; Karaali et al. 2003; AK et al. 2007; Yaz et al. 2010). It is possible that additional observational investigations (some projects aimed at spectroscopic sky surveys such as SEGUE, LAMOST, GAIA) will give more evidence for the metallicity gradient of the Galaxy and therefore provide a powerful clue to the disk and halo formation. ## Acknowledgments We especially thank the anonymous referee for numerous helpful comments and suggestions which have significantly improved this manuscript. This work was supported by the joint fund of Astronomy of the National Nature Science Foundation of China and the Chinese Academy of Science, under Grants 10778720 and 10803007. This work was also supported by the GUCAS president fund. ## References * Ak et al. (2007) AK S., Bilir S., Karaali S., Buser R., 2007, AN, 328, 169 * Ak et al. (2007) AK S., Bilir S., Karaali S., Buser R., Cabrera-Lavers A., 2007, NewA, 12, 605 * Allende et al. (2006) Allende Prieto C., Beers T. C., Wilhelm R., Newberg H. J., Rockosi C. M., Yanny B., Lee Y. S., 2006, ApJ, 636, 804 * Bahcall et al. (1981) Bahcall J. N., Soneira R. M., 1981, ApJS, 47, 357 * Belokurov et al. (2003) Belokurov V. A., Evans N. W. 2003, MNRAS, 341, 569 * Bilir et al. (2008) Bilir S., Cabrera - Lavers A., Karaali S., Ak S., Yaz E., López-Corredoira M., 2008, PASP, 25, 69 * Bullock et al. (2005) Bullock J. S., Johnston K. V., 2005, ApJ, 635, 931 * Du et al. (2003) C.-H. Du, et al. 2003, A&A, 407, 541 * Du et al. (2004) C.-H. Du, et al. 2004, AJ, 128, 2265 * Du et al. (2006) C.-H. Du, et al. 2006, MNRAS, 372, 1304 * Carollo et al. (2007) Carollo D., et al. 2007, Nature., 450, 1020 * Chen et al. (2001) Chen B. et al., 2001, ApJ, 553, 184 * Chen et al. (2003) Chen L., Hou J. L., Wang J. J., 2003, AJ, 125, 1397 * De et al. (2010) De Jong Jelte T. A.; Yanny B., Rix Hans-Walter, Dolphin A. E., Martin N. F., Beers T. C., 2010, ApJ, 714, 663 * Fan et al. (1996) Fan X. H. et al., 1996, AJ, 112, 628 * Freeman et al. (2002) Freeman K., Bland-Hawthorn J., 2002, ARA&A, 40, 487 * Friel (1988) Friel E. D., 1988, AJ, 95, 1727 * Fukugita et al. (1996) Fukugita M., Ichikawa T., Gunn J. E., Doi M., Shimasaku K., Schneider D. P., 1996, AJ, 111, 1748 * Gilmore et al. (1963) Gilmore G., Reid N., 1983, MNRAS, 202, 1025 * Gilmore (1984) Gilmore G., 1984, MNRAS, 207, 223 * Gilmore G & Wyse (1985) Gilmore G., Wyse R. F. G., 1985, AJ, 90, 2015 * Gilmore et al. (1989) Gilmore G., Wyse R. F. G., Kuijken K., 1989, ARA&A, 27, 555 * Gunn et al. (1998) Gunn J. E. et al., 1998, AJ, 116, 3040 * Hogg et al. (2001) Hogg D. W., Finkbeiner D. P., Schlegel D. J., Gunn J. E., 2001, AJ, 122, 2129 * Ivezic Z. et al. (2000) Ivezic Z. et al., 2000, AJ, 120, 963 * Ivezic Z. et al. (2008) Ivezic Z. et al., 2008, ApJ, 684, 287 * Juric et al. (2008) Juric M. et al., 2008, ApJ, 673, 864 * Karaali et al. (2003) Karaali S., AK S. G., Bilir S., Karatas Y., Gilmore G., 2003, MNRAS, 343, 1013 * Karaali et al. (2003) Karaali S., Bilir S., Karatas Y., Ak S. G., 2003, PASA, 20, 165 * Karaali et al. (2005) Karaali S., Bilir S., Tuncel S., 2005, PASA, 22, 24 * Karatas et al. (2009) Karatas Y., Kilic M., Günes O., Limboz F., 2009, PASP, 26, 1 * Klement et al. (2009) Klement R. et al., 2009, ApJ, 698, 865 * Kroupa et al. (1993) Kroupa P., Tout C. A., Gilmore G., 1993, MNRAS, 262, 545 * Lang et al. (1992) Lang K. R., 1992, Astrophysical Data I, Planets and Stars. Spinger-Verlag, Berlin * Belokurov et al. (2003) Belokurov V. A., Evans N. W. 2003, MNRAS, 341, 569 * Lang et al. (1992) Lang K. R., 1992, Astrophysical Data I, Planets and Stars. Spinger-Verlag, Berlin * Lejeune et al. (1997) Lejeune T., Cuisinier F., Buser R., 1997, A&AS, 125, 229 * Lemon et al. (2004) Lemon D. J., Wyse Rosemary F. G., Liske J., Driver S. P., Horne K., 2004, MNRAS, 347, 1043 * López -Corredoira et al. (2002) López -Corredoira M., Cabrera - Lavers A., Garzón F., Hammersley P. L., 2002, A&A, 394, 883 * Lupton et al. (2001) Lupton R. H., Gunn J. E., Ivezic Z., Knapp G. R., Kent S., 2001, ASPC, 238, 269 * Majewski et al. (2003) Majewski S. R., Skrutskie M. F., Weinberg M. D., Ostheimer J. C., 2003, ApJ, 599, 1082 * Majewski et al. (1993) Majewski S. R., 1993, ARA&A, 31, 575 * Newberg et al. (2002) Newberg H. J. et al., 2002, ApJ, 569, 245 * Ojha et al. (1996) Ojha D. K., Bienayme O., Robin A. C., Creze M., Mohan V., 1996, A&A, 311, 456 * Oke et al. (1983) Oke J. B., Gunn J. E., 1983, ApJ, 266, 713 * Ratnatunga et al. (1989) Ratnatunga K. U., Freeman K. C., 1989, ApJ, 339, 126 * Rocha-Pintoet al. (2003) Rocha-Pinto H. J., Majewski S. R., Skrutskie M. F., Crane J. D., 2003, ApJ, 594, 115 * Sandage et al. (1969) Sandage A., 1969, ApJ, 158, 1115 * Siegel et al. (2009) Siegel M. H., Karatas Y., Reid I. N., Yüksel R., I. Neill, 2009, MNRAS, 395, 1569 * Stetson (1987) Stetson P. B., 1987, PASP, 99, 191 * Searle et al. (1978) Searle L., Zinn R., 1978, ApJ, 225, 357 * Trefzger et al. (1995) Trefzger C. F., Pel J. W., Gabi S., 1995, A&A, 304, 381 * Vivas et al. (2001) Vivas A. K. et al., 2001, ApJ, 554, 33 * Xia et al. (2002) Xia L. F. et al., 2002, PASP, 114, 1349 * Yanny et al. (2000) Yanny B. et al., 2000, ApJ, 540, 825 * Yoss et al. (1987) Yoss K. M., Neese C. L., Hartkopf W. I., 1987, AJ, 94, 1600 * Yaz et al. (2010) Yaz E., Karaali S., 2010, NewA, 15, 234 * York et al. (2000) York D. et al., 2000, AJ, 120, 1579 * Zhou et al. (2003) Zhou X. et al., 2003, A&A, 397, 361	arxiv-papers	2011-11-21T02:56:43	2024-09-04T02:49:24.527261	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Xiyan Peng, Cuihua Du, Zhenyu Wu", "submitter": "Cuihua Du", "url": "https://arxiv.org/abs/1111.4719" }
1111.4723	Self-dual interval orders and row-Fishburn matrices Sherry H. F. Yan, Yuexiao Xu Department of Mathematics, Zhejiang Normal University, Jinhua 321004, P.R. China huifangyan@hotmail.com Abstract. Recently, Jelínek derived that the number of self-dual interval orders of reduced size $n$ is twice the number of row-Fishburn matrices of size $n$ by using generating functions. In this paper, we present a bijective proof of this relation by establishing a bijection between two variations of upper-triangular matrices of nonnegative integers. Using the bijection, we provide a combinatorial proof of the refined relations between self-dual Fishburn matrices and row-Fishburn matrices in answer to a problem proposed by Jelínek. Key words: self-dual interval order, self-dual Fishburn matrix, row-Fishburn matrix. AMS Mathematical Subject Classifications: 05A05, 05C30. ## 1 Introduction A poset is said to be an interval order ( also known as $(2+2)$-free poset) if it does not contain an induced subposet that is isomorphic to $2+2$, the union of two disjoint $2$-element chains. Let $P$ be a poset with a strict order relation $\prec$. A strict down-set of an element $x\in P$ is the set $D(x)$ of all the elements of P that are smaller than $y$, i.e., $D(y)=\\{y\in P:y\prec x\\}$. Similarly, the strict up-set of $x$, denoted by $U(x)$, is the set $\\{y\in P:y\succ x\\}$. A poset $P$ is $(2+2)$-free if and only if its sets of strict down-sets, $D(P)=\\{D(x):x\in P\\}$ can be written as $D(P)=\\{D_{1},D_{2},\ldots,D_{m}\\}$ where $\emptyset=D_{1}\subset D_{2}\subset\ldots\subset D_{m}$, see [1, 2]. In such context, we say that $x\in P$ has level $i$ if $D(x)=D_{i}$. An element $x$ is said to be a minimal element if $x$ has level $1$. Following Fishburn [7], we call the number $m$ of distinct strict down-sets the magnitude of P. It turns out that $m$ is also equal to the number of distinct strict up-sets, and we can order the strict up-sets of $P$ into a decreasing chain $D(P)=\\{U_{1},U_{2},\ldots,U_{m}\\}$ where $U_{1}\supset U_{2}\supset\ldots\supset U_{m}=\emptyset$, see [7, 8]. We say that $x$ has up-level $i$ if $U(x)=U_{i}$. An element $x$ is said to be a maximal element if $x$ has up-level $m$. The dual of a poset $P$ is the poset $\overline{P}$ with the same elements as $P$ and an order relation $\overline{\prec}$ defined by $x\overline{\prec}y$ $\Longleftrightarrow$ $y\prec x$. A poset is self-dual if it is isomorphic to its dual. Fishburn [7, 9] did pioneering work on interval orders; for instance, he showed the basic theorem that a poset is an interval order if and only if it is $(2+2)$-free and established a bijection between interval orders and a certain kind of integer matrices, called Fishburn matrices. Recently, Bousquet-Mélou et al. [2] constructed bijections between interval orders and ascent sequences, between ascent sequences and permutations avoiding a certain pattern, between interval orders and regular linearized chord diagrams by Stoimenow [12]. Several other papers have focused on bijections between interval orders and other objects. For instance, Dukes and Parviainen [4] have described a direct bijection between Fishburn matrices and ascent sequences, while the papers of Claesson et al. [3] and Dukes et al. [6] extend the bijection between interval orders and Fishburn matrices to more general combinatorial structures. A Fishburn matrix of size $n$ is an upper-triangular matrix with nonnegative integers which sum to $n$ and each row and each column contains a nonzero entry. Throughout this paper that each matrix has its rows numbered from top to bottom, and columns numbered left-to-right, starting with row and column number one. We let $M_{i,j}$ denote the entry of M in row $i$ and column $j$. The size of a matrix $M$ is the sum of all its entries. Moreover, the dimension of an upper triangular matrix is defined to the number of rows. The dual matrix of $M$, denoted by $\overline{M}$, is obtained from $M$ by transposition along the diagonal running from bottom-left to top-right. More precisely, for $1\leq i,j\leq m$, we have $\overline{M}_{i,j}=M_{m+1-j,m+1-i}$ where $m$ is the dimension of $M$. If a matrix M is equal to $\overline{M}$, we call it self-dual. Fishburn [7, 9] showed that an interval order $P$ of magnitude m corresponds to an $m\times m$ Fishburn matrix $M$ with $M_{i,j}$ being equal to the number of elements of $P$ that have level $i$ and up-level $j$. Jelínek [10] showed that the Fishburn’s bijection turns out to be a bijection between self-dual interval orders of size $n$ and self-dual Fishburn matrices of size $n$. Following the terminologies given in [10], we distinguish three types of cells in a Fishburn matrix $M$ of dimension $k$ : a cell $(i,j)$ is a diagonal cell if $i+j=k+1$, i.e., $(i,j)$ belongs to the north-east diagonal of the matrix. If $i+j<k+1$ (i.e., $(i,j)$ is above and to the left of the diagonal) then $(i,j)$ is a North-West cell, or NW-cell, while if $i+j>k+1$, then $(i,j)$ is an SE-cell. Clearly, NW-cells and diagonal cells together determine a self- dual Fishburn matrix. The reduced size of a self-dual fishburn matrix $M$ is the sum of all diagonal cells and NW-cells. The reduced size of a self-dual interval order $P$ is the reduced size of its corresponding self-dual Fishburn matrix under Fishburn’s bijection. A row-Fishburn matrix of size $n$ is defined to be an upper-triangular matrix with nonnegative integers which sum to $n$ and each row contains a nonzero entry. In a matrix $A$, the sum of a column (resp. row) is defined to the sum of all the entries in this column (resp. row). A column or a row is said to be zero if it contains no nonzero entries. The set of self-dual Fishburn matrices of reduced size $n$ is denoted by $\mathcal{M}(n)$. Denote by $\mathcal{M}(n,k,p)$ be the set of self-dual Fishburn matrices of reduced size $n$ whose first row has sum $k$ and diagonal cells have sum $p$. Let $\mathcal{RM}(n)$ be the set of row-Fishburn matrices of size $n$. The set of row-Fishburn matrices in $\mathcal{RM}(n)$ whose last column has sum $k$ is denote by $\mathcal{RM}(n,k)$. Denote by $\mathcal{RM}(n,k,p)$ be the set of row-Fishburn matrices in $\mathcal{RM}(n,k)$ whose first row has sum $p$. Moreover, the set of self-dual interval orders of reduced size $n$ is denoted by $\mathcal{I}(n)$. Based on the bijection between interval orders and Fishburn matrices, Jelínek [10] presented a new method to derive formulas for the generating functions of interval orders, counted with respect to their size, magnitude, and number of minimal and maximal elements, which generalize previous results on refined enumeration of interval orders obtained by Bousquet-Mélou et al. [2], Kitaev and Remmel [11], and Dukes et al. [5]. Applying the new method, Jelínek [10] obtained formulas for the generating functions of self-dual interval orders with respect to analogous statistics. From the obtained generating functions, relations between self-dual Fishburn matrices and row-Fishburn matrices were derived, that is, $\|\mathcal{M}(n,k,0)\|=\|\mathcal{RM}(n,k)\|,$ (1.1) and for $p\geq 1$ $\|\mathcal{M}(n,k,p)\|=\|\mathcal{RM}(n,k,p)\|.$ (1.2) Combining the bijection between self-dual interval orders and self-dual Fishburn martices, formulas (1.1) and (1.2), Jelínek derived that for $n\geq 1$, $\|\mathcal{I}(n)\|=\|\mathcal{M}(n)\|=2\|\mathcal{RM}(n)\|,$ (1.3) and asked for bijective proofs of (1.1) and (1.2). The main objective of this paper is to present bijective proofs of these formulas by establishing a one- to-one correspondence between two variations of upper-triangular matrices of nonnegative integers. Let $\mathcal{M}(n,k)$ be the set of self-dual Fishburn matrices of reduced size $n$ whose first row has sum $k$. Denote by $\mathcal{EM}(n,k)$ (resp. $\mathcal{OM}(n,k)$ ) be the set of self-dual Fishburn matrices in $\mathcal{M}(n,k)$ whose dimension are even (resp. odd). Using the bijection between two variations of upper-triangular matrices of nonnegative integers, we derive that $\|\mathcal{EM}(n,k)\|=\|\mathcal{OM}(n,k)\|=\|\mathcal{RM}(n,k)\|.$ (1.4) ## 2 The bijective proofs Recall that a self-dual Fishburn matrix is determined by its NW-cells and diagonal cells. Given a self-dual Fishburn matrix $M$, the reduced matrix of $M$, denoted by $R(M)$, is a matrix obtained from $M$ by filling all the SE- cells with zeros. An upper-triangular matrix is said to a super triangular matrix if all its SE-cells are zero. ###### Lemma 2.1 Let $M^{\prime}$ be a super triangular matrix of dimension $m$. Then $M^{\prime}$ is a reduced matrix of a self-dual Fishburn matrix if and only if it satisfies the following two conditions: * (i) for $1\leq i\leq\lceil{m\over 2}\rceil$, each column $i$ contains a nonzero entry; * (ii) for $1\leq i\leq\lceil{m\over 2}\rceil$, either row $i$ or column $m+1-i$ contains a nonzero entry. Proof. Let $M$ be a self-dual Fishburn matrix with $R(M)=M^{\prime}$. Clearly, $M^{\prime}$ is a super triangular matrix. Since the first $\lceil{m\over 2}\rceil$ columns of $M^{\prime}$ are the same as those in $M$, the condition $(i)$ follows immediately. It remains to show that $M^{\prime}$ satisfies condition $(ii)$. Since $M$ is self-dual Fishburn matrix, for all $1\leq i\leq m$, row $i$ must contains a nonzero entry, that is, $\sum_{j=1}^{m}M_{i,j}=\sum_{j=1}^{m-i}M_{i,j}+\sum_{j=m+1-i}^{m}M_{i,j}=\sum_{j=1}^{m-i}M_{i,j}+\sum_{j=1}^{i}M_{j,m+1-i}>0.$ Hence, for $1\leq i\leq\lceil{m\over 2}\rceil$, either row $i$ or column $m+1-i$ of $R(M)$ contains a nonzero entry. Therefore, the condition $(ii)$ holds for $R(M)$. Conversely, given a super triangular matrix $M^{\prime}$ satisfying conditions $(i)$ and $(ii)$, We can recover a self-dual matrix $M$ from $M^{\prime}$ by filling the SE-cell $(m+1-j,m+1-i)$ with $M^{\prime}_{i,j}$. If $1\leq i\leq\lceil{m\over 2}\rceil$, the sum of row $i$ of $M$ is given by $\sum_{j=i}^{m}M_{i,j}=\sum_{j=i}^{m-i}M_{i,j}+\sum_{j=m+1-i}^{m}M_{i,j}=\sum_{j=1}^{m-i}M_{i,j}+\sum_{j=1}^{i}M_{j,m+1-i}=\sum_{j=1}^{m-i}M^{\prime}_{i,j}+\sum_{j=1}^{i}M^{\prime}_{j,m+1-i}.$ By the condition $(ii)$, we have $\sum_{j=i}^{m}M_{i,j}=\sum_{j=1}^{m-i}M^{\prime}_{i,j}+\sum_{j=1}^{i}M^{\prime}_{j,m+1-i}>0$, which implies that row $i$ contains a nonzero entry. If $\lceil{m\over 2}\rceil+1\leq i\leq m$, the sum of row $i$ of $M$ is given by $\sum_{j=i}^{m}M_{i,j}=\sum_{j=i}^{m}M^{\prime}_{m+1-j,m+1-i}=\sum_{j=1}^{m+1-i}M^{\prime}_{j,m+1-i},$ which implies that the sum of row $i$ of $M$ is the same as that of column $m+1-i$ of $M^{\prime}$. By condition $(i)$, row $i$ contains a nonzero entry. Hence $M$ is a self-dual Fishburn matrix with $R(M)=M^{\prime}$. This completes the proof. Denote by $\mathcal{SM}_{k}(n)$ the set of all super triangular matrices of size $n$ and dimension $2k+1$ having the following two properties: 1. $(a)$ for $1\leq i\leq k$, each column $i$ contains a nonzero entry; 2. $(b)$ for $1\leq i\leq k$, either row $k+1-i$ or column $k+1+i$ contains a nonzero entry. Let $\mathcal{SM}(n)=\bigcup_{k\geq 0}\mathcal{SM}_{k}(n)$. Now we proceed to present a map $\alpha$ from $\mathcal{M}(n)$ to $\mathcal{SM}(n)$. Given a nonempty self-dual matrix $M$ of dimension $m$, let $\alpha(M)$ be the matrix obtained from $M$ by the following procedure. * • If $m=2k+1$ for some integer $k\geq 0$, then let $\alpha(M)$ be the matrix obtained from the reduced matrix $R(M)$ of $M$ by interchanging the cell $(i,k+1)$ and the diagonal cell $(i,m+1-i)$ for $1\leq i\leq k$. * • If $m=2k$ for some integer $k\geq 1$, then let $A$ be the matrix obtained from $R(M)$ by adding one zero row and one zero column immediately after column $k$ and row $k$. Define $\alpha(M)$ to be the matrix obtained from $A$ by interchanging the cell $(i,k+1)$ and the diagonal cell $(i,m+1-i)$ of the resulting matrix $A$. Obviously, $\alpha(M)$ is a super triangular matrix of dimension $2k+1$ and size $n$. It easy to check that the map $\alpha$ preserves the first $k$ columns and the total sum of row $i$ and column $m+1-i$ of the reduced matrix $R(M)$. By Lemma 2.1, the matrix $\alpha(M)$ has properties $(a)$ and $(b)$. Hence $\alpha(M)$ is a super triangular matrix in $\mathcal{SM}(n)$. Conversely, given a super triangular matrix $M^{\prime}$ in $\mathcal{SM}(n)$ of dimension $2k+1$, we can recover a matrix $M\in\mathcal{M}(n)$ with $\alpha(M)=M^{\prime}$. First we interchange the cell $(i,k+1)$ with the diagonal cell $(i,m-i)$ for $1\leq i\leq k$. Then we obtain a matrix $A$ by deleting column $k+1$ and row $k+1$ if they are zero. It is easy to check that properties $(a)$ and $(b)$ ensure that the obtained matrix $A$ is the reduced matrix of a self-dual Fishburn matrix. Let $M$ be a self dual Fishburn matrix with $R(M)=A$. Hence $\alpha$ is a bijection between $\mathcal{M}(n)$ and $\mathcal{SM}(n)$. Let $M$ be a super triangular matrix of dimension $2k+1$, then column $k+1$ is called a center column. From the construction of the bijection $\alpha$, we see that the map $\alpha$ transforms the sum of the diagonal cells of a self- dual matrix to the sum of the center column of a super triangular matrix. Hence, we have the following result. ###### Theorem 2.2 The map $\alpha$ is a bijection between $\mathcal{M}(n)$ and $\mathcal{SM}(n)$. Moreover, the bijection $\alpha$ preserves the sum of the first row, and transforms the sum of the diagonal cells of a self-dual matrix to the sum of the center column of a super triangular matrix. ###### Example 2.3 Consider a matrix $A\in\mathcal{M}(5)$, $A=\begin{bmatrix}1&0&1&0&0\\\ 0&1&1&1&0\\\ 0&0&0&1&1\\\ 0&0&0&1&0\\\ 0&0&0&0&1\\\ \end{bmatrix}.$ The reduced matrix of $A$ is given by $R(A)=\begin{bmatrix}1&0&1&0&0\\\ 0&1&1&1&0\\\ 0&0&0&0&0\\\ 0&0&0&0&0\\\ 0&0&0&0&0\\\ \end{bmatrix},$ and we have $\alpha(A)=\begin{bmatrix}1&0&0&0&1\\\ 0&1&1&1&0\\\ 0&0&0&0&0\\\ 0&0&0&0&0\\\ 0&0&0&0&0\\\ \end{bmatrix}.$ Let $\mathcal{B}(n)$ the set of upper-triangular matrices of size $n$ in which each row contains a nonzero entry except for the first row. Given a nonempty matrix $A\in\mathcal{RM}(n)$, we can get two distinct matrices in $\mathcal{B}(n)$ from $A$ by either doing nothing or adding a zero row and a zero column before the first row and the first column. Thus for $n\geq 1$ we have the following relation $\|\mathcal{B}(n)\|=2\|\mathcal{RM}(n)\|.$ (2.1) Now we proceed to construct a bijection between the set $\mathcal{SM}(n)$ and the set $\mathcal{B}(n)$. Before constructing the bijection, we need some definitions. In a matrix $A$ with $m$ rows, the operation of adding column $i$ to column $j$ is defined by increasing $A_{k,j}$ by $A_{k,i}$ for each $k=1,2,\ldots,m$. Let $\mathcal{B}(n,k,p)$ be the set of matrices in $\mathcal{B}(n)$ whose whose first row has sum $p$ and last column has sum $k$. Similarly, let $\mathcal{SM}(n,k,p)$ be the set of matrices in $\mathcal{SM}(n)$ whose first row has sum $k$ and center column has sum $p$. ###### Theorem 2.4 There is a bijection $\beta$ between $\mathcal{SM}(n)$ and $\mathcal{B}(n)$. Moreover, the map $\beta$ is essentially a bijection between $\mathcal{SM}(n,k,p)$ and $\mathcal{B}(n,k,p)$. Proof. Given a nonempty triangular matrix $A\in\mathcal{SM}(n)$ of dimension $2k+1$, we recursively construct a sequence of super triangular matrices $A^{(0)},A^{(1)},\ldots,A^{(l)}$. Let $A^{(0)}=A$ and assume that we have obtained the matrix $A^{(j)}$. Let $A^{(j)}$ be a super triangular matrix of dimension $2r+1$ for some integer $r$. For $1\leq i\leq r$, if each column $r+1+i$ is zero, then let $A^{(l)}=A^{(j)}$. Otherwise, we proceed to generate the matrix $A^{(j+1)}$ by the following insertion algorithm. * • Find the largest value $i$ such that column $r+1+i$ contains a nonzero entry. Then fill the entries of column $r+1+i$ with zeros. * • Insert one column immediately after column $r+1-i$, one zero row immediately after row $r+1-i$, one zero column immediately before column $r+1+i$ and one zero row immediately before row $r+1+i$. Let the entry in row $j$ of the new inserted column after column $r+1-i$ be filled with the entry in row $j$ of column $r+1+i$ of $A^{(j)}$ for $1\leq j\leq 2r+1$. Suppose that $A^{(l)}$ is of dimension $2q+1$. Then the last $q$ rows and $q$ columns of $A^{(l)}$ are zero rows and columns. Let $B$ be an upper-triangular matrix obtained from $A^{(l)}$ by deleting the last $q$ columns and $q$ rows. From the above insertion procedure to generate $A^{(j+1)}$ form $A^{(j)}$ , we see that the inserted column after column $r+1-i$ contains a nonzero entry. This ensures that each matrix $A^{(j)}$ has property $(a)$ with $0\leq j\leq l$. Hence each column $i$ of $A^{(l)}$ contains a nonzero entry with $1\leq i\leq q$. Hence, $B$ is an upper-triangular matrix in which each column contains a nonzero entry except for the last column. Moreover, the insertion algorithm preserves the sum of each nonzero row of $A$, which implies that $B$ is of size $n$. Let $\beta(A)$ be the dual matrix of $B$. Hence we have $\beta(A)\in\mathcal{B}(n)$. Conversely, we can construct a matrix $A=\beta^{\prime}(A^{\prime})$ in $\mathcal{SM}(n)$ from a matrix $A^{\prime}$ of dimension $k+1$ in $\mathcal{B}(n)$. Let $B$ be the dual matrix of $A^{\prime}$. Define $M$ to be a matrix of dimension $2k+1$ obtained from $B$ by adding $k$ consecutive zero rows and $k$ consecutive zero columns immediately after column $k+1$ and row $k+1$. Clearly, the obtained matrix is a super triangular matrix having property $(a)$. If for all $1\leq i\leq k$, either row $k+1-i$ or column $k+1+i$ contains a nonzero entry, then we do nothing for $M$ and let $A=M$. Otherwise, we can construct a new super triangular matrix $A$ by the following removal algorithm. * • Find the least value $i$ such that neither row $k+1-i$ nor column $k+1+i$ contains a nonzero entry. Then we obtain a super triangular matrix by adding column $k+1-i$ to column $k+2+i$ and removing columns $k+1+i$, $k+1-i$ and rows $k+1-i$, $k+1+i$. * • Repeat the above procedure for the resulting matrix until the obtained matrix has property $(b)$. Obviously, the obtained matrix $A$ is a super triangular matrix having properties $(a)$ and $(b)$. Since the algorithm preserves the sums of entries in each non-zero row of $B$, the matrix $A$ is of size $n$ and the sum of the first row of $A$ is the same as that of $B$. The property $(b)$ ensures that the inserted columns in the insertion algorithm are the removed columns in the removal algorithm. Thus the map $\beta^{\prime}$ is the inverse of the map $\beta$. From the construction of the removal algorithm, the sum of the center column of $A$ is equal to the sum of the last column of $B$ as well as the the sum of the first row of $A^{\prime}$. This completes the proof. ###### Example 2.5 Consider a matrix $A\in\mathcal{SM}(6)$, $A=\begin{bmatrix}1&0&0&1&1\\\ 0&1&1&1&0\\\ 0&0&0&0&0\\\ 0&0&0&0&0\\\ 0&0&0&0&0\\\ \end{bmatrix}.$ Let $A^{(0)}=A$. By applying the insertion algorithm, we get $A^{(1)}=\begin{bmatrix}1&{\textbf{1}}&0&0&1&\textbf{0}&0\\\ \textbf{0}&\textbf{0}&\textbf{0}&\textbf{0}&\textbf{0}&\textbf{0}&\textbf{0}\\\ 0&\textbf{0}&1&1&1&\textbf{0}&0\\\ 0&\textbf{0}&0&0&0&\textbf{0}&0\\\ 0&\textbf{0}&0&0&0&\textbf{0}&0\\\ \textbf{0}&\textbf{0}&\textbf{0}&\textbf{0}&\textbf{0}&\textbf{0}&\textbf{0}\\\ 0&\textbf{0}&0&0&0&\textbf{0}&0\\\ \end{bmatrix},$ $A^{(2)}=\begin{bmatrix}1&1&0&\textbf{1}&0&\textbf{0}&0&0&0\\\ 0&0&0&\textbf{0}&0&\textbf{0}&0&0&0\\\ 0&0&1&\textbf{1}&1&\textbf{0}&0&0&0\\\ \bf{0}&\textbf{0}&\textbf{0}&\textbf{0}&\textbf{0}&\textbf{0}&\textbf{0}&\textbf{0}&\textbf{0}\\\ 0&0&0&\textbf{0}&0&\textbf{0}&0&0&0\\\ \textbf{0}&\textbf{0}&\textbf{0 }&\textbf{0}&\textbf{0}&\textbf{0}&\textbf{0}&\textbf{0}&\textbf{0}\\\ 0&0&0&\textbf{0}&0&\textbf{0}&0&0&0\\\ 0&0&0&\textbf{0}&0&\textbf{0}&0&0&0\\\ 0&0&0&\textbf{0}&0&\textbf{0 }&0&0&0\\\ \end{bmatrix},$ where the inserted rows and columns are illustrated in bold at each step of the insertion algorithm. Removing the last $4$ zero rows and $4$ zero columns, we get $B=\begin{bmatrix}1&1&0&1&0\\\ 0&0&0&0&0\\\ 0&0&1&1&1\\\ 0&0&0&0&0\\\ 0&0&0&0&0\\\ \end{bmatrix}.$ Finally, we obtain $A^{\prime}=\beta(A)=\begin{bmatrix}0&0&1&0&0\\\ 0&0&1&0&1\\\ 0&0&1&0&0\\\ 0&0&0&0&1\\\ 0&0&0&0&1\\\ \end{bmatrix}$ Conversely, given $A^{\prime}\in\mathcal{B}(6)$, by applying removal algorithm, we can get $A\in\mathcal{SM}(6)$, where the removed rows and columns are illustrated in bold at each step of the removal algorithm. Combining the bijection between self-dual interval orders and self-dual Fishburn matrices and Theorems 2.2 and 2.4, we get a bijective proof of $(\ref{eq3})$. From Theorems 2.2 and 2.4, we have $\|\mathcal{M}(n,k,p)\|=\|\mathcal{SM}(n,k,p)\|=\|\mathcal{B}(n,k,p)\|.$ Given a matrix $M\in\mathcal{B}(n,k,0)$, we can get a matrix $A\in\mathcal{RM}(n,k)$ by deleting the first row and the first column. Conversely, given a matrix $A^{\prime}\in\mathcal{RM}(n,k)$, we can obtain a matrix $M^{\prime}\in\mathcal{B}(n,k,0)$ by inserting a zero row and a zero column before the first row and the first column. This yields that $\|\mathcal{M}(n,k,0)\|=\|\mathcal{B}(n,k,0)\|=\|\mathcal{RM}(n,k)\|.$ (2.2) If $p>0$, then $\mathcal{B}(n,k,p)$ is the same as $\mathcal{RM}(n,k,p)$. Hence, if $p>0$ then we have $\|\mathcal{M}(n,k,p)\|=\|\mathcal{B}(n,k,p)\|=\|\mathcal{RM}(n,k,p)\|.$ (2.3) Therefore, we get combinatorial proofs of (1.1) and (1.2), in answer to the problem posed by Jelínek [10]. Now we proceed to prove (1.4). Given a matrix $A\in\mathcal{EM}(n,k)$ of dimension $2m$ for some integers $m\geq 1$, let $R(A)$ be its reduced matrix. We obtain a super triangular matrix $A^{\prime}$ from $A$ by inserting a zero column and a zero row immediately after column $m$ and row $m$. By Lemma 2.1, we have $A^{\prime}\in\mathcal{SM}(n,k,0)$. Conversely, given a matrix $A^{\prime}\in\mathcal{SM}(n,k,0)$ of dimension $2m+1$ for some integer $m\geq 1$, we can recover a self-dual matrix $A\in\mathcal{EM}(n,k)$ as follows. First, we get a super triangular matrix $B$ from $A^{\prime}$ by deleting column $m+1$ and row $m+1$. Let $A$ be a matrix with $B=R(A)$. Obviously, we have the matrix $A\in\mathcal{EM}(n,k)$. Hence, we get $\|\mathcal{EM}(n,k)\|=\|\mathcal{SM}(n,k,0)\|.$ By (2.2), we deduce that $\|\mathcal{EM}(n,k)\|=\|\mathcal{SM}(n,k,0)\|=\|\mathcal{RM}(n,k)\|.$ (2.4) From (2.2) and (2.3), we have $\begin{array}[]{lll}\|\mathcal{M}(n,k)\|&=&\|\mathcal{M}(n,k,0)\|+\sum_{p\geq 1}\|\mathcal{M}(n,k,p)\|\\\ &=&\|\mathcal{RM}(n,k)\|+\sum_{p\geq 1}\|\mathcal{RM}(n,k,p)\|\\\ &=&2\|\mathcal{RM}(n,k)\|.\end{array}$ Meanwhile, we have $\|\mathcal{M}(n,k)\|=\|\mathcal{EM}(n,k)\|+\|\mathcal{OM}(n,k)\|$. Hence, (1.4) follows from (2.4). Acknowledgments. This work was supported by the National Natural Science Foundation of China (10901141). ## References * [1] K.P. Bogart, An obvious proof of Fishburn’s interval order theorem, Discrete Math. 118 (1993), 239–242. * [2] M. Bousquet-Mélou, A. Claesson, M. Dukes, S. Kitaev, $(2+2)$-free posets, ascent sequences and pattern avoiding permutations, J. Combin. Theory Ser. A 117 (2010), 884–909. * [3] A. Claesson, M. Dukes, and M. Kubitzke, Partition and composition matrices, J. Combin. Theory, Ser. A, 118 (2011), 1624–1637. * [4] M. Dukes, R. Parviainen, Ascent sequences and upper triangular matrices containing non-negative integers, Electronic J. combin. 17 (2010), R53. * [5] M. Dukes, S. Kitaev, J. Remmel, and E. Steingrímsson, Enumerating (2+2)-free posets by indistinguishable elements, arXiv:1006.2696, 2010\. * [6] M. Dukes, V. Jelínek, and M. Kubitzke, Composition matrices, (2+2)-free posets and their specializations, Electronic J. Combin., 18 (2011), P44. * [7] P. C. Fishburn, Interval lengths for interval orders: A minimization problem, Discrete Mathematics, 47 (1983), 63–82. * [8] P. C. Fishburn, Interval graphs and interval orders, Discrete Mathematics, 55 (1985), 135–149. * [9] P. C. Fishburn, Interval orders and interval graphs: A study of partially ordered sets, John Wiley & Sons, 1985. * [10] V. Jelínek, Counting self-dual interval orders, arXiv:1106.2261, 2011. * [11] S. Kitaev, J. Remmel, Enumerating $(2+2)$-free posets by the number of minimal elements and other statistics, Disctere Appl. Math., 159 (2011), 2098–2108. * [12] A. Stoimenow, Enumumeration of chord diagrams and an upper bound for Vassiliev invariants, J. Knot Theory Ramifications 7 (1998), 93–114. * [13] D. Zagier, Vassiliev invariants and a stange identity related to the Dedeking eta-function, Topology 40 (2001), 945–960.	arxiv-papers	2011-11-21T03:33:39	2024-09-04T02:49:24.535265	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Sherry H. F. Yan, Yuexiao Xu", "submitter": "Sherry H.F. Yan", "url": "https://arxiv.org/abs/1111.4723" }
1111.4831	# Analytical calculation of optimal POVM for unambiguous discrimination of quantum states using KKT method N. Karimi a aDepartment of Physics, Azarbaijan University of Tarbiat Moallem, 53714-161 Tabriz, Iran. E-mail:${na}_{-}karimi@yahoo.com$ ###### Abstract In the present paper, an exact analytic solution for the optimal unambiguous state discrimination(OPUSD) problem involving an arbitrary number of pure linearly independent quantum states with real and complex inner product is presented. Using semidefinite programming and Karush-Kuhn-Tucker convex optimization method, we derive an analytical formula which shows the relation between optimal solution of unambiguous state discrimination problem and an arbitrary number of pure linearly independent quantum states. Keywords: Unambiguous discrimination, Semidefinite programming, linearly independent States. PACs Index: 03.67.Hk, 03.65.Ta ## 1 Introduction Many applications in quantum communication and quantum cryptography are based on transmitting quantum systems that, with given prior probabilities, are prepared in one from a set of known mutually nonorthogonal states[1]. A fundamental aspect of quantum information theory is that nonorthogonal quantum states cannot be perfectly distinguished. Therefore, a central problem in quantum mechanics is to design measurements optimized to distinguish between a collection of nonorthogonal quantum states. The topic of quantum state discrimination was firmly established in the 1970s by the pioneering work of Helstrom [2], who considered a minimum error discrimination of two known quantum states. In this case, the state identification is probabilistic. Another possible discrimination strategy is the so-called unambiguous state discrimination (USD) where the states are successfully identified with nonunit probability, but without error. USD was originally formulated and analyzed by Ivanovic, Dieks, and Peres [3, 4, 5] in 1987. The solution for unambiguous discrimination of two known pure states appearing with arbitrary prior probabilities was obtained by Jaeger and Shimony [6]. Although the two-state problem is well developed, the problem of unambiguous discrimination between multiple quantum states has received considerably less attention. The problem of discrimination among three nonorthogonal states was first considered by Peres and Terno [5]. They developed a geometric approach and applied it numerically on several examples. A different method was considered by Duan and Guo [7] and Sun et al.[8] . Chefles [9] showed that a necessary and sufficient condition for the existence of unambiguous measurements for distinguishing between N quantum states is that the states are linearly independent. He also proposed a simple suboptimal measurement for unambiguous discrimination for which the probability of an inconclusive result is the same regardless of the state of the system. Equivalently, the measurement yields an equal probability of correct detection of each one of the ensemble states. Over the past years, semidefinite programming (SDP) has been recognized as a valuable numerical tool for control system analysis and design. In SDP, one minimizes a linear function subject to the constraint that an affine combination of symmetric matrices is positive semidefinite. SDP has been studied (under various names) as far back as the 1940s. Subsequent research in semidefinite programming during the 1990s was driven by applications in combinatorial optimization [10], communications and signal processing [11, 12, 13] and other areas of engineering [14]. Although semidefinite programming is designed to be applied in numerical methods, it can be used for analytic computations, too. In the context of quantum computation and quantum information, Barnum, Saks, and Szegedy have reformulated quantum query complexity in terms of a semidefinite program [15]. The problem of finding the optimal measurement to distinguish between a set of quantum states was first formulated as a semidefinite program in 1974 by Holevo, who gave optimality conditions equivalent to the complementary slackness conditions[2]. Recently, Eldar, Megretski, and Verghese showed that the optimal measurements can be found efficiently by solving the dual followed by the use of linear programming[16]. Also in Ref. [17], SDP has been used to show that the standard algorithm implements the optimal set of measurements. All of the above mentioned applications indicate that the method of SDP is very useful. The reason why the area has shown relatively slow progress until recently within the rapidly evolving field of quantum information is that it poses quite formidable mathematical challenges. Except for a handful of very special cases, no general exact solution has been available involving more than two arbitrary states and mostly numerical algorithms are proposed for finding optimal measurements for quantum-state discrimination, where the theory of the semidefinite programming provides a simple check of the optimality of the numerically obtained results. In Ref. [18] we obtained the feasible region in terms of the inner product of the reciprocal states and the feasible region in terms of the inner product of the states which enables us to solve the problem without using reciprocal states. Moreover, for the real inner product of states, we obtained an exact analytic solution for OPUSD problem involving an arbitrary number of pure linearly independent quantum states by using KKT convex optimization method. In this paper, an exact analytic solution for the optimal unambiguous state discrimination(OPUSD) problem involving an arbitrary number of pure linearly independent quantum states with real and complex inner product of states,using the Karush- Kuhn-Tucker convex optimization method, is given. The organization of the paper is as follows. First, the definition of the unambiguous quantum state discrimination is given. Then, Semidefinite programming, Karush-Kuhn-Tucker (KKT) theorem and SDP formulation of unambiguous discrimination is studied. Finally for the real and complex inner product of reciprocal states, an exact analytic solution for OPUSD problem involving an arbitrary number of pure linearly independent quantum states is presented by using KKT convex optimization method. The paper ends with a brief conclusion. ## 2 Unambiguous quantum state discrimination In quantum theory, measurements are represented by positive operator valued measures (POVMs). A measurement is described by a collection ${M_{k}}$ of measurement operators. These operators are acting on the state space of the system being measured. The index k refers to the measurement outcomes that may occur in the experiment. In quantum information theory the measurement operators ${M_{k}}$ are often called Kraus operators [19]. If we define the operator $\Pi_{k}=M^{\dagger}_{k}M_{k},$ (2.1) the probability of obtaining the outcome $k$ for a given state $\rho_{i}$ is given by $p(k\|i)=Tr(\Pi_{k}\rho_{i})$. Thus, the set of operators $\Pi_{k}$ is sufficient to determine the measurement statistics. ### 2.1 Definition of the POVM A set of operators $\\{\Pi_{k}\\}$ is named a positive operator valued measure if and only if the following two conditions are met: $(1)$ each operator $\Pi_{k}$ is positive positive $\Leftrightarrow\langle\psi\|\Pi_{k}\|\psi\rangle\geq 0$, $\forall\|\psi\rangle$ and $(2)$ the completeness relation is satisfied, i.e., $\sum_{k}\Pi_{k}=1.$ (2.2) The elements of $\\{\Pi_{k}\\}$ are called effects or POVM elements. On its own, a given POVM $\\{\Pi_{k}\\}$ is enough to give complete knowledge about the probabilities of all possible outcomes; measurement statistics is the only item of interest. Consider a set of known states $\rho_{i},i=1,...,N$ with their prior probabilities $\eta_{i}$. We are looking for a measurement that either identifies a state uniquely (conclusive result) or fails to identify it (inconclusive result). The goal is to minimize the probability of inconclusive results. The measurements involved are typically generalized measurements. A measurement described by a POVM $\\{\Pi_{k}\\}_{k=1}^{N}$ is called an unambiguous state discrimination measurement (USDM) on the set of states $\\{\rho_{i}\\}_{i=1}^{N}$ if and only if the following conditions are satisfied. (1) The POVM contains the elements $\\{\Pi_{k}\\}_{k=0}^{N}$ where $N$ is the number of different signals in the set of states. The element $\Pi_{0}$ is related to an inconclusive result, while the other elements correspond to an identification of one of the states $\rho_{i},i=1,...,N$. (2) No states are wrongly identified, that is, $Tr(\rho_{i}\Pi_{k})=0\ \forall i\neq k,k=1,...,N$. Each USD measurement gives rise to a failure probability, that is, the rate of inconclusive result. This can be calculated as $Q=\sum_{i}\eta_{i}Tr(\eta_{i}\Pi_{0}).$ (2.3) The success probability can be calculated as $P=1-Q=\sum_{i}\eta_{i}Tr(\eta_{i}\Pi_{i}).$ (2.4) A measurement described by a POVM $\\{\Pi_{k}^{opt}\\}$ is called an optimal unambiguous state discrimination measurement (OUSDM) on a set of states $\\{\rho_{i}\\}$ with the corresponding prior probabilities $\\{\eta_{i}\\}$ if and only if the following conditions are satisfied. (1) The POVM $\\{\Pi_{k}^{opt}\\}$ is a USD measurement on $\\{\rho_{i}\\}$. (2) The probability of inconclusive result is minimal, that is, $Q(\\{\Pi_{k}^{opt}\\})=min\ Q(\\{\Pi_{k}\\})$, where the minimum is taken over all USDM. Unambiguous state discrimination is an error-free discrimination. This implies a strong constraint on the measurement. Suppose that a quantum system is prepared in a pure quantum state drawn from a collection of given states $\\{\|\psi_{i}\rangle\\},1\leq i\leq N$ in d-dimensional complex Hilbert space $\mathcal{H}$ with $d\geq N$. These states span a subspace $\mathcal{U}$ of $\mathcal{H}$. In order to detect the state of the system, a measurement is constructed comprising $N+1$ measurement operators $\\{\Pi_{i},0\leq i\leq N\\}$. Given that the state of the system is $\|\psi_{i}\rangle$, the probability of obtaining outcome k is $\langle\psi_{i}\|\Pi_{k}\|\psi_{i}\rangle$. Therefore, in order to ensure that each state is either correctly detected or an inconclusive result is obtained, we must have $\langle\psi_{i}\|\Pi_{k}\|\psi_{i}\rangle=p_{i}\delta_{ij},\ 1\leq i,k\leq N$ (2.5) for some $0\leq p_{i}\leq 1$. Since $\Pi_{0}=I_{d}-\sum_{i=1}^{N}\Pi_{i}$, we have $\langle\psi_{i}\|\Pi_{0}\|\psi_{i}\rangle=1-p_{i}$. So a system with given state $\|\psi_{i}\rangle$ , the state of the system is correctly detected with probability $p_{i}$ and an inconclusive result is obtained with probability $1-p_{i}$. It was shown in Ref. [20] that Eq.(2.5) is satisfied if and only if the vectors $\|\psi_{i}\rangle$ are linearly independent, or equivalently, $dim\ \mathcal{U}=N$. Therefore, we will take this assumption throughout the paper. In this case, we may choose [21] $\Pi_{i}=p_{i}\|\widetilde{\psi_{i}}\rangle\langle\widetilde{\psi_{i}}\|,\ 1\leq i\leq N,$ (2.6) where the vectors $\|\widetilde{\psi_{i}}\rangle\in\mathcal{U}$ are the reciprocal states associated with the states $\|\psi_{i}\rangle$ , i.e., there are unique vectors in $\mathcal{U}$ such that $\langle\widetilde{\psi}_{k}\|\psi_{i}\rangle=\delta_{ij},\ 1\leq i,k\leq N.$ (2.7) With $\Phi$ and $\widetilde{\Phi}$ we denote the matrices such that their columns are $\|\psi_{i}\rangle$ and $\|\widetilde{\psi_{i}}\rangle$, respectively. Then, one can show that $\widetilde{\Phi}$ is $\widetilde{\Phi}=\Phi(\Phi^{\ast}\Phi)^{-1}.$ (2.8) Since the vectors $\|\psi_{i}\rangle,i=1,...,N$ are linearly independent, $\Phi^{\ast}\Phi$ is always invertible. Alternatively, $\widetilde{\Phi}=(\Phi\Phi^{\ast})^{{\ddagger}}\Phi,$ (2.9) so that $\|\widetilde{\psi_{i}}\rangle=(\Phi\Phi^{\ast})^{{\ddagger}}\|\psi_{i}\rangle,$ (2.10) where $(...)^{{\ddagger}}$ denotes the Moore-Penrose pseudoinverse [22, 23]. The inverse is taken on the subspace spanned by the columns of the matrix. If the state $\|\psi_{i}\rangle$ is prepared with prior probability $\eta_{i}$, then the total probability of correctly detecting the state is $P=\sum_{i=1}^{N}\eta_{i}\langle\psi_{i}\|\Pi_{i}\|\psi_{i}\rangle=\sum_{i=1}^{N}\eta_{i}p_{i}$ (2.11) and the probability of the inconclusive result is given by $Q=1-P=\sum_{i=1}^{N}\eta_{i}\langle\psi_{i}\|\Pi_{0}\|\psi_{i}\rangle=1-\sum_{i=1}^{N}\eta_{i}p_{i}$ (2.12) In general, an optimal measurement for a given strategy depends on the quantum states and the prior probabilities of their appearance. In the unambiguous discrimination for a given strategy and a given ensemble of states, the goal is to find a measurement which minimizes the inconclusive result. In fact, it is known that USD (of both pure and mixed states) is a convex optimization problem. Mathematically, this means that the quantity which is to be optimized as well as the constraints on the unknowns, are convex functions. Practically, this implies that the optimal solution can be computed in an extremely efficient way. This is therefore a very useful tool. Nevertheless, our aim is to understand the structure of USD in order to relate it with neat and relevant quantities and to find feasible region for numerical and analytic solutions. ## 3 Semidefinite programming A SDP problem requires minimizing a linear function subject to a linear matrix inequality (LMI) constraint $\mathbf{minimize}\ p=c^{T}x,such\ that\ F(x)\geq 0,$ (3.13) where $c^{T}$ is a given vector, $x=(x_{1},...,x_{n})$, and $F(x)=F_{0}+\sum_{i}x_{i}F_{i}$, for some fixed Hermitian matrices $F_{i}.$ The inequality sign in $F(x)\geq 0$ means that $F(x)$ is positive semidefinite. This problem is called the primal problem. Vectors $x$ whose components are the variables of the problem and satisfy the constraint $F(x)\geq 0$ are called primal feasible points, and if they satisfy $F(x)>$ they are called strictly feasible points. The minimal objective value $c^{T}x$ is by convention denoted by $P^{\ast}$ and is called the primal optimal value. Due to the convexity of set of feasible points, SDP has a nice duality structure with the associated dual program being $\mathbf{maximize}\ -Tr[F_{0}Z],\ Z\geq,\ Tr[F_{i}Z]=c_{i}.$ (3.14) Here the variable is the real symmetric (or Hermitian) matrix $Z$, and the data $c$, $F_{i}$ are the same as in the primal problem. Correspondingly, matrix $Z$ satisfying the constraints are called dual feasible (or strictly dual feasible if $Z>$). The maximal objective value of $-Tr[F_{0}Z]$, i.e., the dual optimal value, is denoted by $d^{\ast}$. The objective value of a primal (dual) feasible point is an upper (lower) bound on $P^{\ast}(d^{\ast})$. The main reason why one is interested in the dual problem is that one can prove that $d^{\ast}\leq P^{\ast}$, and under relatively mild assumptions, we can have $P^{\ast}=d^{\ast}$. If the equality holds, one can prove the following optimality condition on $x$. A primal feasible $x$ and a dual feasible $Z$ are optimal which is denoted by $\hat{x}$ and $\hat{Z}$ if and only if $F(\hat{x})\hat{Z}=\hat{Z}F(\hat{x}).$ (3.15) This latter condition is called the complementary slackness condition. In one way or another, numerical methods for solving SDP problems always exploit the inequality $d\leq d^{\ast}\leq P^{\ast}\leq P,$ where $d$ and $P$ are the objective values for any dual feasible point and primal feasible point, respectively. The difference $P^{\ast}-d^{\ast}=c^{T}x+Tr[F_{0}Z]=Tr[F_{x}Z]\geq 0$ (3.16) is called the duality gap. If the equality holds $d^{\ast}=P^{\ast}$, i.e., the optimal duality gap is zero, then we say that strong duality holds. ### 3.1 Karush-Kuhn-Tucker (KKT) theorem Assuming that functions $g_{i}$, $h_{i}$ are differentiable and that strong duality holds, there exists vectors $\xi\in R^{k}$ and $y\in R^{m}$ such that the gradient of dual Lagrangian $L(x^{\ast}.\xi^{\ast},y^{\ast})=f(x^{\ast})+\sum_{i}\xi_{i}^{\ast}h_{i}(x^{\ast})+\sum_{i}y_{i}^{\ast}g_{i}(x^{\ast})$ over $x$ vanishes at $x^{\ast}$: $h_{i}(x^{\ast})=0\ (primal\ feasible),$ $g_{i}(x^{\ast})\leq 0\ (primal\ feasible),$ $y_{i}^{\ast}\geq 0\ (dual\ feasible),\ y_{i}^{\ast}g_{i}(x^{\ast})=0,$ $\bigtriangledown f(x^{\ast})+\sum_{i}\xi_{i}^{\ast}\bigtriangledown h_{i}(x^{\ast})+\sum_{i}y_{i}^{\ast}\bigtriangledown g_{i}(x^{\ast})=0.$ (3.17) Then $x^{\ast}$ and $(\xi_{i}^{\ast},y_{i}^{\ast})$ are primal and dual optimal with zero duality gap. In summary, for any convex optimization problem with differentiable objective and constraint functions, the points which satisfy the KKT conditions are primal and dual optimal, and have zero duality gap. Necessary KKT conditions satisfied by any primal and dual optimal pair and for convex problems, KKT conditions are also sufficient. If a convex optimization problem with differentiable objective and constraint functions satisfies Slater s condition, then the KKT conditions provide necessary and sufficient conditions for optimality: Slater s condition implies that the optimal duality gap is zero and the dual optimum is attained, so x is optimal if and only if there are $(\xi_{i}^{\ast},y_{i}^{\ast})$ such that they, together with $x$, satisfy the KKT conditions. ### 3.2 Slater s condition Suppose $x^{\ast}$ solves $\mathbf{minimize}\ f(x)g_{i}(x)\geq b_{i},\ i=1,...,m$ (3.18) and the feasible set is nonempty. Then there is a nonnegative vector $\xi$ such that for all $x$ $L(x,\xi)=f(x)+\xi^{T}[b-g(x)]\leq f(x^{\ast})=L(x^{\ast},\xi).$ (3.19) In addition, if $f(...)$, $g_{i}(...)$, $i=1,...,m,$ are continuously differentiable, then $\frac{\partial f(x^{\ast})}{\partial(x_{j})}-\xi\frac{\partial g(x^{\ast})}{\partial(x)}=0.$ (3.20) In the spatial case the vector x is a solution of the linear program $\mathbf{minimize}\ c^{T}x,such\ that\ Ax=bx\geq 0,$ (3.21) if and only if there exist vectors $\xi\in R^{k}$ and $y\in R^{m}$ for which the following conditions hold for $(x,\xi,y)=(x^{\ast},\xi^{\ast},y^{\ast})$: $A^{T}\xi+y=c,Ax=b,x_{i}\geq 0;\ y\geq 0;$ $x_{i}y_{i}=0,\ i=1,...,m.$ (3.22) A solution $(x^{\ast},\xi^{\ast},y^{\ast})$ is called strictly complementary, if $x^{\ast}+y^{\ast}>0$ , i.e., if there exists no index $i\in 1,...m$ such that $x_{i}^{\ast}=y_{i}^{\ast}$. ## 4 SDP Formulation of unambiguous discrimination Eldar, Megretski, and Verghese in Ref.[24] have showed that the unambiguous discrimination problem can be reduced to SDP method and the KKT conditions can be defined as $F(p)=\sum_{i=1}^{N}p_{i}F_{i}+F_{0}\geq 0,\ Tr(\Pi_{i}X)=z_{i}+\eta_{i},$ $z_{i}\geq 0,1\leq i\leq N,\ AF(p)=0,\ X(I_{d}-\sum_{i=1}^{N}p_{i}\Pi_{i})=0,$ $z_{i}p_{i}=0,\ 1\leq i\leq N,\exists p_{i}:\sum_{i=1}^{N}p_{i}F_{i}+F_{0}\geq 0,$ $\exists X,z:X\geq 0,z\geq 0$ (4.23) such that $\|p\rangle=:\left(\begin{array}[]{c}p_{1}\\\ \vdots\\\ p_{N}\\\ \end{array}\right),\|c\rangle=:\left(\begin{array}[]{c}\eta_{1}\\\ \vdots\\\ \eta_{2}\\\ \end{array}\right),$ $F_{0}=:\left(\begin{array}[]{cccc}I_{d}&0&\cdots&0\\\ 0&0&\cdots&0\\\ \vdots&\vdots&\ddots&\vdots\\\ 0&0&\cdots&0\\\ \end{array}\right),F_{1}=:\left(\begin{array}[]{ccccc}-\Pi_{1}&0&0&\cdots&0\\\ 0&1&0&\cdots&0\\\ 0&\cdots&0&0&0\\\ \vdots&\vdots&\vdots&\ddots&\vdots\\\ 0&0&\cdots&0&0\\\ \end{array}\right),F_{2}=:\left(\begin{array}[]{ccccc}-\Pi_{2}&0&0&\cdots&0\\\ 0&0&0&\cdots&0\\\ 0&\cdots&1&0&0\\\ \vdots&\vdots&\vdots&\ddots&\vdots\\\ 0&0&\cdots&0&0\\\ \end{array}\right),...,$ $F_{N}=:\left(\begin{array}[]{ccccc}-\Pi_{N}&0&0&\cdots&0\\\ 0&0&0&\cdots&0\\\ 0&\cdots&0&0&0\\\ \vdots&\vdots&\vdots&\ddots&\vdots\\\ 0&0&\cdots&0&1\\\ \end{array}\right),A=:\left(\begin{array}[]{cc}X_{d}&Y\\\ Y^{T}&\left(\begin{array}[]{ccc}z_{1}&\cdots&0\\\ 0&\ddots&0\\\ 0&\cdots&z_{N}\\\ \end{array}\right)\\\ \end{array}\right),$ (4.24) where $X_{d}$ is the $d\times d$ matrix and $Y$ is the $N\times d$ matrix. ## 5 Analytical solution of unambiguous state discrimination problem When the set of states $\\{\psi_{i}\\}_{i=1}^{N}$ and the prior probabilities $\\{\eta_{i}\\}_{i=1}^{N}$ are given, the POVM elements for the measurement $(\Pi_{i}=\sum_{i=1}^{N}p_{i}\|\widetilde{\psi_{i}}\rangle\langle\widetilde{\psi_{i}}\|,\ \Pi_{0}=I-\sum_{i=1}^{N}\prod_{i})$ depend on only the variables $p_{i}$ and these variables lie in the feasible region. Feasible region that gives the domain of acceptable values of $p_{i}$ is determined by the following equation [25, 26, 27, 28]: $1-\sum_{i}D_{i}p_{i}+\sum_{i<j}D_{ij}p_{i}p_{j}-\sum_{i<j<k}D_{ijk}p_{i}p_{j}p_{k}+...+(-1)^{N}\sum_{i_{1}<i_{2}...<i_{N}}p_{i_{1}}p_{i_{2}}...p_{i_{N}}D_{i_{1}i_{2}...i_{N}}=0,$ (5.25) where the set of $\\{D_{i_{1}i_{2}...i_{N}}\\}$ are the subdeterminants (minor) of matrix D defined by $D=\left(\begin{array}[]{cccc}\tilde{a}_{11}&\tilde{a}_{12}&\cdots&\tilde{a}_{1N}\\\ \tilde{a}_{21}&\tilde{a}_{22}&\cdots&\tilde{a}_{2N}\\\ \vdots&\vdots&\ddots&\vdots\\\ \tilde{a}_{N1}&\tilde{a}_{N2}&\cdots&\tilde{a}_{NN}\\\ \end{array}\right)$ (5.26) with $\tilde{a}_{ij}=\langle\tilde{\psi}_{i}\|\tilde{\psi}_{j}\rangle$. In this section, for the real and complex inner product of reciprocal states, an exact analytic solution for OPUSD problem involving an arbitrary number of pure linearly independent quantum states is presented by using KKT convex optimization method. The KKT conditions for unambiguous discrimination of N linearly independent states are given by $I-\sum_{i=1}^{N}p_{i}\|\widetilde{\psi_{i}}\rangle\langle\widetilde{\psi_{i}}\|\geq 0,\ p_{i}\geq 0,$ $(I-\sum_{i=1}^{N}p_{i}\|\widetilde{\psi_{i}}\rangle\langle\widetilde{\psi_{i}}\|)X=X(I-\sum_{i=1}^{N}p_{i}\|\widetilde{\psi_{i}}\rangle\langle\widetilde{\psi_{i}}\|)=0,$ $z_{i}p_{i}=0,$ $Tr(X\|\widetilde{\psi_{i}}\rangle\langle\widetilde{\psi_{i}}\|)=z_{i}+\eta_{i},\eta_{i}\geq 0,\ i=1,...,N$ (5.27) Then using KKT conditions one can show that $\left(\begin{array}[]{cccc}1-p_{1}\tilde{a}_{11}&-p_{2}\tilde{a}_{12}&\cdots&-p_{N}\tilde{a}_{1N}\\\ -p_{1}\tilde{a}_{21}&1-p_{2}\tilde{a}_{22}&\cdots&-p_{N}\tilde{a}_{2N}\\\ \vdots&\vdots&\ddots&\vdots\\\ -p_{1}\tilde{a}_{N1}&-p_{2}\tilde{a}_{N2}&\cdots&1-p_{N}\tilde{a}_{NN}\\\ \end{array}\right)\times\left(\begin{array}[]{cccc}x_{11}&x_{12}&\cdots&x_{1N}\\\ x_{21}&x_{22}&\cdots&x_{2N}\\\ \vdots&\vdots&\ddots&\vdots\\\ x_{N1}&x_{N2}&\cdots&x_{NN}\\\ \end{array}\right)=0$ (5.28) If $p_{i}\geq 0,(i=1,...,N)$ usin Eq.(5.27) we can conclude that $X$ is the rank one matrix and we have $x_{ij}=e^{i\varphi_{i}}e^{i\varphi_{j}}\sqrt{\eta}_{i}\sqrt{\eta_{j}}$ (5.29) Consequently, the Eq. (5.28) can be written as $\left(\begin{array}[]{cccc}1-p_{1}\tilde{a}_{11}&-p_{2}\tilde{a}_{12}&\cdots&-p_{N}\tilde{a}_{1N}\\\ -p_{1}\tilde{a_{21}}&1-p_{2}\tilde{a_{22}}&\cdots&-p_{N}\tilde{a_{2N}}\\\ \vdots&\vdots&\ddots&\vdots\\\ -p_{1}\tilde{a}_{N1}&-p_{2}\tilde{a}_{N2}&\cdots&1-p_{N}\tilde{a}_{NN}\\\ \end{array}\right)\times\left(\begin{array}[]{c}x_{11}\\\ x_{12}\\\ \vdots\\\ x_{1N}\\\ \end{array}\right)=0$ (5.30) Or, equivalently, $\left(\begin{array}[]{cccc}\tilde{a}_{11}&\tilde{a}_{12}&\cdots&\tilde{a}_{1N}\\\ \tilde{a}_{21}&\tilde{a}_{22}&\cdots&\tilde{a}_{2N}\\\ \vdots&\vdots&\ddots&\vdots\\\ \tilde{a}_{N1}&\tilde{a}_{N2}&\cdots&\tilde{a}_{NN}\\\ \end{array}\right)\times\left(\begin{array}[]{cccc}x_{11}&0&\cdots&0\\\ 0&x_{22}&\cdots&0\\\ \vdots&\vdots&\ddots&\vdots\\\ 0&0&\cdots&x_{NN}\\\ \end{array}\right)\times\left(\begin{array}[]{c}p_{1}\\\ p_{2}\\\ \vdots\\\ p_{N1}\\\ \end{array}\right)=\left(\begin{array}[]{c}x_{11}\\\ x_{12}\\\ \vdots\\\ x_{1N}\\\ \end{array}\right)$ (5.31) If the first condition in Eq. (5.27) is satisfied, using above equation one can show that the optimal solution for $p_{i}$ is given by $p_{i}=\frac{1}{e^{i\varphi_{i}}\sqrt{\eta_{i}}det(D)}\times\det\left(\begin{array}[]{ccccccc}\tilde{a}_{1,1}&\cdots&\tilde{a}_{1,i-1}&e^{i\varphi_{1}}\sqrt{\eta_{1}}&\tilde{a}_{1,i+1}&\cdots&\tilde{a}_{1,N}\\\ \tilde{a}_{2,1}&\cdots&\tilde{a}_{2,i-1}&e^{i\varphi_{2}}\sqrt{\eta_{N}}&\tilde{a}_{2,i+1}&\cdots&\tilde{a_{2,N}}\\\ \vdots&\ddots&\vdots&\vdots&\vdots&\ddots&\vdots\\\ \tilde{a}_{N,1}&\cdots&\tilde{a}_{N,i-1}&e^{i\varphi_{i}}\sqrt{\eta_{i}}&\tilde{a}_{N,i+1}&\cdots&\tilde{a}_{N,N}\\\ \end{array}\right)$ (5.32) Using the fact that the matrix whose components are $\tilde{a}_{ij}$ is inverse matrix of the one whose components are $a_{ij}=\langle\psi_{i}\|\psi_{j}\rangle$, we have $\left(\begin{array}[]{c}x_{11}p_{1}\\\ x_{12}p_{2}\\\ \vdots\\\ x_{1N}p_{N}\\\ \end{array}\right)=\left(\begin{array}[]{cccc}a_{11}&a_{12}&\cdots&a_{1N}\\\ a_{21}&a_{22}&\cdots&a_{2N}\\\ \vdots&\vdots&\ddots&\vdots\\\ a_{N1}&a_{N2}&\cdots&a_{NN}\\\ \end{array}\right)\left(\begin{array}[]{c}x_{11}\\\ x_{12}\\\ \vdots\\\ x_{1N}\\\ \end{array}\right)\Rightarrow x_{ii}p_{i}=\sum_{i=1}^{N}a_{ij}x_{ij}$ Using Eq. (5.29) we have $p_{i}=\sum_{i=1}^{N}\frac{e^{i(\varphi_{i}+\varphi_{j})}}{e^{2i\varphi_{i}}}\sqrt{\frac{\eta_{i}}{\eta_{j}}}a_{ij}$ Then we can rewrite the probability $p_{i}$ in a simpler form $p_{i}=e^{-i\varphi_{i}}\sum_{i=1}^{N}e^{i\varphi_{j}}\sqrt{\frac{\eta_{i}}{\eta_{j}}}a_{ij}$ (5.33) where $e^{-i\varphi_{j}}a_{ij}$ is determined such that $p_{i}$ satisfy the condition $0\leq p_{i}\leq 1$. Then the success probability with respect to $\tilde{a}_{ij}$ components is given by $P=-\frac{1}{\det(D)}\times\det\left(\begin{array}[]{cccc}0&e^{i\varphi_{1}}\sqrt{\eta_{1}}&\cdots&e^{i\varphi_{N}}\sqrt{\eta_{N}}\\\ e^{i\varphi_{1}}\sqrt{\eta_{1}}&&&\\\ \vdots&&D&\\\ e^{i\varphi_{N}}\sqrt{\eta_{N}}&&&\\\ \end{array}\right)$ (5.34) The success probability with respect to $a_{ij}$ components is as follows $P=\sum_{i=1}^{N}\eta_{i}p_{i}=\sum_{i,j=1}^{N}e^{i(\varphi_{i}-\varphi_{j})}\sqrt{\eta_{i}\eta_{j}}a_{ij}=\\|\sum_{j=1}^{N}\sqrt{\eta_{j}}e^{i\varphi_{j}}\|\psi_{j}\rangle\\|^{2}$ (5.35) If the first condition of KKT is not satisfied, for a specified i, $p_{i}$ does not lie in the feasible region $(p_{i}\leq 0\ or\ p_{i}\geq 1)$. In this case, one can omit the jth row and jth column in the square $N\times N$ matrices and jth row in the row matrices in Eqs. (5.32) and (5.34) , and find the optimal solutions with $p_{i}=0$. After this reduction, if any other $p_{i}$ does not lie in the feasible region, the same procedure will be repeated. Noted that a similar result was derived in Ref. [29] with another method. The equations (5.33) and (5.35) gives an analytical relation between the maximum average success probability and the $N$ pure linearly independent quantum states to be discriminated. In general, finding an exact analytic solution for the OPUSD problem involving an arbitrary number of pure linearly independent quantum states is hard, since the explicit expressions of the phases $e^{i\varphi_{j}}(j=1,...,N)$ are not given in equations (5.33) and (5.35). However using (5.33) and (5.35) one can simplify the calculation of the optimal solution in special cases and it may also drive some bounds for the average success probability [18, 29]. Then, the approximated methods are useful for unambiguous discrimination of N linearly independent quantum states. Since we have presented an analytic relation for the feasible region of N linearly independent quantum states, and this region is convex, then one can easily obtain the optimal POVM by some well-known numerical methods such as constrained linear or nonlinear least-squares, interior points, and simplex and quadratic programming methods [30]. ## 6 Conclusion In conclusion, for the real and complex inner product of states, we have been able to obtain an exact analytic solution for OPUSD problem involving an arbitrary number of pure linearly independent quantum states by using KKT convex optimization method. Moreover, Using semidefinite programming and Karush-Kuhn-Tucker convex optimization method, we have been able to obtain an analytical formula which shows the relation between optimal solution of unambiguous discrimination problem and an arbitrary number of pure linearly independent quantum states to be identified. Using this analytical formula one can simplify the calculation of the optimal solution in special cases and it may also drive some bounds for the average success probability. ## References * [1] see, e. g., J. A Bergou, U. Herzog, and M. Hillery, Lect. Notes Phys. 649, 417-465 (Springer, Berlin, 2004). * [2] A. S. Holevo, Probl. Peredachi Inf. 10, 51 (1974); A. S. Holevo, Probl. Inf. Transm. 10, 51 (1974). * [3] I. D. Ivanovic, Phys. Lett. A 123, 257 (1987). * [4] D. Dieks, Phys. Lett. A 126, 303 (1988). * [5] .A. Peres and D. R. Terno, J. Phys. A 31, 7105 (1998). * [6] G. Jaeger and A. Shimony, Phys. Lett. A 197, 83 (1995). * [7] L. M. Duan and G. C. Guo, Phys. Rev. Lett. 80, 4999 (1998). * [8] Y. Sun, M. Hillery, and J. A. Bergou, Phys. Rev. A 64, 022311 (2001). * [9] A. Chefles, Phys. Lett. A 239, 339 (1998). * [10] M. X. Goemans and D. P. Williamson, J. Assoc. Comput. Mach. 42, 1115 (1995). * [11] Z. Q. Luo, Math. Program. 97, 587 (2003). * [12] T. N. Davidson, Z.-Q. Luo, and K. M. Wong, IEEE Trans. Signal Process. 48, 1433 (2000). * [13] Wing-Kin Ma, T. N. Davidson, K. M. Wong, Z.-Q. Luo, and P. C. Ching, IEEE Trans. Signal Process. 50, 912 (2002). * [14] S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan, Studies in Applied Mathematics, SIAM (Philadelphia, PA 1994), Vol.15. * [15] H. Barnum, M. Saks, and M. Szegedy, in Proceedings of the 18th IEEE Annual Conference on Computational Complexity (IEEE Computer Society, New York, 2003), pp. 179 193. * [16] Y. C. Eldar, A. Megretski, and G. C. Verghese, IEEE Trans. Inf. Theory 49, 1007 (2003). * [17] L. Ip, http://www.qcaustralia.org * [18] M. A. Jafarizadeh, M. Rezaei, N. Karimi and A. R. Amiri, Phys. Rev. A 77, 042314(2008). * [19] K. Kraus, States, Effects, and Operations, Lecture Notes in Physics No. 190 (Springer, Berlin, 1983). * [20] A. Chefles, Phys. Lett. A 239, 339 (1998). * [21] Y. C. Eldar, IEEE Trans. Inf. Theory 49, 446 (2003). * [22] G. H. Golub and C. F. Van Loan, Matrix Computations, 3rd ed.(Johns Hopkins University Press, Baltimore, MD, 1996). * [23] L. R. Welch, IEEE Trans. Inf. Theory 20, 397 (1974). * [24] Y. C. Eldar, A. Megretski, and G. C. Verghese, IEEE Trans. Inf. Theory 49, 1007 (2003). * [25] M. Lewenstein and A. Sanpera, Phys. Rev. Lett. 80, 2261(1998). * [26] S. Karnas and M. Lewenstein, J. Phys. A 34, 6919 (2001). * [27] M. A. Jafarizadeh, M. Mirzaee, M. Rezaee, Int. J. Quantum Inf. 2, 541 (2004). * [28] M. A. Jafarizadeh, M. Mirzaee, and M. Rezaee, Quantum Inf. Process. 4, 199 (2005). * [29] Shengshi Pang, Shengjun Wu, Phys. Rev. A 80, 052320(2009). * [30] S. Boyd and L. Vandenberghe, Convex Optimization (Cambridge University Press, Cambridge, 2004).	arxiv-papers	2011-11-21T11:12:53	2024-09-04T02:49:24.543791	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "N. Karimi", "submitter": "Nasser Karimi", "url": "https://arxiv.org/abs/1111.4831" }
1111.5037	# The TW Hya Disk at 870 $\mu$m: Comparison of CO and Dust Radial Structures Sean M. Andrews11affiliation: Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138 , David J. Wilner11affiliation: Harvard- Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138 , A. M. Hughes22affiliation: University of California at Berkeley, Department of Astronomy, 601 Campbell Hall, Berkeley, CA 94720 , Chunhua Qi11affiliation: Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138 , Katherine A. Rosenfeld11affiliation: Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138 , Karin I. Öberg11affiliation: Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138 33affiliation: Hubble Fellow , T. Birnstiel44affiliation: Ludwig-Maximilians-Universität, University Observatory Munich, Scheinerstrasse 1, D-81679 Munich, Germany , Catherine Espaillat11affiliation: Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138 55affiliation: NSF Astronomy & Astrophysics Postdoctoral Fellow , Lucas A. Cieza66affiliation: University of Hawaii Institute for Astronomy, 2680 Woodlawn Drive, Honolulu, HI 96822 , Jonathan P. Williams66affiliation: University of Hawaii Institute for Astronomy, 2680 Woodlawn Drive, Honolulu, HI 96822 , Shin-Yi Lin77affiliation: Center for Astrophysics & Space Science, University of California San Diego, La Jolla, CA 92093 , and Paul T. P. Ho11affiliation: Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138 88affiliation: Academia Sinica Institute of Astronomy and Astrophysics, P.O. Box 23-141, Taipei 106, Taiwan ###### Abstract We present high resolution ($0\farcs 3=16$ AU), high signal-to-noise ratio Submillimeter Array observations of the 870 $\mu$m (345 GHz) continuum and CO $J$=3$-$2 line emission from the protoplanetary disk around TW Hya. Using continuum and line radiative transfer calculations, those data and the multiwavelength spectral energy distribution are analyzed together in the context of simple two-dimensional parametric disk structure models. Under the assumptions of a radially invariant dust population and (vertically integrated) gas-to-dust mass ratio, we are unable to simultaneously reproduce the CO and dust observations with model structures that employ either a single, distinct outer boundary or a smooth (exponential) taper at large radii. Instead, we find that the distribution of millimeter-sized dust grains in the TW Hya disk has a relatively sharp edge near 60 AU, contrary to the CO emission (and optical/infrared scattered light) that extends to a much larger radius of at least 215 AU. We discuss some possible explanations for the observed radial distribution of millimeter-sized dust grains and the apparent CO-dust size discrepancy, and suggest that they may be hallmarks of substructure in the dust disk or natural signatures of the growth and radial drift of solids that might be expected for disks around older pre-main sequence stars like TW Hya. circumstellar matter — protoplanetary disks — planetary systems: formation — stars: individual (TW Hya) ## 1 Introduction The physical conditions of the gas and dust in young circumstellar disks shape the formation and early evolution of planetary systems. The spatial distribution of disk material – the density structure – plays many fundamental roles, and is especially relevant for dictating when and where planets can form and migrate in the disk. Naturally, empirical constraints on protoplanetary disk structures are of significant value in efforts to develop realistic models of the complex processes involved in planet formation. Some recent studies have made progress in extracting the radial (surface) density profiles in these disks from high angular resolution measurements of their millimeter-wave continuum emission (Andrews et al., 2009, 2010b; Isella et al., 2009; Guilloteau et al., 2011). However, such observations are only sensitive to the density structure of the (presumably) trace population of disk solids, and not the gas that wields far more influence over the evolution of disk structure and the formation of planets. The major challenge is that the gas in these disks is primarily cold H2, which is not directly observable. And while there have been many investigations of less abundant molecular species (see Dutrey et al., 2007), limited sensitivity to low optical depth lines and an incomplete understanding of the complicated chemistry in these disks have so far made it difficult to convert observations of the gas into density constraints (see Williams & Cieza, 2011). Given those obstacles, there have not been many attempts to directly compare the gas and dust structures of protoplanetary disks with spatially resolved data. A few studies used simple models to fit the millimeter continuum (dust) and CO line (gas) emission independently, finding inconsistent structures where the dust is much more compact than the gas (Piétu et al., 2005; Isella et al., 2007). Hughes et al. (2008) suggested that this discrepancy was likely an optical depth illusion, an artifact of the assumed sharp outer edge in the density model. They showed that a surface density profile with a smooth taper at large radii could better reproduce the dust and gas observations. However, Panić et al. (2009) argued that similar modifications to the model structure of the IM Lup disk (see Pinte et al., 2008) were insufficient to account for the much different sizes they measure for its gas and dust emission. Recently, Qi et al. (2011) successfully matched their observations of multiple CO transitions and continuum emission from the HD 163296 disk with a single density model, although refinements to that model based on the detailed morphology of the continuum emission were not a priority in that study. In general, there is still substantial uncertainty that the gas and dust trace the same structures in protoplanetary disks. From a practical standpoint, that uncertainty is disconcerting. Our knowledge of the mass contents of these disks is entirely based on their (easy to measure) dust emission, but we are not sure if those dust structures also describe the gas reservoirs that effectively control the key disk evolution and planet formation processes. Although direct measurements of H2 densities are not possible, in principle an indirect comparison of the dust and gas structures in a protoplanetary disk can be made with sufficiently sensitive and high resolution observations of the thermal continuum and emission lines from trace gas species. To meet those requirements, a relatively massive and nearby disk would be an ideal candidate target. TW Hya is an isolated, $\sim$0.8 M⊙ T Tauri star located only $54\pm 6$ pc from the Sun (Rucinski & Krautter, 1983; Wichmann et al., 1998; van Leeuwen, 2007). Despite its advanced age ($\sim$10 Myr; Kastner et al., 1997; Webb et al., 1999), it hosts a massive, gaseous accretion disk with a rich molecular spectrum and strong continuum emission out to centimeter wavelengths (Wilner et al., 2000, 2003, 2005; Qi et al., 2004, 2006, 2008). Those observations and a suite of resolved optical/infrared scattered light images confirm that the disk is well-resolved and viewed nearly face-on (Krist et al., 2000; Trilling et al., 2001; Weinberger et al., 2002; Apai et al., 2004; Roberge et al., 2005). The inner edge of this “transition” disk is truncated at a radius of 4 AU, likely by an unseen (and maybe planetary) companion (Calvet et al., 2002; Hughes et al., 2007). Given its proximity, favorable orientation, and rich gas and dust content, the TW Hya disk is uniquely well- suited for a comparative investigation of the radial behavior of its gas and dust structures. In this article, we present new sub-arcsecond resolution observations of the 870 $\mu$m continuum and CO $J$=3$-$2 emission from the TW Hya disk. Using state-of-the-art radiative transfer modeling tools, we use those measurements to compare the radial distributions of its CO gas and dust. In §2 we describe the observations and data calibration. The basic characteristics of the data are reviewed in §3. A detailed description of the modeling and results are provided in §4. The implications of that analysis are discussed in §5, and our principle conclusions are summarized in §6. ## 2 Observations and Data Reduction TW Hya was observed at 345 GHz (870 $\mu$m) with the Submillimeter Array (SMA; Ho et al., 2004) on 8 occasions in 2006-2010 in the compact (C), extended (E), and very extended (V) array configurations, providing baseline lengths of 16-70 m, 28-226 m, and 68-509 m, respectively. To accomodate various studies of the CO $J$=3$-$2 emission line (at 345.796 GHz), the SMA double sideband receivers and correlator were configured with several different local oscillator (LO) settings and spectral resolutions throughout this campaign. The “default” setup used an LO frequency of 340.755 GHz (880 $\mu$m) with the CO line in the center of the upper sideband, with a resolution of 0.70 km s-1 in one spectral chunk. The other 23 partially-overlapping 104 MHz spectral chunks in each sideband were sampled coarsely with 32 channels each. Some of the 2008 observations utilized a “high” resolution correlator mode, with a large portion of the bandwidth devoted to probing the CO line with 0.044 km s-1 channels (see Hughes et al., 2011). While those data sample the continuum with the default spectral resolution, they have a reduced continuum bandwidth (2.6 GHz; $\sim$70% of the available bandwidth at the time) and a higher LO frequency at 349.935 GHz (857 $\mu$m). The 2010 observations were conducted in a “medium” resolution mode that employed the new expanded bandwidth capabilities at the SMA. In that case, two different 2 GHz IF bands were centered $\pm$5 GHz (the normal setup) and $\pm$7 GHz from the LO frequency, 339.853 GHz (883 $\mu$m). The CO line was observed at a moderate resolution of 0.18 km s-1 in the upper sideband of the first IF band, and the continuum was coarsely sampled across the remaining expanded bandwidth. Finally, a “hybrid” mode was utilized at the end of 2006 to cover the $J$=4$-$3 transition of H13CO+ (see Qi et al., 2008). A summary of the observational setups is provided in Table 1. Table 1: Summary of SMA Observations UT Date \| array \| spectral \| RMS noise \| beam size \| beam PA ---\|---\|---\|---\|---\|--- \| config. \| setup \| [mJy beam-1] \| [″] \| [°] (1) \| (2) \| (3) \| (4) \| (5) \| (6) 2006 Dec 28 \| C \| hybrid \| 3.7 \| $4.2\times 1.7$ \| 4 2008 Jan 23 \| E \| high \| 3.2 \| $0.9\times 0.7$ \| 21 2008 Feb 20 \| E \| high \| 3.1 \| $1.0\times 0.7$ \| $-$20 2008 Feb 21 \| E \| default \| 4.1 \| $1.2\times 0.5$ \| 14 2008 Mar 2 \| C \| high \| 3.6 \| $3.9\times 1.9$ \| $-$17 2008 Mar 9 \| C \| default \| 3.2 \| $3.7\times 1.8$ \| $-$13 2008 Apr 3 \| V \| default \| 2.2 \| $0.6\times 0.3$ \| 19 2010 Feb 9 \| V \| medium \| 1.8 \| $0.5\times 0.3$ \| $-$3 combined \| all \| $\cdots$ \| 2.0 \| $0.8\times 0.6$ \| $-$3 Note. — Col. (1): UT date of observations. Col. (2): SMA array configuration, C = compact, E = extended, and V = very extended. Col. (3): Adopted correlator mode (see §2). Col. (4): Continuum RMS noise in a naturally-weighted map. Col. (5): Naturally-weighted synthesized beam dimensions. Col. (6): Naturally- weighted synthesized beam orientation (major axis position angle, measured east of north). TW Hya was observed in an alternating sequence with the nearby quasar J1037-295 (a projected separation of 7.3°), with a total cycle time of 15 minutes in the C and E configurations and 8-10 minutes in the V configuration. The quasars 3C 279 or J1146-289 were observed every other cycle. For the tracks on 2008 February 21, March 9, and April 3, the target portion of the cycle was shared with the nearby sources HD 98800 and Hen 3-600 (see Andrews et al., 2010a). Planets, satellites, and bright quasars were observed as bandpass and absolute flux calibrators when TW Hya was at low elevations ($<$18°), depending on their availability and the array configuration. Most of these data were obtained in the best atmospheric conditions available at Mauna Kea, with zenith opacities of 0.03-0.05 at 225 GHz (0.6-1.0 mm of precipitable water vapor) and well-behaved phase variations on timescales longer than the calibration cycle. The conditions on 2008 March 2 and 9 were worse, but typical for Mauna Kea (with 1.4-1.8 mm of precipitable water vapor). The data from each individual observation were reduced independently with the IDL-based MIR software package. The bandpass response was calibrated with observations of bright planets and quasars, and broadband continuum channels in each sideband (and IF band, where applicable) were generated by averaging and then combining the central 82 MHz in each line-free spectral chunk. The visibility amplitude scale was derived from observations of Uranus, Titan, Callisto, or Vesta, with a typical systematic uncertainty of 10-15%. The antenna-based complex gain response of the system was determined with reference to J1037-295. The observations of 3C 279 or J1146-289 were used to assess the quality of phase transfer in the gain calibration process. Based on those data, we estimate that the “seeing” generated by atmospheric phase noise and any small baseline errors is small, 0.10-0$\farcs$15\. This result is a testament to the high quality of the observing conditions, especially given the low target elevation and wide projected separations between the calibrators (for reference, 3C 279 is located 39.7° from TW Hya and 38.5° from J1037-295, while the corresponding separations for J1146-289 are 11.0° and 15.1°, respectively). Before the observations could be combined, we had to account for the proper motion of TW Hya over the $\sim$3 year observing baseline ($\mu_{\alpha}\cos{\delta}=-0\farcs 066$ yr-1, $\mu_{\delta}=-0\farcs 014$ yr-1; van Leeuwen, 2007). Fortunately, each individual dataset exhibited bright, symmetric, centrally-peaked continuum emission that we associate with dust in the TW Hya disk. Using elliptical Gaussian fits to those continuum visibilities, we determined centroid positions for each individual set of observations and associated them with the stellar position at that epoch. The measured emission centroids are consistent with the expected stellar positions, well within the $\sim$0$\farcs$1 absolute astrometric accuracy of the SMA (set primarily by small baseline uncertainties). Based on those centroid measurements, each individual dataset was aligned to a common coordinate system. After confirming that the aligned visibility sets showed excellent agreement on all overlapping baselines, all datasets were combined to produce composite 345 GHz continuum and CO $J$=3$-$2 visibilities. The composite visibilities were Fourier inverted, deconvolved with the CLEAN algorithm, and restored with a synthesized beam using the MIRIAD software package. The natural weighting of the continuum data produces an image with a $0\farcs 80\times 0\farcs 58$ beam and an RMS noise level of 2.0 mJy beam-1 (which is dynamic-range limited). Composite CO $J$=3$-$2 channel maps were synthesized with a velocity resolution of 0.2 km s-1 and a circular beam with a FWHM of 1$\farcs$0 (the naturally-weighted resolution is $0\farcs 96\times 0\farcs 73$). The RMS noise level is 0.13 Jy beam-1 in each channel. Ancillary information in the literature was used to construct the TW Hya spectral energy distribution (SED) that will be used with the SMA data. We adopted the optical monitoring results of Mekkaden (1998) and near-infrared photometry from Weintraub et al. (2000) and the 2MASS point source catalog (Cutri et al., 2003). In the thermal infrared, Spitzer flux densities measured by Hartmann et al. (2005) and Low et al. (2005) were supplemented with IRAS and Herschel data (Weaver & Jones, 1992; Thi et al., 2010). We also include a Spitzer IRS spectrum, kindly provided by E. Furlan (see Uchida et al., 2004). At submillimeter wavelengths, we relied on the calibrator flux densities from the SHARC-II111http://www.submm.caltech.edu/$∼$sharc/analysis/calibration.htm and SCUBA222http://www.jach.hawaii.edu/JCMT/continuum/calibration/sens/potentialcalibrators.html instruments at the Caltech Submillimeter Observatory and James Clerk Maxwell Telescope, respectively ($F_{\nu}=6.13\pm 0.68$, $3.9\pm 0.7$, and $1.37\pm 0.01$ Jy at 350, 443, and 869 $\mu$m, respectively). Additional millimeter- wave measurements were collected from Weintraub et al. (1989), Qi et al. (2004), and Wilner et al. (2003). ## 3 Results The astrometrically aligned and combined SMA data are shown together in Figure 1, along with the broadband SED. The synthesized continuum image in Figure 1$a$ has an effective frequency of 344.4 GHz (870 $\mu$m), with an integrated flux density of $1.34\pm 0.13$ Jy and a peak flux density of $0.337\pm 0.034$ Jy beam-1 (a peak signal-to-noise ratio of $\sim$170), including the systematic calibration uncertainties. The continuum emission is regular and symmetric on the angular scales probed here, with a synthesized beam size of $43\times 31$ AU projected on the sky. Essentially no emission is detected outside a $\sim$1″ radius. Given that the integrated flux density determined from the SMA data is in good agreement with single-dish measurements (see §2; Weintraub et al., 1989; Di Francesco et al., 2008), it is clear that this is no optical depth effect: all of the millimeter-wave dust emission from the TW Hya disk is concentrated inside a projected radius of $\sim$60 AU. Figure 1: ($a$) Naturally weighted composite image of the 870 $\mu$m continuum emission from the TW Hya disk. Contours start at 10 mJy beam-1 (5 $\sigma$) and increase in 30 mJy beam-1 (15 $\sigma$) increments. The synthesized beam dimensions are shown in the lower left. ($b$) Azimuthally averaged 870 $\mu$m continuum visibility profile as a function of the deprojected baseline length (real part only; the imaginary terms are effectively zero on all baselines). The uncertainties are typically smaller than the symbol sizes. Note the low- amplitude oscillations beyond $\sim$150 k$\lambda$. ($c$) Complete SED for the TW Hya star+disk system (references are in the text; see §2). The Spitzer IRS spectrum is shown as a thick gray curve. Our adopted stellar photosphere model is overlaid as a thin gray curve (see §4.1). ($d$) CO $J$=3$-$2 channel maps from the TW Hya disk, on the same angular scale as the continuum map in panel $a$. Contours are drawn at 0.4 Jy beam-1 intervals ($\sim$3 $\sigma$) in each 0.2 km s-1-wide channel. The 1″ synthesized beam is shown in the lower left. The central channel represents the TW Hya systemic velocity, at $V_{\rm LSR}=+2.86$ km s-1. While the continuum image appears rather plain, some interesting emission features are apparent in a direct examination of the visibilities. Figure 1$b$ displays the azimuthally averaged 870 $\mu$m continuum real visibilities as a function of the deprojected baseline length (see Andrews et al., 2009), assuming the inclination ($i=6\arcdeg$) and major axis position angle (PA = 335°) derived from high spectral resolution CO emission line data by Hughes et al. (2011). The imaginary visibilities are effectively zero within the noise on all sampled baselines, confirming the accuracy of the astrometric alignment and reinforcing that there are no obvious departures from axisymmetry in the TW Hya disk. The visibility profile in Figure 1$b$ shows a smooth decrease out to $\sim$150 k$\lambda$, followed by low-amplitude oscillations on longer baselines and an apparent null near 500 k$\lambda$. These features are distinct in independent datasets (particularly the “dip” near 180 k$\lambda$, where 4 different observations span that range of baselines), are present regardless of the bin sizes used for profile averaging, and are not noted in the visibility profiles for the test calibrators (J1146-289 or 3C 279). Despite the challenges of calibrating SMA data for low-elevation targets, the persistence of these visibility modulations make us confident that they are real features. Moreover, we will demonstrate in §4 that they can be reproduced with models that incorporate a sharp outer edge in their emission profiles. The null at 500 k$\lambda$ is consistent with the 4 AU-radius inner disk cavity inferred from the SED (see Figure 1$c$; Calvet et al., 2002) and a VLA 7 mm continuum image (Hughes et al., 2007). The panels in Figure 1$d$ show the composite CO $J$=3$-$2 emission line channel maps for the TW Hya disk, resampled to a velocity resolution of 0.2 km s-1 with a circular 1$\farcs$0 (54 AU) synthesized beam. The emission is firmly detected ($>$3 $\sigma$) out to $\pm$1.2 km s-1 from the systemic velocity ($V_{\rm LSR}=2.86$ km s-1), with an integrated intensity of $34.8\pm 3.5$ Jy km s-1 and a peak flux of $4.2\pm 0.4$ Jy beam-1 ($43\pm 4$ K), including the calibration uncertainties. Those values are in good agreement with previous single-dish and SMA measurements (van Zadelhoff et al., 2001; Qi et al., 2004; Hughes et al., 2011). The channel maps in Figure 1$d$ show a clear rotation pattern, from northwest (blueshifted) to southeast (redshifted), with a narrow line-width due to a face-on viewing geometry. Near the systemic velocity, the CO emission subtends $\sim$4″ (215 AU) in radius. ## 4 Modeling Analysis These SMA observations offer some new insights into the TW Hya disk structure. Naturally, as one of the nearest pre-main sequence stars, TW Hya and its associated disk have been the subject of intense observational scrutiny. The global structure of the TW Hya disk has been investigated previously, using the SED (Calvet et al., 2002), optical/infrared scattered light observations (Krist et al., 2000; Trilling et al., 2001; Weinberger et al., 2002; Apai et al., 2004; Roberge et al., 2005), millimeter/radio-wave continuum images (Wilner et al., 2000, 2003, 2005; Hughes et al., 2007), and resolved molecular line maps (Qi et al., 2004, 2006, 2008; Hughes et al., 2008, 2011). However, none of those previous studies had the combination of angular resolution and sensitivity for the optically thin dust and high-quality gas tracers that are available from the SMA datasets presented here. In the following, a technique is described for extracting the structure of the TW Hya disk from these data using radiative transfer models. We focus specifically on enabling a comparison between the radial distributions of the dust and CO tracers. The approach we have adopted to make that comparison consists of three key steps. First, we construct a model of the radial density structure of the dust that is able to reproduce our resolved observations of 870 $\mu$m continuum emission and the broadband SED. Next, we make the assumption of a radially constant (vertically-integrated) dust-to-gas mass ratio and use the model structure we derive from the dust to predict the emission morphology of the CO $J$=3$-$2 line. Then, we show that this assumption implies an inconsistency with the observations, highlighting a clear difference in the radial distributions of millimeter-sized dust grains and CO gas in the TW Hya disk. Some of the potential implications of this inconsistency are discussed further in §5. ### 4.1 Dust Structure The dust disk structure is determined following the technique outlined by Andrews et al. (2011), with some modifications for generality. We assume the dust is spatially distributed with a parametric two-dimensional density structure in cylindrical-polar coordinates {$r$, $z$}, $\rho_{d}(r,z)=\frac{\Sigma_{d}}{\sqrt{2\pi}z_{d}}\exp{\left[-\frac{1}{2}\left(\frac{z}{z_{d}}\right)^{2}\right]},$ (1) where $\Sigma_{d}$ and $z_{d}$ are surface densities and characteric heights, which both vary radially (see below). As will be explained further in §4.3, we investigated two different models for the radial surface density profile. First, we employed the similarity solution for simple viscous accretion disk structures (Lynden-Bell & Pringle, 1974) that we have used successfully to characterize both normal and transition disks in the past (Andrews et al., 2009, 2010a, 2010b, 2011; Hughes et al., 2010). In that case, $\Sigma_{d}(r)=\Sigma_{c}\left(\frac{r}{r_{c}}\right)^{-\gamma}\exp{\left[-\left(\frac{r}{r_{c}}\right)^{2-\gamma}\right]},$ (2) where $\Sigma_{c}$ is a normalization, $r_{c}$ is a characteristic scaling radius, and $\gamma$ is a gradient parameter. As an alternative, we considered a less physically motivated (but perhaps more commonly used) model that incorporates a power-law density profile with a sharp cut-off (see Andrews et al., 2008), $\Sigma_{d}(r)=\Sigma_{0}\left(\frac{r}{r_{0}}\right)^{-p}\,\,\,\,\,\,\,({\rm if}\,\,\,r\leq r_{0};\,\,{\rm else}\,\,\,\Sigma_{d}=0),$ (3) where $\Sigma_{0}$ is a normalization, $r_{0}$ is the outer edge of the disk, and $p$ is a gradient parameter. In either case, the surface densities at small radii are modified to account for the TW Hya disk cavity (Calvet et al., 2002; Hughes et al., 2007). To simplify the inner disk model of Andrews et al. (2011), we set the surface densities to a constant value $\Sigma_{\rm in}$ between the sublimation radius ($r_{\rm sub}$) and a “gap” radius ($r_{\rm gap}$). No dust is present between that gap radius and the cavity edge, $r_{\rm cav}$. In the vertical dimension, the dust is distributed like a Gaussian with a variance $z_{d}^{2}$. The characteristic height varies with radius like $z_{d}=z_{0}(r/r_{0})^{1+\psi}$. Following Andrews et al. (2011), we employ a cavity “wall” to reproduce the infrared spectrum of TW Hya (no such feature was required at the sublimation radius). The local value of $z_{d}$ is scaled up to $z_{\rm wall}$ at $r_{\rm cav}$, and then exponentially joined to the global $z_{d}$ distribution over a small radial width, $\Delta r_{\rm wall}$. This structure model has 11 parameters: three describe the base surface density profile, {$\Sigma_{c}$, $r_{c}$, $\gamma$} or {$\Sigma_{0}$, $r_{0}$, $p$}, five determine the cavity and inner disk properties, {$\Sigma_{\rm in}$, $r_{\rm sub}$, $r_{\rm gap}$, $r_{\rm cav}$, $\Delta r_{\rm wall}$}, and three others characterize the vertical distribution of dust, {$z_{0}$, $z_{\rm wall}$, $\psi$}. To simplify the modeling, we fixed some of the parameters that are of less direct interest here. The sublimation radius was set to $r_{\rm sub}=0.05$ AU, the location where dust temperatures reach 1400 K (see also Eisner et al., 2006). The gap radius was set to $r_{\rm gap}=0.3$ AU and the (constant) inner disk density to $\Sigma_{\rm in}=5\times 10^{-4}$ g cm-2. The cavity edge was fixed at $r_{\rm cav}=4$ AU (see Hughes et al., 2007), the wall height was set to $z_{\rm wall}=0.25$ AU, and the wall width to $\Delta r_{\rm wall}=1$ AU. Since the details of this gap are not the focus, no attempt was made to reconcile the models with infrared interferometric data (but see Eisner et al., 2006; Ratzka et al., 2007; Akeson et al., 2011). After extensive experimentation with modeling the SED, we also fixed the scale height gradient to $\psi=0.25$. The interplay and degeneracies between these free parameters were discussed in detail by Andrews et al. (2011). For our purposes here, it is worth emphasizing that the parameters we have fixed have little quantitative impact on the derived radial structures (i.e., sizes and density gradients). We used the dust composition advocated by Pollack et al. (1994), consisting of a mixture of astronomical silicates, water ice, troilite, and organics with the abundances, optical properties, and sublimation temperatures discussed by D’Alessio et al. (2001). Based on the efforts of Uchida et al. (2004) to faithfully reproduce the details of the Spitzer IRS spectrum, we let 25% of the total silicate abundance inside the disk cavity ($r\leq r_{\rm cav}$) be composed of crystalline forsterite (using optical constants from Jäger et al., 2003). Two grain populations were employed, with a power-law size ($s$) distribution, $n(s)\propto s^{-3.5}$, between $s_{\rm min}=0.005$ $\mu$m and a given $s_{\rm max}$. Outside the cavity wall, 95% of the dust (by mass) has $s_{\rm max}=1$ mm and the remaining 5% has $s_{\rm max}=1$ $\mu$m. The dust in the wall itself and the tenuous inner disk was assumed to have $s_{\rm max}=1$ $\mu$m. No effort was made to distinguish the vertical distributions of these dust populations. Opacity spectra for each population were determined from Mie calculations, assuming segregated spherical grains. For these dust assumptions, the 870 $\mu$m dust opacity in the outer disk is $\kappa_{\rm mm}=3.4$ cm2 g-1. We assumed the central star has a K7 spectral type with $T_{\rm eff}=4110$ K, $R_{\ast}=1.04$ R⊙, and $M_{\ast}=0.8$ M⊙ ($\log{g}=4.3$), based on an effort to match Lejeune et al. (1997) spectral synthesis models to the broadband SED and the detailed optical/infrared spectral analysis work of Yang et al. (2005). The best-fit stellar spectrum template is overlaid on the SED in Figure 1$d$ as a light gray curve. Recently, Vacca & Sandell (2011) have argued instead for a M2.5 spectral type in the near-infrared, and a correspondingly cooler stellar photosphere (3400 K), larger radius (1.29 R⊙), and lower mass (0.4 M⊙). While that stellar model provides a good match to the broadband infrared photometry for TW Hya, it underpredicts the observed optical fluxes by a factor of $\sim$3 (in the $BVR$ bandpasses, and its known variability does not bridge that gap; see Mekkaden, 1998). We prefer the parameters for the warmer photosphere because they produce a template spectrum that better matches the SED across the complete set of optical and infrared bandpasses. For a given set of parameters, we simulated the stellar irradiation and emission output of a model dust structure using the two-dimensional, axisymmetric Monte Carlo radiative transfer code RADMC (see Dullemond & Dominik, 2004a). Assuming the fixed viewing geometry determined by Hughes et al. (2011), a raytracing algorithm was then used to compute a synthetic model SED and set of 870 $\mu$m continuum visibilities sampled at the same spatial frequencies observed with the SMA. For each surface density model, we found the best simultaneous fit to the observed SED and SMA visibilities over a coarse grid of the gradient parameter $\gamma$ or $p$, by varying the parameters {$\Sigma_{c}$ or $\Sigma_{0}$, $r_{c}$ or $r_{0}$, $z_{0}$}. Based on those results, we refined our search and permitted the gradients to vary freely to find the best-fit parameter sets for each model type (see §4.3 for results). ### 4.2 CO Gas Structure Unlike for the dust, the radial density profile of the gas disk cannot be inferred directly from models of the optically thick $J$=3$-$2 transition of CO. Therefore, we make a fundamental assumption that the gas traces the dust in the radial dimension. For any given $\Sigma_{d}$, we define the gas surface density profile as $\Sigma_{g}=\Sigma_{d}/\zeta$, where $\zeta$ is a (radially) constant dust-to-gas mass ratio. However, we have elected to permit some freedom in the vertical distribution of the gas to facilitate a more faithful reproduction of the CO channel maps. Using a multi-transition CO dataset, Qi et al. (2006) noted that models of the TW Hya disk structure had a difficult time reproducing an appropriate vertical temperature gradient of the gas. The intensity of the high-excitation $J$=6$-$5 line indicated that the gas in the disk atmosphere was significantly hotter than the dust, presumably due to substantial X-ray heating from the central star. To be able to reproduce the observed CO spectral images, we have characterized the vertical temperature profile of the gas in parametric form, based on the modeling analysis of Dartois et al. (2003). We assume that $T_{g}(r,z)=T_{a}+(T_{m}-T_{a})\cos{\left(\frac{\pi z}{2z_{q}}\right)}^{2\delta},$ (4) where $T_{a}=T_{1}(r/{\rm 1\,AU})^{-q}$ is a parametric radial temperature profile in the disk atmosphere, $T_{m}$ is the midplane temperature determined from the RADMC simulations of the dust, $\delta$ describes the shape of the vertical profile, and $z_{q}$ defines the height of the atmosphere layer such that $T_{g}(z\geq z_{q})=T_{a}$. In our modeling, we fix $\delta=2$ and $z_{q}=4H_{p}$, where $H_{p}$ is the pressure scale height assuming the midplane temperature at each radius ($H_{p}=c_{s}/\Omega$, the ratio of the midplane sound speed to the Keplerian angular velocity). In practice, Eq. (4) is a reasonable parametric approximation of the vertical temperature profile in an irradiated disk (e.g., D’Alessio et al., 1999). We have grounded the models by forcing $T_{g}=T_{d}$ at the midplane ($z=0$), but allowed the gas temperatures to increase faster with height than the dust to simulate any additional external heating sources. For a given $T_{g}(r,z)$ specified by {$T_{1}$, $q$}, we then calculate the vertical density structure of the gas by numerically integrating the equation of vertical hydrostatic equilibrium, $\frac{\partial\ln{\rho_{g}}}{\partial z}=-\left[\left(\frac{GM_{\ast}z}{(r^{2}+z^{2})^{3/2}}\right)\left(\frac{\mu m_{H}}{kT_{g}}\right)+\frac{\partial\ln{T_{g}}}{\partial z}\right]$ (5) using $\Sigma_{g}$ as a boundary condition, where $G$ is the gravitational constant, $\mu=2.37$ is the mean molecular weight of the gas, $m_{H}$ is the mass of a hydrogen atom, and $k$ is the Boltzmann constant. For reference, a vertical slice of a representative model structure is shown in Figure 2. Figure 2: Schematic demonstration of our model vertical structure, shown as a vertical slice at a fixed radius. (top) The density profile of the gas (red) and dust (blue) as a function of height above the midplane. The latter has a parametric Gaussian distribution with a variance $z_{d}^{2}$, while the former is computed assuming it is in hydrostatic pressure balance for its specified temperature structure. The relative normalizations of each are represented accurately, such that $\Sigma_{d}=\zeta\Sigma_{g}$, where $\zeta$ is a fixed dust-to-gas mass ratio. The hatched region near the midplane ($z\leq z_{m}$) marks where CO is depleted from the gas phase because it is frozen onto dust grain mantles (where $T_{g}\leq T_{\rm frz}$). The comparable surface CO depletion zone due to photodissociation ($z\geq z_{s}$) is well off the right of the plot. (bottom) The corresponding temperature profiles, where $T_{d}$ is computed from the RADMC radiative transfer calculations and $T_{g}$ is determined parametrically, as described in the text. The gas and dust temperatures are equivalent ($T_{g}=T_{d}=T_{m}$) in the midplane, but $T_{g}$ rises more rapidly than the dust before it saturates to a value $T_{a}$ at a height $z_{q}$. To quantify the density of CO molecules from any given gas model structure, we adopt the layered approach of Qi et al. (2008, 2011). Based on the detailed chemical calculations of Aikawa & Nomura (2006), we define two vertical boundaries {$z_{m}$, $z_{s}$} at any given radius such that the CO mass fraction (relative to H2) is $X_{\rm co}$ if $z_{m}(r)\leq z\leq z_{s}(r)$ and $10^{-4}X_{\rm co}$ elsewhere. The “midplane” boundary, $z_{m}$, marks the maximum height where CO molecules are expected to be frozen out of the gas phase and affixed to dust grain mantles. In practice, $z_{m}$ is defined as the minimum height where $T_{g}\geq T_{\rm frz}$, where the freeze-out temperature $T_{\rm frz}$ is a parameter considered to be constant with radius. The “surface” boundary, $z_{s}$, is meant to represent the height where CO molecules can be photodissociated by X-rays or cosmic rays. Following Qi et al. (2008), we define $z_{s}$ such that $N_{\rm pd}=\frac{1}{\mu m_{H}}\int_{\infty}^{z_{s}}\rho_{g}\,\,dz,$ (6) where $N_{\rm pd}$ is a vertically-integrated H2 column density that effectively represents the penetration depth of the photodissociating radiation field: $N_{\rm pd}$ is treated as a radially constant parameter. So, for any dust model a corresponding CO model can be characterized with a set of six additional parameters: two describe the abundances of dust and CO relative to the total gas mass, {$\zeta$, $X_{\rm co}$}, two others characterize the gas temperatures in the disk atmosphere, {$T_{1}$, $q$}, and the last two define the spatial distribution of CO in the gas phase, {$T_{\rm frz}$, $N_{\rm pd}$}. For a given set of these parameters, we generate a two- dimensional grid of $n_{\rm co}(r,z)$ and $T_{g}(r,z)$ values and define a velocity field based on Keplerian rotation around a point mass $M_{\ast}$, assuming a minimal turbulent velocity line width of 10 m s-1 based on the analysis of Hughes et al. (2011). We then feed that information into the radiative transfer modeling code LIME (Brinch & Hogerheijde, 2010) to solve the non-LTE molecular excitation conditions of the model and generate a synthetic high-resolution model cube for the CO $J$=3$-$2 transition. That model cube was then re-sampled at the velocity resolution of the data, and its Fourier transform was sampled at the same spatial frequencies observed by the SMA. In practice, we fixed the dust-to-gas ratio based on the assumed dust composition, where $\zeta=0.014$ (a gas-to-dust mass ratio of 71, see Pollack et al., 1994; D’Alessio et al., 2001). For each dust model, we varied {$T_{1}$, $q$} for a coarse grid of CO abundance layer parameters {$X_{\rm co}$, $T_{\rm frz}$, $N_{\rm pd}$} to find the best available match to the SMA spectral visibilities. Since we are using only a single CO transition in this investigation, the abundance layer parameters do not have a strong, independent effect on the synthetic CO visibilities. For simplicity, we adopt the best-fit models where $X_{\rm co}=2\times 10^{-6}$, $T_{\rm frz}=20$ K, and $N_{\rm pd}=10^{21}$ cm-2 as representative. Table 2: Estimated Model Parameters and Fit Results Model \| $\Sigma_{d}$(10) \| $\gamma$, $p$ \| $r_{c}$, $r_{0}$ \| $z_{d}$(10) \| $T_{a}$(10) \| $q$ \| $\chi_{\rm sed}^{2}$ \| $\chi_{\rm cont}^{2}$ \| $\chi_{\rm co}^{2}$ ---\|---\|---\|---\|---\|---\|---\|---\|---\|--- \| [g cm-2] \| \| [AU] \| [AU] \| [K] \| \| \| \| (1) \| (2) \| (3) \| (4) \| (5) \| (6) \| (7) \| (8) \| (9) \| (10) sA \| 0.29 \| 1.0 \| 35 \| 0.62 \| 104 \| 0.55 \| 201 \| 313,914 \| 319,974 sB \| 0.51 \| 0.5 \| 43 \| 0.58 \| 101 \| 0.54 \| 204 \| 304,303 \| 319,989 sC \| 0.28 \| 0.0 \| 45 \| 0.57 \| 98 \| 0.54 \| 210 \| 302,132 \| 320,013 sD \| 0.20 \| -0.5 \| 42 \| 0.53 \| 103 \| 0.53 \| 216 \| 304,329 \| 320,024 sE \| 0.14 \| -1.0 \| 40 \| 0.50 \| 102 \| 0.53 \| 231 \| 307,785 \| 320,040 pA \| 0.79 \| 1.5 \| 77 \| 0.62 \| 104 \| 0.54 \| 205 \| 309,370 \| 320,056 pB \| 0.46 \| 1.0 \| 67 \| 0.60 \| 99 \| 0.54 \| 211 \| 302,747 \| 320,047 pC \| 0.39 \| 0.75 \| 60 \| 0.58 \| 99 \| 0.54 \| 215 \| 301,785 \| 320,044 pD \| 0.31 \| 0.5 \| 58 \| 0.58 \| 100 \| 0.54 \| 227 \| 302,570 \| 320,035 pE \| 0.23 \| 0.0 \| 51 \| 0.55 \| 104 \| 0.54 \| 249 \| 306,980 \| 320,068 Note. — Col. (1): Model designation, where ‘s’ = similarity solution and ‘p’ = power-law with sharp edge as defined in Eq. 2 and 3, respectively. Col. (2): Dust surface density at $r=10$ AU. Col. (3): Surface density gradient. Col. (4): Characteristic scaling radius (‘s’ models) or outer edge radius (‘p’ models). Col. (5): Characteristic dust height at $r=10$ AU. Col. (6): Gas temperature in the disk atmosphere at $r=10$ AU (see Eq. 4). Col. (7): Radial gradient of the atmosphere gas temperature profile. Col. (8): $\chi^{2}$ statistic for the SED (including Spitzer IRS spectrum). Col. (9): $\chi^{2}$ statistic for the 870 $\mu$m visibilities. Col. (10): $\chi^{2}$ statistic for the CO $J$=3$-$2 spectral visibilities. The quoted parameter values and fit results are valid for a set of additional fixed parameters, including: the gradient of the dust height profile $\psi=0.25$, the sublimation radius $r_{\rm sub}=0.05$ AU, the inner radius of the gap $r_{\rm gap}=0.3$ AU, the cavity edge $r_{\rm cav}=4$ AU, the cavity wall height $z_{\rm wall}=0.25$ AU, the gas temperature profile parameters $\delta=2$ and $z_{q}=4H_{p}$, the dust-to-gas ratio $\zeta=0.014$, the CO/H2 abundance ratio $X_{\rm co}=2\times 10^{-6}$, the CO freezeout temperature $T_{\rm frz}=20$ K, and the CO photodissociation column $N_{\rm pd}=10^{21}$ cm-2. Furthermore, we assume a disk inclination of 6°, major axis position angle of 335°, and stellar mass of 0.8 M⊙. Figure 3: (top) Best-fit gas surface densities (where $\Sigma_{g}=\Sigma_{d}/\zeta$). Similarity solution models are shown on the left and power-law + sharp edge models are shown on the right. (bottom) Temperature profiles at the disk midplane ($T_{m}$) are shown as solid curves, and were determined from RADMC radiative transfer calculations. The parametric atmosphere temperatures ($T_{a}$; at a height $z_{q}$, see §4.2) are overlaid as dashed curves. All models have a gap at $r=4$ AU, marked with a dotted gray line. The CO freezeout temperature, $T_{\rm frz}=20$ K, is marked as a horizontal gray line. The temperature profiles shown here are similar for all models: the midplane values do not change much because the total irradiated dust mass is roughly the same in each case, and the atmosphere temperatures are determined from the same CO emission data. However, each model does have slight variation in the vertical temperature profile, which is not displayed here for the sake of clarity. Figure 4: A comparison of the model structures in Table 2 with observations of the dust continuum emission from the TW Hya disk, including the 870 $\mu$m visibility profile (top, with a zoomed-in view of the emission on longer baselines in the middle) and SED (bottom). The similarity solution surface density models (Eq. 2) are shown on the left, and the power-law models with sharp outer edges (Eq. 3) are shown on the right. The model SEDs are essentially indistinguishable, but there are clear differences in the 870 $\mu$m radial emission profiles. The overall best match to the continuum emission is Model pC, which has a density gradient $p=0.75$ and a sharp outer edge at $r_{0}=60$ AU (for the similarity solution models, the best match is Model sC). ### 4.3 Modeling Results The best-fit parameters for a range of representative surface density gradients of each model type are compiled in Table 2. For clear notation, each model is labeled with a letter designation corresponding to its $\gamma$ or $p$ value, preceded by a lower-case ‘s’ for the similarity solution models (based on Eq. 2) or ‘p’ for the power-law models (based on Eq. 3). The surface densities, characteristic dust heights, and gas atmosphere temperatures are listed for a radius of 10 AU, where the two surface density model types have comparable behavior. The four parameters describing the dust densities, {$\Sigma_{d}(10)$, $\gamma$ or $p$, $r_{c}$ or $r_{0}$, $z_{d}(10)$}, were determined from joint fits to the SED and 870 $\mu$m continuum visibilities. With those parameters fixed, the gas temperature parameters {$T_{a}(10)$, $q$} were estimated from the CO $J$=3$-$2 visibilities only. The individual $\chi^{2}$ values for the SED, continuum visibilities, and CO visibilities are listed in Cols. (8)-(10). There are 77 independent datapoints in the SED (including 45 equally-spaced points across the Spitzer IRS spectrum), 166,796 continuum visibilities, and 83,740 CO visibilities in each of 15 spectral channels (a total of 1,256,100). The $\chi^{2}$ values were calculated with weights that incorporate the quadrature sum of the formal uncertainties and absolute calibration uncertainties on each datapoint (e.g., a 10% systematic uncertainty on the amplitudes). Figure 3 shows the radial structures for each of the disk models in Table 2, including their surface densities (top) and temperature profiles (bottom). Figure 4 directly compares the 870 $\mu$m visibilities and SEDs for these model structures with the observations. All of the model structures provide excellent fits to the broadband SED and Spitzer IRS spectrum. Individual SED model behaviors are indistinguishable, aside from small deviations near the far-infrared turnover where the measurement uncertainties are largest. However, the different structures and model types exhibit distinctive signatures in their resolved 870 $\mu$m emission profiles. The similarity solution models with positive density gradients ($\gamma>0$; Models sA and sB) substantially over-predict the emission on $\sim$100-200 k$\lambda$ scales, while those with negative gradients ($\gamma<0$; Models sD and sE) have visibility nulls at $\sim$150 k$\lambda$ that are clearly not commensurate with the data. With an intermediate $\gamma=0$, Model sC provides the best match to the continuum data for this model type. However, it and all other similarity solution models fail to reproduce the visibility oscillations beyond 200 k$\lambda$. This “ringing” is a classic sign of a sharp edge in the radial emission profile, which is naturally produced with the power-law models described by Eq. (3). For those structures, steep density gradients ($p=1.0$-1.5, Models pA and pB) produce too much emission on 100-200 k$\lambda$ baselines and shallow gradients ($p=0.0$-0.5, Models pD and pE) generate visibility nulls that are not observed. However, an intermediate case with $p=0.75$ (Model pC) is an excellent match to the data, with only a small (albeit significant) departure on 280-380 k$\lambda$ scales. That mismatch is likely related to the shape of the outer edge, although we have not pursued that speculation further. Of all the dust structures explored here, Model pC is clearly favored. Figure 5: Moment maps of the CO $J$=3$-$2 emission from the TW Hya disk and the various disk structure models compiled in Table 2. The leftmost panels show the SMA observations. The top panels make a direct comparison with the similarity solution models, and the bottom panels do the same for the power- law models with sharp edges. In all plots, contours mark the velocity- integrated CO intensities (0th moment) at 0.4 Jy km s-1 ($\sim$3 $\sigma$) intervals and the color scale corresponds to the intensity-weighted line velocities (1st moment). Only Model sA provides a good match to the observed CO emission; all others predict gas distributions that are too small relative to observations. Figure 6: A direct comparison of the observed CO $J$=3$-$2 channel maps with synthetic data from the similarity solution models in Table 2. The top block of panels show the model spectral visibilities synthesized in the same way as the data, and the bottom block displays the imaged residual visibilities. All maps have the same contour intervals as in Figure 1$d$. The viewing geometry of the TW Hya disk is marked with a gray cross in each channel map. Figure 7: Same as Figure 6, but for the power-law models with sharp outer edges. A consistent feature of all the dust-based model structures is their compactness. The similarity solution models require characteristic radii of $r_{c}=35$-45 AU and the power-law models call for sharp outer edges at $r_{0}=51$-77 AU. That compact dust distribution was noted in §3, where we pointed out that all of the 870 $\mu$m dust continuum emission is concentrated inside a radius of roughly 1″ ($\sim$60 AU; see Figure 1$a$). Given the much larger radial extent of the CO $J$=3$-$2 emission – out to radii of at least 4″ (215 AU; see Figure 1$d$) – the best-fit models face major problems reconciling the CO and dust observations. The moment maps in Figure 5 confirm this tension between the dust-based structure models and CO data, highlighting a clear CO-dust size discrepancy in nearly all cases. More direct comparisons of the CO channel maps with each model structure can be made in Figures 6 and 7, which show both the models (top panels) and the imaged residual visibilities (bottom panels) synthesized in the same way as the data. Thanks to the freedom afforded by the parametric treatment of the gas temperatures, all models successfully reproduce the central CO emission core that is generated in the line wings. However, only Model sA has a sufficient amount of mass at large disk radii to account for the observations near the systemic velocity. Unfortunately, Model sA provides a poor match to the resolved 870 $\mu$m continuum emission profile. Before investigating potential physical explanations for the apparent CO-dust size discrepancy, it is important to evaluate the possibility that this is an artifact of some assumption in the modeling (aside from the underlying parametric models themselves; see §5). Admittedly, we adopted a relaxed approach for treating the relative vertical distributions of gas and dust in these models. Only two basic requirements were enforced for consistency: the gas and dust are in thermal equilibrium at the disk midplane, and the gas is distributed vertically according to hydrostatic pressure equilibrium. There are more sophisticated treatments of the gas in disk atmospheres that accurately account for the effects that X-ray heating, chemical differentiation, and gas-dust temperature departures have on the vertical structure of the disk (e.g., van Zadelhoff et al., 2001; Jonkheid et al., 2004; Kamp & Dullemond, 2004; Kamp et al., 2010; Woitke et al., 2009, 2010; Aresu et al., 2011). A nearby disk like TW Hya would benefit from a more detailed analysis with such models, particularly using multiple resolved molecular lines (see Thi et al., 2010, for a start). With a more physically motivated treatment of the vertical structure, we might infer a quantitatively different set of atmosphere properties that would affect the normalization and shape of the CO emission profile (currently controlled by {$T_{1}$, $q$} and other fixed parameters; see §4.2). However, the added complexity of those models would not change the fact that a compact density distribution is required to explain the dust emission. And since we assumed that the CO gas traces the dust in the radial direction, the size discrepancy would remain: these dust-based density profiles do not have enough material to be able to excite CO emission lines at large disk radii (see Figure 3). To summarize, the fundamental conclusion from the radiative transfer modeling analysis of the SMA data is that the spatial distributions of the CO and millimeter-sized dust in the TW Hya disk are different: the dust is systematically inferred to be more compact than the CO gas. ## 5 Discussion We used the SMA to observe the 870 $\mu$m continuum and CO $J$=3$-$2 line emission from the disk around the nearby young star TW Hya. These data represent the most sensitive high spatial resolution (down to scales of 16 AU) probes of the millimeter-wave dust and molecular gas content in any circumstellar disk to date. Along with the SED, these observations were used in concert with continuum and line radiative transfer calculations in an effort to extract the disk density structure. We were unable to identify a consistent model structure that simultaneously accounts for the observed radial distributions of CO and dust. Assuming a radially constant grain size distribution and (vertically integrated) gas-to-dust mass ratio, the millimeter-sized dust structure is significantly more compact than the CO. The resolved continuum emission profile demonstrates that the radial distribution of the millimeter-sized solids in the TW Hya disk has a relatively sharp outer edge near 60 AU, which is considerably smaller than the observed extent of the CO emission (out to at least 215 AU). The TW Hya disk structure has been studied extensively with millimeter-wave observations at lower angular resolution. In a series of investigations focused almost exclusively on spectral line emission, Qi et al. (2004, 2006, 2008) and Hughes et al. (2011) relied on a slightly modified version of the physical structure model that was developed by Calvet et al. (2002) to match the TW Hya SED. While that model has been used quite successfully to explain the molecular gas structure, there has always been some tension between observations and its predicted millimeter/radio continuum emission profile (see Qi et al., 2004; Wilner et al., 2003, 2005). Given the modest quality of the previous continuum data and the fact that this structure model was not designed (or fitted) with access to resolved observations of any kind, the disagreement was understandably dismissed. As might be expected from its assumption of a “steady” accretion disk structure (i.e., with a large, positive surface density gradient), the Calvet et al. (2002) model exhibits the same type of behavior as Models sA/B or pA/B: it agrees with the data on large scales ($<$100 k$\lambda$), but significantly over-predicts the amount of emission on smaller scales. The same is true for the parametric similarity solution models (where $\gamma\approx 1$) explored by Hughes et al. (2008, 2011), and would certainly apply to the analogous models developed by Thi et al. (2010) and Gorti et al. (2011). All of this previous modeling work admirably reproduces the extended molecular line emission that is observed, and in many cases the SED and millimeter-wave continuum emission on large angular scales as well. It is only with the sensitive, high angular resolution data presented here that we recognize a problem: the radial distribution of the millimeter-sized dust grains is much more compact than for the CO. Unlike the thermal radiation at millimeter wavelengths, the optical and near- infrared light that scatters off small ($\leq$1 $\mu$m) grains in the TW Hya disk surface is detected out to large distances from the central star – at least $\sim$4″, comparable to what is inferred from CO spectral images (Krist et al., 2000; Trilling et al., 2001; Weinberger et al., 2002; Apai et al., 2004; Roberge et al., 2005). So some dust traces the molecular gas, even if it is only a limited mass of small grains up in the disk atmosphere. However, these exquisitely detailed scattered light images exhibit subtle structural complexities. Krist et al. (2000) identified four distinct radial zones in the optical scattered light disk, with a prominent steepening of the brightness distribution just outside a radius of 50 AU ($\sim$1″; their Zone 1/2 boundary). Similar infrared behavior is noted in the studies by Weinberger et al. (2002) and Apai et al. (2004), which both suggested a break in the emission profile in the 50-80 AU ($\sim$1.0-1.5″) range. Those results were confirmed in a comprehensive analysis of new data by Roberge et al. (2005), who also called attention to a color change at a similar radius as well as an azimuthal asymmetry out to a slightly larger distance from the star ($\sim$135 AU). The physical origin of these scattered light features has been a mystery, although speculation centered around variations in the dust height (shadowing) and gradients in the dust scattering properties (either mineralogical or size- related). But in light of our discovery of an abrupt drop in the millimeter- wave continuum emission at the same location as these features, it is only natural to suspect that a more fundamental change occurs in the physical structure of the TW Hya dust disk near 60 AU. Perhaps the most straightforward explanation of the apparent CO-dust size discrepancy inferred from the SMA data is that we have used an incomplete description of the disk structure. As an example, consider a modification of either model type that incorporates an abrupt decrease in the surface densities (or millimeter-wave dust opacities) – not the dust-to-gas ratio – outside $r\approx 60$ AU. If that drop in $\Sigma$ (or $\kappa_{\rm mm}$) was not too large (perhaps a factor of $\sim$100), there would still be enough disk material to produce bright emission from the optically thick CO lines and scattered light while also accounting for the sharp edge feature noted in the optically thin 870 $\mu$m emission profile. That “substructure” in the outer dust disk might actually enhance the local gas-phase CO abundance, as ultraviolet radiation can penetrate deeper into the disk interior and photodesorb CO from the (small) reservoir of cold dust grains that remains at large radii (e.g., Hersant et al., 2009). Nevertheless, a physical origin for such a dramatic drop in the dust densities and/or the millimeter-wave dust opacities is not obvious. One possibility is that the disk has been perturbed by a long-period (as yet unseen) companion. If a faint object is embedded in the disk near the apparent edge of the 870 $\mu$m emission distribution, it might open a gap that splits the disk into two distinct reservoirs and generate the warp asymmetry suggested by Roberge et al. (2005). But, a narrow gap alone would not account for the SMA continuum observations. The millimeter-wave luminosity exterior to the gap would still need to be decreased, perhaps because the particles at those larger radii were preferentially unable to grow to millimeter sizes. Weinberger et al. (2002) quote deep limits on $H$-band point sources in the TW Hya disk that suggest there are no companions more massive than $\sim$6 MJup near 60 AU, according to the Baraffe et al. (2003) models (the corresponding mass limit is higher for the models of Marley et al., 2007). Certainly this kind of truncation or other forms of substructure could be invoked to explain the sharp radial edge in the SMA dust observations. But rather than engage in further speculation on the details, it should suffice to point out that the potential for substructure or other anomalies in the TW Hya disk can be tested with a substantial increase in resolution and sensitivity. Moreover, spectral imaging of optically thinner gas tracers (e.g., the CO isotopologues) would make for an ideal test of the origins of the apparent CO-dust size discrepancy. Fortunately, such observations will shortly be available as the Atacama Large Millimeter Array (ALMA) begins routine science operations. There is a compelling alternative explanation that has a more concrete physical motivation. In any protoplanetary disk, the thermal pressure of the gas is thought to cause it to orbit the star at slightly sub-Keplerian rates, generating a small velocity difference relative to the particles embedded in it (Weidenschilling, 1977). Depending on their size, particles can experience a head-wind from this gas drag that decays their orbits and sends them spiraling in toward the central star. This radial drift of particles modifies the radial dust-to-gas mass ratio profile and introduces a pronounced spatial gradient in the particle size distribution – in essence, it causes $\zeta$ and $\kappa_{\rm mm}$ to decrease with radius (Brauer et al., 2007, 2008; Birnstiel et al., 2009). In the outer disk, the drift rates are expected to be largest for millimeter-sized particles (e.g., Takeuchi & Lin, 2002). Since thermal emission peaks at a wavelength comparable to the particle size, the drift process should then naturally produce a millimeter-wave emission profile that is considerably more compact than would be inferred from tracers of the gas (Takeuchi & Lin, 2005). Moreover, the dust particles that reflect light in the disk atmosphere are small enough to be dynamically coupled to the gas – therefore, scattered light images should be extended like the probes of the gas phase. From a qualitative perspective, our analysis of the CO and dust structures in the TW Hya disk are certainly consistent with a scenario where growth and radial drift have had an observable impact. However, it is unclear if realistic models of the growth and migration of solids in a disk like this can quantitatively account for the details. One particular challenge worth highlighting is related to timescales. The Takeuchi & Lin (2005) models suggest that millimeter-wave dust emission should be strongly attenuated on a timescale of $\sim$1 Myr without constant replenishment (presumably from growth and/or fragmentation). Given the advanced age of TW Hya ($\sim$8-20 Myr), this model would require either that replenishment shuts off after several Myr or that drift is inefficient at early times before becoming more important later in the disk evolution process. If this is indeed the mechanism responsible for our results, additional observations of disks at a range of ages could be used to help calibrate models of the long-term evolution of drift rates. One potential way to test this hypothesis relies on the particle size dependence of the radial drift rates. The theory implies that larger particles will end up with more centrally concentrated density distributions. At long and optically thin wavelengths, we expect that the size of the continuum emission region should be anti-correlated with wavelength – longer wavelengths imply more compact emission. In principle, this could be tested in the near future by combining high resolution ALMA and Expanded Very Large Array (EVLA) observations of the TW Hya dust disk that span the millimeter/radio spectrum. In many ways, TW Hya and its disk are unique and may not be representative of the bulk population of pre-main sequence stars and their circumstellar material. Nevertheless, it is worth considering the broader implications of our findings for this specific example – they may prove to be more generally applicable. It is possible that the CO-dust size discrepancy found here is present in most other millimeter-wave disk observations, but it would likely be difficult to identify with current sensitivity and resolution limitations. If that feature is common and its underlying cause is a drop in the dust-to- gas ratio in the outer disk, there would be serious consequences for disk mass estimates based on dust continuum measurements. There could be large and hidden mass reservoirs of molecular gas in the outer reaches of protoplanetary disks, implying that disk masses might be substantially under-estimated. If true, gas densities at large disk radii may be higher than typically assumed, with profound implications for facilitating giant planet formation by gravitational instability (e.g., Boley, 2009; Kratter et al., 2010; Boss, 2011). However, if radial drift is the key process responsible for the apparent CO-dust size discrepancy, it is also possible that any “primordial” dust-to-gas ratio integrated over the entire disk is preserved. This would be the case if the millimeter-sized dust originally present at large radii had its inward migration halted before it was accreted onto the central star. In that scenario, the total disk mass estimates from millimeter-wave luminosities would still be relatively accurate (assuming a proper model for the dust opacities that considers the simultaneous particle size evolution is available), although the densities in the outer disk would remain uncertain without measurements of optically thin gas emission lines. Independent of the apparent CO-dust size discrepancy, our finding that the millimeter-wave continuum emission from the TW Hya disk is so sharply truncated comes as a surprise. Modern interferometric datasets generally do not have sufficient sensitivity to differentiate between dust density models with sharp edges or smooth tapers (e.g., see Isella et al., 2010; Guilloteau et al., 2011). Given that ambiguity, it is possible that the edge feature we have identified in the 870 $\mu$m emission profile of the TW Hya disk could be relatively common. Moreover, it is tempting to associate this $r\approx 60$ AU edge in the distribution of larger solid particles in the TW Hya disk with the similarly abrupt truncation of the classical Kuiper Belt at $r\approx 40$-50 AU in our solar system (Trujillo et al., 2001; Gladman et al., 2001). If this behavior ends up being a generic feature in protoplanetary disks, it may signify an important diagnostic of the radial migration of disk solids and provide new insights into the structural origins and evolution of the outer solar system (e.g., see Kenyon & Luu, 1999; Levison & Morbidelli, 2003). Further speculation on the generality of the features identified in the TW Hya disk is unnecessary. With the recent start of ALMA science operations, the quality of the data presented here will be matched (and exceeded) routinely for large samples, and the basic trends of disk properties like those probed here will be clarified. If the TW Hya disk is not anomalous, it is clear that the general methods used to interpret observations of dust in disks will need to be modified to focus less on their bulk density structures and more on the dynamical evolution of their solid contents. ## 6 Summary We have presented sensitive, high resolution ($0\farcs 3=16$ AU) SMA observations of the 870 $\mu$m continuum and CO $J$=3$-$2 line emission from the disk around the nearby young star TW Hya. Based on two different parametric formulations for the disk densities, we used radiative transfer calculations to compare the predicted radial structures of the dust and CO gas in the TW Hya disk with these SMA observations and ancillary measurements of the spectral energy distribution. The key conclusions from this modeling analysis of these high-quality data include: 1. 1. Under the assumption that the dust-to-gas surface density ratio is constant with radius, we were not able to find any model structure that can simultaneously reproduce the resolved brightness profiles of the 870 $\mu$m continuum and CO line emission. We have identified a clear CO-dust size discrepancy that is present regardless of whether the assumed surface density profile has a sharp outer edge or a smooth (exponential) taper at large radii. 2. 2. The radial distribution of millimeter-sized dust grains in the TW Hya disk is substantially more compact than its CO gas reservoir. The 870 $\mu$m dust emission has a sharp outer edge near 60 AU, while the CO emission (and optical/infrared scattered light from small grains that are dynamically coupled to the gas) extends to a radius of at least 215 AU. 3. 3. The observationally inferred CO-dust size discrepancy could potentially be explained with a more complex dust density profile that exhibits a sudden decrease by a large factor near a radius of 60 AU. That “break” in the density profile might be consistent with a tidal perturbation by a long-period (unseen) companion, although the dust at larger radii would then also have to be preferentially depleted of millimeter-sized grains. 4. 4. Alternatively, the observations might have uncovered some preliminary evidence for a key evolutionary mechanism related to the planet formation process: the growth and inward migration of disk solids. The radial drift of millimeter- sized particles is expected to naturally concentrate long-wavelength thermal emission near the star relative to tracers of the molecular gas reservoir. However, a more detailed exploration of disk evolution models is needed to verify if the observations of the TW Hya disk are quantitatively consistent with this scenario. We thank Mark Gurwell for his assistance with the multi-epoch calibration of the SMA data, Elise Furlan and the Spitzer IRS Disks team for providing a reduced TW Hya spectrum, and an anonymous referee for thoughtful suggestions that contributed significant value to the article. We are especially grateful to Christian Brinch for his generous help with the LIME code, Kees Dullemond for technical support with the RADMC package, and Paola D’Alessio for her valuable advice on dust populations and optical constants. The Submillimeter Array (SMA) is a joint project between the Smithsonian Astrophysical Observatory and the Academia Sinica Institute of Astronomy and Astrophysics and is funded by the Smithsonian Institution and the Academia Sinica. ## References * Aikawa & Nomura (2006) Aikawa, Y., & Nomura, H. 2006, ApJ, 642, 1152 * Akeson et al. (2011) Akeson, R. L., et al. 2011, ApJ, 728, 96 * Andrews et al. (2008) Andrews, S. M., Hughes, A. M., Wilner, D. J., & Qi, C. 2008, ApJ, 678, L133 * Andrews et al. (2009) Andrews, S. M., Wilner, D. J., Hughes, A. M., Qi, C., & Dullemond, C. P. 2009, ApJ, 700, 1502 * Andrews et al. (2010a) Andrews, S. M., Czekala, I., Wilner, D. J., Espaillat, C., Dullemond, C. P., & Hughes, A. M. 2010, ApJ, 710, 462 (2010a) * Andrews et al. (2010b) Andrews, S. M., Wilner, D. J., Hughes, A. M., Qi, C., & Dullemond, C. P. 2010, ApJ, 723, 1241 (2010b) * Andrews et al. (2011) Andrews, S. M., et al. 2011, ApJ, 732, 42 * Apai et al. (2004) Apai, D., et al. 2004, A&A, 415, 671 * Aresu et al. (2011) Aresu, G., Kamp, I., Meijerink, R., Woitke, P., Thi, W.-F., & Spaans, M. 2011, A&A, 526, 163 * Baraffe et al. (2003) Baraffe, I., Chabrier, G., Barman, T. S., Allard, F., & Hauschildt, P. H. 2003, A&A, 402, 701 * Birnstiel et al. (2009) Birnstiel, T., Dullemond, C. P., & Brauer, F. 2009, A&A, 503, L5 * Boley (2009) Boley, A. C. 2009, ApJ, 695, L53 * Boss (2011) Boss, A. P. 2011, ApJ, 731, 74 * Brauer et al. (2007) Brauer, F., Dullemond, C. P., Henning, T., Klahr, H., & Natta, A. 2007, A&A, 469, 1169 * Brauer et al. (2008) Brauer, F., Dullemond, C. P., & Henning, T. 2008, A&A, 859, 480 * Brinch & Hogerheijde (2010) Brinch, C., & Hogerheijde, M. R. 2010, A&A, 523, 25 * Calvet et al. (2002) Calvet, N., D’Alessio, P., Hartmann, L., Wilner, D. J., Walsh, A., & Sitko, M. 2002, 568, 1008 * Cutri et al. (2003) Cutri, R. M., et al. 2003, 2MASS All-Sky Point Source Catalog (Pasadena: IPAC) * D’Alessio et al. (1999) D’Alessio, P., Calvet, N., Hartmann, L., Lizano, S., & Cantó, J. 1999, ApJ, 527, 893 * D’Alessio et al. (2001) D’Alessio, P., Calvet, N., & Hartmann, L. 2001, ApJ, 553, 321 * Dartois et al. (2003) Dartois, E., Dutrey, A., & Guilloteau, S. 2003, A&A, 399, 773 * Di Francesco et al. (2008) Di Francesco, J., Johnstone, D., Kirk, H., MacKenzie, T., & Ledwosinska, E. 2008, ApJS, 175, 277 * Dullemond & Dominik (2004a) Dullemond, C. P., & Dominik, C. 2004, A&A, 417, 159 (2004a) * Dutrey et al. (2007) Dutrey, A., Guilloteau, S., & Ho, P. 2007, in Protostars & Planets V, eds. B. Reipurth, D. Jewitt, & K. Keil (Univ. Arizona Press: Tucson), 495 * Eisner et al. (2006) Eisner, J. A., Chiang, E. I., & Hillenbrand, L. A. 2006, ApJ, 637, L133 * Gladman et al. (2001) Gladman, B., et al. 2001, AJ, 122, 1051 * Gorti et al. (2011) Gorti, U., Hollenbach, D., Najita, J., & Pascucci, I. 2011, ApJ, 735, 90 * Guilloteau et al. (2011) Guilloteau, S., Dutrey, A., Piétu, V., & Boehler, Y. 2011, A&A, 529, 105 * Hartmann et al. (2005) Hartmann, L., et al. 2005, ApJ, 629, 881 * Hersant et al. (2009) Hersant, F., Wakelam, V., Dutrey, A., Guilloteau, S., & Herbst, E. 2009, A&A, 493, L49 * Ho et al. (2004) Ho, P. T. P., Moran, J. M., & Lo, K. Y. 2004, ApJ, 616, L1 * Hughes et al. (2007) Hughes, A. M., Wilner, D. J., Calvet, N., D’Alessio, P., Claussen, M. J., & Hogerheijde, M. R. 2007, ApJ, 664, 536 * Hughes et al. (2008) Hughes, A. M., Wilner, D. J., Qi, C., & Hogerheijde, M. R. 2008, ApJ, 678, 1119 * Hughes et al. (2010) Hughes, A. M., et al. 2010, AJ, 140, 887 * Hughes et al. (2011) Hughes, A. M., Wilner, D. J., Andrews, S. M., Qi, C., & Hogerheijde, M. R. 2010, ApJ, 727, 85 * Isella et al. (2007) Isella, A., Testi, L., Natta, A., Neri, R., Wilner, D., & Qi, C. 2007, A&A, 469, 213 * Isella et al. (2009) Isella, A., Carpenter, J. M., & Sargent, A. I. 2009, ApJ, 701, 260 * Isella et al. (2010) Isella, A., Carpenter, J. M., & Sargent, A. I. 2010, ApJ, 714, 1746 * Jäger et al. (2003) Jäger, C., Dorschner, J., Mutschke, H., Posch, T., & Henning, T. 2003, A&A, 408, 193 * Jonkheid et al. (2004) Jonkheid, B., Faas, F. G. A., van Zadelhoff, G.-J., & van Dishoeck, E. F. 2004, A&A, 428, 511 * Kamp & Dullemond (2004) Kamp, I., & Dullemond, C. P. 2004, ApJ, 615, 991 * Kamp et al. (2010) Kamp, I., Tilling, I., Woitke, P., Thi, W.-F., & Hogerheijde, M. R. 2010, A&A, 510, 18 * Kastner et al. (1997) Kastner, J. H., Zuckerman, B., Weintraub, D. A., & Forveille, T. 1997, Science, 277, 67 * Kenyon & Luu (1999) Kenyon, S. J., & Luu, J. X. 1999, ApJ, 526, 465 * Kratter et al. (2010) Kratter, K. M., Murray-Clay, R. A., & Youdin, A. N. 2010, ApJ, 710, 1375 * Krist et al. (2000) Krist, J. E., Stapelfeldt, K. R., Ménard, F., Padgett, D. L., & Burrows, C. J. 2000, ApJ, 538, 793 * Lejeune et al. (1997) Lejeune, T., Cuisinier, F., & Buser, R. 1997, A&AS, 125, 229 * Levison & Morbidelli (2003) Levison, H. F., & Morbidelli, A. 2003, Nature, 426, 419 * Low et al. (2005) Low, F. J., et al. 2005, ApJ, 631, 1170 * Lynden-Bell & Pringle (1974) Lynden-Bell, D., & Pringle, J. E. 1974, MNRAS, 168, 603 * Marley et al. (2007) Marley, M. S., Fortney, J. J., Hubickyj, O., Bodenheimer, P., & Lissauer, J. J. 2007, ApJ, 655, 541 * Mekkaden (1998) Mekkaden, M. V. 1998, A&A, 340, 135 * Panić et al. (2009) Panić, O., Hogerheijde, M. R., Wilner, D., & Qi, C. 2009, A&A, 501, 269 * Perryman et al. (1997) Perryman, M. A. C., et al. 1997, A&A, 323, L49 * Piétu et al. (2005) Piétu, V., Guilloteau, S., & Dutrey, A. 2005, A&A, 443, 945 * Pinte et al. (2008) Pinte, C., et al. 2008, A&A, 489, 633 * Pollack et al. (1994) Pollack, J. B., Hollenbach, D., Beckwith, S., Simonelli, D. P., Roush, T., & Fong, W. 1994, ApJ, 421, 615 * Qi et al. (2004) Qi, C., et al. 2004, ApJ, 616, L11 * Qi et al. (2006) Qi, C., et al. 2006, ApJ, 636, L157 * Qi et al. (2008) Qi, C., Wilner, D. J., Aikawa, Y., Blake, G. A., & Hogerheijde, M. R. 2008, ApJ, 681, 1396 * Qi et al. (2011) Qi, C., D’Alessio, P., Öberg, K. I., Wilner, D. J., Hughes, A. M., Andrews, S. M., & Ayala, S. 2011, ApJ, 740, 84 * Ratzka et al. (2007) Ratzka, T., Leinert, C., Henning, T., Bouwman, J., Dullemond, C. P., & Jaffe, W. 2007, A&A, 471, 173 * Roberge et al. (2005) Roberge, A., Weinberger, A. J., & Malumuth, E. M. 2005, ApJ, 622, 1171 * Rucinski & Krautter (1983) Rucinski, S. M., & Krautter, J. 1983, A&A, 121, 217 * Takeuchi & Lin (2002) Takeuchi, T., & Lin, D. N. C. 2002, ApJ, 581, 1344 * Takeuchi & Lin (2005) Takeuchi, T., & Lin, D. N. C. 2005, ApJ, 623, 482 * Thi et al. (2010) Thi, W.-F., et al. 2010, A&A, 518, L125 * Trilling et al. (2001) Trilling, D. E., Koerner, D. W., Barnes, J. W., Ftaclas, C., & Brown, R. H. 2001, ApJ, 522, L151 * Trujillo et al. (2001) Trujillo, C. A., Jewitt, D. C., & Luu, J. X. 2001, AJ, 122, 457 * Uchida et al. (2004) Uchida, K. I., et al. 2004, ApJS, 154, 439 * Vacca & Sandell (2011) Vacca, W. D., & Sandell, G. 2011, ApJ, 732, 8 * van Leeuwen (2007) van Leeuwen, F. 2007, A&A, 474, 653 * van Zadelhoff et al. (2001) van Zadelhoff, G.-J., van Dishoeck, E. F., Thi, W.-F., & Blake, G. A. 2001, A&A, 377, 566 * Weaver & Jones (1992) Weaver, W. B., & Jones, G. 1992, ApJS, 78, 239 * Webb et al. (1999) Webb, R. A., et al. 1999, ApJ, 512, L63 * Weidenschilling (1977) Weidenschilling, S. J., MNRAS, 180, 57 * Weinberger et al. (2002) Weinberger, A. J., et al. 2002, ApJ, 566, 409 * Weintraub et al. (1989) Weintraub, D. A., Sandell, G., & Duncan, W. D. 1989, ApJ, 340, L69 * Weintraub et al. (2000) Weintraub, D. A., Saumon, D., Kastner, J. H., & Forveille, T. 2000, ApJ, 530, 867 * Wichmann et al. (1998) Wichmann, R., Bastian, U., Krautter, J., Jankovics, I., & Rucinski, S. M. 1998, MNRAS, 301, L39 * Williams & Cieza (2011) Williams, J. P., & Cieza, L. A. 2011, ARA&A, 49, 67 * Wilner et al. (2000) Wilner, D. J., Ho, P. T. P., Kastner, J. H., & Rodríguez, L. F. 2000, ApJ, 534, L101 * Wilner et al. (2003) Wilner, D. J., Bourke, T. L., Wright, C. M., Jorgensen, J. K., van Dishoeck, E. F., & Wong, T. 2003, ApJ, 596, 597 * Wilner et al. (2005) Wilner, D. J., D’Alessio, P., Calvet, N., Claussen, M. J., & Hartmann, L. 2005, ApJ, 626, L109 * Woitke et al. (2009) Woitke, P., Kamp, I., & Thi, W.-F. 2009, A&A, 501, 383 * Woitke et al. (2010) Woitke, P., et al. 2010, MNRAS, 405, L26 * Yang et al. (2005) Yang, H., Johns-Krull, C. M., & Valenti, J. A. 2005, ApJ, 635, 466	arxiv-papers	2011-11-21T21:39:13	2024-09-04T02:49:24.555643	{ "license": "Public Domain", "authors": "Sean M. Andrews, David J. Wilner, A. M. Hughes, Chunhua Qi, Katherine\n A. Rosenfeld, Karin I. Oberg, T. Birnstiel, Catherine Espaillat, Lucas A.\n Cieza, Jonathan P. Williams, Shin-Yi Lin, and Paul T. P. Ho", "submitter": "Sean Andrews", "url": "https://arxiv.org/abs/1111.5037" }
1111.5042	# Evidence for a Universal Scaling of Length, Time and Energy in the Cuprate High Temperature Superconductors J.D. Rameau Z.-H. Pan H.-B. Yang G.D. Gu P.D. Johnson Brookhaven National Laboratory, Upton, NY 11973, USA ###### Abstract A microscopic scaling relation linking the normal and superconducting states of the cuprates in the presence of a pseudogap is presented using Angle Resolved Photoemission Spectroscopy. This scaling relation, complementary to the bulk universal scaling relation embodied by Homes’ law, explicitly connects the momentum dependent amplitude of the d-wave superconducting order parameter at T$\sim 0$ to quasiparticle scattering mechanisms operative at T$\gtrsim T_{c}$. The form of the scaling is proposed to be a consequence of the Marginal Fermi Liquid phenomenology and the inherently strong dissipation of the normal pseudogap state of the cuprates. ###### pacs: Not long after the discovery of high temperature superconductivity in the cuprates it was hypothesized that the transition temperature $T_{c}$ of these materials might be governed by the onset of phase coherence amongst “preformed” Cooper pairsEmery and Kivelson (1995); Emergy and Kivelson (1994). This scenario, essentially postulating a form of Bose condensation of such pairs, gives rise to a situation in which $T_{c}$ is lower than $T_{pair}$, the temperature at which the pairing amplitude of the superconducting order parameter develops. This point of view was bolstered early on by the observation of Uemura et al.Uemura _et al._ (1989) that underdoped cuprates obey a seemingly universal scaling law, $\rho_{s0}\propto T_{c}$, where $\rho_{s0}$ is the superfluid density, or phase stiffness, at $T=0$, implying that the mechanism for high $T_{c}$ superconductivity does indeed entail a Bose condensation of well defined, preformed pairs rather than the traditional BCS mechanism in which the pairing amplitude of the order parameter and global phase coherence arise simultaneously. Recently however the Uemura relation was shown to be accompanied by another universal scaling law, “Homes’ law”Homes _et al._ (2004, 2005), stating that $\rho_{s0}\propto\sigma_{DC}(T_{c})T_{c}$ where $\sigma_{DC}(T_{c})$ is the DC optical conductivity at $T\gtrsim T_{c}$. While Homes’ law is valid over a much wider swath of the cuprate phase diagram than the Uemura relation, having been shown to apply to optimally and overdoped materials as well as the underdoped variety and even the new Fe base high $T_{c}$ superconductorsHomes _et al._ (2010), a transparent picture of what it portends for the mechanism of high $T_{c}$ superconductivity in these materials has yet to emerge. In this Report it is shown that angle resolved photoemission spectroscopy (ARPES) provides evidence for a complementary scaling relation between the momentum dependent single particle scattering rates of carriers at $T\gtrsim T_{c}$, at the Fermi energy $E_{F}$ on the Fermi surface (FS) ’arcs’, and the magnitude of the superconducting gap at $T\sim 0$ K, respectively. This finding, deriving from an examination of Homes’ lawHomes _et al._ (2004, 2005), extends and clarifies the microscopic origins of that relationship, which was derived originally in the context of optical conductivity. As such, the present work represents a long sought after correlation between _microscopic_ spectral properties of the normal and superconducting states of high $T_{c}$ materials. Homes’ law has previously been interpreted as arising from a universal clean limit superconductivity ($\xi_{0}\ll\ell_{TC}$ where $\xi_{0}$ is Pippard’s coherence length at $T=0$ and $\ell_{TC}$ is the electronic mean free path at $T\sim T_{c}$), universal dirty limit superconductivity Homes _et al._ (2005) ($\xi_{0}\geq\ell_{TC}$), “hard core” boson scattering Lindner and Auerbach (2010) and as indicative of normal state cuprates obeying a quantum critical- like relation of the form $k_{B}T_{c}\approx\hbar/\tau_{TC}$ where $\tau_{TC}$ is the mean free time (here in the sense of transport) of a normal state electron at $T\sim T_{c}$Zaanen (2004)Abdel-Jawad _et al._ (2006). This “Planckian” dissipation, viewable as a limit of the Marginal Fermi Liquid (MFL) phenomenologyVarma _et al._ (1989), signifies that the observed electronic scattering is as rapid as is causally allowed. Separately, it has been suggested that Homes’ law implies $\ell_{TC}\approx 2\xi_{0}$Tallon _et al._ (2006). Altogether this implies the central issue in distinguishing various interpretations of Homes’ law rests upon understanding the ratio $R=\ell_{TC}/\xi_{0}$, or equivalently, $R=\Delta_{0}\tau_{TC}$, a quantity often used to quantify the strength of scattering in a superconductor relative to the robustness of its pairing. To accomplish this in an ARPES experiment we must take full account of the d-wave nature of the superconducting order parameter and generalize from the coherence length and mean free path measured in transport to momentum dependent quantities $\xi_{0}(\phi)$ and $\ell_{TC}(\phi)$, $\phi$ being the FS angle as measured from the node (defined in the inset of Fig.1e). While such a generalization must be treated with care, especially near the node where $\xi_{0}(\phi)$ diverges, the result is nonetheless phenomenologically simple. $\xi_{0}(\phi)$ can be measured in ARPES assuming a generalization of the coherence length, $\xi_{0}(\phi)=\frac{\hbar v_{F}^{0}(\phi)}{\pi\Delta_{0}(\phi)}$, where $v_{F}^{0}(\phi)$ and $\Delta_{0}(\phi)$ are the momentum dependent bare Fermi velocity at $T\sim T_{c}$ and the anisotropic superconducting gap at $T\sim 0$, respectively. Similarly $\ell_{TC}(\phi)=1/\Delta k_{TC}(\phi)$ is the momentum dependent mean free path measured at $T\sim T_{c}$ with $\Delta k_{TC}(\phi)$ being the Lorentzian full width at half maximum of the momentum distribution curve (MDC) at $E=E_{F}$Valla _et al._ (1999). Noting that $\hbar v_{F}^{0}(\phi)\Delta k(\phi)=\hbar/\tau_{TC}(\phi)=\Gamma_{TC}(\phi)=2\Im\Sigma_{TC}$, where $\Im\Sigma$ is the imaginary part of the electron self energy, we find that the quantity of interest from the point of view of ARPES is $R(\phi)=\pi\Delta_{0}(\phi)/\Gamma_{TC}(\phi)$ where we recall that the inverse quasiparticle (QP) lifetime $\Gamma_{TC}(\phi)$ is the full width at half maximum of the peak in the ARPES energy distribution curve (EDC) line shape. Expressing $R(\phi)$ in terms of energy rather than length scales via the MDC equation has the advantage of obviating the need to infer the bare Fermi velocity from measurement. It is evident from our definition of $R(\phi)$ that it can only have meaning when measured over the Fermi arc, understood to be the visible side of a nodal hole pocketYang _et al._ (2011) as it exists at $T_{c}$, because $\Gamma_{TC}(\phi)$ is formally undefined for $\phi>\phi_{c}$ (where $\phi_{c}$ is the FS angle of the arc tip) due to the presence of the pseudogap (PG) at $E_{F}$ in optimal, underdoped and some lightly overdoped materials. Taken altogether the program for examining the quantity $R(\phi)$ using ARPES is to measure the QP lifetime at $E_{F}$ from the node to the tip of the Fermi arc, $\phi_{c}$, in the normal state at $T\gtrsim T_{c}$ (which are the states probed in transport experiments) and to compare these to measurements of the momentum dependent superconducting gap at the same points in k-space, for the same samples, at low temperature. Figure 1: (Color online) Raw ARPES spectra of OP91 a) and UD70 b) at the tip of the Fermi arc, $\phi_{c}$, in the normal state just above $T_{c}$. Low temperature spectra ($T=15$ K, after deconvolution, are shown for the OP91 sample in c) and the UD70 sample in d). e) and f) show the $T_{c}$ and $T=15$ K symmetrized EDC’s for OP91 and UD70, respectively, as well as Lorentzian fits thereof. The energy scale of the low temparture EDC’s in panels e) and f) have been scaled up by a factor of $\pi$ to better illustrate Eq. 1 evaluated at $\phi=\phi_{c}$ and the intensity scaled to half that of the normal state EDC’s. The inset of panel e) illustrates the Fermi surface angle $\phi$ in relation to the normal state Fermi surface and the cuts represented by the current experiment (solid lines). Figure 2: (Color online) a) Superconducting gap versus Fermi surface angle at T=15 K for UD70 (black squares) and OP91 (red circles). The gap is measured as the distance of the coherence peak in the superconducting state to $E_{F}$. b) Inverse lifetime at $T\sim T_{c}$ versus Fermi surface angle for UD70 (black squares) and OP91 (red circles). Fits described in the text are shown as dotted lines extrapolated towards the antinodal regionPushp _et al._ (2009). All error bars are derived from the fits. The experiment described above was carried out at beamline U13UB of the NSLS with a Scienta-2002 electron spectrometer. The end station was equipped with an open flow Helium cryostat. Two samples, high quality single crystals of Bi2Sr2CaCu2O8+x (Bi2212) grown by the floating zone method, were used for measurements of $R(\phi)$: an optimally doped sample with $T_{c}$=91 K (OP91) and an underdoped sample with $T_{c}$=70 K (UD70). The lower $T_{c}$ of the UD70 sample was achieved by annealing in vacuum at 500 C for two days. Transition temperatures for both samples were ascertained prior to the ARPES measurement by SQUID magnetometry. The OP91 and UD70 samples were measured in their normal states at $T=95$ K and $T=70+$ K, respectively, and in their superconducting states at T=15 K. The overall resolution of the experiment was set to 12.5 meV in energy and $0.1^{o}$ in angle. The photon energies used were 18 eV (OP91) and 17.46 eV (UD70). All measurements were acquired within 48 hours of cleaving the samples at a chamber pressure of $1\times 10^{-10}$ Torr. Low temperature spectra used for acquiring $\Delta_{0}(\phi)$ were resolution corrected using the Lucy-Richardson methodRameau _et al._ (2010)Yang _et al._ (2008)Plumb _et al._ (2010) of deconvolution, yielding an effective energy resolution of 4 meV. Values for the superconducting gap were determined by the binding energy of the coherence peak. Raw normal state and deconvolved low temperature data for $\phi=\phi_{c}$ are shown in Fig. 1a)-d). Values of $\Delta_{0}(\phi)$ for UD70 and OP91 are presented in Fig. 2a along with pure d-wave ($\Delta(\phi)=\Delta_{0}^{AN}\sin(2\phi)$) fits, $\Delta_{0}^{AN}$ being determined by extrapolation from the nodal regionPushp _et al._ (2009). Normal state values for $\Gamma_{TC}(\phi)$ were acquired by fitting Lorentzians on a linear background to spectra symmetrized about $E_{F}$. Strictly speaking this procedure is only valid for states residing at $E_{F}$ and $k_{F}$ on the Fermi arc in the normal PG state of the copper oxides due to the presence of the previously observed particle-hole asymmetry in the nodal regionYang _et al._ (2008). This is, by design, where the measurement is carried out. It has similarly been affirmed that the superconducting state is particle-hole symmetric well below $T_{c}$Yang _et al._ (2008)Lee _et al._ (2007) so that symmetrization for our purposes is allowed at low temperature. The angular dependence of the inverse lifetimes, presented in Fig. 2b, were fit with the “offset” d-wave Abdel-Jawad _et al._ (2006) $\Gamma_{TC}(\phi)=\Gamma^{N}_{TC}+\delta\Gamma_{TC}\sin(2\phi)$ where $\delta\Gamma_{TC}=\Gamma_{TC}^{AN}-\Gamma_{TC}^{N}$, $\Gamma_{TC}^{N}$ and $\Gamma_{TC}^{AN}$ being the nodal and antinodal inverse lifetimes at $T_{c}$, respectively. $R(\phi)$, extracted from the data in Fig. 2, is plotted in Fig. 3 along with analytical fits. We emphasize that $R(\phi)$ is obtained using total scattering rate (at each point in k-space) rather than just the anistropic component, as is appropriate to a comparison with the DC optical conductivity used in obtaining Homes’ law. Figure 3: (Color online) a) Plot of $R(\phi)$ for OP91 (red circles) and UD70 Bi2212 (black squares). Error bars are formally propagated from those in Fig. 2. $R(\phi)$ obtained from the fits is superimposed on the data. Figure 4: (Color online) Summary of available ARPES data from the current study as well as Ref. Kondo _et al._ (2009) for $R(\psi)$. $\phi_{c}$ is taken in every case to simply be the last point to which a lifetime at the Fermi level could be reasonably ascertained. Error bars in the abscissa have been suppressed. To augment the present experimental results we reanalyzed data from a previous ARPES experiment performed on the single layer Bi2201 system under similar experimental conditionsKondo _et al._ (2009). In Fig. 4, which constitutes our main finding, the quantity $R(\psi)=\pi\Delta_{0}(\psi)/\Gamma_{TC}(\psi)$ versus the reduced Fermi surface angle $\psi\equiv\phi/\phi_{c}$ is plotted. Plotting $R$ in terms of $\psi$, which ranges between 0 at the node and 1 at the arc tip, rather than $\phi$, serves the purpose of collapsing data from samples with varying $T_{c}$ onto an equal footing. Remarkably, though the $T_{c}$’s of the materials thus investigated range between 25 K and 91 K, including single and bilayer systems, all materials are found to be very well approximated by a simple expression: $\Delta_{0}(\psi)\cong\frac{\psi\Gamma_{TC}(\psi)}{2\pi}.$ (1) While Eq. 1 might be taken as a purely phenomenological expression it can be related to the important length scales of the system, allowing our previous derivation, such that $\xi_{0}(\psi)\cong 2\psi^{-1}\ell_{TC}(\psi).$ (2) Eq. 2 must be treated carefully in order to avoid divergence at the node. Superconductors well described by the BCS theory have long been characterized as being in a “clean” or “dirty” limit based on comparisons of the type represented by Eqs. 1 and 2DeGennes (1999). What such distinctions mean for superconductors possessing anisotropic order parameters is far from clear. In the present case those terms should evidently be eschewed because while the ratio of $\xi_{0}$ to $\ell_{TC}$, for example, is clearly a useful metric for parameterizing BCS superconductors there is no evidence of, or prescription for, a universal relationship between these quantities for a generic system as there is, say, between $T_{c}$ and $\Delta_{0}$. The present findings thus constitute evidence of a fundamental physical process in the cuprate superconductors that is not an obvious consequence of the BCS theory. We postulate that a simple explanation for this behavior can be found by invoking the MFL phenomenology at $T_{c}$, $\Gamma_{TC}\propto T_{c}$, and the BCS superfluid, $\Delta_{0}\propto T_{c}$. If both of these properties hold across the Fermi arc then it is natural to conclude that $\Delta_{0}\propto\Gamma_{TC}$ will also hold across the fermi arc. Indeed, there is mounting evidence from ARPES and Scanning Tunneling Microscopy (STM) that $\Delta_{0}(\phi_{c})\propto T_{c}$ Pushp _et al._ (2009)Kurosawa _et al._ (2010)Lee _et al._ (2007) and it was shown long ago that the MFL phenomenology is maintained in the ARPES spectrum of Bi2212Valla _et al._ (1999)Valla _et al._ (2000). Transport studies have also repeatedly reported observations of a correlation between the anisotropic dissipation of the normal state and $T_{c}$ Abdel-Jawad _et al._ (2006)Zaanen (2004)Homes _et al._ (2005) in the cuprates. It has also been shown recently that spin fluctuations, a leading candidate for the pairing mechanism of the curpates, leads (at least in some cases) to a T-linear scattering rate in the normal pseudogap stateJin _et al._ (2011). Regardless, the physical content of Eq. 1 is to imply that the interaction responsible for the anomalous non-Fermi liquid normal state scattering rate is intimately related to the interaction that gives rise to the pairing strength observed as a single particle gap on the Fermi arcs below $T_{c}$. It is hard to escape this conclusion given the ultimate proportionality between the superconducting gap at low $T$ and the imaginary part of the self energy at $T_{c}$ reported here. The link between the pairing interaction and the electronic dissipation at $T_{c}$ has been remarked upon previously Basov and Chubukov (2011), and indeed is explicitly predicted within the MFL phenomenologyAbrahams and Varma (2000), though the existence of such a direct experimental relationship between the two, on the microscopic level, has not to our knowledge been previously reported. Eq. 2 offers a more intuitive, real space picture of what the maximal dissipation of the normal state as a function of $T$ implies for superconductivity. Evidently, even if carriers were to experience a strong pairing interaction well above $T_{c}$, true pairs could not arise on the Fermi arc because the constituents rescatter before that information can be coherently propagated to a mate. $T_{c}$ appears to occur when all the carriers on the Fermi arc have a pairing amplitude _and_ can propagate that information, implying the relevance of an “intra-pair” phase coherence to the magnitude of $T_{c}$. This length scale, $\xi_{0}$, set by the pairing strength plays a role fundamentally different than in BCS materials. If the scattering length of a single particle is too short relative to the size of the pairing potential it’s in, it won’t sense that potential. Such a dependence of the phase transition temperature $T_{c}$ purely on the relevant length and time scales of the system, rather than the details of the interaction, is the essence of quantum critical phenomenaSachdev (1999). Eq. 2 shows that in the cuprates, the new longer length scale is introduced a priori by the pairing interaction. Additionally, all carriers on the Fermi arc must be able to pair coherently before a gap can open, otherwise the symmetry of the d-wave order parameter would be violated. Finally, we note that states at the Fermi arc tip appear to play a unique role in the phenomenology of the cuprates, on par with the high symmetry points of the nodal and antinodal states. There, Eqs. 1 and 2 reduce to $\Delta_{0}(\phi_{c})=\Gamma_{TC}(\phi_{c})/2\pi$ and $\xi_{0}(\phi_{c})=2\ell_{TC}(\phi_{c})$, respectively. This scaling, in relation to Eq. 1, is illustrated graphically in Fig. 1e)-f). That PG states at higher momenta cannot satisfy this conditional relationship highlights a fundamental, if subtle, difference between the nodal and antinodal region of the Brillouin zone and suggests an inability of carriers tied up in the PG state above $T_{c}$ to ever condense into a true superfluid state. Our findings evidence a universal, microscopic scaling relation between two fundamental properties of the cuprate superconductors: the normal state lifetime of carriers on the Fermi surface and the superconducting gap that arises from those states well below $T_{c}$. This relationship represents a clear departure from BCS theory by itself, yet suggests several key concepts of the BCS superfluid survive the quantum critical nature of the cuprates’ anomalous normal state. The transition temperature is shown to be governed by a competition between length and time scales - pairing and single particle - both of which appear to be modulated by the same interaction. Lastly we note that the successful application of Homes’ law to the pnictide and chalcogenide superconductors raises the intriguing possibility of performing ARPES experiments similar to those presented here in those systems. ###### Acknowledgements. We would like to acknowledge illuminating conversations with C.C. Homes, A.M. Tsvelik, M. Khodas and T.M. Rice. We would also like to acknowledge that the original inspiration for this work arose from conversations with Myron Strongin. This work is supported by the US DOE under Contract No. DE- AC02-98CH10886 and by the Center for Emergent Superconductivity, an Energy Frontier Research Consortium supported by the Office of Basic Energy Science of the Department of Energy. ## References * Emery and Kivelson (1995) V. J. Emery and S. A. Kivelson, Phys. Rev. Lett. 74, 3253 (1995). * Emergy and Kivelson (1994) V. J. Emergy and S. A. Kivelson, Nature (London) 374, 434 (1994). * Uemura _et al._ (1989) Y. J. Uemura, G. M. Luke, B. J. Sternlieb, J. H. Brewer, J. F. Carolan, W. N. Hardy, R. Kadono, J. R. Kempton, R. F. Kiefl, S. R. Kreitzman, P. Mulhern, T. M. Riseman, D. L. Williams, B. X. Yang, S. Uchida, H. Takagi, J. Gopalakrishnan, A. W. Sleight, M. A. Subramanian, C. L. Chien, M. Z. Cieplak, G. Xiao, V. Y. Lee, B. W. Statt, C. E. Stronach, W. J. Kossler, and X. H. Yu, Phys. Rev. Lett. 62, 2317 (1989). * Homes _et al._ (2004) C. C. Homes, S. v. Dordevic, M. Strongin, D. A. Bonn, R. Liang, W. N. Hardy, S. Komiya, Y. Ando, G. Yu, N. Kaneko, X. Zhao, M. Greven, D. N. Basov, and T. Timusk, Nature (London) 430, 539 (2004). * Homes _et al._ (2005) C. C. Homes, S. V. Dordevic, T. Valla, and M. Strongin, Phys. Rev. B 72, 134517 (2005). * Homes _et al._ (2010) C. C. Homes, A. Akrap, J. S. Wen, Z. J. xu, Z. W. L. nd Q. Li, and G. D. Gu, Phys. Rev. B 81, 180508(R) (2010). * Lindner and Auerbach (2010) N. H. Lindner and A. Auerbach, Phys. Rev. B 81, 054512 (2010). * Zaanen (2004) J. Zaanen, Nature (London) 430, 512 (2004). * Abdel-Jawad _et al._ (2006) M. Abdel-Jawad, M. P. Kennett, L. Balicas, A. Carrington, A. P. Mackenzie, R. H. McKenzie, and N. E. Hussey, Nature Physics 2, 821 (2006). * Varma _et al._ (1989) C. M. Varma, P. B. Littlewood, and S. Schmitt-Rink, Phys. Rev. Lett. 63, 1996 (1989). * Tallon _et al._ (2006) J. L. Tallon, J. R. Cooper, S. H. Naqib, and J. W. Loram, Phys. Rev. B 73, 180504(R) (2006). * Valla _et al._ (1999) T. Valla, A. V. Fedorov, P. D. Johnson, B. O. Wells, S. L. Hulbert, Q. Li, G. D. Gu, and N. Koshizuka, Science 285, 2110 (1999). * Yang _et al._ (2011) H.-B. Yang, J. D. Rameau, Z.-H. Pan, G. D. Gu, P. D. Johnson, H. Claus, D. G. Hinks, and T. E. Kidd, Phys. Rev. Lett. 107, 047003 (2011). * Pushp _et al._ (2009) A. Pushp, , C. V. Parker, A. N. Pasupathy, K. K. Gomes, S. Ono, J. Wen, Z. Xu, G. Gu, and A. Yazdani, Science 324, 1689 (2009). * Rameau _et al._ (2010) J. D. Rameau, H.-B. Yang, and P. D. Johnson, J. Elec. Spec. Relat. Phenom. 181, 35 (2010). * Yang _et al._ (2008) H.-B. Yang, J. D. Rameau, P. D. Johnson, T. Valla, A. Tsvelik, and G. D. Gu, Nature (London) 456, 77 (2008). * Plumb _et al._ (2010) N. C. Plumb, T. J. Reber, J. D. Koralek, Z. Sun, J. F. Douglas, Y. Aiura, K. Oka, H. Eisaki, and D. S. Dessau, Phys. Rev. Lett. 105, 046402 (2010). * Lee _et al._ (2007) W. S. Lee, I. M. Vishik, K. Tanaka, D. H. Lu, T. Sasagawa, N. Nagaosa, T. P. Devereaux, Z. Hussain, and Z.-X. Shen, Nature 450, 81 (2007). * Kondo _et al._ (2009) T. Kondo, R. K. T. Takeuchi, J. Schmalian, and A. Kaminski, Nature (London) 457, 296 (2009). * DeGennes (1999) P. G. DeGennes, _Superconductivity of Metals and Alloys_ , 3rd ed. (Westview Press, 1999). * Kurosawa _et al._ (2010) T. Kurosawa, T. Yoneyama, Y. Takano, M. Hagiwara, R. Inoue, N. Hagiwara, K. Kurusu, K. Takeyama, N. Momono, M. Oda, and M. Ido, Phys. Rev. B 81, 094519 (2010). * Valla _et al._ (2000) T. Valla, A. V. Fedorov, P. D. Johnson, Q. Li, G. D. Gu, and N. Koshizuka, Phys. Rev. Lett. 85(4) (2000). * Jin _et al._ (2011) K. Jin, N. P. Butch, K. Kirshenbaum, J. Paglione, and R. L. Greene, Nature (London) 476, 73 (2011). * Basov and Chubukov (2011) D. N. Basov and A. V. Chubukov, Nature Physics 7, 272 (2011). * Abrahams and Varma (2000) E. Abrahams and C. M. Varma, Proc. Natl. Acad. Sci. USA 97, 5714 (2000). * Sachdev (1999) S. Sachdev, _Quantum Phase Transitions_ (Cambridge University Press, 1999).	arxiv-papers	2011-11-21T21:54:11	2024-09-04T02:49:24.568199	{ "license": "Public Domain", "authors": "J. D. Rameau, Z.-H. Pan, H.-B. Yang, G. D. Gu and P. D. Johnson", "submitter": "Jonathan Rameau", "url": "https://arxiv.org/abs/1111.5042" }
1111.5254	# Markov Chains application to the financial-economic time series prediction Vladimir Soloviev vnsoloviev@rambler.ru Cherkasy National University named after B. Khmelnitsky, Cherkassy, Ukraine Vladimir Saptsin saptsin@sat.poltava.ua Kremenchug National University named after M. Ostrogradskii, Kremenchuk, Ukraine Dmitry Chabanenko chdn6026@mail.ru Cherkasy National University named after B. Khmelnitsky, Ukraine ###### Abstract In this research the technology of complex Markov chains is applied to predict financial time series. The main distinction of complex or high-order Markov Chains and simple first-order ones is the existing of aftereffect or memory. The technology proposes prediction with the hierarchy of time discretization intervals and splicing procedure for the prediction results at the different frequency levels to the single prediction output time series. The hierarchy of time discretizations gives a possibility to use fractal properties of the given time series to make prediction on the different frequencies of the series. The prediction results for world’s stock market indices is presented. Prediction, time series, complex Markov chains, discrete time, fractal properties, discrete Fourier prediction. ###### Contents 1. 1 Introduction 2. 2 Analysis of prominent publications relevant to the subject 3. 3 Aims of the paper, problem statement 4. 4 Classical modeling problems of ESE systems dynamics 5. 5 Modern concepts in ESE systems modeling 6. 6 Markov chains prediction technology 7. 7 Prediction construction algorithm 8. 8 States in complex Markov chains and approaches for defining them 9. 9 Step-by-step prediction procedure. Defining the most probable state on the next step, prediction scenarios 10. 10 Time increments hierarchy and splicing procedure 11. 11 Results of stock indices prediction 12. 12 Conclusions and further work ## 1 Introduction Successful modeling and prediction of processes peculiar to complex systems, such as ecological, social, and economical (ESE) ones, remain one of the most relevant problems as applied to the whole complex of natural, human and social sciences (r001SamarskyMihaylov01 ; IvakhnenkoMGUA68 ; r002BogoboyashyKurbanov04 ; AndersenGluzmanSornette2000 ; PincakCurrencyStringTheory2011 ; PincakStringPrediction2011 ). The diversity of possible approaches to modeling such systems and, usually, more than modest success in the dynamics prediction, compel us to look for the reasons of failure, finding them not only in details, but also in the axiomatics, which relates to problem statement, chosen modeling methods, results interpretation, connections with other scientific directions. With the appearance of quantum mechanics and relativity theory in early twentieth century new philosophical ideas on physical values, measuring procedures and system state have been established, the ones that are completely different from Newtonian notions r003ElutinKrivchenko76 ; r004LandauQuantNerelyativistic . For more than 70 years basic concepts of classical and neoclassical economic theories have been discussed by leading scientists, generating new approaches r005SapirEconTheoryNeodnorSyst . The general systems theory has acquired recognition in the middle of the 20th century giving way to development of the new, systemic, emergent, and quantum in essence approach to investigation of complex objects, which postulates the limited nature of any kind of modeling and is based upon fixed and closed system of axioms r006BertalanfyGenSyst62 . However, the development of this new philosophical basis of ESE systems modeling is still accompanied with numerous difficulties, and new principles are often merely declared. Current research is devoted to investigation and application of the new modeling and prediction technology, suggested in r007KurbanovSaptsin07 ; r008SaptsinMarkovPaper09_ , based on concepts of determined chaos, complex Markov chains and hierarchic (in terms of time scale) organization of calculating procedures. ## 2 Analysis of prominent publications relevant to the subject Prediction of financial-economic time series is an extremely urgent task. Modern approaches to the problem can be characterized by the following directions: 1) approximation of a time series using an analytical function and extrapolation of the derived function towards future – so-called trend models r009LukashinAdapt03 ; 2) investigation of the possible influence various factors might have on the index, which is being predicted, as well as development of econometric or more complicated models using the Group Method of Data Handling (GMDH) IvakhnenkoMGUA68 ; r010ZaychenkoMonogr08 ; 3) modeling future prices as the decisions-making results using neuronal networks, genetic algorithms, fuzzy sets r010ZaychenkoMonogr08 ; r011EzhovShumsky98 ; r012Zaencev . Unfortunately, these techniques don’t produce stable forecasts, what can be explained by complexity of the investigated systems, constant changes in their structure. Although we are trying to join these directions in one algorithm, it is the latter option that we prefer, with it consisting in creating a model adequate to the process generating a price time series r013ChabM10_ . This very approach gives a chance to approach the complexity of the system, which generates the observed series, develop the model and use its properties as the prognosis. ## 3 Aims of the paper, problem statement Assume the time series is set by a sequence of discrete levels with constant step of time sampling $\Delta t$. We need to generate variants of the time series continuation (prognosis scenarios) according to the relations between the sequences of absolute and relative changes discovered with the help of complex Markov chains. ## 4 Classical modeling problems of ESE systems dynamics Another peculiar feature of ESE systems, apart from complexity, is a memory, including the long-term one, as well as nonlinear and unstable nature of interactions and components, which makes it harder to predict their future behavior. Unfortunately, mathematical models based on differential equations have no memory (there is no aftereffect), while for models with memory, where integral interrelations are used, it is not always possible to take into account nonlinearity (the integration procedure is linear by definition). In reality, in the Cauchy problem future systems behavior is defined by its initial state and doesn’t depend on the way the system reached its current state. However, it is hardly true that future behavior of a real socio- economic or socio-ecological system can be predicted by giving an immediate time “slice” of a variables set that describe its state. Let us consider possible ways to take into account past events while modeling ESE systems’ dynamics, which goes beyond the boundaries of classical differential and integral equations. Functional differential lagging equation can serve as a simple example of the dynamic model with memory, where present time is defined by the state variable $x(t)$ and depends on the past state $x(t-\tau)$ with constant time lag $\tau=const$: $x(t)=f\left(x(t-\tau)\right);t\geq t_{0},$ (1) where $f(x)$ is the known function, with initial conditions being set for the half-interval $t_{0}-\tau\leq t<t_{0}$ by the function $\phi(t)$: $x(t)=\phi(t);t_{0}-\tau\leq t<t_{0}.$ (2) Given the 2 equation 1 has the only solution, defined by recurrent ratios: $x(t)=\begin{cases}f\left(\phi(t-\tau)\right);\text{ if }t_{0}\leq t<t+\tau;\\\ f\left(f\left(\phi(t-\tau)\right)\right);\text{ if }t_{0}+\tau\leq t<t+2\tau;\\\ f\left(\left(f\left(\phi(t-\tau)\right)\right)\right);\text{ if }t_{0}+2\tau\leq t<t+3\tau;\\\ \cdots\end{cases}$ (3) Using Dirac delta function, as defined by ratios: $\delta(t)=0,\text{ if }x\neq 0;\int\limits_{-\infty}^{+\infty}{\delta(t)dt}=1,$ (4) we can formally rewrite equation 1 in the integral form: $x(t)=\int\limits_{-\infty}^{t}{dt_{1}f\left(x(t_{1})\right)H(t_{1},t)};H(t_{1},t)\equiv\delta\left(t_{1}-(t-\tau)\right);t\geq t_{0}.$ (5) Delta function is not a function in the conventional interpretation and is related to the class of generalized functions that were mathematically described only in the middle of the last century r014SobloevFunctProstr (physics started using this function much earlier). Its classical form is considered to be a limit of the “peak” sequence, with its centre set in the point of origin. The afore-mentioned “peaks” indefinitely converge widthway, indefinitely increase throughout the height and have a unit area. An approximate classic integral analogue of the equation 5 can be derived by substituting $\delta(t)$ with an ordinary function - some specific narrow enough “peak” of a unit area, a certain finite width $\backsim\Delta t$ as well as a finite height . The derivative of the Fermi function is one of the possible examples: $\Phi(t)=\frac{1}{1+exp\left(\frac{-t}{\theta}\right)};\delta(t)\approx\frac{1}{\theta\left(2+exp\left(\frac{-t}{\theta}\right)+exp\left(\frac{t}{\theta}\right)\right)}.$ (6) If the system’s state in the moment $t$, $x(t)$, is defined not by one, as in 1, but $k$ ($k=2,3,4,\cdots$) of her past states $x(t-\tau_{1}),x(t-\tau_{2}),\cdots x(t-\tau_{k})$ in the following moments of time $(t-\tau_{1}),(t-\tau_{2}),\cdots,(t-\tau_{k})$ respectively $(\tau_{1}=const,\tau_{2}=const,\cdots,\tau_{k}=const,\tau_{1}>\tau_{2}>\cdots>\tau_{k}>0)$, then instead of (1), (2), (5) we get: $x(t)=f\left(x(t-\tau_{1});x(t-\tau_{2});\cdots;x(t-\tau_{k})\right);t\geq t_{0};$ (7) $x(t)=\phi(t);t_{0}-\tau_{1}\leq t<t_{0};$ (8) $\begin{array}[]{c}x(t)=\int\limits_{-\infty}^{t}{dt_{1}}\int\limits_{-\infty}^{t}{dt_{2}}\cdots\int\limits_{-\infty}^{t}{dt_{k}f\left(x(t_{1}),x(t_{2}),\cdots,x(t_{k})\right)}.\\\ \delta\left((t_{1}-(t-\tau_{1})\right)\delta\left((t_{2}-(t-\tau_{2})\right)\cdots\delta\left((t_{k}-(t-\tau_{k})\right);t\geq t_{0}.\end{array}$ (9) Therefore if the system’s state in the time moment $t$ depends on the infinite sequence of its past states, the integral analogue of the functional differential lagging equation will, generally speaking, contain an integral of the infinite multiplicity. At the same time the infinite amount of past states can relate to both finite $(t-\tau_{1};t)$ (short-term memory) and infinite $(-\infty;t)$ (long-term memory) time span. Pay attention that the classic integral lagging equation is one of the Volterra type r015HeliFunctDiffury : $x(t)=\int\limits_{-\infty}^{t}{F\left(x(\tilde{t});t;\tilde{t}\right)d\tilde{t}},$ (10) where $F\left(x(\tilde{t});t;\tilde{t}\right)$ \- is an arbitrary (generally nonlinear) function of variables $x(\tilde{t});t;\tilde{t}$, which allows to take into account system’s memory of its past states only in the additive approximation, which becomes evident, if the right section 10 is rewritten in the following way: $\begin{array}[]{c}\int\limits_{-\infty}^{t}{F\left(x(\tilde{t});t;\tilde{t}\right)d\tilde{t}}\equiv\int\limits_{t_{1}}^{t}{F\left(x(\tilde{t});t;\tilde{t}\right)d\tilde{t}}+\int\limits_{t_{2}}^{t_{1}}{F\left(x(\tilde{t});t;\tilde{t}\right)d\tilde{t}}+\cdots=\\\ F\left(x(\tilde{t_{1}});t;\tilde{t_{1}}\right)\cdot(t-t_{1})+F\left(x(\tilde{t_{2}});t;\tilde{t_{2}}\right)\cdot(t_{1}-t_{2})+\cdots;\\\ t>t_{1}>t_{2}>\cdots;\tilde{t_{1}}\in\left[t_{1},t\right];\tilde{t_{2}}\in\left[t_{2},t_{1}\right];\cdots\end{array}$ (11) In connection with it note that the equation 9 in case of an additive dependency of contemporaneity on the past, i.e. in case: $f\left(x(t-\tau_{1});x(t-\tau_{2});\cdots\right)\equiv f_{1}\left(x(t-\tau_{1})\right);f_{2}\left(x(t-\tau_{2}))\right);\cdots$ (12) becomes a particular case of the equation 10 with the following integrand: $F\left(x(\tilde{t});t;\tilde{t}\right)\equiv f_{1}\left(x(\tilde{t})\right)\delta\left(\tilde{t}-(t-\tau_{1})\right)+f_{2}\left(x(\tilde{t})\right)\delta\left(\tilde{t}-(t-\tau_{2})\right)+\cdots$ (13) Meaningful analysis of nonlinear models dynamics with memory, in which the future is defined by the infinite amount of states in the past is generally possible only in case of a discrete representation. The results of such analysis will be approximated, i.e. will contain uncertainty, which has to be considered endogenous, i.e. internal, and peculiar to this very system. With a certain level of time sampling, models with memory both 7 and 10 becomes: $x(n+1)=f\left(x(n);x(n-1);x(n-2)\cdots\right).$ (14) To take into account and quantify the uncertainties, observed in ESE as well as other complex systems probability models are normally used. However their application is based on doubtful hypotheses, while the statistical interpretation of the results is not always informative enough and results might not correspond with the real process occurring within the system. In particular, the well-known problem of $1/f$–noise (look for example r016BukingemShumy ), closely connected to the presence of long-term memory in complex systems, implies the absence of the mean temporary value (as a limit of a certain time span converging to infinity, which serves as the basis for averaging) for any process occurring in such kind of system. Therefore such processes can’t have a rigorous statistical substantiation. ## 5 Modern concepts in ESE systems modeling New approaches to modeling and prediction of complex nonlinear systems dynamics with memory are based on the use of determined chaos and neural networks technologies (cf. e.g. r011EzhovShumsky98 ; r017LorenzNonlinear89 ; r018PetersChaosOrder ). Both investigation and realization of such techniques has become possible only with the appearance of quick-operating computers. Use of the recurrent computational process has become the general feature for all these technologies: $x_{n+1}=f_{n}\left(f_{n-1}\left(\cdots\left(f_{1}(x_{1})\cdots\right)\right)\right),n=1,2,\cdots,$ (15) where $f_{i}(x_{i})$ is a certain nonlinear mapping of a multi-dimensional vector $x_{i}$, $i$ \- discrete, real or fictitious, time. Identification of the model 15 is reduced to the determination of functions $f_{i}(x_{i})$, while the differences between the models of determined chaos and neural networks are connected with the function type and methods of its definition (neural network models normally use rather narrow class of $f_{i}(x_{i})$ mappings r012Zaencev ). Generally speaking, stability or convergence of the process 15 is not required, whereas a single-step set of vector $x_{i}$ components as well as their time dynamics can be of great interest. For the particular case of the model 15, introducing corresponding lagged variables, a model 14 can be transformed. Both determined stable processes, described by integro-differential equations, and random processes, which also include complex Markov chains (CMC), can be formally considered as separate extreme cases of determined chaos models realization 15. Given the sampling scale, which tends to zero, if such a tendency makes sense and corresponding limits exist, we derive classical differential and integral problem statement. Finite $\Delta t$ allows to get models with discrete time, which in the general case in the corresponding phase space (which also includes lagged variables) can produce both measurable sets (discrete or continuous) that allow probabilistic interpretation and those of the special structure – fractals r019FederFractals , that can’t be always interpreted in that way. Various digital generators of so-called random sequences used in imitational modeling can be an example of determined chaos models that allow probabilistic interpretation. Let us note that in reality there are no accurate procedures that would give an opportunity to distinguish a “real” random sequence from the pseudorandom one. ## 6 Markov chains prediction technology Suppose there is a sequence of a certain system discrete states. From this sequence we can determine transitions probabilities between the two states. Simple Markov chain is a random process, in which the next state probability depends solely on the previous state and is independent from the rest of them. Complex Markov chain, unlike the simple one, stands for the random process, in which the next state probability depends not only on the current, but also on the sequence of several previous states (history). The amount of states in history is the order of the Markov chain. Theory of simple Markov chains is widely presented in literature, for example r020TihonovMironovMarkovProcesses . As for the high order Markov chains, modern literature r021KornVMSpravochnik73 can offer us a mere definition. Developing complex or high order Markov chain’s properties is not widely presented in modern scientific publications. It’s necessary to mention the papers RafteryHighOrderMarkovChain ; RafteryTavare94 where properties of complex Markov Chains are developed, but no prediction algorithm is proposed there. The development of prediction method, based on complex Markov chains, is proposed in this paper. Markov chain of the higher order can be brought to a simple Markov chain by introducing the notion of a “generalized state” and including a series of consequent system’s states into it. In this case, tools of simple Markov chains can be applied to the complex ones. Investigated dynamic series is a result of a certain process. It is assumed that this process is determined, which implies the existence of a causal dependence of further states on history. It is impossible to fix and analyze the infinite history, which puts obstacles in the way of an accurate detection of this influence and making precise predictions. The problem consists in the maximal use of information, which is contained in the known segment of the time series, and subsequent modeling of the most probable future dynamics scenario. The observed process is described as a time series of prices $p_{t}$ with the given sampling time span $\Delta t$ $p_{ti}=p(t_{0}+i\Delta t).$ (16) Discrete presentation of the time series is in fact a way of existence of this very system. New prices are formed on the basis of contracts or deals, made on the market in certain discrete moments of time, while the price time series is a series of the averaged price levels during the chosen time intervals. While making a decision each trader, who is an active part of the pricing system, works solely with discrete series of the chosen time interval (e.g. minute, 5-minute, hourly, daily etc.). For $\Delta t\to 0$ the accuracy of data presentations reaches a certain limit, since for relatively small $\Delta t$ the price leaps in the moment of deal, while staying unchanged and equal to the last deal during the time between the two deals. Hence, the discreteness of time series has to be understood not only as a limited presentation of activity of the complex financial system, but also as one of the principles of its operation r007KurbanovSaptsin07 ; r008SaptsinMarkovPaper09_ ; r022SaptsinSoloviev_ ; r022SapSolArxiv_ ; r023SolDerbMonogr2010_ . The time series of initial conditions has to be turned into a sequence of discrete states. Let us denote the amount of chosen states as $s$, each of them being connected to the change in the quantity of the initial signal (returns). For example, consider the classification with two states, first of which corresponds to positive returns as the price increases, while the second one – to negative as it descends. Generally all possible increments of the initial time series are divided into $s$ groups. Ways of division will be discussed further. Next we develop predictions for the time series of sampled states. For the given order of the Markov chain and the last generalized state the most probable state is chosen to be the next one. In case if ambiguity occurs while the state of maximum probability is being evaluated, an algorithm is used that allows reducing the amount of possible prediction scenarios. Therefore we get the series of predicted states that can be turned into a sampled sequence of prognostic values. Evaluation of increments, prediction, and subsequent restoration are conducted for the given hierarchy of time increments $t$. To use the given information as effectively as possible, the prediction is conducted for time increments $t=1,2,4,8,...$, or a more complex hierarchy of increments and subsequent “splicing” of the results derived from different prediction samplings. The procedure of prediction and splicing is iterative and conducted starting from smaller increments, adding a prediction with the bigger time increment on every step. As the sampling time step $t$ increases, the statistics for the investigation of Markov chains decreases, whereas the biggest sampling step, which takes part in the prognostication, limits itself. To supplement the prediction with the low-frequency component the approximation of zero order is being used in the form of a linear trend or a combination of a linear trend and harmonic oscillations r024SapChFourierKharkov ; r025ChabS10_ . ## 7 Prediction construction algorithm Let us consider the consequence of operations, required for the prognostic time series construction. To do this we need to set the following parameters: 1) The type of time increments hierarchy (simple – powers of two, complex – product of powers of the first simple numbers). 2) Values of $s$ – the amount of states and $r$ – the order of the Markov chain. These parameters can be individual for every sampling level; finding of optimal parameters is done experimentally. 3) Threshold values $\delta$, and minimal number of transitions $N_{min}$. Prediction construction algorithm includes the following steps: 1) Generating hierarchy of time increments - $t$ sequence. The maximal of them has to correspond to the length of a prognostic interval $N_{max}$. 2) For every time increment $\Delta t$, as the increments increase, a prediction of states and restoration of the time series along the prognostic states is conducted. Current stage includes following actions: 2.1. Evaluating increments (returns) of the series with $\Delta t$ sampling. 2.2. Transforming the time series of increments into the series of state numbers ($1..s$). 2.3. Calculating transition probabilities for generalized states. 2.4. Constructing the series of prognostic states using the procedure of defining the most probable next state. 2.5. Restoring the value series from the state series with $\Delta t$ sampling. 2.6. Splicing the prediction of $\Delta t$ sampling with the time series derived from splicing of the previous layers (with the lesser step $\Delta t$). In case if the current time series is the first one, the unchanged time series will come as a result of splicing. 3) To splice the last spliced time series with the continuation of the linear trend, created along all previously known points. The time series, spliced with the linear trend, is the result of prediction. Let us consider the stages of the given algorithm in detail. ## 8 States in complex Markov chains and approaches for defining them In everything that concerns current technology, states are connected to the measuring of a prognostic value. There is a number of ways to classify returns in states, from which the following are suggested. One of them is the classification based on the homogeneity principle as concerning the amount of representatives in classes; based on the homogeneity principle of deviation, as well their combinations for different deviation modules. Increment or returns of the time series serves as the basis for states classification r025ChabS10_ ; r026solovievmatheconomics_ . Absolute $r_{a}$ and relative $r_{t}$ increments of the time series are considered: $r_{a}=p_{t}-p_{t-\Delta t},$ (17) $r_{t}=\frac{p_{t}-p_{t-\Delta t}}{p_{t}},$ (18) where $p_{t}$ – is the input time series of price dynamics, $\Delta t$ – sampling interval, which is chosen for subsequent analysis. It is known that mathematical expectation of the returns time series equals zero, whereas variation comes as the measure of time series volatility. Based on returns values $r_{t}$ classification and transformation of values to the time series of discrete states are conducted. One of the classification principles is homogeneity according to the amount of class representatives. This classification divides the set of all increments into $s$ groups equal in number. Calculated with the given sampling, time series increments are then systematized in growth and divided into equal parts. Thus we define limit values $\\{r_{lim,i}\\}$, which are used afterwards during transformation of the returns into class numbers. Large number of identical states can cause certain problems, such as identical bounds of several neighbouring states. It creates a number of states with no representatives, which makes correction of the division a necessary action. In that way we will reach the largest possible homogeneity in state division. Classification is conducted along the following algorithm r027SolSapChDrezden09 ; r028ChabRiga10 : $s_{t}=(i\mid r_{lim,i-1}>r_{t}>r_{lim,i})$ (19) where $s_{t}$ is the number of state, which corresponds to the moment of time $t$, for which the returns level was computed $r_{t}$; i is the number of state $[1\dots s]$, which is characterized by the interval $[r_{lim,i-1},r_{lim,i}]$ corresponding to the calculated returns level $r_{t}$. Apart from the returns interval, given by the aforementioned values $[r_{lim,i-1},r_{lim,i}]$, a mean returns value is chosen for every state $r_{avg,i}$, which will be used in time series values transformation according to the prognostic discrete states. Another way of dividing the time series into states implies dividing the interval of returns values into equal parts, from minimal to maximal deviation. In this case homogeneity according to the amount of representatives in states does not occur. In fact this method differs from the previous one in terms of defining limit values $\\{r_{lim,i}\\}$. Possible combined ways of division, in case of which the limit value, dependent on standard deviation, is used instead of maximal and minimal value, and division is conducted homogenously according to the deviation. Since the real causal dependence is unknown during the process, to find it adequate state classification, which would allow to reveal vital dependencies of the time series, is required. We suggest a couple of ways to divide the time series into states, which in the first place allow to preserve adequate transition probabilities between states, as well as prevent averaged deviations inside the states from affecting the accuracy of the derived prediction. To check the efficiency of division we conduct the sampling procedure and classify the increments according to each hierarchy. Having completed that, we restore the time series using known states for each hierarchy and finish the splicing procedure. Since the state series correspond to the initial time series, we get the curve, with deviation, caused exclusively by the state averaging mistake (quantum mistake). Thus, having set a certain value of state numbers $s$ and carried out sampling, restoring, and sampling procedures (excluding prediction), we get absolute sampling (quantum) mistake. Increasing the number of states, we improve the accuracy of restoration, however one should remember, that the choice of the quantum levels is limited by the fact that the transition probabilities definition with sufficient accuracy is required, which is confirmed by artificial test time series prediction experiments. ## 9 Step-by-step prediction procedure. Defining the most probable state on the next step, prediction scenarios Predicting procedure uses the most probable state as the next one under current circumstances. Probability matrix of state transitions is used for the afore-mentioned purpose. In this case, you have to take into account that probabilities are calculated with a certain mistake. We cannot precisely compute the probabilities, since it is impossible to derive an infinite time series, and only a part of the time series is known – the known part serves as the basis for probabilities. The second important aspect implies the case of several states with maximal probability. To prevent the omission of the states, for which the probabilities are computed with a mistake, one should add a state with maximal probability to the states, which are located in the distance of $\delta$ from the maximal one. The value of parameter $\delta$ depends on the probability evaluation mistake and requires experimental refinement. If $\delta>0$, the number of states with maximal probability increases in comparison to the value $\delta=0$. Let us call a couple of neighbouring states with maximal probability a cluster. Cluster states with average deviation values are supposed to have the largest probability. To predict the dynamics, let us confine ourselves to one or two most probable states. To define them a following algorithm is suggested: 1) If levels (discretized increments) create several clusters (cluster is a group of several neighbouring levels – cluster elements, minimal cluster is a single isolated level) with maximal probability, we choose the largest cluster. 2) If the number of cluster elements is odd, as $k_{max}$ we choose a central element. 3) If the number of cluster elements is even, we consider two central cluster elements and choose as $k_{max}$ the one, which is closer to the centre of distribution. 4) If two central cluster elements are equidistant to the centre of distribution, we consider both cases as possible variants of $k_{max}$ values (bifurcation point). 5) If there are several clusters of maximal size, we consider them as new elements, which can also form clusters that will undergo the same steps 1)-4). This principle is based on the following ideas: 1) If there are two neighbouring states of maximal probability, it is better to take the one, which is closer to the centre of distribution, in order to minimize the risk of occurrence of false linear trends in the prediction. 2) If levels of maximal probability are not the neighbouring ones, at least two variants have to be considered, as it can be connected to the bifurcations that should not be omitted. 3) If the prediction is carried out according to 1) (on all stages of the hierarchy), we receive a certain approximation of the lower limit of the prediction, whereas in case of 2) – we get an approximation of the upper limit. Hence this algorithm can adequately restore the case of possible bimodal probability distribution, it is proposed to consider 2 prediction scenarios. In case of the complex Markov chains, probability of the next state depends not only on the previous state, but also on the sequence of $r$ states, which have occurred before given. In this case, it is necessary to calculate transition probabilities from the sequence of $r$ states into the $r+1$ state. Formally, these probabilities can be written into the rectangular table of $(r^{s},s)$ size. Having generalized the notion of “present state” and included a sequence of $r$ preceding states into it, we can reduce Markov chains of $r$ order to the chain of the first order. Thus transition probabilities can be written into rectangular matrices of $(r^{s},r^{s})$, that come as transition probability matrices for generalized states. The process of prediction implies the following: the last state is chosen (in case of Markov chains of an order $r>1$ a sequence of $r$ latest states is taken). The probability of transition from current state to all possible states is defined. From all possible states a state with maximal probability is chosen. It is possible that several states with maximal probability occur, which can be explained by the bimodal probability distribution. The process of decision-making in this case is described later. The chosen most probable state is taken as the next prognostic state and the procedure is repeated for the next (last added) state. Thus we receive a time series of prognostic states for the given sampling time $\Delta t$. Further according to the received state sequence and known initial value the time series is being restored for the given time sampling $\Delta t$. In this case every state implies $\Delta t$ points of the time series. On the stage of state classification every state was connected to the average increment $r_{avg,i}$, which is added to the value of the last point in the time series, and the next discrete point is computed. Intermediate points are filled as linear interpolation of two known neighbouring points. Algorithm of $y_{t}$ time series values restoration according to the initial price $p_{t}$ and a series of average increments $r_{avg,ik}$, corresponding to the prognostic states $s_{k}$, can be given by a sequence of calculations: $\begin{array}[]{c}y_{t}=p_{t},\\\ y_{t+1}=y_{t}+r_{avg,i1}/\Delta t=p_{t}+r_{avg,i1}/\Delta t,\\\ y_{t+2}=y_{t+1}+r_{avg,i1}/\Delta t=p_{t}+2r_{avg,i1}/\Delta t,\\\ \dots\\\ y_{t+\Delta t-1}=y_{t+\Delta t-2}+r_{avg,i1}/\Delta t=p_{t}+(\Delta t-1)r_{avg,i}/t,\\\ y_{t+\Delta t}=y_{t+\Delta t-1}+r_{avg,i1}/\Delta t=p_{t}+\Delta tr_{avg,i}/\Delta t=p_{t}+r_{avg,i1},\\\ y_{t+\Delta t+1}=y_{t+\Delta t}+r_{avg,i2}/\Delta t=p_{t}+r_{avg,i1}+r_{avg,i2}/\Delta t,\\\ \dots\\\ y_{t+n\Delta t-1}=y_{t+n\Delta t-2}+r_{avg,in}/\Delta t=p_{t}+\sum_{k=1}^{n-1}r_{avg,ik}+\frac{\left(\Delta t-1\right)}{\Delta t}r_{avg,ik},\\\ y_{t+n\Delta t}=y_{t+n\Delta t-1}+r_{avg,in}/\Delta t=p_{t}+\sum_{k=1}^{n}r_{avg,ik}.\\\ \end{array}$ (20) ## 10 Time increments hierarchy and splicing procedure Time series increments will be computed with different steps. For example, analogous to the discrete Fourier transform, time increments are equal the powers of 2 are considered. First, we calculate increments as a remainder of two nearest neighbouring time series values, then next nearest values are considered with the step of 2, 4, 8, 16 etc. Let us mark this difference in time as $\Delta t$. For every $\Delta t$ we conduct an increment time series transformation leading to a time series of states. Further we predict the future sequence of states and restore the time series with the given sampling rate according to the prognostic series of states. Time series, received as a result of restoring for different $\Delta t$, undergo the splicing procedure, which gives out an actual prognostic time series. Thus an increment hierarchy is chosen, where each one is responsible for its own sampling rate, which serves as a basis for predicting, restoring and splicing. The splicing process implies the following. The procedure is iterative. With every next (along with the increasing step) sampling time the series corrects itself, driving the prediction, formed under lower $\Delta t$, to its actual point. Transformations that are conducted during splicing can be written down in the form of the following calculations. Suppose the splicing procedure has been finished for all time increments $\Delta t<Deltat_{i}$, the prediction has been done under the $\Delta t_{i}$ sampling according to formulae 20, and as a result a time series $y_{i}$ has been derived. Let us consider the iterative splicing procedure of the received series $y_{i}$ with the series, acquired during all preceding splicing procedures $g_{i}$. Since the series $y_{i}$ contains system points only in moments aliquot to $\Delta t_{i}$, and other points of the series are interpolated, the process of splicing implies the substitution of these interpolated points with the values of system points from previous $\Delta t<\Delta t_{i}$, which are contained in the series of results of previous splicing procedures $g_{i}$. Splicing algorithm can be written in the sequence of computations: $\begin{array}[]{c}z_{t}=g_{t}=p_{t},\\\ z_{t+1}=g_{t+1}+\left(y_{t+\Delta t_{i}}-g_{t+\Delta ti}\right)/\Delta t_{i},\\\ z_{t+2}=g_{t+2}+2\left(y_{t+\Delta t_{i}}-g_{t+\Delta ti}\right)/\Delta t_{i},\\\ \dots\\\ z_{t+\Delta t-1}=g_{t+\Delta t-1}+\left(\Delta t_{i}-1\right)\left(y_{t+\Delta t_{i}}-g_{t+\Delta ti}\right)/\Delta t_{i},\\\ z_{t+\Delta t}=g_{t+\Delta t}+\left(\Delta t_{i}\right)\left(y_{t+\Delta t_{i}}-g_{t+\Delta ti}\right)/\Delta t_{i}=y_{t+\Delta_{ti}},\\\ z_{t+\Delta t+1}=g_{t+\Delta t+1}+\left(\left(y_{t+2\Delta ti}-g_{t+2\Delta ti}\right)-\left(y_{t+\Delta ti}-g_{t+\Delta ti}\right)\right)/\Delta t_{i},\\\ z_{t+\Delta t+2}=g_{t+\Delta t+2}+2\left(\left(y_{t+2\Delta ti}-g_{t+2\Delta ti}\right)-\left(y_{t+\Delta ti}-g_{t+\Delta ti}\right)\right)/\Delta t_{i},\\\ \dots\\\ z_{t+n\Delta t-1}=g_{t+n\Delta t-1}+\frac{\left(\Delta t-1\right)}{\Delta t}\left(\left(y_{t-n\Delta t}-g_{t-n\Delta t}\right)-\left(y_{t-(n-1)\Delta t}-g_{t-(n-1)\Delta t}\right)\right),\\\ z_{t+n\Delta t}=g_{t+n\Delta t}+\left(\left(y_{t-n\Delta t}-g_{t+n\Delta t}\right)-\left(y_{t+(n-1)\Delta t}-g_{t+(n-1)\Delta t}\right)\right)=\\\ =g_{t+(n-1)\Delta t}-y_{t+(n-1)\Delta t}-y_{t-n\Delta t}.\end{array}$ (21) ## 11 Results of stock indices prediction In this section we offer the results of stock indices prediction. The stock’s indices databases are available from Finance_yahoo . Point 2000 indicates the starting moment of the prognosis: March 24, 2011. The green line on the next figures indicates real indice’s or price’s values. Our software for time series forecasting by the proposed methods is available from our website: http://kafek.at.ua/MarkovChains1_2_20100505.rar. Figure 1: Stock indices prediction. a) Dow Jones Industrial Average - DJI (USA). b) FTSE 100 (Great Britain) Figure 2: Financial companie’s share prices forecasting. a) Morgan Stanley (USA). b) BNP Paribas (France) Prediction time series with different input learning set’s length are shown at the fig.3 and 4. Prediction series for DJI at the figure 3 are more correlated, than FTSE index at the figure 4. At the subplot b) of the above mentioned plots the mean value and standard deviations of the prediction’s series are presented. The time of prediction series beginning on the next figures is the point 1000 and correspond to October 14, 2011. Figure 3: Dow Jones Industrial Average - DJI (USA). a) Prediction series, calculated with different learning set’s length. b) Mean value and standard deviation for prediction series. Figure 4: FTSE 100 index prediction. a) Prediction series, calculated with different learning set’s length. b) Mean value and standard deviation for prediction series. The normalization procedure is proposed in order to compare indices and it’s prediction series with different absolute values. The normalized values calculated with the following formula: $y_{n}(t)=\frac{y(t)-min\left(y(t)\right)}{max\left(y(t)\right)-min\left(y(t)\right)}.$ (22) Normalized prediction time series are shown at the fig.5 (America), fig.7 (Europe, developed countries), fig.7 (Europe, PIIGS), fig.9 (Asian markets). All the figures contain mean time series, which are weighted average of countrie’s stock indices predictions, weihted with GDP values EconomyWatch_com for the corresponding countries. Figure 5: Normalized mean values for the prediction series of America’s stock indices. Brazil (BVSP), Mexico (MXX), Canada (GSPTSE), Argentina (MERV), USA (S&P 500) Figure 6: Normalized mean values for the prediction series of European stock indices. Developed countries: FTSE (Great Britain), DAX (Germany) FCHI (France), Netherlands (AEX) Figure 7: Normalized mean values for the prediction series of European stock indices. Portugal (PSI20), Italy (FTSEMIB), Ireland (ISEQ), Greece (GD) and Spain (IBEX). Figure 8: Normalized mean values for the prediction series of Asian stock indices. China (SSEC, HSI), Korea (KS11), Japan (NIKKEI), India (BSESN), New Zealand (NZ50). Figure 9: Mean values of normalized World’s powerful economies indices prediction series. ## 12 Conclusions and further work Current paper suggests an algorithm of time series prediction based on complex Markov chains. Hierarchy of time increments principle allows to use the information, which is contained in the time series during the prognosis construction, to its fullest. Experimental work on stock market indices time series prediction shows the efficiency of the algorithm and confirms the relevance of further research of the offered method. ## References * [1] A. A. Samarskii and A. P. Mikhailov. Mathematical Modelling: Ideas. Methods. Examples. Fizmatlit, Moscow, 2001. * [2] V. V. Bohoboyaschyy, K. R. Kurbanov, P.B. Paly, and V.M. Shmandiy. Principles of Forecasting in Ecology: Textbook (in Ukrainian). Center navchalnoji literaturi, Kyiv, 2004. * [3] O. G. Ivakhnenko. Grouping method of data handling - the concurrent of stochastic approximation methods (in ukrainian) Automatika, 3(3):58–72, 1968. * [4] J. V. Andersen, S. Gluzman, and D. Sornette. Fundamental framework for “technical analysis” of market prices. European Physical Journal B, 14:579–601, March 2000. * [5] D. Horvath and R. Pincak. From the currency rate quotations onto strings and brane world scenarios. arXiv:1104.4716 , April 2011. * [6] M. Repasan and R. Pincak. The string prediction models as application to financial forex market. arXiv:1109.0435 , September 2011. * [7] P. V. Elyutin and V. D. Krivchenkov. Quantum Mechanics with tasks. Nauka, Moscow, 1976. * [8] L.D. Landau and E.M. Lifshitz. Quantum mechanics: non-relativistic theory. Teoreticheska fizika (Izd. 3-e) (Landau, L. D, 1908-1968). Butterworth-Heinemann, 1977. * [9] J. Sapir. K Ekonomitcheskoj teorii neodnorodnyh sistem - opyt issledovanija decentralizovannoj ekonomiki. GU VShE, Moskov, 2001. * [10] L. von Bertalanffy. General system theory—a critical review. “General Systems”, VII:1–20, 1962. * [11] K. R. Kurbanov and V. M. Saptsin. Markov chains as technology for social, economic and ecological processes forecasting. In “Problemy ta perspectivy rozvitku reg onalno rinkovo ekonom ki” Conference proceedings (in Russian), pages 10 – 14, Kremenchuk, May, 11-13 2007. * [12] V. M. Saptsin. Experience of using genetically complex Markov chains for the neural network technology forecasting. Visnyk Krivorizkogo ekonomichnogo institutu KNEU, 2 (18):56 – 66, 2009. * [13] Y. P. Lukashin. Adaptive Methods of Time Series Forecasting: Textbook. Finance and Statistics, Moscow, 2003. * [14] Y. P. Zaichenko. Fuzzy models and techniques in intelligent systems. Monograph(in Russian). Slovo, Kiev, 2008. * [15] A. A. Ezhov and S. A. Shumsky. Neurocomputing and its application in economics and business (Series “Textbooks” of Economic-Analytical Institute MEPI) / Ed. prof. V. V. Kharitonov. MEPI, Moscow, 1998. * [16] I. V. Zayencev. Neural networks: basic models. Textbook for the course “Neural networks” for 5-th grade students (in Russian) Physical Electronics Department, Faculty of Voronezh State University Voronezh State University, Voronezh. * [17] D. M. Chabanenko. Detection of short- and long-term memory and time series prediction methods of complex Markov chains. In Visnyk Natsionalnogo tehnichnogo universitetu “Kharkivsky politehnichny institut”. Zbirnik Naukovyh pratz. Tematichny vypusk: Informatika i modelyuvannya (in Ukrainian), number 31, pages 184 – 190. NTU KHPI, Kharkov, 2010. http://www.pim.net.ua/ARCH_F/V_pim_10.pdf * [18] S. L. Sobolev. Selected topics from the theory of functional spaces and generalized functions. Nauka, Moscow, 1989. * [19] J. Hale. Theory of Functional Differential Equations. Springer-Verlag, New York, Heidelberg, Berlin, 1977. * [20] M.J. Buckingham. Noise in electronic devices and systems. Ellis Horwood series in electrical and electronic engineering. E. Horwood, 1983. * [21] Hanz-Valter Lorenz. Nonlinear Dynamical Economics and Chaotic Motion. Springer-Verlag, 1989. * [22] E.E. Peters. Chaos and order in the capital markets: a new view of cycles, prices, and market volatility. Wiley finance editions. Wiley, 1996. * [23] J. Feder. Fractals. Physics of solids and liquids. Plenum Press, 1988. * [24] V. I. Tikhonov and V. A. Mironov. Markov Processes. Soviet Radio, Moscow, 1977. * [25] G.A. Korn and T.M. Korn. Mathematical handbook for scientists and engineers: definitions, theorems, and formulas for reference and review. Dover books on mathematics. Dover Publications, 2000. * [26] Andrian E. Raftery. A model for high-order markov chains. Journal of the Royal Statistical Society., 1985. * [27] Adrian Raftery and Simon Tavare. Estimation and modelling repeated patterns in high order markov chains with the mixture transition distribution model. Appl. Statist., 43(1):179–199, 1994. * [28] V. M. Saptsin and V. N. Soloviev. Relativistic Quantum Econophysics. New paradigms of Complex systems modeling: Monograph. http://kafek.at.ua/sol_sap_monogr.rar Brama-Ukraine, Cherkassy, 2009. * [29] V. Saptsin and V. Soloviev. Relativistic quantum econophysics - new paradigms in complex systems modelling. arXiv:0907.1142v1 [physics.soc-ph]. * [30] V. D. Derbentsev, A. A. Serdyuk, V. N. Soloviev, and O. D. Sharapov. Synergetical and econophysical methods for the modeling of dynamic and structural characteristics of economic systems. Monograph (in Ukrainian). Brama-Ukraine, Cherkassy, 2010. http://kafek.at.ua/Monogr.pdf * [31] V. M. Saptsin and D. N. Chabanenko. Fourier-based forecasting of low-frequency components of economical dynamic’s time series. In Problemy ekonomichnoyi kibernetiki: Tezy dopovidey XIV Vseukrayinskoyi Naukovo-praktichnoyi konferentsiyi.(in Ukrainian), page 132, Kharkiv, Oct 8-9, 2009 2009. KhNU imeni VN Karazina. * [32] D. M. Chabanenko. Discrete Fourier-based forecasting of time series. Sistemni tehnologii. Regionalny mizhvuzivsky zbirnik naukovyh pratz (in Ukrainian), 1(66):114 – 121, 2010. http://www.nbuv.gov.ua/portal/natural/syte/2010_1/15.pdf * [33] V. N. Soloviev. Mathematical economics. A textbook for self-study (in Ukrainian). CHNU, Cherkassy, 2008. http://kafek.at.ua/Posibnyk_Soloviev.rar * [34] Vladimir Soloviev, Vladimir Saptsin, and Dmitry Chabanenko. Prediction of financial time series with the technology of high-order markov chains. In Working Group on Physics of Socio-economic Systems (AGSOE), Drezden, 2009. http://www.dpg-verhandlungen.de/2009/dresden/agsoe.pdf * [35] V. Soloviev, V. Saptsin, and D. Chabanenko. Financial time series prediction with the technology of complex markov chains. Computer Modelling and New Technologies, 14(3):63–67, 2010. http://www.tsi.lv/RSR/vol14_3/14_3-7.pdf * [36] Yahoo! finance database. http://finance.yahoo.com * [37] Economic indicators : Gdp per capita (current prices, national currency) 2010. http://www.economywatch.com/economic-statistics/economic-indicators/GDP_Per_Capita_Current_Prices_National_Currency/2010/	arxiv-papers	2011-11-22T17:10:09	2024-09-04T02:49:24.578985	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Vladimir Soloviev and Vladimir Saptsin and Dmitry Chabanenko", "submitter": "Vladimir Saptsin", "url": "https://arxiv.org/abs/1111.5254" }
1111.5289	# HEISENBERG UNCERTAINTY PRINCIPLE AND ECONOMIC ANALOGUES OF BASIC PHYSICAL QUANTITIES Soloviev V., Prof. Dr. Sc vnsoloviev@rambler.ru Cherkasy National University named after B. Khmelnitsky, Cherkassy, Ukraine Saptsin V., PhD saptsin@sat.poltava.ua Kremenchug National University named after M. Ostrogradskii, Kremenchuk, Ukraine ###### Abstract From positions, attained by modern theoretical physics in understanding of the universe bases, the methodological and philosophical analysis of fundamental physical concepts and their formal and informal connections with the real economic measurings is carried out. Procedures for heterogeneous economic time determination, normalized economic coordinates and economic mass are offered, based on the analysis of time series, the concept of economic Plank’s constant has been proposed. The theory has been approved on the real economic dynamic’s time series, including stock indices, Forex and spot prices, the achieved results are open for discussion. quantum econophysics, uncertainty principle, economic dynamics time series, economic time. ###### Contents 1. 1 Introduction 2. 2 About nature and interrelations of basic physical notions 3. 3 Dynamical peculiarities of economic measurements, economical analog of Heisenberg’s uncertainty ratio 4. 4 Experimental results and their discussion 5. 5 Conclusions ## 1 Introduction The instability of global financial systems depending on ordinary and natural disturbances in modern markets and highly undesirable financial crises are the evidence of methodologial crisis in modelling, predicting and interpretation of current socio-economic conditions. In papers r001SolSapSimferopolThesis ; r002SaptsinSoloviev ; r003SapSolArxiv we have suggestsed a new paradigm of complex systems modelling based on the ideas of quantum as well as relativistic mechanics. It has been revealed that the use of quantum-mechanical analogies (such as the uncertainty principle, notion of the operator, and quantum measurement interpretation) can be applied to describing socio-economic processes. In papers r001SolSapSimferopolThesis ; r002SaptsinSoloviev ; r003SapSolArxiv we have suggestsed a new paradigm of complex systems modelling based on the ideas of quantum as well as relativistic mechanics. It has been revealed that the use of quantum- mechanical analogies (such as the uncertainty principle, notion of the operator, and quantum measurement interpretation) can be applied to describing socio-economic processes. It is worth noting that quantum analogies in economy need to be considered as the subject of new inter-disciplinary direction – quantum econophysics (e.g. r004BqaqueQuantumFin2004 ; r005MaslovQuantumEconomics06 ; r006GuevaraQuantumEconophysics06 ; r007 ; r008Goncalez2011 ), which, despite being relatively young, has already become a part of classical econophysics r009MantegnaStanley00 ; r010Romanovsky07 ; r011solovievmatheconomics ; r012SolDerbMonogr2010 . Ideas r001SolSapSimferopolThesis ; r002SaptsinSoloviev ; r003SapSolArxiv were anticipated and further developed in our works on modelling, predicting, and identification of socio-economic systems r013SaptsinGenComplMarkovChains08 ; r014SaptsinMarkovPaper09 ; r015SolSapChDrezden09 ; r016ChabM09 ; r017ChabM10 ; r018ChabRiga10 ; r019SolSapChPsepGlava ; r020KuznetzKoleb2011 (complex Markov chains), r019SolSapChPsepGlava ; r020KuznetzKoleb2011 ; r021Fourier2009 ; r022SapChFourierKharkov ; r023ChabS10 (discrete Fourier Transorm), r024SapDvufaznye05 ; r025OlhovayaKonkur2010 ; r026SapChabAGranUstoych11 ; r027SapSolBatyrNelnConcur2agents10 (multi-agent modelling), r028SolSapDynNetwMaths ; r029SolSapChabAPedZbornik (dynamic network mathematics), r030SaptsinSetPrir ; r031SaptsinMerezhPrirody (network measurements), r032SolSapShokotko ; r033HeisenbergRiga (uncertainty principle in economics) etc. However significant differences between physical and socio-economical phenomena, diversity and complexity of mathematical toolset (which, on account of historical circumstances, has been developed as the language of sciences), as well as lack of deep understanding of quantum ideology among the scientists, working at the joint of different fields require a special approach and attention while using quantum econophysical analogies. Aim of work. Our aim is to conduct detailed methodological and phylosophical analysis of fundamental physical notions and constants, such as time, space and spatial coordinates, mass, Planck’s constant, light velocity from the point of view of modern theoretical physics, and search of adequate and useful analogues in socio-economic phenomena and processes. ## 2 About nature and interrelations of basic physical notions Time, distance and mass are normally considered to be initial, main or basic physical notions, that are not strictly defined. It is thought that they can be matched with certain numerical values. In this case other physical values, e.g. speed, acceleration, pulse, force, energy, electrical charge, current etc. can be conveyed and defined with the help of the three above-listed ones via appropriate physical laws. Let us emphasize that none of the modern physical theories, including relativistic and quantum physics, can exist without basic notions. Nevertheless, we would like to draw attention to the following aspects. As Einstein has shown in his relativity theory, presence of heterogeneous masses leads to the distortion of 4-dimensional time-space in which our world exists. As a result Cartesian coordinates of the 4-dimensional Minkowski space $(x,y,z,ict)$, including three ordinary Cartesian coordinates $(x,y,z)$ and the forth formally introduced time-coordinate $ict$ ($i=\sqrt{-1}$ \- imaginary unit, $c$ \- speed of light in vacuum, $t$ \- time), become curvilinear r034landau1975classical . It is also possible to approach the interpretation of Einstein’s theory from other point of view, considering that the observed heterogeneous mass distribution is the consequence of really existing curvilinear coordinates $(x,y,z,ict)$. Then the existence of masses in our world becomes the consequence of geometrical factors (presence of time-space and its curvature) and can be described in geomatrical terms. If we step away from global macro-phenomena that are described by the general relativity theory, and move to micro-world, where laws of quantum physics operate, we come to the same conclusion about the priority of time-space coordinates in the definition of all other physical values, mass included. To demonstrate it, let us use the known Heisenberg’s uncertainty ratio which is the fundamental consequence of non-relativistic quantum mechanics axioms and appears to be (e.g. r002SaptsinSoloviev ): $\Delta x\cdot\Delta v\geq\frac{\hbar}{2m_{0}},$ (1) where $\Delta x$ and $\Delta v$ are mean square deviations of $x$ coordinate and velocity $v$ corresponding to the particle with (rest) mass $m_{0}$, $\hbar$ \- Planck’s constant. Considering values $\Delta x$ $\Delta v$ to be measurable when their product reaches its minimum, we derive (from (1)): $m_{0}=\frac{\hbar}{2\cdot\Delta x\cdot\Delta v},$ (2) i.e. mass of the particle is conveyed via uncertainties of its coordinate and velocity – time derivative of the same coordinate. Nowadays, scientists from different fields occupy themselves with the investigation of structure and other fundamental properties of spacetime from physical, methodological, psychological, philosophical and other points of view. However, theoretical physics r035CramerTransactional86 ; r036kaku1999introduction , including its most advanced and developing spheres (e.g. string theory r036kaku1999introduction ; r037BalasubramanianWhatWeDontKnow ) is expected to show the most significant progress in understanding the subject, though there is no single concept so far r035CramerTransactional86 ; r036kaku1999introduction ; r037BalasubramanianWhatWeDontKnow ; r038Vladimirov_p1 ; r039Vladimirov_p2 ; r040kaku2006parallel ; r041kaku2008physics . Within fundamental physical science we can mark out two investigational directions: 1) receipt of quantitive patterns, possible to verify experimentally or empirically and 2) interpretation of existing theories or development of new theories, that allow accurate and laconic (involving as little as possible mathematical notions and formalisms) interpretation of basic physical facts. The second direction is especially important when speaking of transferring physical notions and mathematical formalisms into other spheres, e.g. economics. Not claiming to be exhaustive, aiming to make the audience (professional economists included) as wide as possible, we will confine ourselves to the examination of the most typical and clear examples. According to the concept r038Vladimirov_p1 ; r039Vladimirov_p2 , having been developed for the last couple of decades by the Moscow school of theoretical physicists (headed by Y. Vladimirov), space, time, and four fundamental physical interactions (gravitational, electromagnetic, strong and weak) are secondary notions. They share common origins and are generated by the so- called world matrix which has special structure and peculiar symmetrical properties. Its elements are complex numbers which have double transitions in some abstract pre-space. At the same time, physical properties of spacetime in this very point are defined by the nonlocal (“immediate”) interaction of this point with its close and distant neighbourhood, and acquire statistical nature. In other words, according to Vladimirov’s concept, the observed space coordinates and time have statistical nature. It is worth noting that similar ideas as of interpreting quantum mechanics, different from those of the Copenhagen school were proclaimed by John Cramer r035CramerTransactional86 (Transactional interpretation of quantum mechanics). In our opinion the afore-metioned conception of nonlocal statistical origin of time and space coordinates can be qualitatively illustrated on the assuptions of quantum-mechanical uncertainty principle using known ratios (e.g. r002SaptsinSoloviev :) $\Delta p\cdot\Delta x\sim\hbar;$ (3) $\Delta E\cdot\Delta t\sim\hbar;$ (4) $\Delta p\cdot\Delta t\sim\frac{\hbar}{c}.$ (5) Interpreting values $\Delta E,\Delta p,\Delta x,\Delta t$ as uncertainties of particle’s energy $E$, its pulse $p$, coordinate $x$ and time localization $t$ (the latter ratio relates to the relativistic case $E=pc$, and is formally derived from the ratio (4), if $\Delta E=\Delta p\cdot c$, and takes into account maximum speed $c$ limitations in an explicit form), let us conduct the following reasoning. While $\Delta x\to 0$ uncertainty of pulse, and thus particle energy, uncertainty, formally becomes as big as possible, which can be provided only by its significant and nonlocal energetical interaction with the rest of the neighbourhood 3. On the other side, while $\Delta p\to 0$ the particle gets smeared along the whole space (according to (3) $\Delta x\to\infty$), i.e. becomes delocalized. It might be supposed that the fact of “delocalized” state of the particle takes place in any other, not necessarily marginal $\Delta x$ and $\Delta p$ value ratios. Similar results can be acquired while analyzing ratios (4)-(5), and for temporary localization $\Delta t$. Vladimirov’s concept probably becomes more graphic (at least for those, who are familiar with the basics of the band theory), if one remembers that so- called “electrones” and “holes” are considered to be really existing charge bearers in semiconductors. These “particles” have negative and positive charge respectively, accurate to the decimal place, which corresponds to the charge of a free electron, and are characterised by effective masses $m_{e}$ and $m_{h}$, different from the mass of a free electron (generally $m_{e}$ and $m_{h}$ can also be tensor values). However, in reality, these particles are virtual results of the whole semicondoctor crystal – so-called quasi-particles – and don’t exist beyond its bounds. Drawing the analogy with crystal it can be supposed that all structural formations of our Universes are such “quasi-particles”, caused by nonlocal interaction and non-existent beyond its spacetime bounds. In conclusion we would like to note that the conept of nonlocal interaction is quite capable of giving the logical explanation to the empirical fact of indistinguishability and identity of all microparticles of this kind, which always takes place during the observation (identification) regardless of spacetime localization of this very observation. ## 3 Dynamical peculiarities of economic measurements, economical analog of Heisenberg’s uncertainty ratio Main physical laws are normally distinguished with the presence of constants, that have been staying unchanged for the past $\sim 10^{11}$ years (the age of our Universe since so-called “big bang”- the most widespread hypothesis of its origin). Gravitational constant, speed of light in vaccuum, Planck’s constant are among the above-listed. Speaking of economic laws, based on the results of both physical (e.g. quantities of material resources) and economical (e.g. their value) dynamic measurements, the situation will appear to be somewhat different. Adequacy of the formalisms used for mathematical descriptions has to be constantly checked and corrected if necessary. The reason is that measurements always imply a comparison with something, considered to be a model, while there are no constant standards in economics (they change not only quantitavely, but also qualitatively – new standards and models appear). Thus, economic measurements are fundamentally relative, are local in time, space and other socio-economic coordinates, and can be carried out via consequent and/or parallel comparisons “here and now”, “here and there”, “yesterday and today”, “a year ago and now” etc. (see r030SaptsinSetPrir ; r031SaptsinMerezhPrirody for further information on the subject). Due to these reasons constant monitoring, analysis, and time series prediction (time series imply data derived from the dynamics of stock indices, exchange rates, spot prices and other socio-economic indicators) becomes relevant for evaluation of the state, tendencies, and perspectives of global, regional, and national economies. Let us proceed to the description of structural elements of our work and building of the model. Suppose there is a set of $M$ time series, each of $N$ samples, that correspond to the single distance $T$, with an equal minimal time step $\Delta t_{\min}$: $X_{i}(t_{n}),\begin{array}[]{c}{}\hfil\end{array}t_{n}=\Delta t_{\min}n;\begin{array}[]{c}{}\hfil\end{array}n=0,1,2,...N-1;\begin{array}[]{c}{}\hfil\end{array}i=1,2,...M.$ (6) To bring all series to the unified and non-dimentional representation, accurate to the additive constant, we normalize them, having taken a natural logarithm of each term of the series: $x_{i}(t_{n})=\ln X_{i}(t_{n}),\begin{array}[]{c}{}\hfil\end{array}t_{n}=\Delta t_{\min}n;\begin{array}[]{c}{}\hfil\end{array}n=0,1,2,...N-1;\begin{array}[]{c}{}\hfil\end{array}i=1,2,...M.$ (7) Let us consider that every new series $x_{i}(t_{n})$ is a one-dimensional trajectory of a certain fictitious or abstract particle numbered $i$, while its coordinate is registered after every time span $\Delta t_{\min}$, and evaluate mean square deviations of its coordinate and speed in some time window $\Delta T$: $\Delta T=\Delta N\cdot\Delta t_{\min}=\Delta N,\begin{array}[]{c}{}\hfil\end{array}1<<\Delta N<<N.$ (8) The “immediate” speed of $i$ particle at the moment $t_{n}$ is defined by the ratio: $v_{i}\left(t_{n}\right)=\frac{x_{i}(t_{n+1})-x_{i}(t_{n})}{\Delta t_{\min}}=\frac{1}{\Delta t_{\min}}\ln\frac{X_{i}(t_{n+1})}{X_{i}(t_{n})},$ (9) its variance $D_{v_{i}}$: $D_{v_{i}}=\frac{1}{(\Delta t_{\min})^{2}}\left(<\ln^{2}\frac{X_{i}(t_{n+1})}{X_{i}(t_{n})}>_{n,\Delta N}-\left(<\ln\frac{X_{i}(t_{n+1})}{X_{i}(t_{n})}>_{n,\Delta N}\right)^{2}\right),$ (10) and mean square deviation $\Delta v_{i}$: $\Delta v_{i}=\sqrt{D_{v_{i}}}=\frac{1}{(\Delta t_{\min})}\left(<\ln^{2}\frac{X_{i}(t_{n+1})}{X_{i}(t_{n})}>_{n,\Delta N}-\left(<\ln\frac{X_{i}(t_{n+1})}{X_{i}(t_{n})}>_{n,\Delta N}\right)^{2}\right)^{\frac{1}{2}},$ (11) where $<...>_{n,\Delta N}$ means averaging on the time window of $\Delta T=\Delta N\cdot\Delta t_{\min}$ length. Calculated according to (11) value of $\Delta v_{i}$ has to be ascribed to the time, corresponding with the middle of the avaraging interval $\Delta T$. To evaluate dispersion $D_{x_{i}}$ coordinates of the $i$ particle are used in an approximated ratio: $2D_{x_{i}}\approx D_{\Delta x_{i}},$ (12) where $D_{\Delta x_{i}}=\begin{array}[]{c}{}\hfil\end{array}<\left(x_{i}(t_{n+1})-x_{i}(t_{n})\right)^{2}>_{n,\Delta N}-\left(<x_{i}(t_{n+1})-x_{i}(t_{n})>_{n,\Delta N}\right)^{2}=$ $=\begin{array}[]{c}{}\hfil\end{array}<\ln^{2}\frac{X_{i}(t_{n+1})}{X_{i}(t_{n})}>_{n,\Delta N}-\left(<\ln\frac{X_{i}(t_{n+1})}{X_{i}(t_{n})}>_{n,\Delta N}\right)^{2},$ (13) which is derived from the supposition that $x$ coordinates neighbouring subject to the time of deviation from the average value $\bar{x}$ are weakly correlated: $<\left(x_{i}\left(t_{n}\right)-\bar{x}\right)\left(x_{i+1}\left(t_{n}\right)-\bar{x}\right)>_{n,\Delta N}\approx 0.$ (14) Thus, taking into account 12 and 13 we get: $\Delta x_{i}=\sqrt{\frac{D_{\Delta x_{i}}}{2}}=\frac{1}{\sqrt{2}}\left(<\ln^{2}\frac{X_{i}(t_{n+1})}{X_{i}(t_{n})}>_{n,\Delta N}-\left(<\ln\frac{X_{i}(t_{n+1})}{X_{i}(t_{n})}>_{n,\Delta N}\right)^{2}\right)^{\frac{1}{2}}.$ (15) Pay attention that it was not necessary for us to prove the connection 12, as it was possible to postulate statement (15) as the definition of $\Delta x_{i}$. It is also worth noting that the value $\left\|v_{i}\left(t_{n}\right)\right\|\cdot\Delta t_{\min}=\left\|\ln\frac{X_{i}(t_{n+1})}{X_{i}(t_{n})}\right\|,$ which, accurate to multiplier $\Delta t_{\min}$ coincides with $\left\|v_{i}\left(t_{n}\right)\right\|$ (see (9)), is commonly named absolute returns, while dispersion of a random value $\ln\left({X_{i}(t_{n+1})\mathord{\left/{\vphantom{X_{i}(t_{n+1})X_{i}(t_{n})}}\right.\kern-1.2pt}X_{i}(t_{n})}\right)$, which differs from $D_{v_{i}}$ by $(\Delta t_{\min})^{2}$ (see (13)) – volatility. The chaotic nature of real time series allows to $x_{i}(t_{n})$ as the trajectory of a certain abstract quantum particle (observed at $\Delta t_{\min}$ time spans). Analogous to (1) we can write an uncertainty ratio for this trajectory: $\Delta x_{i}\cdot\Delta v_{i}\sim\frac{h}{m_{i}},$ (16) or, taking into account (11) and (15): $\frac{1}{\Delta t_{\min}}\left(<\ln^{2}\frac{X_{i}(t_{n+1})}{X_{i}(t_{n})}>_{n,\Delta N}-\left(<\ln\frac{X_{i}(t_{n+1})}{X_{i}(t_{n})}>_{n,\Delta N}\right)^{2}\right)\sim\frac{h}{m_{i}},$ (17) where $m_{i}$ \- economic “mass” of an $i$ series, $h$ \- value which comes as an economic Planck’s constant. Having rewritten the ration 17: $\Delta t_{\min}\cdot\frac{m_{i}}{(\Delta t_{\min})^{2}}\left(<\ln^{2}\frac{X_{i}(t_{n+1})}{X_{i}(t_{n})}>_{n,\Delta N}-\left(<\ln\frac{X_{i}(t_{n+1})}{X_{i}(t_{n})}>_{n,\Delta N}\right)^{2}\right)\sim h$ (18) and interpreting the multiplier by $\Delta t_{\min}$ in the left part as the uncertainty of an “economical” energy (accurate to the constant multiplier), we get an economic analog of the ratio (4). Since the analogy with physical particle trajectory is merely formal, $h$ value, unlike the physical Planck’s constant $\hbar$, can, generally speaking, depend on the historical period of time, for which the series are taken, and the length of the averaging interval (e.g. economical processes are different in the time of crisis and recession), on the series number $i$ etc. Whether this analogy is correct or not depends on particular series’ roperties. Let us generalize the ratios (17), (18) for the case, when economic measurements on the time span $T$, used to derive the series (6), are conducted with the time step $\Delta t=k\cdot\Delta t_{\min}$, where $k\geq 1$ \- is a certain given integer positive number. From the formal point of view it would mean that all terms, apart from those numbered $n=0,k,2k,3k,...$. are discarded from the initial series (6). As a result the ratios would be the following: $\frac{1}{k\Delta t_{\min}}\left(<\ln^{2}\frac{X_{i}(t_{n+k})}{X_{i}(t_{n})}>_{n,\Delta N}-\left(<\ln\frac{X_{i}(t_{n+k})}{X_{i}(t_{n})}>_{n,\Delta N}\right)^{2}\right)\sim\frac{h}{m_{i}},$ (19) $k\Delta t_{\min}\cdot\frac{1}{(k\Delta t_{\min})^{2}}\left(<\ln^{2}\frac{X_{i}(t_{n+k})}{X_{i}(t_{n})}>_{n,\Delta N}-\left(<\ln\frac{X_{i}(t_{n+k})}{X_{i}(t_{n})}>_{n,\Delta N}\right)^{2}\right)\sim\frac{h}{m_{i}}$ (20) and would be dependent on $k$. Let us proceed to the analysis of the acquired results, that have to be considered as intermediate. In case of $h=const$, the formal analogy with the physical particle would be complete, and in this case, as appears from (19), variance of a random $i$-numbered value: $\ln\frac{X_{i}(t_{n+k})}{X_{i}(t_{n})}\approx\frac{X_{i}(t_{n+k})-X_{i}(t_{n})}{X_{i}(t_{n})}$ \- practically coinciding with the relative increment of terms of the $i$ initial series – would keep increasing in a linear way with $k\Delta t_{\min}$ (interval between the observations) growing. Such dynamics is peculiar to the series with statistically independent increments. However, both in cases of a real physical particle and its formal economic analogue any kind of change influences on the result. Therefore statistic properties of the “thinned” series, used to create the ratio (19), have to depend on real measurements in the intermediate points if there were any. Besides, presence of “long” and “heavy” “tails” increasing along the amplitude with decreasing $\Delta t$ on distributions of corresponding returns ${\Delta X\mathord{\left/{\vphantom{\Delta XX}}\right.\kern-1.2pt}X}$, are in our opinion the evidence of this thesis (see for example r031SaptsinMerezhPrirody ). Thus, generalizing everything said above, ${h\mathord{\left/{\vphantom{hm_{i}}}\right.\kern-1.2pt}m_{i}}$ratio on the right side of (19) (or (20)) has to be considered a certain unknown function of the series number $i$, size of the averaging window$\Delta N$, time $\bar{n}$ (centre of the averaging window), and time step of the observation (registration) $k$. To get at least an approximate, yet obvious, formula of this function and track the nature of dependencies, we postulate the following model presentation of the right side (19): $\frac{h}{m_{i}}\simeq\frac{\tau\left(\bar{n},\Delta N_{\tau}\right)\cdot H_{i}\left(k,\bar{n},\Delta N_{H}\right)}{\Delta t_{\min}\cdot m_{i}},$ (21) where $\frac{1}{m_{i}}=\begin{array}[]{c}{}\hfil\end{array}<\varphi_{i}\left(n,1\right)>_{(0\leq n\leq N-2)},$ (22) $m_{i}$ is a non-dimentional economic mass of an $i$-numbered series, $\tau\left(\bar{n}\right)=\frac{<\varphi_{i}\left(n,1,\Delta N_{\tau}\right)>_{(\bar{n}-{\Delta N_{\tau}\mathord{\left/{\vphantom{\Delta N_{\tau}2\begin{array}[]{c}{}\hfil\end{array}<n<\begin{array}[]{c}{}\hfil\end{array}\bar{n}+{\Delta N_{\tau}\mathord{\left/{\vphantom{\Delta N_{\tau}2}}\right.\kern-1.2pt}2}}}\right.\kern-1.2pt}2\begin{array}[]{c}{}\hfil\end{array}<n<\begin{array}[]{c}{}\hfil\end{array}\bar{n}+{\Delta N_{\tau}\mathord{\left/{\vphantom{\Delta N_{\tau}2}}\right.\kern-1.2pt}2}}),\begin{array}[]{c}{}\hfil\end{array}(1\leq i\leq M)}}{<\left(<\varphi_{i}\left(n,1,\Delta N_{\tau}\right)>_{(\bar{n}-{\Delta N_{\tau}\mathord{\left/{\vphantom{\Delta N_{\tau}2\begin{array}[]{c}{}\hfil\end{array}<n<\begin{array}[]{c}{}\hfil\end{array}\bar{n}+{\Delta N_{\tau}\mathord{\left/{\vphantom{\Delta N_{\tau}2}}\right.\kern-1.2pt}2}}}\right.\kern-1.2pt}2\begin{array}[]{c}{}\hfil\end{array}<n<\begin{array}[]{c}{}\hfil\end{array}\bar{n}+{\Delta N_{\tau}\mathord{\left/{\vphantom{\Delta N_{\tau}2}}\right.\kern-1.2pt}2}}),\begin{array}[]{c}{}\hfil\end{array}(1\leq i\leq M)}\right)>_{\bar{n}}}$ (23) \- local physical time compression ($\tau\left(\bar{n}\right)<1$) or magnification ($\tau\left(\bar{n}\right)>1$) ratio, which allows to introduce the notion of heterogenous economic time (for a homogenous $\tau\left(\bar{n}\right)=1$), $H_{i}\left(k,\bar{n}\right)=\frac{<\varphi_{i}\left(n,k,\Delta N_{H}\right)>_{\bar{n}-{\Delta N_{H}\mathord{\left/{\vphantom{\Delta N_{H}2\begin{array}[]{c}{}\hfil\end{array}<n<\begin{array}[]{c}{}\hfil\end{array}\bar{n}+{\Delta N_{H}\mathord{\left/{\vphantom{\Delta N_{H}2}}\right.\kern-1.2pt}2}}}\right.\kern-1.2pt}2\begin{array}[]{c}{}\hfil\end{array}<n<\begin{array}[]{c}{}\hfil\end{array}\bar{n}+{\Delta N_{H}\mathord{\left/{\vphantom{\Delta N_{H}2}}\right.\kern-1.2pt}2}}}}{<\varphi_{i}\left(n,1,\Delta N_{H}\right)>_{\bar{n}-{\Delta N_{H}\mathord{\left/{\vphantom{\Delta N_{H}2\begin{array}[]{c}{}\hfil\end{array}<n<\begin{array}[]{c}{}\hfil\end{array}\bar{n}+{\Delta N_{H}\mathord{\left/{\vphantom{\Delta N_{H}2}}\right.\kern-1.2pt}2}}}\right.\kern-1.2pt}2\begin{array}[]{c}{}\hfil\end{array}<n<\begin{array}[]{c}{}\hfil\end{array}\bar{n}+{\Delta N_{H}\mathord{\left/{\vphantom{\Delta N_{H}2}}\right.\kern-1.2pt}2}}}};\begin{array}[]{c}{}\hfil\end{array}k=1,2,...k_{\max}$ (24) \- non-dimentional coefficient of the order of unit, which indicates differences in the dependence of variance $D_{\Delta x_{i}}$ (see (13) taking into account the case of $k\geq 1$) on the law $D_{\Delta x_{i}}\sim k$ for the given $i$ and $\bar{n}$. $\varphi_{i}\left(n,k,\tilde{N}\right)=\frac{1}{k}\left(\ln^{2}\frac{X_{i}(t_{n+k})}{X_{i}(t_{n})}-\left(<\ln\frac{X_{i}(t_{n+k})}{X_{i}(t_{n})}>_{n,\tilde{N}}\right)^{2}\right)$ (25) (index $\tilde{N}=N,\Delta N_{\tau},\Delta N_{H}$ in the last formula indicates the averaging parameters according to $n$ and formulae (22),(23),(24), averaging windows $\Delta N_{\tau},\Delta N_{H}$ are chosen with thew following the conditions taken into consideration: $k_{\max}<\Delta N_{\tau}<\Delta N_{H}<N.$ (26) According to the definitions (23),(24) for coefficients $\tau\left(\bar{n}\right)$ and $H_{i}\left(k,\bar{n}\right)$ following conditions of the normalization take place: $<\tau\left(\bar{n}\right)>_{\bar{n},N}=1;\begin{array}[]{c}{}\hfil\end{array}H_{i}\left(1,\bar{n}\right)=1,$ (27) and the multiplier ${1\mathord{\left/{\vphantom{1\Delta t_{\min}}}\right.\kern-1.2pt}\Delta t_{\min}}$ on the right side (21) can be considered as an invariant component of an economic Planck’s constant $h$: $\bar{h}={1\mathord{\left/{\vphantom{1\Delta t_{\min}}}\right.\kern-1.2pt}\Delta t_{\min}},$ (28) As you can see, $\bar{h}$ has a natural dimension ¡time¿ to the negative first power. It is also worth noting that average economic mass of the whole set of series (or any separate group of the series) can be introduced with the help of the following formula: $\frac{1}{m}=\frac{1}{M}\sum_{i=1}^{M}\frac{1}{m_{i}}.$ (29) Acquired with the help of series (6) ratios (7,(19)-(28) also allow different interpretations. For example, it can be considered that normalized series (7) depict the trajectory of a certain hypothetical economic quantum quasi- particle in an abstract $M$-dimensional space of economic indices, and $m_{i}^{-1}$ are the main components of inverse mass tensor of the quasi- particle (the analogy with quasi-particles as free carriers of the electric charge in semiconductors), which has already been used in the previous chapter. In the final part of this chapter we would like to pay attention to the chosen variant of the theory, which is probably the simplest one, because of the following reasons. Carrying out various $n$ (discrete time) and $i$ (series number) averagings, we didn’t take into account at least two fairly important factors: 1) amounts of financial and material resources (their movement is reflected by each series) and 2) possible correlation between the series. However generalization of the theory and introduced notions is not so difficult in this case. It is enough to form a row $\left(\alpha_{1},\alpha_{2},...\alpha_{M}\right)$ of positive weight coefficients with the following condition of normalization: $\sum_{i=1}^{M}\alpha_{i}=M,$ (30) with each of them taking into account the importance of separate series in terms of a certain criterion, while for random values $\phi_{i}\left(n,k\right)=\sqrt{\frac{\alpha_{i}}{k}}\ln\frac{X_{i}(t_{n+k})}{X_{i}(t_{n})},\begin{array}[]{c}{}\hfil\end{array}i=1,2,...M$ (31) instead of a one-dimensional massive (25) we should introduce a covariance matrix: $\Psi=\left[\psi_{ij}\right],$ (32) where $\psi_{ij}=\psi_{ij}\left(k,\tilde{N}\right)=\begin{array}[]{c}{}\hfil\end{array}<\left(\phi_{i}\left(n,k\right)-\bar{\phi}_{i}\left(n,k\right)\right)\cdot\left(\phi_{j}\left(n,k\right)-\bar{\phi}_{j}\left(n,k\right)\right)>_{n,\tilde{N}},$ (33) $\bar{\phi}_{i}\left(j,k,\tilde{N}\right)=\begin{array}[]{c}{}\hfil\end{array}<\phi_{i}\left(n,k\right)>_{n,\tilde{N}}$ (34) (with $\alpha_{i}=1$ and absence of correlations $\psi_{ij}=\varphi_{i}\delta_{ij}$). Using a standard algorithm of characteristic constants $\lambda_{i},\begin{array}[]{c}{}\hfil\end{array}i=1,2,...M$ and corresponding orthonormal vectors $C_{i}=\left(c_{i1},c_{i2},...c_{iM}\right)$ search in $\Psi$ matrix, we proceed to the new basis, where “renormalized” series $y_{i}(t_{n})=\sum_{j=1}^{M}c_{ij}x_{j}\left(t_{n}\right)$ (new basis vectors) aren’t correlated any more. However the presence of zero characteristic constants or $\lambda_{i}$, which are distinguished with relatively low values in absolute magnitude, will mean that the real dimension of the set of series (7) is in fact less than $M$ (initial series (7) or their parts are strongly correlated). In this case renormalized series $y_{i}(t_{n})$ with zero or low characteristic constants have to be discarded. The remaining renormalized series will undergo all above-listed procedures. ## 4 Experimental results and their discussion To test the suggested ratios and definitions we have chosen 9 economic series with $\Delta t_{\min}$ in one day for the period from April 27, 1993 to March 31, 2010. The chosen series correspond to the following groups that differ in their origin: 1) stock market indices: USA (S&P500), Great Britain (FTSE 100) and Brazil (BVSP); 2) currency dollar cross-rates (chf, jpy, gbp); 3) commodity market (gold, silver, and oil prices). On Fig. 1-3 normalized plots of the corresponding series, divided by groups, are introduced, while $\Delta t_{\min}$is taken equal to the unit. As you can see from the Fig. 1-3, all time series include visually noticeable chaotic component and obviously differ from each other, which allows us to hope for the successful application the afore-mentioned theory to the interpretation and analysis of real series. Let us confine to its elementary variant. As an example on fig. 4 we suggest absolute values of immediate speeds (or absolute returns according to the general terminology used in literature), calculated with the help of the formula evaluation (9), and their variance (volatility), calculated with the help of the formula evaluation (13) for the series of Japanese yen (jpy) US dollar cross-rates. As we can see from the plots, the dependence of immediate speed or returns on time is of chaotic nature, while the dependence of volatility is smooth but not monotonous. For the rest of initial series, the dependencies of volatility and returns are similar to the depicted on the fig. 4 ones. Figure 1: USA (S&P500), Great Britain (FTSE 100), and Brazil (BVSP) daily stock indices from April 27, 1993 to March 31, 2010. Figure 2: Daily currency dollar cross-rates (chf, jpy, gbp) Figure 3: Commodity market. Daily gold, silver, and oil prices. Figure 4: Absolute values of immediate speeds (abs returns) and their dispersions (volatility). Fig. 5 shows averaged coefficients of time $\tau(t)$ compression-expansion (formula (23)) for three groups of incoming series: currency (forex), stock, and commodity markets. Figure 5: Coefficients of time compression-expansion, market “temperature”. The explanation is in the text. The formulae (11),(23),(25) show that $\tau$(t) exists in proportion to the averaged square speed (according to the chosen time span and series), i.e. average “energy” of the economical “particle” (as it is in our analogy), and can be thus interpreted as the series “temperature”. Crises are distinguished with the intensification of economic processes (the “temperature” is rising), while during the crisis-free period their deceleration can be observed (the “temperature” is falling), what can be interpreted as the heterogenous flow of economic time. $\tau(t)$ dependences shown on the fig. 5 illustrate all afore- mentioned. Note that local time acceleration-deceleration can be rather significant. Transition to heterogenous economic time allows to make the observed economic series more homogenous, which can simplify both analysis and prediction r042 . In table we give the values of a non-dimentional economic mass of the $m_{i}$ series, calculated using (22) for all 9 incoming series, as well as average masses of each group (formula (29)). Table. Economic series masses Incoming series \| Economic mass \| Average economic mass of the group ---\|---\|--- Commodity market \| gold \| $2,816\cdot 10^{4}$ \| $4,983\cdot 10^{3}$ silver \| $4,843\cdot 10^{3}$ oil \| $2,777\cdot 10^{3}$ Currency market \| jpy \| $2,148\cdot 10^{4}$ \| $2,499\cdot 10^{4}$ gbp \| $3,523\cdot 10^{4}$ chf \| $2,180\cdot 10^{4}$ Stock market \| S&P 500 \| $6,251\cdot 10^{3}$ \| $4,748\cdot 10^{3}$ FTSE 100 \| $6,487\cdot 10^{3}$ BVSP \| $1,507\cdot 10^{3}$ As you can see from the table, the stock market is distinguished with the lowest mass value, while the currency one shows the maximum number. Oil price series has the lowest mass on the commodity market, gold – the highest one. As for the currency market, British pound (gbp) have the highest value and Japanese yen rates (jpy) demonstrates the minimum mass of the group, although the dispersion is lower than that of the commodity market. The smallest spread is peculiar to the currency market. Dynamic and developing Brazilian market (BVSP) has the lowest mass, while the maximum value, just like in the previous case, corresponds to Great Britain (FTSE 100). It is explained by the well- known fact: Britain has been always known for its relatively “closed” economy as comraped with the rest of the European and non-European countries. The last group of experimental data corresponds to the dependence of Planck’s economic constant (calculated for different series) on time $\Delta t=k\Delta t_{\min}$ (time between the neighbouring registered observations), which is characterised by $H_{i}\left(k,\bar{n}\right)$ coefficient (see formula (24)). On fig. 6-8 integral dependencies $H_{i}\left(k\right)$ are depicted. The following are averaged on the whole period of time 1993-2010 and calculated for commodity, stock, and currency markets. As you can see there are no obvious regularities, which can be explained by various crises and recessions of the world and national economies that took place during the investigated period. To decide whether it is possible for local regularities of Planck’s economic constant dependence on $\Delta t$ to appear, we have chosen relatively small averaging fragments, $\Delta N=500$, which approximately equals two years. Corresponding results for some of these fragments on commodity, currency, and stock markets are given on fig. 9-11. Evidently, all three figures show clear tendencies of $H_{i}\left(k,\bar{n}\right)$ recession and rise for each type of the market (unlike integral dependences $H_{i}\left(k\right)$). Figure 6: Integral coefficient $H_{i}\left(k\right)$ dependences for commodity market. Figure 7: Integral coefficient $H_{i}\left(k\right)$ dependences for stock market. Figure 8: Integral coefficient $H_{i}\left(k\right)$ dependences for currency market. Figure 9: Local coefficient $H_{i}\left(k,\bar{n}\right)$ dependeces for commodity market (averaging time span from 27.04.1993 to 12.06.1995, 500 daily values). Figure 10: Local coefficient $H_{i}\left(k,\bar{n}\right)$ for currency market (averaging time span from 12.06.1995 to 15.07.1997, 500 daily values). Figure 11: Local coefficient $H_{i}\left(k,\bar{n}\right)$ dependences for stock market (averaging time span from 12.06.1995 to 15.07.1997, 500 daily values). ## 5 Conclusions We have conducted methodological and philosophical analysis of physical notions and their formal and informal connections with real economic measurements. Basic ideas of the general relativity theory and relativistic qantum mechanics concerning spacetime properties and physical dimensions peculiarities were used as well. We have suggested procedures of detecting normalized economic coordinates, economic mass and heterogenous economic time. The afore-mentioned procedures are based on socio-economic time series analysis and economical interpretation of Heisenberg’s uncertainty principle. The notion of economic Planck’s constant has also been introduced. The theory has been tested on real economic time series, including stock indices, currency rates, and commodity prices. Acquired results indicate the availability of further investigations. ## References * [1] V. N. Soloviev and V. M. Saptsin. Quantum econophysics - a physical substantiation of system concepts in socio-economic processes modeling. In Analysis, modeling, management, development of economic systems Proceedings of the II International Symposium of the School-AMUR-2008 Sebastopol Sept. 12-18. 2008 / ed. O. Koroleva, A. Segal., Simferopol, 2008. * [2] V. M. Saptsin and V. N. Soloviev. Relativistic Quantum Econophysics. New paradigms of Complex systems modeling: Monograph. http://kafek.at.ua/sol_sap_monogr.rar Brama-Ukraine, Cherkassy, 2009. * [3] V. Saptsin and V. Soloviev. Relativistic quantum econophysics - new paradigms in complex systems modelling. arXiv:0907.1142v1 [physics.soc-ph]. * [4] B. E. Baaquie. Quantum Finance. Cambridge University Press, Cambridge, 2004. * [5] V. Maslov. Quantum Economics. Science, Moscow, 2006. * [6] E. Guevara. Quantum Econophysics. arXiv:physics/0609245 , September 2006. * [7] W.-X. Zhou and D. Sornette. Self-organizing Ising model of financial markets. Eur. Phys. J. B, 55(2):175–181, 2007. * [8] C. P. Goncalves. Quantum Financial Economics - Risk and Returns. arXiv:1107.2562 , July 2011. * [9] R. N. Mantegna and H. E. Stanley. An Introduction to Econophysics: Correlations and Complexity in Finance. Cambridge Univ. Press, Cambridge UK, 2000. * [10] M. Yu Romanovsky and Yu. M. Romanovsky. An introduction to econophysics. Statistical and dynamical models. IKI, Moscow, 2007. * [11] V. N. Soloviev. Mathematical economics. A textbook for self-study (in Ukrainian). CHNU, Cherkassy, 2008. http://kafek.at.ua/Posibnyk_Soloviev.rar * [12] V. D. Derbentsev, A. A. Serdyuk, and V. N. Solovievand O. D. Sharapov. Synergetical and econophysical methods for the modeling of dynamic and structural characteristics of economic systems. Monograph (in Ukrainian). Brama-Ukraine, Cherkassy, 2010. http://kafek.at.ua/Monogr.pdf * [13] V. M. Saptsin. Genetically complex Markov chains – a new neural network approach for forecasting of socio-economic and other poorly formalized processes. In Monitoring, modelyuvannya i management emerdzhentnoyi ekonomiki: Conference Proceedings., pages 196 – 198, Cherkassy, 13-15 may 2008\. Brama. * [14] V. M. Saptsin. Experience of using genetically complex Markov chains for the neural network technology forecasting. Visnyk Krivorizkogo ekonomichnogo institutu KNEU, 2 (18):56 – 66, 2009. * [15] Vladimir Soloviev, Vladimir Saptsin, and Dmitry Chabanenko. Prediction of financial time series with the technology of high-order Markov chains. In Working Group on Physics of Socio-economic Systems (AGSOE), Drezden, 2009. http://www.dpg-verhandlungen.de/2009/dresden/agsoe.pdf * [16] D. M. Chabanenko. Financial time series prediction algorithm, based on complex Markov chains. Visnyk Cherkasskogo Universitetu (in Ukrainian), 1(173):90 – 102, 2009. http://www.nbuv.gov.ua/portal/Soc_Gum/Vchu/N173/N173p090-102.pdf * [17] D. M. Chabanenko. Detection of short- and long-term memory and time series prediction methods of complex Markov chains. In Visnyk Natsionalnogo tehnichnogo universitetu “Kharkivsky politehnichny institut”. Zbirnik Naukovyh pratz. Tematichny vypusk: Informatika i modelyuvannya (in Ukrainian), number 31, pages 184 – 190. NTU KHPI, Kharkov, 2010. http://www.pim.net.ua/ARCH_F/V_pim_10.pdf * [18] V. Soloviev, V. Saptsin, and D. Chabanenko. Financial time series prediction with the technology of complex Markov chains. Computer Modelling and New Technologies, 14(3):63 – 67, 2010. http://www.tsi.lv/RSR/vol14_3/14_3-7.pdf * [19] V. N. Soloviev, V. M. Saptsin, and D. N. Chabanenko. Prognozuvannya sotsialno-ekonomichnih protsesiv: suchasni pidhody ta perspektivy. Monograph (in Ukrainian), chapter Financial and economic time series forecasting using Markov chains and Fourier-continuation, pages 141–155. 2011\. * [20] V. N. Soloviev, V. M. Saptsin, and D. N. Chabanenko. Modern methods of analysis and prediction of oscillatory processes in complex systems. The scientific heritage of Simon Kuznets and prospects of the global and national economies in the XXI Century, 2011. CD-ROM. * [21] V. N. Soloviev, V. M. Saptsin, and D. N. Chabanenko. Fourier-based forecasting of economic dynamics. In Informatsiyni tehnologiyi modelyuvannya v ekonomitsi: Conference proceedings (in Ukrainian), page 204, Cherkassy, 19-21 may 2009. Brama-Ukraine. * [22] V. M. Saptsin and D. N. Chabanenko. Fourier-based forecasting of low-frequency components of economical dynamic’s time series. In Problemy ekonomichnoyi kibernetiki: Tezy dopovidey XIV Vseukrayinskoyi Naukovo-praktichnoyi konferentsiyi.(in Ukrainian), page 132, Kharkiv, Oct 8-9, 2009 2009. KhNU imeni VN Karazina. * [23] D. M. Chabanenko. Discrete Fourier-based forecasting of time series. Sistemni tehnologii. Regionalny mizhvuzivsky zbirnik naukovyh pratz (in Ukrainian), 1(66):114 – 121, 2010. http://www.nbuv.gov.ua/portal/natural/syte/2010_1/15.pdf * [24] V. M. Saptsin. Two-phase nonlinear discrete second-order system of logistical type: the local stability condition and classification. In Kompyuterne modeluvannya ta informatsiyni tehnologii v nautsi, ekonomitsi i osviti. Conference procceedings (in Ukrainian), pages 194 – 195, Krivy Rig, 2005. KEI KNEU. * [25] Y. A. Olkhovaya and V. M. Saptsin. The relationship of concurence, cooperation and dominance in network models with discrete-time. In Electromechanical and energetic systems, simulation and optimization methods: Conference proceedings (in Russian), pages 302–305., Kremenchuk, Apr 8-9 2010. KDU. * [26] V. M. Saptsin and A. N. Chabanenko. Analysis of the stability limit of the nonlinear two-agent model with dominance in the space of parameters. In Kompyuterne modelyuvannya i informatsiyni tehnologiyi v nautsi, ekonomitsi i osviti: Conference proceedings (in Ukrainian), pages 143–146, Cherkassy, 2011. Brama. * [27] V. M. Saptsin, V. N. Soloviev, and A. Batyr. Nonlinear concurence in two-agent systems. In 2 volumes, editor, Tezy dopovidey VII Vseukryinskoyi Naukova-praktichnoyi koferentsiyi “Informatsiyni tehnologiyi in osviti, nautsi i tehnitsi” (ITONT 2010)., volume 1, page 101, Cherkassy, 2010. Cherkassy State Technological University. * [28] V. N. Soloviev and V. M. Saptsin. Dynamical network mathematics - a new look at the problems of complex systems mathematical modeling. In Analysis, modeling, management, development of economic systems: collection of scientific papers of the IV International Symposium school - AMUR-2010, (in Russian), pages 340–342, Simferopol, Sep 13-19 2010\. TNU named after V. I. Vernadsky. * [29] V. N. Soloviev, V. M. Saptsin, and A. N Chabanenko. Dynamical network mathematics - a new look at the problems of complex systems mathematical modeling. In Visnyk Cherkaskogo universitetu. Seriya pedagogichni nauki, Vypusk 191. Part 1., pages 121–127. CHNU, 2010. * [30] V. M. Saptsin. Econophysical analysis of the network nature of the low-frequency noise in socio-economic systems. In “Monitoring, modelyuvannya i management emerdzhentnoyi ekonomiki”, conference proceedings (in Ukrainian), pages 184–189, Cherkassy, 2010. Brama-Ukraine. * [31] V. M. Saptsin. Econophysical analysis of network nature of low-frequency noise in socio-economic systems. Visnyk Cherkaskogo universitetu, 187:108–115., 2010. * [32] V. N Soloviev, V. M. Saptsin, and Shokotko L. N. The Heisenberg’s principle and financial markets. In Systemny analiz. Informatika. Upravlinnya (SAIU-2011): Conference proceedings, pages 197–198, Zaporizhzhya, 2011. Classical Private University. * [33] V.N. Soloviev, V. M. Saptsin, and L. N. Shokotko. Heisenberg uncertainty principle and financial markets. In The 9-th International conference “Information technologies and management 2011”, pages 135–136, Riga, Latvia, April 14-15 2011. Information Systems Management Institute. * [34] L.D. Landau and E.M. Lifshit is. The classical theory of fields. Course of theoretical physics. Butterworth Heinemann, 1975. * [35] John G. Cramer. The transactional interpretation of quantum mechanics. Rev. Mod. Phys., 58:647 – 687, Jul 1986. * [36] M. Kaku. Introduction to superstrings and M-theory. Graduate texts in contemporary physics. Springer, 1999. * [37] V. Balasubramanian. What we don’t know about time. Foundations of Physics, page 139, sep 2011. * [38] Y. S. Vladimirov. A relational theory of space-time interactions. Part 1. MGU, Moscow, 1996. * [39] Y. S. Vladimirov. A relational theory of space-time interactions. Part 2. MGU, Moscow, 1996. * [40] M. Kaku. Parallel worlds: a journey through creation, higher dimensions, and the future of the cosmos. Anchor Books, 2006. * [41] M. Kaku. Physics of the impossible: a scientific exploration into the world of phasers, force fields, teleportation, and time travel. Doubleday, 2008. * [42] V. M. Saptsin and D. M. Chabanenko. The complexity problem and non-linear time in socio-economic processes forecasting. In Problemy ekonomichnoyi kibernetiki: Tezi dopovidey XIV Vseukrayinskoyi Naukova-praktichnoyi konferentsiyi, pages 130–131, Kharkiv, 2009\. KNU imeni V. N. Karazina.	arxiv-papers	2011-11-10T17:47:21	2024-09-04T02:49:24.588977	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Vladimir Soloviev and Vladimir Saptsin", "submitter": "Vladimir Saptsin", "url": "https://arxiv.org/abs/1111.5289" }
1111.5340	# On the Expected Complexity of Random Convex Hulls††thanks: This work has been supported by a grant from the U.S.–Israeli Binational Science Foundation. This work is part of the author’s Ph.D. thesis, prepared at Tel-Aviv University under the supervision of Prof. Micha Sharir. Sariel Har-Peled Department of Computer Science; University of Illinois; 201 N. Goodwin Avenue; Urbana, IL, 61801, USA; sariel@uiuc.edu; http://www.uiuc.edu/~sariel/. ###### Abstract In this paper we present several results on the expected complexity of a convex hull of $n$ points chosen uniformly and independently from a convex shape. (i) We show that the expected number of vertices of the convex hull of $n$ points, chosen uniformly and independently from a disk is $O(n^{1/3})$, and $O(k\log{n})$ for the case a convex polygon with $k$ sides. Those results are well known (see [RS63, Ray70, PS85]), but we believe that the elementary proof given here are simpler and more intuitive. (ii) Let ${\cal D}$ be a set of directions in the plane, we define a generalized notion of convexity induced by ${\cal D}$, which extends both rectilinear convexity and standard convexity. We prove that the expected complexity of the ${\cal D}$-convex hull of a set of $n$ points, chosen uniformly and independently from a disk, is $O\left({n^{1/3}+\sqrt{n\alpha({\cal D})}}\right)$, where $\alpha({\cal D})$ is the largest angle between two consecutive vectors in ${\cal D}$. This result extends the known bounds for the cases of rectilinear and standard convexity. (iii) Let ${\cal B}$ be an axis parallel hypercube in ${\rm I\\!R}^{d}$. We prove that the expected number of points on the boundary of the quadrant hull of a set $S$ of $n$ points, chosen uniformly and independently from ${\cal B}$ is $O(\log^{d-1}n)$. Quadrant hull of a set of points is an extension of rectilinear convexity to higher dimensions. In particular, this number is larger than the number of maxima in $S$, and is also larger than the number of points of $S$ that are vertices of the convex hull of $S$. Those bounds are known [BKST78], but we believe the new proof is simpler. ## 1 Introduction Let $C$ be a fixed compact convex shape, and let $X_{n}$ be a random sample of $n$ points chosen uniformly and independently from $C$. Let $Z_{n}$ denote the number of vertices of the convex hull of $X_{n}$. Rényi and Sulanke [RS63] showed that $E[Z_{n}]=O(k\log{n})$, when $C$ is a convex polygon with $k$ vertices in the plane. Raynaud [Ray70] showed that expected number of facets of the convex hull is $O(n^{(d-1)/(d+1)})$, where $C$ is a ball in ${\rm I\\!R}^{d}$, so $E[Z_{n}]=O(n^{1/3})$ when $C$ is a disk in the plane. Raynaud [Ray70] showed that the expected number of facets of ${\mathop{\mathrm{CH}}}(X_{n})=ConvexHull(X_{n})$ is $O\left({(\log(n))^{(d-1)/2}}\right)$, where the points are chosen from ${\rm I\\!R}^{d}$ by a $d$-dimensional normal distribution. See [WW93] for a survey of related results. All these bounds are essentially derived by computing or estimating integrals that quantify the probability of two specific points of $X_{n}$ to form an edge of the convex hull (multiplying this probability by $\binom{n}{2}$ gives $E[Z_{n}]$). Those integrals are fairly complicated to analyze, and the resulting proofs are rather long, counter-intuitive and not elementary. Efron [Efr65] showed that instead of arguing about the expected number of vertices directly, one can argue about the expected area/volume of the convex hull, and this in turn implies a bound on the expected number of vertices of the convex hull. In this paper, we present a new argument on the expected area/volume of the convex hull (this method can be interpreted as a discrete approximation to the integral methods). The argument goes as follows: Decompose $C$ the into smaller shapes (called tiles). Using the topology of the tiling and the underlining type of convexity, we argue about the expected number of tiles that are exposed by the random convex hull, where a tile is exposed if it does not lie completely in the interior of the random convex hull. Resulting in a lower bound on the area/volume of the random convex hull. We apply this technique to the standard case, and also for more exotic types of convexity. In Section 2, we give a rather simple and elementary proofs of the aforementioned bounds $E[Z_{n}]=O(n^{1/3})$ for $C$ a disk, and $E[Z_{n}]=O\left({k\log{n}}\right)$ for $C$ a convex $k$-gon. We believe that these new elementary proofs are indeed simpler and more intuitive111Preparata and Shamos [PS85, pp. 152] comment on the older proof for the case of a disk: “Because the circle has no corners, the expected number of hull vertices is comparatively high, although we know of no elementary explanation of the $n^{1/3}$ phenomenon in the planar case.” It is the author’s belief that the proof given here remedies this situation. than the previous integral-based proofs. The question on the expected complexity of the convex hull remains valid, even if we change our type of convexity. In Section 3, we define a generalized notion of convexity induced by ${\cal D}$, a given set of directions. This extends both rectilinear convexity, and standard convexity. We prove that the expected complexity of the ${\cal D}$-convex hull of a set of $n$ points, chosen uniformly and independently from a disk, is $O\left({n^{1/3}+\sqrt{n\alpha({\cal D})}}\right)$, where $\alpha({\cal D})$ is the largest angle between two consecutive vectors in ${\cal D}$. This result extends the known bounds for the cases of rectilinear and standard convexity. Finally, in Section 4, we deal with another type convexity, which is an extension of the generalized convexity mentioned above for the higher dimensions, where the set of the directions is the standard orthonormal basis of ${\rm I\\!R}^{d}$. We prove that the expected number of points that lie on the boundary of the quadrant hull of $n$ points, chosen uniformly and independently from the axis-parallel unit hypercube in ${\rm I\\!R}^{d}$, is $O(\log^{d-1}n)$. This readily imply $O(\log^{d-1}n)$ bound on the expected number of maxima and the expected number of vertices of the convex hull of such a point set. Those bounds are known [BKST78], but we believe the new proof is simpler and more intuitive. ## 2 On the Complexity of the Convex Hull of a Random Point Set In this section, we show that the expected number of vertices of the convex hull of $n$ points, chosen uniformly and independently from a disk, is $O(n^{1/3})$. Applying the same technique to a convex polygon with $k$ sides, we prove that the expected number of vertices of the convex hull is $O(k\log{n})$.222As already noted, these results are well known ([RS63, Ray70, PS85]), but we believe that the elementary proofs given here are simpler and more intuitive. The following lemma, shows that the larger the expected area outside the random convex hull, the larger is the expected number of vertices of the convex hull. ###### Lemma 2.1 Let $C$ be a bounded convex set in the plane, such that the expected area of the convex hull of $n$ points, chosen uniformly and independently from $C$, is at least $\left({1-f(n)}\right)Area(C)$, where $1\geq f(n)\geq 0$, for $n\geq 0$. Then the expected number of vertices of the convex hull is $\leq nf(n/2)$. ###### Proof. Let $N$ be a random sample of $n$ points, chosen uniformly and independently from $C$. Let $N_{1}$ (resp. $N_{2}$) denote the set of the first (resp. last) $n/2$ points of $N$. Let $V_{1}$ (resp. $V_{2}$) denote the number of vertices of $H={\mathop{CH}}(N_{1}\cup N_{2})$ that belong to $N_{1}$ (resp. $N_{2}$), where ${\mathop{CH}}(N_{1}\cup N_{2})={\mathrm{ConvexHull}}(N_{1}\cup N_{2})$. Clearly, the expected number of vertices of $C$ is $E[V_{1}]+E[V_{2}]$. On the other hand, $E\left[{V_{1}\,\left\|\,{N_{2}}\rule[-5.69046pt]{0.0pt}{11.38092pt}\right.{\rule[-5.69046pt]{0.0pt}{0.0pt}}}\right]\leq\frac{n}{2}\left({\frac{Area(C)-Area({\mathop{CH}}(N_{2}))}{Area(C)}}\right),$ since $V_{1}$ is bounded by the expected number of points of $N_{1}$ falling outside ${\mathop{CH}}(N_{2})$. We have $\displaystyle E[V_{1}]$ $\displaystyle=$ $\displaystyle E_{N_{2}}\left[{E[V_{1}\|N_{2}]{\rule[-5.69046pt]{0.0pt}{0.0pt}}}\right]\leq E\left[{\frac{n}{2}\left({\frac{Area(C)-Area({\mathop{CH}}(N_{2}))}{Area(C)}}\right){\rule[-5.69046pt]{0.0pt}{0.0pt}}}\right]$ $\displaystyle\leq$ $\displaystyle\frac{n}{2}f(n/2),$ since $E[X]=E_{Y}[E[X\|Y]]$ for any two random variables $X,Y$. Thus, the expected number of vertices of $H$ is $E[V_{1}]+E[V_{2}]\leq nf(n/2)$. ∎ ###### Remark 2.2 Lemma 2.1 is known as Efron’s Theorem. See [Efr65]. ###### Theorem 2.3 The expected number of vertices of the convex hull of $n$ points, chosen uniformly and independently from the unit disk, is $O(n^{1/3})$. ###### Proof. We claim that the expected area of the convex hull of $n$ points, chosen uniformly and independently from the unit disk, is at least $\pi-O\left({n^{-2/3}}\right)$. Indeed, let $D$ denote the unit disk, and assume without loss of generality, that $n=m^{3}$, where $m$ is a positive integer. Partition $D$ into $m$ sectors, ${\cal S}_{1},\ldots,{\cal S}_{m}$, by placing $m$ equally spaced points on the boundary of $D$ and connecting them to the origin. Let $D_{1},\ldots,D_{m^{2}}$ denote the $m^{2}$ disks centered at the origin, such that (i) $D_{1}=D$, and (ii) $Area(D_{i-1})-Area(D_{i})=\pi/m^{2}$, for $i=2,\ldots,m^{2}$. Let $r_{i}$ denote the radius of $D_{i}$, for $i=1,\ldots,m^{2}$. Let $S_{i,j}=(D_{i}\setminus D_{i+1})\cap{\cal S}_{j}$, and $S_{m^{2},j}=D_{m^{2}}\cap{\cal S}_{j}$, for $i=1,\ldots,m^{2}-1$, $j=1,\ldots,m$. The set $S_{i,j}$ is called the $i$-th tile of the sector ${\cal S}_{j}$, and its area is $\pi/n$, for $i=1,\ldots,m^{2}$, $j=1,\ldots,m$. Let $N$ be a random sample of $n$ points chosen uniformly and independently from $D$. Let $X_{j}$ denote the first index $i$ such that $N\cap S_{i,j}\neq\emptyset$, for $j=1,\ldots,m$. For a fixed $j\in\left\\{{1,\ldots,m}\right\\}$, the probability that $X_{j}=k$ is upper- bounded by the probability that the tiles $S_{1,j},\ldots,S_{(k-1),j}$ do not contain any point of $N$; namely, by $\left({1-\frac{k-1}{n}}\right)^{n}$. Thus, $P[X_{j}=k]\leq\left({1-\frac{k-1}{n}}\right)^{n}\leq e^{-(k-1)}$, since $1-x\leq e^{-x}$, for $x\geq 0$. Thus, $E\left[{X_{j}{\rule[-5.69046pt]{0.0pt}{0.0pt}}}\right]=\sum_{k=1}^{m^{2}}kP[X_{j}=k]\leq\sum_{k=1}^{m^{2}}ke^{-(k-1)}=O(1),$ for $j=1,\ldots,m$. Let $K_{o}$ denote the convex hull of $N\cup\left\\{{o}\right\\}$, where $o$ is the origin. The tile $S_{i,j}$ is exposed by a set $K$, if $S_{i,j}\setminus K\neq\emptyset$. We claim that at most $X_{j-1}+X_{j+1}+O(1)$ tiles are exposed by $K_{o}$ in the sector ${\cal S}_{j}$, for $j=1,\ldots,m$ (where we put $X_{0}=X_{m}$, $X_{m+1}=X_{1}$). $s$${\cal S}_{j+1}$${\cal S}_{j-1}$${\cal S}_{j}$$p$$q$$o$$T$${\cal S}_{i,j}$ Figure 1: Illustrating the proof that bounds the number of tiles exposed by $T$ inside ${\cal S}_{j}$ Indeed, let $w=w(N,j)=max(X_{j-1},X_{j+1})$, and let $p,q$ be the two points in $S_{j-1,w},S_{j+1,w}$, respectively, such that the number of sets exposed by the triangle $T=\triangle{opq}$, in the sector ${\cal S}_{i}$, is maximal. Both $p$ and $q$ lie on ${\partial}{D_{w+1}}$ and on the external radii bounding ${\cal S}_{j-1}$ and ${\cal S}_{j+1}$, as shown in Figure 1. Clearly, any tile which is exposed in ${\cal S}_{j}$ by $K_{o}$ is also exposed by $T$. Let $s$ denote the segment connecting the middle of the base of $T$ to its closest point on ${\partial}{D_{w}}$. The number of tiles in ${\cal S}_{j}$ exposed by $T$ is bounded by $\max\left({X_{j-1},X_{j+1}}\right)$, plus the number of tiles intersecting the segment $s$. The length of $s$ is $\|oq\|-\|oq\|\cos\left({\frac{3}{2}\cdot\frac{2\pi}{m}}\right)\leq 1-\cos\left({\frac{3}{2}\cdot\frac{2\pi}{m}}\right)\leq\frac{1}{2}\left({\frac{3\pi}{m}}\right)^{2}=\frac{4.5\pi^{2}}{m^{2}},$ since $\cos(x)\geq 1-x^{2}/2$, for $x\geq 0$. On the other hand, $r_{i+1}-r_{i}\geq r_{i}-r_{i-1}\geq 1/(2m^{2})$, for $i=2,\ldots,m^{2}$. Thus, the segment $s$ intersects at most $\left\lceil{\|\|s\|\|/(1/(2m^{2}))}\right\rceil=\left\lceil{9\pi^{2}}\right\rceil=89$ tiles, and we have that the number of tiles exposed in the sector ${\cal S}_{i}$ by $K_{o}$ is at most $\max\left({X_{j-1},X_{j+1}}\right)+89\leq X_{j-1}+X_{j+1}+89$, for $j=1,\ldots,m$. Thus, the expected number of tiles exposed by $K_{o}$ is at most $E\left[{\sum_{i=1}^{m}\left({X_{j-1}+X_{j+1}+89}\right){\rule[-5.69046pt]{0.0pt}{0.0pt}}}\right]=O(m).$ The area of $K={\mathop{CH}}(N)$ is bounded from below by the area of tiles which are not exposed by $K$. The probability that $K\subsetneq K_{o}$ (namely, the origin is not inside $K$, or, equivalently, all points of $N$ lie in some semidisk) is at most $2\pi/2^{n}$, as easily verified. Hence, $E[Area(K)]\geq E[Area(C)]-P\left[{C\neq K{\rule[-5.69046pt]{0.0pt}{0.0pt}}}\right]\pi=\pi-O(m)\frac{\pi}{n}-\frac{2\pi}{2^{n}}\pi=\pi-O\left({n^{-2/3}}\right).$ The assertion of the theorem now follows from Lemma 2.1. ∎ ###### Lemma 2.4 The expected number of vertices of the convex hull of $n$ points, chosen uniformly and independently from the unit square, is $O(\log{n})$. $R_{5}(1)$$R_{5}(2)$$R_{5}(3)$$R_{5}(4)$$R_{5}^{\prime}(1)$$R_{5}^{\prime}(2)$$R_{5}^{\prime}(3)$$R_{5}^{\prime}(4)$$R_{5}^{\prime}(5)$$R_{5}^{\prime}(6)$ Figure 2: Illustrating the proof that bounds the number of tiles exposed by ${\mathop{CH}}(N)$ inside the $j$-th column, by using a non-uniform tiling of the strips to the left and to the right of the $j$-th column. The area of such a larger tile is at least $1/n$. ###### Proof. We claim that the expected area of the convex hull of $n$ points, chosen uniformly and independently from the unit square, is at least $1-O\left({\log(n)/n}\right)$. Let $S$ denote the unit square. Partition $S$ into $n$ rows and $n$ columns, such that $S$ is partitioned into $n^{2}$ identical squares. Let $S_{i,j}=[(i-1)/n,i/n]\times[(j-1)/n,j/n]$ denote the $j$-th square in the $i$-th column, for $1\leq i,j\leq n$. Let ${\cal S}_{i}=\cup_{j=1}^{n}S_{i,j}$ denote the $i$-th column of $S$, for $i=1,\ldots,n$, and let ${\cal S}(l,k)=\cup_{i=l}^{k}{\cal S}_{i}$, for $1\leq l\leq k\leq n$. Let $N$ be a random sample of $n$ points chosen uniformly and independently from $S$. Let $X_{j}$ denote the first index $i$ such that $N\cap(\cup_{l=1}^{j-1}S_{l,i})\neq\emptyset$, for $j=2,\ldots,n-1$; namely, $X_{j}$ is the index of the first row in ${\cal S}(1,j-1)$ that contains a point from $N$. Symmetrically, let $X_{j}^{\prime}$ be the index of the first row in ${\cal S}(j+1,n)$ that contains a point of $N$. Clearly, $E[X_{j}]=E[X_{n-j+1}^{\prime}]$, for $j=2,\ldots,n-1$. Let $Z_{j}$ denote the number of squares $S_{i,j}$ in the bottom of the $j$-th column that are exposed by ${\mathop{CH}}(N)$, for $j=2,\ldots,n-1$. Arguing as in the proof of Theorem 2.3, we have that $Z_{j}\leq\max(X_{j},X_{j}^{\prime})\leq X_{j}+X_{j}^{\prime}$. Thus, in order to bound $E[Z_{j}]$, we first bound $E[X_{j}]$ by covering the strips ${\cal S}(1,j-1),{\cal S}(j+1,n)$ by tiles of area $\geq 1/n$. In particular, let $h(l)=\left\lceil{n/(l-1)}\right\rceil$, and let $R_{j}(m)=[0,(j-1)/n]\times[h(n-j+1)(m-1)/n,h(j)m/n]$, and let $R_{j}^{\prime}(m)=[(j+1)/n,1]\times[h(j)(m-1)/n,h(j)m/n]$, for $j=2,\ldots,n-1$. See Figure 2. Let $Y_{j}$ denote the minimal index $i$ such that $R_{j}(i)\cap N\neq\emptyset$. The area of $R_{j}(i)$ is at least $1/n$, for any $i$ and $j$. Arguing as in the proof of Theorem 2.3, it follows that $E[Y_{j}]=O(1)$. On the other hand, $E[X_{j}]\leq h(j)E[Y_{j}]=O(n/(j-1))$. Symmetrically, $E[X_{j}^{\prime}]=O(n/(n-j))$. Thus, by applying the above argument to the four directions (top, bottom, left, right), we have that the expected number of squares $S_{i,j}$ exposed by ${\mathop{CH}}(N)$ is bounded by $4n-4+4{\sum_{j=2}^{n-1}E[Z_{j}]}<4n+4{\sum_{j=2}^{n-1}(E[X_{j}]+E[X_{j}^{\prime}])}=4n+8{\sum_{j=2}^{n-1}O\left({\frac{n}{j-1}}\right)}=O(n\log{n}),$ where $4n-4$ is the number of squares adjacent to the boundary of $S$. Since the area of each square is $1/n^{2}$, it follows that the expected area of ${\mathop{CH}}(N)$ is at least $1-O(\log(n)/n)$. By Lemma 2.1, the expected number of vertices of the convex hull is $O(\log n)$. ∎ ###### Lemma 2.5 The expected number of vertices of the convex hull of $n$ points, chosen uniformly and independently from a triangle, is $O(\log{n})$. ###### Proof. We claim that the expected area of the convex hull of $n$ points, chosen uniformly and independently from a triangle $T$, is at least $(1-O\left({\log(n)/n}\right))Area(T)$. We adapt the tiling used in Lemma 2.4 to a triangle. Namely, we partition $T$ into $n$ equal-area triangles, by segments emanating from a fixed vertex, each of which is then partitioned into $n$ equal-area trapezoids by segments parallel to the opposite side, such that each resulting trapezoid has area $1/n^{2}$. See Figure 3. Notice that this tiling has identical topology to the tiling used in Lemma 2.4. Thus, the proof of Lemma 2.4 can be applied directly to this case, repeating the tiling process three times, once for each vertex of $T$. This readily implies the asserted bound. ∎ $R_{5}(1)$$R_{5}(2)$$R_{5}(3)$$R_{5}^{\prime}(1)$$R_{5}^{\prime}(2)$$R_{5}^{\prime}(3)$$R_{5}^{\prime}(4)$$R_{5}^{\prime}(5)$ Figure 3: Illustrating the proof of Lemma 2.4 for the case of a triangle. ###### Theorem 2.6 The expected number of vertices of the convex hull of $n$ points, chosen uniformly and independently from a polygon $P$ having $k$ sides, is $O(k\log{n})$. ###### Proof. We triangulate $P$ in an arbitrary manner into $k$ triangles $T_{1},\ldots,T_{k}$. Let $N$ be a random sample of $n$ points, chosen uniformly and independently from $P$. Let $Y_{i}=\|T_{i}\cap N\|$, $N_{i}=T_{i}\cap N$, and $Z_{i}=\|{\mathop{CH}}(N_{i})\|$, for $i=1,\ldots,k$. Notice that the distribution of the points of $N_{i}$ inside $T_{i}$ is identical to the distribution of $Y_{i}$ points chosen uniformly and independently from $T_{i}$. In particular, $E[Z_{i}\|Y_{i}]=O(\log{Y_{i}})$, by Lemma 2.5, and $E[Z_{i}]=E_{Y_{i}}[E[Z_{i}\|Y_{i}]]=O(\log{n})$, for $i=1,\ldots,k$. Thus, $E[\|{\mathop{CH}}(N)\|]\leq E\left[{\sum_{i=1}^{k}\|{\mathop{CH}}(N_{i})\|{\rule[-5.69046pt]{0.0pt}{0.0pt}}}\right]\leq\sum_{i=1}^{k}E[Z_{i}]=O(k\log{n})$. ∎ ## 3 On the Expected Complexity of a Generalized Convex Hull Inside a Disk In this section, we derive a bound on the expected complexity on a generalized convex hull of a set of points, chosen uniformly and independently for the unit disk. The new bound matches the known bounds, for the case of standard convexity and maxima. The bound follows by extending the proof of Theorem 2.3. We begin with some terminology and some initial observations, most of them taken or adapted from [MP97]. A set ${\cal D}$ of vectors in the plane is a set of directions, if the length of all the vectors in ${\cal D}$ is $1$, and if $v\in{\cal D}$ then $-v\in{\cal D}$. Let ${\cal D}_{{\rm I\\!R}}$ denote the set of all possible directions. A set $C$ is ${\cal D}$-convex if the intersection of $C$ with any line with a direction in ${\cal D}$ is connected. By definition, a set $C$ is convex (in the standard sense), if and only if it is ${\cal D}_{{\rm I\\!R}}$-convex. For a set $C$ in the plane, we denote by ${\mathop{\cal{CH}}}_{D}(C)$ the ${\cal D}$-convex hull of $C$; that is, the smallest ${\cal D}$-convex set that contains $C$. While this seems like a reasonable extension of the regular notion of convexity, its behavior is counterintuitive. For example, let ${\cal D}_{Q}$ denote the set of all rational directions (the slopes of the directions are rational numbers). Since ${\cal D}_{Q}$ is dense in ${\cal D}_{{\rm I\\!R}}$, one would expect that ${\mathop{\cal{CH}}}_{{\cal D}_{Q}}(C)={\mathop{\cal{CH}}}_{{\cal D}_{\rm I\\!R}}(C)={\mathop{\mathrm{CH}}}(C)$. However, if $C$ is a set of points such that the slope of any line connecting a pair of points of $C$ is irrational, then ${\mathop{\cal{CH}}}_{D_{Q}}(C)=C$. See [OSSW85, RW88, RW87] for further discussion of this type of convexity. ###### Definition 3.1 Let $f$ be a real function defined on a ${\cal D}$-convex set $C$. We say that $f$ is ${\cal D}$-convex if, for any $x\in C$ and any $v\in{\cal D}$, the function $g(t)=f(x+tv)$ is a convex function of the real variable $t$. (The domain of $g$ is an interval in ${\rm I\\!R}$, as $C$ is assumed to be ${\cal D}$-convex.) Clearly, any convex function, in the standard sense, defined over the whole plane satisfies this condition. ###### Definition 3.2 Let $C\subseteq{\rm I\\!R}^{2}$. The set ${\mathop{\cal{CH}}}^{\cal D}(C)$, called the functional ${\cal D}$-convex hull of $C$, is defined as ${\mathop{\cal{CH}}}^{\cal D}(C)=\left\\{{x\in{\rm I\\!R}^{2}\,\left\|\,{f(x)\leq\sup_{y\in C}f(y)\text{ for all ${\cal D}$-convex }f:{\rm I\\!R}^{2}\rightarrow{\rm I\\!R}}\rule[-5.69046pt]{0.0pt}{11.38092pt}\right.}\right\\}$ A set $C$ is functionally ${\cal D}$-convex if $C={\mathop{\cal{CH}}}^{{\cal D}}(C)$. ###### Definition 3.3 Let ${\cal D}$ be a set of directions. A pair of vectors $v_{1},v_{2}\in{\cal D}$, is a ${{\cal D}\text{-pair}}$, if $v_{2}$ is counterclockwise from $v_{1}$, and there is no vector in ${\cal D}$ between $v_{1}$ and $v_{2}$. Let ${{\cal D}}_{\text{pairs}}$ denote the set of all ${{\cal D}\text{-pair}}$s. Let $\mathop{pspan}(u_{1},u_{2})$ denote the portion of the plane that can be represented as a positive linear combination of $u_{1},u_{2}\in{\cal D}$. Thus $\mathop{pspan}(u_{1},u_{2})$ is the open wedge bounded by the rays emanating from the origin in directions $u_{1},u_{2}$. We define by $(v_{1},v_{2})_{L}=\mathop{pspan}(-v_{1},v_{2})$ and $(v_{1},v_{2})_{R}=\mathop{pspan}(v_{1},-v_{2})$: these are two of the four quadrants of the plane induced by the lines containing $v_{1}$ and $v_{2}$. Similarly, for $v\in{\cal D}$ we denote by ${{v}_{L}}$ and ${{v}_{R}}$ the two open half-planes defined by the line passing through $v$. Let ${\cal{Q}}({\cal D})=\left\\{{{{v}_{L}},{{v}_{R}}\,\left\|\,{v\in{\cal D}}\rule[-5.69046pt]{0.0pt}{11.38092pt}\right.}\right\\}\cup\left\\{{(v_{1},v_{2})_{R},(v_{1},v_{2})_{L}\,\left\|\,{(v_{1},v_{2})\in{D}_{\text{pairs}}}\rule[-5.69046pt]{0.0pt}{11.38092pt}\right.}\right\\}.$ ###### Definition 3.4 For a set $S\subseteq{\rm I\\!R}^{2}$ we denote by $T(S)$ the set of translations of $S$ in the plane, that is $T(S)=\left\\{{S+p\,\left\|\,{p\in{\rm I\\!R}^{2}}\rule[-5.69046pt]{0.0pt}{11.38092pt}\right.}\right\\}$. Given a set of directions ${\cal D}$, let ${\cal T}({\cal D})=\bigcup_{Q\in{\cal{Q}}({\cal D})}T(Q)$. For ${\cal D}_{\rm I\\!R}$, the set ${\cal T}({\cal D}_{\rm I\\!R})$ is the set of all open half-planes. The standard convex hull of a planar point set $S$ can be defined as follows: start from the whole plane, and remove from it all the open half-planes $H^{+}$ such that $H^{+}\cap S=\emptyset$. We extend this definition to handle ${\cal D}$-convexity for an arbitrary set of directions ${\cal D}$, as follows: $\mathop{{{\cal D}}\text{-}{\cal CH}}(S)={\rm I\\!R}^{2}\setminus\left({\bigcup_{I\in{\cal T}({\cal D}),I\cap S=\emptyset}I}\right);$ that is, we remove from the plane all the translations of quadrants and halfplanes in ${\cal{Q}}({\cal D})$ that do not contain a point of $S$. See Figures 4, 5. (a)(b)(c) Figure 4: (a) A set of directions ${\cal D}$, (b) the set of quadrants ${\cal{Q}}({\cal D})$ induced by ${\cal D}$, and (c) the $\mathop{{{\cal D}}\text{-}{\cal CH}}$ of three points. (a)(b)(c) Figure 5: (a) A set of directions ${\cal D}$, such that $\alpha({\cal D})>\pi/2$, (b) the set of quadrants ${\cal{Q}}({\cal D})$ induced by ${\cal D}$, and (c) the $\mathop{{{\cal D}}\text{-}{\cal CH}}$ of a set of points which is not connected. For the case ${\cal D}_{xy}=\left\\{{(0,1),(1,0),(0,-1),(-1,0)}\right\\}$, Matoušek and Plecháč [MP97] showed that if $\mathop{{{\cal D}_{xy}}\text{-}{\cal CH}}(S)$ is connected, then ${\mathop{\cal{CH}}}^{{\cal D}_{xy}}(S)=\mathop{{{\cal D}_{xy}}\text{-}{\cal CH}}(S)$. ###### Definition 3.5 For a set of directions ${\cal D}$, we define the density of ${\cal D}$ to be $\alpha({\cal D})=\max_{(v_{1},v_{2})\in{{\cal D}}_{\text{pairs}}}\alpha(v_{1},v_{2}),$ where $\alpha(v_{1},v_{2})$ denotes the counterclockwise angle from $v_{1}$ to $v_{2}$. See Figure 5, for an example of a set of directions with density larger than $\pi/2$. ###### Corollary 3.6 Let ${\cal D}$ be a set of directions in the plane. Then: * • The set $\mathop{{{\cal D}}\text{-}{\cal CH}}(A)$ is ${\cal D}$-convex, for any $A\subseteq{\rm I\\!R}^{2}$. * • For any $A\subseteq B\subseteq{\rm I\\!R}^{2}$, one has $\mathop{{{\cal D}}\text{-}{\cal CH}}(A)\subseteq\mathop{{{\cal D}}\text{-}{\cal CH}}(B)$. * • For two sets of directions ${\cal D}_{1}\subseteq{\cal D}_{2}$ we have $\mathop{{{\cal D}_{1}}\text{-}{\cal CH}}(S)\subseteq\mathop{{{\cal D}_{2}}\text{-}{\cal CH}}(S)$, for any $S\subseteq{\rm I\\!R}^{2}$. * • Let $S$ be a bounded set in the plane, and let ${\cal D}_{1}\subseteq{\cal D}_{2}\subseteq{\cal D}_{3}\cdots$ be a sequence of sets of directions, such that $\lim_{i\rightarrow\infty}\alpha(D_{i})=0$. Then, $\mathop{int}{{\mathop{\mathrm{CH}}}(S)}\subseteq\lim_{i\rightarrow\infty}\mathop{{{\cal D}_{i}}\text{-}{\cal CH}}(S)\subseteq{\mathop{\mathrm{CH}}}(S)$. ###### Lemma 3.7 Let ${\cal D}$ a set of directions, and let $S$ be a finite set of points in the plane. Then $C=\mathop{{{\cal D}}\text{-}{\cal CH}}(S)$ is a polygonal set whose complexity is $O(\|S\cap{\partial}{C}\|)$. ###### Proof. It is easy to show that $C$ is polygonal. We charge each vertex of $C$ to some point of $S^{\prime}=S\cap{\partial}{C}$. Let $C^{\prime}$ be a connected component of $C$. If $C^{\prime}$ is a single point, then this is a point of $S^{\prime}$. Otherwise, let $e$ be an edge of $C^{\prime}$, and let $I$ be a set in ${\cal T}({\cal D})$ such that $e\subseteq{\partial}{I}$, and $I\cap S=\emptyset$. Since $e$ is an edge of $C^{\prime}$, there is no $q\in{\rm I\\!R}^{2}$ such that $e\subseteq q+I$, and $(q+I)\cap S=\emptyset$. This implies that there must be a point $p$ of $S$ on ${\partial}{I}\cap l_{e}$, where $l_{e}$ is the line passing through $e$. However, $C$ is a ${\cal D}$-convex set, and the direction of $e$ belongs to ${\cal D}$. It follows that $l_{e}$ intersects $C$ along a connected set (i.e., the segment $e$), and $p\in l_{e}\cap C=e$. We charge the edge $e$ to $p$. We claim that a point $p$ of $S^{\prime}$ can be charged at most 4 times. Indeed, for each edge $e^{\prime}$ of $C$ incident to $p$, there is a supporting set in ${\cal T}({\cal D})$, such that $p$ and $e^{\prime}$ lie on its boundary. Only two of those sets can have angle less than $\pi/2$ at $p$ (because such a set corresponds to a ${{\cal D}\text{-pair}}(v_{1},v_{2})$ with $\alpha(v_{1},v_{2})>\pi/2$). Thus, a point of $S^{\prime}$ is charged at most $\max(2\pi/(\pi/2),\pi/(\pi/2)+2)=4$ times. ∎ ###### Lemma 3.8 Let ${\cal D}$ be a set of directions, and let $K$ be a bounded convex body in the plane, such that the expected area of $\mathop{{{\cal D}}\text{-}{\cal CH}}(N)$ of a set $N$ of $n$ points, chosen uniformly and independently from $K$, is at least $\left({1-f(n)}\right)Area(K)$, where $1\geq f(n)\geq 0$, for $n\geq 1$. Then, the expected number of vertices of $C=\mathop{{{\cal D}}\text{-}{\cal CH}}(N)$ is $O(nf(n/2))$. ###### Proof. By Lemma 3.7, the complexity of $C$ is proportional to the number of points of $N$ on the boundary of $C$. Using this observation, it is easy to verify that the proof of Lemma 2.1 can be extended to this case. ∎ We would like to apply the proof of Theorem 2.3 to bound the expected complexity of a random ${\cal D}$-convex hull inside a disk. Unfortunately, if we try to concentrate only on three consecutive sectors (as in Figure 1) it might be that there is a quadrant $I$ of ${\cal T}({\cal D})$ that intersects the middle the middle sector from the side (i.e. through the two adjacent sectors). This, of course, can not happen when working with the regular convexity. Thus, we first would like to decompose the unit disk into “safe” regions, where we can apply a similar analysis as the regular case, and the “unsafe” areas. To do so, we will first show that, with high probability, the $\mathop{{{\cal D}}\text{-}{\cal CH}}$ of a random point set inside a disk, contains a “large” disk in its interior. Next, we argue that this implies that the random $\mathop{{{\cal D}}\text{-}{\cal CH}}$ covers almost the whole disk, and the desired bound will readily follows from the above Lemma. ###### Definition 3.9 For $r\geq 0$, let $B_{r}$ denote the disk of radius of $r$ centered at the origin. ###### Lemma 3.10 Let ${\cal D}$ be a set of directions, such that $0\leq\alpha({\cal D})\leq\pi/2$. Let $N$ be a set of $n$ points chosen uniformly and independently from the unit disk. Then, with probability $1-n^{-10}$ the set $\mathop{{{\cal D}}\text{-}{\cal CH}}(N)$ contains $B_{r}$ in its interior, where $r=1-c\sqrt{\log{n}/n}$, for an appropriate constant $c$. ###### Proof. Let $r^{\prime}=1-c\sqrt{(\log{n})/n}$, where $c$ is a constant to be specified shortly. Let $q$ be any point of $B_{r^{\prime}}$. We bound the probability that $q$ lies outside $C=\mathop{{{\cal D}}\text{-}{\cal CH}}(N)$ as follows: Draw $8$ rays around $q$, such that the angle between any two consecutive rays is $\pi/4$. This partitions $q+B_{r^{\prime\prime}}$, where $r^{\prime\prime}=c\sqrt{(\log{n})/n}$, into eight portions $R_{1},\ldots,R_{8}$, each having area $\pi c^{2}\log{n}/(8n)$. Moreover, $R_{i}\subseteq q+B_{r^{\prime\prime}}\subseteq B_{1}$, for $i=1,\ldots,8$. The probability of a point of $N$ to lie outside $R_{i}$ is $1-c^{2}\log{n}/(8n)$. Thus, the probability that all the points of $N$ lie outside $R_{i}$ is $P\left[{N\cap R_{i}=\emptyset{\rule[-5.69046pt]{0.0pt}{0.0pt}}}\right]\leq\left({1-\frac{c^{2}\log{n}}{8n}}\right)^{n}\leq e^{-(c^{2}\log{n})/8}=n^{-c^{2}/8},$ since $1-x\leq e^{-x}$, for $x\geq 0$. Thus, the probability that one of the $R_{i}$’s does not contain a point of $N$ is bounded by $8n^{-c^{2}/8}$. We claim that if $R_{i}\cap N\neq\emptyset$, for every $i=1,\ldots,8$, then $q\in C$. Indeed, if $q\notin C$ then there exists a set $Q\in{\cal{Q}}({\cal D})$, such that $(q+Q)\cap N=\emptyset$. Since $\alpha({\cal D})\leq\pi/2$ there exists an $i$, $1\leq i\leq 8$, such that $R_{i}\subseteq q+Q$; see Figure 6. This is a contradiction, since $R_{i}\cap N\neq\emptyset$. Thus, the probability that $q$ lies outside $C$ is $\leq 8n^{-c^{2}/8}$. $R_{1}$$R_{2}$$F_{3}$$R_{4}$$R_{5}$$R_{6}$$R_{7}$$R_{8}$$q$$q+Q$ Figure 6: Since $\alpha({\cal D})\leq\pi/2$, any quadrant $Q\in{\cal{Q}}({\cal D})$, when translated by $q$, must contain one of the $R_{i}$’s. Let $N^{\prime}$ denote a set of $n^{10}$ points spread uniformly on the boundary of $B_{r^{\prime}}$. By the above analysis, all the points of $N^{\prime}$ lie inside $C$ with probability at least $1-8n^{10-c^{2}/8}$. Furthermore, arguing as above, we conclude that $B_{r}\subseteq\mathop{{{\cal D}}\text{-}{\cal CH}}(N^{\prime})$, where $r=1-2c\sqrt{(\log{n})/n}$. Hence, with probability at least $1-8n^{10-c^{2}/8}$, $\mathop{{{\cal D}}\text{-}{\cal CH}}(C)$ contains $B_{r}$. The lemma now follows by setting $c=20$, say. ∎ Since the set of directions may contain large gaps, there are points in $B_{1}\setminus B_{r}$ that are “unsafe”, in the following sense: ###### Definition 3.11 Let ${\cal D}$ be a set of directions, and let $0\leq r\leq 1$ be a prescribed constant, such that $0\leq\alpha({\cal D})\leq\pi/2$. A point $p$ in $B_{1}$ is safe, relative to $B_{r}$, if $op\subseteq\mathop{{{\cal D}}\text{-}{\cal CH}}(B_{r}\cup\left\\{{p}\right\\})$. See Figure 7 for an example how the unsafe area looks like. The behavior of the $\mathop{{{\cal D}}\text{-}{\cal CH}}$ inside the unsafe areas is somewhat unpredictable. Fortunately, those areas are relatively small. $B_{r}$$B_{1}$$T$$\overrightarrow{v_{1}}$$\overrightarrow{v_{2}}$$o$ Figure 7: The dark areas are the unsafe areas for a consecutive pairs of directions $v_{1},v_{2}\in{\cal D}$. ###### Lemma 3.12 Let ${\cal D}$ be a set of directions, such that $0\leq\alpha({\cal D})\leq\pi/2$, and let $r=1-O\left({\sqrt{(\log{n})/n}}\right)$. The unsafe area in $B_{1}$, relative to $B_{r}$, can be covered by a union of $O(1)$ caps. Furthermore, the length of the base of such a cap is $O(((\log{n})/n)^{1/4})$, and its height is $O\left({\sqrt{(\log{n})/n}}\right)$. ###### Proof. Let $p$ be an unsafe point of $B_{1}$. Let $\overrightarrow{v_{1}},\overrightarrow{v_{2}}$ be the consecutive pair of vectors in ${\cal D}$, such that the vector $\overrightarrow{po}$ lies between them. If $\mathop{ray}(p,\overrightarrow{v_{1}})\cap B_{r}\neq\emptyset$, and $\mathop{ray}(p,\overrightarrow{v_{2}})\cap B_{r}\neq\emptyset$ then $po\subseteq{\mathop{\mathrm{CH}}}\left({\left\\{{p,o,p_{1},p_{2}}\right\\}}\right)\subseteq\mathop{{{\cal D}}\text{-}{\cal CH}}(B_{r}\cup\left\\{{p}\right\\})$, for any pair of points $p_{1}\in B_{r}\cap\mathop{ray}(p,\overrightarrow{v_{1}}),p_{2}\in B_{r}\cap\mathop{ray}(p,\overrightarrow{v_{2}})$. Thus, $p$ is unsafe only if one of those two rays miss $B_{r}$. Since $p$ is close to $B_{r}$, the angle between the two tangents to $B_{r}$ emanating from $p$ is close to $\pi$. This implies that the angle between $\overrightarrow{v_{1}}$ and $\overrightarrow{v_{2}}$ is at least $\pi/4$ (provided $n$ is a at least some sufficiently large constant), and the number of such pairs is at most $8$. The area in the plane that sees $o$ in a direction between $\overrightarrow{v_{1}}$ and $\overrightarrow{v_{2}}$, is a quadrant $Q$ of the plane. The area in $Q$ which is is safe, is a parallelogram $T$. Thus, the unsafe area in $B_{1}$ that induced by the pair $\overrightarrow{v_{1}}$ and $\overrightarrow{v_{2}}$ is $(B_{1}\cap Q)\setminus T$. Since $\alpha({\cal D})\leq\pi/2$, this set can covered with two caps of $B_{1}$ with their base lying on the boundary of $B_{r}$. See Figure 7. The height of such a cap is $1-r=O\left({\sqrt{\frac{\log{n}}{n(\pi-\alpha)}}}\right)$, and the length of the base of such a cap is $2\sqrt{1-r^{2}}=O\left({\left({\frac{\log{n}}{n(\pi-\alpha)}}\right)^{1/4}}\right)$. ∎ The proof of Lemma 3.12 is where our assumption that $\alpha({\cal D})\leq\pi/2$ plays a critical role. Indeed, if $\alpha({\cal D})>\pi/2$, then the unsafe areas in $B_{1}\setminus B_{r}$ becomes much larger, as indicated by the proof. ###### Theorem 3.13 Let ${\cal D}$ be a set of directions, such that $0\leq\alpha({\cal D})\leq\pi/2$. The expected number of vertices of $\mathop{{{\cal D}}\text{-}{\cal CH}}(N)$, where $N$ is a set of $n$ points, chosen uniformly and independently from the unit disk, is $O\left({n^{1/3}+\sqrt{n\alpha({\cal D})}}\right)$. ###### Proof. We claim that the expected area $\mathop{{{\cal D}}\text{-}{\cal CH}}(N)$ is at least $\pi-O\left({n^{-2/3}+\sqrt{\alpha/n}}\right)$, where $\alpha=\alpha({\cal D})$. The theorem will then follow from Lemma 3.8. Indeed, let $m$ be an integer to be specified later, and assume, without loss of generality, that $m$ divides $n$. Partition $B$ into $m$ congruent sectors, ${\cal S}_{1},\ldots,{\cal S}_{m}$. Let $B^{1},\ldots,B^{\mu}$ denote the $\mu=n/m$ disks centered at the origin, such that (i) $B^{1}=B_{1}$, and (ii) $Area(B^{i-1})-Area(B^{i})=\pi/\mu$, for $i=2,\ldots,\mu$. Let $r_{i}$ denote the radius of $B^{i}$, for $i=1,\ldots,\mu$. Note333 $Area(B^{1})-Area(B^{2})=\pi(1-r_{2}^{2})=\pi/\mu$, thus $r_{2}^{2}=1-1/\mu$. We have $r_{2}\leq 1-1/(2\mu)$, and $r_{1}-r_{2}\geq 1-(1-1/(2\mu))=1/(2\mu)$., that $r_{i}-r_{i+1}\geq r_{i-1}-r_{i}\geq 1/(2\mu)$, for $i=2,\ldots,\mu-1$. Let $r=1-O\left({\sqrt{(\log{n})/n}}\right)$, and let $U$ be the set of sectors that either intersect an unsafe area of $B$ relative to $B_{r}$, or their neighboring sectors intersect the unsafe area of $B$. By Lemma 3.12, the number of sectors in $U$ is $O(1)\cdot O\left({\frac{((\log{n})/n)^{1/4}}{(2\pi/m)}}\right)=O(m((\log{n})/n)^{1/4})$. Let $S_{i,j}=(B^{i}\setminus B^{i+1})\cap{\cal S}_{j}$, and $S_{\mu,j}=B^{\mu}\cap{\cal S}_{j}$, for $i=1,\ldots,\mu-1$, and $j=1,\ldots,m$. The set $S_{i,j}$ is called the $i$-th tile of the sector ${\cal S}_{j}$, and its area is $\pi/n$, for $i=1,\ldots,\mu$, and $j=1,\ldots,m$. Let $X_{j}$ denote the first index $i$ such that $N\cap S_{i,j}\neq\emptyset$, for $j=1,\ldots,m$. The probability that $X_{j}=k$ is upper-bounded by the probability that the tiles $S_{1,j},\ldots,S_{(k-1),j}$ do not contain any point of $N$; namely, by $\left({1-\frac{k-1}{n}}\right)^{n}$. Thus, $P[X_{j}=k]\leq\left({1-\frac{k-1}{n}}\right)^{n}\leq e^{-(k-1)}$. Thus, $E\left[{X_{j}{\rule[-5.69046pt]{0.0pt}{0.0pt}}}\right]=\sum_{k=1}^{\mu}kP[X_{j}=k]\leq\sum_{k=1}^{\mu}ke^{-(k-1)}=O(1),$ for $j=1,\ldots,m$. Let $C$ denote the set $\mathop{{{\cal D}}\text{-}{\cal CH}}(N\cup B_{r})$. The tile $S_{i,j}$ is exposed by a set $K$, if $S_{i,j}\setminus K\neq\emptyset$. We claim that the expected number of tiles exposed by $C$ in a section $S_{j}\notin U$ is at most $X_{j-1}+X_{j+1}+O(\mu/m^{2}+\alpha\mu/m)$, for $j=1,\ldots,m$ (where we put $X_{0}=X_{m}$, $X_{m+1}=X_{1}$). Indeed, let $w=max(X_{j-1},X_{j+1})$, and let $p,q$ be the two points in $S_{j-1,w},S_{j+1,w}$, respectively, such that the number of sets exposed by the triangle $T=\triangle{opq}$, in the sector ${\cal S}_{j}$, is maximal. Both $p$ and $q$ lie on ${\partial}{B^{w+1}}$ and on the external radii bounding ${\cal S}_{j-1}$ and ${\cal S}_{j+1}$, as shown in Figure 1. Let $s$ denote the segment connecting the midpoint $\rho$ of the base of $T$ to its closest point on ${\partial}{B^{w}}$. The number of tiles in ${\cal S}_{j}$ exposed by $T$ is bounded by $w$, plus the number of tiles intersecting the segment $s$. The length of $s$ is $\|oq\|-\|oq\|\cos\left({\frac{3}{2}\cdot\frac{2\pi}{m}}\right)\leq 1-\cos\left({\frac{3\pi}{m}}\right)\leq\frac{1}{2}\left({\frac{3\pi}{m}}\right)^{2}=\frac{4.5\pi^{2}}{m^{2}},$ since $\cos{x}\geq 1-x^{2}/2$, for $x\geq 0$. On the other hand, the segment $s$ intersects at most $\left\lceil{\|\|s\|\|/(1/(2\mu))}\right\rceil=O(\mu/m^{2})$ tiles, and we have that the number of tiles exposed in the sector ${\cal S}_{i}$ by $T$ is at most $w+O(\mu/m^{2})$, for $j=1,\ldots,m$. Since ${\cal S}_{j}\notin U$, the points $p,q$ are safe, and $op,oq\subseteq C$. This implies that the only additional tiles that might be exposed in ${\cal S}_{j}$ by $C$, are exposed by the portion of the boundary of $C$ between $p$ and $q$ that lie inside $T$. Let $V$ be the circular cap consisting of the points in $T$ lying between $pq$ and a circular arc $\gamma\subseteq T$, connecting $p$ to $q$, such that for any point $p^{\prime}\in\gamma$ one has $\angle{pp^{\prime}q}=\pi-\alpha$. See Figure 8. $T$$V$$p$$q$$\geq\pi-\alpha$ Figure 8: The portion of $T$ that can be removed by a quadrant $Q$ of ${\cal T}({\cal D})$, is covered by the darkly-shaded circular cap, such that any point on its bounding arc creates an angle $\pi-\alpha$ with $p$ and $q$. Let $Q\in{\cal T}({\cal D})$ be any quadrant of the plane induce by ${\cal D}$, such that $Q\cap N=\emptyset$ (i.e. $C\cap Q=\emptyset$), and $Q\cap T\neq\emptyset$. Then, $Q\cap op=\emptyset,Q\cap oq=\emptyset$ since $p$ and $q$ are safe. Moreover, the angle of $Q$ is at least $\pi-\alpha$, which implies that $Q\cap T\subseteq V$. See Figure 8. Let $s^{\prime}$ be the segment $o\rho\cap V$, where $\rho$ is as above, the midpoint of $pq$. The length of $s^{\prime\prime}$ is $\|s^{\prime}\|\leq\sin\left({\frac{3}{2}\cdot\frac{2\pi}{m}}\right)\tan{\frac{\alpha}{2}}\leq\frac{3\pi}{m}\frac{\sqrt{2}\alpha}{2}\leq\frac{3\pi\alpha}{m},$ since $\sin{x}\leq x$, for $x\geq 0$, and $1/\sqrt{2}\leq\cos{(\alpha/2)}$ (because $0\leq\alpha\leq\pi/2$). Thus, the expected number of tiles exposed by $C$, in a sector ${\cal S}_{j}\notin U$, is bounded by $X_{j-1}+X_{j+1}+O\left({\frac{\mu}{m^{2}}}\right)+O\left({\frac{3\pi\alpha/m}{1/(2\mu)}}\right)=X_{j-1}+X_{j+1}+O\left({\frac{\mu}{m^{2}}}\right)+O\left({\frac{\alpha\mu}{m}}\right).$ Thus, the expected number of tiles exposed by $C$, in sectors that do not belong to $U$, is at most $E\left[{\sum_{j=1}^{m}\left({X_{j-1}+X_{j+1}+O\left({\frac{\mu}{m^{2}}}\right)+O\left({\frac{\alpha\mu}{m}}\right)}\right){\rule[-5.69046pt]{0.0pt}{0.0pt}}}\right]=O\left({m+\frac{\mu}{m}+\alpha\mu}\right).$ Adding all the tiles that lie outside $B_{r}$ in the sectors that belong to $U$, it follows that the expected number of tiles exposed by $C$ is at most $\displaystyle O$ $\displaystyle\left({m+\frac{\mu}{m}+\alpha\mu+\|U\|\cdot\frac{1-r}{1/2\mu}}\right)=O\left({m+\frac{\mu}{m}+\alpha\mu+m\left({\frac{\log{n}}{n}}\right)^{1/4}\cdot\mu\sqrt{\left({\frac{\log{n}}{n}}\right)}}\right)$ $\displaystyle=$ $\displaystyle O\left({m+\frac{n}{m^{2}}+\frac{\alpha n}{m}+n\left({\frac{\log{n}}{n}}\right)^{3/4}}\right)=O\left({m+\frac{n}{m^{2}}+\frac{\alpha n}{m}+n^{1/4}\log^{3/4}{n}}\right).$ Setting $m=\max{\left({n^{1/3},\sqrt{n\alpha}}\right)}$, we conclude that the expected number of tiles exposed by $C$ is $O\left({n^{1/3}+\sqrt{n\alpha}}\right)$. The area of $C^{\prime}=\mathop{{{\cal D}}\text{-}{\cal CH}}(N)$ is bounded from below by the area of the tiles which are not exposed by $C^{\prime}$. The probability that $C^{\prime}\neq C$ (namely, that the disk $B_{r}$ is not inside $C^{\prime}$) is at most $n^{-10}$, by Lemma 3.10. Hence the expected area of $C^{\prime}$ is at least $E[Area(C)]-Prob\left[{C\neq C^{\prime}{\rule[-5.69046pt]{0.0pt}{0.0pt}}}\right]\pi=\pi-O\left({n^{1/3}+\sqrt{n\alpha}}\right)\frac{\pi}{n}-n^{-10}\pi=\pi-O\left({n^{-2/3}+\sqrt{\frac{\alpha}{n}}\;}\right).$ The assertion of the theorem now follows from Lemma 3.8. ∎ The expected complexity of the $\mathop{{{\cal D}_{xy}}\text{-}{\cal CH}}$ of $n$ points, chosen uniformly and independently from the unit square, is $O(\log{n})$ (Lemma 2.4). Unfortunately, this is a degenerate case for a set of directions with $\alpha({\cal D})=\pi/2$, as the following corollary testifies: ###### Corollary 3.14 Let ${\cal D}_{xy}^{\prime}$ be the set of directions resulting from rotating ${\cal D}_{xy}$ by 45 degrees. Let $N$ be a set of $n$ points, chosen independently and uniformly from the unit square ${S^{\prime}}$. The expected complexity of $\mathop{{{\cal D}_{xy}^{\prime}}\text{-}{\cal CH}}(N)$ is $\Omega\left({\sqrt{n}}\right)$. ###### Proof. Without loss of generality, assume that $n=m^{2}$ for some integer $m$. Tile ${S^{\prime}}$ with $n$ translated copies of a square of area $1/n$. Let ${\cal S}_{1},\ldots,{\cal S}_{m}$ denote the squares in the top raw of this tiling, from left to right. Let $A_{j}$ denote the event that ${\cal S}_{j}$ contains a point of $N$, and neither of the two adjacent squares $S_{j-1},S_{j+1}$ contains a point of $N$, for $j=2,\ldots,m-1$. We have $Prob\left[{A_{j}{\rule[-5.69046pt]{0.0pt}{0.0pt}}}\right]=Prob\left[{{\cal S}_{j+1}\cap N=\emptyset\text{ and }{\cal S}_{j-1}\cap N=\emptyset{\rule[-5.69046pt]{0.0pt}{0.0pt}}}\right]-Prob\left[{({\cal S}_{j-1}\cup{\cal S}_{j}\cup{\cal S}_{j+1})\cap N=\emptyset{\rule[-5.69046pt]{0.0pt}{0.0pt}}}\right],$ for $j=2,\ldots,m-1$. Hence, $\lim_{n\rightarrow\infty}Prob\left[{A_{j}{\rule[-5.69046pt]{0.0pt}{0.0pt}}}\right]=\lim_{n\rightarrow\infty}\left({\left({1-\frac{2}{n}}\right)^{n}-\left({1-\frac{3}{n}}\right)^{n}}\right)=e^{-2}-e^{-3}\approx 0.0855$ $q$$q+Q_{top}$${\cal S}_{j-1}$${\cal S}_{j}$${\cal S}_{j+1}$ Figure 9: If $A_{j}$ happens, then the squares ${\cal S}_{j-1},{\cal S}_{j+1}$ do not contain a point of $N$. Thus, if $q$ is the highest point in ${\cal S}_{j}$, then $q+Q_{top}$ can not contain a point of $N$, and $q$ is a vertex of $\mathop{{{\cal D}_{xy}^{\prime}}\text{-}{\cal CH}}(N)$. This implies, that for $n$ large enough, $Prob\left[{A_{j}{\rule[-5.69046pt]{0.0pt}{0.0pt}}}\right]\geq 0.01$. Thus, the expected value of $Y$ is $\Omega(m)=\Omega\left({\sqrt{n}}\right)$, where $Y$ is the number of $A_{j}$’s that have occurred, for $j=2,\ldots,m-1$. However, if $A_{j}$ occurs, then $C=\mathop{{{\cal D}_{xy}^{\prime}}\text{-}{\cal CH}}(N)$ must have a vertex at ${\cal S}_{j}$. Indeed, let $Q_{top}$ denote the quadrant of ${\cal{Q}}({\cal D}_{xy}^{\prime})$ that contains the positive $y$-axis. If we translate $Q_{top}$ to the highest point in $S_{j}\cap N$, then it does not contain a point of $N$ in its interior, implying that this point is a vertex of $C$, see Figure 9. This implies that the expected complexity of $\mathop{{{\cal D}_{xy}^{\prime}}\text{-}{\cal CH}}(N)$ is $\Omega\left({\sqrt{n}}\right)$ ∎ ## 4 On the Expected Number of Points on the Boundary of the Quadrant Hull Inside a Hypercube In this section, we show that the expected number of points on the boundary of the quadrant hull of a set $S$ of $n$ points, chosen uniformly and independently from the unit cube is $O(\log^{d-1}n)$. Those bounds are known [BKST78], but we believe the new proof is simpler. ###### Definition 4.1 ([MP97]) Let ${\cal Q}$ be a family of subsets of ${\rm I\\!R}^{d}$. For a set $A\subseteq{\rm I\\!R}^{d}$, we define the ${\cal Q}$-hull of $A$ as $\mathrm{\mathop{{{\cal Q}}\text{-}{co}}}(A)=\bigcap\left\\{{Q\in{\cal Q}\,\left\|\,{A\subseteq Q}\rule[-5.69046pt]{0.0pt}{11.38092pt}\right.}\right\\}.$ ###### Definition 4.2 ([MP97]) For a sign vector $s\in\left\\{{-1,+1}\right\\}^{d}$, define $q_{s}=\left\\{{x\in{\rm I\\!R}^{d}\,\left\|\,{\mathop{\mathrm{sign}}(x_{i})=s_{i},\text{ for }i=1,\ldots,d}\rule[-5.69046pt]{0.0pt}{11.38092pt}\right.}\right\\},$ and for $a\in{\rm I\\!R}^{d}$, let $q_{s}(a)=q_{s}+a$. We set ${\cal Q}_{sc}=\left\\{{{\rm I\\!R}^{d}\setminus q_{s}(a)\,\left\|\,{a\in{\rm I\\!R}^{d},s\in\left\\{{-1,+1}\right\\}^{d}}\rule[-5.69046pt]{0.0pt}{11.38092pt}\right.}\right\\}$. We shall refer to $\mathrm{\mathop{{{\cal Q}_{sc}}\text{-}{co}}}(A)$ as the quadrant hull of $A$. These are all points which cannot be separated from $A$ by any open orthant in space (i.e., quadrant in the plane). ###### Definition 4.3 Given a set of points $S\subseteq{\rm I\\!R}^{d}$, a point $p\in{\rm I\\!R}^{d}$ is ${\cal Q}_{sc}$-exposed, if there is $s\in\left\\{{-1,+1}\right\\}^{d}$, such that $q_{s}(p)\cap S=\emptyset$. A set $C$ is ${\cal Q}_{sc}$-exposed, if there exists a point $p\in C$ which is ${\cal Q}_{sc}$-exposed. ###### Definition 4.4 For a set $S\subseteq{\rm I\\!R}^{d}$, let $n_{sc}(S)$ denote the number of points of $S$ on the boundary of $\mathrm{\mathop{{{\cal Q}_{sc}}\text{-}{co}}}(S)$. ###### Theorem 4.5 Let ${\cal C}$ be a unit axis parallel hypercube in ${\rm I\\!R}^{d}$, and let $S$ be a set of $n$ points chosen uniformly and independently from ${\cal C}$. Then, the expected number of points of $S$ on the boundary of $H=\mathrm{\mathop{{{\cal Q}_{sc}}\text{-}{co}}}(S)$ is $O(\log^{d-1}(n))$. ###### Proof. We partition ${\cal C}$ into equal size tiles, of volume $1/n^{d}$; that is $C(i_{1},i_{2},\ldots,i_{d})=[(i_{1}-1)/n,i_{1}/n]\times\cdots\times[(i_{d}-1)/n,i_{d}/n]$, for $1\leq i_{1},i_{2},\ldots,i_{d}\leq n$. We claim that the expect number of tiles in our partition of ${\cal C}$ which are exposed by $S$ is $O(n^{d-1}\log^{d-1}n)$. Indeed, let $q=q_{(-1,-1,\ldots,-1)}$ be the “negative” quadrant of ${\rm I\\!R}^{d}$. Let $X(i_{2},\ldots,i_{d})$ be the maximal integer $k$, for which $C(k,i_{2},\ldots,i_{d})$ is exposed by $q$. The probability that $X(i_{2},\ldots,i_{d})\geq k$ is bounded by the probability that the cubes $C(l_{1},l_{2},\ldots,l_{d})$ does not contain a point of $S$, where $l_{1}<k,l_{2}<i_{2},\ldots,l_{d}<i_{d}$. Thus, $\displaystyle\Pr\left[{X(i_{2},\ldots,i_{d})\geq k{\rule[-5.69046pt]{0.0pt}{0.0pt}}}\right]$ $\displaystyle\leq$ $\displaystyle\left({1-\frac{(k-1)(i_{2}-1)\cdots(i_{d}-1)}{n^{d}}}\right)^{n}$ $\displaystyle\leq$ $\displaystyle\exp\left({-\frac{{(k-1)(i_{2}-1)\cdots(i_{d}-1)}}{n^{d-1}}}\right),$ since $1-x\leq e^{-x}$, for $x\geq 0$. Hence, the probability that $\Pr\left[{X(i_{2},\ldots,i_{d})\geq i\cdot m+1{\rule[-5.69046pt]{0.0pt}{0.0pt}}}\right]\leq e^{-i}$, where $m=\left\lceil{\frac{n^{d-1}}{(i_{2}-1)\cdots(i_{d}-1)}}\right\rceil$. Thus, $\displaystyle E\left[{X(i_{2},\ldots,i_{d}){\rule[-5.69046pt]{0.0pt}{0.0pt}}}\right]$ $\displaystyle=$ $\displaystyle\sum_{i=1}^{\infty}i\Pr\left[{X(i_{2},\ldots,i_{d})=i{\rule[-5.69046pt]{0.0pt}{0.0pt}}}\right]=\sum_{i=0}^{\infty}\sum_{j=im+1}^{(i+1)m}j\Pr\left[{X(i_{2},\ldots,i_{d})=j{\rule[-5.69046pt]{0.0pt}{0.0pt}}}\right]$ $\displaystyle\leq$ $\displaystyle\sum_{i=0}^{\infty}(i+1)m\Pr\left[{X(i_{2},\ldots,i_{d})\geq im+1{\rule[-5.69046pt]{0.0pt}{0.0pt}}}\right]\leq\sum_{i=0}^{\infty}(i+1)me^{-i}=O(m).$ Let $r$ denote the expected number of tiles exposed by $q$ in ${\cal C}$. If $C(i_{1},\ldots,i_{d})$ is exposed by $q$, then $X(i_{2},\ldots,i_{d})\geq i_{1}$. Thus, one can bound $r$ by the number of tiles on the boundary of ${\cal C}$, plus the sum of the expectations of the variables $X(i_{2},\ldots,i_{d})$. We have $\displaystyle r$ $\displaystyle=$ $\displaystyle O(n^{d-1})+\sum_{i_{2}=2}^{n-1}\sum_{i_{3}=2}^{n-1}\cdots\sum_{i_{d}=2}^{n-1}O\left({\frac{n^{d-1}}{(i_{2}-1)(i_{3}-1)\cdots(i_{d}-1)}}\right)$ $\displaystyle=$ $\displaystyle O\left({n^{d-1}}\right)\sum_{i_{2}=2}^{n-1}\frac{1}{i_{2}-1}\sum_{i_{3}=2}^{n-1}\frac{1}{i_{3}-1}\cdots\sum_{i_{d}=2}^{n-1}\frac{1}{i_{d}-1}=O\left({n^{d-1}\log^{d-1}{n}}\right).$ The set ${\cal Q}_{sc}$ contains translation of $2^{d}$ different quadrants. This implies, by symmetry, that the expected number of tiles exposed in ${\cal C}$ by $S$ is $O\left({2^{d}n^{d-1}\log^{d-1}{n}}\right)=O\left({n^{d-1}\log^{d-1}{n}}\right)$. However, if a tile is not exposed by any $q_{s}$, for $s\in\left\\{{-1,+1}\right\\}^{d}$, then it lies in the interior of $H$. Implying that the expected volume of $H$ is at least $\frac{n^{d}-O\left({n^{d-1}\log^{d-1}{n}}\right)}{n^{d}}=1-O\left({\frac{\log^{d-1}n}{n}}\right).$ We now apply an argument similar to the one used in Lemma 2.1 (Efron’s Theorem), and the theorem follows. ∎ ###### Remark 4.6 A point $p$ of $S$ is a maxima, if there is no point $p^{\prime}$ in $S$, such that $p_{i}\leq p^{\prime}_{i}$, for $i=1,\ldots,d$. Clearly, a point which is a maxima, is also on the boundary of $\mathrm{\mathop{{{\cal Q}_{sc}}\text{-}{co}}}(S)$. By Theorem 4.5, the expected number of maxima in a set of $n$ points chosen independently and uniformly from the unit hypercube in ${\rm I\\!R}^{d}$ is $O(\log^{d-1}n)$. This was also proved in [BKST78], but we believe that our new proof is simpler. Also, as noted in [BKST78], a vertex of the convex hull of $S$ is a point of $S$ lying on the boundary of the $\mathrm{\mathop{{{\cal Q}_{sc}}\text{-}{co}}}(S)$. Hence, the expected number of vertices of the convex hull of a set of $n$ points chosen uniformly and independently from a hypercube in ${\rm I\\!R}^{d}$ is $O(\log^{d-1}n)$. ### Acknowledgments I wish to thank my thesis advisor, Micha Sharir, for his help in preparing this manuscript. I also wish to thank Pankaj Agarwal, and Imre Bárány for helpful discussions concerning this and related problems. ## References * [BKST78] J. L. Bentley, H. T. Kung, M. Schkolnick, and C. D. Thompson. On the average number of maxima in a set of vectors and applications. Journal of the ACM, 25:536–543, 1978. * [Efr65] B. Efron. The convex hull of a random set of points. Biometrika, 52(3):331–343, 1965. * [MP97] J. Matoušek and P. Plecháč. On functional separately convex hulls. To appear in Discrete Comput. Geom., 1997. * [OSSW85] T. Ottmann, E. Soisalon-Soininen, and D. Wood. On the definition and computation of rectilinear convex hulls. Inform. Sci., 33:157–171, 1985. * [PS85] F. P. Preparata and M. I. Shamos. Computational Geometry: An Introduction. Springer-Verlag, 1985. * [Ray70] H. Raynaud. Sur l’enveloppe convex des nuages de points aleatoires dans $R^{n}$. J. Appl. Probab., 7:35–48, 1970. * [RS63] A. Rényi and R. Sulanke. Über die konvexe Hülle von $n$ zufällig gerwähten Punkten I. Z. Wahrsch. Verw. Gebiete, 2:75–84, 1963. * [RW87] G. J. E. Rawlins and D. Wood. Optimal computation of finitely oriented convex hulls. Inform. Comput., 72:150–166, 1987. * [RW88] G. J. E. Rawlins and D. Wood. Computational geometry with restricted orientations. In Proc. 13th IFIP Conf. System Modelling and Optimization, volume 113 of Lecture Notes in Control and Information Science, pages 375–384. Springer-Verlag, 1988. * [WW93] W. Weil and J. A. Wieacker. Stochastic geometry. In P. M. Gruber and J. M. Wills, editors, Handbook of Convex Geometry, volume B, chapter 5.2, pages 1393–1438. North-Holland, 1993.	arxiv-papers	2011-11-22T21:17:34	2024-09-04T02:49:24.601234	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Sariel Har-Peled", "submitter": "Sariel Har-Peled", "url": "https://arxiv.org/abs/1111.5340" }
1111.5385	# The elliptic flow in Au+Au collisions at $\sqrt{\mathrm{s}_{{}_{NN}}}$ = 7.7, 11.5 and 39 GeV at STAR††thanks: Presented at the conference ‘Strangeness in Quark Matter 2011’, Cracow, Poland, September 18-24, 2011 Shusu Shi (for the STAR collaboration) Institute of Particle Physics, Central China Normal University, Wuhan, Hubei, 430079, China The Key Laboratory of Quark and Lepton Physics (Central China Normal University), Ministry of Education, Wuhan, Hubei, 430079, China ###### Abstract We present elliptic flow, $v_{2}$, measurements for charged and identified particles at midrapidity in Au+Au collisions at $\sqrt{\mathrm{s}_{{}_{NN}}}$ = 7.7, 11.5 and 39 GeV at STAR. We compare the inclusive charged hadron $v_{2}$ to those from high energies at RHIC ($\sqrt{\mathrm{s}_{{}_{NN}}}$ = 62.4 and 200 GeV), at LHC ($\sqrt{\mathrm{s}_{{}_{NN}}}$ = 2.76 TeV). A significant difference in $v_{2}$ between baryons and anti-baryons is observed and the difference increases with decreasing beam energy. We observed the $v_{2}$ of $\phi$ meson is systematically lower than other particles in Au+Au collisions at $\sqrt{\mathrm{s}_{{}_{NN}}}$ = 11.5 GeV. 25.75.Ld, 25.75.Dw ## 1 Introduction The main physics motivation for the Beam Energy Scan at RHIC-STAR experiment is searching for the phase boundary and critical point predicted by QCD theory. The elliptic flow, $v_{2}$, which is generated by the initial anisotropy in the coordinate space, is defined by $v_{2}=\langle\cos 2(\phi-\Psi_{R})\rangle$ (1) , where $\phi$ is azimuthal angle of an outgoing particle, $\Psi_{R}$ is the azimuthal angle of the impact parameter, and angular brackets denote an average over many particles and events. Due to the self-quenching effect, it is sensitive to the early stage of the heavy ion collisions [1]. The number of constituent quark (NCQ) scaling observed in the top energy of RHIC Au+Au and Cu+Cu collisions [2, 3, 5] reflects that the collectivity has been built up at the partonic stage. Especially, the NCQ scaling for multi-strange hadrons, $\phi$ ($s\overline{s}$) and $\Omega$ ($sss$) supports the deconfinement and partonic collectivity picture [4, 6]. A study based on a multi-phase transport model (AMPT) indicates the NCQ scaling is related to the degrees of freedom in the system [7]. The scaling and no scaling in $v_{2}$ reflects the partonic and hadronic degrees of freedom respectively. The importance of $\phi$ meson has been emphasized, where the $\phi$ meson $v_{2}$ could be small or zero without partonic phase [8]. Thus, the measurements of elliptic flow with the Beam Energy Scan data offer us the opportunity to investigate the QCD phase boundary. Figure 1: (Color online) The top panels show the $v_{2}\\{4\\}$ vs. $p_{T}$ at midrapidity for various beam energies ($\sqrt{\mathrm{s}_{{}_{NN}}}$ = 7.7 GeV to 2.76 TeV). The results for $\sqrt{\mathrm{s}_{{}_{NN}}}$ = 7.7 to 200 GeV are for $\mathrm{Au+Au}$ collisions and those for 2.76 TeV are for Pb + Pb collisions. The red line shows the fit to the results from $\mathrm{Au+Au}$ collisions at $\sqrt{\mathrm{s}_{{}_{NN}}}$ = 200 GeV. The bottom panels show the ratio of $v_{2}\\{4\\}$ vs. $p_{T}$ for all $\sqrt{\mathrm{s}_{{}_{NN}}}$ with respect to this fitted line. The results are shown for three collision centrality classes: $10-20\%$ (a1), $20-30\%$ (b1) and $30-40\%$ (c1). In this proceedings, we present the $v_{2}$ results of charged and identified hadrons by the STAR experiment from Au+Au collisions in $\sqrt{\mathrm{s}_{{}_{NN}}}$ = 7.7, 11.5 and 39 GeV. STAR’s Time Projection Chamber (TPC) [9] is used as the main detector for event plane determination. The centrality was determined by the number of tracks from the pseudorapidity region $\|\eta\|\leq 0.5$. The particle identification for $\pi^{\pm}$, $K^{\pm}$ and $p~{}(\overline{p})$ is achieved via the energy loss in the TPC and the time of flight information from the multi-gap resistive plate chamber detector [10]. Strange hadrons are reconstructed with the decay channels: ${K}^{0}_{S}$ $\rightarrow\pi^{+}+\pi^{-}$, $\phi\rightarrow K^{+}+K^{-}$, $\Lambda$ $\rightarrow p+\pi^{-}$ ($\overline{\Lambda}\rightarrow\overline{p}+\pi^{+}$), and $\Xi^{-}\rightarrow$ $\Lambda$ $+\ \pi^{-}$ ($\overline{\Xi}^{+}\rightarrow$ $\overline{\Lambda}$\+ $\pi^{+}$)). The detailed description of the procedure can be found in Refs. [2, 3, 11]. The event plane method [12] and cumulant method [13, 14] are used for the $v_{2}$ measurement. ## 2 Results and Discussions Figure 2: (Color online) The difference of $v_{2}$ for particles and anti- particles ($v_{2}(X)-v_{2}(\overline{X})$) divided by particle $v_{2}$ ($v_{2}(X)$) as a function of beam energy in Au+Au collisions (0-80%). Figure 3: (Color online) The number of constituent quark ($n_{\rm cq}$) scaled $v_{2}$ as a function of transverse kinetic energy over $n_{\rm cq}$ ($(m_{T}-m_{0})/n_{\rm cq}$) for various identified particles in Au+Au (0-80%) collisions at $\sqrt{\mathrm{s}_{{}_{NN}}}$ = 11.5 and 39 GeV. The Beam Energy Scan data from RHIC-STAR experiment, offer a opportunity to study the beam energy dependence of $v_{2}$ in a wide range of beam energy. Figure 1 shows the results of transverse momentum ($p_{T}$) dependence of $v_{2}\\{4\\}$ for charged hadrons from $\sqrt{\mathrm{s}_{{}_{NN}}}$ = 7.7 GeV to 2.76 TeV in $10-20\%$ (a1), $20-30\%$ (b1) and $30-40\%$ (c1) centrality bins, where the ALICE results in Pb + Pb collisions at $\sqrt{s_{NN}}$ = 2.76 TeV are taken from [15]. At low $p_{T}$ ($p_{T}<2~{}\mbox{$\mathrm{GeV}/c$}$), the $v_{2}$ values increases with increase in beam energy. Beyond $p_{T}=2~{}\mbox{$\mathrm{GeV}/c$}$ the $v_{2}$ results show comparable values with in statistic errors. There is no saturation signal of $v_{2}$ up to collisions at $\sqrt{\mathrm{s}_{{}_{NN}}}$ = 2.76 TeV. Figure 2 shows excitation function for the relative difference of $v_{2}$ between particles and anti-particles. Here, to reduce the non-flow effect, the $\eta$-sub event plane method is used to calculate $v_{2}$. The $\eta$-sub event plane method is similar to the event plane method, except one defines the flow vector for each particle based on particles measured in the opposite hemisphere in pseudorapidity. An $\eta$ gap of $\|\eta\|<0.05$ is used between negative/positive $\eta$ sub-event to guarantee that non-flow effects are reduced by enlarging the separation between the correlated particles. The difference for baryon and anti-baryon (protons and $\Lambda$s) could be observed from 7.7 to 62.4 GeV. The difference of $v_{2}$ for baryons is within 10% at $\sqrt{\mathrm{s}_{{}_{NN}}}$ = 39 and 62.4 GeV, while a significant difference is observed below $\sqrt{\mathrm{s}_{{}_{NN}}}$ = 39 GeV. For example, the difference of protons versus anti-protons is around 60%. There is no obvious difference for $\pi^{+}$ versus $\pi^{-}$ (within 3%) and $K^{+}$ versus $K^{-}$ (within 2%) when $\sqrt{\mathrm{s}_{{}_{NN}}}$ = 39 GeV. As the decrease of beam energy, $\pi^{+}$ versus $\pi^{-}$ and $K^{+}$ versus $K^{-}$ start to show the difference. The $v_{2}$ of $\pi^{-}$ is larger than that of $\pi^{+}$ and the $v_{2}$ of $K^{+}$ is larger than that of $K^{-}$. This difference between particles and anti-particles might be due to the baryon transport effect to midrapidity [16] or absorption effect in the hadronic stage. The results could indicate the hadronic interaction become more dominant in lower beam energy. The immediate consequence of the significant difference between baryon and anti-baryon $v_{2}$ is that the NCQ scaling is broken between particles and anti-particles at $\sqrt{\mathrm{s}_{{}_{NN}}}$ = 7.7 and 11.5 GeV. The transverse momentum dependence of $v_{2}$ for the selected identified particles is shown in Fig. 3. The $v_{2}$ and $m_{T}-m_{0}$ has been divided by number of constituent quark in each hadron. In Au+Au collisions at $\sqrt{\mathrm{s}_{{}_{NN}}}$ = 39 GeV, the similar scaling behavior at $\sqrt{\mathrm{s}_{{}_{NN}}}$ =200 GeV is observed. Especially, the $\phi$ mesons which is insensitive to the later hadronic rescatterings follows the same trend of other particles. It suggests that the partonic collectivity has been built up in collisions at $\sqrt{\mathrm{s}_{{}_{NN}}}$ = 39 GeV. However, the $v_{2}$ for $\phi$ mesons falls off from other particles at $\sqrt{\mathrm{s}_{{}_{NN}}}$ = 11.5 GeV. The mean deviation to the $v_{2}$ pions is 2.6 $\sigma$. It indicates that the hadronic interaction are dominant in collisions at $\sqrt{\mathrm{s}_{{}_{NN}}}$ = 11.5 GeV. ## 3 Summary In summary, we present the $v_{2}$ measurement for charged hadrons and identified hadrons in Au+Au collisions at $\sqrt{\mathrm{s}_{{}_{NN}}}$ = 7.7, 11.5 and 39 GeV. The magitude of $v_{2}$ increases as increasing of the beam energy from 7.7 GeV to 2.76 TeV. The difference between the $v_{2}$ of particles and anti-particles is observed. The baryon and anti-baryon show significant difference at $\sqrt{\mathrm{s}_{{}_{NN}}}$ = 7.7 and 11.5 GeV. The ongoing analysis with 19.6 and 27 GeV data collected in 2011 will fill the gap between 11.5 and 39 GeV. The pions and kaons are almost consistent at $\sqrt{\mathrm{s}_{{}_{NN}}}$ = 39 GeV. This difference increase with decreasing of the beam energy. The $v_{2}$ of $\phi$ meson falls off from other particles in collisions at $\sqrt{\mathrm{s}_{{}_{NN}}}$ = 11.5 GeV. Experimental data suggests the hadronic interactions are dominant when $\sqrt{\mathrm{s}_{{}_{NN}}}$ $\leq$ 11.5 GeV. ## 4 Acknowledgments This work was supported in part by the National Natural Science Foundation of China under grant No. 11105060, 10775060 and 11135011. ## References * [1] S. A. Voloshin, A. M. Poskanzer and R. Snellings, arXiv:0809.2949. * [2] J. Adams et al. (STAR Collaboration), Phys. Rev. Lett. 92, 052302 (2004). * [3] J. Adams et al. (STAR Collaboration), Phys. Rev. Lett. 95, 122301 (2005). * [4] B. I. Abelev et al., (STAR Collaboration), Phys. Rev. Lett. 99 112301 (2007). * [5] B. I. Abelev et al., (STAR Collaboration), Phys. Rev. C 81, 044902 (2010). * [6] S. Shi (for the STAR collaboration), Nucl. Phys. A 830, 187c (2009). * [7] F. Liu, K.J. Wu, and N. Xu, J. Phys. G 37 094029(2010). * [8] B. Mohanty and N. Xu, J. Phys. G 36, 064022(2009). * [9] K. H. Ackermann et al. (STAR Collaboration), Nucl. Instrum. Methods A 499, 624 (2003). * [10] W. J. Llope (STAR TOF Group), Nucl. Instr. and Meth. B 241, 306 (2005). * [11] C. Adler et al. (STAR Collaboration), Phys. Rev. Lett. 89, 132301 (2002). * [12] A. M. Poskanzer and S. A. Voloshin, Phys. Rev. C 58 1671 (1998). * [13] N. Borghini, P. M. Dinh, and J.-Y. Ollitrault, Phys. Rev. C 63, 054906 (2001). * [14] N. Borghini, P. M. Dinh, and J.-Y. Ollitrault, Phys. Rev. C 64, 054901 (2001). * [15] K. Aamodt et al. (ALICE Collaboration), Phys. Rev. Lett. 105, 252302 (2010). * [16] J. Dunlop, M.A. Lisa and P. Sorensen, arXiv:1107.3078.	arxiv-papers	2011-11-23T02:07:41	2024-09-04T02:49:24.612319	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Shusu Shi (for the STAR collaboration)", "submitter": "Shusu Shi", "url": "https://arxiv.org/abs/1111.5385" }
1111.5410	# Forward Jets and Forward-Central Jets at CMS Niladri Sen, on behalf of the CMS Collaboration ###### Abstract We report on cross section measurements for inclusive forward jet production and for the simultaneous production of a forward and a central jet in $\sqrt{s}=$7 TeV $pp$-collisions. Data collected in 2010, corresponding to an integrated luminosity of 3.14 pb-1, is used for the measurements. Jets in the transverse momentum range pT = 35 - 140 GeV/c are reconstructed with the anti- kT (R = 0.5) algorithm. The extended coverage of large pseudo-rapidities is provided by the Hadronic Forward calorimeter (3.2 $<\|\eta\|<$ 4.7), while central jets are limited to $\|\eta\|<$ 2.8, covered by the main detector components. The two differential cross sections are presented as a function of the jet transverse momentum. Comparisons to next-to-leading order perturbative calculations, and predictions from event generators based on different parton showering mechanisms (pythia and herwig) and parton dynamics (cascade) are shown. ###### Keywords: CMS, forward physics, jets, QCD ###### : 13.60.Hb, 13.85.Fb,13.87.-a ## 1 Introduction Jet production in hadron-hadron collisions is sensitive to underlying partonic processes, initial- and final-state radiation (isr and fsr), and to the parton density functions (pdfs) of the colliding hadrons. Measurements of jet cross sections at previous colliders are well described over several orders of magnitude by perturbative calculations Collaboration (2008a) Collaboration (2008b). However, the jets were limited to central pseudo-rapidities ($\|\eta\|<$2.4), where the momentum fraction of the incoming partons ($x_{1}$, $x_{2}$) were of the same order. Jets at large pseudo-rapidities (i.e., forward or backward jets, $\|\eta\|>$3 ) result from interactions between colliding partons with differing momentum fractions (e.g. $x_{1}<<x_{2}$), allowing us to investigate QCD-effects at small-$x$. At such small-$x$ values, pdfs are well constrained by dis data, but there might be additional effects that play a role. We expect signals of parton dynamics beyond the standard dglap evolution (e.g., BFKL or CCFM), and saturation effects are foreseen. Moreover, forward jets are of interest in vector-boson-fusion processes, which is one of the mechanisms for Higgs boson production. The Large Hadron Collidor (lhc) is a proton-proton collider with a beam energy of 7 TeV and a design luminosity of $L=$34 cm-2s-1, designed to explore a new energy scale. At the collider, momentum fraction of the proton ($x$) carried by the partons can become very small and the parton densities become large. Additionally, the probability of more than one partonic interaction per event increases for collision energies produced at the lhc, and probing the forward region opens up opens up a large phase space for qcd emissions. Thus, the lhc provides the perfect opportunity to study small-$x$ physics and qcd effects through measurements of forward and forward-central jet production. Here, a measurement of the cross section of inclusive forward jets Collaboration (2011a) and of the production of a forward jet in conjunction with a central jet Collaboration (2011b), using data collected by the Compact Muon Solenoid (cms), in $\sqrt{s}=$ 7 TeV $pp$-collisions at the lhc is presented. ## 2 The CMS Detector The CMS experiment Collaboration (2008c), located on the French-side of CERN, operates a multi-puprpse detector to study $pp$ collisions at the LHC. The detector covers a solid angle approaching 4$\pi$ and incorporates vertex, calorimeter and muon chambers. The tracking system covers the pseudo-rapidity range -2.5 $<\eta<$ 2.5 and the calorimeter system covers the range \- 5 $<\eta<$ 5\. The Hadronic Forward Calorimeter (HF), made of iron absorbers embedded with radiation hard quartz fibres, is used for measuring forward jets and energy, providing an almost hermetic coverage upto $\|\eta\|\sim$ 5. ## 3 Cross Section Measurement For Inclusive Forward Jet Production Jets are built from calorimeter information using the infrared and collinear safe anti-kT (R=0.5) jet clustering algorithm M. Cacciari (2008). A single jet trigger (p${}_{T}>$ 15 GeV), fully efficient in region of the measurement (p${}_{T}>$ 35 GeV) selects jets from the data sample. Jet quality criteria are applied to remove unphysical energy deposits, and the selection requirements ensure that the jets are well contained within the fiducial acceptance, i.e. their reconstructed axis is within 3.2 $<\|\eta\|<$ 4.7 Collaboration (2007). The jet cross section is fully corrected for detector reconstruction effects via a bin-by-bin correction factor calculated from simulated samples. The number of jets, Njets, are binned in transverse momentum (pT) and pseudo- rapidity ($\eta$). The final differential inclusive jet cross-section is: $\frac{d^{2}\sigma}{dp_{T}d\eta}=\frac{C_{unfold}}{L}\cdot\frac{N_{jets}}{\Delta p_{T}\cdot\Delta{\eta}}$ (1) where Cunfold is the correction factor accounting for detector effects (e.g. migrations, resolution), and $\Delta p_{T}$ and $\Delta{\eta}$ are the bin widths in $p_{T}$ and $\eta$ respectively. Both experimental and theoretical sources of uncertainties have been considered. The dominant experimental systematic uncertainty is the accuracy of the jet energy scale (jes), which is 20%–30% in all pT bins of the measured cross section. Additional sources include the pT resolution for bin-by-bin corrections (3%–6%) and the luminosity measurement (4%). The theoretical uncertainties are estimated by checking non-perturbative effects through pythia and herwig comparisons , ascertaining the impact of different pdfs and the variation of $\mu_{r}$ and $\mu{f}$ by a factor of 2. Figure 1 shows the fully corrected jet cross section as a function of pT, in comparison to various theoretical models. The yellow-band indicates the experimental uncertainty. Figure 2 shows the fractional difference between the experimental jet cross-section and the theoretical predictions. Within the uncertainties, the hadron-level predictions are in good agreement with the data. The exception is cascade, where shape variations at high pT are observed between the measured forward jet spectra and the prediction. Figure 1: Inclusive jet cross section measured at forward pseudo-rapidities (3.2 $<\|\eta\|<$ 4.7), fully corrected and unfolded, compared to various hadron-level predictions. The error band represents the experimental systematic uncertainty. Figure 2: Fractional differences between forward jet spectra and theoretical predictions. The error-bars on the data points show statistical uncertainties. The error bands represent the systematical and theoretical uncertainties. ## 4 Cross Section Measurement for Simultaneous Production of a Forward and a Central Jet The jet clustering algorithm, widths in the pT spectrum, and selection criteria are the same in this measurement as for the inclusive jet cross section analysis. The only difference is the additional requirement of a well- reconstructed central ($\|\eta\|<$ 2.8) jet . That is, an event is accepted if there are at least two reconstructed jets with p${}_{T}>$ 35 GeV, one with its axis within the central region ($\|\eta\|<$ 2.8), and the other within the hf (3.2 $<\|\eta\|<$ 4.7). If there is more than one jet present in either of the regions, the leading jet is considered. The jet cross-sections are obtained using a bin-by-bin correction as for the inclusive jet production analysis. In this case, however, the simulated samples were re-weighted at hadron level to describe the measured data distributions. The jes remains as the dominant systematic uncertainty ($\sim$ 25%). The model dependence is the second largest uncertainty, estimated by the variation of bin-by-bin correction factors from different mcs (5% – 15%). The fully corrected cross section for simultaneous production of at least one central and one forward jet is measured as a function of jet pT. Figures 3– 4 present the measurement with the corresponding hadron-level predictions in contrast, for central jets and forward jets respectively. The yellow bands indicate the experimental systematic uncertainties summed in quadrature and the error bars represent the statistical uncertainty. The plots on the left show some discrepancies between pythia and data, especially in the central region. herwig, which uses angular-ordered parton showers, describes the data better, for both regions of pseudo-rapidity. Figure 3: Fully corrected pT-differential jet cross-section for the central region ($\|\eta\|<$ 2.8) compared to various event generators, pythia and cascade (left), herwig and hej (right). The error-bars on the data points show statistical uncertainties. The error bands represent the systematical. Figure 4: Fully corrected pT-differential jet cross-section for the forward region (3.2 $<\|\eta\|<$ 4.7) compared to various event generators, pythia and cascade (left), herwig and hej (right). The error-bars on the data points show statistical uncertainties. The error bands represent the systematical uncertainties. ## 5 Conclusion We present a measurement of jet production in the forward pseudo-rapidity range 3.2 $<\|\eta\|<$ 4.7, and the cross section for the simultaneous production of one central and one forward jet, using 3.14 pb-1 of cms data collected during the early $\sqrt{s}=$ 7 TeV pp-collisions. Within the current experimental and theoretical uncertainties, perturbative calculations reproduce the measured inclusive forward jet cross section well. The data- model comparison of the forward-central jet measurement show that only some calculations are in reasonable agreement with data. Both measurements are a first test of perturbative qcd calculations in the forward region, providing the basis for further investigation of this interesting region of phase space. ## 6 Acknowledgements Copyright CERN for the benefit of the CMS Collaboration. ## References * Collaboration (2008a) CDF Collaboration, _Phys. Rev._ D78 (2008a). * Collaboration (2008b) D0 Collaboration, _Phys. Rev. Lett._ 101 (2008b). * Collaboration (2011a) CMS Collaboration, Measurement of forward jets in proton-proton collisions at 7 TeV (2011a), CMS Physics Analysis Summary FWD-10-003. * Collaboration (2011b) CMS Collaboration, Cross section measurement for simultaneous production of a central and a forward jet in proton-proton collisions at 7 TeV (2011b), CMS Physics Analysis Summary FWD-10-006. * Collaboration (2008c) CMS Collaboration, _The CMS experiment at the CERN LHC_ , JINST 3:S08004, 2008c. * M. Cacciari (2008) G.Soyez, M. Cacciari, G.P. Salam, _JHEP_ 04 (2008). * Collaboration (2007) CMS Collaboration, Performance of jet algorithms in CMS (2007), CMS Physics Analysis Summary JME-07-003.	arxiv-papers	2011-11-23T06:31:44	2024-09-04T02:49:24.617544	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Niladri Sen (for the CMS Collaboration)", "submitter": "Niladri Sen", "url": "https://arxiv.org/abs/1111.5410" }
1111.5550	11institutetext: National Laboratory of Solid State Microstructure, Department of Physics, Nanjing University - Nanjing 210093, China Center for Statistical and Theoretical Condensed Matter Physics, Zhejiang Normal University - Jinhua 321004, China Quantum Hall effects Spin polarized transport in semiconductors Magnetoelectronics; spintronics: devices exploiting spin polarized transport or integrated magnetic fields # Transversal Propagation of Helical Edge States in Quantum Spin Hall Systems Feng Lu 11 Yuan Zhou 11 Jin An and Chang-De Gong 1122111122 ###### Abstract The transversal propagation of the edge states in a two-dimensional quantum spin Hall (QSH) system is classified by the characteristic parameter $\lambda$. There are two different types of the helical edge states, the normal and special edge states, exhibiting distinct behaviors. The penetration depth of the normal edge state is momentum dependent, and the finite gap for edge band decays monotonously with sample width, leading to the normal finite size effect. In contrast, the penetration depth maintains a uniform minimal value in the special edge states, and consequently the finite gap decays non- monotonously with sample width, leading to the anomalous finite size effect. To demonstrate their difference explicitly, we compared the real materials in phase diagram. An intuitive way to search for the special edge states in the two-dimensional QSH system is also proposed. ###### pacs: 73.43.-f ###### pacs: 72.25.Dc ###### pacs: 85.75.-d ## 1 Introduction Owing to the linear dispersion and $Z_{2}$ topological invariant, the anomalous transport properties of the helical edge states in a quantum spin Hall (QSH)system are predicted[1, 2, 3, 4, 5, 6]. Well localized edge states were usually treated as ideal one-dimensional channels to investigate the exotic properties[2, 5]. However, their behaviors are significantly changed due to the transversal broadening of these edge states in real samples. The penetration depth of helical edge states had been discussed in both HgTe quantum well system and Bi thin film[7]. In those two systems, its value are determined by the inverse of momentum space distance between the edge state and the absorption point into the bulk. The finite penetration depth also leads to the so-called finite size effect in two-dimensional (2D) QSH system[8]. A gap opens at $\Gamma$ point when the opposite helical edge channels overlap each other, which had been used to confirm the intrinsic spin Hall effect in HgTe quantum well system[9]. Recently, several electric devices had been designed to manipulate the charge and spin transport with such finite size effect[10, 11]. An anomalous finite size effect was further reported in three-dimensional ($3$D) topological insulator $Bi_{2}Se_{3}$, much shorter penetration depth and oscillatory finite size gap had been revealed[12]. This gap oscillation had also been used to search new candidate of topological non- trivial systems[13, 14]. These previous discussions were all based on the specific materials. The general comprehension of the transversal propagation behaviors is expected in QSH system, especially the intrinsic difference between the normal and anomalous finite size effects. In this paper, the transversal propagation behaviors of the helical edge states in 2D QSH system are investigated. The helical edge states can be classified into two modes, the normal and special edge states, according to the decay characteristic quantity $\lambda$. In normal edge states, the penetration depth shows clear momentum-dependence, and the finite gap for edge states decays monotonously with the sample width. While in special edge states, the penetration depth keeps unchanged in the momentum space, and its finite gap decays oscilatorily with sample width. The normal and anomalous finite size effect can be found in the respective edge states. These facts give explicit explanations on the difference between the real materials. Based on the theoretical calculations, the search of the special edge states in the $2$D case is proposed. The paper is organized as follows: In Sec.II, we specify two different transversal propagation modes of the helical edge states without specific boundary condition. A semi-finite boundary condition is adopted to show the distinct evolution of the penetration depth in the normal and special edge states in Sec.III. As a consequent effect, the normal and anomalous finite size effects are discussed in Sec.IV. In Sec.V, the role of particle-hole asymmetry and the comparison of real materials are further discussed. The conclusion is drawn in Sec.VI. ## 2 Transversal Modes of 2D QSH Model The QSH effect was theoretically predicted in $HgTe$ quantum well[15, 3], and soon confirmed experimentally by König el at.[16]. We start from the effective 4$\times$4 model for a 2D QSH system proposed by Bernevig, Hughes, and Zhang[5, 3]. Very recently, it was also adopted as an effective 2D model for the 3D topological insulator in ultrathin limit[17]. The model Hamiltonian is expressed as $H(k)=\left[\begin{array}[]{cc}h(k)&0\\\ 0&h^{}(-k)\end{array}\right],$ (1) where $h(k)=\varepsilon_{k}\mathbf{I}_{2\times 2}+\mathbf{d}_{k}\cdot{\bm{\sigma}}$, with $\varepsilon_{k}=C-D(k^{2}_{x}+k^{2}_{y})$. The k-dependent effective field $\mathbf{d}_{k}=(Ak_{x},-Ak_{y},M_{k})$, where $M_{k}=M-B(k^{2}_{x}+k^{2}_{y})$. $\bm{\sigma}$ is the Pauli matrices. $A$, $B$, $C$, $D$, and $M$ are determined by the quantum well geometry in real materials. Here, we treat them as independent parameters and study their respective role first. Keep in mind that, the interested topological non- trivial QSH phase[18] emerges only when $MB>0$. In HgTe quantum well system, such condition is controlled by the thickness of the quantum well[3, 16]. The properties of counter-part $h^{}(-k)$ can be conveniently obtained by applying the time reversal operation to $h_{k}$. To focus on the edge properties, $k_{y}$ needs to be replaced by $-i\partial_{y}$, while $k_{x}$ remains a good quantum number due to the translational symmetry. A trial solution of $\Psi_{k_{x}}(y)=C_{k_{x}}e^{-\lambda y}$ can be introduced, and the decay characteristic quantities are subsequently obtained as[8] $\lambda_{1,2}^{2}(k_{x},E)=k_{x}^{2}+F(E)\pm\sqrt{F^{2}(E)+\frac{E^{2}-M^{2}}{B^{2}-D^{2}}},$ (2) where $F(E)=\frac{A^{2}-2(MB+DE)}{2(B^{2}-D^{2})}$ is a function of energy $E$. The transversal propagation behaviors of the states are determined by both of $\lambda_{1,2}$. However, it is clear in Eqs. (2) that, $\lambda(k_{x},E)$ has a definite distribution in momentum space, independent of the boundary condition. Hence, we can directly discuss the transversal propagation behaviors of the states from Eqs. (2). There are four modes of the states specified by different combinations of $\lambda_{1,2}$, as illustrated in TABLE 2. [hbtp] Combination for $\lambda_{1,2}$ Mode* Condition $\lambda_{1}$ $\lambda_{2}$ Edge1 $\lambda_{1}^{2}>\lambda_{2}^{2}\geq 0$ Real Real Bulk1** $\lambda_{1}^{2}\geq 0>\lambda_{2}^{2}$ Real Imaginary Bulk2 $0>\lambda_{1}^{2}>\lambda_{2}^{2}$ Imaginary Imaginary Edge2 $\lambda_{1}^{2}=(\lambda_{2}^{2})^{}$ Complex Complex * For bulk states, at least one of $\lambda_{1,2}$ is purely imaginary. While for edge states, both of $\lambda_{1,2}$ should have a non-zero real part. * ** Trivial edge states are also included due to the real $\lambda_{1}$. Since only the non-trivial edge states are interested in this paper, it is natural to ask whether there is something different between the two edge modes in TABLE 2. For convenience, we specify the Edge1 state with both $\lambda_{1,2}$ real as the normal edge state (NES); and the Edge2 state with $\lambda_{1,2}$ complex conjugates as the special edge state (SES). To fulfill the condition of conjugate, the term under square root in Eqs. (2) must be negative, which restricts SES existing in a specific regime in momentum space confined by $E^{SES}_{\pm}=D\left(\frac{A^{2}-2MB}{2B^{2}}\right)\pm\|\frac{A}{2B}\|\sqrt{\gamma\left(4MB-A^{2}\right)}$ (3) Here a screen factor denoting the particle-hole asymmetry $\gamma=1-\frac{D^{2}}{B^{2}}$ is introduced. The system undergoes a phase transition from an insulator to a semimetal when $D\geq B$[19], which is not interested for us. Eqs. (3) naturally requires $4MB\geq A^{2}\geq 0$, implying a non-trivial QSH state. There is no special restriction for the NES. Present classification is a natrual consequence of the breaking of periodic boundary condition, which leads to a definite distribution of transversal propagation modes in momentum space. An explicit boundary condition just creates a specific spectrum onto such distribution. The emergence of SES is determined by the $k_{x}$-independent $E_{\pm}^{SES}$. For given parameters, $E^{SES}_{\pm}$ will squish the bulk band, leading to a flat valence band top (or conduct band bottom), as in Fig. 1(c)(e)(f). Conversely, such feature can be viewed as a sufficient condition for SES in QSH system, even without the explicit knowledge of material parameters. Moreover, the squished bulk band also gives rise to the Bulk2 states in TABLE 2, which exhibit a larger density of states than Bulk1 states in our numerical results. Such classification is also significant in the problems of interference tunneling and restricted edge transport[20, 11]. More differences of the edge modes will be discussed in the following sections. [width=13.5cm]spectra.eps Figure 1: Helical edge spectra for different effective parameters. $A=0.4$ and $B=-1.0$. The upper/lower panels are spectra with/without particle-hole symmetry. $\|M\|$ increases from left to right. The upper, and lower part in each panel is the edge spectrum, and corresponding $\lambda_{1,2}$ respectively. The bold solid line in black/blue represents the edge spectrum of NES/SES. The lighter/darker gray regimes describe the Bulk1/Bulk2 states in Table 2. The light grey dash line are the confines of SES given by Eqs. (3). The red solid lines in lower part of each panel is $\Re\lambda_{2}$, which give the inverse of $\ell$. The blue dash lines are the corresponding $\Im\lambda_{1,2}$. In panel (c), three states are specified, and corresponding transversal propagation behaviors in real space are shown in Fig. 2. ## 3 Penetration Depth of Helical Edge States The penetration depth distribution of the helical edge states is distinct from that of the chiral edge states in integer quantum Hall effect[21, 22]. The latter is determined by the universal magnetic length, which is related to the external magnetic field. In contrast, the penetration depth in QSH system is $k$-dependent, originated from the band structure[5]. To address this, the semi-infinite boundary condition[7] is adopted here. We restrict $\Re\lambda_{1,2}\geq 0$ to obtain an evanescent edge state localized near the boundary. The boundary condition $\Psi_{k_{x}}(0)=0$ gives the linear dispersion relation[7, 8] $E=-A\sqrt{\gamma}k_{x}-\frac{MD}{B}$ (4) The penetration depth $\ell=max\\{\Re\lambda_{1,2}^{-1}\\}$[6] behaves differently in NES and SES. The NES situation had been discussed in previous work as in Fig. 1(a) and (d). The $k_{x}$-dependent $\ell_{NES}$ reaches its minimum at $k_{x}=\frac{A^{2}}{4}(1-\gamma^{-1})$. The edge state is absorbed by the bulk when $\Re\lambda_{2}=0$. In contrast, the penetration depth in SES maintains a uniform minimal value across the whole regime of SES, given by $\ell_{SES}=2/(\lambda_{1}+\lambda_{2})=\|\frac{2\sqrt{\gamma}B}{A}\|,$ (5) which is also independent of $E$, $k_{x}$ and $M$, as shown in Fig. 1. Here $\Re\lambda_{1,2}$ governs the transversal decay behavior. In fact, although the relation $\lambda_{1}+\lambda_{2}=\frac{A}{\sqrt{\gamma}B}$ keeps unchanged even in NES, the penetration depth in NES is merely determined by the minimum of real $\lambda_{1,2}$. In Fig. 2, the transversal propagation behaviors of three selected helical edge states are plotted in real space. The wave function of SES(1) and SES(2) exhibit an evanescent oscillation with different periods. However, they share the same penetration depth. In contrast, the NES(3) shows no oscillation but much longer penetration depth. The non-monotonous decay behavior of edge state was also reported in the lattice model[23], which can be naturally attributed to the SES. The HgTe quantum well[3] and the ultra-thin $Bi_{2}Se_{3}$ film[17] correspond to the situation in Fig. 1(d), where the SES is absent. The penetration depth is estimated to be about 50 nm[8]. In previous studies,[7] the $Bi\\{111\\}$ thin film was compared with the HgTe quantum well system, and remarkable difference was found in the behaviors of the penetration depth. We notice that, a flat valence band top emerges in $Bi\\{111\\}$ spectrum[7], implying the existence of SES. Therefore, such difference can be well understood within present discussion. Due to the similarity of $\lambda$ at $\Gamma$ point, the topological surface states (TSS) of the 3D topological insulator can be equivalently discussed within our framework, corresponding to the situation in Fig. 1(f), where the edge states are SES dominated. The penetration depth of TSS in 3D $Bi_{2}Se_{3}$ was also reported in previous work, with a shorter $\ell$ of about 10 nm[12]. However, they concluded $\ell$ is proportional to the inverse of $\left\|M\right\|$, distinguished from present discussion. In fact, this situation does not belong to NES, but SES, since both the decay characteristic quantities $\lambda_{1,2}$ have image part as they stressed, too. Therefore, the penetration depth should be independent of $\left\|M\right\|$. The difference between NES and SES will be further discussed in the next section. [width=8.5cm]propagation.eps Figure 2: Transversal propagation behaviors of edge state wave function in real space. Corresponding states are marked in Fig. 1(c). The dash line roughly gives the penetration depth behaviors with function $exp(-y/\ell)$. The black dash line takes $\ell_{SES}$ given by Eqs. (5), while the blue one takes $\ell=\Re\lambda_{2}^{-1}$. ## 4 Normal And Anomalous Finite Size Effects The finite size effect in QSH system arises from the overlap of the opposite channel due to the decreasing sample width, leading to the finite energy gap opening for the energy dispersion of edge state near Dirac point[8]. Since the penetration depth of NES and SES is quite different, the consequent finite size effect is also expected to be distinct. We now turn to the ribbon geometry with the boundary condition of $\Psi_{k_{x}}(-L/2)=\Psi_{k_{x}}(L/2)=0$, where $L$ is the width of the ribbon. Our numerical results reveal that, the relative gap $\delta\Delta(k_{x})=\Delta(k_{x})-2\|Ak_{x}\|$ reaches its maximum at $\Gamma$ point and decays exponentially with $\|k_{x}\|$. Hence, we just focus on the situation at $\Gamma$ point where $\delta\Delta(0)=\Delta(0)$. We follow the previous discussions[8, 14] to evaluate the finite size gap in different situations. [width=8cm]finite-size.eps Figure 3: Normal and anomalous finite size effects. (a), and (b) show the edge bands and gap behaviors varying with $L$ in NES, and SES, respectively. (c) gives the $M$-dependent $\Delta(0)$ with the same parameter adopted in Fig. 1 (in logarithmic scale). The normal (left), and anomalous (right) finite size effect is divided by the critical $M_{c}$ (black dash line). (d), and (e) present the conductance at finite temperature corresponding to the situation of (a), and (b), respectively, with $1/k_{B}T=400$ and $G_{0}=\frac{e^{2}}{h}$. When the NES dominates the Dirac point, as discussed previously in the HgTe quantum well[8], the gap was estimated to be $\Delta(0)\simeq\frac{4\|AM\gamma\|}{\sqrt{A^{2}-4MB\gamma}}e^{-\lambda_{2}L}.$ (6) Here we assume $\lambda_{1}L>>1$ and $\lambda_{1}>>\lambda_{2}$. This is the normal finite size effect as shown in Fig. 3(a). When Dirac point locates inside the SES regime, the gap turns to be $\Delta(0)\simeq\frac{8\|AM\gamma\sin(\Im\lambda_{2}L)\|}{\sqrt{A^{2}-4MB\gamma}}e^{-\ell_{SES}^{-1}L},$ (7) here $\Im\lambda_{2}=\sqrt{\frac{M}{B}-\frac{A^{2}}{4\gamma B^{2}}}$ is the imaginary part of $\lambda_{2}$. The gap exhibits an oscillatory behavior with $L$, as described in Fig. 3(b). This oscillation was also predicted in 3D topological insulator[12, 14], referred as the anomalous finite size effect. We numerically investigate the $M$-dependent evolution of $\Delta(0)$ to distinguish the difference between NES and SES as shown in Fig. 3(c). Here $D=0$ is applied to avoid the mismatch between the Dirac point and the regime of SES. For small $\|M\|$, the Dirac point is NES, and the corresponding $\Delta(0)$ evolves monotonously with $\|M\|$. For large $L$, the Dirac point turns to be SES, $\Delta(0)$ is oscillatory. A critical $\|M_{c}\|=0.04$ is obtained with the same parameters taken in Fig. 1. The number of oscillatory periods increase with $\|M\|$, owing to a decreasing $\Im\lambda$. Considerable gap always opens at $L\sim 35$ (arb. units) for all $\|M\|>\|M_{c}\|$, which coincides with the uniform minimal $\ell_{SES}$ discussed above. Here we emphasize that, the uniform minimal $\ell_{SES}$ found in Fig. 1 is protected by the linear dispersion of Eqs. (4). This linear relation is not preserved when the finite size gap opens, then $\ell$ becomes momentum-dependent again even in SES. The essence of such differences can be understood based on present results. For SES, a $y$-dependent phase factor emerges due to the finite $\Im\lambda$, which is absent in NES. The edge band is renormalized, together with gap opening, due to the overlap of opposite edge states. Meanwhile, the transversal phase coherence of the opposite edge states contributes to the oscillation of $\Delta$ for SES. Hence, the interference-fringe-like picture can be obtained as shown in Fig. 3(c). Such effect can be detected in transport measurements at low temperature[9]. The conductance at finite temperature is simply given by[8] $G(\mu,T)=(2e^{2}/h)\left[f(\Delta/2-\mu)-f(-\Delta/2-\mu)+1\right]$ (8) when $\mu$ locates inside the bulk gap. Here $f(E)$ is the Fermi distribution function and $\Delta$ is the finite size gap. Fig. 3(d) and Fig. 3(e) present the conductance for NES and SES respectively. Recently, a non-monotonous gap evolution had been observed in ultra-thin $Bi_{2}Se_{3}$, which is noted as a possible anomalous finite size effect for TSS[24]. ## 5 Discussion [width=8.5cm]comparison.eps Figure 4: Phase diagrams in parameter space. (a), and (b) are with, and without the particle-hole symmetry respectively. The black solid line ($A^{2}=4MB$) divides the region of edge state into SES and NES. The dash line ($A^{2}=4\gamma MB$) in (b) indicates the mismatch between SES and the Dirac point as described in the text, and moves along the direction indicated by arrow when $\gamma$ decreases. The intensity stands for the inverse of penetration depth $\ell^{-1}$ at Dirac point. Different materials are compared in (c), where $A$, $B$ and $M$ are unified into the units of $meV$ and $nm$, so that $\ell$ has a common unit of $nm$. Here the logarithmic scale is adopted. Materials with/without SES Dirac point are marked with cycles/squares. The intensity in (c) is for $\gamma=1$. For $Bi_{2}Se_{3}$ TSS, and $Sb_{2}Te_{3}$, $\|D/B\|$ is $0.13$, and $0.63$, respectively. Up to now, we have discussed the penetration depth and the finite size effect in NES and SES. The particle-hole asymmetry factor $\gamma$ also plays a subtle role on these properties. The Dirac point moves upward, and the SES regime also shifts, leading to the possible mismatch as shown in Fig. 1(e). The existence of SES Dirac point requires $\frac{A^{2}}{B^{2}}\leq\gamma\frac{4M}{B}$. In Fig. 4(b), the regime between the solid line and the dash line describes the mismatch: the SES exists, but the Dirac point moves outside. It should be pointed out that such mismatch is not sensitive with selected $\|D/B\|$ unless it approaches to $1$. The penetration depth of the edge states at Dirac point and finite size effect are discussed together in the phase space of relative parameters $\frac{A}{B}$ and $\frac{M}{B}$, as shown in Fig. 4. In these phase diagrams, since the solid line divides the parameter space into two regimes, $\ell^{-l}$ at Dirac point reveals two distinct evolutions. $\ell^{-1}$ at Dirac point increases with $M/B$ and decreases with $\|A/B\|$ in NES. In contrast, it remains unchanged with $M/B$ but increases with $\|A/B\|$ in SES. The recently discovered topological non-trivial systems: 2D HgTe quantum well[8], Ultrathin $Bi_{2}Se_{3}$ film[17], 3D $Bi_{2}Se_{3}$, $Sb_{2}Te_{3}$ and $Bi_{2}Te_{3}$ [25, 6] are compared in the same phase space. The former two effective 2D systems are described by the same model of Eqs. (1). Although the 3D topological insulators have a different effective model[25], the situation at Dirac point is equivalent to the two 2D systems under proper parameter substitution[14, 12]. As in Fig. 4(c), the helical edge states in the two 2D systems contain only NES, therefore, large size is required to avoid the normal finite size effect. In contrast, the 2D TSS of 3D $Bi_{2}Se_{3}$ and $Sb_{2}Te_{3}$ implies a shorter penetration depth and a possible anomalous finite size effect[12, 24]. Interestingly, the SES exits in $Bi_{2}Te_{3}$, however, its Dirac point moves into bulk state due to strong particle-hole asymmetry mentioned above. This may be true as compared with the angle resolved photoemission spectroscopy measurements[26, 6]. Similar behavior may also can be found in the bulk $HgTe$ under uniaxial strain[27]. We expect that, these special effects can be electrically detected in other QSH systems with smaller $A/B$ or larger $M/B$ as shown in Fig. 4. Recently, several designs had been performed based on the finite size effect[10, 11]. The future applications could be quite sensitive to these properties. In this sense, present work provides a theoretical prediction on the possible finite size effects in new materials. ## 6 Conclusion In conclusion, two different transversal propagation modes of the helical edge states, i.e., NES and SES, in the QSH system are specified by the decay characteristic quantities $\lambda$. The emergence of the flat bulk band implies the special edge state, which gives a sufficient criterion to distinguish the two modes. The penetration depth of SES keeps a uniform minimal value, independent of the selected $E$, $k_{x}$ and $M$. In contrast, it is much larger and shows clear momentum dependence in NES. Different finite size effects are studied in respective edge states. Especially, the oscillatory gap for edge band is found in SES. Some real materials are compared in the phase diagram to demonstrate the difference between NES and SES. We also give clues to search possible QSH materials with SES for future applications. ###### Acknowledgements. We would like to thank Y. F. Wang, and L. Xu for helpful discussions. This work is supported by NSFC Project No. 10804047, and A Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions. J. An acknowledges NSFC Project No. 10804073. C. D. Gong also acknowledges 973 Projects No. 2009CB929504. ## References * [1] Kane C. L. Mele E. J. Phys. Rev. Lett.952005146802. * [2] Wu C., Bernevig B. A. Zhang S.-C. Phys. Rev. Lett.962006106401. * [3] Bernevig B. A., Hughes T. L. Zhang S.-C. Science31420061757. * [4] Kane C. L. Mele E. J. Phys. Rev. Lett.952005226801. * [5] König M., Buhmann H., Molenkamp L. W., Hughes T., Liu C.-X., Qi X.-L. Zhang S.-C. Journal of Physical Society of Japan772008031007. * [6] Qi X.-L. Zhang S.-C. Rev. Mod. Phys.8320111057. * [7] Wada M., Murakami S., Freimuth F. Bihlmayer G. Phys. Rev. B832011121310. * [8] Zhou B., Lu H.-Z., Chu R.-L., Shen S.-Q. Niu Q. Phys. Rev. Lett.1012008246807. * [9] Brüne C., Roth A., Novik E. G., König M., Buhmann H., Hankiewicz E. M., Hanke W., Sinova J. Molenkamp L. W. Nat. Phys.62010448. * [10] Zhang L.-B., Cheng F., Zhai F. Chang K. Phys. Rev. B832011081402. * [11] Krueckl V. Richter K. Phys. Rev. Lett.1072011086803. * [12] Linder J., Yokoyama T. Sudbø A. Phys. Rev. B802009205401. * [13] Liu C.-X., Zhang H., Yan B., Qi X.-L., Frauenheim T. , Dai X. , Fang Z. Zhang S.-C. Phys. Rev. B812010041307. * [14] Shan W.-Y., Lu H.-Z. Shen S.-Q. New Journal of Physics122010043048. * [15] Bernevig B. A. Zhang S.-C. Phys. Rev. Lett.962006106802. * [16] König M., Wiedmann S., Brüne C., Roth A., Buhmann H., Molenkamp L. W., Qi X.-L. Zhang S.-C. Science3182007766. * [17] Lu H.-Z. , Shan W.-Y., Yao W., Niu Q. Shen S.-Q. Phys. Rev. B812010115407. * [18] Murakami S. Progress of Theoretical Physics Supplement1762008279. * [19] Mao S. Kuramoto Y. Phys. Rev. B832011085114. * [20] Guigou M., Recher P., Cayssol J. Trauzettel B. Phys. Rev. B842011094534. * [21] Jackiw R. Rebbi C. Phys. Rev. D1319763398. * [22] Thouless D. J., Kohmoto M., Nightingale M. P. den Nijs M. Phys. Rev. Lett.491982405. * [23] Ohyama Y., Tsuchiura H. Sakuma A. Journal of Physics: Conference Series2662011012103. * [24] Sakamoto Y., Hirahara T., Miyazaki H., Kimura S. Hasegawa S. Phys. Rev. B812010165432. * [25] Zhang H., Liu C.-X., Qi X.-L., Dai X., Fang Z. Zhang S.-C. Nat. Phys.52009438. * [26] Li Y.-Y., Wang G., Zhu X.-G., Liu M.-H., Ye C., Chen X., Wang Y.-Y., He K., Wang L.-L., Ma X.-C., Zhang H.-J., Dai X., Fang Z., Xie X.-C., Liu Y., Qi X.-L., Jia J.-F., Zhang S.-C., Xue Q.-K. Adv. Mat.2220104002. * [27] Dai X., Hughes T. L., Qi X.-L., Fang Z., and Zhang S.-C. Phys. Rev. B772008125319.	arxiv-papers	2011-11-23T16:54:56	2024-09-04T02:49:24.625528	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Feng Lu, Yuan Zhou, Jin An, and Chang-De Gong", "submitter": "Yuan Zhou", "url": "https://arxiv.org/abs/1111.5550" }
1111.5836	LPT Orsay 11-60 Enhanced Higgs Mediated Lepton Flavour Violating Processes in the Supersymmetric Inverse Seesaw Model Asmaa Abada, Debottam Das and Cédric Weiland Laboratoire de Physique Théorique, CNRS – UMR 8627, Université Paris-Sud 11, F-91405 Orsay Cedex, France We study the impact of the inverse seesaw mechanism on several low-energy flavour violating observables such as $\tau\rightarrow\mu\mu\mu$ in the context of the Minimal Supersymmetric Standard Model. As a consequence of the inverse seesaw, the contributions of the right-handed sneutrinos significantly enhance the Higgs-mediated penguin diagrams. We find that different flavour violating branching ratios can be enhanced by as much as two orders of magnitude. We also comment on the impact of the Higgs-mediated processes on the leptonic $B$-meson decays and on the Higgs flavour violating decays. KEYWORDS: Supersymmetry, Lepton Flavour Violation, Inverse Seesaw ## 1 Introduction Neutrino oscillations have provided indisputable evidence for flavour violation in the neutral lepton sector. In the absence of any fundamental principle that prevents charged lepton flavour violation, one expects that extensions of the Standard Model (SM) accommodating neutrino masses and mixings should also allow for lepton flavour violation (LFV) in the charged lepton sector. Indeed, the additional new flavour dynamics and new field content present in many extensions of the SM may provide contributions to charged LFV (cLFV) processes such as radiative (e.g. $\mu\to e\gamma$) and three-body lepton decays (for instance $\tau\to\mu\mu\mu$). These decays generally arise from higher order processes, and so their branching ratios (Brs) are expected to be small, making them difficult to observe. Thus, any cLFV signal would provide clear evidence for new physics: mixings in the lepton sector and probably the presence of new particles, possibly shedding light on the origin of neutrino mass generation. The search for manifestations of charged LFV constitutes the goal of several experiments [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], dedicated to look for signals of processes such as rare radiative decays, three-body decays and muon-electron conversion in nuclei. Despite the fact that a cLFV signal could provide clear evidence for new physics, the underlying mechanism of lepton mixing might be difficult to unravel. In parallel to the low-energy searches for new physics, i.e. via indirect effects of possible new particles, the LHC has started to search directly for these new particles in its quest to unveil the mechanism of electroweak symmetry breaking, thus possibly providing a solution to the SM hierarchy problem. Among the many possible extensions of the SM, supersymmetry (SUSY) is a well motivated solution for the hierarchy problem, providing many other appealing aspects such as gauge coupling unification and dark matter candidates. If the LHC experiments indeed discover SUSY, it is then extremely interesting to consider supersymmetric models that can also explain neutrino masses and mixings. Furthermore, it is only natural to expect that such models might also give rise to cLFV. If SUSY is indeed realised in nature, cLFV (mediated by new sparticles) would provide a new probe to explore the origin of lepton mixings, playing a complementary rôle to other searches of new physics, i.e. LHC direct searches and neutrino dedicated experiments. One of the most economical possibilities is to embed a seesaw mechanism [14, 15, 16] in the framework of SUSY models (i.e. the SUSY seesaw) [17]. For any seesaw realisation, the neutrino Yukawa couplings could leave their imprints in the SUSY soft-breaking slepton mass matrices, and consequently induce flavour violation at low energies due to the renormalisation group (RG) evolution of the SUSY soft-breaking parameters. The caveat of these scenarios is that, in order to have sufficiently large Yukawa couplings (as required to account for large cLFV Brs), the typical scale of the extra particles (such as right handed neutrinos, scalar or fermionic isospin triplets) is in general very high, potentially very close to the gauge coupling unification scale. However, such a high (seesaw) scale would be impossible to probe experimentally. On the other hand, the so-called inverse seesaw [18] constitutes a very appealing alternative to the ”standard” seesaw realisations. Embedding an inverse seesaw mechanism in the Minimal Supersymmetric extension of the SM (MSSM) requires the inclusion of two additional gauge singlet superfields, with opposite lepton numbers ($+1$ and $-1$). When compared to other SUSY seesaw realisations, cLFV observables are enhanced in this framework , and such an enhancement can be attributed to large neutrino Yukawa couplings ($Y_{\nu}\sim O(1)$), compatible with a seesaw scale $M$, close to the electroweak one, thus within LHC reach. The differences between the inverse seesaw and the standard one can be conceptually understood from an effective point of view and linked to the distinct properties of the lepton number violating dimension-5 (Weinberg) operator (responsible for neutrino masses and mixings) and the total lepton number conserving dimension-6 operator, which is at the origin of cLFV. Contrary to what occurs in the standard seesaw, these two operators are de- correlated in the inverse seesaw, implying that the suppression of the coefficient of the dimension-5 operator will not affect the size of the coefficient of the dimension-6 operator. In both seesaws, the latter operator is proportional to $\left(Y_{\nu}^{\dagger}\frac{1}{\left\|M\right\|^{2}}Y_{\nu}\right)$; however, in the case of a type I seesaw, the dimension-5 operator is proportional to $\left(Y_{\nu}^{\dagger}\frac{1}{M}Y_{\nu}\right)$, while in the case of an inverse seesaw, it has a further suppression of $\frac{\mu}{M}$ ($\mu$ being a dimensionful parameter, linked to the mass of the sterile singlets). The dimension-6 operator will thus be extremely suppressed in the case of a type I seesaw, since in this case $M$ is very large to accommodate natural $Y_{\nu}$. In contrast, in the inverse seesaw, small neutrino masses can easily be accommodated via tiny values of $\mu$, which will not affect the dimension 6 operator. Furthermore, such small values of $\mu$ are natural in the sense of ’t Hooft since in the limit where $\mu\to 0$, the total lepton number symmetry is restored [19]. In view of the strong potential of the inverse seesaw mechanism regarding cLFV, several phenomenological studies have recently been carried out [20, 21, 22, 23, 24, 25, 26, 27, 28, 29]. While a non-supersymmetric inverse seesaw requires two pairs of singlets to explain neutrino oscillation data [26], the supersymmetric generalization can accommodate neutrino data [28] with just one pair of singlets. The latter scenario is also known as the minimal supersymmetric inverse seesaw model (MSISM). This model can also comply with the dark matter relic abundance of the Universe [23]. The extra TeV scale singlet neutrinos may significantly contribute to cLFV observables, irrespective of the supersymmetric states [30]. Supersymmetric realisations of the inverse seesaw may enhance these cLFV rates even further (e.g. the contributions to $l_{i}\rightarrow l_{j}\gamma$, which has been analysed in [20]). Furthermore, this seesaw model can have LHC signatures: the extra singlets can participate in the decay chains, leading to effects which can be important, particularly in the case in which one of the singlets is the lightest supersymmetric particle (LSP) [28]. In this paper, we focus on contributions to cLFV observables, such as $\tau\to\mu\mu\mu$, arising from a Higgs-mediated effective vertex. We explore the contributions which are due to the presence of comparatively light right- handed neutrinos and sneutrinos (which are usually negligible in the framework of a type I SUSY-seesaw), while still having large neutrino Yukawa couplings. We find that all these contributions can lead to a significant enhancement of several cLFV observables. The paper is organised as follows. In Section 2, we define the model, providing a brief overview on the implementation of the inverse seesaw in the MSSM. In Section 3, we discuss the implications of this model regarding low- energy cLFV observables, particular emphasis being given to the Higgs-mediated processes. In Section 4, we study the Higgs-mediated contributions to several lepton flavour violating observables and compare our results to present bounds and to future experimental sensitivities in Section 5. Then we draw some generic conclusions on the viability of an inverse seesaw as the underlying mechanism of LFV. We finally conclude in Section 6. ## 2 Inverse Seesaw Mechanism in the MSSM The model consists of the MSSM extended by three pairs of singlet superfields, $\widehat{\nu}^{c}_{i}$ and $\widehat{X}_{i}$ ($i=1,2,3$)111 $\widetilde{\nu}^{c}=\widetilde{\nu}_{R}^{}$ with lepton numbers assigned to be $-1$ and $+1$, respectively. The supersymmetric inverse seesaw model is defined by the superpotential $\displaystyle{\mathcal{W}}$ $\displaystyle=$ $\displaystyle\varepsilon_{ab}\left[Y^{ij}_{d}\widehat{D}_{i}\widehat{Q}_{j}^{b}\widehat{H}_{d}^{a}+Y^{ij}_{u}\widehat{U}_{i}\widehat{Q}_{j}^{a}\widehat{H}_{u}^{b}+Y^{ij}_{e}\widehat{E}_{i}\widehat{L}_{j}^{b}\widehat{H}_{d}^{a}\right.$ (2.1) $\displaystyle+$ $\displaystyle\left.Y^{ij}_{\nu}\widehat{\nu}^{c}_{i}\widehat{L}^{a}_{j}\widehat{H}_{u}^{b}-\mu\widehat{H}_{d}^{a}\widehat{H}_{u}^{b}\right]+M_{R_{i}}\widehat{\nu}^{c}_{i}\widehat{X}_{i}+\frac{1}{2}\mu_{X_{i}}\widehat{X}_{i}\widehat{X}_{i}~{},$ where $i,j=1,2,3$ denote generation indices. In the above, $\widehat{H}_{d}$ and $\widehat{H}_{u}$ are the down- and up-type Higgs superfields, $\widehat{L}_{i}$ denotes the SU(2) doublet lepton superfields. $M_{R_{i}}$ represents the right-handed neutrino mass term which conserves lepton number. Due to the presence of non-vanishing $\mu_{X_{i}}$, the total lepton number $L$ is no longer a good quantum number; nevertheless, notice that in our formulation $(-1)^{L}$ is still a good symmetry. Without loss of generality, the terms $\widehat{\nu}^{c}_{i}\widehat{X}_{i}$ and $\widehat{X}_{i}\widehat{X}_{i}$ are taken to be diagonal in generation space. Clearly, as $\mu_{X_{i}}\rightarrow 0$, lepton number conservation is restored, since $M_{R}$ does not violate lepton number. Although in the present study we consider three generations of $\widehat{\nu}^{c}$ and $\widehat{X}$, we recall that in the minimal version of the SUSY inverse seesaw (where only one generation of $\widehat{\nu}^{c}$ and $\widehat{X}$ is included), neutrino data can be accommodated [28]. The soft SUSY breaking Lagrangian can be written as $-{\mathcal{L}}_{\rm soft}=-{\mathcal{L}}^{\rm MSSM}_{\rm soft}+m^{2}_{\widetilde{\nu}^{c}}\widetilde{\nu}^{c\dagger}_{i}\widetilde{\nu}^{c}_{i}+m^{2}_{X}\widetilde{X}^{\dagger}_{i}\widetilde{X}_{i}+\left(A_{\nu}Y^{ij}_{\nu}\varepsilon_{ab}\widetilde{\nu}^{c}_{i}\widetilde{L}^{a}_{j}H_{u}^{b}+B_{M_{R_{i}}}\widetilde{\nu}^{c}_{i}\widetilde{X}_{i}+\frac{1}{2}B_{\mu_{X_{i}}}\widetilde{X}_{i}\widetilde{X}_{i}+{\rm h.c.}\right),$ (2.2) where ${\mathcal{L}}^{\rm MSSM}_{\rm soft}$ denotes the soft SUSY breaking terms of the MSSM. In the above, the singlet scalar states $\widetilde{X}_{i}$ and $\widetilde{\nu}^{c}_{i}$ are assumed to have flavour universal masses, i.e. $m^{2}_{X_{i}}=m^{2}_{X}$ and $m^{2}_{\widetilde{\nu}^{c}_{i}}=m^{2}_{\widetilde{\nu}^{c}}$. The parameters $B_{M_{R_{i}}}$ and $B_{\mu_{X_{i}}}$ are the new terms involving the scalar partners of the sterile neutrino states (notice that while the former conserves lepton number, the latter gives rise to a lepton number violating $\Delta L=2$ term). Working under the assumption of a flavour-blind mechanism for SUSY breaking, we will assume universal boundary conditions222In our subsequent numerical analysis, we will relax some of these universality conditions, considering non-universal soft breaking terms for the Higgs sector. In what concerns the right-handed sneutrino sector, we will assume that the corresponding soft-breaking masses hardly run between the GUT and the low-energy scale. for the soft SUSY breaking parameters at some very high energy scale (e.g. the gauge coupling unification scale $\sim 10^{16}$ GeV), $m_{\phi}=m_{0}\,,M_{\text{gaugino}}=M_{1/2}\,,A_{i}=A_{0}\,.$ (2.3) Before addressing neutrino mass generation, a few comments on the nature of the superpotential are in order. As can be seen from Eq. (2.1), the two singlets $\widehat{\nu}^{c}_{i}$ and $\widehat{X}_{i}$ are differently treated in the sense that a $\Delta L=2$ Majorana mass term is present for $\widehat{X}_{i}$ ($\mu_{X_{i}}\widehat{X}_{i}\widehat{X}_{i}$), while no $\mu_{\nu^{c}_{i}}\widehat{\nu}^{c}_{i}\widehat{\nu}^{c}_{i}$ is present in ${\mathcal{W}}$. Although a generic superpotential with $(-1)^{L}$ should contain the latter term, let us notice that similar to what occurs for $\mu_{X_{i}}$, the absence of $\mu_{\nu^{c}_{i}}$ also enhances the symmetry of the model; moreover, we emphasise that it is the magnitude of $\mu_{X_{i}}$ (and not that of $\mu_{\nu^{c}_{i}}$) which controls the size of the light neutrino mass [24, 29]. In view of this, and for the sake of simplicity, we have assumed $\mu_{\nu^{c}_{i}}=0$ (considering non-vanishing, yet small values of $\mu_{\nu^{c}_{i}}$ would not change the qualitative features of the model). Although in our formulation we treat $\mu_{X_{i}}$ as an effective parameter, its origin can be explained either dynamically or in a framework of SUSY Grand Unified Theories (GUT) [24, 29, 25]. Furthermore $\mu_{\nu^{c}_{i}}\ll\mu_{X_{i}}$ can also be realised in extended frameworks [24]. In order to illustrate the pattern of light neutrino masses in the inverse seesaw model and how it is related to the lepton number violating parameter $\mu_{X_{i}}$, we consider the one-generation case. In the $\\{\nu,{\nu^{c}},X\\}$ basis the $(3\times 3)$ neutrino mass matrix can be written as $\displaystyle{\cal M}$ $\displaystyle=$ $\displaystyle\left(\begin{array}[]{ccc}0&m_{D}&0\\\ m_{D}&0&M_{R}\\\ 0&M_{R}&\mu_{X}\\\ \end{array}\right)\ ,$ (2.7) with $m_{D}=Y_{\nu}v_{u}$, yielding the mass eigenvalues ($m_{1}\ll m_{2,3}$): $\displaystyle m_{1}=\frac{m_{D}^{2}\mu_{X}}{m_{D}^{2}+M_{R}^{2}}\,,~{}~{}~{}~{}m_{2,3}=\mp\sqrt{M_{R}^{2}+ùm_{D}^{2}}+\frac{M_{R}^{2}\mu_{X}}{2(m_{D}^{2}+M_{R}^{2})}\,.$ (2.8) The above equation clearly reveals that the lightness of the smallest eigenvalue $m_{1}$ is indeed due to the smallness of $\mu_{X}$ ($\mu_{X}\simeq m_{1}$). Thus the lepton number conserving mass parameters ($m_{D}$ and $M_{R}$) are completely unconstrained in this model. Finally, it is worth noticing that the effective right-handed sneutrino mass term (Dirac-like) is given by $M^{2}_{\widetilde{\nu}^{c}_{i}}=m^{2}_{\widetilde{\nu}^{c}}+M_{R_{i}}^{2}+\sum_{j}{\|Y^{ij}_{\nu}\|^{2}v_{u}^{2}}$. Assuming $M_{R}\sim{\mathcal{O}}$(TeV), the effective mass term will not be very large, in clear contrast to what occurs in the standard (type I) SUSY seesaw. In our analysis, we will be particularly interested in the rôle of such a light sneutrino (i.e. $M^{2}_{\widetilde{\nu}^{c}}\sim M^{2}_{\text{SUSY}}$) in the enhancement of Higgs mediated contributions to lepton flavour violating observables. ## 3 Lepton flavour violation: Higgs-mediated contributions In the SUSY seesaw framework, the only source of flavour violation is encoded in the neutrino Yukawa couplings (which are necessarily non-diagonal to account for neutrino oscillation data); even under the assumption of universal soft breaking terms at the GUT scale, radiative effects proportional to $Y_{\nu}$ induce flavour violation in the slepton mass matrices, which in turn give rise to slepton mediated cLFV observables [31, 32]. As an example, in the leading logarithmic approximation, the RGE corrections to the left-handed slepton soft-breaking masses are given by $\displaystyle(\Delta m_{\widetilde{L}}^{2})_{ij}$ $\displaystyle\simeq$ $\displaystyle-\frac{1}{8\pi^{2}}(3m_{0}^{2}+A_{0}^{2})(Y_{\nu}^{\dagger}LY_{\nu})_{ij}\,,~{}~{}L=\ln\frac{M_{GUT}}{M_{R}}\,$ (3.1) $\displaystyle=$ $\displaystyle\xi(Y^{\dagger}_{\nu}Y_{\nu})_{ij}.$ (For simplicity, in the above we are implicitly assuming a degenerate right- handed neutrino spectrum, $M_{R_{i}}=M_{R}$.) The RGE-induced flavour violating entries, $(\Delta m_{\widetilde{L}}^{2})_{ij}$, give rise to the dominant contributions to low-energy flavour violating observables in the charged lepton sector, such as $\ell_{i}\to\ell_{j}\gamma$ (mediated by chargino-sneutrino and neutralino-slepton loops) and $\ell_{i}\to\ell_{j}\ell_{k}\ell_{m}$ (from photon, $Z$ and Higgs mediated penguin diagrams). Compared to the standard (type I) SUSY seesaw, where $M_{R}\sim 10^{14}$ GeV, the inverse seesaw is characterised by a right handed neutrino mass scale $M_{R}\sim\mathcal{O}(\text{TeV})$ and this in turn leads to an enhancement of the factor $\xi$, (see Eq. (3.1)), and hence to all low-energy cLFV observables, in the latter framework. Furthermore, having right-handed sneutrinos whose mass is of the same order of the other sfermions, i.e. $M^{2}_{\widetilde{\nu}^{c}}\sim M^{2}_{\text{SUSY}}$, the $\widetilde{\nu}^{c}$-mediated processes are no longer suppressed, and might even significantly contribute to the low-energy flavour violating observables. Here, we focus on the impact of such a light $\widetilde{\nu}^{c}$ in the Higgs mediated processes which are expected to be important in the large $\tan\beta$ regime. Although at tree level Higgs-mediated neutral currents are flavour conserving, non-holomorphic Yukawa interactions of the type $\bar{D}_{R}Q_{L}H_{u}^{}$ can be induced at the one-loop level, as first noticed in [33]. In the large $\tan\beta$ regime, in addition to providing significant corrections to the masses of the $b$-quark, these non-holomorphic couplings have an impact on $B^{0}-\bar{B}^{0}$ mixing and flavour violating decays, in particular $B_{s}\rightarrow\mu^{+}\mu^{-}$ [34, 35, 36, 37, 38]. Similarly, in the lepton sector, the origin of the Higgs-mediated flavour violating couplings can be traced to a non-holomorphic Yukawa term of the form $\bar{E}_{R}LH_{u}^{}$ [39]. Other than the corrections to the $\tau$ lepton mass, these new couplings give rise to additional contributions to several cLFV processes mediated by Higgs exchange. In particular $B_{s}\rightarrow\mu\tau$, $B_{s}\rightarrow e\tau$ (the so-called double penguin processes) were considered in [38], while $\tau\rightarrow\mu\eta$ was studied in [40]. A detailed analysis of the several $\mu-\tau$ lepton flavour violating processes, namely $\tau\rightarrow\mu X$ ($X=\gamma,e^{+}e^{-},\mu^{+}\mu^{-},\rho,\pi,\eta,\eta^{\prime}$) can be found in [41]. Even though the flavour violating processes in the quark and lepton sectors have a similar diagrammatic origin, the source of flavour violation is different in each case. In the quark sector, trilinear soft SUSY breaking couplings involving up-type squarks provide the dominant source of flavour violation [35], while in the lepton case, LFV stems from the radiatively induced non-diagonal terms in the slepton masses (see Eq. (3.1)) [39]. In the standard SUSY seesaw (type I), the term ${\widetilde{\nu}^{c}_{i}}H_{u}{\widetilde{L}_{Lj}}$ is usually neglected, as it is suppressed by the very heavy right handed sneutrino masses (${M_{\widetilde{\nu}^{c}_{i}}}\sim 10^{14}$GeV). However, in scenarios such as the inverse SUSY seesaw, where ${M_{\widetilde{\nu}^{c}_{i}}}\sim\mathcal{O}$(TeV), this term may provide the dominant contributions to Higgs mediated lepton flavour violation. The effective Lagrangian describing the couplings of the neutral Higgs fields to the charged leptons is given by $\displaystyle-{\cal L}^{\text{eff}}=\bar{E}^{i}_{R}Y_{e}^{ii}\left[\delta_{ij}H_{d}^{0}+\left(\epsilon_{1}\delta_{ij}+\epsilon_{2ij}(Y_{\nu}^{\dagger}Y_{\nu})_{ij}\right)H_{u}^{0\ast}\right]E^{j}_{L}+\text{h.c.}\,.$ (3.2) In the above, the first term corresponds to the usual Yukawa interaction, while the coefficient $\epsilon_{1}$ encodes the corrections to the charged lepton Yukawa couplings. In the basis where the charged lepton Yukawa couplings are diagonal, the last term in Eq. (3.2), i.e. $\epsilon_{2ij}(Y_{\nu}^{\dagger}Y_{\nu})_{ij}$, is in general non-diagonal, thus providing a new source of charged lepton flavour violation through Higgs mediation. Its origin can be diagrammatically understood from Fig.1, where flavour violation is parametrized via a mass insertion $(\Delta m_{\widetilde{L}}^{2})_{ij}$ (see Eq. (3.1)). \| ---\|--- \| Figure 1: Diagrams contributing to $\epsilon_{2}$. Crosses on scalar lines represent LFV mass insertions $(\Delta m_{\widetilde{L}}^{2})_{ij}$, while those on fermion lines denote chirality flips. The coefficient $\epsilon_{2}$ can be estimated as $\displaystyle\epsilon_{2ij}$ $\displaystyle=$ $\displaystyle\frac{\alpha^{\prime}}{8\pi}\xi\mu M_{1}\left[2F_{2}\left(M_{1}^{2},m_{\widetilde{E}_{Lj}}^{2},m_{\widetilde{E}_{Li}}^{2},m_{\widetilde{E}_{Ri}}^{2}\right)-F_{2}\left(\mu^{2},m^{2}_{\widetilde{E}_{Lj}},m^{2}_{\widetilde{E}_{Li}},M_{1}^{2}\right)\right]+$ (3.4) $\displaystyle\frac{\alpha_{2}}{8\pi}\xi\mu M_{2}\left[F_{2}\left(\mu^{2},m^{2}_{\widetilde{E}_{Lj}},m^{2}_{\widetilde{E}_{Li}},M_{2}^{2}\right)+2F_{2}\left(\mu^{2},m_{\widetilde{\nu}_{Lj}}^{2},m_{\widetilde{\nu}_{Li}}^{2},M_{2}^{2}\right)\right]\,,$ where $\displaystyle F_{2}\left(x,y,z,w\right)=-\frac{x\ln x}{(x-y)(x-z)(x-w)}-\frac{y\ln y}{(y-x)(y-z)(y-w)}+(x\leftrightarrow z,y\leftrightarrow w)\,.$ (3.5) Here, $M_{1}$ and $M_{2}$ are the masses of the electroweak gauginos at low energies. On the other hand, the flavour conserving loop-induced form factor $\epsilon_{1}$ (notice that the diagrams of Fig.1 contribute to this form factor, but without the slepton flavour mixings in the internal lines) can be expressed as [39, 38] $\displaystyle\epsilon_{1}$ $\displaystyle=$ $\displaystyle\frac{\alpha^{\prime}}{8\pi}\mu M_{1}\left[2F_{1}\left(M_{1}^{2},m_{\widetilde{E}_{L}}^{2},m_{\widetilde{E}_{R}}^{2}\right)-F_{1}\left(M_{1}^{2},\mu^{2},m^{2}_{\widetilde{E}_{L}}\right)+2F_{1}\left(M_{1}^{2},\mu^{2},m^{2}_{\widetilde{E}_{R}}\right)\right]$ (3.6) $\displaystyle+\frac{\alpha_{2}}{8\pi}\mu M_{2}\left[F_{1}\left(\mu^{2},m^{2}_{\widetilde{E}_{L}},M_{2}^{2}\right)+2F_{1}\left(\mu^{2},m_{\widetilde{\nu}_{L}}^{2},M_{2}^{2}\right)\right],$ with $\displaystyle F_{1}\left(x,y,z\right)$ $\displaystyle=$ $\displaystyle-\frac{xy\ln(x/y)+yz\ln(y/z)+zx\ln(z/x)}{(x-y)(y-z)(z-x)}\,.$ (3.7) In the standard seesaw mechanism, the diagrams of Fig. 1 provide the only source for Higgs-mediated lepton flavour violation. However, in the framework of the inverse SUSY seesaw, there is an additional diagram that may even account for the dominant Higgs-mediated lepton flavour violation contribution: the sneutrino-chargino mediated loop, depicted in Fig. 2. (Due to the large masses of $\widetilde{\nu}^{c}$ in the standard (type I) seesaw, this process provides negligible contributions, and is hence not taken into account.) Figure 2: Right-handed sneutrino contribution to $\epsilon^{\prime}_{2}$. This contribution is particularly relevant when $\widetilde{\nu}^{c}$ is light. The effective Lagrangian terms encoding lepton flavour violation is accordingly modified as $\displaystyle-{\cal L}^{\text{LFV}}=\bar{E}^{i}_{R}Y_{e}^{ii}\epsilon^{\text{tot}}_{2ij}(Y_{\nu}^{\dagger}Y_{\nu})_{ij}H_{u}^{0\ast}E^{j}_{L}+\text{h.c.}\,,$ (3.8) where $\epsilon^{\text{tot}}_{2}=\epsilon_{2}+\epsilon_{2}^{\prime}$, $\epsilon_{2}^{\prime}$ being the contribution from the new diagram. This contribution can be expressed as $\displaystyle\epsilon^{\prime}_{2ij}=\frac{1}{16\pi^{2}}\mu A_{\nu}F_{1}(\mu^{2},m^{2}_{\widetilde{\nu}_{i}},M^{2}_{\widetilde{\nu}^{c}_{j}}).$ (3.9) In the above, we have parametrized the soft trilinear term for the neutral leptons as $A_{\nu}Y_{\nu}$, where $A_{\nu}$ is a flavour independent real mass term. Below, we provide an approximate estimate of the relative contributions of the terms $\epsilon_{2}$ and $\epsilon^{\prime}_{2}$: for simplicity we take $M_{\widetilde{\nu}^{c}}\sim\mathcal{O}$(TeV) and assume common values for the masses of all SUSY particles and dimensionful terms $A_{\nu}$ at low energies, symbolically denoted by $A_{\nu}\sim\langle\widetilde{m}\rangle\sim M_{\text{SUSY}}$. In this limit, the loop functions are given by $F_{2}\left(x,x,x,x\right)=\frac{1}{6x^{2}}$ and $F_{1}\left(x,x,x\right)=\frac{1}{2x}$. This leads to $\displaystyle\epsilon_{2}=\frac{1}{8\pi}\xi{\widetilde{m}}^{2}\left(\frac{\alpha^{\prime}}{6{\widetilde{m}}^{4}}+3\frac{\alpha_{2}}{{6\widetilde{m}}^{4}}\right)\simeq-0.0007\,,$ (3.10) while $\displaystyle\epsilon^{\prime}_{2}=\frac{1}{16\pi^{2}}{\widetilde{m}}^{2}\frac{1}{2{\widetilde{m}}^{2}}\simeq 0.003\,.$ (3.11) In this illustrative (leading order) calculation, we have assumed that at $M_{\text{GUT}}$, one has $A_{0}=0$, taking for the gauge couplings $\alpha_{2}=0.03$ and $\alpha^{\prime}=0.008$. Following Eq. (3.1), and assuming $M_{R}=10^{3}~{}$GeV, one gets $\xi\sim-1.1\,m_{0}^{2}$. Thus, at the leading order in the inverse seesaw, the lepton flavour violation coefficient becomes $\|\epsilon^{\text{tot}}_{2}\|=\|\epsilon_{2}+\epsilon^{\prime}_{2}\|\simeq 2\times 10^{-3}$. For completeness, let us notice that in the standard seesaw model (where sizable Yukawa couplings are typically associated to a right-handed neutrino mass scale $\sim 10^{14}$ GeV), assuming the same amount of flavour violation as parametrized by $\xi$, one finds $\|\epsilon^{\text{tot}}_{2}\|=\|\epsilon_{2}\|\simeq 2\times 10^{-4}$. This clearly reveals that in the inverse SUSY seesaw, $\epsilon^{\text{tot}}_{2}$ is enhanced by a factor of order $\sim 10$ compared to the standard seesaw. The large enhancement of $\epsilon^{\text{tot}}_{2}$ will have an impact regarding all Higgs-mediated lepton flavour violating observables. The computation of the cLFV observables requires specifying the couplings of the physical Higgs bosons to the leptons, in particular $\bar{E}^{i}_{R}E^{j}_{L}H_{k}$ (where $H_{k}=h,H,A$). The effective Lagrangian describing this interaction can be derived from Eq. (3.2), and reads [39, 38] as $\displaystyle-{\cal L}^{\text{eff}}_{i\neq j}=(2G_{F}^{2})^{1/4}\,\frac{m_{E_{i}}\kappa^{E}_{ij}}{\cos^{2}\beta}\left(\bar{E}^{i}_{R}\,E^{j}_{L}\right)\left[\cos(\alpha-\beta)h+\sin(\alpha-\beta)H-iA\right]+\text{h.c.}\,,\,\,\,$ (3.12) where $\alpha$ is the CP-even Higgs mixing angle and $\tan\beta=v_{u}/v_{d}$, and $\displaystyle\kappa^{E}_{ij}$ $\displaystyle=$ $\displaystyle\frac{\epsilon^{\text{tot}}_{2ij}(Y^{\dagger}_{\nu}Y_{\nu})_{ij}}{\left[1+\left(\epsilon_{1}+\epsilon^{\text{tot}}_{2ii}(Y^{\dagger}_{\nu}Y_{\nu})_{ii}\right)\tan\beta\right]^{2}}\ .$ (3.13) As clear from the above equation, large values of $\epsilon^{\text{tot}}_{2}$ lead to an augmentation of $\kappa^{E}_{ij}$. Given that the cLFV branching ratios are proportional to $({\kappa^{E}_{ij}})^{2}$, a sizeable enhancement, as large as two orders of magnitude, is expected for all Higgs-mediated LFV observables. ## 4 Higgs-mediated lepton flavour violating observables Here we focus our attention on the cLFV observables where the dominant contribution to flavour violation arises from the Higgs penguin diagrams, in particular those involving $\tau$-leptons (due to the comparatively large value of $Y_{\tau}$). In what follows, we discuss some of these LFV decays in detail. • $\tau\rightarrow 3\mu$ In the large $\tan\beta$ regime, Higgs-mediated flavour violating diagrams would be particularly important in this decay mode. The branching ratio can be expressed as [39, 38] $\displaystyle\text{Br}(\tau\to 3\mu)$ $\displaystyle=$ $\displaystyle\frac{G_{F}^{2}\,m_{\mu}^{2}\,m_{\tau}^{7}\,\tau_{\tau}}{1536\,\pi^{3}\cos^{6}\beta}\,\|\kappa_{\tau\mu}^{E}\|^{2}\left[\left(\frac{\sin(\alpha-\beta)\cos\alpha}{M_{H}^{2}}-\frac{\cos(\alpha-\beta)\sin\alpha}{M_{h}^{2}}\right)^{2}+\frac{\sin^{2}\beta}{M_{A}^{4}}\right]$ (4.1) $\displaystyle\approx$ $\displaystyle\frac{G_{F}^{2}\,m_{\mu}^{2}\,m_{\tau}^{7}\,\tau_{\tau}}{768\,\pi^{3}\,M_{A}^{4}}\|\kappa_{\tau\mu}^{E}\|^{2}\tan^{6}\beta\,.$ (4.2) In the above, $\tau_{\tau}$ is the $\tau$ life time and the approximate result has been obtained in the large $\tan\beta$ regime. For other Higgs-mediated lepton flavour violating 3-body decays, $\tau\rightarrow e\mu\mu$, $\tau\rightarrow 3e$ or $\mu\rightarrow 3e$, their corresponding branching ratios can easily be obtained with the appropriate kinematic factors and the flavour changing factor $\kappa$. While $\text{Br}(\tau\rightarrow e\mu\mu)$ can be as large as $\text{Br}(\tau\rightarrow 3\mu)$ when $(Y^{\dagger}_{\nu}Y_{\nu})_{13}\sim O(1)$ (which is possible in the case of an inverted hierarchical light neutrino spectrum), other flavour violating decays with final state electrons such as $\mu\rightarrow 3e$ are considerably suppressed due to the smallness of the Yukawa couplings. * • $B_{s}\to\ell_{i}\ell_{j}$ $B$ mesons can also have Higgs-mediated LFV decays, which are significantly enhanced in the large $\tan\beta$ regime. The branching fraction is given by $\displaystyle\text{Br}(B_{s}\to\ell_{i}\ell_{j})$ $\displaystyle=$ $\displaystyle\frac{G_{F}^{4}\,M^{4}_{W}}{8\,\pi^{5}}\,\|V_{tb}^{}V_{ts}\|^{2}\,M_{B_{s}}^{5}\,f_{B_{s}}^{2}\,\tau_{B_{s}}\biggl{(}\frac{m_{b}}{m_{b}+m_{s}}\biggr{)}^{2}$ (4.3) $\displaystyle\times$ $\displaystyle\sqrt{\biggl{[}1-\frac{(m_{\ell_{i}}+m_{\ell_{j}})^{2}}{M_{B_{s}}^{2}}\biggr{]}\biggl{[}1-\frac{(m_{\ell_{i}}-m_{\ell_{j}})^{2}}{M_{B_{s}}^{2}}\biggr{]}}$ $\displaystyle\times$ $\displaystyle\Biggl{\\{}\biggl{(}1-\frac{(m_{\ell_{i}}+m_{\ell_{j}})^{2}}{M_{B_{s}}^{2}}\biggr{)}\|c^{ij}_{S}\|^{2}+\biggl{(}1-\frac{(m_{\ell_{i}}-m_{\ell_{j}})^{2}}{M_{B_{s}}^{2}}\biggr{)}\|c_{P}^{ij}\|^{2}\Biggr{\\}}\,,$ where $V_{ij}$ represents the Cabibbo-Kobayashi-Maskawa (CKM) matrix, $M_{B_{s}}$ and $\tau_{B_{s}}$ respectively denote the mass and lifetime of the $B_{s}$ meson, while $f_{B_{s}}=230\pm 30$ MeV [42] is the ${B_{s}}$ meson decay constant and $c_{P}^{ij}$, $c_{S}^{ij}$ are the form factors. As an example, the lepton flavour violating (double-penguin) $B_{s}\to\mu\tau$ decay can be computed with the following form factors [38]: $\displaystyle c_{S}^{\mu\tau}=c_{P}^{\mu\tau}$ $\displaystyle=$ $\displaystyle\frac{\sqrt{2}\,\pi^{2}}{G_{F}\,M^{2}_{W}}\frac{m_{\tau}\,\kappa_{bs}^{d}\,\kappa_{\tau\mu}^{E\ast}}{\cos^{4}\beta\,\bar{\lambda}^{t}_{bs}}\left[\frac{\sin^{2}(\alpha-\beta)}{M^{2}_{H}}+\frac{\cos^{2}(\alpha-\beta)}{M^{2}_{h}}+\frac{1}{M^{2}_{A}}\right]$ (4.4) $\displaystyle\approx$ $\displaystyle\frac{8\,\pi^{2}\,m_{\tau}\,m_{t}^{2}}{M^{2}_{W}}\frac{\epsilon_{Y}~{}\kappa_{\tau\mu}^{E}~{}\tan^{4}\beta}{\left[1+(\epsilon_{0}+\epsilon_{Y}Y^{2}_{t})\tan\beta\right]\left[1+\epsilon_{0}\tan\beta\right]}\frac{1}{M^{2}_{A}}\,.$ Here, $\kappa_{bs}^{d}$ represents the flavour mixing in the quark sector while $\bar{\lambda}^{t}_{bs}=V^{}_{tb}V_{ts}$. Similarly, $\epsilon_{0}$ and $\epsilon_{Y}$ are the down type quark form factors mediated by gluino and squark exchange diagrams. The final result was, once again, derived in the large $\tan\beta$ regime. The branching fractions of other flavour violating decays such as $\text{Br}(B_{d,s}\rightarrow\tau e)$, would receive identical contribution from the Higgs penguins. Likewise, the $\text{Br}(B_{d,s}\rightarrow\mu e)$ can be calculated using the appropriate form factors and lepton masses; as expected, these will be suppressed when compared to $\text{Br}(B_{d,s}\rightarrow\tau\mu)$. * • $\tau\to\mu P$ Similar to what occurred in the previous processes, virtual Higgs exchange could also induce decays such as $\tau\to\mu P$, where $P$ denotes a neutral pseudoscalar meson ($P=\pi,\eta,\eta^{\prime})$. In the large $\tan\beta$ limit, where the pseudoscalar Higgs couplings to down-type quarks are enhanced, CP-odd Higgs boson exchanges provide the dominant contribution to the $\tau\to\mu P$ decay. The coupling can be written as $\displaystyle-i(\sqrt{2}\,G_{F})^{1/2}\tan\beta~{}A(\xi_{d}\,m_{d}\,\bar{d}\,d+\xi_{s}\,m_{s}\,\bar{s}\,s+\xi_{b}\,m_{b}\,\bar{b}\,b)+{\rm h.c.}.$ (4.5) Here, the parameters $\xi_{d},\,\xi_{s},\,\xi_{b}$ are of order $\mathcal{O}(1)$. Since we are mostly interested in the Higgs-mediated contributions, we estimate the amplitude of these processes in the limit when both $\tau\rightarrow 3\mu$ and $\tau\to\mu P$ are indeed dominated by the exchange of the scalar fields. Accordingly, and following [41], one can write $\displaystyle\frac{\text{Br}(\tau\to\mu\eta)}{\text{Br}(\tau\to 3\mu)}$ $\displaystyle\simeq$ $\displaystyle 36\,\pi^{2}\left(\frac{f^{8}_{\eta}\,m^{2}_{\eta}}{m_{\mu}\,m^{2}_{\tau}}\right)^{2}(1-x_{\eta})^{2}\left[\xi_{s}+\frac{\xi_{b}}{3}\left(1+\sqrt{2}\,\frac{f^{0}_{\eta}}{f^{8}_{\eta}}\right)\right]^{2},$ (4.6) $\displaystyle\frac{\text{Br}(\tau\to\mu\eta^{\prime})}{\text{Br}(\tau\to\mu\eta)}$ $\displaystyle\simeq$ $\displaystyle\frac{2}{9}\left(\frac{f^{0}_{\eta^{\prime}}}{f^{8}_{\eta}}\right)^{2}\frac{m^{4}_{\eta^{\prime}}}{m^{4}_{\eta}}\left(\frac{1-x_{\eta^{\prime}}}{1-x_{\eta}}\right)^{2}\left[\frac{1+\frac{3}{\sqrt{2}}\,\frac{f^{8}_{\eta^{\prime}}}{f^{0}_{\eta^{\prime}}}\left(\frac{\xi_{s}}{\xi_{b}}+\frac{1}{3}\right)}{\frac{\xi_{s}}{\xi_{b}}+\frac{1}{3}+\frac{\sqrt{2}}{3}\,\frac{f^{0}_{\eta}}{f^{8}_{\eta}}}\right]^{2},$ (4.7) $\displaystyle\frac{\text{Br}(\tau\to\mu\pi)}{\text{Br}(\tau\to\mu\eta)}$ $\displaystyle\simeq$ $\displaystyle\frac{4}{3}\left(\frac{f_{\pi}}{f^{8}_{\eta}}\right)^{2}\,\frac{m^{4}_{\pi}}{m^{4}_{\eta}}~{}(1-x_{\eta})^{-2}\left[\frac{\frac{\xi_{d}}{\xi_{b}}\,\frac{1}{1+z}+\frac{1}{2}\,(1+\frac{\xi_{s}}{\xi_{b}})\frac{1-z}{1+z}}{\frac{\xi_{s}}{\xi_{b}}+\frac{1}{3}+\frac{\sqrt{2}}{3}\,\frac{f^{0}_{\eta}}{f^{8}_{\eta}}}\right]^{2}\,,$ (4.8) where $z=m_{u}/m_{d}$, $m_{\pi},\ f_{\pi}$ are the pion mass and decay constant, $m_{\eta,\eta^{\prime}}$ are the masses of $\eta,\ \eta^{\prime}$, $x_{\eta,\eta^{\prime}}=m_{\eta,\eta^{\prime}}^{2}/m_{\pi}^{2}$, and $f^{8}_{\eta,\eta^{\prime}}$ and $f^{0}_{\eta,\eta^{\prime}}$ are evaluated from the corresponding matrix elements. As first discussed in [40], and taking $\xi_{s},\xi_{b}\sim 1$ and fixing the other parameters as in [41], one finds $\frac{\text{Br}(\tau\to\mu\eta)}{\text{Br}(\tau\to 3\mu)}\simeq 5$. The other branching fractions such as $\text{Br}(\tau\to\mu\eta^{\prime},\mu\pi)$ are considerably suppressed compared to $\text{Br}(\tau\to\mu\eta)$. While the ratio $\frac{\text{Br}(\tau\to\mu\eta^{\prime})}{\text{Br}(\tau\to\mu\eta)}$ can be as large as $6\times 10^{-3}$, $\frac{\text{Br}(\tau\to\mu\pi)}{\text{Br}(\tau\to\mu\eta)}$ would approximately lie in the range $10^{-3}-4\times 10^{-3}$ [41]. Since all these ratios are independent of $\kappa_{\tau\mu}^{E}$, the above quoted numbers can also be applied to the present framework. However, an enhancement in the $\text{Br}(\tau\to 3\mu)$, due to the large values of $\kappa_{\tau\mu}^{E}$, would also imply sizeable values of $\text{Br}(\tau\to\mu\eta)$. * • $H_{k}\to\mu\tau\;(H_{k}=h,H,A)$ The branching ratios of flavour violating Higgs decays provide another interesting probe of lepton flavour violation. Following [43], the branching fraction $H_{k}\to\mu\tau$ (normalised to the flavour conserving one $H_{k}\to\tau\tau$) can be cast as: $\displaystyle{\text{Br}(H_{k}\to\mu\tau)}=\tan^{2}\beta~{}(\|\kappa_{\tau\mu}^{E}\|^{2})~{}C_{\Phi}~{}{\text{Br}(H_{k}\to\tau\tau)}\,,$ (4.9) where we approximated $1/\cos^{2}\beta\simeq\tan^{2}\beta$. The coefficients $C_{\Phi}$ are given by: $\displaystyle C_{h}=\left[\frac{\cos(\beta-\alpha)}{\sin\alpha}\right]^{2},~{}~{}~{}~{}C_{H}=\left[\frac{\sin(\beta-\alpha)}{\cos\alpha}\right]^{2},~{}~{}~{}~{}C_{A}=1.$ (4.10) ## 5 Results and Discussion As discussed in Section 3, in the inverse supersymmetric seesaw, Higgs- mediated contributions can lead to an enhancement of several LFV observables by as much as two orders of magnitude, compared to what is expected in the standard SUSY seesaw. As expected from the analytical study of Section 4, $m_{A}$ and $\tan\beta$ are the most relevant parameters in the Higgs-mediated flavour violating processes. To better illustrate this, in Fig. 3 we study the dependence of Br($\tau\rightarrow 3\mu$) on the aforementioned parameters. We have assumed a common value for the squark masses, $m_{\widetilde{q}}\sim\text{TeV}$, while for left- and right-handed sleptons we take $m_{\widetilde{\ell}}\sim 400~{}\text{GeV}$ and $M_{\widetilde{\nu}^{c}}\sim 3~{}\text{TeV}$ for the right handed sneutrinos. The contours correspond to different values of the branching ratios (the purple region has already been experimentally excluded). From this figure one can easily identify the regimes for $m_{A}$ and $\tan\beta$ which are associated to values of the LFV observables within reach of the present and future experiments. Figure 3: Branching ratio of the process $\tau\rightarrow 3\mu$ as a function of $m_{A}$ (GeV) and $\tan\beta$. From left to right, the contours correspond to $\text{Br}(\tau\rightarrow 3\mu)=2.1\times 10^{-8}$, $10^{-9}$, $10^{-10}$, $10^{-11}$. The purple region has already been experimentally excluded[44]. In what follows, we numerically evaluate some LFV observables. Concerning the mSUGRA parameters (and instead of scanning over the parameter space), we have selected a few benchmark points [45] that already take into account the most recent LHC constraints [46]. We have also considered the case in which the GUT scale universality conditions are relaxed for the Higgs sector, i.e. scenarios of Non-Universal Higgs Masses (NUHM), as this allows to explore the impact of a light CP-odd Higgs boson. In Table 1, we list the chosen points: CMSSM-A and CMSSM-B respectively correspond to the 10.2.2 and 40.1.1 benchmark points in [45], while NUHM-C is an example of a non-universal scenario. Point \| $\tan\beta$ \| $m_{1/2}$ \| $m_{0}$ \| $m^{2}_{H_{U}}$ \| $m^{2}_{H_{D}}$ \| $A_{0}$ \| $\mu$ \| $m_{A}$ ---\|---\|---\|---\|---\|---\|---\|---\|--- CMSSM-A \| 10 \| 550 \| 225 \| $(225)^{2}$ \| $(225)^{2}$ \| 0 \| 690 \| 782 CMSSM-B \| 40 \| 500 \| 330 \| $(330)^{2}$ \| $(330)^{2}$ \| -500 \| 698 \| 604 NUHM-C \| 15 \| 550 \| 225 \| $(652)^{2}$ \| $-(570)^{2}$ \| 0 \| 478 \| 150 Table 1: Benchmark points used in the numerical analysis (dimensionful parameters in GeV). CMSSM-A and CMSSM-B correspond to 10.2.2 and 40.1.1 benchmark points of [45]. For each point considered, the low-energy SUSY parameters were obtained using SuSpect [47]. In what concerns the evolution of the soft-breaking right-handed sneutrino masses $m_{\tilde{\nu}^{c}}^{2}$, we have assumed that the latter hardly run between the GUT scale and the low-energy one. The flavour-violating charged slepton parameters (e.g. $(\Delta m_{\widetilde{L}}^{2})_{ij}$ or $\xi$), were estimated at the leading order using Eq. (3.1). Concerning NUHM, we use the same value of $\xi$ as for CMSSM-A. Here, we are particularly interested to study the effect of light CP-odd Higgs boson and this naive approximation will serve our purpose. Furthermore, we use the mass insertion approximation, assuming that mixing between left and right chiral slepton states are relatively small. In computing the branching fractions and the flavour violating factor $\kappa^{E}_{ij}$ we have assumed (physical) right- handed sneutrino masses $M_{\widetilde{\nu}^{c}}\approx 3$ TeV and $\left(Y_{\nu}^{\dagger}Y_{\nu}\right)=0.7$, in agreement with low-energy neutrino data as well as other low-energy constraints, which are particularly relevant in the inverse seesaw case such as Non-Standard Neutrino Interactions bounds [48]. Moreover, in our numerical analysis, we have fixed the trilinear soft breaking parameter $A_{\nu}=-500$ GeV (at the SUSY scale). We now proceed to present our results for the flavour violating observables discussed in Section 4. In Table 2, we collect the values of the different branching ratios, as obtained for the considered benchmark points of Table 1. We have also presented the corresponding current experimental bounds and future sensitivity. LFV Process \| Present Bound \| Future Sensitivity \| CMSSM-A \| CMSSM-B \| NUHM-C ---\|---\|---\|---\|---\|--- $\tau\rightarrow\mu\mu\mu$ \| $2.1\times 10^{-8}$[44] \| $8.2\times 10^{-10}$ [52] \| $1.4\times 10^{-15}$ \| $3.9\times 10^{-11}$ \| $8.0\times 10^{-12}$ $\tau^{-}\rightarrow e^{-}\mu^{+}\mu^{-}$ \| $2.7\times 10^{-8}$[44] \| $\sim 10^{-10}$ [52] \| $1.4\times 10^{-15}$ \| $3.4\times 10^{-11}$ \| $8.0\times 10^{-12}$ $\tau\rightarrow eee$ \| $2.7\times 10^{-8}$[44] \| $2.3\times 10^{-10}$ [52] \| $3.2\times 10^{-20}$ \| $9.2\times 10^{-16}$ \| $1.9\times 10^{-16}$ $\mu\rightarrow eee$ \| $1.0\times 10^{-12}$[1] \| \| $6.3\times 10^{-22}$ \| $1.5\times 10^{-17}$ \| $3.7\times 10^{-18}$ $\tau\rightarrow\mu\eta$ \| $2.3\times 10^{-8}$[49] \| $\sim 10^{-10}$ [52] \| $8.0\times 10^{-15}$ \| $3.3\times 10^{-10}$ \| $4.6\times 10^{-11}$ $\tau\rightarrow\mu\eta^{\prime}$ \| $3.8\times 10^{-8}$[49] \| $\sim 10^{-10}$ [52] \| $4.3\times 10^{-16}$ \| $1.1\times 10^{-10}$ \| $3.1\times 10^{-12}$ $\tau\rightarrow\mu\pi^{0}$ \| $2.2\times 10^{-8}$[49] \| $\sim 10^{-10}$ [52] \| $1.8\times 10^{-17}$ \| $8.5\times 10^{-13}$ \| $1.0\times 10^{-13}$ $B^{0}_{d}\rightarrow\mu\tau$ \| $2.2\times 10^{-5}$[50] \| \| $2.7\times 10^{-15}$ \| $8.5\times 10^{-10}$ \| $2.7\times 10^{-11}$ $B^{0}_{d}\rightarrow e\mu$ \| $6.4\times 10^{-8}$[51] \| $1.6\times 10^{-8}$[53] \| $1.2\times 10^{-17}$ \| $3.1\times 10^{-12}$ \| $1.2\times 10^{-13}$ $B^{0}_{s}\rightarrow\mu\tau$ \| \| \| $7.7\times 10^{-14}$ \| $2.5\times 10^{-8}$ \| $7.8\times 10^{-10}$ $B^{0}_{s}\rightarrow e\mu$ \| $2.0\times 10^{-7}$[51] \| $6.5\times 10^{-8}$[53] \| $3.4\times 10^{-16}$ \| $8.9\times 10^{-11}$ \| $3.4\times 10^{-12}$ $h\rightarrow\mu\tau$ \| \| \| $1.3\times 10^{-8}$ \| $2.6\times 10^{-7}$ \| $2.3\times 10^{-6}$ $A,H\rightarrow\mu\tau$ \| \| \| $3.4\times 10^{-6}$ \| $1.3\times 10^{-4}$ \| $5.0\times 10^{-6}$ Table 2: Higgs-mediated contributions to the branching ratios of several lepton flavour violating processes, for the different benchmark points of Table 1. We also present the current experimental bounds and future sensitivities for the LFV observables. From Table 2, one can verify that from an experimental point of view, the most promising channel in the supersymmetric inverse seesaw is $\tau\rightarrow\mu\eta$ which could be tested at the next generation of $B$ factories. The $B^{0}_{d,s}\rightarrow\mu\tau$ decay might also be interesting, but little conclusions can be drawn due to lack of information concerning the future sensitivities. It is important to stress that the numerical results summarised in Table 2 correspond to considering only Higgs-mediated contributions. In the low $\tan\beta$ regime, photon- and $Z$-penguin diagrams may induce comparable or even larger contributions to the observables, and potentially enhance the branching fractions. Thus, the results for small $\tan\beta$ should be interpreted as conservative estimates, representing only partial contributions. For large $\tan\beta$ values, Higgs penguins do indeed provide the leading contributions. Comparing our results with those obtained for a type I SUSY seesaw at high scales (or even with a TeV scale SUSY seesaw), we find a large enhancement of the branching fractions in the inverse seesaw framework. Another interesting property of the Higgs-mediated processes is that the corresponding amplitude strongly depends on the chirality of the heaviest lepton (be it the decaying lepton, or the heaviest lepton produced in $B$ decays). Considering the decays of a left-handed lepton $\ell^{i}_{L}\rightarrow\ell^{j}_{R}X$, one finds that the corresponding branching ratios would be suppressed by a factor $\frac{m_{\ell^{j}}^{2}}{m_{\ell^{i}}^{2}}$ compared to those of the right- handed lepton $\ell^{i}_{R}\rightarrow\ell^{j}_{L}X$. This can induce an asymmetry that potentially allows to identify if Higgs mediation is the dominant contribution to the LFV observables. Furthermore this asymmetry would be more pronounced in the inverse-seesaw framework. Due to its strong enhancement of the Higgs-penguin contributions, if realised in Nature, the inverse seesaw offers a unique framework to test Higgs effects in LFV processes. In fact, and as discussed in [32], if photon penguins provide the dominant contribution to both $\text{Br}(\tau\rightarrow 3\mu)$ and $\text{Br}(\tau\rightarrow\mu\gamma)$, then the latter observables are strongly correlated, $\frac{\text{Br}(\tau\rightarrow 3\mu)}{\text{Br}(\tau\rightarrow\mu\gamma)}\sim 0.003$ (see [32]). On the other hand, if the dominant contribution to the three-body decays arises from Higgs penguins, the correlation no longer holds, and the latter ratio can be significantly enhanced. This would be the case of the present framework. ## 6 Conclusions If observed, charged lepton flavour violation clearly signals the presence of new physics. In this work, we have studied Higgs-mediated LFV processes in the framework of the supersymmetric inverse seesaw. TeV scale right-handed neutrinos (and hence light right-handed sneutrinos) offer the possibility to enhance the Higgs-mediated contributions to several LFV processes. As shown in this work, in the inverse SUSY seesaw, LFV branching ratios can be enhanced by as much as two orders of magnitude when compared to the standard (type I) SUSY seesaw. ### Acknowledgements The authors are thankful to A. Vicente for many enlightening discussions. D.D. acknowledges financial support from the CNRS. This work has been partly done under the ANR project CPV-LFV-LHC NT09-508531. ## References * [1] U. Bellgardt et al. [SINDRUM Collaboration], Nucl. Phys. B 299 (1988) 1. * [2] M. L. Brooks et al. [MEGA Collaboration], Phys. Rev. Lett. 83 (1999) 1521 [arXiv:hep-ex/9905013]. * [3] J. Aysto et al., arXiv:hep-ph/0109217. * [4] J. P. Lees et al. [BaBar Collaboration], Phys. Rev. D 81 (2010) 111101 [arXiv:1002.4550 [hep-ex]]. * [5] A. G. Akeroyd et al. [SuperKEKB Physics Working Group], arXiv:hep-ex/0406071. * [6] B. Aubert et al. [BABAR Collaboration], Phys. Rev. Lett. 95 (2005) 041802 [arXiv:hep-ex/0502032]. * [7] B. Aubert et al. [BABAR Collaboration], Phys. Rev. Lett. 96 (2006) 041801 [arXiv:hep-ex/0508012]. * [8] Y. Kuno, Nucl. Phys. Proc. Suppl. 149 (2005) 376. * [9] S. Ritt [MEG Collaboration], Nucl. Phys. Proc. Suppl. 162 (2006) 279. * [10] M. Bona et al., arXiv:0709.0451 [hep-ex]. * [11] K. Hayasaka et al. [Belle Collaboration], Phys. Lett. B 666 (2008) 16 [arXiv:0705.0650 [hep-ex]]. * [12] O. A. Kiselev [MEG Collaboration], Nucl. Instrum. Meth. A 604 (2009) 304. * [13] The PRIME working group, unpublished; LOI to J-PARC 50-GeV PS, LOI-25, http://psux1.kek.jp/jhf-np/LOIlist/LOIlist.html * [14] P. Minkowski, Phys. Lett. B 67 (1977) 421; M. Gell-Mann, P. Ramond and R. Slansky, in Complex Spinors and Unified Theories eds. P. Van. Nieuwenhuizen and D. Z. Freedman, Supergravity (North-Holland, Amsterdam, 1979), p.315 [Print-80-0576 (CERN)]; T. Yanagida, in Proceedings of the Workshop on the Unified Theory and the Baryon Number in the Universe, eds. O. Sawada and A. Sugamoto (KEK, Tsukuba, 1979), p.95; S. L. Glashow, in Quarks and Leptons, eds. M. Lévy et al. (Plenum Press, New York, 1980), p.687; R. N. Mohapatra and G. Senjanović, Phys. Rev. Lett. 44 (1980) 912. * [15] R. Barbieri, D. V. Nanopolous, G. Morchio and F. Strocchi, Phys. Lett. B 90 (1980) 91; R. E. Marshak and R. N. Mohapatra, Invited talk given at Orbis Scientiae, Coral Gables, Fla., Jan. 14-17, 1980, VPI-HEP-80/02; T. P. Cheng and L. F. Li, Phys. Rev. D 22 (1980) 2860; M. Magg and C. Wetterich, Phys. Lett. B 94 (1980) 61; G. Lazarides, Q. Shafi and C. Wetterich, Nucl. Phys. B 181 (1981) 287; J. Schechter and J. W. F. Valle, Phys. Rev. D 22 (1980) 2227; R. N. Mohapatra and G. Senjanović, Phys. Rev. D 23 (1981) 165. * [16] E. Ma, Phys. Rev. Lett. 81 (1998) 1171 [arXiv:hep-ph/9805219]; R. Foot, H. Lew, X. G. He and G. C. Joshi, Z. Phys. C 44 (1989) 441. * [17] S. F. King, Phys. Lett. B 439 (1998) 350 [arXiv:hep-ph/9806440]; S. Davidson and S. F. King, Phys. Lett. B 445 (1998) 191 [arXiv:hep-ph/9808296]. * [18] R. N. Mohapatra and J. W. F. Valle, Phys. Rev. D 34 (1986) 1642. * [19] G. ’t Hooft, Lecture given at Cargese Summer Inst., Cargese, France, Aug 26 - Sep 8, 1979. * [20] F. Deppisch and J. W. F. Valle, Phys. Rev. D 72 (2005) 036001 [arXiv:hep-ph/0406040]. * [21] F. Deppisch, T. S. Kosmas and J. W. F. Valle, Nucl. Phys. B 752 (2006) 80 [arXiv:hep-ph/0512360]. * [22] J. Garayoa, M. C. Gonzalez-Garcia and N. Rius, JHEP 0702 (2007) 021 [arXiv:hep-ph/0611311]. * [23] C. Arina, F. Bazzocchi, N. Fornengo, J. C. Romao and J. W. F. Valle, Phys. Rev. Lett. 101 (2008) 161802 [arXiv:0806.3225 [hep-ph]]. * [24] E. Ma, Phys. Rev. D 80 (2009) 013013 [arXiv:0904.4450 [hep-ph]]. * [25] P. S. B. Dev and R. N. Mohapatra, Phys. Rev. D 81 (2010) 013001 [arXiv:0910.3924 [hep-ph]]. * [26] M. Malinsky, T. Ohlsson, Z. z. Xing and H. Zhang, Phys. Lett. B 679 (2009) 242 [arXiv:0905.2889 [hep-ph]]. * [27] F. Bazzocchi, D. G. Cerdeno, C. Munoz and J. W. F. Valle, Phys. Rev. D 81 (2010) 051701 [arXiv:0907.1262 [hep-ph]]. * [28] M. Hirsch, T. Kernreiter, J. C. Romao and A. Villanova del Moral, JHEP 1001 (2010) 103 [arXiv:0910.2435 [hep-ph]]. * [29] F. Bazzocchi, Phys. Rev. D83 (2011) 093009. [arXiv:1011.6299 [hep-ph]]. * [30] J. Bernabeu, A. Santamaria, J. Vidal, A. Mendez and J. W. F. Valle, Phys. Lett. B 187 (1987) 303; G. C. Branco, M. N. Rebelo and J. W. F. Valle, Phys. Lett. B 225 (1989) 385; N. Rius and J. W. F. Valle, Phys. Lett. B 246 (1990) 249; M. C. Gonzalez-Garcia and J. W. F. Valle, Mod. Phys. Lett. A 7 (1992) 477. * [31] F. Borzumati and A. Masiero, Phys. Rev. Lett. 57 (1986) 961. * [32] J. Hisano, T. Moroi, K. Tobe and M. Yamaguchi, Phys. Rev. D 53 (1996) 2442 [arXiv:hep-ph/9510309]; J. Hisano and D. Nomura, Phys. Rev. D 59 (1999) 116005 [arXiv:hep-ph/9810479]. * [33] L. J. Hall, R. Rattazzi and U. Sarid, Phys. Rev. D 50 (1994) 7048 [arXiv:hep-ph/9306309]. * [34] S. R. Choudhury and N. Gaur, Phys. Lett. B 451 (1999) 86 [arXiv:hep-ph/9810307]. * [35] K. S. Babu and C. F. Kolda, Phys. Rev. Lett. 84 (2000) 228 [arXiv:hep-ph/9909476]. * [36] G. Isidori and A. Retico, JHEP 0111 (2001) 001 [arXiv:hep-ph/0110121]; JHEP 0209 (2002) 063 [arXiv:hep-ph/0208159]. * [37] A. J. Buras, P. H. Chankowski, J. Rosiek and L. Slawianowska, Nucl. Phys. B 619 (2001) 434 [arXiv:hep-ph/0107048]; Phys. Lett. B 546 (2002) 96 [arXiv:hep-ph/0207241]; G. D’Ambrosio, G. F. Giudice, G. Isidori and A. Strumia, Nucl. Phys. B 645 (2002) 155 [arXiv:hep-ph/0207036]. * [38] A. Dedes, J. R. Ellis and M. Raidal, Phys. Lett. B 549 (2002) 159 [arXiv:hep-ph/0209207]. * [39] K. S. Babu and C. Kolda, Phys. Rev. Lett. 89 (2002) 241802 [arXiv:hep-ph/0206310]. * [40] M. Sher, Phys. Rev. D 66 (2002) 057301 [arXiv:hep-ph/0207136]. * [41] A. Brignole and A. Rossi, Nucl. Phys. B 701 (2004) 3 [arXiv:hep-ph/0404211]. * [42] C. W. Bernard, Nucl. Phys. Proc. Suppl. 94 (2001) 159 [arXiv:hep-lat/0011064]. * [43] A. Brignole and A. Rossi, Phys. Lett. B 566 (2003) 217 [arXiv:hep-ph/0304081]. * [44] K. Hayasaka, K. Inami, Y. Miyazaki, K. Arinstein, V. Aulchenko, T. Aushev, A. M. Bakich, A. Bay et al., Phys. Lett. B687 (2010) 139-143. [arXiv:1001.3221 [hep-ex]] * [45] S. S. AbdusSalam, B. C. Allanach, H. K. Dreiner, J. Ellis, U. Ellwanger, J. Gunion, S. Heinemeyer, M. Kraemer et al., [arXiv:1109.3859 [hep-ph]]. * [46] S. Chatrchyan et al. [ CMS Collaboration ], [arXiv:1107.1870 [hep-ex]]; ATLAS Collaboration, arXiv:1109.6606 [hep-ex]. * [47] A. Djouadi, J. L. Kneur and G. Moultaka, [arXiv:hep-ph/0211331]. * [48] S. Antusch, C. Biggio, E. Fernandez-Martinez, M. B. Gavela and J. Lopez-Pavon, JHEP 0610 (2006) 084 [hep-ph/0607020]; S. Antusch, J. P. Baumann and E. Fernandez-Martinez, Nucl. Phys. B 810 (2009) 369 [arXiv:0807.1003 [hep-ph]]. * [49] K. Hayasaka [Belle Collaboration], PoSICHEP 2010 (2010) 241 [arXiv:1011.6474 [hep-ex]]. * [50] B. Aubert et al. [BABAR Collaboration], Phys. Rev. D 77 (2008) 091104 [arXiv:0801.0697 [hep-ex]]. * [51] T. Aaltonen et al. [CDF Collaboration], Phys. Rev. Lett. 102 (2009) 201801 [arXiv:0901.3803 [hep-ex]]. * [52] B. O’Leary et al. [SuperB Collaboration], arXiv:1008.1541 [hep-ex]. * [53] W. Bonivento and N. Serra, CERN-LHCB-2007-028.	arxiv-papers	2011-11-24T19:06:55	2024-09-04T02:49:24.638009	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Asmaa Abada, Debottam Das, C\\'edric Weiland", "submitter": "C\\'edric Weiland", "url": "https://arxiv.org/abs/1111.5836" }
1111.5994	# Possibility and Impossibility of the Entropy Balance in Lattice Boltzmann Collisions Alexander N. Gorban and Dave Packwood Department of Mathematics, University of Leicester, United Kingdom ###### Abstract We demonstrate that in the space of distributions operated on by lattice Boltzmann methods that there exists a vicinity of the equilibrium where collisions with entropy balance are possible and, at the same time, there exist an area of nonequilibrium distributions where such collisions are impossible. We calculate and graphically represent these areas for some simple entropic equilibria using single relaxation time models. Therefore it is shown that the definition of an entropic LBM is incomplete without a strategy to deal with certain highly nonequilibrium states. Such strategies should be explicitly stated as they may result in the production of additional entropy. ## I Introduction Lattice Boltzmann schemes are a type of discrete algorithm which can be used to simulate fluid dynamics and more Succi ; Benzi . Although such a method can be derived as a discretization of the fully continuous Boltzmann equation, some thermodynamics properties may be lost in this process. The Entropic lattice Boltzmann method (ELBM) was invented first in 1998 as a tool for the construction of single relaxation time lattice Boltzman models which respects a $H$-theorem Htheorem ; SucciRevModPhys . For this purpose, instead of the mirror image with a local equilibrium as the reflection center, the entropic involution was proposed, which preserves the entropy value. Later, it was called the Karlin-Succi involution gorban06 . Nevertheless, controlling the proper entropy balance remained until recently a challenging problem for many lattice Boltzmann models Nonexist . Some discussions of modern ELBM implementations and results were published recently KarlinSucciComment . The distribution functions at the centre of lattice Boltzmann methods are often referred to and understood as particle densities. Of course for such an interpretation to be meaningful the distribution function should be strictly positive. Despite this some lattice Boltzmann implementations may, as a numerical scheme, tolerate negative population values. An ELBM usually involves an evaluation of a Boltzmann type entropy function, which does not exist for negative populations, hence such an ELBM cannot ever tolerate a negative population value. Due to this there are population values for which an entropic involution cannot be performed. A complete definition of an ELBM must include a strategy for what to do in such a situation. The choice of such a strategy should be explicitly given in any definition of an ELBM as it may have side-effects with modification of dissipation which should be understood separately from the influence of the proper entropy balance. In this paper we study the regions in the spaces of distributions (populations) where collisions with entropy preservation are possible (near the equilibrium) and where they are impossible (sufficiently far from the equilibrium) and demonstrate that both such areas always exist apart some trivial degenerated cases. ## II Single Relaxation Time LB Schemes For fluids, LB systems can be derived as a discretization of the Boltzmann Equation ${\partial}_{t}f+\mathbf{v}\cdot{\partial}_{\mathbf{x}}f=Q(f)$ (1) where $f\equiv f(\mathbf{x},\mathbf{v},t)$ is a one particle distribution function over space, velocity space and time and $Q(f)$ represents the interaction between particles, sometimes called a collision operation. A particular example of the interaction $Q(f)$ is the Bhatnagar-Gross-Krook equation $Q(f)=-\frac{1}{\tau}(f-f^{\mathrm{eq}}).$ (2) The BGK operation represents a relaxation towards the local equilibrium $f^{\mathrm{eq}}$ with rate $1/\tau$. The distribution $f^{\mathrm{eq}}$ is given by the Maxwell Boltzmann distribution, $f^{\mathrm{eq}}=\frac{\rho}{(2\pi T)^{D/2}}\exp\left(\frac{-(\mathbf{v}-\mathbf{u})^{2}}{2T}\right).$ (3) The macroscopic quantities are available as integrals over velocity space of the distribution function, $\rho=\int f\;\mathrm{d}\mathbf{v},\;\rho\mathbf{u}=\int\mathbf{v}f\;\mathrm{d}\mathbf{v},\;\rho\mathbf{u}^{2}+\rho T=\int\mathbf{v}^{2}f\;\mathrm{d}\mathbf{v}.$ A discrete approximation to these integrals is the first ingredient to discretize this system. The scalar field of the population function (over space, vector space and time) becomes a sequence of vector fields (over space) in time $f_{i}(\mathbf{x},n_{t}\;\epsilon),n_{t}\in\mathbb{Z}$, where the elements of the vector each correspond with an element of the quadrature. Explicitly the macroscopic moments are given by, $\rho=\sum_{i=1}^{n}f_{i},\;\rho\mathbf{u}=\sum_{i=1}^{n}\mathbf{v}_{i}f_{i},\;\rho\mathbf{u}^{2}+\rho T=\sum_{i=1}^{n}\mathbf{v}_{i}^{2}f_{i}.$ The complete discrete scheme is given by $f_{i}(\mathbf{x}+\epsilon\mathbf{v}_{i},t+\epsilon)=f_{i}(\mathbf{x},t)+\omega(f^{\mathrm{eq}}_{i}(\mathbf{x},t)-f_{i}(\mathbf{x},t))$ (4) where $\epsilon$ is the time step. For this system a discrete equilibrium must be used. The choice of the velocity set $\\{\mathbf{v}_{1},\ldots,\mathbf{v}_{n}\\}$ and the discrete equilibrium distribution $f^{\mathrm{eq}}_{i}$ should provide the best approximation of the transport equations for the moments by the discrete scheme (4). ## III ELBM In the continuous case the Maxwellian distribution maximizes entropy, as measured by the Boltzmann $H$ function, and therefore also has zero entropy production. In the context of lattice Boltzmann methods a discrete form of the $H$-theorem has been suggested as a way to introduce thermodynamic control to the system Htheorem ; Boghosian . A variation on the LBGK is the ELBGK ELBM . In this family of methods, the equilibria are defined as the conditional entropy maximizers under given values of macroscopic variables (entropic equilibria). The entropies have been constructed in a lattice dependent fashion in LatticeEntropies . A slightly different notation is used for the lattice Boltzmann algorithm, $f_{i}(\mathbf{x}+\epsilon\mathbf{v}_{i},t+\epsilon)=f_{i}(\mathbf{x},t)+\alpha\beta(f^{\mathrm{eq}}_{i}(\mathbf{x},t)-f_{i}(\mathbf{x},t)).$ (5) The single parameter $\omega$ is replaced by a composite parameter $\alpha\beta$. In this case $\beta$ controls the viscosity and $\alpha$ is varied to ensure a constant entropy condition according to the discrete $H$-theorem. With knowledge of the entropy function $S$, $\alpha$ is found as the non-trivial root of the equation $S(\mathbf{f})=S(\mathbf{f}+\alpha(\mathbf{f}^{\ast}-\mathbf{f})).$ (6) The trivial root $\alpha=0$ returns the entropy value of the original populations. ELBGK then finds the non-trivial $\alpha$ such that (6) holds. This version of the BGK collision one calls entropic BGK (or EBGK) collision. A solution of (6) must be found at every time step and lattice site. The EBGK collision obviously respects the Second Law (if $\beta\leq 1$), and simple analysis of entropy dissipation gives the proper evaluation of viscosity. In general the entropy function is based upon the lattice. For example, in the case of the simple one dimensional lattice with velocities $\mathbf{v}=(-c,0,c)$ and corresponding populations $\mathbf{f}=(f_{-},f_{0},f_{+})$ an explicit Boltzmann style entropy function is known LatticeEntropies : $S(\mathbf{f})=-f_{-}\log(f_{-})-f_{0}\log(f_{0}/4)-f_{+}\log(f_{+}).$ (7) ## IV Regions of Existence and Non-existence of Entropic Involution Let us study the entropic involution in the distribution simplex $\Sigma$ given by $\sum f_{i}=const>0$, $f_{i}\geq 0$. Figure 1: The simplex $\Sigma$ is given by the white background. (1) Populations relax through the equilibrium given by the single point to an equal entropy point, if possible. The boundary of this possibility is given. (2) The regions $A$ (the entropic involution is possible) and $B$ (the involution is impossible) as subsets of the simplex divided by this boundary are presented. Figure 2: The simplex $\Sigma$ is given by the white background. (1) Populations relax through the their corresponding equilibrium point along the line given by constant $u$ to an equal entropy point, if possible. The boundary of this possibility is given. (2) The regions $A$ and $B$ separated by this boundary are presented. Let us prove that under very natural assumptions about some properties of the entropy that the simplex of distributions can be split into two subsets $A$ and $B$: in the set $A$ the entropic involution exists, and for distributions from the set $B$ equation (6) has no non-trivial solutions. Both sets $A$ and $B$ have non-empty interior (apart of a trivial symmetric degenerated case). Let the entropy $S$ be a strictly concave continuous function in the distribution simplex $\Sigma$. We assume also that $S$ is twice differentiable, the Hessian of $S$, $\partial^{2}S/\partial f_{i}\partial f_{j}$, is negative definite in the interior of the simplex, $\Sigma_{+}$, where $\sum f_{i}=const$, $f_{i}>0$ and the global maximizer of $S$, the equilibrium, belongs to the interior of the simplex. For example, the relative Boltzmann entropy, $S=-\sum f_{i}(\ln(f_{i}/W_{i})-1)$, $W_{i}>0$, satisfies these conditions, because $f\ln f\to 0$ when $f\to 0$ and $\partial^{2}S/\partial f_{i}\partial f_{j}=-\delta_{ij}/f_{i}$, whereas the relative Burg entropy $S=\sum W_{i}(\ln(f_{i}/W_{i}))$ does not satisfy these conditions because it does not exist on the border of the simplex. Macroscopic variables are linear functions of $\mathbf{f}$. The sets with given values of the macroscopic variables in the simplex $\Sigma$ are polyhedra, intersections of $\Sigma$ with linear manifolds with the given values of moments. We assume that in any such a polyhedron the entropy achieves its (conditionally) global maximum at an internal point. This assumption holds for the Boltzmann relative entropy because of the logarithmic singularity of the “chemical potentials” $\mu_{i}=\ln(f_{i}/W_{i})$ on the border of positivity. These maximizers are equilibria. If $\mathbf{f}$ is sufficiently close to a positive equilibrium then, due to the implicit function theorem, the nontrivial solution to equation (6) exists and it gives $\alpha=2+o(\mathbf{f}-\mathbf{f}^{})$. The value $\alpha=2$ corresponds to the mirror image, the small term $o(\mathbf{f}-\mathbf{f}^{})$ gives the corrections to the value $\alpha=2$. Therefore, in some vicinity of the equilibrium the entropic involution exists. To prove the existence of the area where entropic involution is impossible, let us consider one polyhedron with given values of the macroscopic variables and a positive equilibrium. The local minima of the entropy in this polyhedron are situated at the vertices. At least one of them is a global minimum. Let this vertex be $\mathbf{f}^{\mathbf{v}}$. Let us draw a straight line $l$ through points $\mathbf{f}^{\mathbf{v}}$ and $\mathbf{f}^{}$. The intersection $l\cap\Sigma$ is an interval and $S$ achieves its global minimum on this interval at the point $\mathbf{f}^{\mathbf{v}}$. If the dimension of the polyhedron is more than one then the opposite end of this interval is not even a local minimum of $S$ in the polyhedron and the entropic involution does not exists for $\mathbf{f}^{\mathbf{v}}$ and some vicinity around it. A special degeneration is possible when the polyhedra are one-dimensional, i.e. intervals, and the values of the entropy at both ends of each interval coincide. For example, for two-dimensional distributions, $f_{+},f_{-}$, the entropy $S==f_{+}\ln f_{+}-f_{-}\ln f_{-}$ and the macroscopic variable $\rho=f_{+}+f_{-}$. Apart from such symmetric one-dimensional cases there exists an area near the maximally non-equilibrium vertex $\mathbf{f}^{\mathbf{v}}$ where the entropic involution cannot be defined. Such an area may also exist near some other vertices, where local entropy minima are reached. For the Burg entropy, the entropic involution is always possible Boghosian because it tends to $-\infty$ at the border of positivity. The same is true, for the relative entropy of the form $S=-\beta^{-1}\sum W_{i}((W_{i}/f_{i})^{\beta}-1)$ that tends to the Burg entropy when $\beta\to 0$ GoGoJudge2011 . This negative brunch of the relative Tsallis entropy is less known. The standard Tsallis entropy Tsallis is finite at the border of positivity, hence, collisions with entropy preservation are not always possible for it. We now demonstrate the population function values where the involution cannot be performed for some simple examples. We use the standard 1-D lattice described in Section III with the discrete equilibrium given in Eq 7. We begin with an LBM with only one conserved moment in collision, namely density. The equilibrium is $f_{-}^{}=\frac{\rho}{6},\;\;f_{0}^{}=\frac{2\rho}{3},\;\;f_{+}^{}=\frac{\rho}{6}.$ In Fig. 1, the simplex $\Sigma$ of positive populations with a fixed density $\rho=1$ is the triangle given by the intersection of three half-planes, $f^{+}>0,\,f^{-}>0$, and $1-f^{+}-f^{-}>0$. Within that region we plot several entropy level contours $S(\mathbf{f})=c$ and the unique equilibrium point. The region is divided into the parts where the entropic involution is possible (around the equilibrium) and where it is impossible. A more common use of lattice Boltzmann involves a second fixed moment, momentum. The entropic equilibria used by the ELBGK are available explicitly as the maximum of the entropy function (7), $f_{\mp}^{}=\frac{\rho}{6}(\mp 3u-1+2\sqrt{1\\!+\\!3u^{2}}),\;f_{0}^{}=\frac{2\rho}{3}(2-\sqrt{1\\!+\\!3u^{2}}).$ In this case the dimension of the equilibrium is one greater. In Fig. 2 all relaxation occurs parallel to the lines of constant $u$. The region where entropic involution is possible is again given. In each experiment the region is discretized into many individual points. For each point a value for $\alpha$ is attempted to be found. The method used is simply to begin with a guess of $\alpha=1$ and then add increments of $10^{-3}$ until a solution of Eq. 6 occurs, or the edge of the positivity domain is reached. This method would be inappropriate to use in a usual ELBM, due to the very large computational cost, but it is very robust and hence useful for this experiment with many higly non-equilibrium distributions. Another approach (with the same result) implies calculation of the entropic involution for all the boundary points where it exists. In this method we draw a straight line $l$ through a boundary point $\mathbf{f}$ and the equilibrium and find the intersection $l\cap\Sigma$ which consists of all points on $l$ with non-negative coordinates. One end of this interval is $\mathbf{f}$, another end is also a boundary point, $\mathbf{f}^{\prime}$. The entropic involution for $\mathbf{f}$ exits if and only if $S(\mathbf{f})^{\prime}\leq S(\mathbf{f})$. After we check this inequality, we can solve Eq. (6). The images of these involutions form the border that separates sets $A$ and $B$ (see Figs). ## V Conclusion The entropic involution is not always possible to perform. We have demonstrated that apart some special one-dimensional spaces of distributions with additional symmetry there exist domains where collisions with the preservation of entropy are not possible. We illustrated this statement by some simple and well known examples of ELBGK systems for which we directly calculated the areas where entropic collisions exist and where they do not exist. Such phenomena should be observable in all ELBM schemes with the classical entropies: there exists a vicinity of the equilibrium where the entropic involution is possible but for some areas of non-equilibrium distributions there exists no non-trivial root of equation (6). A collision which preserves entropy does not exist for this area. Therefore, for the regimes close to equilibrium (the vicinities $A$ of equilibria, Figs 1,2), ELBM schemes guaranty the precise balance of the entropy and for more nonequilibrium regimes, when at some sites the distribution belonges to sets $B$, ELBM schemes work as limiters Limiters . with additional dissipation. It is necessary for any complete definition of an ELBM algorithm to prescribe what to do when the involution is not possible. A reasonable choice would be to over-relax the maximum amount possible while maintaining positive population values. Such a technique is independently in use as a stabilizer for lattice Boltzmann schemes, sometimes called the ‘positivity limiter’ BGJ ; Limiters ; Li ; Servan ; Tosi . An effect of this operation is a local increase in viscosity/entropy production. Hence, if an ELBM were to apply such a scheme it would necessarily break the proper entropy balance. In this sense, ELBM belongs to a large family of add-ons that regularise LBM by the management of the addtional dissipation Add-ons . ## References * (1) S. Ansumali and I. V. Karlin, Phys. Rev. E, 62 (6), 7999–8003 (2000). * (2) R. Benzi, S. Succi, and M. Vergassola, Phys. Reports, 222, 145–197 (1992). * (3) B. M. Boghosian, J. Yepez, P. V. Coveney, A. Wagner, Proc. R. Soc. Lond. A, 457, 717–766 (2001). * (4) R. A. Brownlee, A. N. Gorban, and J. Levesley, Phys. Rev. E, 75, 036711 (2007). * (5) R. A. Brownlee, A. N. Gorban, and J. Levesley, Physica A, 387 (2-3), 385–406 (2008). * (6) R. A. Brownlee, J. Levesley, D. Packwood, A.N. Gorban, arXiv:1110.0270 [physics.comp-ph]. * (7) A. N. Gorban, in _Model Reduction and Coarse-Graining Approaches for Multiscale Phenomena_ , (Springer, Berlin-Heidelberg, New York 2006), 117–176; arXiv:cond-mat/0602024 [cond-mat.stat-mech]. * (8) A. N. Gorban, P. A. Gorban, G. Judge, arXiv:1003.1377 [physics.data-an]. * (9) I. V. Karlin, A. Ferrante, and H. C. Öttinger, Europhys. Lett. 47, 182–188 (1999). * (10) I. V. Karlin, A. N. Gorban, S. Succi, and V. Boffi, Phys. Rev. Lett., 81, 6–9 (1998). * (11) I. V. Karlin, S. Succi, arXiv:1107.3025 [cond-mat.stat-mech] * (12) Y. Li, R. Shock, R. Zhang, H. Chen. J. Fluid Mech., 519, 273–300 (2004) * (13) B. Servan-Camas, FT-C. Tsai. J Comput Physics 228 (1), 236–256 (2009). * (14) S. Succi, _The lattice Boltzmann equation for fluid dynamics and beyond_ (Oxford University Press, New York 2001). * (15) S. Succi, I. V. Karlin, and H. Chen, Rev. Mod. Phys. 74, 1203 (2002). * (16) F. Tosi, S. Ubertini, S. Succi, H. Chen, I.V. Karlin, Math Comput Simulation, 72, 227–231 (2006). * (17) C. Tsallis, J. Stat. Phys., 52, 479–487 (1988). * (18) W.-A. Yong and L.-S. Luo, Phys. Rev. E, 67, 051105 (2003).	arxiv-papers	2011-11-25T14:18:54	2024-09-04T02:49:24.652497	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "A. N. Gorban, D. Packwood", "submitter": "Alexander Gorban", "url": "https://arxiv.org/abs/1111.5994" }
1111.6213	# version1.4 Study of triangular flow $v_{3}$ in Au+Au and Cu+Cu collisions with a multiphase transport model Kai Xiao Institute of Particle Physics, Central China Normal University, Wuhan, Hubei, 430079, China The Key Laboratory of Quark and Lepton Physics (Central China Normal University), Ministry of Education, Wuhan, Hubei, 430079, China Na Li nli@mail.hust.edu.cn Department of Physics, Huazhong University of Science and Technology, Wuhan 430074, China Shusu Shi sss@iopp.ccnu.edu.cn Institute of Particle Physics, Central China Normal University, Wuhan, Hubei, 430079, China The Key Laboratory of Quark and Lepton Physics (Central China Normal University), Ministry of Education, Wuhan, Hubei, 430079, China Feng Liu Institute of Particle Physics, Central China Normal University, Wuhan, Hubei, 430079, China The Key Laboratory of Quark and Lepton Physics (Central China Normal University), Ministry of Education, Wuhan, Hubei, 430079, China ###### Abstract We studied the relation between the initial geometry anisotropy and the anisotropic flow in a multiphase transport model (AMPT) for both Au+Au and Cu+Cu collisions at $\sqrt{s_{{}_{NN}}}$=200 GeV. It is found that unlike the elliptic flow $v_{2}$, little centrality dependence of the triangular flow $v_{3}$ is observed. After removing the initial geometry effect, $v_{3}/\varepsilon_{3}$ increases with the transverse particle density, which is similar to $v_{2}/\varepsilon_{2}$. The transverse momentum ($p_{T}$) dependence of $v_{3}$ from identified particles is qualitatively similar to the $p_{T}$ dependence of $v_{2}$. ###### pacs: 25.75.Ld, 25.75.Dw ## I Introduction A novel state of matter called quark-gluon plasma (QGP) composed by deconfined partons is believed to be created experimentally in heavy ion collisions at RHIC QGP . The discovery of large elliptic flow indicates that the partonic collectivity is built up during the collisions, and the number-of-quark scaling suggests that the partonic degrees of freedom are active nqscaling . The anisotropic flow is usually described by a Fourier decomposition of the azimuthal distribution with respect to the reaction plane ArtPRC . The second harmonic coefficient, $v_{2}$, so called elliptic flow, has the biggest magnitude at high energy collisions largeflow . It is believed that the observed anisotropy in the momentum space is caused by the anisotropy in the coordinate space in the initial condition. Lots of attention has been put on the relation between $v_{2}$ and spatial eccentricity to see the hydrodynamics behavior of the created system v2ecc ; PRCrun4 ; cucu_STAR . Recent studies show that the event-by-event fluctuation of the initial geometry v3ecc may play an important role in the study of collective flow. The triangular shape in the initial geometry will be transferred to the momentum space as the system expands, and finally leads to the none zero value of the third harmonic coefficient, $v_{3}$. It is found that the triangular flow $v_{3}$ is responsible for the ridge and shoulder structures and the broad away-side of two-particle azimuthal correlation v3ridge . Besides, it is also considered to be a good probe to study the viscous hydrodynamics behavior of the colliding system v3hydro . Lots of properties of the triangular flow $v_{3}$ have been studied in hydrodynamic and transport models v3hydro ; v3ampt . However, since the reaction plane can not be directly measured in the experiment, those anisotropic parameters can not be directly obtained. It is found that different methods may cause up to $20\%$ discrepancy on $v_{2}$ Artv2Review , thus it should also be carefully evaluated for the $v_{3}$ study. Besides, $v_{3}$ is directly related with the initial fluctuation, it is interesting to see its system size dependence. In this paper, we will study the triangular flow $v_{3}$ in both Au+Au and Cu+Cu collisions in a multiphase transport model (AMPT) AMPT . The relation between $v_{3}$ and $\varepsilon_{3}$ is studied as a function of number of participants and transverse momentum. The paper is organized as follows: In Sec. II, the observables and technical methods are introduced. A brief description of AMPT model is given in Sec. III. The results and discussions are presented in Sec. IV. Finally, a summary is given in Sec. V. ## II OBSERVABLES In a non-central collisions, the overlap region of two nuclei is an almond shape. Since the position of nucleons may fluctuate event by event, as discussed in Ref epart2 ; definition ; partecc , those initial geometric irregularities of the colliding system can be described by $\varepsilon_{n}$: $\varepsilon_{n}=\frac{{\sqrt{\left\langle{r^{2}\cos(n\varphi)}\right\rangle^{2}+\left\langle{r^{2}\sin(n\varphi)}\right\rangle^{2}}}}{{\left\langle{r^{2}}\right\rangle}},$ (1) where $r$ and $\varphi$ are the polar coordinate position of participating nucleons and $\langle\cdots\rangle$ is the average over all the participants in an event. $n$ refers to the $n$-th harmonic, i.e., $\varepsilon_{2}$ describes the elliptic shape and $\varepsilon_{3}$ describes the triangular shape. As the system evolves, the anisotropy in the coordinate space is transferred to the anisotropy in the momentum space due to the pressure gradient. The particle distribution then can be written as $\frac{dN}{d\phi}\propto 1+2\sum_{n=1}v_{n}\cos[n(\phi-\Psi_{n})],$ (2) where $\phi$ is the azimuthal angle, and $\Psi_{n}$ is the $n$-th event plane angle reconstructed by the final state particles:__ $\Psi_{n}=\frac{1}{n}\left[\tan^{-1}\frac{\sum\limits_{i}\sin(n\phi_{i})}{\sum\limits_{i}\cos(n\phi_{i})}\right].$ (3) The observed anisotropic flow is defined as the $n$-th Flourier coefficient $v_{n}$: $v^{\rm obs}_{n}=\left\langle\cos[n(\phi-\Psi_{n})]\right\rangle.$ (4) Here $\langle\cdots\rangle$ is taking the average over all the particles in the sample. This is the so-called event plane method of calculating $v_{n}$. The reconstructed event plane fluctuates around the reaction plane. The observed signals need to be revised by the corresponding resolution ArtPRC : $v_{n}=\frac{v^{\rm obs}_{n}}{\mathscr{R}_{n}}.$ (5) Due to the finite multiplicity of final state particles, the resolution $\mathscr{R}_{n}=\langle\cos[n(\Psi_{n}-\Psi_{\rm nR})]\rangle$ (6) is usually smaller than 1. $\Psi_{\rm nR}$ represents the nth real event plane angle. ## III AMPT Model There are four main components in AMPT model: the initial conditions, parton interactions, hadronization and hadron interactions. The initial conditions are obtained from the HIJING model HIJING , which includes the spatial and momentum distributions of minijet partons from hard processes and strings from soft processes. The time evolution of partons is then treated according to the ZPC ZPC parton cascade model. After partons stop interacting, a combined coalescence and string fragmentation model are used for the hadronization of partons. The scattering among the resulting hadrons is described by a relativistic transport (ART) model ART which includes baryon-baryon, baryon- meson and meson-meson elastic and inelastic scattering. In our study, we analyzed the events from AMPT with the parton cross section equals to 3 mb and 10 mb. As all the conclusions are independent on the parton cross section, only the results from 3 mb are shown in this paper. There are about 8 million events in $\mathrm{Au}+\mathrm{Au}$ collisions and 19 million events in $\mathrm{Cu}+\mathrm{Cu}$ collisions at $\sqrt{s_{{}_{NN}}}$= 200 used. The string melting AMPT version is used since the previous study shows that the string melting AMPT version agrees with the experimental results better AMPT . The centrality is defined by the impact parameter. ## IV Results and Discussions Figure 1: The second and third harmonic event plane resolution calculated by the particles with pseudo-rapidty region of $\|\eta\|>2$ as a function of centrality in both Au+Au and Cu+Cu collisions at $\sqrt{s_{{}_{NN}}}$=200 GeV in AMPT model. Figure 2: $v_{n}$ as a function of centrality in both Au+Au and Cu+Cu collisions at $\sqrt{s_{{}_{NN}}}$=200 GeV in AMPT model. In order to be comparable with the experimental data, the event plane method is used to calculate $v_{n}$. The procedure is slightly different between ours and Ref. v3ampt , in which the event plane was reconstructed by initial partons. Charged particles with $p_{T}\leq 2$ GeV/$c$, $\|\eta\|>2$ are chosen to reconstruct the event plane according to the Eq. 3. The particles used for the $v_{n}$ measurements are within the $\|\eta\|<1$. The $\eta$ gap used here is to reduce the auto-correlation between the particles used to reconstruct the event plane and the particles of interest. In the following, the observed $v_{n}$ are all corrected by the corresponding resolution. Fig. 1 shows the resolution of $v_{2}$ and $v_{3}$ in both Au+Au and Cu+Cu collisions. The resolution of $v_{2}$ shows a peak in mid-central collisions which is consistent with the experimental result PRCrun2 . This is because the resolution of $v_{2}$ is affected by both of the $v_{2}$ signal and the multiplicity used to reconstruct the event plane. While the resolution of $v_{3}$ only depends on the multiplicity, and keeps decreasing as the multiplicity drops. In Fig. 2, $v_{2}$ and $v_{3}$ are shown as functions of centrality in both Au+Au and Cu+Cu collisions. We can see that $v_{2}$ shows strong centrality dependence since it is mainly coming from the elliptic anisotropy in the initial geometry. Unlike $v_{2}$, the dependence of $v_{3}$ on centrality and system size are much smaller. The trend of $v_{3}$ observed is the same as that in Ref v3ampt . However, the event plane angle $\Psi_{n}$ in Ref v3ampt is obtained from the initial parton distribution, which is not observed in the experiment, while in our study it is from the final state particle distribution. The results indicate that the triangular flow is less sensitive to the centrality and system size compared with the elliptic flow. It could be understood as a result of combined effects from initial geometrical fluctuation and collective dynamics which requires the size of bulk to interact among themselves. Figure 3: (Color online) $v_{2}$ as a function of $\varepsilon_{2}$ in Au+Au collisions at $\sqrt{s_{{}_{NN}}}$=200 GeV in AMPT model. The black points are the average $v_{2}$ in the selected $\varepsilon_{2}$ bin. The pad in the right down corner is the average of $v_{2}$ with smaller scale. Figure 4: (Color online) $v_{3}$ as a function of $\varepsilon_{3}$ in Au+Au collisions at $\sqrt{s_{{}_{NN}}}$=200 GeV in AMPT model. The black points are the average $v_{3}$ in the selected $\varepsilon_{3}$ bin. The pad in the right down corner is the average of $v_{3}$ with smaller scale. It is commonly assumed that the harmonic flow coefficients $v_{n}$ linearly depends on the $\varepsilon_{n}$. This assumption is supported by hydrodynamic simulations v3hydro as long as one probes deformed initial profiles with only a single non-vanishing harmonic eccentricity coefficient. In Fig. 3 and Fig. 4, we investigate the feasibility of this assumption for $v_{2}$ and $v_{3}$ respectively. The relations between $v_{n}$ and $\varepsilon_{n}$ are drawn event by event in the two-dimension plots. The black points are the average values of $v_{n}$ in an selected $\varepsilon_{n}$ bin, and the curves are the connection of points to guide our eyes. The pads in the right down corners are the average values of $v_{n}$ with smaller scale. In Fig. 3, the $v_{2}$ increases with $\varepsilon_{2}$, which is consistent with the ideal hydrodynamic calculation Ulrichv3hydro . While in Fig. 4, the triangular flow $v_{3}$ firstly increases with $\varepsilon_{3}$ up to 0.17, and then decreases. Based on our study, the higher $\varepsilon_{3}$ bin corresponds to the more peripheral collisions. It is known that $v_{3}$ is caused by initial geometrical fluctuation, and built up by the interactions of constituents. The less interactions in the higher $\varepsilon_{3}$ bin may cause the less converting efficiency from $\varepsilon_{3}$ to $v_{3}$. That could be the reason of decreasing trend of $v_{3}$ when $\varepsilon_{3}$ is larger than 0.17. Both the trend and the value of $v_{3}$ show discrepancy to the ideal hydrodynamic calculations Ulrichv3hydro . As discussed in Ref. v3hydro , the viscosity causes the decrease of $v_{3}$, however, the effects to $v_{3}$ versus $\varepsilon_{3}$ is not shown in the viscous hydrodynamic calculation. Figure 5: $v_{n}/\varepsilon_{n}$ as a function of transverse particle density in both Au+Au and Cu+Cu collisions at $\sqrt{s_{{}_{NN}}}$=200 GeV in AMPT model. Figure 6: $v_{3}$ as a function of transverse momentum in (a) Au+Au and (b) Cu+Cu collisions at $\sqrt{s_{{}_{NN}}}$=200 GeV in AMPT model. The ratio of elliptic flow to eccentricity $v_{2}/\varepsilon_{2}$ gains lots of interests by comparing with the hydrodynamic model definition ; Ollitrault_etaOverS . Recently, the behavior of triangular flow $v_{3}$ in ideal hydrodynamics is also discussed v3hydro . In Fig. 5, we study the $v_{n}/\varepsilon_{n}$ as a function of transverse particle density. From the plot we can see that $v_{n}/\varepsilon_{n}$ from Au+Au and Cu+Cu are consistent with each other very well. As the transverse particle density increases, $v_{3}/\varepsilon_{3}$ rises with smaller value than $v_{2}/\varepsilon_{2}$. It implies that as the particle density increases, the initial geometry asymmetry transfers to momentum asymmetry more efficiently while the system expands. Besides, the second order harmonic is more efficient than the third order. At last, the transverse momentum dependence of $v_{3}$ for $\pi$, $K$, p and $\Lambda$ is also studied in Au+Au and Cu+Cu collisions. In Fig. 6, we can see that $v_{3}$ shows quite similar trend to $v_{2}$. At low $p_{T}$, the mass ordering phenomena is observed. The lighter particles are found with larger $v_{3}$. It indicates that although $v_{3}$ is driven by $\varepsilon_{3}$, its transverse momentum dependence is dominated by the hydrodynamics behavior of the system. While when $p_{T}\geq 1.5$ GeV/$c$, baryons and mesons are separated into two groups. The $p_{T}$ dependence of $v_{3}$ from identified particles is qualitatively similar to the $p_{T}$ dependence of $v_{2}$ PRCrun4 ; cucu_STAR . The $v_{3}$ results of identified particles from AMPT model are similar to the STAR preliminary results Yadav . ## V Summary In summary, we studied the relation between initial geometry parameter $\varepsilon_{n}$ and anisotropic flow $v_{n}$ in Au+Au and Cu+Cu collisions using the AMPT Monte-Carlo model. We find that the triangular flow $v_{3}$ is less sensitive to the centrality and system size compared with the elliptic flow $v_{2}$. The $v_{2}$ displays an increasing trend as a function of $\varepsilon_{2}$, which is qualitatively consistent with hydrodynamic calculation. We found that $v_{3}$ shows an increasing trend when $\varepsilon_{3}$ is less than 0.17, and then decreases beyond $\varepsilon_{3}$ = 0.17. It may be because of the lower converting efficiency from $\varepsilon_{3}$ to $v_{3}$ in the higher $\varepsilon_{3}$ bin. This decreasing trend is in contrast to the results of ideal hydrodynamic calculation. Both $v_{2}/\varepsilon_{2}$ and $v_{3}/\varepsilon_{3}$ increase with the transverse particle density, and the second harmonic asymmetry in the initial geometry seems to transfer to the momentum asymmetry more efficiently than the third harmonic. The triangular flow $v_{3}$ of identified particles shows a mass ordering in low $p_{T}$ and meson-baryon splitting at intermediate $p_{T}$ in both Au+Au and Cu+Cu collisions which is similar to the $p_{T}$ dependence of $v_{2}$. ## VI Acknowledgments We wish to thank Prof. Fuqiang Wang for useful suggestions, and Dr. Kejun Wu for useful discussions on the AMPT model. This work was supported in part by the National Natural Science Foundation of China under grant No. 10775060, 11105060, 11135011, 11147146 and ‘the Fundamental Research Funds for the Central Universities’, Grant No. HUST: 2011QN195. ## References * (1) I. Arsene et al. (BRAHMS Collaboration), Nucl. Phys. A 757, 1 (2005); B. B. Back et al. (PHOBOS Collaboration), Nucl. Phys. A 757, 28 (2005); J. Adams et al. (STAR Collaboration), Nucl. Phys. A 757, 102 (2005); S. S. Adcox et al. (PHENIX Collaboration), Nucl. Phys. A 757, 184 (2005). * (2) J. Adams et al (STAR Collaboration), Phys. Rev. Lett. 95 122301 (2005). * (3) A. M. Poskanzer and S. A. Voloshin, Phys. Rev. C 58, 1671 (1998). * (4) K. H. Achermann et al. (STAR Collaboration), Phys. Rev. Lett. 86, 402 (2001). * (5) B. I. Abelev et al (STAR Collaboration), Phys. Rev. C 77, 54901 (2008). * (6) B. I. Abelev et al (STAR Collaboration), Phys. Rev. C 81, 044902 (2010). * (7) S. A. Voloshin and A. M. Poskanzer, Phys. Lett. B 474, 27 (2000). P. F. Kolb, J. Sollfrank and U. W. Heinz, Phys. Rev. C 62, 054909 (2000). * (8) B. Alver et al., Phys. Rev. Lett. 98, 242302 (2007); B. Alver et al. PHOBOS Collaboration, Phys. Rev. C 81, 034915 (2010). * (9) B. Alver and G. Roland, Phys. Rev. C 81, 054905 (2010); Jun Xu, Che Ming Ko, Phys. Rev. C 83, 021903 (2011);G.-L. Ma, X.-N. Wang, Phys. Rev. Lett. 106, 162301 (2011). * (10) B. H. Alver, C. Gombeaud, M. Luzum, and J. Y. Ollitrault, Phys. Rev. C 82, 034913 (2010). * (11) L.-X. Han et al., arXiv:1105.5415 * (12) Z.-W. Lin et al., Phys. Rev. C59, 2716 (1999). * (13) B. Alver et al. (PHOBOS Collaboration), Phys. Rev. C 77, 014906 (2008). * (14) J. Y. Ollitrault, Phys. Rev. D46 229 (1992). * (15) B. Alver et al. (PHOBOS Collaboration), Phys. Rev. Lett. 98, 242302 (2007). * (16) S. A. Voloshin, A. M. Poskanzer, R. Snellings arXiv:0809.2949 (2008) and Voloshin, Sergei A., Poskanzer, Arthur M., Snellings, Raimond: Collective Phenomena in Non-Central Nuclear Collisions. Stock, R. (ed.). SpringerMaterials - The Landolt-Bornstein Database (http://www.springermaterials.com). Springer-Verlag Berlin Heidelberg, 2010. $DOI:10.1007/978-3-642-01539-7_{1}0$ * (17) X.-N. Wang, Phys. Rev. D 43 104 (1991). * (18) B. Zhang, Comput. Phys. Commun. 109, 193 (1998). * (19) B.A. Li and C.M. Ko, Phys. Rev. C 52 2037 (1995). * (20) J. Adams et al (STAR Collaboration), Phys. Rev. C 72, 14904 (2005). * (21) Z. Qiu and U. Heinz, Phys. Rev. C 84, 024911 (2005). * (22) H. J. Drescher, A. Dumitru, C. Gombeaud and J. Y. Ollitrault, Phys. Rev. C76 024905 (2007). * (23) Y. Pandit for the STAR collaboration, poster, RHIC and AGS Annual Users’ Meeting, Upton, NY, Jun. 20-24, 2011.	arxiv-papers	2011-11-27T01:31:48	2024-09-04T02:49:24.662529	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Kai Xiao, Na Li, Shusu Shi, Feng Liu", "submitter": "Shusu Shi", "url": "https://arxiv.org/abs/1111.6213" }
1111.6515	# On the interplay of direct and indirect $C\\!P$ violation in the charm sector M Gersabeck1, M Alexander2, S Borghi2,3, VV Gligorov1 and C Parkes3,1 1 European Organization for Nuclear Research (CERN), Geneva, Switzerland 2 School of Physics and Astronomy, University of Glasgow, Glasgow, United Kingdom 3 School of Physics and Astronomy, University of Manchester, Manchester, United Kingdom marco.gersabeck@cern.ch ###### Abstract Charm mixing and $C\\!P$ violation observables are examined in the light of the recently reported evidence from LHCb for $C\\!P$ violation in the charm sector. If the result is confirmed as being due to direct $C\\!P$ violation at the $1\%$ level, its effect will need to be taken into account in the interpretation of $C\\!P$ violation observables. The contributions of direct and indirect $C\\!P$ violation to the decay rate asymmetry difference $\Delta A_{C\\!P}$ and the ratios of effective lifetimes $A_{\Gamma}$ and $y_{CP}$ are considered here. Terms relevant to the interpretation of future high precision measurements which have been neglected in previous literature are identified. ###### pacs: 13.25.Ft ††: J. Phys. G: Nucl. Part. Phys. Charm, after strange and beauty, is the last system of neutral flavoured mesons where $C\\!P$ violation remains to be discovered. While neutral $\mathrm{B}$ mesons are characterised by their mass splitting leading to fast oscillations and neutral kaons by their width splitting which results in a short-lived and a long-lived state, neutral charm mesons have a very small splitting in both mass and width. For charm, compared to the beauty sector, this leads to rather subtle mixing-related effects in time-dependent as well as in time-integrated charm measurements, which are examined in detail here. First evidence for $C\\!P$ violation in the charm sector has recently been reported by the LHCb collaboration in the study of the difference of the time- integrated asymmetries of $D^{0}\to K^{+}K^{-}$ and $D^{0}\to\pi^{+}\pi^{-}$ decay rates through the parameter $\Delta A_{C\\!P}$ [1]. This measurement is primarily sensitive to the difference in direct $C\\!P$ violation between the two final states as discussed further below. Direct $C\\!P$ violation depends on the final state and is the asymmetry of the rates of particle and antiparticle decays. It can be caused by a difference in the magnitude of the decay rates or by a difference in their phase. Indirect $C\\!P$ violation is considered universal, i.e. final-state independent, and is an asymmetry in the mixing rate or in its weak phase. Indirect $C\\!P$ violation can be measured in time-dependent analyses. To date, two types of measurements were used to search for indirect $C\\!P$ violation in the charm sector. One uses the asymmetry of the lifetimes, $A_{\Gamma}$, measured in $D^{0}$ and $\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ decays to the $C\\!P$ eigenstates $K^{+}K^{-}$ or $\pi^{+}\pi^{-}$ [2, 3, 4]. The other is a time-dependent Dalitz plot analysis of $D^{0}$ and $\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ decays to $K^{0}_{\rm\scriptstyle S}\pi^{+}\pi^{-}$ or $K^{0}_{\rm\scriptstyle S}K^{+}K^{-}$ [5, 6]. Another observable studied related to $A_{\Gamma}$ is $y_{CP}$, which is given by the deviation from one of the ratio of the lifetimes measured in decays to a Cabibbo-allowed, $C\\!P$ averaged, and a Cabibbo-suppressed, $C\\!P$ eigenstate, final state. Any deviation of a measurement of $y_{CP}$ from that of the mixing parameter $y$ would signal $C\\!P$ violation. In the interpretation of $A_{\Gamma}$ and $y_{CP}$, direct $C\\!P$ violation is commonly neglected [7]. In the light of the new evidence this assumption is no longer justified. The relevance of direct $C\\!P$ violation to a measurement of $A_{\Gamma}$ has previously been pointed out in [8]. However, a closer look at both $A_{\Gamma}$ and $y_{CP}$ is necessary to examine the contribution of direct and indirect $C\\!P$ violation in these observables as well as their connection to $\Delta A_{C\\!P}$. The mass eigenstates of neutral $D$ mesons, $\|D_{1,2}\rangle$, with masses $m_{1,2}$ and widths $\Gamma_{1,2}$ can be written as linear combinations of the flavour eigenstates $\|D_{1,2}\rangle=p\|D^{0}\rangle\pm{}q\|\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}\rangle$, with complex coefficients $p$ and $q$ which satisfy $\|p\|^{2}+\|q\|^{2}=1$. The average mass and width are defined as $m\equiv(m_{1}+m_{2})/2$ and $\Gamma\equiv(\Gamma_{1}+\Gamma_{2})/2$. The $D$ mixing parameters are defined using the mass and width difference as $x\equiv(m_{2}-m_{1})/\Gamma$ and $y\equiv(\Gamma_{2}-\Gamma_{1})/2\Gamma$. The phase convention of $p$ and $q$ is chosen such that $C\\!P\|D^{0}\rangle=-\|\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}\rangle$. According to [8], the time dependent decay rates of $D^{0}$ and $\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ decays to the final state $f$, which is a $C\\!P$ eigenstate with eigenvalue $\eta_{C\\!P}$, can be expressed as $\displaystyle\Gamma(D^{0}(t)\to f)=\frac{1}{2}\rme^{-\tau}\left\|A_{f}\right\|^{2}\Big{\\{}$ $\displaystyle\left(1+\|\lambda_{f}\|^{2}\right)\cosh(y\tau)+\left(1-\|\lambda_{f}\|^{2}\right)\cos(x\tau)$ $\displaystyle+2\Re(\lambda_{f})\sinh(y\tau)-2\Im(\lambda_{f})\sin(x\tau)\Big{\\}},$ $\displaystyle\Gamma(\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}(t)\to f)=\frac{1}{2}\rme^{-\tau}\left\|\bar{A}_{f}\right\|^{2}\Big{\\{}$ $\displaystyle\left(1+\|\lambda^{-1}_{f}\|^{2}\right)\cosh(y\tau)+\left(1-\|\lambda^{-1}_{f}\|^{2}\right)\cos(x\tau)$ (1) $\displaystyle+2\Re(\lambda^{-1}_{f})\sinh(y\tau)-2\Im(\lambda^{-1}_{f})\sin(x\tau)\Big{\\}},$ where $\tau\equiv\Gamma t$, $\kern-1.99997pt\stackrel{{\scriptstyle\kern 1.39998pt\textsf{(---)}}}{{A}}_{\kern-2.10002ptf}\kern-3.00003pt$ are the decay amplitudes and $\lambda_{f}$ is given by $\lambda_{f}=\frac{q\bar{A}_{f}}{pA_{f}}=-\eta_{C\\!P}\left\|\frac{q}{p}\right\|\left\|\frac{\bar{A}_{f}}{A_{f}}\right\|\rme^{i\phi},$ (2) where $\eta_{C\\!P}$ is the $C\\!P$ eigenvalue of the final state $f$ and $\phi$ is the $C\\!P$ violating relative phase between $q/p$ and $\bar{A}_{f}/A_{f}$. Introducing $\|q/p\|^{\pm 2}\approx 1\pm A_{m}$ and $\|\bar{A}_{f}/A_{f}\|^{\pm 2}\approx 1\pm A_{d}$, one can write $\|\lambda_{f}^{\pm 1}\|^{2}\approx(1\pm A_{m})(1\pm A_{d}),$ (3) where $A_{m}$ represents a $C\\!P$ violation contribution from mixing and $A_{d}$ from direct $C\\!P$ violation and where both $A_{m}$ and $A_{d}$ are assumed to be small. Expanding (On the interplay of direct and indirect $C\\!P$ violation in the charm sector) up to second order in $\tau$ one can write the effective lifetimes, i.e. those measured as a single exponential, as $\displaystyle\hat{\Gamma}(\kern-1.00006pt\stackrel{{\scriptstyle\kern 0.70004pt\textsf{(---)}}}{{D}}\kern-3.00003pt(t)\to f)\approx$ $\displaystyle\Gamma\Bigg{\\{}1+\left[1\pm\frac{1}{2}(A_{m}+A_{d})-\frac{1}{8}(A_{m}^{2}-2A_{m}A_{d})\right]\eta_{C\\!P}(y\cos\phi\mp x\sin\phi)$ (4) $\displaystyle\quad\mp A_{m}(x^{2}+y^{2})\pm 2A_{m}y^{2}\cos^{2}\phi\mp 4xy\cos\phi\sin\phi\Bigg{\\}},$ where terms below order $10^{-5}$ have been ignored. The experimental constraints [9] give $x$, $y$, and $A_{d}$ for the final states $K^{+}K^{-}$ and $\pi^{+}\pi^{-}$ of order $10^{-2}$, and $A_{m}$ and $\sin\phi$ of order $10^{-1}$. The sum of measurements of $D^{0}$ and $\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ decays leads to the definition of the observable $y_{CP}$ which is given by $y_{CP}=\frac{\hat{\Gamma}+\hat{\bar{\Gamma}}}{2\Gamma}-1\approx\eta_{C\\!P}\Bigg{\\{}\left[1-\frac{1}{8}(A_{m}^{2}-2A_{m}A_{d})\right]y\cos\phi-\frac{1}{2}(A_{m}+A_{d})x\sin\phi\Bigg{\\}}.$ (5) The difference of measurements of $D^{0}$ and $\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ decays leads to the parameter $A_{\Gamma}$ which is defined as $\displaystyle A_{\Gamma}=(\hat{\Gamma}-\hat{\bar{\Gamma}})(\hat{\Gamma}+\hat{\bar{\Gamma}})^{-1}\approx$ $\displaystyle\bigg{[}\frac{1}{2}(A_{m}+A_{d})y\cos\phi-\left(1-\frac{1}{8}A_{m}^{2}\right)x\sin\phi- A_{m}(x^{2}+y^{2})$ (6) $\displaystyle+2A_{m}y^{2}\cos^{2}\phi-4xy\cos\phi\sin\phi\bigg{]}\frac{\eta_{C\\!P}}{1+y_{CP}}.$ The weak phase $\phi$ has not been assumed to be universal. When averaging measurements from different channels, a potential decay-dependent weak phase of the amplitude ratio has to be taken into account [8]. Expanding only up to order $10^{-4}$ leads to $y_{CP}\approx\eta_{C\\!P}\left[\left(1-\frac{1}{8}A_{m}^{2}\right)y\cos\phi-\frac{1}{2}(A_{m})x\sin\phi\right],$ (7) and $A_{\Gamma}\approx\bigg{[}\frac{1}{2}(A_{m}+A_{d})y\cos\phi-x\sin\phi\bigg{]}\frac{\eta_{C\\!P}}{1+y_{CP}}\approx\eta_{C\\!P}\left[\frac{1}{2}(A_{m}+A_{d})y\cos\phi-x\sin\phi\right].$ (8) The difference of $y_{CP}$ evaluated in (7) from the expression used in literature [7] so far is the term $\eta_{C\\!P}\frac{1}{8}A_{m}^{2}y\cos\phi$, which can be of similar order as $\frac{1}{2}A_{m}x\sin\phi$ and should therefore not be ignored. Equation (8) shows that there can be a significant contribution to $A_{\Gamma}$ from direct $C\\!P$ violation. Assuming $y=1\%$ and $\cos\phi=1$, direct $C\\!P$ violation at the level of $A_{d}/2=1\%$ would lead to a contribution to $A_{\Gamma}$ of $10^{-4}$. Current measurements yield a sensitivity of a few $10^{-3}$ [2, 3, 4]. Future measurements at LHCb and future $\mathrm{B}$ factory experiments are expected to reach uncertainties of the level of $10^{-4}$, i.e. that of the direct $C\\!P$ violation contribution. More precise measurements may well change the approximations made in (7) and (8), in particular a measurement of $A_{m}\lesssim A_{d}$. In time-integrated measurements the rate asymmetry is measured which is defined as $A_{C\\!P}\equiv\frac{\Gamma(D^{0}\to f)-\Gamma(\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}\to f)}{\Gamma(D^{0}\to f)+\Gamma(\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}\to f)}.$ (9) Introducing $a_{C\\!P}^{dir}\equiv\frac{\|A_{f}\|^{2}-\|\bar{A}_{f}\|^{2}}{\|A_{f}\|^{2}+\|\bar{A}_{f}\|^{2}}=\frac{1-\left\|\frac{\bar{A}_{f}}{A_{f}}\right\|^{2}}{1+\left\|\frac{\bar{A}_{f}}{A_{f}}\right\|^{2}}=\frac{-A_{d}}{2+A_{d}}\approx-\frac{1}{2}A_{d},$ (10) and using (On the interplay of direct and indirect $C\\!P$ violation in the charm sector), (9) becomes $A_{C\\!P}\approx a_{C\\!P}^{dir}-A_{\Gamma}(1-(a_{C\\!P}^{dir})^{2})\frac{\langle t\rangle}{\tau}\approx a_{C\\!P}^{dir}-A_{\Gamma}\frac{\langle t\rangle}{\tau},$ (11) where $\langle t\rangle$ denotes the average decay time of the observed candidates. Terms in $\langle t\rangle^{2}$ are below order $10^{-4}$, given current experimental constraints, and have been ignored. A common way to reduce experimental systematic uncertainties is to measure the difference in time-integrated asymmetries in related final states. For the two-body final states $K^{+}K^{-}$ and $\pi^{+}\pi^{-}$, this difference is given by $\displaystyle\Delta A_{C\\!P}$ $\displaystyle\equiv A_{C\\!P}(K^{+}K^{-})-A_{C\\!P}(\pi^{+}\pi^{-})$ (12) $\displaystyle=a_{C\\!P}^{dir}(K^{+}K^{-})-a_{C\\!P}^{dir}(\pi^{+}\pi^{-})$ $\displaystyle\quad-A_{\Gamma}(K^{+}K^{-})\frac{\langle t(K^{+}K^{-})\rangle}{\tau}+A_{\Gamma}(\pi^{+}\pi^{-})\frac{\langle t(\pi^{+}\pi^{-})\rangle}{\tau}.$ Assuming the $C\\!P$ violating phase $\phi$ to be universal [10] this can be rewritten as $\displaystyle\Delta A_{C\\!P}\approx\Delta a_{C\\!P}^{dir}\left(1+y\cos\phi\frac{\overline{\langle t\rangle}}{\tau}\right)+\left(a_{C\\!P}^{ind}+\overline{a_{C\\!P}^{dir}}y\cos\phi\right)\frac{\Delta\langle t\rangle}{\tau}$ (13) where $\Delta X\equiv X(K^{+}K^{-})-X(\pi^{+}\pi^{-})$, $\Delta X\equiv X(K^{+}K^{-})-X(\pi^{+}\pi^{-})$, and $a_{C\\!P}^{ind}=-(A_{m}/2)y\cos\phi+x\sin\phi$. The ratio $\overline{\langle t\rangle}/\tau$ is equal to one for the lifetime-unbiased $\mathrm{B}$ factory measurements [11, 12] and is $2.083\pm 0.001$ for LHCb [1] and $2.53\pm 0.02$ for CDF [13], thus leading to a correction of $\Delta a_{C\\!P}^{dir}$ of the order of $10^{-2}$. The factor $\Delta\langle t\rangle/\tau$ multiplying the indirect $C\\!P$ violation is zero for the $\mathrm{B}$ factory measurements and ranges from $0.098\pm 0.003$ to $0.26\pm 0.01$ for LHCb and CDF, respectively. Therefore, $\Delta A_{C\\!P}$ is largely a measure of direct $C\\!P$ violation while an obvious contribution from indirect $C\\!P$ violation exists. The contribution from direct $C\\!P$ violation to $A_{\Gamma}$ pointed out in (8) leads to a term proportional to $y$. This term may be of similar size as the term proportional to $\Delta\langle t\rangle$ and should therefore be taken into account. In summary, the mixing and $C\\!P$ violation parameters $y_{CP}$, $A_{\Gamma}$ and $\Delta A_{C\\!P}$ have been discussed in the light of the recent evidence for $C\\!P$ violation in the $D^{0}$ sector. The parameter $y_{CP}$ is least affected by direct $C\\!P$ violation, however, it contains a term which has been neglected in the literature so far and which can be of the same order as the constribution proportional to $x$. A measurement of $A_{\Gamma}$ can exhibit a contribution of direct $C\\!P$ violation at the level of $10^{-4}$, comparable to the expected future experimental sensitivity. The direct $C\\!P$ violation term in the $\Delta A_{C\\!P}$ measurement contains a contribution proportional to $y$. The interpretation of future high precision measurements of these observables will need to take account of these contributions. The authors would like to thank Tim Gershon, Bostjan Golob, Alex Kagan and Vincenzo Vagnoni for helpful discussions and comments. Furthermore, the authors thank the LHCb collaboration whose work inspired this paper. MG and VVG are supported by a Marie Curie Action: “Cofunding of the CERN Fellowship Programme (COFUND-CERN)” of the European Community’s Seventh Framework Programme under contract number (PCOFUND-GA-2008-229600). MA, SB and CP acknowledge the support of the STFC (United Kingdom). ## References ## References * [1] Aaij R et al. [LHCb collaboration]. Evidence for $C\\!P$ violation in time-integrated $D^{0}\to h^{-}h^{+}$ decay rates. 2011\. Preprint hep-ex 1112.0938. * [2] Staric M et al. [Belle collaboration]. Evidence for $D^{0}$\- $\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ Mixing. Phys. Rev. Lett., 98:211803, 2007. * [3] Aubert B et al. [BaBar collaboration]. Measurement of $D^{0}$\- $\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ Mixing using the Ratio of Lifetimes for the Decays $D^{0}\to K^{-}\pi^{+}$ and $K^{+}K^{-}$. Phys. Rev., D80:071103, 2009. * [4] Aaij R et al. [LHCb collaboration]. Measurement of mixing and $C\\!P$ violation parameters in two-body charm decays. 2011\. Preprint hep-ex 1112.4698. * [5] Abe K et al. [Belle collaboration]. Measurement of $D^{0}$-$\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ Mixing Parameters in $D^{0}\to K^{0}_{\rm\scriptstyle S}\pi^{+}\pi^{-}$ decays. Phys. Rev. Lett., 99:131803, 2007. * [6] Del Amo Sanchez P et al. [BaBar collaboration]. Measurement of $D^{0}$-$\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ mixing parameters using $D^{0}\to K^{0}_{\rm\scriptstyle S}\pi^{+}\pi^{-}$ and $D^{0}\to K^{0}_{\rm\scriptstyle S}K^{+}K^{-}$ decays. Phys. Rev. Lett., 105:081803, 2010. * [7] Bergmann S, Grossman Y, Ligeti Z, Nir Y, and Petrov A A. Lessons from CLEO and FOCUS measurements of $D^{0}$-$\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ mixing parameters. Phys. Lett., B486:418–425, 2000. * [8] Kagan A L and Sokoloff M D. On Indirect $C\\!P$ Violation and Implications for $D^{0}$\- $\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ and $\mathrm{B}_{(s)}$ \- $\kern 1.99997pt\overline{\kern-1.99997pt\mathrm{B}}{}_{(s)}$ mixing. Phys.Rev., D80:076008, 2009. * [9] Asner D et al. [Heavy Flavor Averaging Group]. Averages of b-hadron, c-hadron, and $\tau$-lepton Properties. 2010\. online update at http://www.slac.stanford.edu/xorg/hfag/index.html. * [10] Grossman Y, Kagan A L, and Nir Y. New physics and $C\\!P$ violation in singly Cabibbo suppressed D decays. Phys.Rev., D75:036008, 2007. * [11] Aubert B et al. [BaBar collaboration]. Search for $C\\!P$ violation in the decays $D^{0}\to K^{-}K^{+}$ and $D^{0}\to\pi^{-}\pi^{+}$. Phys. Rev. Lett., 100:061803, 2008. * [12] Staric M et al. [Belle collaboration]. Measurement of $C\\!P$ asymmetry in Cabibbo suppressed $D^{0}$ decays. Phys. Lett., B670:190–195, 2008. * [13] Aaltonen T et al. [CDF collaboration]. Measurement of $C\\!P$–violating asymmetries in $D^{0}\to\pi^{+}\pi^{-}$ and $D^{0}\to K^{+}K^{-}$ decays at CDF. Preprint hep-ex 1111.5023.	arxiv-papers	2011-11-28T17:03:29	2024-09-04T02:49:24.677351	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Marco Gersabeck (1), Michael Alexander (2), Silvia Borghi (2 and 3),\n Vladimir V Gligorov (1), Chris Parkes (3 and 1) ((1) CERN, Geneva,\n Switzerland, (2) School of Physics and Astronomy, University of Glasgow,\n Glasgow, UK, (3) School of Physics and Astronomy, University of Manchester,\n Manchester, UK)", "submitter": "Marco Gersabeck", "url": "https://arxiv.org/abs/1111.6515" }
1111.6550	# Sensitivity to eV-scale Neutrinos of Experiments at a Very Low Energy Neutrino Factory J.H. Cobb C.D. Tunnell Corresponding author: tunnell@fnal.gov Subdepartment of Particle Physics, University of Oxford, Denys Wilkinson Building, Keble Road, Oxford, OX1 3RH, UK A.D. Bross Fermilab, P.O. Box 500, Batavia, IL 60510-0500, USA ###### Abstract The results of LSND have yet to be confirmed at the $5\sigma$-level. An experiment is proposed utilizing a 3 GeV muon storage ring that would allow for both disappearance and appearance channels to be explored at short- baselines. The appearance channel could provide well over $5\sigma$ confirmation or rejection of the LSND result. Other physics could also be performed at such a facility such as the measurement of electron-neutrino cross sections. The sensitivity of experiments at a Very Low Energy Neutrino Factory (VLENF) to neutrinos at the eV-scale is presented. Suggested keywords ###### pacs: 14.60.Pq, 14.60.St ††preprint: APS/123-QED Despite the experimental success of the past decade in establishing that neutrinos are massive and mix, whether they be produced in the Sun Aharmim _et al._ (2008), reactors Ahn _et al._ (2006), accelerators Adamson _et al._ (2008), or the atmosphere Wendell _et al._ (2010), there exist nevertheless some anomalies. All oscillation experiments to date can be explained with three neutrinos except LSND — and various experiments could be made to agree with either. The LEP collider experiments lep (2006) showed that additional light neutrinos cannot couple to the $Z$. The MiniBooNe experiment has been unable to deny or confirm the LSND result since their data that uses the same anti-neutrino beam polarity as LSND agrees with the background-only hypothesis at 0.5% and agrees with short-baseline oscillations at 8.7% Aguilar-Arevalo _et al._ (2010). An experiment with a sensitivity greater than $5\sigma$ is needed in order to refute or confirm evidence of neutrinos at the eV-scale. The LSND data suggests a new mass splitting Athanassopoulos _et al._ (1998) that is potentially observed elsewhere. Recently, a recalculation of reactor fluxes resulted in the _reactor neutrino anomaly_ Mueller _et al._ (2011); Huber (2011) which, along with the _Gallium anomaly_ Acero _et al._ (2008), are possibly caused by sterile neutrinos. Global fits reconcile these anomalies into a framework that introduces a sterile neutrino at the eV-scale, _e.g._ , Giunti and Laveder (2011a); Kopp _et al._ (2011). Additional indications come from cosmology where WMAP favors more than three neutrinos Komatsu _et al._ (2011). The Very Low Energy Neutrino Factory (VLENF) would utilize a muon storage ring to study eV-scale oscillation physics and measure cross sections (including $\nu_{e}$). Pions are collected from a target and injected into a storage ring where they decay to muons. The storage ring is optimized for 2 GeV muons where the energy is optimized for the needs of both oscillation and cross section physics. The muons decay according to $\mu^{+}\to e^{+}\bar{\nu}_{\mu}\nu_{e}$. Straight sections in the storage ring result in neutrinos directed at a near- and far-detector. The storage ring could be a fixed field alternating gradient (FFAG) lattice for large momentum acceptance, which is important given the large momentum spectrum after the target. The pricing and engineering of FFAGs are well- understood because of experience building them c:f (2005). The design would require only normal conducting magnets which simplifies construction, commissioning, and operations. Design work for the injection into the storage ring and the particle collection downstream of the target are underway. The near detector will be placed at 20-50 meters from the end of the straight and will measure neutrino-nucleon cross sections of interest to future long- baseline experiments including the first precision measurement of $\nu_{e}$ cross sections. The far detector at 800 m would measure disappearance and wrong-sign muon appearance channels. The detector would need to be magnetized for the wrong-sign muon appearance channel. Numerous possibilities exist for detector technologies that include liquid argon, MINOS inspired, and totally active scintillating detectors. For the purposes of this study, a detector inspired by MINOS but with thinner plates is assumed. The experiment will take advantage of the “golden channel” of oscillation appearance $\nu_{e}\to\nu_{\mu}$ where the resulting final state is a wrong-sign muon. The probability $\nu_{e}\to\nu_{\mu}$ depends on the mixing matrix, $U$. Let $R_{ij}$ be a rotation between the $i$th and $j$th mass eigenstates without CP violation. For $N$ neutrinos, $R_{ij}$ has dimension $N\times N$. By convention, the three neutrino mixing matrix is $U_{\text{PMNS}}=R_{23}R_{13}R_{12}$. In the (3+1) model of neutrino oscillations, extra rotations can be introduced such that the mixing matrix is $U_{\text{(3+1)}}=R_{34}R_{24}R_{14}U_{\text{PMNS}}$. Given that $\Delta m^{2}_{41}>>\Delta m^{2}_{31}$, $U_{\text{PMNS}}$ can be approximated by the identity matrix (_ie._ the “short-baseline approximation”). It then follows that $\|U_{e4}\|^{2}=\sin(\theta_{14})$ and $\|U_{\mu 4}\|^{2}=\sin(\theta_{24})\cos(\theta_{14})$. The oscillation probabilities for appearance and disappearance, respectively, are: $\displaystyle\text{P}_{\nu_{e}\to\nu_{\mu}}=$ $\displaystyle 4\|U_{e4}\|^{2}\|U_{\mu 4}\|^{2}\sin^{2}\left(\frac{\Delta m^{2}_{41}L}{4E}\right),$ (1) $\displaystyle\text{P}_{\nu_{\alpha}\to\nu_{\alpha}}=$ $\displaystyle 1-\left[4\|U_{\alpha 4}\|^{2}(1-\|U_{\alpha 4}\|^{2})\right]\sin^{2}\left(\frac{\Delta m^{2}_{41}L}{4E}\right).$ (2) Disappearance measurements will constrain $\|U_{\mu 4}\|^{2}$ and $\|U_{e4}\|^{2}$ while the appearance channel measures their product $\|U_{e4}\|^{2}\|U_{\mu 4}\|^{2}$. These parameters are over-constrained. Only the appearance channel will be assumed since the physics sensitivity of disappearance measurements is well-understood for short-baseline experiments. Without oscillations there would be $\sim 10^{5}$ charge current (CC) events in the far detector. The disappearance oscillation probability, if LSND is correct, would be on the order of a few percent for this experiment; the statistical errors and systematic errors are expected to be about half a percent from the presence of a near detector and beam instrumentation. The oscillation sensitivities have been computed using the GLoBES software (version 3.1.10) Huber _et al._ (2005, 2007). Since GLoBES, by default, only allows for a $3\times 3$ mixing matrix, the SNU (version 1.1) add-on Kopp (2008); Kopp _et al._ (2008) is used to extend computations in GLoBES to $4\times 4$ mixing matrices. Unlike previous analyses, the far detector approximation of the source and detector being treated as point-sources cannot be used since the size of the detector and accelerator straight are comparable to the baseline. This study computes the neutrino flux by integrating the phase space of the stored muons using Monte Carlo (MC) integration. The beam occupies a 6D phase space ($x$, $y$, $z$, $p_{x}$, $p_{y}$, $p_{z}$) and the detector has a $6\text{ m}\times 6\text{ m}$ cross section. A random point is chosen within the beam phase space and within the detector volume. The transverse phase space is represented by the Twiss parameters $\alpha=0$ and $\beta=25\text{ m}$ where the $1\sigma$ Gaussian emittance is assumed to be $15\text{ mm}$. It follows that the spread in, for example, $x$ is $\sigma_{x}=\sqrt{\beta\epsilon}$ and the angular divergence in $x$ is $\sigma_{x^{\prime}}=\sqrt{\epsilon/\beta}$. The longitudinal phase space ($z$ and $p_{z}$) is described by assuming a uniform distribution in $z$ and $p_{z}=(2\pm 20\%)\text{ GeV}$. The code for the analysis herein is available online Tunnell (2011) under the GPL license gpl . This analysis assumes $2\times 10^{17}$ decays of $\mu^{+}$ in each straight, normalized to an exposure similar to MiniBooNe of $10^{21}$ protons on target (POT). The far detector at 800 m consists of a kilotonne of fiducial target mass. The background rejection for the wrong-sign muons is assumed to be $10^{-4}$ for the $\bar{\nu}_{\mu}\to\bar{\nu}_{\mu}$ neutral current (NC) events and $10^{-5}$ for the charge misidentification of $\bar{\nu}_{\mu}\to\bar{\nu}_{\mu}$ CC events. The backgrounds from $\nu_{e}\to\nu_{e}$ CC and NC are negligible. The $\nu_{e}\to\nu_{\mu}$ CC events have a 90% detection efficiency. These efficiencies were extrapolated from previous studies Laing (2010) and reconfirmation of these calculations for energies below 2 GeV is still required. Figure 1: The oscillation probability for the “golden channel” $\nu_{e}\to\nu_{\mu}$ using the (3+1) oscillation parameters in TABLE 1. A baseline of 800 meters is assumed. This numerical treatment of the binned oscillation probability agrees with Eq. 1. To initially gauge the sensitivity of this experimental setup, some oscillation parameters must be assumed that include the LSND effect. The best- fit data for (3+1) sterile neutrinos is used Giunti and Laveder (2011b) where the MB $\bar{\nu}$ and LSND $\bar{\nu}$ data was fit (TABLE 1). Lower energy neutrinos contain the most information about the mixing matrix (FIG. 1). Table 1: Best-fit oscillation parameters for the (3+1) sterile neutrino scenario for MB $\bar{\nu}$ and LSND $\bar{\nu}$ data Giunti and Laveder (2011b). Parameter \| Value ---\|--- $\Delta m^{2}_{41}$ [$\text{eV}^{2}$] \| 0.89 $\|U_{e4}\|^{2}$ \| 0.025 $\|U_{\mu 4}\|^{2}$ \| 0.023 Figure 2: The spectrum of signal and background events at the far-detector. The number of wrong-sign muon appearance signal events is 27 which arises through the “golden channel” oscillation probability $\nu_{e}\to\nu_{\mu}$. Roughly 2 background events are in the signal box and correspond to both $\bar{\nu}_{\mu}$ CC with charge misidentification and $\bar{\nu}_{\mu}$ NC events where a short-lived pion has faked a wrong-sign muon. An exposure of $10^{21}$ POT is assumed. The best-fit values allow for an estimation of the event rates. The true number of events without efficiencies is shown in TABLE 2. Applying the assumed efficiencies of $10^{-5}$ and $10^{-4}$ for $\bar{\nu}_{\mu}\to\bar{\nu}_{\mu}$ CC and $\bar{\nu}_{X}\to\bar{\nu}_{X}$ NC, respectively, reveals the event spectrum (FIG. 2). After cuts, there are 27 signal events and 2 background events. Table 2: A table of the raw event rates. The first row corresponds to the “golden channel” appearance signal. The other rows are potential backgrounds to the signal. The background that drives the analysis is $\bar{\nu}_{\mu}\to\bar{\nu}_{\mu}$. Channel \| Interaction \| Pre-cuts ---\|---\|--- $\nu_{e}\to\nu_{\mu}$ \| CC \| 30 $\bar{\nu}_{X}\to\bar{\nu}_{X}$ \| NC \| 16850 $\bar{\nu}_{\mu}\to\bar{\nu}_{\mu}$ \| CC \| 42545 $\nu_{e}\to\nu_{e}$ \| CC \| 78974 Figure 3: Optimization of the mean stored muon energy and far detector baseline. Above muon energies of 1 GeV, appearance physics is not sensitive to the muon energy. For a fixed neutrino energy, the boosted muon frame will result in more events the more it is boosted. The optimal baseline is around 800 meters. In practice the muon energy is fixed by the needs of the cross section physics. However, given a fixed fiducial mass, the baseline and muon energy can be optimized for short-baseline physics. The metric for comparison is the $\chi^{2}$ between simulations for the (3+1) best-fit mentioned earlier and the background-only hypothesis. The short-baseline oscillation physics reach is comparable at energies above 2 GeV and the optimal baseline is around 800 meters (FIG. 3). Having chosen a baseline and energy, it is possible to investigate the physics reach of this facility if we appropriately define the desired $\chi^{2}$ value. The design requirement is to measure the LSND effect at “$5\sigma$” but that can be ambiguously defined. Herein “$n\sigma$” is defined according to the Gaussian probability of the respective deviation. Let $\text{CDF}_{X}$ be the cumulative distribution function of some distribution $X$. If $G$ is a Gaussian with $\mu=0$ and $\sigma=1$, then the $p$-value of a $5\sigma$ effect is $1-\text{CDF}_{G}(5)\simeq 3\times 10^{-7}$. Similarly, if $\chi^{2}_{2}$ is the chi-squared distribution with two degrees of freedom, then the required value of the $\chi^{2}$ to be an $n\sigma$ effect is $\text{CDF}^{-1}_{\chi^{2}_{2}}(\text{CDF}_{G}(n))$. Figure 4: Sensitivity to sterile parameters. The orange band corresponds to the LSND $\bar{\nu}$ and MiniBooNe $\bar{\nu}$ sterile neutrino fit performed by Giunti and Laveder Giunti and Laveder (2011a) where they assumed only one extra massive neutrino. The contours are the sensitivities of an experiment at a 3 GeV FFAG storage ring using parameters defined in the text. The entire 99% confidence interval for the LSND $\bar{\nu}$ and MiniBooNe $\bar{\nu}$ results would be confirmed or excluded at $7\sigma$. The $\chi^{2}$ is computed using the pull-method, which allows for systematics to be included. The signal and background normalization errors are assigned to be 2% and 20%, respectively, and these systematic uncertainties are marginalized over. Spectral information is also used. The sensitivity to new physics is shown in FIG. 4. The 99% confidence interval of a fit in a (3+1) scheme to the MiniBooNe $\bar{\nu}$ and LSND $\bar{\nu}$ data is shown and excluded at $7\sigma$. This meets and exceeds the requested criterion for future eV-scale experiments. Some effects are ignored that could reduce the significance of the result. For example, not included are backgrounds associated with cosmic muons. This could be problematic since muons fill the storage ring which results in a large duty factor. One solution, if it is an issue, would be to put RF cavities in the storage ring to bunch the beam. This merits further study. It has been demonstrated that a VLENF has sensitivity to eV-scale oscillations. Further optimizations of the target horn, collection, injection, and the dynamic aperture of the storage ring will increase the neutrino flux. But previously unthought-of backgrounds may arise and detector charge identification performance is difficult at 1 GeV and below. A design study must be performed to ensure enough contingency in the sensitivity to guarantee a firm confirmation or refutation of the LSND effect. Further study is required. _Thanks_ for guidance and useful discussions with Joachim Kopp, David Neuffer, Kenneth Long, and Alain Blondel. Also thanks to Nick Ryder for a careful reading. ## References * Aharmim _et al._ (2008) B. Aharmim _et al._ (SNO), Phys. Rev. Lett. 101, 111301 (2008), arXiv:0806.0989 [nucl-ex] . * Ahn _et al._ (2006) M. H. Ahn _et al._ (K2K), Phys. Rev. D74, 072003 (2006), arXiv:hep-ex/0606032 . * Adamson _et al._ (2008) P. Adamson _et al._ (MINOS), Phys. Rev. Lett. 101, 131802 (2008), arXiv:0806.2237 [hep-ex] . * Wendell _et al._ (2010) R. Wendell _et al._ (Kamiokande), Phys. Rev. D81, 092004 (2010), arXiv:1002.3471 [hep-ex] . * lep (2006) Phys. Rept. 427, 257 (2006), arXiv:hep-ex/0509008 . * Aguilar-Arevalo _et al._ (2010) A. A. Aguilar-Arevalo _et al._ (MiniBooNE Collaboration), Phys. Rev. Lett. 105, 181801 (2010). * Athanassopoulos _et al._ (1998) C. Athanassopoulos _et al._ (LSND Collaboration), Phys. Rev. Lett. 81, 1774 (1998). * Mueller _et al._ (2011) T. A. Mueller _et al._ , Phys. Rev. C83, 054615 (2011), arXiv:1101.2663 [hep-ex] . * Huber (2011) P. Huber, Phys. Rev. C84, 024617 (2011), arXiv:1106.0687 [hep-ph] . * Acero _et al._ (2008) M. A. Acero, C. Giunti, and M. Laveder, Phys. Rev. D78, 073009 (2008), arXiv:0711.4222 [hep-ph] . * Giunti and Laveder (2011a) C. Giunti and M. Laveder, (2011a), arXiv:1107.1452 [hep-ph] . * Kopp _et al._ (2011) J. Kopp, M. Maltoni, and T. Schwetz, Phys. Rev. Lett. 107, 091801 (2011), arXiv:1103.4570 [hep-ph] . * Komatsu _et al._ (2011) E. Komatsu _et al._ (WMAP), Astrophys. J. Suppl. 192, 18 (2011), arXiv:1001.4538 [astro-ph.CO] . * c:f (2005) http://hadron.kek.jp/FFAG/FFAG05_HP/index.htm (2005), see talks including the summary talk. * Huber _et al._ (2005) P. Huber, M. Lindner, and W. Winter, Comput.Phys.Commun. 167, 195 (2005), arXiv:hep-ph/0407333 [hep-ph] . * Huber _et al._ (2007) P. Huber, J. Kopp, M. Lindner, M. Rolinec, and W. Winter, Comput.Phys.Commun. 177, 432 (2007), arXiv:hep-ph/0701187 [hep-ph] . * Kopp (2008) J. Kopp, Int. J. Mod. Phys. C19, 523 (2008), erratum ibid. C19 (2008) 845, arXiv:physics/0610206 . * Kopp _et al._ (2008) J. Kopp, M. Lindner, T. Ota, and J. Sato, Phys. Rev. D77, 013007 (2008), arXiv:0708.0152 [hep-ph] . * Tunnell (2011) C. D. Tunnell, https://code.launchpad.net/~c-tunnell1/+junk/vlenf_tools (2011), questions should be directed to the corresponding author. * (20) http://www.gnu.org/licenses/gpl-3.0.html. * Laing (2010) A. B. Laing, “Optimisation of detectors for the golden channel at a neutrino factory,” (2010). * Giunti and Laveder (2011b) C. Giunti and M. Laveder, (2011b), arXiv:1109.4033 [hep-ph] .	arxiv-papers	2011-11-28T18:57:56	2024-09-04T02:49:24.684118	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Christopher D. Tunnell and John H. Cobb and Alan D. Bross", "submitter": "Christopher Tunnell", "url": "https://arxiv.org/abs/1111.6550" }
1111.6671	# The dynamics of the 3D radial NLS with the combined terms Changxing Miao Changxing Miao: Institute of Applied Physics and Computational Mathematics, P. O. Box 8009, Beijing, China, 100088, miao_changxing@iapcm.ac.cn , Guixiang Xu Guixiang Xu Institute of Applied Physics and Computational Mathematics, P. O. Box 8009, Beijing, China, 100088, xu_guixiang@iapcm.ac.cn and Lifeng Zhao Lifeng Zhao University of Science and Technology of China, Hefei, China, zhaolifengustc@yahoo.cn ###### Abstract. In this paper, we show the scattering and blow-up result of the radial solution with the energy below the threshold for the nonlinear Schrödinger equation (NLS) with the combined terms $\displaystyle iu_{t}+\Delta u=-\|u\|^{4}u+\|u\|^{2}u$ (CNLS) in the energy space $H^{1}({\mathbb{R}}^{3})$. The threshold is given by the ground state $W$ for the energy-critical NLS: $iu_{t}+\Delta u=-\|u\|^{4}u$. This problem was proposed by Tao, Visan and Zhang in [37]. The main difficulty is the lack of the scaling invariance. Illuminated by [17], we need give the new radial profile decomposition with the scaling parameter, then apply it into the scattering theory. Our result shows that the defocusing, $\dot{H}^{1}$-subcritical perturbation $\|u\|^{2}u$ does not affect the determination of the threshold of the scattering solution of (CNLS) in the energy space. ###### Key words and phrases: Blow up; Dynamics; Nonlinear Schrödinger Equation; Scattering; Threshold Energy. ###### 2000 Mathematics Subject Classification: Primary: 35L70, Secondary: 35Q55 ## 1\. Introduction We consider the dynamics of the radial solutions for the nonlinear Schrödinger equation (NLS) with the combined nonlinearities in $H^{1}({\mathbb{R}}^{3})$ $\left\\{\begin{aligned} iu_{t}+\Delta u=&\;f_{1}(u)+f_{2}(u),,\quad(t,x)\in{\mathbb{R}}\times{\mathbb{R}}^{3},\\\ u(0)=&\;u_{0}(x)\in H^{1}({\mathbb{R}}^{3}).\end{aligned}\right.$ (1.1) where $u:{\mathbb{R}}\times{\mathbb{R}}^{3}\mapsto{\mathbb{C}}$ and $f_{1}(u)=-\|u\|^{4}u$, $f_{2}(u)=\|u\|^{2}u$. As we known, $f_{1}$ has the $\dot{H}^{1}$-critical growth, $f_{2}$ has the $\dot{H}^{1}$-subcritical growth. The equation has the following mass and Hamiltonian quantities $\displaystyle M(u)(t)=$ $\displaystyle\frac{1}{2}\int_{{\mathbb{R}}^{3}}\|u(t,x)\|^{2}\;dx;\quad E(u)(t)=\int_{{\mathbb{R}}^{3}}\frac{1}{2}\|\nabla u(t,x)\|^{2}\;dx+F_{1}(u(t))+F_{2}(u(t))$ where $F_{1}(u(t))=\displaystyle-\frac{1}{6}\int_{{\mathbb{R}}^{3}}\|u(t,x)\|^{6}\;dx,\;\;F_{2}(u(t))=\frac{1}{4}\int_{{\mathbb{R}}^{3}}\|u(t,x)\|^{4}\;dx.$ They are conserved for the sufficient smooth solutions of (1.1). In [37], Tao, Visan and Zhang made the comprehensive study of $\displaystyle iu_{t}+\Delta u=\|u\|^{4}u+\|u\|^{2}u$ in the energy space. They made use of the interaction Morawetz estimate established in [6] and the stability theory for the scattering solution. Their result is based on the scattering result of the defocusing, energy-critical NLS in the energy space, which is established by Bourgain [3, 4] for the radial case, I-team [7], Ryckman-Visan [34] and Visan [38] for the general data. Since the classical interaction Morawetz estimate in [6] fails for (1.1), Tao, et al., leave the scattering and blow-up dichotomy of (1.1) below the threshold as an open problem in [37]. For other results, please refer to [15, 16, 30, 31, 32, 39, 40]. For the focusing, energy-critical NLS $\displaystyle iu_{t}+\Delta u=-\|u\|^{4}u.$ (1.2) Kenig and Merle first applied the concentration compactness in [2, 21, 22] into the scattering theory of the radial solution of (1.2) in [19] with the energy below that of the ground state of $\displaystyle-\Delta W=\|W\|^{4}W.$ (1.3) In this paper, we will also make use of the concentration compactness argument and the stability theory to study the dichotomy of the radial solution of (1.1) with the energy below the threshold, which will be shown to be the energy of the ground state $W$ for (1.2). For the applications of the concentration compactness in the scattering theory and rigidity theory of the critical NLS, NLW, NLKG and Hartree equations, please see [8, 9, 10, 11, 12, 13, 17, 20, 23, 24, 25, 26, 27, 28, 29]. We now show the differences between (1.1) and (1.2). On one hand, there is an explicit solution $W$ for (1.2), which is the ground state of (1.3) and does not scatter. The threshold of the scattering solution of (1.2) is determined by the energy of $W$. While for (1.1), there is no such explicit solution, whose energy is the threshold of the scattering solution of (1.1). We need look for a mechanism to determine the threshold of the scattering solution of (1.1). It turns out that the constrained minimization of the energy as (1.5) is appropriate111The similar constrained minimization of the energy as (1.5) is not appropriate for the focusing perturbation: $iu_{t}+\Delta u=-\|u\|^{4}u-\|u\|^{2}u$, since the threshold $m$ in this way equals to $0$ and it is not the desired result.. On the other hand, for (1.2), it is $\dot{H}^{1}$-scaling invariant, which gives us many conveniences, especially in the nonlinear profile decomposition about (1.2). While for (1.1), it is the lack of scaling invariance. We need give the new profile decomposition with the scaling parameter of (1.1) in $H^{1}(R^{3})$, take care of the role of the scaling parameter in the linear and nonlinear profile decompositions, then apply them into the scattering theory. Now for $\varphi\in H^{1}$, we denote the scaling quantity $\varphi^{\lambda}_{3,-2}$ by $\displaystyle\varphi^{\lambda}_{3,-2}(x)=e^{3\lambda}\varphi(e^{2\lambda}x).$ We denote the scaling derivative of $E$ by $K(\varphi)$ $\displaystyle K(\varphi)=\mathcal{L}E(\varphi):=\dfrac{d}{d\lambda}\Big{\|}_{\lambda=0}E(\varphi^{\lambda}_{3,-2})=\int_{{\mathbb{R}}^{3}}\left(\frac{4}{2}\|\nabla\varphi\|^{2}-\frac{12}{6}\|\varphi\|^{6}+\frac{6}{4}\|\varphi\|^{4}\right)\;dx,$ (1.4) which is connected with the Virial identity, and then plays the important role in the blow-up and scattering of the solution of (1.1). Now the threshold $m$ is determined by the following constrained minimization222In fact, the following minimization of the static energy $\inf\\{M(\varphi)+E(\varphi)\;\|\;\varphi\in H^{1}({\mathbb{R}}^{3}),\;\varphi\not=0,\;K(\varphi)=0\\}$ also equals to $m$. of the energy $E(\varphi)$ $\displaystyle m=\inf\\{E(\varphi)\;\|\;\varphi\in H^{1}({\mathbb{R}}^{3}),\;\varphi\not=0,\;K(\varphi)=0\\}.$ (1.5) Since we consider the $\dot{H}^{1}$-critical growth with the $\dot{H}^{1}$-subcritical perturbation, we will use the modified energy later $\displaystyle E^{c}(u)=$ $\displaystyle\int_{{\mathbb{R}}^{3}}\left(\frac{1}{2}\|\nabla u(t,x)\|^{2}-\frac{1}{6}\|u(t,x)\|^{6}\right)\;dx.$ As the nonlinearity $\|u\|^{2}u$ is the defocusing, $\dot{H}^{1}$-subcritical perturbation, one think that the focusing, $\dot{H}^{1}$-critical term plays the decisive role of the threshold of the scattering solution of (1.1) in the energy space. The first result is to characterize the threshold energy $m$ as following ###### Proposition 1.1. There is no minimizer for (1.5). But for the threshold energy $m$, we have $\displaystyle m=E^{c}(W),$ where $W\in\dot{H}^{1}({\mathbb{R}}^{3})$ is the ground state of the massless equation $\displaystyle-\Delta W=\|W\|^{4}W.$ As the dynamics of the solution of (1.1) with the energy less than the threshold $m$, the conjecture is ###### Conjecture 1.2. Let $u_{0}\in H^{1}({\mathbb{R}}^{3})$ with $E(u_{0})<m,$ (1.6) and $u$ be the solution of (1.1) and $I$ be its maximal interval of existence. Then 1. (a) If $K(u_{0})\geq 0$, then $I={\mathbb{R}}$, and $u$ scatters in both time directions as $t\rightarrow\pm\infty$ in $H^{1}$; 2. (b) If $K(u_{0})<0$, then $u$ blows up both forward and backward at finite time in $H^{1}$. In this paper, we verify the conjecture in the radial case. ###### Theorem 1.3. Conjecture 1.2 holds whenever $u$ is spherically symmetric. ###### Remark 1.4. Our consideration of the radial case is based on the following facts: 1. (1) It is an open problem that the scattering result of (1.2) in dimension three, except for the radial case in [19]. Our result is based on the corresponding scattering result of (1.2). 2. (2) It seems to be hard to lower the regularity of the critical element to $L^{\infty}\dot{H}^{s}$ for some $s<0$ by the double Duhamel argument in dimension three to obtain the compactness of the critical element in $L^{2}$, which is used to control the spatial center function $x(t)$ of the critical element. ###### Remark 1.5. We can remove the radial assumption under the stronger constraint that $\displaystyle M(u_{0})+E(u_{0})<m,$ which can help us to obtain the compactness of the critical element in $L^{2}$ and control the spatial center function $x(t)$ of the critical element. Of course, we need the precondition333By the relation between the sharp Sobolev constant and the ground state $W$, we know that the constrained condition $\displaystyle\int_{{\mathbb{R}}^{3}}\left(\big{\|}\nabla u_{0}\big{\|}^{2}-\big{\|}u_{0}\big{\|}^{6}\right)\;dx\geq 0,\quad\int_{{\mathbb{R}}^{3}}\left(\frac{1}{2}\big{\|}\nabla u_{0}\big{\|}^{2}-\frac{1}{6}\big{\|}u_{0}\big{\|}^{6}\right)\;dx<E^{c}(W)$ is equivalent to the constrained condition $\displaystyle\big{\\|}\nabla u_{0}\big{\\|}^{2}_{L^{2}}\leq\big{\\|}\nabla W\big{\\|}^{2}_{L^{2}},\quad\int_{{\mathbb{R}}^{3}}\left(\frac{1}{2}\big{\|}\nabla u_{0}\big{\|}^{2}-\frac{1}{6}\big{\|}u_{0}\big{\|}^{6}\right)\;dx<E^{c}(W).$ We use the former in this paper while the latter is given by Kenig-Merle in [19]. that the global wellposedness and scattering result of (1.2) holds for $u_{0}\in\dot{H}^{1}({\mathbb{R}}^{3})$ with $\displaystyle\int_{{\mathbb{R}}^{3}}\left(\big{\|}\nabla u_{0}\big{\|}^{2}-\big{\|}u_{0}\big{\|}^{6}\right)\;dx\geq 0,\quad\int_{{\mathbb{R}}^{3}}\left(\frac{1}{2}\big{\|}\nabla u_{0}\big{\|}^{2}-\frac{1}{6}\big{\|}u_{0}\big{\|}^{6}\right)\;dx<$ $\displaystyle m.$ ###### Remark 1.6. From the assumption in Theorem 1.3, we know that the solution starts from the following subsets of the energy space, $\displaystyle\mathcal{K}^{+}=$ $\displaystyle\Big{\\{}\varphi\in H^{1}({\mathbb{R}}^{3})\;\;\Big{\|}\;\;\varphi\;\text{is radial},\;E(\varphi)<m,\;K(\varphi)\geq 0\Big{\\}},$ $\displaystyle\mathcal{K}^{-}=$ $\displaystyle\Big{\\{}\varphi\in H^{1}({\mathbb{R}}^{3})\;\;\Big{\|}\;\;\varphi\;\text{is radial},\;E(\varphi)<m,\;K(\varphi)<0\Big{\\}}.$ By the scaling argument, we know that $\mathcal{K}^{\pm}\not=\emptyset$ (we can also know that $\mathcal{K}^{+}\not=\emptyset$ by the small data theory). In fact, let $\chi(x)$ be a radial smooth cut-off function satisfying $0\leq\chi\leq 1$, $\chi(x)=1$ for $\|x\|\leq 1$ and $\chi(x)=0$ for $\|x\|\geq 2$. If we take $\chi_{R}(x)=\chi(x/R)$ and $\displaystyle\varphi(x)=\theta\lambda^{-1/2}\chi_{R}(x/\lambda)W(x/\lambda),$ where $\theta,\lambda,R$ is determined later and the cutoff function $\chi_{R}$ is not needed for dimension $d\geq 5$ since $W\in H^{1}$. Then we have $\displaystyle\big{\\|}\nabla\varphi\big{\\|}^{2}_{L^{2}}=$ $\displaystyle\theta^{2}\left(\big{\\|}\nabla W\big{\\|}^{2}_{L^{2}}+\int\left((\chi_{R}^{2}-1)\big{\|}\nabla W\big{\|}^{2}+\|\nabla\chi_{R}\|^{2}\|W\|^{2}+2\chi_{R}\nabla\chi_{R}\cdot W\nabla W\right)\;dx\right),$ $\displaystyle\big{\\|}\varphi\big{\\|}^{6}_{L^{6}}=$ $\displaystyle\theta^{6}\left(\big{\\|}W\big{\\|}^{6}_{L^{6}}+\int(\chi_{R}^{6}-1)\|W\|^{6}\;dx\right),\quad\big{\\|}\varphi\big{\\|}^{4}_{L^{4}}=\lambda\cdot\theta^{4}\big{\\|}\chi_{R}W\big{\\|}^{4}_{L^{4}},$ $\displaystyle\big{\\|}\varphi\big{\\|}^{2}_{L^{2}}=$ $\displaystyle\lambda^{2}\cdot\theta^{2}\big{\\|}\chi_{R}W\big{\\|}^{2}_{L^{2}}.$ Therefore, taking $R$ sufficiently large, $\theta=1+\epsilon$ and $\lambda=\epsilon^{3}$ , we have $\displaystyle E(\varphi)=$ $\displaystyle\frac{\theta^{2}}{2}\big{\\|}\nabla W\big{\\|}^{2}_{L^{2}}-\frac{\theta^{6}}{6}\big{\\|}W\big{\\|}^{6}_{L^{6}}$ $\displaystyle+\frac{\theta^{2}}{2}\int\left((\chi_{R}^{2}-1)\big{\|}\nabla W\big{\|}^{2}+\|\nabla\chi_{R}\|^{2}\|W\|^{2}+2\chi_{R}\nabla\chi_{R}\cdot W\nabla W\right)\;dx$ $\displaystyle-\frac{\theta^{6}}{6}\int(\chi_{R}^{6}-1)\|W\|^{6}\;dx+\lambda\cdot\frac{\theta^{4}}{4}\big{\\|}\chi_{R}W\big{\\|}^{4}_{L^{4}}$ $\displaystyle=$ $\displaystyle m-6\epsilon^{2}m+o(\epsilon^{2}),$ $\displaystyle K(\varphi)=$ $\displaystyle 2\theta^{2}\big{\\|}\nabla W\big{\\|}^{2}_{L^{2}}-2\theta^{6}\big{\\|}W\big{\\|}^{6}_{L^{6}}$ $\displaystyle+2\theta^{2}\int\left((\chi_{R}^{2}-1)\big{\|}\nabla W\big{\|}^{2}+\|\nabla\chi_{R}\|^{2}\|W\|^{2}+2\chi_{R}\nabla\chi_{R}\cdot W\nabla W\right)\;dx$ $\displaystyle-2\theta^{6}\int(\chi_{R}^{6}-1)\|W\|^{6}\;dx+\lambda\cdot\frac{3\theta^{4}}{2}\big{\\|}\chi_{R}W\big{\\|}^{4}_{L^{4}}$ $\displaystyle=$ $\displaystyle-24\epsilon m+o(\epsilon^{2}).$ If taking $\epsilon<0$ and $\|\epsilon\|$ sufficient small, then we have $\varphi\in\mathcal{K}^{+}$; If taking $\epsilon>0$ and sufficient small, then we have $\varphi\in\mathcal{K}^{-}$. ### Acknowledgements. The authors are partly supported by the NSF of China (No. 10801015, No. 10901148, No. 11171033). The authors would like to thank Professor K. Nakanishi for his valuable communications. ∎ ## 2\. Preliminaries In this section, we give some notation and some wellknown results. ### 2.1. Littlewood-Paley decomposition and Besov space Let $\Lambda_{0}(x)\in\mathcal{S}({\mathbb{R}}^{3})$ such that its Fourier transform $\widetilde{\Lambda}_{0}(\xi)=1$ for $\|\xi\|\leq 1$ and $\widetilde{\Lambda}_{0}(\xi)=0$ for $\|\xi\|\geq 2$. Then we define $\Lambda_{k}(x)$ for any $k\in{\mathbb{Z}}\backslash\\{0\\}$ and $\Lambda_{(0)}(x)$ by the Fourier transforms: $\displaystyle\widetilde{\Lambda}_{k}(\xi)=\widetilde{\Lambda}_{0}(2^{-k}\xi)-\widetilde{\Lambda}_{0}(2^{-k+1}\xi),\quad\widetilde{\Lambda}_{(0)}(\xi)=\widetilde{\Lambda}_{0}(\xi)-\widetilde{\Lambda}_{0}(2\xi).$ Let $s\in{\mathbb{R}}$, $1\leq p,q\leq\infty$. The inhomogeneous Besov space $B^{s}_{p,q}$ is defined by $\displaystyle B^{s}_{p,q}=\left\\{f\;\big{\|}\;f\in\mathcal{S}^{\prime},\Big{\\|}2^{ks}\big{\\|}\Lambda_{k}f\big{\\|}_{L^{p}_{x}}\Big{\\|}_{l^{q}_{k\geq 0}}<\infty\right\\},$ where $\mathcal{S}^{\prime}$ denotes the space of tempered distributions. The homogeneous Besov space $\dot{B}^{s}_{p,q}$ can be defined by $\displaystyle\dot{B}^{s}_{p,q}=\left\\{f\;\Big{\|}\;f\in\mathcal{S}^{\prime},\left(\sum_{k\in{\mathbb{Z}}\backslash\\{0\\}}2^{qks}\big{\\|}\Lambda_{k}f\big{\\|}^{q}_{L^{p}_{x}}+\big{\\|}\Lambda_{(0)}f\big{\\|}_{L^{p}_{x}}\Big{\\|}^{q}\right)^{1/q}<\infty\right\\}.$ ### 2.2. Linear estimates We say that a pair of exponents $(q,r)$ is Schröidnger $\dot{H}^{s}$-admissible in dimension three if $\displaystyle\dfrac{2}{q}+\dfrac{3}{r}=\dfrac{3}{2}-s$ and $2\leq q,r\leq\infty$. If $I\times{\mathbb{R}}^{3}$ is a space-time slab, we define the $\dot{S}^{0}(I\times{\mathbb{R}}^{3})$ Strichartz norm by $\displaystyle\big{\\|}u\big{\\|}_{\dot{S}^{0}(I\times{\mathbb{R}}^{3})}:=\sup\big{\\|}u\big{\\|}_{L^{q}_{t}L^{r}_{x}(I\times{\mathbb{R}}^{3})}$ where the sup is taken over all $L^{2}$-admissible pairs $(q,r)$. We define the $\dot{S}^{s}(I\times{\mathbb{R}}^{3})$ Strichartz norm to be $\displaystyle\big{\\|}u\big{\\|}_{\dot{S}^{s}(I\times{\mathbb{R}}^{3})}:=\big{\\|}D^{s}u\big{\\|}_{\dot{S}^{0}(I\times{\mathbb{R}}^{3})}.$ We also use $\dot{N}^{0}(I\times{\mathbb{R}}^{3})$ to denote the dual space of $\dot{S}^{0}(I\times{\mathbb{R}}^{3})$ and $\displaystyle\dot{N}^{k}(I\times{\mathbb{R}}^{3}):=\\{u;D^{k}u\in\dot{N}^{0}(I\times{\mathbb{R}}^{3})\\}.$ By definition and Sobolev’s inequality, we have ###### Lemma 2.1. For any $\dot{S}^{1}$ function $u$ on $I\times{\mathbb{R}}^{3}$, we have $\displaystyle\big{\\|}\nabla u\big{\\|}_{L^{\infty}_{t}L^{2}_{x}}+\big{\\|}u\big{\\|}_{L^{10}_{t}\dot{B}^{1/3}_{90/19,2}(I\times{\mathbb{R}}^{3})}+\big{\\|}u\big{\\|}_{L^{\infty}_{t}L^{6}_{x}}+\big{\\|}u\big{\\|}_{L^{12}_{t}L^{9}_{x}}+\big{\\|}u\big{\\|}_{L^{10}_{t,x}}\lesssim\big{\\|}u\big{\\|}_{\dot{S}^{1}}.$ For any $\dot{S}^{1/2}$ function $u$ on $I\times{\mathbb{R}}^{3}$, we have $\displaystyle\big{\\|}u\big{\\|}_{L^{\infty}_{t}\dot{H}^{1/2}_{x}}+\big{\\|}u\big{\\|}_{L^{6}_{t}\dot{B}^{1/2}_{18/7,2}(I\times{\mathbb{R}}^{3})}+\big{\\|}u\big{\\|}_{L^{\infty}_{t}L^{3}_{x}}+\big{\\|}u\big{\\|}_{L^{6}_{t}L^{9/2}_{x}}+\big{\\|}u\big{\\|}_{L^{5}_{t,x}}\lesssim\big{\\|}u\big{\\|}_{\dot{S}^{1/2}}.$ Now we state the standard Strichartz estimate. ###### Lemma 2.2 ([5, 18, 36]). Let $I$ be a compact time interval, $k\in[0,1]$, and let $u:I\times{\mathbb{R}}^{3}\rightarrow{\mathbb{C}}$ be an $\dot{S}^{k}$ solution to the forced Schrödinger equation $\displaystyle iu_{t}+\Delta u=F$ for a function $F$. Then we have $\displaystyle\big{\\|}u\big{\\|}_{\dot{S}^{k}(I\times{\mathbb{R}}^{3})}\lesssim\big{\\|}u(t_{0})\big{\\|}_{\dot{H}^{k}({\mathbb{R}}^{d})}+\big{\\|}F\big{\\|}_{\dot{N}^{k}(I\times{\mathbb{R}}^{3})},$ for any time $t_{0}\in I$. We shall also need the following exotic Strichartz estimate, which is important in the application of the stability theory. ###### Lemma 2.3 ([14]). For any $F\in L^{2}_{t}\left(I;\dot{B}^{1/3}_{18/11,2}\right)$, we have $\displaystyle\left\\|\int^{t}_{0}e^{i(t-s)\Delta}F(s)\;ds\right\\|_{L^{10}_{t}\dot{B}^{1/3}_{90/19,2}}\lesssim\big{\\|}F\big{\\|}_{L^{2}_{t}\dot{B}^{1/3}_{18/11,2}}.$ ### 2.3. Local wellposedness and Virial identity Let $ST(I):=L^{10}_{t}\dot{B}^{1/3}_{90/19,2}\cap L^{12}_{t}L^{9}_{x}\cap L^{6}_{t}\dot{B}^{1/2}_{18/7,2}\cap L^{5}_{t,x}(I\times{\mathbb{R}}^{3}).$ By the definition of admissible pair, we know that $L^{10}_{t}\dot{B}^{1/3}_{90/19,2}\cap L^{12}_{t}L^{9}_{x}$ is the $\dot{H}^{1}$-admissible space, $L^{6}_{t}\dot{B}^{1/2}_{18/7,2}\cap L^{5}_{t,x}$ is the $\dot{H}^{1/2}$-admissible space. Now we have ###### Theorem 2.4 ([37]). Let $u_{0}\in H^{1}$, then for every $\eta>0$, there exists $T=T(\eta)$ such that if $\displaystyle\big{\\|}e^{it\Delta}u_{0}\big{\\|}_{ST([-T,T])}\leq\eta,$ then (1.1) admits a unique strong $H^{1}_{x}$-solution $u$ defined on $[-T,T]$. Let $(-T_{min},T_{max})$ be the maximal time interval on which $u$ is well-defined. Then, $u\in S^{1}(I\times{\mathbb{R}}^{d})$ for every compact time interval $I\subset(-T_{min},T_{max})$ and the following properties hold: 1. (1) If $T_{max}<\infty$, then $\displaystyle\big{\\|}u\big{\\|}_{ST((0,T_{max})\times{\mathbb{R}}^{d})}=\infty.$ Similarly, if $T_{min}<\infty$, then $\displaystyle\big{\\|}u\big{\\|}_{ST((-T_{min},0)\times{\mathbb{R}}^{d})}=\infty.$ 2. (2) The solution $u$ depends continuously on the initial data $u_{0}$ in the following sense: The functions $T_{min}$ and $T_{max}$ are lower semicontinuous from $\dot{H}^{1}_{x}\cap\dot{H}^{1/2}_{x}$ to $(0,+\infty]$. Moreover, if $u^{(m)}_{0}\rightarrow u_{0}$ in $\dot{H}^{1}_{x}\cap\dot{H}^{1/2}_{x}$ and $u^{(m)}$ is the maximal solution to (1.1) with initial data $u^{(m)}_{0}$, then $u^{(m)}\rightarrow u$ in $ST(I\times{\mathbb{R}}^{3})$ and every compact subinterval $I\subset(-T_{min},T_{max})$. ###### Proof. The proof is based on the Strichartz estimate and exotic Strichartz estimate and the following nonlinear estimates. $\displaystyle\big{\\|}\|u\|^{4}u\big{\\|}_{L^{2}\dot{B}^{1/3}_{18/11,2}}\lesssim\big{\\|}u\big{\\|}_{L^{10}_{t}\dot{B}^{1/3}_{90/19,2}}\big{\\|}u\big{\\|}^{4}_{L^{10}_{t,x}},\quad$ $\displaystyle\big{\\|}\|u\|^{2}u\big{\\|}_{L^{2}\dot{B}^{1/3}_{18/11,2}}\lesssim\big{\\|}u\big{\\|}_{L^{10}_{t}\dot{B}^{1/3}_{90/19,2}}\big{\\|}u\big{\\|}^{2}_{L^{5}_{t,x}},$ $\displaystyle\big{\\|}\|u\|^{4}u\big{\\|}_{L^{2}\dot{B}^{1/2}_{6/5,2}}\lesssim\big{\\|}u\big{\\|}_{L^{6}_{t}\dot{B}^{1/2}_{18/7,2}}\big{\\|}u\big{\\|}^{4}_{L^{12}_{t}L^{9}_{x}},\quad$ $\displaystyle\big{\\|}\|u\|^{2}u\big{\\|}_{L^{2}\dot{B}^{1/3}_{6/5,2}}\lesssim\big{\\|}u\big{\\|}_{L^{6}_{t}\dot{B}^{1/2}_{18/7,2}}\big{\\|}u\big{\\|}^{2}_{L^{6}_{t}L^{9/2}_{x}}.$ ∎ ###### Lemma 2.5. Let $\phi\in C^{\infty}_{0}({\mathbb{R}}^{3})$, radially symmetric and $u$ be the radial solution of (1.1). Then we have $\displaystyle\partial_{t}\int_{{\mathbb{R}}^{3}}\phi(x)\big{\|}u(t,x)\big{\|}^{2}\;dx=$ $\displaystyle-2\Im\int_{{\mathbb{R}}^{3}}\nabla\phi\cdot\nabla\bar{u}\;u\;dx$ $\displaystyle\partial^{2}_{t}\int_{{\mathbb{R}}^{3}}\phi(x)\big{\|}u(t,x)\big{\|}^{2}\;dx=$ $\displaystyle 4\int_{{\mathbb{R}}^{3}}\phi^{\prime\prime}(r)\big{\|}\nabla u\big{\|}^{2}\;dx-\int_{{\mathbb{R}}^{3}}\Delta^{2}\phi\big{\|}u(t,x)\big{\|}^{2}\;dx$ $\displaystyle-\frac{4}{3}\int_{{\mathbb{R}}^{3}}\Delta\phi\big{\|}u(t,x)\big{\|}^{6}\;dx+\int_{{\mathbb{R}}^{3}}\Delta\phi\big{\|}u(t,x)\big{\|}^{4}\;dx,$ where $r=\|x\|$. ###### Proof. By the simple computation, we have $\displaystyle\partial^{2}_{t}\int_{{\mathbb{R}}^{3}}\phi(x)\big{\|}u(t,x)\big{\|}^{2}\;dx=$ $\displaystyle 4\int_{{\mathbb{R}}^{3}}\phi_{jk}\cdot\Re(u_{k}\overline{u}_{j})\;dx-\int_{{\mathbb{R}}^{3}}\Delta^{2}\phi\cdot\big{\|}u(t,x)\big{\|}^{2}\;dx$ $\displaystyle-\frac{4}{3}\int_{{\mathbb{R}}^{3}}\Delta\phi\cdot\big{\|}u(t,x)\big{\|}^{6}\;dx+\int_{{\mathbb{R}}^{3}}\Delta\phi\cdot\big{\|}u(t,x)\big{\|}^{4}\;dx.$ Then the result comes from the following fact $\displaystyle\partial^{2}_{jk}\phi(x)=\phi^{\prime\prime}(r)\frac{x_{j}x_{k}}{r^{2}}+\frac{\phi^{\prime}(r)}{r}\left(\delta_{jk}-\frac{x_{j}x_{k}}{r^{2}}\right)$ holds for any radial symmetric function $\phi(x)$. ∎ ### 2.4. Variational characterization In this subsection, we give the threshold energy $m$ (Proposition 1.1) by the variational method, and various estimates for the solutions of (1.1) with the energy below the threshold. There is no the radial assumption on the solution. We first give some notation before we show the behavior of $K$ near the origin. Let us denote the quadratic and nonlinear parts of $K$ by $K^{Q}$ and $K^{N}$, that is, $\displaystyle K(\varphi)=K^{Q}(\varphi)+K^{N}(\varphi),$ where $K^{Q}(\varphi)=\displaystyle 2\;\int_{{\mathbb{R}}^{3}}\|\nabla\varphi\|^{2}\;dx,$ and $K^{N}(\varphi)=\displaystyle\int_{{\mathbb{R}}^{3}}\left(-2\|\varphi\|^{6}+\frac{3}{2}\varphi\|^{4}\right)\;dx$. ###### Lemma 2.6. For any $\varphi\in H^{1}({\mathbb{R}}^{3})$, we have $\displaystyle\lim_{\lambda\rightarrow-\infty}K^{Q}(\varphi^{\lambda}_{3,-2})=0.$ (2.1) ###### Proof. It is obvious by the definition of $K^{Q}$. ∎ Now we show the positivity of $K$ near 0 in the energy space. ###### Lemma 2.7. For any bounded sequence $\varphi_{n}\in H^{1}({\mathbb{R}}^{3})\backslash\\{0\\}$ with $\displaystyle\lim_{n\rightarrow+\infty}K^{Q}(\varphi_{n})=0,$ then for large $n$, we have $\displaystyle K(\varphi_{n})>0.$ ###### Proof. By the fact that $K^{Q}(\varphi_{n})\rightarrow 0$, we know that $\displaystyle\lim_{n\rightarrow+\infty}\big{\\|}\nabla\varphi_{n}\big{\\|}^{2}_{L^{2}}=0.$ Then by the Sobolev and Gagliardo-Nirenberg inequalities, we have for large $n$ $\displaystyle\big{\\|}\varphi_{n}\big{\\|}^{6}_{L^{6}_{x}}\lesssim\big{\\|}\nabla\varphi_{n}\big{\\|}^{6}_{L^{2}_{x}}=$ $\displaystyle o(\big{\\|}\nabla\varphi_{n}\big{\\|}^{2}_{L^{2}}),$ $\displaystyle\big{\\|}\varphi_{n}\big{\\|}^{4}_{L^{4}_{x}}\lesssim\big{\\|}\varphi_{n}\big{\\|}_{L^{2}}\big{\\|}\nabla\varphi_{n}\big{\\|}^{3}_{L^{2}}$ $\displaystyle=o(\big{\\|}\nabla\varphi_{n}\big{\\|}^{2}_{L^{2}}),$ where we use the boundedness of $\big{\\|}\varphi_{n}\big{\\|}_{L^{2}}$. Hence for large $n$, we have $\displaystyle K(\varphi_{n})=$ $\displaystyle\int_{{\mathbb{R}}^{3}}\left(2\|\nabla\varphi_{n}\|^{2}-2\|\varphi_{n}\|^{6}+\frac{3}{2}\|\varphi_{n}\|^{4}\right)\;dx\thickapprox\int_{{\mathbb{R}}^{3}}\|\nabla\varphi_{n}\|^{2}\;dx>0.$ This concludes the proof. ∎ By the definition of $K$, we denote two real numbers by $\displaystyle\bar{\mu}=\max\\{4,0,6\\}=6,\quad\underline{\mu}=\min\\{4,0,6\\}=0.$ Next, we show the behavior of the scaling derivative functional $K$. ###### Lemma 2.8. For any $\varphi\in H^{1}$, we have $\displaystyle\left(\bar{\mu}-\mathcal{L}\right)E(\varphi)=$ $\displaystyle\int_{{\mathbb{R}}^{3}}\left(\big{\|}\nabla\varphi\big{\|}^{2}+\big{\|}\varphi\big{\|}^{6}\right)\;dx,$ $\displaystyle\mathcal{L}\left(\bar{\mu}-\mathcal{L}\right)E(\varphi)=$ $\displaystyle\int_{{\mathbb{R}}^{3}}\left(4\big{\|}\nabla\varphi\big{\|}^{2}+12\big{\|}\varphi\big{\|}^{6}\right)\;dx.$ ###### Proof. By the definition of $\mathcal{L}$, we have $\displaystyle\mathcal{L}\big{\\|}\nabla\varphi\big{\\|}^{2}_{L^{2}}=4\big{\\|}\nabla\varphi\big{\\|}^{2}_{L^{2}},\quad\mathcal{L}\big{\\|}\varphi\big{\\|}^{6}_{L^{6}}=12\big{\\|}\varphi\big{\\|}^{6}_{L^{6}},\quad\mathcal{L}\big{\\|}\varphi\big{\\|}^{4}_{L^{4}}=6\big{\\|}\varphi\big{\\|}^{4}_{L^{4}},$ which implies that $\displaystyle\left(\bar{\mu}-\mathcal{L}\right)E(\varphi)=6E(\varphi)-K(\varphi)=$ $\displaystyle\int_{{\mathbb{R}}^{3}}\left(\big{\|}\nabla\varphi\big{\|}^{2}+\big{\|}\varphi\big{\|}^{6}\right)\;dx,$ $\displaystyle\mathcal{L}\left(\bar{\mu}-\mathcal{L}\right)E(\varphi)=\mathcal{L}\big{\\|}\nabla\varphi\big{\\|}^{2}_{L^{2}}+\mathcal{L}\big{\\|}\varphi\big{\\|}^{6}_{L^{6}}$ $\displaystyle=\int_{{\mathbb{R}}^{3}}\left(4\big{\|}\nabla\varphi\big{\|}^{2}+12\big{\|}\varphi\big{\|}^{6}\right)\;dx.$ This completes the proof. ∎ According to the above analysis, we will replace the functional $E$ in (1.5) with a positive functional $H$, while extending the minimizing region from “$K(\varphi)=0$” to “$K(\varphi)\leq 0$”. Let $\displaystyle H(\varphi):=\left(1-\frac{\mathcal{L}}{\bar{\mu}}\right)E(\varphi)=$ $\displaystyle\int_{{\mathbb{R}}^{3}}\left(\frac{1}{6}\big{\|}\nabla\varphi\big{\|}^{2}+\frac{1}{6}\big{\|}\varphi\big{\|}^{6}\right)\;dx,$ then for any $\varphi\in H^{1}\backslash\\{0\\}$, we have $\displaystyle H(\varphi)>0,\quad\mathcal{L}H(\varphi)\geq 0.$ Now we can characterization the minimization problem (1.5) by use of $H$. ###### Lemma 2.9. For the minimization $m$ in (1.5), we have $\displaystyle m=$ $\displaystyle\inf\\{H(\varphi)\;\|\;\varphi\in H^{1}({\mathbb{R}}^{3}),\;\varphi\not=0,\;K(\varphi)\leq 0\\}$ $\displaystyle=$ $\displaystyle\inf\\{H(\varphi)\;\|\;\varphi\in H^{1}({\mathbb{R}}^{3}),\;\varphi\not=0,\;K(\varphi)<0\\}.$ (2.2) ###### Proof. For any $\varphi\in H^{1}$, $\varphi\not=0$ with $K(\varphi)=0$, we have $E(\varphi)=H(\varphi)$, this implies that $\displaystyle m=$ $\displaystyle\inf\\{E(\varphi)\;\|\;\varphi\in H^{1}({\mathbb{R}}^{3}),\;\varphi\not=0,\;K(\varphi)=0\\}$ $\displaystyle\geq$ $\displaystyle\inf\\{H(\varphi)\;\|\;\varphi\in H^{1}({\mathbb{R}}^{3}),\;\varphi\not=0,\;K(\varphi)\leq 0\\}.$ (2.3) On the other hand, for any $\varphi\in H^{1}$, $\varphi\not=0$ with $K(\varphi)<0$, by Lemma 2.6, Lemma 2.7 and the continuity of $K$ in $\lambda$, we know that there exists a $\lambda_{0}<0$ such that $\displaystyle K(\varphi^{\lambda_{0}}_{3,-2})=0,$ then by $\mathcal{L}H\geq 0$, we have $\displaystyle E(\varphi^{\lambda_{0}}_{3,-2})=H(\varphi^{\lambda_{0}}_{3,-2})\leq H(\varphi^{0}_{3,-2})=H(\varphi).$ Therefore, $\displaystyle\inf\\{E(\varphi)\;\|\;\varphi\in H^{1}({\mathbb{R}}^{3}),\;\varphi\not=0,\;K(\varphi)=0\\}$ $\displaystyle\leq\inf\\{H(\varphi)\;\|\;\varphi\in H^{1}({\mathbb{R}}^{3}),\;\varphi\not=0,\;K(\varphi)<0\\}.$ (2.4) By (2.3) and (2.4), we have $\displaystyle\inf\\{H(\varphi)\;\|\;\varphi\in H^{1}({\mathbb{R}}^{3}),\;\varphi\not=0,\;K(\varphi)\leq 0\\}$ $\displaystyle\leq m\leq\inf\\{H(\varphi)\;\|\;\varphi\in H^{1}({\mathbb{R}}^{3}),\;\varphi\not=0,\;K(\varphi)<0\\}.$ In order to show (2.2), it suffices to show that $\displaystyle\inf\\{H(\varphi)\;\|\;\varphi\in H^{1}({\mathbb{R}}^{3}),\;\varphi\not=0,\;K(\varphi)\leq 0\\}$ $\displaystyle\geq\inf\\{H(\varphi)\;\|\;\varphi\in H^{1}({\mathbb{R}}^{3}),\;\varphi\not=0,\;K(\varphi)<0\\}.$ (2.5) For any $\varphi\in H^{1}$, $\varphi\not=0$ with $K(\varphi)\leq 0$. By Lemma 2.8, we know that $\displaystyle\mathcal{L}K(\varphi)=\bar{\mu}K(\varphi)-\int_{{\mathbb{R}}^{3}}\left(4\big{\|}\nabla\varphi\big{\|}^{2}+12\big{\|}\varphi\big{\|}^{6}\right)\;dx<0,$ then for any $\lambda>0$ we have $\displaystyle K(\varphi^{\lambda}_{3,-2})<0,$ and as $\lambda\rightarrow 0$ $\displaystyle H(\varphi^{\lambda}_{3,-2})=\int_{{\mathbb{R}}^{3}}\left(\frac{e^{4\lambda}}{6}\big{\|}\nabla\varphi\big{\|}^{2}+\frac{e^{12\lambda}}{6}\big{\|}\varphi\big{\|}^{6}\right)\;dx\longrightarrow H(\varphi).$ This shows (2.5), and completes the proof. ∎ Next we will use the ($\dot{H}^{1}$-invariant) scaling argument to remove the $L^{4}$ term (the lower regularity quantity than $\dot{H}^{1}$) in $K$, that is, to replace the constrained condition $K(\varphi)<0$ with $K^{c}(\varphi)<0$, where $\displaystyle K^{c}(\varphi):=\int_{{\mathbb{R}}^{3}}\left(2\|\nabla\varphi\|^{2}-2\|\varphi\|^{6}\right)\;dx.$ In fact, we have ###### Lemma 2.10. For the minimization $m$ in (1.5), we have $\displaystyle m=$ $\displaystyle\inf\\{H(\varphi)\;\|\;\varphi\in H^{1}({\mathbb{R}}^{3}),\;\varphi\not=0,\;K^{c}(\varphi)<0\\}$ $\displaystyle=$ $\displaystyle\inf\\{H(\varphi)\;\|\;\varphi\in H^{1}({\mathbb{R}}^{3}),\;\varphi\not=0,\;K^{c}(\varphi)\leq 0\\}.$ ###### Proof. Since $K^{c}(\varphi)\leq K(\varphi)$, it is obvious that $\displaystyle m=$ $\displaystyle\inf\\{H(\varphi)\;\|\;\varphi\in H^{1}({\mathbb{R}}^{3}),\;\varphi\not=0,\;K(\varphi)<0\\}$ $\displaystyle\geq$ $\displaystyle\inf\\{H(\varphi)\;\|\;\varphi\in H^{1}({\mathbb{R}}^{3}),\;\varphi\not=0,\;K^{c}(\varphi)<0\\}.$ Hence in order to show the first equality, it suffices to show that $\displaystyle\inf\\{H(\varphi)\;\|\;\varphi\in H^{1}({\mathbb{R}}^{3}),\;\varphi\not=0,\;K(\varphi)<0\\}$ $\displaystyle\leq\inf\\{H(\varphi)\;\|\;\varphi\in H^{1}({\mathbb{R}}^{3}),\;\varphi\not=0,\;K^{c}(\varphi)<0\\}.$ (2.6) To do so, for any $\varphi\in H^{1}$, $\varphi\not=0$ with $K^{c}(\varphi)<0$, taking $\displaystyle\varphi^{\lambda}_{1,-2}(x)=e^{\lambda}\varphi(e^{2\lambda}x),$ we have $\varphi^{\lambda}_{1,-2}\in H^{1}$ and $\varphi^{\lambda}_{1,-2}\not=0$ for any $\lambda>0$. In addition, we have $\displaystyle K(\varphi^{\lambda}_{1,-2})=\int_{{\mathbb{R}}^{3}}\left(2\big{\|}\nabla\varphi\big{\|}^{2}-2\big{\|}\varphi\big{\|}^{6}+\frac{3}{2}e^{-2\lambda}\big{\|}\varphi\big{\|}^{4}\right)\;dx$ $\displaystyle\longrightarrow K^{c}(\varphi),$ $\displaystyle H(\varphi^{\lambda}_{1,-2})=\int_{{\mathbb{R}}^{3}}\left(\frac{1}{6}\big{\|}\nabla\varphi\big{\|}^{2}+\frac{1}{6}\big{\|}\varphi\big{\|}^{6}\right)\;dx=$ $\displaystyle H(\varphi),$ as $\lambda\rightarrow+\infty$. This gives (2.6), and completes the proof of the first equality. For the second equality, it is obvious that $\displaystyle\inf\\{H(\varphi)\;\|\;\varphi\in H^{1}({\mathbb{R}}^{3}),\;\varphi\not=0,\;K^{c}(\varphi)<0\\}$ $\displaystyle\geq\inf\\{H(\varphi)\;\|\;\varphi\in H^{1}({\mathbb{R}}^{3}),\;\varphi\not=0,\;K^{c}(\varphi)\leq 0\\},$ hence we only need to show that $\displaystyle\inf\\{H(\varphi)\;\|\;\varphi\in H^{1}({\mathbb{R}}^{3}),\;\varphi\not=0,\;K^{c}(\varphi)<0\\}$ $\displaystyle\leq\inf\\{H(\varphi)\;\|\;\varphi\in H^{1}({\mathbb{R}}^{3}),\;\varphi\not=0,\;K^{c}(\varphi)\leq 0\\}.$ (2.7) To do this, we use the ($L^{2}$-invariant) scaling argument. For any $\varphi\in H^{1}$, $\varphi\not=0$ with $K^{c}(\varphi)\leq 0$, we have $\varphi^{\lambda}_{3,-2}\in H^{1}$, $\varphi^{\lambda}_{3,-2}\not=0$. In addition, by $\displaystyle\mathcal{L}K^{c}(\varphi)=$ $\displaystyle\int_{{\mathbb{R}}^{3}}\left(8\big{\|}\nabla\varphi\big{\|}^{2}-24\big{\|}\varphi\big{\|}^{6}\right)\;dx=4K^{c}(\varphi)-16\big{\\|}\varphi\big{\\|}^{6}_{L^{6}}<0,$ $\displaystyle H(\varphi^{\lambda}_{3,-2})=\int_{{\mathbb{R}}^{3}}\left(\frac{e^{4\lambda}}{6}\big{\|}\nabla\varphi\big{\|}^{2}+\frac{e^{12\lambda}}{6}\big{\|}\varphi\big{\|}^{6}\right)\;dx,$ we have $K^{c}(\varphi^{\lambda}_{3,-2})<0$ for any $\lambda>0$, and $\displaystyle H(\varphi^{\lambda}_{3,-2})\rightarrow H(\varphi),\;\;\text{as}\;\;\lambda\rightarrow 0.$ This implies (2.7) and completes the proof. ∎ After these preparations, we can now make use of the sharp Sobolev constant in [1, 35] to compute the minimization $m$ of (1.5), which also shows Proposition 1.1. ###### Lemma 2.11. For the minimization $m$ in (1.5), we have $\displaystyle m=E^{c}(W).$ ###### Proof. By Lemma 2.10, we have $\displaystyle m=$ $\displaystyle\inf\left\\{\frac{1}{6}\int_{{\mathbb{R}}^{3}}\left(\|\nabla\varphi\|^{2}+\|\varphi\|^{6}\right)\;dx\;\Big{\|}\;\varphi\in H^{1},\;\varphi\not=0,\;\big{\\|}\nabla\varphi\big{\\|}^{2}_{L^{2}}\leq\big{\\|}\varphi\big{\\|}^{6}_{L^{6}}\right\\}$ $\displaystyle\geq$ $\displaystyle\inf\left\\{\int_{{\mathbb{R}}^{3}}\frac{1}{6}\left(\|\nabla\varphi\|^{2}+\|\varphi\|^{6}\right)+\frac{1}{6}\left(\|\nabla\varphi\|^{2}-\|\varphi\|^{6}\right)\;dx\;\Big{\|}\;\varphi\in H^{1},\;\varphi\not=0,\;\big{\\|}\nabla\varphi\big{\\|}^{2}_{L^{2}}\leq\big{\\|}\varphi\big{\\|}^{6}_{L^{6}}\right\\}$ where the equality holds if and only if the minimization is taken by some $\varphi$ with $\big{\\|}\nabla\varphi\big{\\|}^{2}_{L^{2}}=\big{\\|}\varphi\big{\\|}^{6}_{L^{6}}$. While $\displaystyle\inf\left\\{\int_{{\mathbb{R}}^{3}}\frac{1}{3}\|\nabla\varphi\|^{2}\;dx\;\big{\|}\;\varphi\in H^{1},\;\varphi\not=0,\;\big{\\|}\nabla\varphi\big{\\|}^{2}_{L^{2}}\leq\big{\\|}\varphi\big{\\|}^{6}_{L^{6}}\right\\}$ $\displaystyle=\inf\left\\{\frac{1}{3}\big{\\|}\nabla\varphi\big{\\|}^{2}_{L^{2}}\left(\frac{\big{\\|}\nabla\varphi\big{\\|}^{2}_{L^{2}}}{\big{\\|}\varphi\big{\\|}^{6}_{L^{6}}}\right)^{1/2}\;\Big{\|}\;\varphi\in H^{1},\;\varphi\not=0\right\\}$ $\displaystyle=\inf\left\\{\frac{1}{3}\left(\frac{\big{\\|}\nabla\varphi\big{\\|}_{L^{2}}}{\big{\\|}\varphi\big{\\|}_{L^{6}}}\right)^{3}\;\Big{\|}\;\varphi\in H^{1},\;\varphi\not=0\right\\}$ $\displaystyle=\inf\left\\{\frac{1}{3}\left(\frac{\big{\\|}\nabla\varphi\big{\\|}_{L^{2}}}{\big{\\|}\varphi\big{\\|}_{L^{6}}}\right)^{3}\;\Big{\|}\;\varphi\in\dot{H}^{1},\;\varphi\not=0\right\\}=\frac{1}{3}\big{(}C^{}_{3}\big{)}^{-3}.$ where we use the density property $H^{1}\hookrightarrow\dot{H}^{1}$ in the last second equality and that $C^{}_{3}$ is the sharp Sobolev constant in ${\mathbb{R}}^{3}$, that is, $\displaystyle\big{\\|}\varphi\big{\\|}_{L^{6}_{x}}\leq C^{}_{3}\big{\\|}\nabla\varphi\big{\\|}_{L^{2}_{x}},\;\;\forall\;\varphi\in\dot{H}^{1}({\mathbb{R}}^{3}),$ and the equality can be attained by the ground state $W$ of the following elliptic equation $\displaystyle-\Delta W=\|W\|^{4}W.$ This implies that $\frac{1}{3}\big{(}C^{}_{3}\big{)}^{-3}=E^{c}(W)$. The proof is completed. ∎ After the computation of the minimization $m$ in (1.5), we next give some variational estimates. ###### Lemma 2.12. For any $\varphi\in H^{1}$ with $K(\varphi)\geq 0$, we have $\displaystyle\int_{{\mathbb{R}}^{3}}\left(\frac{1}{6}\big{\|}\nabla\varphi\big{\|}^{2}+\frac{1}{6}\big{\|}\varphi\big{\|}^{6}\right)dx\leq E(\varphi)\leq\int_{{\mathbb{R}}^{3}}\left(\frac{1}{2}\big{\|}\nabla\varphi\big{\|}^{2}+\frac{1}{4}\big{\|}\varphi\big{\|}^{4}\right)dx.$ (2.8) ###### Proof. On one hand, the right hand side of (2.8) is trivial. On the other hand, by the definition of $E$ and $K$, we have $\displaystyle E(\varphi)=\int_{{\mathbb{R}}^{3}}\left(\frac{1}{6}\big{\|}\nabla\varphi\big{\|}^{2}+\frac{1}{6}\big{\|}\varphi\big{\|}^{6}\right)\;dx+\frac{1}{6}K(\varphi),$ which implies the left hand side of (2.8). ∎ At the last of this section, we give the uniform bounds on the scaling derivative functional $K(\varphi)$ with the energy $E(\varphi)$ below the threshold $m$, which plays an important role for the blow-up and scattering analysis in Section 3 and Section 6. ###### Lemma 2.13. For any $\varphi\in H^{1}$ with $E(\varphi)<m$. 1. (1) If $K(\varphi)<0$, then $\displaystyle K(\varphi)\leq-6\big{(}m-E(\varphi)\big{)}.$ (2.9) 2. (2) If $K(\varphi)\geq 0$, then $\displaystyle K(\varphi)\geq\min\left(6(m-E(\varphi)),\frac{2}{3}\big{\\|}\nabla\varphi\big{\\|}^{2}_{L^{2}}+\frac{1}{2}\big{\\|}\varphi\big{\\|}^{4}_{L^{4}}\right).$ (2.10) ###### Proof. By Lemma 2.8, for any $\varphi\in H^{1}$, we have $\displaystyle\mathcal{L}^{2}E(\varphi)=\bar{\mu}\mathcal{L}E(\varphi)-4\big{\\|}\nabla\varphi\big{\\|}^{2}_{L^{2}}-12\big{\\|}\varphi\big{\\|}^{6}_{L^{6}}.$ Let $j(\lambda)=E(\varphi^{\lambda}_{3,-2})$, then we have $\displaystyle j^{\prime\prime}(\lambda)=\bar{\mu}j^{\prime}(\lambda)-4e^{4\lambda}\big{\\|}\nabla\varphi\big{\\|}^{2}_{L^{2}}-12e^{12\lambda}\big{\\|}\varphi\big{\\|}^{6}_{L^{6}}.$ (2.11) Case I: If $K(\varphi)<0$, then by (2.1), Lemma 2.7 and the continuity of $K$ in $\lambda$, there exists a negative number $\lambda_{0}<0$ such that $K(\varphi^{\lambda_{0}}_{3,-2})=0$, and $\displaystyle K(\varphi^{\lambda}_{3,-2})<0,\;\;\forall\;\;\lambda\in(\lambda_{0},0).$ By (1.5), we obtain $j(\lambda_{0})=E(\varphi^{\lambda_{0}}_{3,-2})\geq m$. Now by integrating (2.11) over $[\lambda_{0},0]$, we have $\displaystyle\int^{0}_{\lambda_{0}}j^{\prime\prime}(\lambda)\;d\lambda\leq\bar{\mu}\int^{0}_{\lambda_{0}}j^{\prime}(\lambda)\;d\lambda,$ which implies that $\displaystyle K(\varphi)=j^{\prime}(0)-j^{\prime}(\lambda_{0})\leq\bar{\mu}\left(j(0)-j(\lambda_{0})\right)\leq-\bar{\mu}\big{(}m-E(\varphi)\big{)},$ which implies (2.9). Case II: $K(\varphi)\geq 0$. We divide it into two subcases: When $2\bar{\mu}K(\varphi)\geq 12\big{\\|}\varphi\big{\\|}^{6}_{L^{6}}$. Since $\displaystyle 12\int_{{\mathbb{R}}^{3}}\big{\|}\varphi\big{\|}^{6}\;dx=-6K(\varphi)+\int_{{\mathbb{R}}^{3}}\left(12\big{\|}\nabla\varphi\big{\|}^{2}+9\big{\|}\varphi\big{\|}^{4}\right)\;dx,$ then we have $\displaystyle 2\bar{\mu}K(\varphi)\geq-6K(\varphi)+\int_{{\mathbb{R}}^{3}}\left(12\big{\|}\nabla\varphi\big{\|}^{2}+9\big{\|}\varphi\big{\|}^{4}\right)\;dx,$ which implies that $\displaystyle K(\varphi)\geq\frac{2}{3}\big{\\|}\nabla\varphi\big{\\|}^{2}_{L^{2}}+\frac{1}{2}\big{\\|}\varphi\big{\\|}^{4}_{L^{4}}.$ When $2\bar{\mu}K(\varphi)\leq 12\big{\\|}\varphi\big{\\|}^{6}_{L^{6}}$. By (2.11), we have for $\lambda=0$ $\displaystyle 0<$ $\displaystyle 2\bar{\mu}j^{\prime}(\lambda)<12e^{12\lambda}\big{\\|}\varphi\big{\\|}^{6}_{L^{6}},$ $\displaystyle j^{\prime\prime}(\lambda)=\bar{\mu}j^{\prime}(\lambda)-$ $\displaystyle 4e^{4\lambda}\big{\\|}\nabla\varphi\big{\\|}^{2}_{L^{2}}-12e^{12\lambda}\big{\\|}\varphi\big{\\|}^{6}_{L^{6}}\leq-\bar{\mu}j^{\prime}(\lambda).$ (2.12) By the continuity of $j^{\prime}$ and $j^{\prime\prime}$ in $\lambda$, we know that $j^{\prime}$ is an accelerating decreasing function as $\lambda$ increases until $j^{\prime}(\lambda_{0})=0$ for some finite number $\lambda_{0}>0$ and (2.12) holds on $[0,\lambda_{0}]$. By $K(\varphi^{\lambda_{0}}_{3,-2})=j^{\prime}(\lambda_{0})=0,$ we know that $\displaystyle E(\varphi^{\lambda_{0}}_{3,-2})\geq m.$ Now integrating (2.12) over $[0,\lambda_{0}]$, we obtain that $\displaystyle-K(\varphi)=j^{\prime}(\lambda_{0})-j^{\prime}(0)\leq-\bar{\mu}\big{(}j(\lambda_{0})-j(0)\big{)}\leq-\bar{\mu}(m-E(\varphi)).$ This completes the proof. ∎ ## 3\. Part I: Blow up for $\mathcal{K}^{-}$ In this section, we prove the blow-up result of Theorem 1.3. We can also refer to [33]. Now let $\phi$ be a smooth, radial function satisfying $\partial^{2}_{r}\phi(r)\leq 2$, $\phi(r)=r^{2}$ for $r\leq 1$, and $\phi(r)$ is constant for $r\geq 3$. For some $R$, we define $\displaystyle V_{R}(t):=\int_{{\mathbb{R}}^{3}}\phi_{R}(x)\|u(t,x)\|^{2}\;dx,\quad\phi_{R}(x)=R^{2}\phi\left(\frac{\|x\|}{R}\right).$ By Lemma 2.5, $\Delta\phi_{R}(r)=6$ for $r\leq R,$ and $\Delta^{2}\phi_{R}(r)=0$ for $r\leq R,$ we have $\displaystyle\partial^{2}_{t}V_{R}(t)=$ $\displaystyle\;4\int_{{\mathbb{R}}^{3}}\phi_{R}^{\prime\prime}(r)\big{\|}\nabla u(t,x)\big{\|}^{2}\;dx-\int_{{\mathbb{R}}^{3}}(\Delta^{2}\phi_{R})(x)\|u(t,x)\|^{2}\;dx$ $\displaystyle-\frac{4}{3}\int_{{\mathbb{R}}^{3}}(\Delta\phi_{R})\|u(t,x)\|^{6}\;dx+\int_{{\mathbb{R}}^{3}}(\Delta\phi_{R})\|u(t,x)\|^{4}\;dx$ $\displaystyle\leq$ $\displaystyle\;4\int_{{\mathbb{R}}^{3}}\left(2\|\nabla u(t)\|^{2}-2\|u(t)\|^{6}+\frac{3}{2}\|u(t)\|^{4}\right)\;dx$ $\displaystyle+\frac{c}{R^{2}}\int_{R\leq\|x\|\leq 3R}\big{\|}u(t)\big{\|}^{2}\;dx+c\int_{R\leq\|x\|\leq 3R}\left(\big{\|}u(t)\big{\|}^{4}+\big{\|}u(t)\big{\|}^{6}\right)\;dx.$ By the Gagliardo-Nirenberg and radial Sobolev inequalities, we have $\displaystyle\big{\\|}f\big{\\|}^{4}_{L^{4}(\|x\|\geq R)}\leq$ $\displaystyle\frac{c}{R^{2}}\big{\\|}f\big{\\|}^{3}_{L^{2}(\|x\|\geq R)}\big{\\|}\nabla f\big{\\|}_{L^{2}(\|x\|\geq R)},$ $\displaystyle\big{\\|}f\big{\\|}_{L^{\infty}(\|x\|\geq R)}\leq$ $\displaystyle\frac{c}{R}\big{\\|}f\big{\\|}^{1/2}_{L^{2}(\|x\|\geq R)}\big{\\|}\nabla f\big{\\|}^{1/2}_{L^{2}(\|x\|\geq R)}.$ Therefore, by mass conservation and Young’s inequality, we know that for any $\epsilon>0$ there exist sufficiently large $R$ such that $\displaystyle\partial^{2}_{t}V_{R}(t)\leq$ $\displaystyle 4K(u(t))+\epsilon\big{\\|}\nabla u(t,x)\big{\\|}^{2}_{L^{2}}+\epsilon^{2}.$ $\displaystyle=$ $\displaystyle 48E(u)-\big{(}16-\epsilon\big{)}\big{\\|}\nabla u(t)\big{\\|}^{2}_{L^{2}}-6\big{\\|}u(t)\big{\\|}^{4}_{L^{4}}+\epsilon^{2}$ (3.1) By $K(u)<0$, mass and energy conservations, Lemma 2.13 and the continuity argument, we know that for any $t\in I$, we have $\displaystyle K(u(t))\leq-6\left(m-E(u(t))\right)<0.$ By Lemma 2.9, we have $\displaystyle m\leq H(u(t))<\frac{1}{3}\big{\\|}u(t)\big{\\|}^{6}_{L^{6}}.$ where we have used the fact that $K(u(t))<0$ in the second inequality. By the fact $m=\frac{1}{3}\left(C^{}_{3}\right)^{-3}$ and the Sharp Sobolev inequality, we have $\displaystyle\big{\\|}\nabla u(t)\big{\\|}^{6}_{L^{2}}\geq\left(C^{}_{3}\right)^{-6}\big{\\|}u(t)\big{\\|}^{6}_{L^{6}}>\left(C^{}_{3}\right)^{-9},$ which implies that $\big{\\|}\nabla u(t)\big{\\|}^{2}_{L^{2}}>3m$. In addition, by $E(u_{0})<m$ and energy conservation, there exists $\delta_{1}>0$ such that $E(u(t))\leq(1-\delta_{1})m$. Thus, if we choose $\epsilon$ sufficiently small, we have $\displaystyle\partial^{2}_{t}V_{R}(t)\leq 48(1-\delta_{1})m-3\big{(}16-\epsilon\big{)}m+\epsilon^{2}\leq-24\delta_{1}m,$ which implies that $u$ must blow up at finite time. ∎ ## 4\. Perturbation theory In this part, we give the perturbation theory of the solution of (1.1) with the global space-time estimate. First we denote the space-time space $ST(I)$ on the time interval $I$ by $\displaystyle ST(I):=\left(L^{10}_{t}\dot{B}^{1/3}_{90/19,2}\cap L^{12}_{t}L^{9}_{x}\cap L^{6}_{t}\dot{B}^{1/2}_{18/7,2}\cap L^{5}_{t,x}\right)$ $\displaystyle(I\times{\mathbb{R}}^{3}),$ $\displaystyle ST^{}(I):=\left(L^{2}_{t}\dot{B}^{1/3}_{18/11,2}\cap L^{2}_{t}\dot{B}^{1/2}_{6/5,2}\right)(I\times{\mathbb{R}}^{3})$ . The main result in this section is the following. ###### Proposition 4.1. Let $I$ be a compact time interval and let $w$ be an approximate solution to (1.1) on $I\times{\mathbb{R}}^{3}$ in the sense that $\displaystyle i\partial_{t}w+\Delta w=-\|w\|^{4}w+\|w\|^{2}w+e$ for some suitable small function $e$. Assume that for some constants $L,E_{0}>0$, we have $\displaystyle\big{\\|}w\big{\\|}_{ST(I)}\leq L,\quad\big{\\|}w(t_{0})\big{\\|}_{H^{1}_{x}({\mathbb{R}}^{3})}\leq E_{0}$ for some $t_{0}\in I$. Let $u(t_{0})$ close to $w(t_{0})$ in the sense that for some $E^{\prime}>0$, we have $\displaystyle\big{\\|}u(t_{0})-w(t_{0})\big{\\|}_{H^{1}_{x}}\leq E^{\prime}.$ Assume also that for some $\varepsilon$, we have $\displaystyle\left\\|e^{i(t-t_{0})\Delta}\big{(}u(t_{0})-w(t_{0})\big{)}\right\\|_{ST(I)}$ $\displaystyle\leq\varepsilon,\quad\big{\\|}e\big{\\|}_{ST^{}(I)}\leq\varepsilon,$ (4.1) where $0<\varepsilon\leq\varepsilon_{0}=\varepsilon_{0}(E_{0},E^{\prime},L)$ is a small constant. Then there exists a solution $u$ to (1.1) on $I\times{\mathbb{R}}^{3}$ with initial data $u(t_{0})$ at time $t=t_{0}$ satisfying $\displaystyle\big{\\|}u-w\big{\\|}_{ST(I)}\leq$ $\displaystyle C(E_{0},E^{\prime},L)\;\varepsilon,\quad\text{and}\quad\big{\\|}u\big{\\|}_{ST(I)}\leq C(E_{0},E^{\prime},L).$ ###### Proof. Since $w\in ST(I)$, there exists a partition of the right half of $I$ at $t_{0}$: $\displaystyle t_{0}<t_{1}<\cdots<t_{N},\quad I_{j}=(t_{j},t_{j+1}),\quad I\cap(t_{0},\infty)=(t_{0},t_{N}),$ such that $N\leq C(L,\delta)$ and for any $j=0,1,\ldots,N-1$, we have $\displaystyle\big{\\|}w\big{\\|}_{ST(I_{j})}\leq\delta\ll 1.$ (4.2) The estimate on the left half of $I$ at $t_{0}$ is analogue, we omit it. Let $\displaystyle\gamma(t,x)=$ $\displaystyle u(t,x)-w(t,x),$ $\displaystyle\gamma_{j}(t,x)=$ $\displaystyle e^{i(t-t_{j})\Delta}\Big{(}u(t_{j},x)-w(t_{j},x)\Big{)},$ then $\gamma$ satisfies the following difference equation $\displaystyle i\gamma_{t}+\Delta\gamma=O(w^{4}\gamma+w^{3}\gamma^{2}+w^{2}\gamma^{3}+w\gamma^{4}+\gamma^{5}+w^{2}\gamma+w\gamma^{2}+\gamma^{3})-e,$ which implies that $\displaystyle\gamma(t)=$ $\displaystyle\gamma_{j}(t)-i\int^{t}_{t_{j}}e^{i(t-s)\Delta}\Big{(}O(w^{4}\gamma+w^{3}\gamma^{2}+w^{2}\gamma^{3}+w\gamma^{4}+\gamma^{5}+w^{2}\gamma+w\gamma^{2}+\gamma^{3})-e\Big{)}\;ds,$ $\displaystyle\gamma_{j+1}(t)=$ $\displaystyle\gamma_{j}(t)-i\int^{t_{j+1}}_{t_{j}}e^{i(t-s)\Delta}\Big{(}O(w^{4}\gamma+w^{3}\gamma^{2}+w^{2}\gamma^{3}+w\gamma^{4}+\gamma^{5}+w^{2}\gamma+w\gamma^{2}+\gamma^{3})-e\Big{)}\;ds.$ By Lemma 2.2, we have $\displaystyle\big{\\|}\gamma-\gamma_{j}\big{\\|}_{L^{6}_{t}\left(I_{j};\dot{B}^{1/2}_{18/7,2}\right)\cap L^{5}_{t,x}\left(I_{j}\right)}+\big{\\|}\gamma_{j+1}-\gamma_{j}\big{\\|}_{L^{6}_{t}\left({\mathbb{R}};\dot{B}^{1/2}_{18/7,2}\right)\cap L^{5}_{t,x}\left({\mathbb{R}}\times{\mathbb{R}}^{3}\right)}$ (4.3) $\displaystyle\lesssim\big{\\|}O(w^{4}\gamma+w^{3}\gamma^{2}+w^{2}\gamma^{3}+w\gamma^{4}+\gamma^{5}\big{\\|}_{L^{2}_{t}\left(I_{j};\dot{B}^{1/2}_{6/5,2}\right)}$ $\displaystyle\quad+\big{\\|}w^{2}\gamma+w\gamma^{2}+\gamma^{3})\big{\\|}_{L^{2}_{t}\left(I_{j};\dot{B}^{1/2}_{6/5,2}\right)}+\big{\\|}e\big{\\|}_{L^{2}_{t}\left(I_{j};\dot{B}^{1/2}_{6/5,2}\right)}$ $\displaystyle\lesssim\big{\\|}w\big{\\|}^{4}_{L^{12}_{t}\left(I_{j};L^{9}_{x}\right)}\big{\\|}\gamma\big{\\|}_{L^{6}_{t}\left(I_{j};\dot{B}^{1/2}_{18/7,2}\right)}+\big{\\|}w\big{\\|}^{3}_{L^{12}_{t}\left(I_{j};L^{9}_{x}\right)}\big{\\|}w\big{\\|}_{L^{6}_{t}\left(I_{j};\dot{B}^{1/2}_{18/7,2}\right)}\big{\\|}\gamma\big{\\|}_{L^{12}_{t}\left(I_{j};L^{9}_{x}\right)}$ $\displaystyle\quad+\big{\\|}w\big{\\|}^{3}_{L^{12}_{t}\left(I_{j};L^{9}_{x}\right)}\big{\\|}\gamma\big{\\|}_{L^{12}_{t}\left(I_{j};L^{9}_{x}\right)}\big{\\|}\gamma\big{\\|}_{L^{6}_{t}\left(I_{j};\dot{B}^{1/2}_{18/7,2}\right)}+\big{\\|}w\big{\\|}^{2}_{L^{12}_{t}\left(I_{j};L^{9}_{x}\right)}\big{\\|}w\big{\\|}_{L^{6}_{t}\left(I_{j};\dot{B}^{1/2}_{18/7,2}\right)}\big{\\|}\gamma\big{\\|}^{2}_{L^{12}_{t}\left(I_{j};L^{9}_{x}\right)}$ $\displaystyle\quad+\big{\\|}w\big{\\|}^{2}_{L^{12}_{t}\left(I_{j};L^{9}_{x}\right)}\big{\\|}\gamma\big{\\|}^{2}_{L^{12}_{t}\left(I_{j};L^{9}_{x}\right)}\big{\\|}\gamma\big{\\|}_{L^{6}_{t}\left(I_{j};\dot{B}^{1/2}_{18/7,2}\right)}+\big{\\|}w\big{\\|}_{L^{12}_{t}\left(I_{j};L^{9}_{x}\right)}\big{\\|}w\big{\\|}_{L^{6}_{t}\left(I_{j};\dot{B}^{1/2}_{18/7,2}\right)}\big{\\|}\gamma\big{\\|}^{3}_{L^{12}_{t}\left(I_{j};L^{9}_{x}\right)}$ $\displaystyle\quad+\big{\\|}w\big{\\|}_{L^{12}_{t}\left(I_{j};L^{9}_{x}\right)}\big{\\|}\gamma\big{\\|}^{3}_{L^{12}_{t}\left(I_{j};L^{9}_{x}\right)}\big{\\|}\gamma\big{\\|}_{L^{6}_{t}\left(I_{j};\dot{B}^{1/2}_{18/7,2}\right)}+\big{\\|}w\big{\\|}_{L^{6}_{t}\left(I_{j};\dot{B}^{1/2}_{18/7,2}\right)}\big{\\|}\gamma\big{\\|}^{4}_{L^{12}_{t}\left(I_{j};L^{9}_{x}\right)}$ $\displaystyle\quad+\big{\\|}\gamma\big{\\|}^{4}_{L^{12}_{t}\left(I_{j};L^{9}_{x}\right)}\big{\\|}\gamma\big{\\|}_{L^{6}_{t}\left(I_{j};\dot{B}^{1/2}_{18/7,2}\right)}$ $\displaystyle\quad+\big{\\|}w\big{\\|}^{2}_{L^{6}\left(I_{j};L^{9/2}_{x}\right)}\big{\\|}\gamma\big{\\|}_{L^{6}_{t}\left(I_{j};\dot{B}^{1/2}_{18/7,2}\right)}+\big{\\|}w\big{\\|}_{L^{6}\left(I_{j};L^{9/2}_{x}\right)}\big{\\|}w\big{\\|}_{L^{6}_{t}\left(I_{j};\dot{B}^{1/2}_{18/7,2}\right)}\big{\\|}\gamma\big{\\|}_{L^{6}\left(I_{j};L^{9/2}_{x}\right)}$ $\displaystyle\quad+\big{\\|}w\big{\\|}_{L^{6}\left(I_{j};L^{9/2}_{x}\right)}\big{\\|}\gamma\big{\\|}_{L^{6}\left(I_{j};L^{9/2}_{x}\right)}\big{\\|}\gamma\big{\\|}_{L^{6}_{t}\left(I_{j};\dot{B}^{1/2}_{18/7,2}\right)}+\big{\\|}w\big{\\|}_{L^{6}_{t}\left(I_{j};\dot{B}^{1/2}_{18/7,2}\right)}\big{\\|}\gamma\big{\\|}^{2}_{L^{6}\left(I_{j};L^{9/2}_{x}\right)}$ $\displaystyle\quad+\big{\\|}\gamma\big{\\|}^{2}_{L^{6}\left(I_{j};L^{9/2}_{x}\right)}\big{\\|}\gamma\big{\\|}_{L^{6}_{t}\left(I_{j};\dot{B}^{1/2}_{18/7,2}\right)}+\big{\\|}e\big{\\|}_{L^{2}_{t}\left(I_{j};\dot{B}^{1/2}_{6/5,2}\right)}.$ At the same time, by Lemma 2.3, we have $\displaystyle\big{\\|}\gamma-\gamma_{j}\big{\\|}_{L^{10}_{t}\left(I_{j};\dot{B}^{1/3}_{90/19,2}\right)\cap L^{12}_{t}\left(I_{j};L^{9}_{x}\right)}+\big{\\|}\gamma_{j+1}-\gamma_{j}\big{\\|}_{L^{10}_{t}\left({\mathbb{R}};\dot{B}^{1/3}_{90/19,2}\right)\cap L^{12}_{t}\left({\mathbb{R}};L^{9}_{x}\right)}$ (4.4) $\displaystyle\lesssim$ $\displaystyle\left\\|O(w^{4}\gamma+w^{3}\gamma^{2}+w^{2}\gamma^{3}+w\gamma^{4}+\gamma^{5}+w^{2}\gamma+w\gamma^{2}+\gamma^{3})\right\\|_{L^{2}(I_{j};\dot{B}^{1/3}_{\frac{18}{11},2})}+\big{\\|}e\big{\\|}_{L^{2}(I_{j};\dot{B}^{1/3}_{\frac{18}{11},2})}$ $\displaystyle\lesssim$ $\displaystyle\big{\\|}w\big{\\|}^{4}_{L^{10}_{t,x}(I_{j})}\big{\\|}\gamma\big{\\|}_{L^{10}_{t}(I_{j};\dot{B}^{1/3}_{90/19,2})}+\big{\\|}w\big{\\|}^{3}_{L^{10}_{t,x}(I_{j})}\big{\\|}w\big{\\|}_{L^{10}_{t}(I_{j};\dot{B}^{1/3}_{90/19,2})}\big{\\|}\gamma\big{\\|}_{L^{10}_{t,x}(I_{j})}$ $\displaystyle+\big{\\|}w\big{\\|}^{3}_{L^{10}_{t,x}(I_{j})}\big{\\|}\gamma\big{\\|}_{L^{10}_{t,x}(I_{j})}\big{\\|}\gamma\big{\\|}_{L^{10}_{t}(I_{j};\dot{B}^{1/3}_{90/19,2})}+\big{\\|}w\big{\\|}^{2}_{L^{10}_{t,x}(I_{j})}\big{\\|}w\big{\\|}_{L^{10}_{t}(I_{j};\dot{B}^{1/3}_{90/19,2})}\big{\\|}\gamma\big{\\|}^{2}_{L^{10}_{t,x}(I_{j})}$ $\displaystyle+\big{\\|}w\big{\\|}^{2}_{L^{10}_{t,x}(I_{j})}\big{\\|}\gamma\big{\\|}^{2}_{L^{10}_{t,x}(I_{j})}\big{\\|}\gamma\big{\\|}_{L^{10}_{t}(I_{j};\dot{B}^{1/3}_{90/19,2})}+\big{\\|}w\big{\\|}_{L^{10}_{t,x}(I_{j})}\big{\\|}w\big{\\|}_{L^{10}_{t}(I_{j};\dot{B}^{1/3}_{90/19,2})}\big{\\|}\gamma\big{\\|}^{3}_{L^{10}_{t,x}(I_{j})}$ $\displaystyle+\big{\\|}w\big{\\|}_{L^{10}_{t,x}(I_{j})}\big{\\|}\gamma\big{\\|}^{3}_{L^{10}_{t,x}(I_{j})}\big{\\|}\gamma\big{\\|}_{L^{10}_{t}(I_{j};\dot{B}^{1/3}_{90/19,2})}+\big{\\|}w\big{\\|}_{L^{10}_{t}(I_{j};\dot{B}^{1/3}_{90/19,2})}\big{\\|}\gamma\big{\\|}^{4}_{L^{10}_{t,x}(I_{j})}$ $\displaystyle+\big{\\|}\gamma\big{\\|}^{4}_{L^{10}_{t,x}(I_{j})}\big{\\|}\gamma\big{\\|}_{L^{10}_{t}(I_{j};\dot{B}^{1/3}_{90/19,2})}$ $\displaystyle+\big{\\|}w\big{\\|}^{2}_{L^{5}_{t,x}(I_{j})}\big{\\|}\gamma\big{\\|}_{L^{10}_{t}(I_{j};\dot{B}^{1/3}_{90/19,2})}+\big{\\|}w\big{\\|}_{L^{5}_{t,x}(I_{j})}\big{\\|}w\big{\\|}_{L^{10}_{t}(I_{j};\dot{B}^{1/3}_{90/19,2})}\big{\\|}\gamma\big{\\|}_{L^{5}_{t,x}(I_{j})}$ $\displaystyle+\big{\\|}w\big{\\|}_{L^{5}_{t,x}(I_{j})}\big{\\|}\gamma\big{\\|}_{L^{5}_{t,x}(I_{j})}\big{\\|}\gamma\big{\\|}_{L^{10}_{t}(I_{j};\dot{B}^{1/3}_{90/19,2})}+\big{\\|}w\big{\\|}_{L^{10}_{t}(I_{j};\dot{B}^{1/3}_{90/19,2})}\big{\\|}\gamma\big{\\|}^{2}_{L^{5}_{t,x}(I_{j})}$ $\displaystyle+\big{\\|}\gamma\big{\\|}^{2}_{L^{5}_{t,x}(I_{j})}\big{\\|}\gamma\big{\\|}_{L^{10}_{t}(I_{j};\dot{B}^{1/3}_{90/19,2})}+\big{\\|}e\big{\\|}_{L^{2}(I_{j};\dot{B}^{1/3}_{18//11,2})}.$ By the interpolation, we have $\displaystyle\big{\\|}f\big{\\|}_{L^{6}\left(I_{j};L^{9/2}_{x}\right)}\lesssim\big{\\|}f\big{\\|}_{L^{6}_{t}\left(I_{j};\dot{B}^{1/2}_{18/7,2}\right)},\quad\big{\\|}f\big{\\|}_{L^{10}_{t,x}(I_{j})}\lesssim\big{\\|}f\big{\\|}_{L^{10}_{t}(I_{j};\dot{B}^{1/3}_{90/19,2})}.$ Therefore, assuming that $\displaystyle\big{\\|}\gamma\big{\\|}_{ST(I_{j})}\leq\delta\ll 1,\quad\forall\;j=0,1,\ldots,N-1,$ (4.5) then by (4.2), (4.3) and (4.4), we have $\displaystyle\big{\\|}\gamma\big{\\|}_{ST(I_{j})}+\big{\\|}\gamma_{j+1}\big{\\|}_{ST(t_{j+1},t_{N})}\leq C\big{\\|}\gamma_{j}\big{\\|}_{ST(t_{j},t_{N})}+\varepsilon,$ for some absolute constant $C>0$. By (4.1) and iteration on $j$, we get $\displaystyle\big{\\|}\gamma\big{\\|}_{ST(I)}\leq(2C)^{N}\varepsilon\leq\frac{\delta}{2},$ if we choose $\varepsilon_{0}$ sufficiently small. Hence the assumption (4.5) is justified by continuity in $t$ and induction on $j$. then repeating the estimate (4.3) and (4.4) once again, we can obtain the $ST$-norm estimate on $\gamma$, which implies the Strichartz estimate on $u$. ∎ ## 5\. Profile decomposition In this part, we will use the method in [2, 17, 21] to show the linear and nonlinear profile decompositions of the sequences of radial, $H^{1}$-bounded solutions of (1.1), which will be used to construct the critical element (minimal energy non-scattering solution) and show its properties, especially the compactness. In order to do it, we now introduce the complex-valued function $\overrightarrow{v}(t,x)$ by $\displaystyle\overrightarrow{v}(t,x)=\left<\nabla\right>v(t,x),\quad v(t,x)=\left<\nabla\right>^{-1}\overrightarrow{v}(t,x).$ Given $(t^{j}_{n},h^{j}_{n})\in{\mathbb{R}}\times(0,1]$, let $\tau^{j}_{n}$, $T^{j}_{n}$ denote the scaled time drift, the scaling transformation, defined by $\displaystyle\tau^{j}_{n}=-\frac{t^{j}_{n}}{\left(h^{j}_{n}\right)^{2}},\quad T^{j}_{n}\varphi(x)=\frac{1}{(h^{j}_{n})^{3/2}}\varphi\left(\frac{x}{h^{j}_{n}}\right).$ We also introduce the set of Fourier multipliers on ${\mathbb{R}}^{3}$. $\displaystyle\mathcal{MC}=\\{\mu=\mathcal{F}^{-1}\widetilde{\mu}\mathcal{F}\;\|\;\widetilde{\mu}\in C({\mathbb{R}}^{3}),\;\exists\lim_{\|\xi\|\rightarrow+\infty}\widetilde{\mu}(\xi)\in{\mathbb{R}}\\}.$ ### 5.1. Linear profile decomposition In this subsection, we show the profile decomposition with the scaling parameter of a sequence of the radial, free Schrödinger solutions in the energy space $H^{1}({\mathbb{R}}^{3})$, which implies the profile decomposition of a sequence of radial initial data. ###### Proposition 5.1. Let $\overrightarrow{v}_{n}(t,x)=e^{it\Delta}\overrightarrow{v}_{n}(0)$ be a sequence of the radial solutions of the free Schrödinger equation with bounded $L^{2}$ norm. Then up to a subsequence, there exist $K\in\\{0,1,2,\ldots,\infty\\}$, radial functions $\\{\varphi^{j}\\}_{j\in[0,K)}\subset L^{2}({\mathbb{R}}^{3})$ and $\\{t^{j}_{n},h^{j}_{n}\\}_{n\in{\mathbb{N}}}\subset{\mathbb{R}}\times(0,1]$ satisfying $\displaystyle\overrightarrow{v}_{n}(t,x)=\sum^{k-1}_{j=0}\overrightarrow{v}^{j}_{n}(t,x)+\overrightarrow{w}^{k}_{n}(t,x),$ (5.1) where $\overrightarrow{v}^{j}_{n}(t,x)=e^{i(t-t^{j}_{n})\Delta}T^{j}_{n}\varphi^{j}$, and $\displaystyle\lim_{k\rightarrow K}\varlimsup_{n\rightarrow+\infty}\big{\\|}\overrightarrow{w}^{k}_{n}\big{\\|}_{L^{\infty}_{t}({\mathbb{R}};B^{-3/2}_{\infty,\infty}({\mathbb{R}}^{3}))}=0,$ (5.2) and for any Fourier multiplier $\mu\in\mathcal{MC}$, any $l<j<k\leq K$ and any $t\in{\mathbb{R}}$, $\displaystyle\lim_{n\rightarrow+\infty}\left(\log\left\|\dfrac{h^{j}_{n}}{h^{l}_{n}}\right\|+\left\|\frac{t^{j}_{n}-t^{l}_{n}}{(h^{l}_{n})^{2}}\right\|\right)=\infty,$ (5.3) $\displaystyle\lim_{n\rightarrow+\infty}\left<\mu\overrightarrow{v}^{l}_{n}(t)\;,\;\mu\overrightarrow{v}^{j}_{n}(t)\right>_{L^{2}_{x}}=\lim_{n\rightarrow+\infty}\left<\mu\overrightarrow{v}^{j}_{n}(t)\;,\;\mu\overrightarrow{w}^{k}_{n}(t)\right>_{L^{2}_{x}}=0.$ (5.4) Moreover, each sequence $\\{h^{j}_{n}\\}_{n\in{\mathbb{N}}}$ is either going to $0$ or identically $1$ for all $n$. ###### Remark 5.2. We call $\overrightarrow{v}^{j}_{n}$ and $\overrightarrow{w}^{k}_{n}$ the free concentrating wave and the remainder, respectively. From (5.4), we have the following asymptotic orthogonality $\displaystyle\lim_{n\rightarrow+\infty}\left(\big{\\|}\mu\overrightarrow{v}_{n}(t)\big{\\|}^{2}_{L^{2}}-\sum^{k-1}_{j=0}\big{\\|}\mu\overrightarrow{v}^{j}_{n}(t)\big{\\|}^{2}_{L^{2}}-\big{\\|}\mu\overrightarrow{w}^{k}_{n}(t)\big{\\|}^{2}_{L^{2}}\right)$ $\displaystyle=0.$ (5.5) ###### Proof of Proposition 5.1. Let $\displaystyle\nu:=\varlimsup_{n\rightarrow\infty}\big{\\|}\overrightarrow{v}_{n}\big{\\|}_{L^{\infty}_{t}B^{-3/2}_{\infty,\infty}}=\varlimsup_{n\rightarrow\infty}\sup_{(t,x)\in{\mathbb{R}}\times{\mathbb{R}}^{3},\atop k\geq 0}2^{-3k/2}\big{\|}\Lambda_{k}\overrightarrow{v}_{n}(t,x)\big{\|}.$ If $\nu=0$, then we have done with $K=0$. Otherwise, $\displaystyle\nu=\varlimsup_{n\rightarrow\infty}\big{\\|}\overrightarrow{v}_{n}\big{\\|}_{L^{\infty}_{t}B^{-3/2}_{\infty,\infty}}>0$. By the radial Gagliardo-Nirenberg inequality and the Bernstein inequality, we have $\displaystyle\sup_{t\in{\mathbb{R}},\|2^{k}x\|\geq R,\atop k\geq 0}2^{-3k/2}\big{\|}\Lambda_{k}\overrightarrow{v}_{n}(t,x)\big{\|}\lesssim$ $\displaystyle\sup_{k\geq 0}\frac{2^{k}2^{-3k/2}}{R}\big{\\|}\Lambda_{k}\overrightarrow{v}_{n}(t,x)\big{\\|}^{1/2}_{L^{\infty}_{t}L^{2}_{x}}\cdot\big{\\|}\nabla\Lambda_{k}\overrightarrow{v}_{n}(t,x)\big{\\|}^{1/2}_{L^{\infty}_{t}L^{2}_{x}}$ $\displaystyle\lesssim$ $\displaystyle\sup_{k\geq 0}\frac{1}{R}\big{\\|}\overrightarrow{v}_{n}(t,x)\big{\\|}_{L^{\infty}_{t}L^{2}_{x}}\lesssim\frac{1}{R}.$ If taking $R$ sufficiently large, we have $\displaystyle\sup_{t\in{\mathbb{R}},\|2^{k}x\|\geq R,k\geq 0}2^{-3k/2}\big{\|}\Lambda_{k}\overrightarrow{v}_{n}(t,x)\big{\|}\leq\frac{1}{2}\nu.$ thus, there exists a sequence $(t_{n},x_{n},k_{n})$ with $k_{n}\geq 0$ and $\|2^{k_{n}}x_{n}\|\leq R$ such that for large $n$, $\displaystyle\frac{1}{2}\varlimsup_{n\rightarrow\infty}\big{\\|}\overrightarrow{v}_{n}\big{\\|}_{L^{\infty}_{t}B^{-3/2}_{\infty,\infty}}=\frac{1}{2}\nu\leq 2^{-3k_{n}/2}\big{\|}\Lambda_{k_{n}}\overrightarrow{v}_{n}(t_{n},x_{n})\big{\|}.$ Now we define $h_{n}$ and $\psi_{n}$ by $h_{n}=2^{-k_{n}}\in(0,1]$ and $\displaystyle\overrightarrow{v}_{n}(t_{n},x)=$ $\displaystyle\left(T_{n}\psi_{n}\right)(x-x_{n})=\frac{1}{(h_{n})^{3/2}}\psi_{n}\left(\frac{x-x_{n}}{h_{n}}\right)$ (5.6) $\displaystyle=$ $\displaystyle T_{n}\left(\psi_{n}\Big{(}x-\frac{x_{n}}{h_{n}}\Big{)}\right).$ Since $\big{\\|}\psi_{n}\big{\\|}_{L^{2}}=\big{\\|}T_{n}\psi_{n}\big{\\|}_{L^{2}}=\big{\\|}\overrightarrow{v}_{n}(t_{n})\big{\\|}_{L^{2}}\leq C,$ then there exists some $\psi\in L^{2}$, such that, up to a subsequence, we have as $n\rightarrow+\infty$ $\displaystyle\frac{x_{n}}{h_{n}}\rightarrow x^{0},\;\;\text{and}\;\;\psi_{n}\rightharpoonup\psi\quad\text{weakly in}\;\;L^{2}.$ (5.7) On the other hand, if $k_{n}=0$, we have $\displaystyle 2^{-3k_{n}/2}\big{\|}\Lambda_{k_{n}}\overrightarrow{v}_{n}(t_{n},x_{n})\big{\|}=$ $\displaystyle\displaystyle\int_{{\mathbb{R}}^{3}}\Lambda_{0}(y)\;\;2^{-3k_{n}/2}\overrightarrow{v}_{n}\left(t_{n},x_{n}-\frac{y}{2^{k_{n}}}\right)\;dy$ $\displaystyle=$ $\displaystyle\int_{{\mathbb{R}}^{3}}\Lambda_{0}(y)\;\psi_{n}(-y)\;dy$ $\displaystyle\longrightarrow$ $\displaystyle\int_{{\mathbb{R}}^{3}}\Lambda_{0}(y)\;\psi(-y)\;dy\lesssim\big{\\|}\psi\big{\\|}_{L^{2}}.$ By the same way, if $k_{n}\geq 1$, we have $\displaystyle 2^{-3k_{n}/2}\big{\|}\Lambda_{k_{n}}\overrightarrow{v}_{n}(t_{n},x_{n})\big{\|}=$ $\displaystyle\int_{{\mathbb{R}}^{3}}\Lambda_{(0)}(y)\;2^{-3k_{n}/2}\overrightarrow{v}_{n}\left(t_{n},x_{n}-\frac{y}{2^{k_{n}}}\right)\;dy$ $\displaystyle=$ $\displaystyle\int_{{\mathbb{R}}^{3}}\Lambda_{(0)}(y)\;\psi_{n}(-y)\;dy$ $\displaystyle\longrightarrow$ $\displaystyle\int_{{\mathbb{R}}^{3}}\Lambda_{(0)}(y)\;\psi(-y)\;dy\lesssim\big{\\|}\psi\big{\\|}_{L^{2}}.$ If $h_{n}\rightarrow 0$, then we take $\displaystyle(t^{0}_{n},h^{0}_{n})=(t_{n},h_{n}),\quad\varphi^{0}(x)=\psi\left(x-x^{0}\right),$ otherwise, up to a subsequence, we may assume that $h_{n}\rightarrow h_{\infty}$ for some $h_{\infty}\in(0,1]$, and take $\displaystyle(t^{0}_{n},h^{0}_{n})=(t_{n},1),\quad\varphi^{0}(x)=\frac{1}{(h_{\infty})^{3/2}}\psi\left(\frac{x}{h_{\infty}}-x^{0}\right),$ then $\displaystyle T_{n}\left(\psi\Big{(}x-\frac{x_{n}}{h_{n}}\Big{)}\right)-T^{0}_{n}\varphi^{0}(x)\longrightarrow 0\quad\text{strongly in}\;\;L^{2}.$ (5.8) In addition, since $\overrightarrow{v}_{n}(t_{n},x)=\left(T_{n}\psi_{n}\right)(x-x_{n})$ is radial, so is $\varphi^{0}(x)$. Let $\overrightarrow{v}^{0}_{n}(t,x)=e^{i(t-t^{0}_{n})\Delta}T^{0}_{n}\varphi^{0}$, we define $\overrightarrow{w}^{1}_{n}$ by $\displaystyle\overrightarrow{v}_{n}(t,x)=$ $\displaystyle\overrightarrow{v}^{0}_{n}(t,x)+\overrightarrow{w}^{1}_{n}(t,x),$ (5.9) then by (5.7) and (5.8), we have $\displaystyle(T^{0}_{n})^{-1}\overrightarrow{w}^{1}_{n}(t^{0}_{n})=(T^{0}_{n})^{-1}T_{n}\left(\psi_{n}\Big{(}x-\frac{x_{n}}{h_{n}}\Big{)}\right)-\varphi^{0}\rightharpoonup 0\quad\text{weakly in}\;\;L^{2},$ which implies that $\displaystyle\left<\mu\overrightarrow{v}^{0}_{n}(t),\mu\overrightarrow{w}^{1}_{n}(t)\right>=$ $\displaystyle\left<\mu\overrightarrow{v}^{0}_{n}(t^{0}_{n}),\mu\overrightarrow{w}^{1}_{n}(t^{0}_{n})\right>=\left<\mu^{0}_{n}\varphi^{0},\mu^{0}_{n}(T^{0}_{n})^{-1}\overrightarrow{w}^{1}_{n}(t^{0}_{n})\right>\longrightarrow 0,$ where we used the conservation law in the first equality and the dominated convergence theorem and $\mu^{0}_{n}(D)=\mu\left(\frac{D}{h^{0}_{n}}\right)$ in the last equality. It is the decomposition for $k=1$. Next we apply the above procedure to the sequence $\overrightarrow{w}^{1}_{n}$ in place of $\overrightarrow{v}_{n}$, then either $\displaystyle\varlimsup_{n\rightarrow\infty}\big{\\|}\overrightarrow{w}^{1}_{n}\big{\\|}_{L^{\infty}_{t}B^{-3/2}_{\infty,\infty}}=0$ or we can find the next concentrating wave $\overrightarrow{v}^{1}_{n}$ and the remainder $\overrightarrow{w}^{2}_{n}$, such that for some $(t^{1}_{n},h^{1}_{n})$ with $h^{1}_{n}\in(0,1]$ and radial function $\varphi^{1}\in L^{2}({\mathbb{R}}^{3})$, $\displaystyle\overrightarrow{w}^{1}_{n}(t,x)=\overrightarrow{v}^{1}_{n}(t,x)+$ $\displaystyle\overrightarrow{w}^{2}_{n}(t,x)=e^{i(t-t^{1}_{n})\Delta}T^{1}_{n}\varphi^{1}(x)+\overrightarrow{w}^{2}_{n}(t,x),$ (5.10) and $\displaystyle\varlimsup_{n\rightarrow+\infty}\big{\\|}\overrightarrow{w}^{1}_{n}\big{\\|}_{L^{\infty}_{t}B^{-3/2}_{\infty,\infty}}\lesssim$ $\displaystyle\;\big{\\|}\varphi^{1}\big{\\|}_{L^{2}}=\big{\\|}\overrightarrow{v}^{1}_{n}\big{\\|}_{L^{2}},$ (5.11) $\displaystyle(T^{1}_{n})^{-1}\overrightarrow{w}^{2}_{n}(t^{1}_{n})\rightharpoonup 0\quad\text{weakly in}\;\;L^{2}$ $\displaystyle\Longrightarrow\left<\mu\overrightarrow{v}^{1}_{n}(t),\mu\overrightarrow{w}^{2}_{n}(t)\right>\longrightarrow 0.$ Iterating the above procedure, we can obtain the decomposition (5.1). It remains to show the properties (5.2), (5.3) and (5.4). We first assume that (5.4) holds, then by (5.5) and the Cauchy criterion, we have $\displaystyle\lim_{n\rightarrow+\infty}\big{\\|}\overrightarrow{w}^{k}_{n}\big{\\|}_{L^{\infty}_{t}B^{-3/2}_{\infty,\infty}}\lesssim\big{\\|}\varphi^{k}\big{\\|}_{L^{2}}=\big{\\|}\overrightarrow{v}^{k}_{n}\big{\\|}_{L^{2}}\longrightarrow 0\quad\text{as}\;\;k\rightarrow+\infty.$ (5.12) which implies (5.2). Now we show (5.3) by contradiction. Suppose that (5.3) fails, then there exists a minimal $(l,j)$ which violates (5.3). By extracting a subsequence, We may assume that $h^{l}_{n}\rightarrow h^{l}_{\infty}$ and $h^{l}_{n}/h^{j}_{n}$ and $(t^{l}_{n}-t^{j}_{n})/(h^{l}_{n})^{2}$ all converge. Now consider $\displaystyle\left(T^{l}_{n}\right)^{-1}\overrightarrow{w}^{l+1}_{n}(t^{l}_{n})=$ $\displaystyle\sum^{j}_{m=l+1}\left(T^{l}_{n}\right)^{-1}\overrightarrow{v}^{m}_{n}(t^{l}_{n})+\left(T^{l}_{n}\right)^{-1}\overrightarrow{w}^{j+1}_{n}(t^{l}_{n})$ $\displaystyle=$ $\displaystyle\sum^{j}_{m=l+1}\left(T^{l}_{n}\right)^{-1}e^{i(t^{l}_{n}-t^{m}_{n})\Delta}T^{m}_{n}\varphi^{m}+\left(T^{l}_{n}\right)^{-1}\overrightarrow{w}^{j+1}_{n}(t^{l}_{n})$ $\displaystyle=$ $\displaystyle\sum^{j-1}_{m=l+1}S^{l,m}_{n}\varphi^{m}+S^{l,j}_{n}\varphi^{j}+\left(T^{l}_{n}\right)^{-1}\overrightarrow{w}^{j+1}_{n}(t^{l}_{n}),$ where $\displaystyle S^{l,m}_{n}=\left(T^{l}_{n}\right)^{-1}e^{i(t^{l}_{n}-t^{m}_{n})\Delta}T^{m}_{n}=e^{i\frac{t^{l}_{n}-t^{m}_{n}}{(h^{l}_{n})^{2}}\Delta}\left(T^{l}_{n}\right)^{-1}T^{m}_{n}:=e^{it^{l,m}_{n}\Delta}T^{l,m}_{n}$ with the sequence $\displaystyle t^{l,m}_{n}=\frac{t^{l}_{n}-t^{m}_{n}}{(h^{l}_{n})^{2}},\quad h^{l,m}_{n}=\frac{h^{m}_{n}}{h^{l}_{n}}.$ (5.13) By the procedure of constructing (5.1), as $n\rightarrow+\infty$, we have $\displaystyle\left(T^{l}_{n}\right)^{-1}\overrightarrow{w}^{l+1}_{n}(t^{l}_{n})\rightharpoonup 0$ $\displaystyle\quad\text{weakly in}\;L^{2},$ $\displaystyle\left(T^{j}_{n}\right)^{-1}\overrightarrow{w}^{j+1}_{n}(t^{j}_{n})\rightharpoonup 0$ $\displaystyle\quad\text{weakly in}\;L^{2},$ and by the asymptotic orthogonality (5.3) between $m$ and $l$ with $m\in[l+1,j-1]$ $\displaystyle S^{l,m}_{n}\varphi^{m}\rightharpoonup 0,\;\;\forall\;m\in[l+1,j-1],$ and by the convergence of $h^{l}_{n}/h^{j}_{n}$ and $(t^{l}_{n}-t^{j}_{n})/(h^{l}_{n})^{2}$, we have $S^{l,j}_{n}\varphi^{j}\rightarrow S^{l,j}_{\infty}\varphi^{j}$ and $\displaystyle\left(T^{l}_{n}\right)^{-1}\overrightarrow{w}^{j+1}_{n}(t^{l}_{n})=$ $\displaystyle S^{l,j}_{n}\left(T^{j}_{n}\right)^{-1}\overrightarrow{w}^{j+1}_{n}(t^{j}_{n})\rightharpoonup 0\quad\text{weakly in}\;L^{2}.$ Then $\varphi^{j}=0$, it is a contradiction. Thus we obtain the orthogonality (5.3). Last we show (5.4). For $j\not=l$, we have $\displaystyle\left<\mu\overrightarrow{v}^{l}_{n}(t)\;,\;\mu\overrightarrow{v}^{j}_{n}(t)\right>_{L^{2}_{x}}$ $\displaystyle=$ $\displaystyle\left<\mu\overrightarrow{v}^{l}_{n}(0)\;,\;\mu\overrightarrow{v}^{j}_{n}(0)\right>_{L^{2}_{x}}=\left<\mu e^{-it^{l}_{n}\Delta}T^{l}_{n}\varphi^{l}\;,\;\mu e^{-it^{j}_{n}\Delta}T^{j}_{n}\varphi^{j}\right>_{L^{2}_{x}}$ $\displaystyle=$ $\displaystyle\left<e^{-it^{l}_{n}\Delta}T^{l}_{n}\mu^{l}_{n}\varphi^{l}\;,\;e^{-it^{j}_{n}\Delta}T^{j}_{n}\mu^{j}_{n}\varphi^{j}\right>_{L^{2}_{x}}=\left<\left(T^{j}_{n}\right)^{-1}e^{i(t^{j}_{n}-t^{l}_{n})\Delta}T^{l}_{n}\mu^{l}_{n}\varphi^{l}\;,\;\mu^{j}_{n}\varphi^{j}\right>_{L^{2}_{x}}$ $\displaystyle=$ $\displaystyle\left<e^{i\frac{t^{j}_{n}-t^{l}_{n}}{(h^{j}_{n})^{2}}\Delta}\left(T^{j}_{n}\right)^{-1}T^{l}_{n}\mu^{l}_{n}\varphi^{l}\;,\;\mu^{j}_{n}\varphi^{j}\right>_{L^{2}_{x}}=\left<S^{j,l}_{n}\mu^{l}_{n}\varphi^{l}\;,\;\mu^{j}_{n}\varphi^{j}\right>_{L^{2}_{x}}\rightarrow 0\quad\text{as}\;\;n\rightarrow+\infty$ where $\widetilde{\mu}^{l}_{n}(\xi)=\widetilde{\mu}\left(\xi/h^{l}_{n}\right)$ and we used the fact that $S^{j,l}_{n}\rightharpoonup 0$ weakly in $L^{2}$ as $n\rightarrow+\infty$ by (5.3). In addition, we have $\displaystyle\left<\mu\overrightarrow{v}^{j}_{n}(t)\;,\;\mu\overrightarrow{w}^{k}_{n}(t)\right>_{L^{2}_{x}}=\left<\mu\overrightarrow{v}^{j}_{n}(t)\;,\;\mu\Big{(}\overrightarrow{w}^{j+1}_{n}(t)-\sum^{k-1}_{m=j+1}\overrightarrow{v}^{m}_{n}(t)\Big{)}\right>_{L^{2}_{x}}\longrightarrow 0$ as $n\rightarrow+\infty$. This completes the proof of (5.4). ∎ After the orthogonality’s proof of the linear energy, we begin with the orthogonal analysis for the nonlinear energy. ###### Lemma 5.3. Let $\overrightarrow{v}_{n}$ be a sequence of the radial solutions of the free Schrödinger equation. Let $\displaystyle\overrightarrow{v}_{n}(t,x)=\sum^{k-1}_{j=0}\overrightarrow{v}^{j}_{n}(t,x)+\overrightarrow{w}^{k}_{n}(t,x)$ be the linear profile decomposition given by Proposition 5.1. Then we have $\displaystyle\lim_{k\rightarrow K}\varlimsup_{n\rightarrow+\infty}\left\|M(v_{n}(0))-\sum^{k-1}_{j=0}M(v^{j}_{n}(0))-M(w^{k}_{n}(0))\right\|=0,$ $\displaystyle\lim_{k\rightarrow K}\varlimsup_{n\rightarrow+\infty}\left\|E(v_{n}(0))-\sum^{k-1}_{j=0}E(v^{j}_{n}(0))-E(w^{k}_{n}(0))\right\|=0,$ $\displaystyle\lim_{k\rightarrow K}\varlimsup_{n\rightarrow+\infty}\left\|K(v_{n}(0))-\sum^{k-1}_{j=0}K(v^{j}_{n}(0))-K(w^{k}_{n}(0))\right\|=0.$ ###### Proof. We can show that the quadratic terms in $M$, $E$ and $K$ have the orthogonal decomposition by taking $\mu=\frac{1}{\left<\nabla\right>}$ and $\mu=\frac{\|\nabla\|}{\left<\nabla\right>}$ in Remark 5.2, thus it suffices to show that $\displaystyle\lim_{k\rightarrow K}\varlimsup_{n\rightarrow+\infty}\left\|F_{i}\left(v_{n}(0)\right)-\sum_{j<k}F_{i}\left(v^{j}_{n}(0)\right)-F_{i}\left(w^{k}_{n}(0)\right)\right\|=0,\quad i=1,2,$ where $F_{1}$ and $F_{2}$ are denoted by $\displaystyle F_{1}(u(t))=\int_{{\mathbb{R}}^{3}}\|u(t,x)\|^{6}\;dx,\;\;F_{2}(u(t))=\int_{{\mathbb{R}}^{3}}\|u(t,x)\|^{4}\;dx.$ In order to do so, we need re-arrange the linear concentrating wave with respect to its dispersive decay (whether $\tau^{j}_{n}$ goes to $\pm\infty$ or not for all $j$). Let $v^{<k}_{n}(0)=\displaystyle\sum_{j<k}v^{j}_{n}(0)=\sum_{j<k,\tau^{j}_{n}\rightarrow\tau^{j}_{\infty}}v^{j}_{n}(0)+\sum_{j<k,\tau^{j}_{n}\rightarrow\pm\infty}v^{j}_{n}(0)$ for some finite numbers $\tau^{j}_{\infty}$’s, then we have $\displaystyle\Big{\|}F_{i}\left(v_{n}(0)\right)-$ $\displaystyle\sum_{j<k}F_{i}\left(v^{j}_{n}(0)\right)-F_{i}\left(w^{k}_{n}(0)\right)\Big{\|}$ $\displaystyle\leq$ $\displaystyle\left\|F_{i}\left(v_{n}(0)\right)-F_{i}\left(v^{<k}_{n}(0)\right)\right\|+\left\|F_{i}\left(v^{<k}_{n}(0)\right)-F_{i}\left(\sum_{j<k,\tau^{j}_{n}\rightarrow\tau^{j}_{\infty}}v^{j}_{n}(0)\right)\right\|$ $\displaystyle+\left\|F_{i}\left(\sum_{j<k,\tau^{j}_{n}\rightarrow\tau^{j}_{\infty}}v^{j}_{n}(0)\right)-\sum_{j<k,\tau^{j}_{n}\rightarrow\tau^{j}_{\infty}}F_{i}\left(v^{j}_{n}(0)\right)\right\|$ (5.14) $\displaystyle+\left\|\sum_{j<k,\tau^{j}_{n}\rightarrow\pm\infty}F_{i}\left(v^{j}_{n}(0)\right)\right\|+\left\|F_{i}\left(w^{k}_{n}(0)\right)\right\|.$ First, by (5.2) and interpolation, we have that $\displaystyle\lim_{k\rightarrow K}\varlimsup_{n\rightarrow+\infty}\big{\\|}w^{k}_{n}(0)\big{\\|}_{L^{p}_{x}}=0,\quad\forall\;\;2<p\leq 6.$ which implies that $\displaystyle\lim_{k\rightarrow K}\varlimsup_{n\rightarrow+\infty}\left\|F_{i}\left(v_{n}(0)\right)-F_{i}\left(v^{<k}_{n}(0)\right)\right\|$ $\displaystyle=0,$ $\displaystyle\lim_{k\rightarrow K}\varlimsup_{n\rightarrow+\infty}\left\|F_{i}\left(w^{k}_{n}(0)\right)\right\|=0.$ Second by the dispersive estimate for $v^{j}_{n}(0)$ with $\tau^{j}_{n}\rightarrow\pm\infty$, we have $\displaystyle\lim_{k\rightarrow K}\varlimsup_{n\rightarrow+\infty}\left\|F_{i}(v^{<k}_{n}(0))-F_{i}\left(\sum_{j<k,\tau^{j}_{n}\rightarrow\tau^{j}_{\infty}}v^{j}_{n}(0)\right)\right\|$ $\displaystyle=0,$ $\displaystyle\lim_{k\rightarrow K}\varlimsup_{n\rightarrow+\infty}\left\|\sum_{j<k,\tau^{j}_{n}\rightarrow\pm\infty}F_{i}\left(v^{j}_{n}(0)\right)\right\|=0.$ Last we will use the approximation argument in [17] to show that every non- dispersive concentrating wave will get away from the others, which contributes to the orthogonality of (5.14). Let $\psi^{j}:=e^{i\tau^{j}_{\infty}\Delta}\varphi^{j}\in L^{2}$, we have $\displaystyle\left\|F_{i}\left(\sum_{j<k,\tau^{j}_{n}\rightarrow\tau^{j}_{\infty}}v^{j}_{n}(0)\right)-\sum_{j<k,\tau^{j}_{n}\rightarrow\tau^{j}_{\infty}}F_{i}\left(v^{j}_{n}(0)\right)\right\|$ (5.15) $\displaystyle\leq\left\|F_{i}\left(\sum_{j<k,\tau^{j}_{n}\rightarrow\tau^{j}_{\infty}}v^{j}_{n}(0)\right)-F_{i}\left(\sum_{j<k,\tau^{j}_{n}\rightarrow\tau^{j}_{\infty}}\left<\nabla\right>^{-1}T^{j}_{n}\psi^{j}\right)\right\|$ $\displaystyle\quad+\left\|\sum_{j<k,\tau^{j}_{n}\rightarrow\tau^{j}_{\infty}}F_{i}\left(v^{j}_{n}(0)\right)-\sum_{j<k,\tau^{j}_{n}\rightarrow\tau^{j}_{\infty}}F_{i}\left(\left<\nabla\right>^{-1}T^{j}_{n}\psi^{j})\right)\right\|$ $\displaystyle\quad+\left\|F_{i}\left(\sum_{j<k,\tau^{j}_{n}\rightarrow\tau^{j}_{\infty}}\left<\nabla\right>^{-1}T^{j}_{n}\psi^{j}\right)-\sum_{j<k,\tau^{j}_{n}\rightarrow\tau^{j}_{\infty}}F_{i}\left(\left<\nabla\right>^{-1}T^{j}_{n}\psi^{j})\right)\right\|.$ (5.16) For those $v^{j}_{n}(0)$ with $\tau^{j}_{n}\rightarrow\tau^{j}_{\infty}$, by the continuity of the operator $e^{it\Delta}$ in $t$ in $H^{1}$, we have $\displaystyle v^{j}_{n}(0)=$ $\displaystyle\left<\nabla\right>^{-1}e^{-it^{j}_{n}\Delta}T^{j}_{n}\varphi^{j}=\left<\nabla\right>^{-1}T^{j}_{n}e^{i\tau^{j}_{n}\Delta}\varphi^{j}\longrightarrow\left<\nabla\right>^{-1}T^{j}_{n}\psi^{j}\quad\text{in}\;\;H^{1}({\mathbb{R}}^{3}),$ which implies that $\displaystyle\left\|F_{i}\left(\sum_{j<k,\tau^{j}_{n}\rightarrow\tau^{j}_{\infty}}v^{j}_{n}(0)\right)-F_{i}\left(\sum_{j<k,\tau^{j}_{n}\rightarrow\tau^{j}_{\infty}}\left<\nabla\right>^{-1}T^{j}_{n}\psi^{j}\right)\right\|\rightarrow 0,$ $\displaystyle\left\|\sum_{j<k,\tau^{j}_{n}\rightarrow\tau^{j}_{\infty}}F_{i}\left(v^{j}_{n}(0)\right)-\sum_{j<k,\tau^{j}_{n}\rightarrow\tau^{j}_{\infty}}F_{i}\left(\left<\nabla\right>^{-1}T^{j}_{n}\psi^{j})\right)\right\|\rightarrow 0.$ Now we consider (5.16) for $i=1,2$, separately. First for $i=2$, we compute as following, $\displaystyle\left\|F_{2}\left(\sum_{j<k,\tau^{j}_{n}\rightarrow\tau^{j}_{\infty}}\left<\nabla\right>^{-1}T^{j}_{n}\psi^{j}\right)-\sum_{j<k,\tau^{j}_{n}\rightarrow\tau^{j}_{\infty}}F_{2}\left(\left<\nabla\right>^{-1}T^{j}_{n}\psi^{j})\right)\right\|$ $\displaystyle\leq\left\|F_{2}\left(\sum_{j<k,\tau^{j}_{n}\rightarrow\tau^{j}_{\infty}}\left<\nabla\right>^{-1}T^{j}_{n}\psi^{j}\right)-F_{2}\left(\sum_{j<k,\tau^{j}_{n}\rightarrow\tau^{j}_{\infty},h^{j}_{n}=1}\left<\nabla\right>^{-1}T^{j}_{n}\psi^{j}\right)\right\|$ $\displaystyle\quad+\left\|\sum_{j<k,\tau^{j}_{n}\rightarrow\tau^{j}_{\infty}}F_{2}\left(\left<\nabla\right>^{-1}T^{j}_{n}\psi^{j})\right)-\sum_{j<k,\tau^{j}_{n}\rightarrow\tau^{j}_{\infty},h^{j}_{n}=1}F_{2}\left(\left<\nabla\right>^{-1}T^{j}_{n}\psi^{j})\right)\right\|$ $\displaystyle\quad+\left\|F_{2}\left(\sum_{j<k,\tau^{j}_{n}\rightarrow\tau^{j}_{\infty},h^{j}_{n}=1}\left<\nabla\right>^{-1}T^{j}_{n}\psi^{j}\right)-\sum_{j<k,\tau^{j}_{n}\rightarrow\tau^{j}_{\infty},h^{j}_{n}=1}F_{2}\left(\left<\nabla\right>^{-1}T^{j}_{n}\psi^{j})\right)\right\|.$ For $h^{j}_{n}\rightarrow 0$, we have $\displaystyle\left<\nabla\right>^{-1}T^{j}_{n}\psi^{j}\rightarrow 0\quad\text{in}\;\;L^{p},\;\;\forall\;2\leq p<6,$ which implies that $\displaystyle\left\|F_{2}\left(\sum_{j<k,\tau^{j}_{n}\rightarrow\tau^{j}}\left<\nabla\right>^{-1}T^{j}_{n}\psi^{j}\right)-F_{2}\left(\sum_{j<k,\tau^{j}_{n}\rightarrow\tau^{j},h^{j}_{n}=1}\left<\nabla\right>^{-1}T^{j}_{n}\psi^{j}\right)\right\|\rightarrow 0,$ $\displaystyle\left\|\sum_{j<k,\tau^{j}n_{n}\rightarrow\tau^{j}}F_{2}\left(\left<\nabla\right>^{-1}T^{j}_{n}\psi^{j})\right)-\sum_{j<k,\tau^{j}_{n}\rightarrow\tau^{j},h^{j}_{n}=1}F_{2}\left(\left<\nabla\right>^{-1}T^{j}_{n}\psi^{j})\right)\right\|\rightarrow 0.$ In addition, by the orthogonality (5.3), we know that there is at most one term $\left<\nabla\right>^{-1}T^{j}_{n}\psi^{j}$ with $\tau^{j}_{n}\rightarrow\tau^{j}_{\infty},h^{j}_{n}=1$, hence $\displaystyle\left\|F_{2}\left(\sum_{j<k,\tau^{j}_{n}\rightarrow\tau^{j}_{\infty},h^{j}_{n}=1}\left<\nabla\right>^{-1}T^{j}_{n}\psi^{j}\right)-\sum_{j<k,\tau^{j}_{n}\rightarrow\tau^{j}_{\infty},h^{j}_{n}=1}F_{2}\left(\left<\nabla\right>^{-1}T^{j}_{n}\psi^{j})\right)\right\|=0.$ Now we consider the case $i=1$, Let $\widehat{\psi}^{j}=\|\nabla\|^{-1}\psi^{j}$ if $h^{j}_{n}\rightarrow 0$, and $\widehat{\psi}^{j}=\left<\nabla\right>^{-1}\psi^{j}$ if $h^{j}_{n}\equiv 1$, then we have $\widehat{\psi}^{j}\in L^{6}_{x}$, and $\displaystyle\left\|F_{1}\left(\sum_{j<k,\tau^{j}_{n}\rightarrow\tau^{j}_{\infty}}\left<\nabla\right>^{-1}T^{j}_{n}\psi^{j}\right)-\sum_{j<k,\tau^{j}_{n}\rightarrow\tau^{j}_{\infty}}F_{1}\left(\left<\nabla\right>^{-1}T^{j}_{n}\psi^{j})\right)\right\|$ $\displaystyle\leq\left\|F_{1}\left(\sum_{j<k,\tau^{j}_{n}\rightarrow\tau^{j}_{\infty}}\left<\nabla\right>^{-1}T^{j}_{n}\psi^{j}\right)-F_{1}\left(\sum_{j<k,\tau^{j}_{n}\rightarrow\tau^{j}_{\infty}}h^{j}_{n}T^{j}_{n}\widehat{\psi}^{j}\right)\right\|$ $\displaystyle\quad+\left\|\sum_{j<k,\tau^{j}_{n}\rightarrow\tau^{j}_{\infty}}F_{1}\left(\left<\nabla\right>^{-1}T^{j}_{n}\psi^{j})\right)-\sum_{j<k,\tau^{j}_{n}\rightarrow\tau^{j}_{\infty}}F_{1}\left(h^{j}_{n}T^{j}_{n}\widehat{\psi}^{j}\right)\right\|$ $\displaystyle\quad+\left\|F_{1}\left(\sum_{j<k,\tau^{j}_{n}\rightarrow\tau^{j}_{\infty}}h^{j}_{n}T^{j}_{n}\widehat{\psi}^{j}\right)-\sum_{j<k,\tau^{j}_{n}\rightarrow\tau^{j}_{\infty}}F_{1}\left(h^{j}_{n}T^{j}_{n}\widehat{\psi}^{j}\right)\right\|.$ Since $\displaystyle\big{\\|}\left<\nabla\right>^{-1}T^{j}_{n}\psi^{j}-h^{j}_{n}T^{j}_{n}\widehat{\psi}^{j}\big{\\|}_{L^{6}_{x}}=$ $\displaystyle\begin{cases}\big{\\|}\left<\nabla\right>^{-1}T^{j}_{n}\psi^{j}-h^{j}_{n}T^{j}_{n}\|\nabla\|^{-1}\psi^{j}\big{\\|}_{L^{6}_{x}}\quad\text{if}\;\;h^{j}_{n}\rightarrow 0\\\ \big{\\|}\left<\nabla\right>^{-1}T^{j}_{n}\psi^{j}-h^{j}_{n}T^{j}_{n}\left<\nabla\right>^{-1}\psi^{j}\big{\\|}_{L^{6}_{x}}\quad\text{if}\;\;h^{j}_{n}\equiv 1\end{cases}$ $\displaystyle=$ $\displaystyle\begin{cases}\big{\\|}(\left<\nabla\right>^{j}_{n})^{-1}\psi^{j}-\|\nabla\|^{-1}\psi^{j}\big{\\|}_{L^{6}_{x}}\quad\text{if}\;\;h^{j}_{n}\rightarrow 0\\\ 0\quad\text{if}\;\;h^{j}_{n}\equiv 1\end{cases}$ $\displaystyle\longrightarrow 0,\quad\text{as}\;\;n\rightarrow+\infty,$ which shows that $\displaystyle\left\|F_{1}\left(\sum_{j<k,\tau^{j}_{n}\rightarrow\tau^{j}_{\infty}}\left<\nabla\right>^{-1}T^{j}_{n}\psi^{j}\right)-F_{1}\left(\sum_{j<k,\tau^{j}_{n}\rightarrow\tau^{j}_{\infty}}h^{j}_{n}T^{j}_{n}\widehat{\psi}^{j}\right)\right\|\longrightarrow 0,$ $\displaystyle\left\|\sum_{j<k,\tau^{j}_{n}\rightarrow\tau^{j}_{\infty}}F_{1}\left(\left<\nabla\right>^{-1}T^{j}_{n}\psi^{j})\right)-\sum_{j<k,\tau^{j}_{n}\rightarrow\tau^{j}_{\infty}}F_{1}\left(h^{j}_{n}T^{j}_{n}\widehat{\psi}^{j}\right)\right\|\longrightarrow 0.$ We further replace each $\widehat{\psi}^{j}$ by the non-overlap terms $\widetilde{\psi}^{j}_{n}$ with each other $\displaystyle\widetilde{\psi}^{j}_{n}=\widehat{\psi}^{j}\times\begin{cases}0;\quad\exists\;l<j,\;\text{such that}\;h^{l}_{n}<h^{j}_{n}\;\;\text{and}\;\;\frac{x}{h^{j,l}_{n}}\in\text{supp}\;{\widehat{\psi}^{l}},\\\ 1;\quad\text{otherwise},\end{cases}$ where $h^{j,l}_{n}$ is determined by (5.13). By (5.3), we know that $h^{j,l}_{n}\rightarrow 0$, therefore as $n\rightarrow+\infty$ $\displaystyle\widetilde{\psi}^{j}_{n}\rightarrow\widehat{\psi}^{j},$ $\displaystyle\quad a.e.\;x\in{\mathbb{R}}^{3},\quad\text{and}\quad\widetilde{\psi}^{j}_{n}\rightarrow\widehat{\psi}^{j},\quad\text{in}\;\;L^{6}_{x},$ which implies that $\displaystyle\left\|F_{1}\left(\sum_{j<k,\tau^{j}_{n}\rightarrow\tau^{j}_{\infty}}h^{j}_{n}T^{j}_{n}\widehat{\psi}^{j}\right)-F_{1}\left(\sum_{j<k,\tau^{j}_{n}\rightarrow\tau^{j}_{\infty}}h^{j}_{n}T^{j}_{n}\widetilde{\psi}^{j}_{n}\right)\right\|\longrightarrow 0,$ $\displaystyle\left\|\sum_{j<k,\tau^{j}_{n}\rightarrow\tau^{j}_{\infty}}F_{1}\left(h^{j}_{n}T^{j}_{n}\widehat{\psi}^{j}\right)-\sum_{j<k,\tau^{j}_{n}\rightarrow\tau^{j}_{\infty}}F_{1}\left(h^{j}_{n}T^{j}_{n}\widetilde{\psi}^{j}_{n}\right)\right\|\longrightarrow 0.$ On the other hand, by the support property of $\widetilde{\psi}^{j}_{n}$, we know that $\displaystyle F_{1}\left(\sum_{j<k,\tau^{j}_{n}\rightarrow\tau^{j}_{\infty}}h^{j}_{n}T^{j}_{n}\widetilde{\psi}^{j}_{n}\right)=\sum_{j<k,\tau^{j}_{n}\rightarrow\tau^{j}_{\infty}}F_{1}\left(h^{j}_{n}T^{j}_{n}\widetilde{\psi}^{j}_{n}\right).$ Therefore, we have $\displaystyle\left\|F_{1}\left(\sum_{j<k,\tau^{j}_{n}\rightarrow\tau^{j}_{\infty}}h^{j}_{n}T^{j}_{n}\widehat{\psi}^{j}\right)-\sum_{j<k,\tau^{j}_{n}\rightarrow\tau^{j}_{\infty}}F_{1}\left(h^{j}_{n}T^{j}_{n}\widehat{\psi}^{j}\right)\right\|$ $\displaystyle\leq\left\|F_{1}\left(\sum_{j<k,\tau^{j}_{n}\rightarrow\tau^{j}_{\infty}}h^{j}_{n}T^{j}_{n}\widehat{\psi}^{j}\right)-F_{1}\left(\sum_{j<k,\tau^{j}_{n}\rightarrow\tau^{j}_{\infty}}h^{j}_{n}T^{j}_{n}\widetilde{\psi}^{j}_{n}\right)\right\|$ $\displaystyle\quad+\left\|\sum_{j<k,\tau^{j}_{n}\rightarrow\tau^{j}_{\infty}}F_{1}\left(h^{j}_{n}T^{j}_{n}\widehat{\psi}^{j}\right)-\sum_{j<k,\tau^{j}_{n}\rightarrow\tau^{j}_{\infty}}F_{1}\left(h^{j}_{n}T^{j}_{n}\widetilde{\psi}^{j}_{n}\right)\right\|\longrightarrow 0.$ This completes the proof. ∎ ###### Lemma 5.4. Let $k\in{\mathbb{N}}$ and radial functions $\varphi_{0},\ldots,\varphi_{k}\in H^{1}({\mathbb{R}}^{3})$, $m$ be determined by (1.5). Assume that there exist some $\delta$, $\varepsilon>0$ with $4\varepsilon<3\delta$ such that $\displaystyle\sum^{k}_{j=0}E(\varphi_{j})-\varepsilon\leq E\left(\sum^{k}_{j=0}\varphi_{j}\right)<m-\delta,\;\;\text{and}\;\;-\varepsilon\leq K\left(\sum^{k}_{j=0}\varphi_{j}\right)\leq\sum^{k}_{j=0}K(\varphi_{j})+\varepsilon.$ Then $\varphi_{j}\in\mathcal{K}^{+}$ for all $j=0,\ldots,k$. ###### Proof. Suppose that $K(\varphi_{l})<0$ for some $l$. Then by Lemma 2.9, we have $\displaystyle H(\varphi_{l})\geq\inf\left\\{H(\varphi)\;\|\;\varphi\in H^{1}({\mathbb{R}}^{3}),\;\varphi\not=0,\;K(\varphi)\leq 0\right\\}=m.$ By the nonnegativity of $H(\varphi_{j})$ for $j\geq 0$, we have $\displaystyle m\leq$ $\displaystyle H(\varphi_{l})\leq\sum^{k}_{j=0}H(\varphi_{j})=\sum^{k}_{j=0}\left(E(\varphi_{j})-\frac{1}{6}K(\varphi_{j})\right)$ $\displaystyle\leq$ $\displaystyle E\left(\sum^{k}_{j=0}\varphi_{j}\right)+\varepsilon-\frac{1}{6}K\left(\sum^{k}_{j=0}\varphi_{j}\right)+\frac{1}{6}\varepsilon$ $\displaystyle\leq$ $\displaystyle m-\delta+\varepsilon+\frac{1}{3}\varepsilon<m.$ It is a contradiction. Hence for any $j\in\\{0,\ldots,k\\}$, we have $\displaystyle K(\varphi_{j})\geq 0,$ which implies that $\displaystyle E(\varphi_{j})=H(\varphi_{j})+\frac{1}{6}K(\varphi_{j})\geq 0,$ and $\displaystyle E(\varphi_{j})\leq\sum^{k}_{i=0}E(\varphi_{i})<m-\delta+\varepsilon<m,$ which means that $\varphi_{j}\in\mathcal{K}^{+}$ for all $j$. ∎ According to the above results, we conclude as following. ###### Proposition 5.5. Let $\overrightarrow{v}_{n}(t,x)$ be a sequence of the radial solutions of the free Schrödinger equation satisfying $\displaystyle v_{n}(0)\in\mathcal{K}^{+}\;\;\text{and}\;\;E(v_{n}(0))<m.$ Let $\displaystyle\overrightarrow{v}_{n}(t,x)=\sum^{k-1}_{j=0}\overrightarrow{v}^{j}_{n}(t,x)+\overrightarrow{w}^{k}_{n}(t,x),$ be the linear profile decomposition given by Proposition 5.1. Then for large $n$ and all $j<K$, we have $\displaystyle v^{j}_{n}(0)\in\mathcal{K}^{+},\;\;\;\;w^{K}_{n}(0)\in\mathcal{K}^{+},$ and $\displaystyle\lim_{k\rightarrow K}\varlimsup_{n\rightarrow+\infty}\left\|M(v_{n}(0))-\sum_{j<k}M(v^{j}_{n}(0))-M(w^{k}_{n}(0))\right\|=0,$ $\displaystyle\lim_{k\rightarrow K}\varlimsup_{n\rightarrow+\infty}\left\|E(v_{n}(0))-\sum_{j<k}E(v^{j}_{n}(0))-E(w^{k}_{n}(0))\right\|=0,$ $\displaystyle\lim_{k\rightarrow K}\varlimsup_{n\rightarrow+\infty}\left\|K(v_{n}(0))-\sum_{j<k}K(v^{j}_{n}(0))-K(w^{k}_{n}(0))\right\|=0.$ Moreover for all $j<K$, we have $\displaystyle 0\leq\varliminf_{n\rightarrow+\infty}E(v^{j}_{n}(0))\leq\varlimsup_{n\rightarrow+\infty}E(v^{j}_{n}(0))\leq\varlimsup_{n\rightarrow+\infty}E(v_{n}(0)),$ where the last inequality becomes equality only if $K=1$ and $w^{1}_{n}\rightarrow 0$ in $L^{\infty}_{t}\dot{H}^{1}_{x}$. ### 5.2. Nonlinear profile decomposition After the linear profile decomposition of a sequence of initial data in the last subsection, we now show the nonlinear profile decomposition of a sequence of radial solutions of (1.1) with the same initial data in the energy space $H^{1}({\mathbb{R}}^{3})$. First we introduce some notation $\displaystyle\left<\nabla\right>^{j}_{n}=\sqrt{\left(h^{j}_{n}\right)^{2}-\Delta},\;\;\left<\nabla\right>^{j}_{\infty}=\sqrt{\left(h^{j}_{\infty}\right)^{2}-\Delta}\;.$ Now let $v_{n}(t,x)$ be a sequence of radial solutions for the free Schrödinger equation with initial data in $\mathcal{K}^{+}$, that is, $v_{n}\in H^{1}({\mathbb{R}}^{3})$ is radial and $\displaystyle\left(i\partial_{t}+\Delta\right)v_{n}=0,\quad v_{n}(0)\in\mathcal{K}^{+}.$ Let $\displaystyle\overrightarrow{v}_{n}(t,x)=\left<\nabla\right>v_{n}(t,x),$ then by Proposition 5.1, we have a sequence of the radial, free concentrating wave $\overrightarrow{v}^{j}_{n}(t,x)$ with $\overrightarrow{v}^{j}_{n}(t^{j}_{n})=T^{j}_{n}\varphi^{j}$, $v^{j}_{n}(0)\in\mathcal{K}^{+}$ for $j=0,\ldots,K$, such that $\displaystyle\overrightarrow{v}_{n}(t,x)=$ $\displaystyle\sum^{k-1}_{j=0}\overrightarrow{v}^{j}_{n}(t,x)+\overrightarrow{w}^{k}_{n}(t,x)=\sum^{k-1}_{j=0}e^{i(t-t^{j}_{n})\Delta}T^{j}_{n}\varphi^{j}+\overrightarrow{w}^{k}_{n}$ $\displaystyle=$ $\displaystyle\sum^{k-1}_{j=0}T^{j}_{n}e^{i\left(\frac{t-t^{j}_{n}}{(h^{j}_{n})^{2}}\right)\Delta}\varphi^{j}+\overrightarrow{w}^{k}_{n}.$ Now for any concentrating wave $\overrightarrow{v}^{j}_{n}$, $j=0,\ldots,K$, we undo the group action, i.e., the scaling transformation $T^{j}_{n}$, to look for the linear profile $V^{j}$. Let $\displaystyle\overrightarrow{v}^{j}_{n}(t,x)=$ $\displaystyle T^{j}_{n}\overrightarrow{V}^{j}\left(\frac{t-t^{j}_{n}}{(h^{j}_{n})^{2}}\right),$ then we have $\displaystyle\left(i\partial_{t}+\Delta\right)\overrightarrow{V}^{j}=0,\quad\overrightarrow{V}^{j}(0)=\varphi^{j}.$ Now let $u^{j}_{n}(t,x)$ be the nonlinear solution of (1.1) with initial data $v^{j}_{n}(0)$, that is $\displaystyle\left(i\partial_{t}+\Delta\right)\overrightarrow{u}^{j}_{n}(t,x)=$ $\displaystyle\left<\nabla\right>f_{1}(\left<\nabla\right>^{-1}\overrightarrow{u}^{j}_{n})+\left<\nabla\right>f_{2}(\left<\nabla\right>^{-1}\overrightarrow{u}^{j}_{n}),$ $\displaystyle\quad\overrightarrow{u}^{j}_{n}(0)=$ $\displaystyle\overrightarrow{v}^{j}_{n}(0)=T^{j}_{n}\overrightarrow{V}^{j}(\tau^{j}_{n}),\quad u^{j}_{n}(0)\in\mathcal{K}^{+},$ where $\tau^{j}_{n}=-t^{j}_{n}/(h^{j}_{n})^{2}$. In order to look for the nonlinear profile $\overrightarrow{U}^{j}_{\infty}$ associated to the radial, free concentrating wave $\left(\overrightarrow{v}^{j}_{n};\;h^{j}_{n},t^{j}_{n}\right)$, we also need undo the group action. We denote $\displaystyle\overrightarrow{u}^{j}_{n}(t,x)=$ $\displaystyle T^{j}_{n}\overrightarrow{U}^{j}_{n}\left(\frac{t-t^{j}_{n}}{(h^{j}_{n})^{2}}\right),$ then we have $\displaystyle\left(i\partial_{t}+\Delta\right)\overrightarrow{U}^{j}_{n}=$ $\displaystyle\left(\left<\nabla\right>^{j}_{n}\right)f_{1}\left(\left(\left<\nabla\right>^{j}_{n}\right)^{-1}\overrightarrow{U}^{j}_{n}\right)+h^{j}_{n}\cdot\left(\left<\nabla\right>^{j}_{n}\right)f_{2}\left(\left(\left<\nabla\right>^{j}_{n}\right)^{-1}\overrightarrow{U}^{j}_{n}\right),$ $\displaystyle\overrightarrow{U}^{j}_{n}(\tau^{j}_{n})=$ $\displaystyle\overrightarrow{V}^{j}(\tau^{j}_{n}).$ Up to a subsequence, we may assume that there exist $h^{j}_{\infty}\in\\{0,1\\}$ and $\tau^{j}_{\infty}\in[-\infty,\infty]$ for every $j=\\{0,\ldots,K\\}$, such that $\displaystyle h^{j}_{n}\rightarrow\;h^{j}_{\infty},\;\;\text{and}\;\;\tau^{j}_{n}\rightarrow\;\tau^{j}_{\infty}.$ As $n\rightarrow+\infty$, the limit equation of $\overrightarrow{U}^{j}_{n}$ is given by $\displaystyle\left(i\partial_{t}+\Delta\right)\overrightarrow{U}^{j}_{\infty}=$ $\displaystyle\left(\left<\nabla\right>^{j}_{\infty}\right)f_{1}\left(\left(\left<\nabla\right>^{j}_{\infty}\right)^{-1}\overrightarrow{U}^{j}_{\infty}\right)+h^{j}_{\infty}\cdot\left(\left<\nabla\right>^{j}_{\infty}\right)f_{2}\left(\left(\left<\nabla\right>^{j}_{\infty}\right)^{-1}\overrightarrow{U}^{j}_{\infty}\right),$ $\displaystyle\overrightarrow{U}^{j}_{\infty}(\tau^{j}_{\infty})=$ $\displaystyle\overrightarrow{V}^{j}(\tau^{j}_{\infty})\in L^{2}({\mathbb{R}}^{3}).$ Let $\displaystyle\widehat{U}^{j}_{\infty}:=\left(\left<\nabla\right>^{j}_{\infty}\right)^{-1}\overrightarrow{U}^{j}_{\infty},$ then $\displaystyle\left(i\partial_{t}+\Delta\right)\widehat{U}^{j}_{\infty}=$ $\displaystyle f_{1}\left(\widehat{U}^{j}_{\infty}\right)+h^{j}_{\infty}\cdot f_{2}\left(\widehat{U}^{j}_{\infty}\right),$ (5.17) $\displaystyle\widehat{U}^{j}_{\infty}(\tau^{j}_{\infty})=$ $\displaystyle\left(\left<\nabla\right>^{j}_{\infty}\right)^{-1}\overrightarrow{V}^{j}(\tau^{j}_{\infty}).$ (5.18) The unique existence of a local radial solution $\overrightarrow{U}^{j}_{\infty}$ around $\tau^{j}_{\infty}$ is known in all cases, including $h^{j}_{\infty}=0$ and $\tau^{j}_{\infty}=\pm\infty$. $\overrightarrow{U}^{j}_{\infty}$ on the maximal existence interval is called the nonlinear profile associated with the radial, free concentrating wave $\left(\overrightarrow{v}^{j}_{n};\;h^{j}_{n},t^{j}_{n}\right)$. The nonlinear concentrating wave $u^{j}_{(n)}$ associated with $\left(\overrightarrow{v}^{j}_{n};\;h^{j}_{n},t^{j}_{n}\right)$ is defined by $\displaystyle\overrightarrow{u}^{j}_{(n)}(t,x)=T^{j}_{n}\overrightarrow{U}^{j}_{\infty}\left(\frac{t-t^{j}_{n}}{(h^{j}_{n})^{2}}\right),$ then we have $\displaystyle\left(i\partial_{t}+\Delta\right)\overrightarrow{u}^{j}_{(n)}=$ $\displaystyle\left(\sqrt{\|\nabla\|^{2}+\left(\frac{h^{j}_{\infty}}{h^{j}_{n}}\right)^{2}}\right)f_{1}\left(\left(\sqrt{\|\nabla\|^{2}+\left(\frac{h^{j}_{\infty}}{h^{j}_{n}}\right)^{2}}\right)^{-1}\overrightarrow{u}^{j}_{(n)}\right)$ $\displaystyle+\frac{h^{j}_{\infty}}{h^{j}_{n}}\cdot\left(\sqrt{\|\nabla\|^{2}+\left(\frac{h^{j}_{\infty}}{h^{j}_{n}}\right)^{2}}\right)f_{2}\left(\left(\sqrt{\|\nabla\|^{2}+\left(\frac{h^{j}_{\infty}}{h^{j}_{n}}\right)^{2}}\right)^{-1}\overrightarrow{u}^{j}_{(n)}\right)$ $\displaystyle=$ $\displaystyle\left<\nabla\right>^{j}_{\infty}f_{1}\left(\left(\left<\nabla\right>^{j}_{\infty}\right)^{-1}\overrightarrow{u}^{j}_{(n)}\right)+h^{j}_{\infty}\cdot\left<\nabla\right>^{j}_{\infty}f_{2}\left(\left(\left<\nabla\right>^{j}_{\infty}\right)^{-1}\overrightarrow{u}^{j}_{(n)}\right),$ $\displaystyle\overrightarrow{u}^{j}_{(n)}(0)=$ $\displaystyle T^{j}_{n}\overrightarrow{U}^{j}_{\infty}(\tau^{j}_{n}),$ which implies that $\displaystyle\big{\\|}\overrightarrow{u}^{j}_{(n)}(0)-\overrightarrow{u}^{j}_{n}(0)\big{\\|}_{L^{2}}=$ $\displaystyle\big{\\|}T^{j}_{n}\overrightarrow{U}^{j}_{\infty}(\tau^{j}_{n})-T^{j}_{n}\overrightarrow{V}^{j}(\tau^{j}_{n})\big{\\|}_{L^{2}}=\big{\\|}\overrightarrow{U}^{j}_{\infty}(\tau^{j}_{n})-\overrightarrow{V}^{j}(\tau^{j}_{n})\big{\\|}_{L^{2}}$ $\displaystyle\leq$ $\displaystyle\big{\\|}\overrightarrow{U}^{j}_{\infty}(\tau^{j}_{n})-\overrightarrow{U}^{j}_{\infty}(\tau^{j}_{\infty})\big{\\|}_{L^{2}}+\big{\\|}\overrightarrow{V}^{j}(\tau^{j}_{n})-\overrightarrow{V}^{j}(\tau^{j}_{\infty})\big{\\|}_{L^{2}}\rightarrow 0.$ We denote $\displaystyle\overrightarrow{u}^{j}_{(n)}=\left<\nabla\right>u^{j}_{(n)}.$ If $h^{j}_{\infty}=1$, we have $h^{j}_{n}\equiv 1$, then $u^{j}_{(n)}\in H^{1}({\mathbb{R}}^{3})$ is radial and satisfies $\displaystyle\left(i\partial_{t}+\Delta\right)u^{j}_{(n)}=f_{1}(u^{j}_{(n)})+f_{2}(u^{j}_{(n)}).$ If $h^{j}_{\infty}=0$, then $u^{j}_{(n)}\in H^{1}({\mathbb{R}}^{3})$ is radial and satisfies $\displaystyle\left(i\partial_{t}+\Delta\right)u^{j}_{(n)}=\frac{\|\nabla\|}{\left<\nabla\right>}f_{1}\left(\frac{\left<\nabla\right>}{\|\nabla\|}u^{j}_{(n)}\right).$ Let $u_{n}$ be a sequence of (local) radial solutions of (1.1) with initial data in $\mathcal{K}^{+}$ at $t=0$, and let $v_{n}$ be the sequence of the radial, free solutions with the same initial data. We consider the linear profile decomposition given by Proposition 5.1 $\displaystyle\overrightarrow{v}_{n}(t,x)=\sum^{k-1}_{j=0}\overrightarrow{v}^{j}_{n}(t,x)+\overrightarrow{w}^{k}_{n}(t,x),\quad\overrightarrow{v}^{j}_{n}(t^{j}_{n})=T^{j}_{n}\varphi^{j},\quad v^{j}_{n}(0)\in\mathcal{K}^{+}.$ With each free concentrating wave $\\{\overrightarrow{v}^{j}_{n}\\}_{n\in{\mathbb{N}}}$, we associate the nonlinear concentrating wave $\\{\overrightarrow{u}^{j}_{(n)}\\}_{n\in{\mathbb{N}}}$. A nonlinear profile decomposition of $u_{n}$ is given by $\displaystyle\overrightarrow{u}^{<k}_{(n)}(t,x):=\sum^{k-1}_{j=0}\overrightarrow{u}^{j}_{(n)}(t,x)=\sum^{k-1}_{j=0}T^{j}_{n}\overrightarrow{U}^{j}_{\infty}\left(\frac{t-t^{j}_{n}}{(h^{j}_{n})^{2}}\right).$ (5.19) Since the smallness condition (5.2) and the orthogonality condition (5.3) ensure that every nonlinear concentrating wave and the remainder interacts weakly with the others, we will show that $\overrightarrow{u}^{<k}_{(n)}+\overrightarrow{w}^{k}_{n}$ is a good approximation for $\overrightarrow{u}_{n}$ provided that each nonlinear profile has the finite global Strichartz norm. Now we define the Strichartz norms for the nonlinear profile decomposition. Let $ST(I)$ and $ST^{}(I)$ be the function spaces on $I\times{\mathbb{R}}^{3}$ defined as Section 4 $\displaystyle ST(I):=\left(L^{10}_{t}\dot{B}^{1/3}_{90/19,2}\cap L^{12}_{t}L^{9}_{x}\cap L^{6}_{t}\dot{B}^{1/2}_{18/7,2}\cap L^{5}_{t,x}\right)$ $\displaystyle(I\times{\mathbb{R}}^{3}),$ $\displaystyle ST^{}(I):=\left(L^{2}_{t}\dot{B}^{1/3}_{18/11,2}\cap L^{2}_{t}\dot{B}^{1/2}_{6/5,2}\right)(I\times{\mathbb{R}}^{3})$ . The Strichartz norm for the nonlinear profile $\widehat{U}^{j}_{\infty}$ depends on the scaling $h^{j}_{\infty}$. $\displaystyle ST^{j}_{\infty}(I):=\begin{cases}ST(I),\quad&\text{for}\;h^{j}_{\infty}=1,\\\ \left(L^{10}_{t}\dot{B}^{1/3}_{90/19,2}\cap L^{12}_{t}L^{9}_{x}\right)(I\times{\mathbb{R}}^{3}),\quad&\text{for}\;h^{j}_{\infty}=0.\end{cases}$ ###### Lemma 5.6. In the nonlinear profile decomposition (5.19). Suppose that for each $j<K$, we have $\displaystyle\big{\\|}\widehat{U}^{j}_{\infty}\big{\\|}_{ST^{j}_{\infty}({\mathbb{R}})}+\big{\\|}\overrightarrow{U}^{j}_{\infty}\big{\\|}_{L^{\infty}_{t}L^{2}_{x}({\mathbb{R}}^{3})}<\infty.$ Then for any finite interval $I$, any $j<K$ and any $k\leq K$, we have $\displaystyle\varlimsup_{n\rightarrow+\infty}\big{\\|}u^{j}_{(n)}\big{\\|}_{ST(I)}\lesssim$ $\displaystyle\big{\\|}\widehat{U}^{j}_{\infty}\big{\\|}_{ST^{j}_{\infty}({\mathbb{R}})},$ (5.20) $\displaystyle\varlimsup_{n\rightarrow+\infty}\big{\\|}u^{<k}_{(n)}\big{\\|}^{2}_{ST(I)}\lesssim$ $\displaystyle\varlimsup_{n\rightarrow+\infty}\sum_{j<k}\big{\\|}u^{j}_{(n)}\big{\\|}^{2}_{ST(I)},$ (5.21) where the implicit constants do not depend on $I,j$ or $k$. We also have $\displaystyle\lim_{n\rightarrow+\infty}\left\\|f_{1}\left(u^{<k}_{(n)}\right)-\sum_{j<k}\frac{\left<\nabla\right>^{j}_{\infty}}{\left<\nabla\right>}f_{1}\left(\frac{\left<\nabla\right>}{\left<\nabla\right>^{j}_{\infty}}u^{j}_{(n)}\right)\right\\|_{ST^{}(I)}=0,$ (5.22) $\displaystyle\lim_{n\rightarrow+\infty}\left\\|f_{2}\left(u^{<k}_{(n)}\right)-\sum_{j<k}h^{j}_{\infty}\frac{\left<\nabla\right>^{j}_{\infty}}{\left<\nabla\right>}f_{2}\left(\frac{\left<\nabla\right>}{\left<\nabla\right>^{j}_{\infty}}u^{j}_{(n)}\right)\right\\|_{ST^{}(I)}=0.$ (5.23) ###### Proof. Proof of (5.20). By the definitions of $u^{j}_{(n)}$ and $\widehat{U}^{j}_{\infty}$, we know that $\displaystyle u^{j}_{(n)}(t,x)=$ $\displaystyle\left<\nabla\right>^{-1}\overrightarrow{u}^{j}_{(n)}(t,x)=\left<\nabla\right>^{-1}T^{j}_{n}\overrightarrow{U}^{j}_{\infty}\left(\frac{t-t^{j}_{n}}{(h^{j}_{n})^{2}}\right)$ $\displaystyle=$ $\displaystyle\left<\nabla\right>^{-1}T^{j}_{n}\left<\nabla\right>^{j}_{\infty}\widehat{U}^{j}_{\infty}\left(\frac{t-t^{j}_{n}}{(h^{j}_{n})^{2}}\right)=h^{j}_{n}T^{j}_{n}\frac{\left<\nabla\right>^{j}_{\infty}}{\left<\nabla\right>^{j}_{n}}\widehat{U}^{j}_{\infty}\left(\frac{t-t^{j}_{n}}{(h^{j}_{n})^{2}}\right).$ For the case $h^{j}_{\infty}=1$, we have $u^{j}_{(n)}(t,x)=\widehat{U}^{j}_{\infty}(t-t^{j}_{n},x)$, hence (5.20) is trivial. For the case $h^{j}_{\infty}=0$, by the above relation between $u^{j}_{(n)}$ and $\widehat{U}^{j}_{\infty}$, we have $\displaystyle\big{\\|}u^{j}_{(n)}\big{\\|}_{\left(L^{10}_{t}\dot{B}^{1/3}_{90/19,2}\cap L^{12}_{t}L^{9}_{x}\right)(I\times{\mathbb{R}}^{3})}\lesssim$ $\displaystyle\left\\|\frac{\|\nabla\|}{\left<\nabla\right>^{j}_{n}}\widehat{U}^{j}_{\infty}\right\\|_{\left(L^{10}_{t}\dot{B}^{1/3}_{90/19,2}\cap L^{12}_{t}L^{9}_{x}\right)({\mathbb{R}}\times{\mathbb{R}}^{3})}$ $\displaystyle\lesssim$ $\displaystyle\big{\\|}\widehat{U}^{j}_{\infty}\big{\\|}_{\left(L^{10}_{t}\dot{B}^{1/3}_{90/19,2}\cap L^{12}_{t}L^{9}_{x}\right)({\mathbb{R}}\times{\mathbb{R}}^{3})},$ and $\displaystyle\big{\\|}u^{j}_{(n)}\big{\\|}_{L^{6}_{t}\dot{B}^{1/2}_{18/7,2}(I\times{\mathbb{R}}^{3})}\lesssim$ $\displaystyle\|I\|^{\frac{1}{12}}\big{\\|}u^{j}_{(n)}\big{\\|}_{L^{12}_{t}\dot{B}^{1/2}_{18/7,2}}\lesssim\|I\|^{\frac{1}{12}}(h^{j}_{n})^{\frac{1}{3}}\big{\\|}\widehat{U}^{j}_{\infty}\big{\\|}_{L^{12}_{t}\dot{B}^{\frac{5}{6}}_{18/7,2}}\rightarrow 0,$ $\displaystyle\big{\\|}u^{j}_{(n)}\big{\\|}_{L^{5}_{t,x}(I\times{\mathbb{R}}^{3})}\lesssim$ $\displaystyle\|I\|^{\frac{7}{60}}\big{\\|}u^{j}_{(n)}\big{\\|}_{L^{12}_{t}L^{5}_{x}}\lesssim\|I\|^{\frac{7}{60}}(h^{j}_{n})^{\frac{4}{15}}\big{\\|}\widehat{U}^{j}_{\infty}\big{\\|}_{L^{12}_{t}\dot{B}^{\frac{4}{15}}_{5,2}}\rightarrow 0.$ where we use the fact that the boundedness of $\widehat{U}^{j}_{\infty}$ in $L^{10}_{t}\dot{B}^{1/3}_{90/19,2}\cap L^{12}_{t}L^{9}_{x}\cap L^{\infty}\dot{H}^{1}$ implies its boundedness in $L^{12}_{t}\dot{B}^{\frac{5}{6}}_{18/7,2}\cap L^{12}_{t}\dot{B}^{\frac{4}{15}}_{5,2}$ by (5.17). Proof of (5.21). We estimate the left hand side of (5.21) by $\displaystyle\big{\\|}u^{<k}_{(n)}\big{\\|}^{2}_{ST(I)}=$ $\displaystyle\left\\|\sum_{j<k,h^{j}_{\infty}=1}u^{j}_{(n)}+\sum_{j<k,h^{j}_{\infty}=0}u^{j}_{(n)}\right\\|^{2}_{ST(I)}$ $\displaystyle\lesssim$ $\displaystyle\left\\|\sum_{j<k,h^{j}_{\infty}=1}u^{j}_{(n)}\right\\|^{2}_{ST(I)}+\left\\|\sum_{j<k,h^{j}_{\infty}=0}u^{j}_{(n)}\right\\|^{2}_{ST(I)}.$ For the case $h^{j}_{\infty}=1$. Define $\widehat{U}^{j}_{\infty,R}$ and $u^{j}_{(n),R}$ by $\displaystyle\widehat{U}^{j}_{\infty,R}(t,x)=\chi_{R}(t,x)\widehat{U}^{j}_{\infty}(t,x),\quad u^{j}_{(n),R}(t,x)=T^{j}_{n}\widehat{U}^{j}_{\infty,R}(t-t^{j}_{n}),$ where $\chi_{R}$ is the cut-off function as in Remark 1.6. Then we have $\displaystyle\left\\|\sum_{j<k,h^{j}_{\infty}=1}u^{j}_{(n)}\right\\|^{2}_{ST(I)}\lesssim$ $\displaystyle\left\\|\sum_{j<k,h^{j}_{\infty}=1}u^{j}_{(n),R}\right\\|^{2}_{ST(I)}+\left\\|\sum_{j<k,h^{j}_{\infty}=1}u^{j}_{(n)}-\sum_{j<k,h^{j}_{\infty}=1}u^{j}_{(n),R}\right\\|^{2}_{ST(I)}.$ On one hand, we know that $\displaystyle\left\\|\sum_{j<k,h^{j}_{\infty}=1}u^{j}_{(n)}-\sum_{j<k,h^{j}_{\infty}=1}u^{j}_{(n),R}\right\\|_{ST(I)}\leq\sum_{j<k,h^{j}_{\infty}=1}\big{\\|}(1-\chi_{R})u^{j}_{(n)}\big{\\|}_{ST(I)}\rightarrow 0,$ as $R\rightarrow+\infty$. On the other hand, by (5.3) and the similar orthogonality analysis as in [17], we know that $\displaystyle\varlimsup_{n\rightarrow+\infty}\left\\|\sum_{j<k,h^{j}_{\infty}=1}u^{j}_{(n),R}\right\\|^{2}_{ST(I)}\lesssim\varlimsup_{n\rightarrow+\infty}\sum_{j<k,h^{j}_{\infty}=1}\left\\|u^{j}_{(n),R}\right\\|^{2}_{ST(I)}\lesssim\varlimsup_{n\rightarrow+\infty}\sum_{j<k,h^{j}_{\infty}=1}\left\\|u^{j}_{(n)}\right\\|^{2}_{ST(I)}.$ For the case $h^{j}_{\infty}=0$, On one hand, by $h^{j}_{n}\rightarrow 0$, we have $\displaystyle\varlimsup_{n\rightarrow+\infty}\left\\|\sum_{j<k,h^{j}_{\infty}=0}u^{j}_{(n)}\right\\|_{\left(L^{6}_{t}\dot{B}^{1/2}_{18/7,2}\cap L^{5}_{t,x}\right)(I\times{\mathbb{R}}^{3})}=0.$ On the other hand, by (5.3) and the analogue approximation analysis as in [17], we have $\displaystyle\varlimsup_{n\rightarrow+\infty}\left\\|\sum_{j<k,h^{j}_{\infty}=0}u^{j}_{(n)}\right\\|^{2}_{L^{10}_{t}\dot{B}^{1/3}_{90/19,2}(I\times{\mathbb{R}}^{3})}\lesssim$ $\displaystyle\varlimsup_{n\rightarrow+\infty}\sum_{j<k,h^{j}_{\infty}=0}\left\\|u^{j}_{(n)}\right\\|^{2}_{L^{10}_{t}\dot{B}^{1/3}_{90/19,2}(I\times{\mathbb{R}}^{3})},$ $\displaystyle\varlimsup_{n\rightarrow+\infty}\left\\|\sum_{j<k,h^{j}_{\infty}=0}u^{j}_{(n)}\right\\|^{2}_{L^{12}_{t}L^{9}_{x}(I\times{\mathbb{R}}^{3})}\lesssim$ $\displaystyle\varlimsup_{n\rightarrow+\infty}\sum_{j<k,h^{j}_{\infty}=0}\left\\|u^{j}_{(n)}\right\\|^{2}_{L^{12}_{t}L^{9}_{x}(I\times{\mathbb{R}}^{3})}.$ Proof of (5.22). Let $\displaystyle u^{<k}_{<n>}(t,x):=\sum_{j<k}u^{j}_{<n>}(t,x)$ where $\displaystyle u^{j}_{<n>}(t,x):=\frac{\left<\nabla\right>}{\left<\nabla\right>^{j}_{\infty}}u^{j}_{(n)}=\frac{1}{\left<\nabla\right>^{j}_{\infty}}\overrightarrow{u}^{j}_{(n)}=\frac{1}{\left<\nabla\right>^{j}_{\infty}}T^{j}_{n}\overrightarrow{U}^{j}_{\infty}\left(\frac{t-t^{j}_{n}}{(h^{j}_{n})^{2}}\right)=h^{j}_{n}T^{j}_{n}\widehat{U}^{j}_{\infty}\left(\frac{t-t^{j}_{n}}{(h^{j}_{n})^{2}}\right),$ and $\displaystyle u^{j}_{(n)}(t,x)=h^{j}_{n}T^{j}_{n}\frac{\left<\nabla\right>^{j}_{\infty}}{\left<\nabla\right>^{j}_{n}}\widehat{U}^{j}_{\infty}\left(\frac{t-t^{j}_{n}}{(h^{j}_{n})^{2}}\right).$ Then we have $\displaystyle\left\\|f_{1}\left(u^{<k}_{(n)}\right)-\sum_{j<k}\frac{\left<\nabla\right>^{j}_{\infty}}{\left<\nabla\right>}f_{1}\left(\frac{\left<\nabla\right>}{\left<\nabla\right>^{j}_{\infty}}u^{j}_{(n)}\right)\right\\|_{ST^{}}$ $\displaystyle\leq\left\\|f_{1}\left(u^{<k}_{(n)}\right)-f_{1}\left(u^{<k}_{\left<n\right>}\right)\right\\|_{ST^{}}+\left\\|f_{1}\left(u^{<k}_{\left<n\right>}\right)-\sum_{j<k}f_{1}\left(u^{j}_{\left<n\right>}\right)\right\\|_{ST^{}}$ $\displaystyle\quad+\left\\|\sum_{j<k}f_{1}\left(u^{j}_{\left<n\right>}\right)-\sum_{j<k}\frac{\left<\nabla\right>^{j}_{\infty}}{\left<\nabla\right>}f_{1}\left(u^{j}_{\left<n\right>}\right)\right\\|_{ST^{}}$ $\displaystyle\leq\left\\|f_{1}\left(u^{<k}_{(n)}\right)-f_{1}\left(u^{<k}_{\left<n\right>}\right)\right\\|_{ST^{}}+\left\\|f_{1}\left(u^{<k}_{\left<n\right>}\right)-\sum_{j<k}f_{1}\left(u^{j}_{\left<n\right>}\right)\right\\|_{ST^{}}$ $\displaystyle\quad+\left\\|\sum_{j<k,h^{j}_{\infty}=0}f_{1}\left(u^{j}_{\left<n\right>}\right)-\sum_{j<k,h^{j}_{\infty}=0}\frac{\|\nabla\|}{\left<\nabla\right>}f_{1}\left(u^{j}_{\left<n\right>}\right)\right\\|_{ST^{}}.$ By (5.3) and the approximation argument in [17], we have $\displaystyle\left\\|f_{1}\left(u^{<k}_{(n)}\right)-f_{1}\left(u^{<k}_{\left<n\right>}\right)\right\\|_{ST^{}}+\left\\|f_{1}\left(u^{<k}_{\left<n\right>}\right)-\sum_{j<k}f_{1}\left(u^{j}_{\left<n\right>}\right)\right\\|_{ST^{}}\longrightarrow 0$ as $n\rightarrow+\infty$. In addition, by $h^{j}_{n}\rightarrow 0$ as $n\rightarrow+\infty$, we have $\displaystyle\left\\|\sum_{j<k,h^{j}_{\infty}=0}\left(1-\frac{\|\nabla\|}{\left<\nabla\right>}\right)f_{1}\left(u^{j}_{\left<n\right>}\right)\right\\|_{L^{2}_{t}\dot{B}^{1/3}_{18/11,2}}=$ $\displaystyle\left\\|\left(1-\frac{\|\nabla\|}{\left<\nabla\right>^{j}_{n}}\right)\sum_{j<k,h^{j}_{\infty}=0}f_{1}\left(\widehat{U}^{j}_{\infty}\right)\right\\|_{L^{2}_{t}\dot{B}^{1/3}_{18/11,2}}\longrightarrow 0,$ $\displaystyle\left\\|\sum_{j<k,h^{j}_{\infty}=0}\left(1-\frac{\|\nabla\|}{\left<\nabla\right>}\right)f_{1}\left(u^{j}_{\left<n\right>}\right)\right\\|_{L^{2}_{t}\dot{B}^{1/2}_{6/5,2}}=$ $\displaystyle\left(h^{j}_{n}\right)^{1/2}\left\\|\left(1-\frac{\|\nabla\|}{\left<\nabla\right>^{j}_{n}}\right)\sum_{j<k,h^{j}_{\infty}=0}f_{1}\left(\widehat{U}^{j}_{\infty}\right)\right\\|_{L^{2}_{t}\dot{B}^{1/2}_{6/5,2}}\longrightarrow 0,$ as $n\rightarrow+\infty$. Therefore, we have $\displaystyle\lim_{n\rightarrow+\infty}\left\\|f_{1}\left(u^{<k}_{(n)}\right)-\sum_{j<k}\frac{\left<\nabla\right>^{j}_{\infty}}{\left<\nabla\right>}f_{1}\left(\frac{\left<\nabla\right>}{\left<\nabla\right>^{j}_{\infty}}u^{j}_{(n)}\right)\right\\|_{ST^{}}=0.$ Proof of (5.23). Note that $\displaystyle\left\\|f_{2}\left(u^{<k}_{(n)}\right)-\sum_{j<k}h^{j}_{\infty}\frac{\left<\nabla\right>^{j}_{\infty}}{\left<\nabla\right>}f_{2}\left(\frac{\left<\nabla\right>}{\left<\nabla\right>^{j}_{\infty}}u^{j}_{(n)}\right)\right\\|_{ST^{}}$ $\displaystyle\leq\left\\|f_{2}\left(u^{<k}_{(n)}\right)-f_{2}\left(u^{<k}_{\left<n\right>}\right)\right\\|_{ST^{}}+\left\\|f_{2}\left(u^{<k}_{\left<n\right>}\right)-\sum_{j<k}f_{2}\left(u^{j}_{\left<n\right>}\right)\right\\|_{ST^{}}+\left\\|\sum_{j<k,h^{j}_{\infty}=0}f_{2}\left(u^{j}_{\left<n\right>}\right)\right\\|_{ST^{}}.$ By the analogue analysis, we have $\displaystyle\left\\|f_{2}\left(u^{<k}_{(n)}\right)-f_{2}\left(u^{<k}_{\left<n\right>}\right)\right\\|_{ST^{}}+\left\\|f_{2}\left(u^{<k}_{\left<n\right>}\right)-\sum_{j<k}f_{2}\left(u^{j}_{\left<n\right>}\right)\right\\|_{ST^{}}\longrightarrow 0,$ and $\displaystyle\left\\|\sum_{j<k,h^{j}_{\infty}=0}f_{2}\left(u^{j}_{\left<n\right>}\right)\right\\|_{ST^{}}\longrightarrow 0$ as $n\rightarrow+\infty$. Hence, we obtain $\displaystyle\lim_{n\rightarrow+\infty}\left\\|f_{2}\left(u^{<k}_{(n)}\right)-\sum_{j<k}h^{j}_{\infty}\frac{\left<\nabla\right>^{j}_{\infty}}{\left<\nabla\right>}f_{2}\left(\frac{\left<\nabla\right>}{\left<\nabla\right>^{j}_{\infty}}u^{j}_{(n)}\right)\right\\|_{ST^{}}=0.$ These complete the proof. ∎ After this preliminaries, we now show that $\overrightarrow{u}^{<k}_{(n)}+\overrightarrow{w}^{k}_{n}$ is a good approximation for $\overrightarrow{u}_{n}$ provided that each nonlinear profile has finite global Strichartz norm. ###### Proposition 5.7. Let $u_{n}$ be a sequence of local, radial solutions of (1.1) around $t=0$ in $\mathcal{K}^{+}$ satisfying $\displaystyle M\left(u_{n}\right)<\infty,\quad\varlimsup_{n\rightarrow\infty}E(u_{n})<m.$ Suppose that in the nonlinear profile decomposition (5.19), every nonlinear profile $\widehat{U}^{j}_{\infty}$ has finite global Strichartz and energy norms we have $\displaystyle\big{\\|}\widehat{U}^{j}_{\infty}\big{\\|}_{ST^{j}_{\infty}({\mathbb{R}})}+\big{\\|}\overrightarrow{U}^{j}_{\infty}\big{\\|}_{L^{\infty}_{t}L^{2}_{x}({\mathbb{R}}^{3})}<\infty.$ Then $u_{n}$ is bounded for large $n$ in the Strichartz and the energy norms $\displaystyle\varlimsup_{n\rightarrow\infty}\big{\\|}u_{n}\big{\\|}_{ST({\mathbb{R}})}+\big{\\|}\overrightarrow{u}_{n}\big{\\|}_{L^{\infty}_{t}L^{2}_{x}({\mathbb{R}})}<\infty.$ ###### Proof. We only need to verify the condition of Proposition 4.1. Note that $u^{<k}_{(n)}+w^{k}_{n}$ satisfies that $\displaystyle\left(i\partial_{t}+\Delta\right)\left(u^{<k}_{(n)}+w^{k}_{n}\right)=f_{1}\left(u^{<k}_{(n)}+w^{k}_{n}\right)+f_{2}\left(u^{<k}_{(n)}+w^{k}_{n}\right)$ $\displaystyle\qquad+f_{1}\left(u^{<k}_{(n)}\right)-f_{1}\left(u^{<k}_{(n)}+w^{k}_{n}\right)+f_{2}\left(u^{<k}_{(n)}\right)-f_{2}\left(u^{<k}_{(n)}+w^{k}_{n}\right)$ $\displaystyle\qquad+\sum_{j<k}\frac{\left<\nabla\right>^{j}_{\infty}}{\left<\nabla\right>}f_{1}\left(\frac{\left<\nabla\right>}{\left<\nabla\right>^{j}_{\infty}}u^{j}_{(n)}\right)-f_{1}\left(u^{<k}_{(n)}\right)+\sum_{j<k}h^{j}_{\infty}\frac{\left<\nabla\right>^{j}_{\infty}}{\left<\nabla\right>}f_{2}\left(\frac{\left<\nabla\right>}{\left<\nabla\right>^{j}_{\infty}}u^{j}_{(n)}\right)-f_{2}\left(u^{<k}_{(n)}\right).$ First, by the construction of $\overrightarrow{u}^{<k}_{(n)}$, we know that $\displaystyle\left\\|\left(\overrightarrow{u}^{<k}_{(n)}(0)+\overrightarrow{w}^{k}_{n}(0)\right)-\overrightarrow{u}_{n}(0)\right\\|_{L^{2}_{x}}\leq\sum_{j<k}\left\\|\overrightarrow{u}^{j}_{(n)}(0)-\overrightarrow{u}^{j}_{n}(0)\right\\|_{L^{2}_{x}}\rightarrow 0,$ as $n\rightarrow+\infty$, which also implies that for large $n$, we have $\displaystyle\left\\|\overrightarrow{u}^{<k}_{(n)}(0)+\overrightarrow{w}^{k}_{n}(0)\right\\|_{L^{2}_{x}}\leq E_{0}.$ Next, by the linear profile decomposition in Proposition 5.1, we know that $\displaystyle\big{\\|}u_{n}(0)\big{\\|}^{2}_{L^{2}}=$ $\displaystyle\left\\|v_{n}(0)\right\\|^{2}_{L^{2}_{x}}=\sum_{j<k}\left\\|v^{j}_{n}(0)\right\\|^{2}_{L^{2}_{x}}+\left\\|w^{k}_{n}(0)\right\\|^{2}_{L^{2}_{x}}+o_{n}(1)$ $\displaystyle\geq$ $\displaystyle\sum_{j<k}\left\\|v^{j}_{n}(0)\right\\|^{2}_{L^{2}_{x}}+o_{n}(1)=\sum_{j<k}\left\\|u^{j}_{(n)}(0)\right\\|^{2}_{L^{2}_{x}}+o_{n}(1),$ $\displaystyle\left\\|u_{n}(0)\right\\|^{2}_{\dot{H}^{1}_{x}}=$ $\displaystyle\left\\|v_{n}(0)\right\\|^{2}_{\dot{H}^{1}_{x}}=\sum_{j<k}\left\\|v^{j}_{n}(0)\right\\|^{2}_{\dot{H}^{1}_{x}}+\left\\|w^{k}_{n}(0)\right\\|^{2}_{\dot{H}^{1}_{x}}+o_{n}(1)$ $\displaystyle\geq$ $\displaystyle\sum_{j<k}\left\\|v^{j}_{n}(0)\right\\|^{2}_{\dot{H}^{1}_{x}}+o_{n}(1)=\sum_{j<k}\left\\|u^{j}_{(n)}(0)\right\\|^{2}_{\dot{H}^{1}_{x}}+o_{n}(1),$ which means except for a finite set $J\subset{\mathbb{N}}$, the energy of $u^{j}_{(n)}$ with $j\not\in J$ is smaller than the iteration threshold, hence we have $\displaystyle\big{\\|}u^{j}_{(n)}\big{\\|}_{ST({\mathbb{R}})}\lesssim\big{\\|}\overrightarrow{u}^{j}_{(n)}(0)\big{\\|}_{L^{2}_{x}},$ thus, for any finite interval $I$, by Lemma 5.6, we have $\displaystyle\sup_{k}\varlimsup_{n\rightarrow+\infty}\big{\\|}u^{<k}_{(n)}\big{\\|}^{2}_{ST(I)}\lesssim$ $\displaystyle\sup_{k}\varlimsup_{n\rightarrow+\infty}\sum_{j<k}\big{\\|}u^{j}_{(n)}\big{\\|}^{2}_{ST(I)}$ $\displaystyle=$ $\displaystyle\sup_{k}\varlimsup_{n\rightarrow+\infty}\left[\sum_{j<k,j\in J}\big{\\|}u^{j}_{(n)}\big{\\|}^{2}_{ST(I)}+\sum_{j<k,j\not\in J}\big{\\|}u^{j}_{(n)}\big{\\|}^{2}_{ST(I)}\right]$ $\displaystyle\lesssim$ $\displaystyle\sum_{j<k,j\in J}\big{\\|}\widehat{U}^{j}_{\infty}\big{\\|}^{2}_{ST^{j}_{\infty}(I)}+\sup_{k}\varlimsup_{n\rightarrow+\infty}\sum_{j<k,j\not\in J}\big{\\|}\overrightarrow{u}^{j}_{(n)}(0)\big{\\|}^{2}_{L^{2}_{x}}$ $\displaystyle<$ $\displaystyle\infty.$ This together with the Strichartz estimate for $w^{k}_{n}$ implies that $\displaystyle\sup_{k}\varlimsup_{n\rightarrow+\infty}\big{\\|}u^{<k}_{(n)}+w^{k}_{n}\big{\\|}^{2}_{ST(I)}<\infty.$ Last we need show the nonlinear perturbation is small in some sense. By Proposition 5.1 and Lemma 5.6, we have $\displaystyle\left\\|f_{1}\left(u^{<k}_{(n)}\right)-f_{1}\left(u^{<k}_{(n)}+w^{k}_{n}\right)\right\\|_{ST^{}(I)}\rightarrow 0,$ $\displaystyle\left\\|f_{2}\left(u^{<k}_{(n)}\right)-f_{2}\left(u^{<k}_{(n)}+w^{k}_{n}\right)\right\\|_{ST^{}(I)}\rightarrow 0,$ and $\displaystyle\left\\|\sum_{j<k}\frac{\left<\nabla\right>^{j}_{\infty}}{\left<\nabla\right>}f_{1}\left(\frac{\left<\nabla\right>}{\left<\nabla\right>^{j}_{\infty}}u^{j}_{(n)}\right)-f_{1}\left(u^{<k}_{(n)}\right)\right\\|_{ST^{}(I)}\rightarrow 0,$ $\displaystyle\left\\|\sum_{j<k}h^{j}_{\infty}\frac{\left<\nabla\right>^{j}_{\infty}}{\left<\nabla\right>}f_{2}\left(\frac{\left<\nabla\right>}{\left<\nabla\right>^{j}_{\infty}}u^{j}_{(n)}\right)-f_{2}\left(u^{<k}_{(n)}\right)\right\\|_{ST^{}(I)}\rightarrow 0,$ as $n\rightarrow+\infty$. Therefore, by Proposition 4.1, we can obtain the desired result, which concludes the proof. ∎ ## 6\. Part II: GWP and Scattering for $\mathcal{K}^{+}$ After the stability analysis of the scattering solution of (1.1) and the compactness analysis (linear and nonlinear profile decompositions) of a sequence of the radial solutions of (1.1) in the energy space. We now use them to show the scattering result of Theorem 1.3 by contradiction. Let $E^{}$ be the threshold for the uniform Strichartz norm bound, i.e., $\displaystyle E^{}:=\sup\\{A>0,ST(A)<\infty\\}$ where $ST(A)$ denotes the supremum of $\big{\\|}u\big{\\|}_{ST(I)}$ for any strong radial solution $u$ of (1.1) in $\mathcal{K}^{+}$ on any interval $I$ satisfying $E(u)\leq A$, $M(u)<\infty$. The small solution scattering theory gives us $E^{}>0$. Now we are going to show that $E^{}\geq m$ by contradiction. From now on, suppose that $E^{}\geq m$ fails, that is, we assume that $\displaystyle E^{}<m.$ (6.1) ### 6.1. Existence of a critical element In this subsection, by the profile decomposition and the stability theory of the scattering solution of (1.1), we show the existence of the critical element, which is the radial, energy solution of (1.1) with the smallness energy $E^{}$ and infinite Strichartz norm. By the definition of $E^{}$ and the fact that $E^{}<m$, there exist a sequence of radial solutions $\\{u_{n}\\}_{n\in{\mathbb{N}}}$ of (1.1) in $\mathcal{K}^{+}$, which have the maximal existence interval $I_{n}$ and satisfy that $\displaystyle M(u_{n})<\infty,\quad E(u_{n})\rightarrow E^{}<m,\quad\big{\\|}u_{n}\big{\\|}_{ST(I_{n})}\rightarrow+\infty,\quad\text{as}\;\;n\rightarrow+\infty,$ then we have $\big{\\|}u_{n}\big{\\|}_{H^{1}}<\infty$ by Lemma 2.12. By the compact argument (profile decomposition) and the stability theory, we can show that ###### Theorem 6.1. Let $u_{n}$ be a sequence of radial solutions of (1.1) in $\mathcal{K}^{+}$ on $I_{n}\subset{\mathbb{R}}$ satisfying $\displaystyle M(u_{n})<\infty,\quad E(u_{n})\rightarrow E^{}<m,\quad\big{\\|}u_{n}\big{\\|}_{ST(I_{n})}\rightarrow+\infty,\quad\text{as}\;\;n\rightarrow+\infty.$ Then there exists a global, radial solution $u_{c}$ of (1.1) in $\mathcal{K}^{+}$ satisfying $\displaystyle E(u_{c})=E^{}<m,\quad K(u_{c})>0,\quad\big{\\|}u_{c}\big{\\|}_{ST({\mathbb{R}})}=\infty.$ In addition, there are a sequence $t_{n}\in{\mathbb{R}}$ and radial function $\varphi\in L^{2}({\mathbb{R}}^{3})$ such that, up to a subsequence, we have as $n\rightarrow+\infty$, $\displaystyle\left\\|\frac{\|\nabla\|}{\left<\nabla\right>}\Big{(}\overrightarrow{u}_{n}(0,x)-e^{-it_{n}\Delta}\varphi(x)\Big{)}\right\\|_{L^{2}}\rightarrow 0.$ (6.2) ###### Proof. By the time translation symmetry of (1.1), we can translate $u_{n}$ in $t$ such that $0\in I_{n}$ for all $n$. Then by the linear and nonlinear profile decomposition of $u_{n}$, we have $\displaystyle e^{it\Delta}\overrightarrow{u}_{n}(0,x)=$ $\displaystyle\sum_{j<k}\overrightarrow{v}^{j}_{n}(t,x)+\overrightarrow{w}^{k}_{n}(t,x),\quad\overrightarrow{v}^{j}_{n}(t,x)=e^{i(t-t^{j}_{n})\Delta}T^{j}_{n}\varphi^{j},$ $\displaystyle\overrightarrow{u}^{<k}_{(n)}(t,x)=$ $\displaystyle\sum_{j<k}\overrightarrow{u}^{j}_{(n)}(t,x),\quad\overrightarrow{u}^{j}_{(n)}(t,x)=T^{j}_{n}\overrightarrow{U}^{j}_{\infty}\left(\frac{t-t^{j}_{n}}{(h^{j}_{n})^{2}}\right),$ $\displaystyle\big{\\|}\overrightarrow{u}^{j}_{(n)}(0)-\overrightarrow{v}^{j}_{n}(0)\big{\\|}_{L^{2}}\rightarrow 0.$ By Proposition 5.5 and the following observations that 1. (1) Every radial solution of (1.1) in $\mathcal{K}^{+}$ with the energy less than $E^{}$ has global finite Strichartz norm by the definition of $E^{}$. 2. (2) Lemma 5.7 precludes that all the nonlinear profiles $\overrightarrow{U}^{j}_{\infty}$ have finite global Strichartz norm. we deduce that there is only one radial profile and $\displaystyle E(u^{0}_{(n)}(0))\rightarrow E^{},\quad u^{0}_{(n)}(0)\in\mathcal{K}^{+},\quad\big{\\|}\widehat{U}^{0}_{\infty}\big{\\|}_{ST^{0}_{\infty}(I)}=\infty,\quad\big{\\|}w^{1}_{n}\big{\\|}_{L^{\infty}_{t}\dot{H}^{1}_{x}}\rightarrow 0.$ If $h^{0}_{n}\rightarrow 0$, then $\widehat{U}^{0}_{\infty}=\|\nabla\|^{-1}\overrightarrow{U}^{0}_{\infty}$ solves the $\dot{H}^{1}$-critical NLS $\displaystyle\left(i\partial_{t}+\Delta\right)\widehat{U}^{0}_{\infty}=f_{1}(\widehat{U}^{0}_{\infty})$ and satisfies $\displaystyle E^{c}\left(\widehat{U}^{0}_{\infty}(\tau^{0}_{\infty})\right)=E^{}<m,\;\;K^{c}\left(\widehat{U}^{0}_{\infty}(\tau^{0}_{\infty})\right)\geq 0,\;\;\big{\\|}\widehat{U}^{0}_{\infty}\big{\\|}_{\left(L^{10}_{t}\dot{B}^{1/3}_{90/19,2}\cap L^{12}_{t}L^{9}_{x}\right)(I\times{\mathbb{R}}^{3})}=\infty.$ However, it is in contradiction with Kenig-Merle’s result444By the global $L^{10}_{t,x}$ estimate of solution $u$ of (1.2), we can obtain the global $L^{q}_{t}\dot{W}^{1,r}_{x}$ estimate of $u$ for any Schrödinger $L^{2}$-admissible pair $(q,r)$. in [19]. Hence $h^{0}_{n}\equiv 1$, which implies (6.2). Now we show that $\widehat{U}^{0}_{\infty}=\left<\nabla\right>^{-1}\overrightarrow{U}^{j}_{\infty}$ is a global solution, which is the consequence of the compactness of (6.2). Suppose not, then we can choose a sequence $t_{n}\in{\mathbb{R}}$ which approaches the maximal existence time. Since $\widehat{U}^{0}_{\infty}(t+t_{n})$ satisfies the assumption of this theorem, then applying the above argument to it, we obtain that for some $\psi\in L^{2}$ and another sequence $t^{\prime}_{n}\in{\mathbb{R}}$, as $n\rightarrow+\infty$ $\displaystyle\left\\|\frac{\|\nabla\|}{\left<\nabla\right>}\left(\overrightarrow{U}^{0}_{\infty}(t_{n})-e^{-it^{\prime}_{n}\Delta}\psi(x)\right)\right\\|_{L^{2}}\rightarrow 0.$ (6.3) Let $\overrightarrow{v}(t):=e^{it\Delta}\psi.$ For any $\varepsilon>0$, there exist $\delta>0$ with $I=[-\delta,\delta]$ such that $\displaystyle\big{\\|}\left<\nabla\right>^{-1}\overrightarrow{v}(t-t^{\prime}_{n})\big{\\|}_{ST(I)}\leq\varepsilon,$ which together with (6.3) implies that for sufficiently large $n$ $\displaystyle\big{\\|}\left<\nabla\right>^{-1}e^{it\Delta}\overrightarrow{U}^{0}_{\infty}(t_{n})\big{\\|}_{ST(I)}\leq\varepsilon.$ If $\varepsilon$ is small enough, this implies that the solution $\widehat{U}^{0}_{\infty}$ exists on $[t_{n}-\delta,t_{n}+\delta]$ for large $n$ by the small data theory. This contradicts the choice of $t_{n}$. Hence $\widehat{U}^{0}_{\infty}$ is a global solution and it is just the desired critical element $u_{c}$. By Proposition 1.1, we know that $K(u_{c})>0$. ∎ ### 6.2. Compactness of the critical element In order to preclude the critical element, we need obtain some useful properties about the critical element. In the following subsections, we establish some properties about the critical element by its minimal energy with infinite Strichartz norm, especially its compactness and its consequence. Since (1.1) is symmetric in $t$, we may assume that $\displaystyle\big{\\|}u_{c}\big{\\|}_{ST(0,+\infty)}=\infty,$ (6.4) we call it a forward critical element. ###### Proposition 6.2. Let $u_{c}$ be a forward critical element. Then the set $\displaystyle\\{u_{c}(t,x);0<t<\infty\\}$ is precompact in $\dot{H}^{s}$ for any $s\in(0,1]$. ###### Proof. By the conservation of the mass, it suffices to prove the precompactness of $u_{c}(t_{n})\\}$ in $\dot{H}^{1}$ for any positive time $t_{1},t_{2},\ldots$. If $t_{n}$ converges, then it is trivial from the continuity in $t$. If $t_{n}\rightarrow+\infty$. Applying Theorem 6.1 to the sequence of solutions $\overrightarrow{u}_{c}(t+t_{n})$, we get another sequence $t^{\prime}_{n}\in{\mathbb{R}}$ and radial function $\varphi\in L^{2}$ such that $\displaystyle\frac{\|\nabla\|}{\left<\nabla\right>}\left(\overrightarrow{u}_{c}(t_{n},x)-e^{-it^{\prime}_{n}\Delta}\varphi(x)\right)\rightarrow 0\quad\text{in}\;\;L^{2}.$ 1. (1) If $t^{\prime}_{n}\rightarrow-\infty$, then we have $\displaystyle\big{\\|}\left<\nabla\right>^{-1}e^{it\Delta}\overrightarrow{u}_{c}(t_{n})\big{\\|}_{ST(0,+\infty)}=\big{\\|}\left<\nabla\right>^{-1}e^{it\Delta}\varphi\big{\\|}_{ST(-t^{\prime}_{n},+\infty)}+o_{n}(1)\rightarrow 0.$ Hence $u_{c}$ can solve (1.1) for $t>t_{n}$ with large $n$ globally by iteration with small Strichartz norms, which contradicts (6.4). 2. (2) If $t^{\prime}_{n}\rightarrow+\infty$, then we have $\displaystyle\big{\\|}\left<\nabla\right>^{-1}e^{it\Delta}\overrightarrow{u}_{c}(t_{n})\big{\\|}_{ST(-\infty,0)}=\big{\\|}\left<\nabla\right>^{-1}e^{it\Delta}\varphi\big{\\|}_{ST(-\infty,-t^{\prime}_{n})}+o_{n}(1)\rightarrow 0$ Hence $u_{c}$ can solve (1.1) for $t<t_{n}$ with large $n$ with vanishing Strichartz norms, which implies $u_{c}=0$ by taking the limit, which is a contradiction. Thus $t^{\prime}_{n}$ is bounded, which implies that $t^{\prime}_{n}$ is precompact, so is $u_{c}(t_{n},x)$ in $\dot{H}^{1}$. ∎ As a consequence, the energy of $u_{c}$ stays within a fixed radius for all positive time, modulo arbitrarily small rest. More precisely, we define the exterior energy by $\displaystyle E_{R}(u;t)=\int_{\|x\|\geq R}\Big{(}\big{\|}\nabla u(t,x)\big{\|}^{2}+\big{\|}u(t,x)\big{\|}^{4}+\big{\|}u(t,x)\big{\|}^{6}\Big{)}\;dx$ for any $R>0$. Then we have ###### Corollary 6.3. Let $u_{c}$ be a forward critical element. then for any $\varepsilon$, there exist $R_{0}(\varepsilon)>0$ such that $\displaystyle E_{R_{0}}(u_{c};t)\leq\varepsilon E(u_{c}),\;\text{for any}\;t>0.$ ### 6.3. Death of the critical element We are in a position to preclude the soliton-like solution by a truncated Virial identity. ###### Theorem 6.4. The critical element $u_{c}$ of (1.1) cannot be a soliton in the sense of Theorem 6.1. ###### Proof. We still drop the subscript $c$. Now let $\phi$ be a smooth, radial function satisfying $0\leq\phi\leq 1$, $\phi(x)=1$ for $\|x\|\leq 1$, and $\phi(x)=0$ for $\|x\|\geq 2$. For some $R$, we define $\displaystyle V_{R}(t):=\int_{{\mathbb{R}}^{3}}\phi_{R}(x)\|u(t,x)\|^{2}\;dx,\quad\phi_{R}(x)=R^{2}\phi\left(\frac{\|x\|^{2}}{R^{2}}\right).$ On one hand, we have $\displaystyle\partial_{t}V_{R}(t)=4\Im\int_{{\mathbb{R}}^{3}}\phi^{\prime}\left(\frac{\|x\|^{2}}{R^{2}}\right)x\cdot\nabla u(t,x)\;\overline{u(t,x)}\;dx.$ Therefore, we have $\displaystyle\big{\|}\partial_{t}V_{R}(t)\big{\|}\lesssim R$ (6.5) for all $t\geq 0$ and $R>0$. On the other hand, by Lemma 2.5 and Hölder’s inequality, we have $\displaystyle\partial^{2}_{t}V_{R}(t)=4\int_{{\mathbb{R}}^{3}}\phi_{R}^{\prime\prime}(r)\big{\|}\nabla u(t,x)\big{\|}^{2}\;dx-\int_{{\mathbb{R}}^{3}}(\Delta^{2}\phi_{R})(x)\|u(t,x)\|^{2}\;dx$ $\displaystyle\qquad\quad-\frac{4}{3}\int_{{\mathbb{R}}^{3}}(\Delta\phi_{R})(x)\|u(t,x)\|^{6}\;dx+\int_{{\mathbb{R}}^{3}}(\Delta\phi_{R})(x)\|u(t,x)\|^{4}\;dx$ $\displaystyle=$ $\displaystyle 4\int_{{\mathbb{R}}^{3}}\left(2\|\nabla u(t,x)\|^{2}-2\|u(t,x)\|^{6}+\frac{3}{2}\|u(t,x)\|^{4}\right)\;dx$ $\displaystyle+$ $\displaystyle O\left(\int_{\|x\|\geq R}\left(\|\nabla u(t,x)\|^{2}+\|u(t,x)\|^{6}+\|u(t,x)\|^{4}\right)\;dx+\left(\int_{R\leq\|x\|\leq 2R}\|u(t,x)\|^{6}\;dx\right)^{1/3}\right)$ $\displaystyle=$ $\displaystyle 4K\left(u(t)\right)+O\left(\int_{\|x\|\geq R}\left(\|\nabla u(t,x)\|^{2}+\|u(t,x)\|^{4}\right)\;dx+\left(\int_{R\leq\|x\|\leq 2R}\|u(t,x)\|^{6}\;dx\right)^{1/3}\right).$ By Lemma 2.13, we have $\displaystyle 4K\left(u(t)\right)=$ $\displaystyle\;4\int_{{\mathbb{R}}^{3}}\left(2\|\nabla u(t,x)\|^{2}-2\|u(t,x)\|^{6}+\frac{3}{2}\|u(t,x)\|^{4}\right)\;dx$ $\displaystyle\gtrsim$ $\displaystyle\min\left(6(m-E(u(t))),\frac{2}{3}\big{\\|}\nabla u(t)\big{\\|}^{2}_{L^{2}}+\frac{1}{2}\big{\\|}u(t)\big{\\|}^{4}_{L^{4}}\right)$ $\displaystyle\gtrsim$ $\displaystyle\big{\\|}\nabla u(t)\big{\\|}^{2}_{L^{2}}+\big{\\|}u(t)\big{\\|}^{4}_{L^{4}}$ $\displaystyle\gtrsim$ $\displaystyle E(u(t)),$ Thus, choosing $\eta>0$ sufficiently small and $\displaystyle R:=C(\eta)$ and by Corollary 6.3, we obtain $\displaystyle\partial^{2}_{t}V_{R}(t)\gtrsim E(u(t))=E(u_{0}),$ which implies that for all $T_{1}>T_{0}$ $\displaystyle(T_{1}-T_{0})E(u_{0})\lesssim R=C(\eta).$ Taking $T_{1}$ sufficiently large, we obtain a contradiction unless $u\equiv 0$. But $u\equiv 0$ is not consistent with the fact that $\big{\\|}u\big{\\|}_{ST({\mathbb{R}})}=\infty$. ∎ ## References * [1] T. Aubin, _Problémes isopérimétriques et espaces de Sobolev_ , J. Diff. Geom., 11 (1976), 573–598. * [2] H. Bahouri and P. Gérard, _High frequency approximation of solutions to critical nonlinear wave equations_ , Amer. J. Math., 121 (1999), 131–175. * [3] J. Bourgain, _Global well-posedness of defocusing 3D critical NLS in the radial case_ , J. Amer. Math. Soc., 12(1999), 145–171. * [4] J. Bourgain, _Global solutions of nonlinear Schrödinger equations_ , Amer. Math. Soc. Colloq. Publ. 46, Amer. Math. Soc., Providence, 1999. * [5] T. Cazenave, _Semilinear Schrödinger equations_. Courant Lecture Notes in Mathematics, Vol. 10. New York: New York University Courant Institute of Mathematical Sciences, 2003. * [6] J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, _Global existence and scattering for rough solution of a nonlinear Schrödinger equation on $\mathbb{R}^{3}$_, Comm. Pure Appl. Math., 57(2004), 987–1014. * [7] J. Colliander, M. Keel, G. Staffilani, H. Takaoka, and T. Tao, _Global well-posedness and scattering for the energy-cirtical nonlinear Schrödinger equation in $\mathbb{R}^{3}$_, Ann. of Math., 166(2007), 1–100. * [8] B. Dodson, _Global well-posedness and scattering for the defocusing, $L^{2}$-critical, nonlienar Schrödinger equation when $d\geq 3$_, arXiv:0912.2467v3. * [9] B. Dodson, _Global well-posedness and scattering for the defocusing, $L^{2}$-critical, nonlienar Schrödinger equation when $d=2$_, arXiv:1006.1375. * [10] B. Dodson, _Global well-posedness and scattering for the defocusing, $L^{2}$-critical, nonlienar Schrödinger equation when $d=1$_, arXiv:1010.0040. * [11] B. Dodson, _Global well-posedness and scattering for the mass critical nonlienar Schrödinger equation with mass below the mass of the ground state_ , arXiv:1104.1114. * [12] T. Duyckaerts and F. Merle, _Dynamic of threshold solutions for energy-critical NLS._ GAFA., 18:6(2009), 1787–1840. * [13] T. Duyckaerts and S. Roudenko, _Threshold solutions for the focusing 3D cubic Schrödinger equation._ Revista. Math. Iber., 26:1(2010), 1–56. * [14] D. Foschi, _Inhomogeneous Strichartz estimates_ , J. Hyper. Diff. Equat., 2(2005), 1–24. * [15] J. Ginibre and G. Velo, _Scattering theory in the energy space for a class of nonlinear Schrödinger equation_ , J. Math. Pures Appl., 64(1985), 363–401. * [16] J. Ginibre and G. Velo, _Time decay of finite energy solutions of the nonlinear Klein-Gordon and Schrödinger equation_ , Ann. Inst. Henri Poincaré, Physique théorique, 43:4(1985), 399–442. * [17] S. Ibrahim, N. Masmoudi and K. Nakanishi, _Scattering threshold for the focusing nonlinear Klein-Gordon equation_ , to appear in Analysis $\&$ PDE. * [18] M. Keel and T. Tao, _Endpoint Strichartz estimates_ , Amer. J. Math. 120:5(1998), 955–980. * [19] C. E. Kenig and F. Merle, _Global well-posedness, scattering and blow up for the energy-critical, focusing, nonlinear Schrödinger equation in the radial case_ , Invent. Math., 166(2006), 645–675. * [20] C. E. Kenig and F. Merle, _Global well-posedness, scattering and blow-up for the energy critical focusing non-linear wave equation_ , Acta Math., 201:2(2008), 147–212. * [21] S. Keraani, _On the defect of compactness for the Strichartz estimates of the Schrödinger equations._ J. Diff. Equat., 175(2001) 353–392. * [22] S. Keraani, _On the blow up phenomenon of the critical Schrödinger equation_ , J. Funct. Anal., 265(2006), 171–192. * [23] R. Killip, T. Tao and M. Visan, _The cubic nonlinear Schrödinger equation in two dimensions with radial data_ , J. Eur. Math. Soc., 11:6(2009), 1203–1258. * [24] R. Killip and M. Visan, _The focusing energy-critical nonlinear Schrödinger equation in dimensions five and higher_ , Amer. J. Math., 132:2(2010), 361–424. * [25] R. Killip, M. Visan and X. Zhang, _The mass-critical nonlinear Schrödinger equation with radial data in dimensions three and higher_ , Analysis $\&$ PDE. 1:2(2008), 229–266. * [26] D. Li and X. Zhang, _Dynamics for the energy critical nonlinear Schrödinger equation in high dimensions_ , J. Funct. Anal., 256:6(2009), 1928–1961. * [27] C. Miao, Y. Wu and G. Xu, _Dynamics for the focusing, energy-critical nonlinear Hartree equation_ , preprint. * [28] C. Miao , G. Xu and L. Zhao, _Global well-posedness, scattering and blow-up for the energy-critical, focusing Hartree equation in the radial case_ , Colloquium Mathematicum, 114(2009), 213–236. * [29] C. Miao , G. Xu and L. Zhao, _Global well-posedness and scattering for the mass-critical Hartree equation with radial data_ , J. Math. Pures Appl., 91(2009), 49–79. * [30] K. Nakanishi, _Energy scattering for nonlinear Klein-Gordon and Schrödinger equations in spatial dimensions $1$ and $2$_, J. Funct. Anal., 169(1999), 201–225. * [31] K. Nakanishi, _Remarks on the energy scattering for nonlinear Klein-Gordon and Schrödinger equations_ , Tohoku Math. J., 53(2001), 285–303. * [32] K. Nakanishi and W. Schlag, _Global dynamics above the ground state energy for the cubic NLS equation in $3D$_, to appear in Calc. of Variations and PDE. * [33] T. Ogawa and Y. Tsutsumi, _Blow-up of $H^{1}$ solution for the nonlinear Schrödinger equation_, J. Diff. Equat., 92(1991), 317–330. * [34] E. Ryckman and M. Visan, _Global well-posedness and scattering for the defocusing energy-critical nonlinear Schrödinger equation in $\mathbb{R}^{1+4}$_, Amer. J. Math., 129(2007), 1–60. * [35] G. Talenti, _Best constant in Sobolev inequality_ , Ann. Mat. Pura. Appl. 110(1976), 353–372. * [36] T. Tao, _Nonlinear dispersive equations, local and global analysis_ , CBMS. Regional conference Series in Mathematics, 106. Published for the Conference Board of the Mathematical Science, Washington, DC; by the American Mathematical Society Providence, RI, 2006. * [37] T. Tao, M. Visan and X. Zhang, _The nonlinear Schrödinger equation with combined power-type nonlinearities_ , Comm. PDEs, 32(2007), 1281–1343. * [38] M. Visan, _The defocusing energy-critical nonlinear Schrödinger equation in higher dimensions_ , Duke Math. J., 138:2(2007), 281–374. * [39] J. Zhang, _Sharp conditions of global existence for nonlinear Schrödinger and Klein-Gordon equations_ , Nonlinear Anal. T. M. A., 48:2(2002), 191–207. * [40] X. Zhang, _On Cauchy problem of 3D energy critical Schrödinger equation with subcritical perturbations_ , J. Diff. Equat., 230:2(2006), 422–445.	arxiv-papers	2011-11-29T02:37:43	2024-09-04T02:49:24.694440	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Changxing Miao, Guixiang Xu and Lifeng Zhao", "submitter": "Changxing Miao", "url": "https://arxiv.org/abs/1111.6671" }
1111.6746	F. Iwamuro et al.FMOS image-based reduction package instrumentation: spectrographs — methods: data analysis # FIBRE-pac: FMOS image-based reduction package Fumihide Iwamuro11affiliation: Department of Astronomy, Kyoto University, Kitashirakawa, Kyoto, Japan Yuuki Moritani11affiliationmark: Kiyoto Yabe11affiliationmark: Masanao Sumiyoshi11affiliationmark: Kaori Kawate11affiliationmark: Naoyuki Tamura22affiliation: Subaru Telescope, National Astronomical Observatory of Japan, Hilo, HI, USA Masayuki Akiyama33affiliation: Astronomical Institute, Tohoku University, Aramaki, Sendai, Japan Masahiko Kimura22affiliationmark: Naruhisa Takato22affiliationmark: Philip Tait22affiliationmark: Kouji Ohta11affiliationmark: Tomonori Totani11affiliationmark: Yuji Suzuki11affiliationmark: and Motonari Tonegawa11affiliationmark: iwamuro@kusastro.kyoto-u.ac.jp ###### Abstract The FIBRE-pac (FMOS image-based reduction package) is an IRAF-based reduction tool for the fiber multiple-object spectrograph (FMOS) of the Subaru telescope. To reduce FMOS images, a number of special techniques are necessary because each image contains about 200 separate spectra with airglow emission lines variable in spatial and time domains, and with complicated throughput patterns for the airglow masks. In spite of these features, almost all of the reduction processes except for a few steps are carried out automatically by scripts in text format making it easy to check the commands step by step. Wavelength- and flux-calibrated images together with their noise maps are obtained using this reduction package. ## 1 Introduction FMOS (Kimura et al., 2010) is the near-infrared fiber multi-object spectrograph that has been in operation as one of the open-use instruments on the Subaru telescope since 2009. It can configure 400 fibers of $1\farcs 2$ aperture in a 30′ diameter field of view at the primary focus. The 400 infrared spectra in two groups are taken by two spectrographs called IRS1 and IRS2 (infrared spectrograph 1 and 2) in either of two modes: a low-resolution mode with a spectral resolution of $\Delta\lambda\sim$ 20 Å in the 0.9-1.8 $\mu$m wavelength range, and a high-resolution mode of $\Delta\lambda\sim$ 5 Å in one of the quarter wavelength ranges of 0.90-1.14, 1.01-1.25, 1.11-1.35, 1.40-1.62, 1.49-1.71, or 1.59-1.80 $\mu$m. The bright OH-airglow emission lines are masked by the mask mirror (Iwamuro et al., 2001) installed in these spectrographs. The individual tools for image reduction in this package were developed during engineering and guaranteed-time observations since 2008, in conjunction with other operation and reduction software. Although these tools were not developed as part of the public reduction software of FMOS, they are now stable, thus they have earned the name of “FIBRE-pac” (FMOS Image-Based REduction package). The basic concepts underlying the package are as follows: 1. 1. Most of the reduction is processed by IRAF. 2. 2. Several complicated steps are processed by original software written in C using the cfitsio library (Pence, 1999). 3. 3. Almost all of the reduction processes are automated using script files in text format. 4. 4. Modification of the text files is done by general UNIX commands. 5. 5. The original 2-dimensional information is kept as far as possible throughout the reduction processes. These concepts enable easy implementation and open processing without the inconvenience of licensed or black box parts and ensure traceable operation with visual confirmation. The 2-dimensional information has advantages not only in filtering out unexpected noise using their small sizes but also on detection of faint emission-features. In this paper, we describe the reduction process of the FMOS images based on its Aug.9-2011 version, taking care of complex conditions in the infrared, using multiple fibers and the OH- suppressed spectrograph. ## 2 The images The FMOS images are acquired by a uniform interval non-destructive readout technique called “Up-the-Ramp sampling” (hereafter “ramp sampling” for short). A typical exposure of 900 seconds consists of 54 images. After an exposure is finished, a final frame (treated as a “raw image” in the reduction process) is created where the count in each pixel is calculated by performing a least squares fit to the signal count of 54 images. In addition to suppressing readout noise, the advantages of the ramp sampling are 1) saturated pixels can be estimated from the counts prior to saturation, and 2) cosmic-ray events can be detected as an unexpected jump in the counts and removed from the final frame (Fixsen et al., 2000). The detection threshold of the cosmic-ray events has been currently set to 10$\sigma$ in an empirical way based on the real FMOS images. For IRS1, the fit and the cosmic-ray rejection are done during the ramp sampling, so that the final frame is ready as the exposure finishes. For IRS2, however, the nonlinear bias variation prevents fitting the slope during the exposure. Instead, the ramp fitting is executed after the simple background subtraction (cf. §3.2) has been performed for all of the images taken during the sampling. For example, 54 background-subtracted ramp images are prepared prior to fitting. Consequently the nonlinear bias component is subtracted together with the background photons. (80mm,80mm)Figure1.eps Figure 1: Raw image taken in the IRS1 low-resolution mode with a ramp readout of 54 times using the HAWAII-2 2k$\times$2k detector (corresponding to a 900-s exposure). The left half of the image consists of spectra in the $J$ band, while the right half is in the $H$ band. The typical FWHM of the each fiber spectrum is 5 pixels with a pitch of 10 pixels between the spectra. Figure 1 shows the resulting “raw image” after ramp sampling 54 times using IRS1. Raw images for IRS2 do not exist for the reasons explained above. The raw images have the following components: 1. 1. Thermal background ($\sim$300 e/900 s) 2. 2. Suppressed OH-airglow ($\sim$600 e/900 s on average) 3. 3. Remaining cosmic ray events ($\sim$200 e/event) 4. 4. Read noise ($\sim$10 e rms/54 ramp readout) 5. 5. Object ($\sim$30 e/900 s for a 20 mag(AB) object) 6. 6. Dark current ($<$10 e/900 s) 7. 7. Bias offset ($<$10 e/exposure) 8. 8. Cross-talk between quadrants ($\sim$0.15% of the count) 9. 9. Bad pixels having no efficiency, unusual large dark current, or too large noise ($\sim$0.2% of the pixels) 10. 10. On-chip amplifier glow at the corners of the four quadrants ($\sim$2000 e/54 ramp readout) 11. 11. Unknown external noise in a stripe pattern (quite rare and the amplitude is $\sim$100 e/900 s at most) 12. 12. Point spread function (PSF) of the fiber of 5 pixels with a pitch of 10 pixels between spectra (having almost no cross talk of the spectra on either side) 13. 13. Position of an individual fiber along the slit 14. 14. Optical distortion 15. 15. Quantum efficiency and its pixel-to-pixel variation (“flat” pattern) 16. 16. System throughput variation with wavelength 17. 17. Atmospheric transmittance 18. 18. Intrinsic absorption and emission of the objects The thermal background can be removed in the initial subtraction described in subsection 3.2. The 2nd strongest component, residual airglow, can be subtracted by interpolation of the background of other fibers after the optical distortion is corrected using a dome-flat image taken in the same observation mode. The remaining cosmic-ray and other strong noise features having a size smaller than the PSF of a fiber are removed in three subsequent stages with different threshold counts. Besides the science frames, the following images are necessary for this reduction process. Detector-flat image : The homogeneous thermal image of the black board attached to the entrance window of the camera dewar. This image gives the pixel-to-pixel quantum efficiency ratio of the detector. More than 30 sets of the ramp sampling frames are averaged to ensure the high signal-to-noise (S/N) ratio. This image is included in the reduction package together with the other standard images such as a bad pixel mask. Bad pixel mask : The distribution of bad pixels in the detector selected through the reduction process of the detector-flat image. Dome-flat image : The dome-flat spectra taken before or after observation. This image is used for measurements of the distortion parameters as well as the relative throughput correction of the scientific images. Th-Ar spectral image : The emission-line spectra for wavelength calibration taken just before the exposures of dome-flat frames. The details of the reduction processes using these images are described in the following section. ## 3 Reduction Process ### 3.1 Preparatory processing Before performing a reduction of the science frames, a preparatory process is applied to the dome-flat and Th-Ar spectral images to determine the optical distortion and the wavelength calibration of the image. First, Flat fielding: The dome-flat image is divided by the detector-flat image to remove differences in the quantum efficiency between pixels. Second, Bad pixel correction: The registered and temporally prominent bad pixels, which are picked up by subtraction of 3$\times$3 median filtered image, are replaced with an interpolated value from the surrounding pixels. Third, Correction of the spatial distortion: The $y$-axis of the image is converted using $y^{\prime}=y+(a_{1}y+a_{2})x^{2}+(b_{1}y+b_{2})x$. (Here, the origin is at the center of the image.) The four parameters are chosen to make the PSF amplitude in the image projected along the $x$-axis maximum (cf. figure 2). In this modification process, the dome-flat spectrum from each fiber with 9 pixels in width is converted into a parallel line to make a “combined 1D image” (as in figure 3) that includes the pattern of the airglow mask. (80mm,79mm)Figure2a.eps (80mm,80mm)Figure2b.eps Figure 2: Correction of the spatial distortion of the dome-flat image. The $y$-axis of the left-hand image is converted to straighten the spectra, as shown at right. The thick lines indicate the shape of the spectra at the top and bottom edges of the detector. The left and right half regions are separately corrected in this low-resolution mode, because the corresponding mask mirrors are separated (Kimura et al., 2010). The projected images of each half region along the $x$-axis are shown along the sides of the image. Fourth, Correction of the spectral distortion: The $x$-axis of the combined 1D image is converted using $x^{\prime}=x+(ax+b)$, in which the two parameters are determined for each line so as to minimize the shift and the magnification difference in the mask pattern. The 200 resulting $a$ and $b$ parameters are fitted with fourth-order polynomials in $y$ to remove the local matching error in the pattern. This conversion process makes the mask pattern straight in the column direction (cf. figure 3), which is necessary for better subtraction of the residual airglow lines in the science frames. These parameters obtained from the dome-flat image are also applied to the Th- Ar spectral image to make a combined 1D image, as shown in figure 4. Although the slit image of each emission line and of the airglow masks should be parallel, alignment errors cause small differences between them. Finally, Wavelength calibration: The correlation between the observed wavelength and the pixels of each spectrum are determined, as represented by $\lambda=px^{3}+qx^{2}+rx+s$. The resulting $p$, $q$, $r$, and $s$ coefficients are used to calculate the corresponding wavelengths in each spectrum without changing them, because the $x$ positions of the individual fibers in the slit may exhibit some scatter due to imperfect fiber alignment along the slit. The results of the wavelength calibration are confirmed by comparing the reduced Th-Ar spectra with an artificial image based on the known wavelengths of the Th-Ar emission lines (cf. figure 4). The typical calibration error of the spectra is less than 1 pixel, corresponding to 5 Å or 1.2 Å in the low- or high-resolution mode, respectively. (80mm,18mm)Figure3.eps Figure 3: Correction of the spectral distortion of the combined 1D image of the dome flat. The $x$-axis of the combined 1D image (at top) is converted using $x^{\prime}=x+(ax+b)$, in which $a$ and $b$ are coefficients of a fourth-order polynomial function in $y$. The images of the airglow masks are consequently straightened in the corrected image (at bottom). The thick lines indicate the shape of the airglow mask at the left and the right edges of the detector. (80mm,18mm)Figure4.eps Figure 4: Combined 1D image of the Th-Ar spectra (at bottom) compared with an artificial image based on the known wavelengths of the Th-Ar emission lines (at top). The agreement in the position of the lines shows that the conversion from the column numbers to wavelengths is correct. ### 3.2 Initial background subtraction Since the usual FMOS observations are carried out using an $ABAB$ nodding pattern of the telescope, $A-B$ simple sky subtractions can be performed using two different sky images: $A_{n}-B_{n-1}$ and $A_{n}-B_{n}$ (where $A_{n}$ denotes the image taken at position $A$ of the $n$-th pair). For IRS2, these sky-subtracted frames are calculated by the ramp fitting algorithm applied to the sky-subtracted sub-frames. When the brightness of the OH airglow varies monotonically, most of it can be canceled by merging these two images according to $A_{n}-B=(A_{n}-B_{n})w+(A_{n}-B_{n-1})(1-w)$. The weight $w$ is chosen to make the sum of the absolute count $\|A_{n}-B\|$ a minimum within the range $-1<w<2$. Figure 5 shows a pair of sky-subtracted images and the merged one. Typically, the weight $w$ is equal to about 0.5. (50mm,50mm)Figure5a.eps (50mm,50mm)Figure5b.eps (50mm,50mm)Figure5c.eps Figure 5: Canceling the residual OH-airglow emission. Left panel: Sky- subtracted image using the previous image as the sky image ($A_{n}-B_{n-1}$). Center panel: Sky-subtracted image using the next image as the sky image ($A_{n}-B_{n}$). Right panel: Merged image ($A_{n}-B$) with a weight of $w=0.472$ in which the negative spectra cannot be used because they have merged information with various weights. ### 3.3 Corrections of detector cross talk, bias difference, and bad pixels After the initial background subtraction, cross talk is removed by subtracting 0.15% for IRS1 and 1% for IRS2 from each quadrant. Next, the bias difference between the quadrants (as indicated in figure 5, the lower left quadrant tends to show a higher bias level) is corrected to make the average over each quadrant equal. After flat fielding using the detector flat image, the registered and temporally prominent bad pixels are rejected, together with the four adjacent pixels by interpolating the surrounding pixels. ### 3.4 Distortion correction and residual sky subtraction The processed image is converted into a combined 1D image based on the distortion parameters obtained in the preparatory reduction process. However, only one line of each spectrum (9 pixels in width) is extracted instead of summing the 9 lines. One can thereby make a set of 9 combined 1D images each of which consists of a different part of each spectrum (cf. figure 6). In other words, the PSF of a fiber is divided into 9 pieces, only one of which is used in a combined 1D image. The flexure or temperature change in the spectrograph causes a small difference between the dome-flat and the scientific images along the vertical direction in position. This difference is corrected using the vertical position of the spectra of bright stars. In this way, the counts of the residual sky becomes smooth along the columns after the relative throughput correction of the fibers. The residual airglow lines are fitted and subtracted in these images, and then the relative throughput difference is multiplied to restore the noise level back to the original state. The residual subtracted images are recombined to form an image in which the PSF of the fiber is determined. Medium-level bad pixels having smaller size than the PSF of a fiber are replaced at the end of this process. (50mm,50mm)Figure6a.eps (50mm,50mm)Figure6b.eps (50mm,50mm)Figure6c.eps Figure 6: Residual background subtraction and recomposition process. Left panel: A set of 9 combined 1D images from different part of the PSF. Center panel: Residual subtracted images by performing fits along the columns in each image. Right panel: Recomposed image from the 9 residual subtracted images. ### 3.5 Combine and residual background subtraction There are two ways to observe scientific targets with the $ABAB$ nodding pattern: “normal beam switching” (NBS), in which all targets are observed at position $A$ (all the fibers are supposed to observe “blank” sky at position $B$), and “cross beam switching” (CBS), in which less than half of the fibers are allocated to the targets at position $A$ while the others are at position $B$. In the CBS observation mode, the same targets appear in both images collected in positions $A$ or $B$. However, the target spectra in position $B$ are merged to minimize the absolute flux of the residual airglow lines in the initial background subtraction process. Thus the same reduction process has to be applied to the images in position $B$, replacing the negative spectra of position $A$ with the corresponding parts of the position $B$ image (as shown in figure 7). The merged images (or all of the position $A$ images in the NBS mode) are then combined into one averaged image. Finally, the combined image is divided into a set of 9 combined 1D images again, in order to perform a fine subtraction of the residual background. The recomposed image also goes through a bad pixel rejection process. (50mm,50mm)Figure7a.eps (50mm,50mm)Figure7b.eps (50mm,50mm)Figure7c.eps Figure 7: Merging the CBS images from positions $A$ and $B$. Left panel: Positive spectra in the reduced $A_{n}-B$ image in which the removed negative (position $B$) spectra contain merged information (cf. figure 5). Center panel: Negative spectra in the reduced $A-B_{n}$ image in which the removed positive (position $A$) spectra are merged one. Right panel: Merged $A_{n}-B_{n}$ image. ### 3.6 Mask edge correction and CBS combine Although the correction for spectral distortions was performed to straighten the images of the airglow masks (as in figure 3), there remain small differences among the throughput patterns of the spectra because of the local shape errors of the mask elements or because of the presence of dust on the mask mirror. These differences are corrected by dividing the averaged image by a “relative” dome-flat image in which the common spectral features have been removed by normalizing the count along lines and columns (cf. figure 8). After correcting the relative differences of the throughput patterns among the spectra, the corresponding spectra from positions $A$ (positive) and $B$ (negative) taken in the CBS observation mode are ready to be combined. The negative spectra in the image are then inverted and rearranged so that they can be combined with the corresponding positive spectra (as shown in figure 9). (80mm,18mm)Figure8.eps Figure 8: Relative dome-flat image to correct the throughput differences between spectra. Top panel: Flux-normalized dome-flat spectra in which the throughput differences of the fiber have been removed. Bottom panel: Relative dome-flat image created by normalizing the top image along the columns. This image is displayed within a range of 0.8 (black) to 1.2 (white) to emphasize the relative differences. (50mm,50mm)Figure9a.eps (50mm,50mm)Figure9b.eps (50mm,50mm)Figure9c.eps Figure 9: Combining a pair of target spectra taken in the CBS observation mode. Left panel: Final $A-B$ image in which the positions of the targets observed only at position $B$ are masked. Center panel: Inverted and rearranged negative spectrum that will be combined with the corresponding positive spectrum. Right panel: Combined spectrum. The S/N ratios of the positive spectra are 1.4 times higher than the ratio of the corresponding negative spectra in the combined image. ### 3.7 Object mask In the process of fitting and subtract the residual background, some fraction of the object flux can be subtracted. The best way to retain the object flux is to mask the objects during the fit. Faint objects to be masked are therefore selected by a human eye on the combined image and the reduction process is repeated with these masks applied. The mask density should not be too high to make the residual background subtraction work well: the maximum density allowed is roughly 75% of the image. ### 3.8 Square-noise map Before the mask edge correction process, the distribution of noise in an image is truncated Poisson distribution caused by the bad pixel rejection process in three subsequent stages. Although the noise level in an image is almost homogeneous at this stage, the noise level map becomes complicated during the throughput correction and CBS combine. To estimate the noise level of each pixel, a frame is defined that consists of the squared noise level measured along columns just before the mask edge correction. Here, the noise level is measured by a 3$\sigma$ clipping algorithm iterated ten times for each column, while some extra contribution proportional to the count is added to the clipped pixels. This square-noise frame is divided by the squared image of the relative dome flat, and reduced to half wherever a pair of spectra is averaged during the CBS combine. Note that the noise level of the bright objects is probably underestimated because the systematic uncertainty (i.e. sub-pixel shifts of the spectra in raw images and tiny variation of the throughput pattern caused by the instrument status such as temperature) is the major contribution for them. ## 4 Flux calibration and check process ### 4.1 Outline To calibrate the flux of the scientific targets, at least one bright ($J(AB)=15$–$18$mag) star in each science frames is needed as a spectral reference, because it must have almost the same atmospheric absorption feature as that for the scientific targets in the same field of view. All the spectra are divided by the reference spectrum, and then multiplied by the expected spectrum of the reference star whose flux and spectral type are known or can be determined by the observed values in two different wavelength regions. If the cataloged or estimated spectrum of the reference star is correct, all the observed spectra will then be calibrated accurately. In the next subsection, the method to estimate the reference star spectrum is described. ### 4.2 Template stellar spectra Since the flux and slope of the spectrum of a reference star are determined from the measured counts in an image, one needs the slope-removed template spectra including the intrinsic absorption features of the star. We analyzed 128 stellar spectra in the IRTF Spectral Library111http://irtfweb.ifa.hawaii.edu/spex/IRTF_Spectral_Library/References.html (Rayner et al., 2009) split into seven groups: F0-F9IV/V (19 objects), G0-G8IV/V (14 objects), K0-K7IV/V (10 objects), F0-F9I/II/III (21 objects), G0-G9I/II/III (31 objects), K0-K3I/II/III (17 objects), and K4-K7I/II/III (16 objects). In each group, the spectra were averaged to improve the S/N ratio and divided by the results of the linear fitting, so that slope-removed template spectra of seven different types were prepared (cf. figure 10). Here, only F, G, and K type stars were used to make the slope-removed template spectra, because 1) their slope-removed spectra become roughly straight in $\lambda$ \- $F_{\nu}$ plot, 2) these stars are quite popular and easy to select in the target field, and 3) A and earlier type spectra are neither available in the IRTF Spectral Library, nor in other database with similar qualities. Next, the correlations between the spectral types and the slopes of the spectra have to be established. Figure 11 shows the distribution of measured value of 128 stellar spectra from figure 10. The stellar types are numbered from 0 (F0) to 29 (K9) while the slopes are defined by $slope=\frac{F_{\nu}(1.55)-F_{\nu}(1.21)}{(1.55-1.21)F_{\nu}(1.31)}$ (1) in $\mu{\rm m}^{-1}$. The correlations between the spectral types and the slopes are determined by second-order polynomial fits to the distributions of stellar types III and V: $slope_{\rm III}=0.00132type^{2}+0.0229type-0.666,$ $slope_{\rm V}=0.000456type^{2}+0.0253type-0.654.$ (2) As a result, any spectrum from F0 to K9 can be synthesized by multiplying a linear spectrum having a defined slope with the intrinsic absorption spectrum from the interpolation of the nearest two slope-removed template spectra. (80mm,80mm)Figure10a.ps (80mm,80mm)Figure10b.ps Figure 10: Stellar spectra in the IRTF Spectral Library. The slope-removed averaged spectrum of each stellar type (thick black lines) is used as the intrinsic absorption template. (80mm,80mm)Figure11.ps Figure 11: Correlation between a stellar type and the slope of its spectrum. The stellar types are numbered from 0 (F0) to 29 (K9), and the slope is defined by equation (1). The open symbols are measured from figure 10: type I (triangles), type II (diamonds), type III (squares), type IV (pentagons), and type V (circles). The distributions of types III and V are fit to second-order polynomials represented by the thick solid and dashed lines, respectively. The thick dotted line indicate the intermediate correlation between types III and V used when the stellar type of a reference star is unknown. ### 4.3 Calibration process The first step in the calibration process is the relative throughput correction of fibers in which the averaged throughput of position $A$ and $B$ is used for the merged spectra in the CBS combine process. The next step is to remap the pixels with an increment of 5 Å/pixel (1.25 Å/pixel in the high- resolution mode) based on the correlation between the observed wavelength and the pixels determined in the preparatory reduction process. In this remapping process, the observed wavelengths are multiplied by the count of each pixel in order to convert the value from photon count to $F_{\nu}$. Next, the converted count is divided by the atmospheric transmittance function of the airmass present during the observation, so as to roughly correct for the effects of atmospheric absorption. After this correction, the flux of the reference star is estimated from the converted count at around 1.31$\mu$m, assuming that the total system efficiency under good seeing condition is 2.5% including losses at the entrance of the fibers. The slope of the spectrum is then measured with a fixed efficiency ratio between 1.21 and 1.55 $\mu$m. (The value of this ratio will be confirmed in the check process, along with the total system efficiency of 2.5%.) Finally, the observed scientific spectra are divided by the reference spectrum and multiplied by the stellar spectra from the measured flux and slope. Here, the type V and III absorption templates are adopted as the reference spectra, respectively bluer than G5 and redder than K1 (cf. figure 11). As a consequence, two of the intrinsic absorption templates of FV, GV, K1III, and K4III (in figure 10) are used to interpolate the absorption of the reference spectrum. If the stellar type of the reference star is known, the corresponding synthesized spectrum is used instead of this predicted spectrum. A resulting wavelength- and flux-calibrated image is shown in figure 12. The size of this image is 1800$\times$1800 pixels, the wavelength in $\mu$m is $\lambda=0.9+0.0005\times(ColumnNumber-1)\ $, the $n$-th spectrum is located between $y=9\times(LineNumber)-8$ and $y=9\times(LineNumber)$, and the count is in $\mu$Jy. The square-noise frame is converted in a similar way except that all factors are multiplied twice. The 1D spectrum of each object is extracted from this image with a user-defined mask of 9 pixels, together with the square-noise frame. An example of the final 1D spectrum is shown in figure 13. (80mm,80mm)Figure12a.eps (80mm,80mm)Figure12b.eps Figure 12: Calibrated spectral image (at left) and a square-noise frame (at right) with a size of 1800$\times$1800 pixels. The wavelength range is 0.9-1.8 $\mu$m with an increment of 5 Å/pixel. Each of the 200 spectra has a width of 9 pixels. (80mm,80mm)Figure13.ps Figure 13: Example of a processed 1D spectrum. The spectrum and the corresponding 1$\sigma$ noise level are represented by the thick continuous line and the thin dotted line, respectively. The wavelength has been converted to the rest wavelength using the cataloged redshift value ($z=0.844$). ### 4.4 Check process The results are checked by comparing the resulting flux in the $J$ and $H$ bands with the photometric data in the catalog. If all the factors contributing to the system efficiency are normal, the flux should be consistent with the catalog value. The accuracy of the efficiency ratio between 1.21 and 1.55 $\mu$m can also be confirmed by these diagrams for most cases. Figure 14 shows an example of such a comparison. However, weather conditions, telescope and instrument focus, extended (non-stellar) morphology of targets, and position accuracy of the catalog may cause the lower observed flux of some targets than that expected from the catalog. Under sufficiently good conditions, the observed flux should match the catalog magnitude as shown in the thick line in figure 14 with some downward scatter due to small allocation error of the fibers. Also, if the estimated efficiency ratio between 1.21 and 1.55 is not correct, the points of $J$ and $H$ will have small offset in this figure. (80mm,80mm)Figure14.ps Figure 14: Comparison of the observed flux with the catalog magnitude. The triangles and circles represent the values in the $J$ and $H$ band, respectively. The thick solid lines indicate the ideal values without loss of flux. ## 5 Summary and Conclusions The reduction process for FMOS images includes two special processing steps. One is the segmented processing of the spectra to handle a given part of the PSF as a unit, while the other is the automatic modeling of the reference spectrum to calibrate the scientific targets. The segmented processing enable to keep the original 2-dimensional information which has a large effect on bad pixel filtering and detection of faint emission-lines. Most of the processes are carried out automatically, but the object mask preparation and the reference star selection require user judgement. The FIBRE-pac is available from the FMOS instrument page222http://www.naoj.org/Observing/Instruments/FMOS/ of the Subaru web site, together with the sample dataset presented in this paper. The base reduction platform is IRAF and the reduction scripts and sources are free and open, so that users can check what is happening in each step by sending the commands one at a time. This work was supported by a Grant-in-Aid for Scientific Research (B) of Japan (22340044) and by a Grant-in-Aid for the Global COE Program ”The Next Generation of Physics, Spun from Universality and Emergence” from the Ministry of Education, Culture, Sports, Science, and Technology (MEXT) of Japan. IRAF is distributed by the National Optical Astronomy Observatories, which are operated by the Association of Universities for Research in Astronomy, Inc., under cooperative agreement with the National Science Foundation. ## References * Fixsen et al. (2000) Fixsen, D. J., Offenberg, J. D., Hanisch, R. J., et al. 2000, PASP, 112, 1350 * Iwamuro et al. (2001) Iwamuro, F., Motohara, K., Maihara, T., Hata, R., & Harashima, T. 2001, PASJ, 53, 355 * Kimura et al. (2010) Kimura, M., et al. 2010, PASJ, 62, 1135 * Pence (1999) Pence, W. 1999, Astronomical Data Analysis Software and Systems VIII, 172, 487 * Rayner et al. (2009) Rayner, J. T., Cushing, M. C., & Vacca, W. D. 2009, ApJS, 185, 289	arxiv-papers	2011-11-29T10:11:12	2024-09-04T02:49:24.715880	{ "license": "Public Domain", "authors": "F. Iwamuro, Y. Moritani, K. Yabe, M. Sumiyoshi, K. Kawate, N. Tamura,\n M. Akiyama, M. Kimura, N. Takato, P. Tait, K. Ohta, T. Totani, Y. Suzuki, and\n M. Tonegawa", "submitter": "Fumihide Iwamuro", "url": "https://arxiv.org/abs/1111.6746" }
1111.6805	# UCAC3 Proper Motion Survey. II. DISCOVERY OF NEW PROPER MOTION STARS IN UCAC3 WITH 0$\farcs$40 yr-1 $>$ $\mu$ $\geq$ 0$\farcs$18 yr-1 BETWEEN DECLINATIONS $-$47$\arcdeg$ and 00$\arcdeg$ Charlie T. Finch, Norbert Zacharias finch@usno.navy.mil U.S. Naval Observatory, Washington DC 20392–5420 Mark R. Boyd, Todd J. Henry Georgia State University, Atlanta, GA 30302–4106 Nigel C. Hambly Scottish Universities Physics Alliance, Institute for Astronomy, University of Edinburgh, Royal Observatory, Blackford Hill, Edinburgh EH9 3HJ, Scotland, UK ###### Abstract We present 474 new proper motion stellar systems in the southern sky having no previously known components, with 0$\farcs$40 yr-1 $>$ $\mu$ $\geq$ 0$\farcs$18 yr-1 between declinations $-$47$\arcdeg$ and 00$\arcdeg$. In this second paper utilizing the U.S. Naval Observatory third CCD Astrograph Catalog (UCAC3) we complete our sweep of the southern sky for objects in the proper motion range targeted by this survey with R magnitudes ranging from 9.80 to 19.61. The new systems contribute a $\sim$16% increase in the number of new stellar systems for the same region of sky reported in previous SuperCOSMOS RECONS (SCR) surveys. Among the newly discovered stellar systems are 16 multiples, plus an additional 10 components that are new common proper motion companions to previously known objects. A comparison of UCAC3 proper motions to those from Hipparcos, Tycho-2, Southern Proper Motion (SPM4), and SuperCOSMOS indicates that all proper motions are consistent to $\sim$10 mas/yr, with the exception of SuperCOSMOS. Distance estimates are derived for all stellar systems having SuperCOSMOS Sky Survey (SSS) $B_{J}$, $R_{59F}$, and $I_{IVN}$ plate magnitudes and Two-Micron All Sky Survey (2MASS) infrared photometry. We find five new red dwarf systems estimated to be within 25 pc. These discoveries support results from previous proper motion surveys suggesting that more nearby stellar systems are to be found, particularly in the fainter, slower moving samples. In this second paper utilizing the U.S. Naval Observatory third CCD Astrograph Catalog (UCAC3) we complete our sweep of the southern sky for objects in the proper motion range targeted by this survey with R magnitudes ranging from 9.80 to 19.61. solar neighborhood — stars: distances — stars: statistics — surveys — astrometry ## 1 INTRODUCTION The third U.S. Naval Observatory (USNO) CCD Astrograph Catalog (UCAC3) (Zacharias et al., 2010) proper motion survey, addresses the possibility that proper motion surveys using digitized scans of photographic plates might overlook some proper motion systems. The UCAC3 obtained accurate proper motions by combining CCD observations with early epoch photographic data. This survey utilizes the UCAC3 proper motions to discover new systems that have been missed in previous efforts. The first paper in this series (Finch et al. 2010b, ) (hereafter, U3PM1), confirmed this suspicion by revealing an additional 25.3% stellar systems having a proper motion of 0$\farcs$40 yr-1 $>$ $\mu$ $\geq$ 0$\farcs$18 yr-1 between declinations $-$90$\arcdeg$ and $-$47$\arcdeg$ over those found by the Research Consortium On Nearby Stars (RECONS)111www.recons.org group using SuperCOSMOS Sky Survey (SSS) data. These new discoveries provided the impetus for this second paper of the series, which completes the sweep of the southern sky for systems with 0$\farcs$40 yr-1 $>$ $\mu$ $\geq$ 0$\farcs$18 yr-1 found in the UCAC3. The data obtained from proper motion surveys aid astronomers in determining accurate stellar luminosity and mass functions, thereby revealing how the Galaxy’s stellar mass is divided among different types of stars. Our main goal — identifying the Sun’s nearest neighbors — provides a vast sample of red dwarf, subdwarf, and white dwarf stellar systems for studies of multiplicity, activity, ages, and exoplanet searches. Because of their proximity, the nearby stars offer the most accessible measurements of each of these characteristics. Our UCAC3 proper motion survey is currently focused on the southern hemisphere, which has not been surveyed as systematically as the northern sky, where the pioneering surveys of Giclas (Giclas et al., 1971, 1978) and Luyten (Luyten, 1979, 1980) were primarily carried out. Historically, proper motion studies have been focused on blinking photographic plates taken at different epochs to determine source motions. Recent surveys that complement the classic efforts utilize various techniques, plate sets, modern computers, and carefully tailored algorithms to effectively blink digitized images of photographic plates. In the southern sky, such surveys include (Wroblewski & Torres, 1994), (Wroblewski & Costa, 1999), (Scholz et al., 2000, 2002), (Oppenheimer et al., 2001), (Pokorny et al., 2003), (Lépine, 2005, 2008), (Deacon et al., 2005; Deacon & Hambly, 2007), and (Deacon et al., 2009). In an effort to understand the stellar population of the solar neighborhood, the RECONS group has been targeting the neglected southern sky to reveal new stellar proper motion systems. To date, these discoveries have been reported in six papers in The Solar Neighborhood (TSN) series (Hambly et al., 2004), (Henry et al., 2004), (Subasavage et al. 2005a, ), (Subasavage et al. 2005b, ), (Finch et al., 2007), (Boyd et al., 2011). These new systems are discovered using the SuperCOSMOS Sky Survey (SSS) data (Hambly et al. 2001a, ) and given the name SCR (SuperCOSMOS-RECONS). Followup observations of intriguing systems are performed at the Cerro Tololo Inter-American Observatory (CTIO) 0.9m telescope, where RECONS operates a trigonometric parallax program focusing on stars within 25 pc. Our UCAC3 survey uses an approach fundamentally different from plate blinking to reveal proper motion systems. We take advantage of observations reported in many catalogs ranging in epochs from the early nineteenth to the early twenty- first centuries, rather than directly using specific sets of digitized images from photographic plates. In this investigation we focus on stellar systems in the UCAC3 found between declinations $-$47$\arcdeg$ and 00$\arcdeg$ that have 0$\farcs$40 yr-1 $>$ $\mu$ $\geq$ 0$\farcs$18 yr-1, completing a sweep of the southern sky. The search region and proper motion range matches that in (Boyd et al., 2011), hereafter TSN25, in which the lower proper motion cutoff was chosen to match that of the NLTT catalog. TSN25 reports 2817 new SCR systems, substantially adding to the number of new SCR systems found previously. In Table 1, we summarize the number of new stellar systems discovered, highlighting those estimated to be within 25 pc, for both the RECONS and UCAC3 surveys. In this paper we will focus in particular on the two SCR searches (TSN18 and TSN25) that correspond to the same proper motion and declination ranges as this UCAC3 survey (U3PM1 and this paper). New stellar objects from this search are given USNO Proper Motion (UPM) names. ## 2 Method ### 2.1 UCAC3 The USNO CCD Astrograph Catalog (UCAC) project finished observations in late 2004 and has been producing astrometric catalogs since October 2000\. This astrometric survey was conceived to densify the optical reference frame to high accuracy beyond the Hipparcos and Tycho magnitudes. UCAC is the first all-sky survey performed with a CCD detector utilizing the high level of precision achievable with this technology. The first release, UCAC1, (Zacharias et al., 2000), was a partial catalog covering 80% of the southern sky. The second catalog, UCAC2, (Zacharias et al., 2004), contains roughly 80% of the entire sky and includes improved proper motions from the use of early epoch plates paired with the Astrograph CCD data. UCAC3 (Zacharias et al., 2010), released in August 2009, is the first in the series to contain coverage of the entire sky. UCAC3 also includes double star fitting and has a slightly deeper limiting magnitude than UCAC2 due to a complete re-reduction of the pixel data (Zacharias, 2010). In addition, data from the Two-Micron All Sky Survey (2MASS) were used in UCAC3 to probe for and reduce systematic errors in UCAC observations, providing a greater number of reference stars to stack up residuals as a function of many parameters, such as observing site and exposure time. A detailed description of the astrometric reductions of UCAC3 can be found in (Finch et al. 2010a, ). A detailed introduction to the UCAC3 can be found in the release paper (Zacharias et al., 2010) and the README file of the data distribution. A new edition, UCAC4, (Zacharias et al., 2011) is scheduled to be released later this year. ### 2.2 PROPER MOTIONS The UCAC3 contains roughly 95 million calculated absolute proper motions. The majority of these are derived proper motions from the use of early epoch catalogs paired with the Astrograph CCD data. Earlier epoch data are all reduced to the International Celestial Reference Frame (ICRF). UCAC3 mean positions and proper motions are calculated using a weighted, least-squares adjustment procedure. Bright stars with R$\sim$8–12 in UCAC3 are combined with ground-based photographic and transit circle catalogs. These include all catalogs used for the production of the Tycho-2 project (Høg et al., 2000), unpublished measures of over 5000 astrograph plates digitized on the StarScan machine (Zacharias et al., 2008), new reductions of Southern Proper Motion (SPM) (Girard et al., 2011) data, and data from the SuperCOSMOS project (Hambly et al. 2001a, ). About 1.2 million star positions to about B$=$ 12 entered UCAC from digitizing the AGK2 plates (epoch about 1930). The Hamburg Zone Astrograph and USNO Black Birch Astrographs contributed another 7.3 million star positions, mainly in the V$=$12 – 14 magnitude range, in fields covering about 30% of the sky, and the Lick Astrograph plates taken around 1990 yielded over one million star positions to V$=$16 in selected fields. For all catalogs used to derive UCAC3 proper motions a systematic error estimate was added to the root mean square (RMS) of the individual stars random errors. The largest error floor added was 100 mas for the SuperCOSMOS data due to zonal systematic errors ranging from 50–200 mas when compared to 2MASS data. To identify previously known high proper motion (HPM) stars in the UCAC3, a source list was compiled using the VizieR on-line data tool, along with targeted supplements from published literature. In the north we used the LSPM- North catalog (Lépine, 2005) containing 61977 new and previously found stars having proper motions greater than 0$\farcs$15 yr-1. For the south we utilized many surveys, notably including the Revised NLTT Catalog (Salim & Gould, 2003), which produced 17730 stars with proper motions greater than 0$\farcs$15 yr-1, and the RECONS efforts (SCR stars). For a full list of catalogs used, see the UCAC3 README file. While this list is not comprehensive, this effort tagged roughly 51000 known HPM stars in UCAC3 over the entire sky. These previously identified HPM stars were given a mean position (MPOS) number greater than 140 million and do not have derived UCAC3 proper motions. We instead used the proper motion data from the catalogs themselves (see $\S$4.5). Proper motion errors in the UCAC3 catalog for stars brighter than R$\sim$12 are only $\sim$1–3 mas/yr in part because of the large epoch spread of roughly 100 years in some cases. The errors of the fainter stars range from $\sim$2–3 mas/yr if found in SPM4 and $\sim$6–8 mas/yr if SuperCOSMOS data are used in lieu of SPM4 data. ### 2.3 SEARCH CRITERIA In this second paper we survey the southern sky between declinations $-$47$\arcdeg$ and 0$\arcdeg$ using the same proper motion range as in U3PM1, 0$\farcs$40 yr-1 $>$ $\mu$ $\geq$ 0$\farcs$18 yr-1. In this area of the sky we identify an initial sample of 212356 proper motion candidates. We utilize the same search criteria as in U3PM1, using UCAC3 flags with values indicative of real proper motion objects. A visual check from a sample of stars confirmed that these flags still hold true in the region of the sky being surveyed. All stars must (1) be in the 2MASS catalog with an e2mpho (2MASS photometry error) less than or equal to 0.05 magnitudes in all three bands, (2) have a UCAC fit model magnitude between 7 and 17 mag, (3) have a double star flag (dsf) equal to 0, 1, 5 or 6, meaning a single star or fitted double, (4) have an object flag (objt) between $-$2 and 2 to exclude positions that used only overexposed images in the fit, (5) have an MPOS number less than 140 million, to exclude already known high proper motion stars, and (6) have a LEDA galaxy flag of zero, meaning that the source is not in the LEDA galaxy catalog. After all these cuts, there remain 17516 “good” candidate list, fewer than expected for this region of the sky, when compared to 9248 in U3PM1. A total of 7641 candidates were excluded from the “good” candidates due to being marked as previously known in the UCAC3 catalog (MPOS number greater than 140 million). These candidates were then cross-checked via VizieR and SIMBAD to determine if they were previously known. All cross-checks are performed using a 90$\arcsec$ search radius, with one exception (the NLTT catalog). A larger search radius of 180$\arcsec$ was used when comparing UPM candidates to the NLTT and LHS catalog, which have been found to have inaccurate positions as reported in (Bakos et al., 2002). Thus, UCAC3 proper motion candidates with positions differing from Luyten’s or any other known object by less than 90$\arcsec$ are considered known. Those differing from Luyten’s by 90–180$\arcsec$ are considered new discoveries but are noted as possible NLTT stars in the tables. Those differing by more than 180$\arcsec$ from Luyten are considered new discoveries. All candidates matched to known stars had a final check to determine if the proper motion and magnitudes matched — those that match are considered known and not reported in this sample. As in U3PM1, it is not a goal of this paper to revise the NLTT catalog and assign proper identifications and accurate positions to NLTT entries; rather, the goal is to identify new high proper motion stars. After this, in effect, second cross-check for previously known stars, the list was reduced to a manageable 3736 candidate proper motion objects. The 13780 known objects found during this cross-check shows how incomplete the UCAC3 catalog can be in identifying previously known high proper motion objects with the given search criteria. Each of these candidates was then visually inspected to confirm proper motion by blinking the $B_{J}$ and $R_{59F}$ SuperCOSMOS digitized plate images. During blinking, we noticed that for declinations between roughly $-$33$\arcdeg$ and 0$\arcdeg$ the epoch spread was insufficient ($\sim$3–5 years) to visually verify proper motion for all candidates. For those candidates, a second sweep was done by blinking the $POSS-IR$ and $R_{59F}$ SuperCOSMOS digitized plate images. Nearly 87% of the candidates were found to have no verifiable proper motions and were discarded. The final counts of new discoveries are 500 proper motion objects in 474 systems. Among these are 25 multiple systems (24 doubles and one triple), of which ten were found to have CPM to previously known primaries. For this search we find a successful hit rate — defined as the number of new and known proper motion stars (21921) divided by the total “good” candidates extracted (25157, including stars with an MPOS number $>$ 140 million) — of 87.1%, which is higher than the 81.4% hit rate found in TSN25. After looking into the calculation used in U3PM1 to determine the successful hit rate a counting error was found. The number of real objects excluded the known proper motion objects tagged in the UCAC3 catalog (stars with an MPOS number $>$ 140 million). If we add these stars in the total for the U3PM1 count, we get a total of 7975 real objects giving a new successful hit rate of 86.2%, which is comparable to this paper. At least three factors mentioned in U3PM1 have been identified that can lead to false detections in the UCAC3 proper motion survey. First, some real objects are discarded during the sifting mentioned above, particularly because of the 2MASS criterion which states that $JHK_{s}$ photometry errors must be less than 0.05 mag. Second, the UCAC3 contains many phantom proper motion objects due to incorrect matches during proper motion calculations. Third, other misidentifications arise from blended images, where a single source in an earlier epoch catalog can be matched with two stars in the UCAC3 data. ## 3 RESULTS In Table 2, we list the 474 new proper motion stellar systems (500 objects) discovered during this search. We highlight the five red dwarf systems estimated to be within 25 pc in Table 3. In both tables we list names, coordinates, proper motions, 1$\sigma$ errors in the proper motions, plate magnitudes from SuperCOSMOS, near-IR photometry from 2MASS, the computed $R_{59F}-J$ color, a distance estimate, and notes. ### 3.1 Positions and Proper Motions All positions on the ICRF system, proper motions, and errors are taken directly from UCAC3, unless otherwise noted. For a few stars that were found during visual inspection without any UCAC3 data, information has been obtained from alternate sources (see $\S$3.4). For this sample, the average positional errors reported in the UCAC3 catalog are 51 mas in RA and 50 mas in Dec. For proper motions, the average errors reported in the UCAC3 for this sample are 8.0 mas/yr in $\mu_{\alpha}\cos\delta$ and 7.7 mas/yr in $\mu_{\delta}$. ### 3.2 Photometry In Tables 2 and 3, we give photographic magnitudes from the SuperCOSMOS and 2MASS surveys. From SuperCOSMOS, magnitudes are given from three plate emulsions, $B_{J}$, $R_{59F}$, and $I_{IVN}$. Magnitude errors are typically less than 0.3 mag for stars fainter than $\sim$15, with errors increasing for brighter sources. From 2MASS, $JHK_{s}$ infrared photometry is given, with errors typically 0.05 mag or less due to the search criteria. Additional objects found during visual inspection are typically fainter with larger photometric errors. The $R_{59F}-J$ color has been computed to indicate the star’s color. While SuperCOSMOS magnitudes are reported in the UCAC3, this sample was checked against the SuperCOSMOS catalog to rectify some mismatches found in the UCAC3 catalog. In some cases, SuperCOSMOS magnitudes are not given in the tables, due to blending, no source detection, high chi-square or other problems where no reliable magnitude is available. 2MASS magnitudes are given for all but one object which was found visually that is not present in the 2MASS catalog, as indicated in the notes. ### 3.3 Distances Plate photometric distance estimates are computed using the same method as in U3PM1 and previous SCR searches. Using the relations in (Hambly et al., 2004), 11 distance estimates are generated based on colors computed from the six-band photometry. This method assumes all objects are main sequence stars, and provides distances accurate to 26%, determined from the mean differences between the true distances for stars with accurate (errors less than 10 mas) trigonometric parallaxes and distances estimated from the relations. Errors are higher for stars with missing photometry, resulting in fewer than 11 relations, and stars that are not single, main sequence red dwarfs, e.g. cool subdwarfs and white dwarfs. It is possible to produce a distance with only one relation; however, six are needed to be considered “reliable” because that allows for one magnitude dropout. Stars having fewer than six relations are identified in the notes to Tables 2 and 3. If a star is identified as a possible subdwarf, the distance estimate is expected to be too large and is given in brackets. ### 3.4 Additional Objects In Table 2 we include 17 additional proper motion objects found during visual inspection of the candidate fields. These objects are CPM companion candidates that either have fainter limiting magnitudes than implemented for this search, were eliminated from the candidate list by the search criteria, or have UCAC3 proper motions less than 0$\farcs$18 yr-1. These new visual discoveries have all been cross-checked with VizieR and SIMBAD using the same methods described above for the main search. Proper motions have been obtained from UCAC3, SPM4, PPMXL (Roeser et al., 2010), or SuperCOSMOS, in that order. For stars that were not found in the UCAC3 data, positions were computed using the epoch, coordinates, and proper motion obtained from the corresponding catalog. Magnitudes are obtained using the 2MASS and SuperCOSMOS catalogs to compute distance estimates. ## 4 ANALYSIS ### 4.1 Color-Magnitude Diagram In Figure 1 we show a color-magnitude diagram of the 334 new UPM proper motion objects (solid circles) and seven known objects (open triangles, companions to UPM objects) from this search having $R_{59F}-J$ colors. Symbols that fall below $R_{59F}\sim 17$ are CPM companion candidates noticed during visual inspection. The brightest new object, UPM 0747-2537A, has $R_{59F}$ = 9.80 and is estimated to be at a distance of 40.6 pc. The reddest object found in this search is UPM 1848-0252 with $R_{59F}-J$ = 5.06, $R_{59F}$ = 16.57, at an estimated distance of 26.9 pc. The subdwarf population is not as well defined as in TSN18 and TSN25 because there are far fewer new objects. Nonetheless, a separation can be seen below the concentration of main sequence stars. ### 4.2 Reduced Proper Motion Diagram In Figure 2, we show the reduced proper motion (RPM) diagram for all objects also plotted in Figure 1, with similar symbols for new and known objects. The RPM diagram is a good method to help separate white dwarfs and subdwarfs from main-sequence stars, under the assumption that objects with larger distances tend to have smaller proper motions. Using the same method as in U3PM1 and TSN25 we obtain $H_{R_{59F}}$ via a modified distance modulus equation, in which $\mu$ is substituted for distance: $H_{R_{59F}}=R_{59F}+5+5\log\mu.$ The solid line seen in Figure 2 is used to separate white dwarfs from subdwarfs. This is the same empirical line used in U3PM1 and previous TSN papers. No white dwarf candidates have been found during this latest search. Subdwarf candidates have been selected using the same method as in U3PM1 and TSN25 — stars with $R_{59F}-J>$ 1.0 and within 4.0 mag in $H_{R}$ of the empirical line separating the white dwarfs are considered subdwarfs. From this survey there are 17 subdwarf candidates, all with distance estimates greater than 122 pc, with the exception of one, UPM 1712-4432, with an estimated distance of 33.9 (see $\S$4.4). Because the relations used to estimate distances assume that stars are on the main sequence, underluminous cool subdwarfs and white dwarfs have large distances, which can, in fact, be used to identify such objects. The distance estimates for these stars are presumably erroneous and are given in brackets in Tables 2, 3 and 4. Follow-up spectroscopic observations will be needed to confirm all subdwarf candidates. ### 4.3 New Common Proper Motion Systems In this search, we find 25 CPM candidate systems consisting of 24 binaries and one triple. Included in these CPM systems are 16 new systems and nine known systems with newly discovered components. One binary system, UPM 0800-0617AB is a possible subdwarf binary system. The lone triple is an SCR system with two newly discovered components. In Table 4, we list the CPM system primaries and companions, their proper motions, and the companions’ separations and position angles relative to the primaries (defined to be the brightest star in each system using the UCAC bandpass, or an alternate bandpass if a UCAC value is not available). We also provide distance estimates for each component, where possible. Components were determined to be potentially physically associated using distance estimates in conjunction with the proper motions and visual inspections. However, most of the companions were found during visual inspection, meaning that proper motions, 2MASS and/or SuperCOSMOS magnitudes may be missing or suspect, as identified in the notes. For systems with data missing in Table 4, the physical connection of the system components should be considered tentative. In Figure 3, we show comparisons of the proper motions in each coordinate for the 19 CPM systems for which both components have a listed proper motion. CPM candidates having proper motions from the UCAC3 are represented by solid circles while those with proper motions from other sources are represented by open circles. If a proper motion was not present in the UCAC3, data were obtained manually from the SPM4, PPMXL or SuperCOSMOS databases, in that order. ### 4.4 Notes on Specific Stars UPM 0443-4129AB is a possible CPM binary. However, UPM 0443-4129A has a suspect proper motion and the companion’s distance estimate uses fewer than 6 relations. It is possible that this pair is a case of a chance alignment. See Table 4 for more details. BD-04 2807AB is a possible CPM binary. However, the primary has a suspect proper motion, a distance estimate that uses fewer than 6 relations, and there is no distance estimate for the secondary. It is possible that this pair is a case of a chance alignment. See Table 4 for more details. UPM 0747-2537A is the brightest new discovery from this search with $R_{59F}=$ 9.80 and an estimated distance of 40.6 pc. However, only one relation was viable, making the distance estimate unreliable. UPM 0800-0617AB is a possible candidate for a binary subdwarf system. The primary is a possible subdwarf at an estimated distance of 175.5 pc. The secondary is at a separation of 5.8$\arcsec$ at position angle 297.2∘ from the primary. Color information is insufficient for a reliable distance estimate. UPM 1226-3516B and C are in a candidate triple system with SCR 1226-3515A. The A and B components are separated by 49.8$\arcsec$ at a position angle of 191.3∘. The C component has a separation of 97.0$\arcsec$ at a position angle of 146.9∘ from the primary. The C component has a suspect proper motion and the distance estimates for all there components are inconsistent. In particular, the C component may not be a part of the system. See Table 4 for more details. UPM 1712-4432 is a subdwarf candidate with $R_{59F}=$ 13.04 and $R_{59F}-J=$ 1.01 at a distance of 33.9 pc. However, only three relations were viable, making the distance estimate unreliable. SuperCOSMOS magnitudes are indicative of a blended image, meaning this is likely not what it seems. UPM 1718-2245B has an estimated distance of only 13.2 pc based on 7 relations, making it the nearest candidate in the sample. However, the primary has a distance estimate of 25.4 pc based on 10 relations so we favor the larger distance for the system. UPM 1848-0252 is the reddest new discovery from this search, with $R_{59F}-J=$ 5.06 and an estimated distance of 26.9 pc. ### 4.5 Comparison to Previous Proper-Motion Surveys During production of the UCAC3 catalog, we made an effort to tag previously known HPM stars. For these stars, proper motions were taken from their respective catalogs rather than calculated using UCAC3 methodology, which made comparisons to other catalogs/surveys difficult. However, during the present search we have found 104 stars in both the Hipparcos and Tycho-2 catalogs that are not tagged as HPM stars in the UCAC3 catalog — these stars are proper motion candidates that were found to be in Tycho-2 during cross-checking. A 2.5$\arcsec$ radius was used to match these stars to sources in the Hipparcos catalog so that we can compare the bright end of the UCAC3 proper motion stars ($R$ $\sim$ 7.13-13.66) to stars in both the Tycho-2 and Hipparcos catalogs. In Figure 4, we compare proper motions in RA and Dec for these stars as given in UCAC3, Hipparcos, and Tycho-2. These plots show that the differences in proper motions are small, in general less than 10 mas/yr, and no significant systematic errors as a function of declination are seen. The RMS differences between UCAC3 proper motions in $\Delta\mu_{\alpha}\cos\delta$ and $\Delta\mu_{\delta}$ and those from Hipparcos are 5.7 and 9.1 mas/yr, respectively. Comparisons to Tycho-2 yield RMS differences of 5.2 and 8.3 mas/yr, respectively. Lower RMS differences of 3.0 mas/yr in $\Delta\mu_{\alpha}\cos\delta$ and 3.2 mas/yr in $\Delta\mu_{\delta}$ are seen when comparing the Hipparcos to Tycho-2 proper motions. To investigate the fainter end of UCAC3, we compare results for 77 stars ($R$ $\sim$ 10.88-16.69) that are in both the SPM4 and SuperCOSMOS catalogs that were not tagged as HPM stars in the UCAC3 catalog — these stars are proper motion candidates that were found to be SCR stars during cross-checking. A 2.5$\arcsec$ radius was used to match these stars to sources in the SPM4 catalog. The SPM4 catalog only covers Dec = $-$90 to $-$20 sky area, limiting the area included for this comparison. In Figure 5, we compare proper motions in RA and Dec for these stars as given in UCAC3, SuperCOSMOS, and SPM4. These plots show that differences in proper motions are similar to those found for brighter stars when comparing UCAC3 and SPM4, but the differences are much larger for the SuperCOSMOS results. The RMS differences between UCAC3 proper motions in $\Delta\mu_{\alpha}\cos\delta$ and $\Delta\mu_{\delta}$ and those in SPM4 are 6.0 and 5.7 mas/yr respectively. Comparisons to SuperCOSMOS yield RMS differences of 16.5 and 14.1 mas/yr, respectively. In Figure 5, we also see that proper motions in Dec appear to be systematically shifted in the SuperCOSMOS data. These high RMS results and the systematic shift are also seen in the comparison of the SPM4 to the SuperCOSMOS proper motions, yielding RMS differences of 15.6 and 15.2 mas/yr in $\Delta\mu_{\alpha}\cos\delta$ and $\Delta\mu_{\delta}$, respectively. The higher RMS differences for the SuperCOSMOS proper motions are in agreement with the findings of TSN18 and U3PM1 where SCR proper motions were found to have higher RMS differences when compared to other external catalogs. It is worth noting that the SuperCOSMOS proper motion RMS reported here are not representative of the entire catalog. Objects having an R$\sim$16–19 with $\mu>$ 0$\farcs$10 yr-1 in the SuperCOSMOS catalog should have an RMS no greater than 10 mas/yr, and considerably better for fields with decades between the epochs (See Tables 1 and 3 from (Hambly et al. 2001b, )). Random and systematic differences of order 10 mas/yr in proper motions between the various catalogs, particularly at the faint end, are expected because of different data quality, measurements, reductions and epoch differences. SuperCOSMOS for example uses Schmidt plates for both early and recent epoch which typically show large errors. The proper motions of faint stars in UCAC3 are based on early epoch Schmidt plates for the sky area north of $-$20 deg Dec and CCD observations for recent epoch data. A combination of CCD data and early astrograph data (SPM plates) is used south of $-$20 deg, with significantly smaller errors. The SPM4 proper motions are derived entirely on SPM astrograph plates from 2 epochs. At the bright end proper motions are more reliable due to higher quality of Hipparcos and Tycho data as well as availability of many other star catalogs, most of which have been used in common between Tycho-2 and UCAC3. However, there can be large differences between Hipparcos and Tycho-2 for some stars because the Hipparcos PMs are based on only about 3.5 years of observing (although with high quality), while Tycho-2 PMs are based on typically 100 years epoch difference. Multiplicity and residual orbital motions sometimes render Hipparcos PMs inferior in spite of their small formal astrometric errors. In TSN25 a total of 3073 objects were reported, all of which fit within the proper motion and declination constraints of this paper. During this UCAC3 search, only 770 of the 3073 objects reported in TSN25 were recovered, or a low 25.1% recovery rate. This is primarily due to the UCAC3 catalog having no proper motion or a reported proper motion not meeting the criteria of this paper (0$\farcs$40 yr-1 $>$ $\mu$ $\geq$ 0$\farcs$18 yr-1) for $\sim$70% of the new discoveries listed in TSN25. The Hipparcos catalog contains 118218 total objects, of which 1690 meet the proper motion and declination constraints of this paper. Tycho-2 contains 2539913 total objects in the main catalog, of which 3187 meet similar limits. We recover 1316 Hipparcos stars and 2543 Tycho-2 stars using the search criteria of this paper, yielding recovery rates of 77.9% and 79.8% respectively. Objects missed in this UCAC3 survey are primarily due to UCAC3 lacking a source detection for $\sim$15% of the Tycho-2 objects. The relatively high recovery rates of UCAC3, when compared to the Hipparcos and Tycho-2 catalogs, implies the UCAC3 can be used as a reliable source to search for new proper motion stars with $\mu$ = 0.18–0.40$\arcsec$ yr-1 for other portions of the sky. ## 5 DISCUSSION We have completed a sweep of the southern sky for new proper motion systems using the UCAC3 catalog. So far, we have uncovered 916 new proper motion systems, of which 474 are described in this paper. These systems constitute an increase of 19.4% over the total number of SCR systems discovered in the southern sky and an increase of 20.7% over SCR systems in the southern sky with 0$\farcs$40 yr-1 $>$ $\mu$ $\geq$ 0$\farcs$18 yr-1. This UCAC3 proper motion survey has added 3.8% to the list of entries in the NLTT catalog south of Dec $=$ 0$\arcdeg$ with 974 new proper motion objects from U3PM1 and this paper. In Figure 6, we show the sky distribution of systems found to date during the UCAC3 proper motion survey. Plus signs represent objects from U3PM1 and solid circles represent objects described in this paper. Overall, the distribution of new objects is similar to that seen in Figure 6 of TSN25, including the discovery of many new proper motion systems along the Galactic plane. In Figure 7, we show a histogram of the number of proper motion systems discovered to date during the UCAC3 proper motion survey, in 0$\farcs$01 yr-1 bins, and highlighting the number of those having distance estimates within 50 pc. Predictably, this plot shows that the slowest proper motion bins have the most new systems. This confirms the trend reported in TSN18, TSN25 and U3PM1, and suggests once again that more nearby stars are yet to be found at slower proper motions. We have found a total of 57 CPM candidate systems during this UCAC3 proper motion survey, including 55 binaries and two triples. These systems have separations of 1–359$\arcsec$ and will need further investigation to confirm which of the systems are, in fact, gravitationally linked. In addition, we have revealed a total of 48 subdwarf candidates, each of which is worthy of followup observations, given the scarcity of nearby subdwarfs. Finally, we have found 20 red dwarf systems likely to be within 25 pc. We plan to obtain CCD photometry through $VRI$ filters for stars having estimated distances within 25 pc in order to make more reliable distance estimates using the $VRIJHK$ relations presented in (Henry et al., 2004). Stars estimated to be within 10 pc will then be put on the CTIO parallax program, potentially to join the ranks of the few hundred systems known to be so close to the Sun (Henry et al., 2006). We thank the entire UCAC team for making this proper motion survey possible, and the USNO summer students, who helped with tagging HPM stars in the UCAC3 catalog. Special thanks go to members of the RECONS team at Georgia State University for their support, and John Subasavage in particular for assistance with the SCR searches. This work has made use of the SIMBAD, VizieR, and Aladin databases operated at the CDS in Strasbourg, France. We have also made use of data from the Two-Micron All Sky Survey, SuperCOSMOS Science Archive and the Southern Proper Motion catalog. ## References * Bakos et al. (2002) Bakos, G. Á., Sahu, K. C., & Németh, P. 2002, ApJS, 141, 187 * Boyd et al. (2011) Boyd, M. R., Winters, J. G., Henry, T. J., Jao, W.-C., Finch, C. T., Subasavage, J. P., & Hambly, N. C. 2011, AJ, 142, 10 * Finch et al. (2007) Finch, C. T., Henry, T. J., Subasavage, J. P., Jao, W.-C., & Hambly, N. C. 2007, AJ, 133, 2898 * (4) Finch, C. T., Zacharias, N., & Wycoff, G. L. 2010, AJ, 139, 2200 * (5) Finch, C. T., Zacharias, N., & Henry, T. J. 2010, AJ, 140, 844 * Deacon et al. (2005) Deacon, N. R., Hambly, N. C., & Cooke, J. A. 2005, A&A, 435, 363 * Deacon & Hambly (2007) Deacon, N. R., & Hambly, N. C. 2007, A&A, 468, 163 * Deacon et al. (2009) Deacon, N. R., et al. 2009, MNRAS, 397, 1685 * Giclas et al. (1971) Giclas, H. L., Burnham, R., & Thomas, N. G. 1971, Flagstaff, Arizona: Lowell Observatory, 1971 * Giclas et al. (1978) Giclas, H. L., Burnham, R., & Thomas, N. G. 1978, Lowell Observatory Bulletin, 8, 89 * Girard et al. (2011) Girard, T. M., et al. 2011, AJ, 142, 15 * (12) Hambly, N. C., MacGillivray, H. T., Read, M. A., Tritton, S. B., Thomson, E. B., Kelly, B. D., Morgan, D. H., Smith, R. E., Driver, S. P., Williamson, J., Parker, Q. A., Hawkins, M. R. S., Williams, P. M., Lawrence, A. 2001, MNRAS, 326, 1279 * (13) Hambly, N. C., Davenhall, A. C., Irwin, M. J., & MacGillivray, H. T. 2001, MNRAS, 326, 1315 * Hambly et al. (2004) Hambly, N. C., Henry, T. J., Subasavage, J. P., Brown, M. A., & Jao, W. 2004, AJ, 128, 437 * Henry et al. (2006) Henry, T. J., Jao, W.-C., Subasavage, J. P., Beaulieu, T. D., Ianna, P. A., Costa, E., & Méndez, R. A. 2006, AJ, 132, 2360 * Henry et al. (2004) Henry, T. J., Subasavage, J. P., Brown, M. A., Beaulieu, T. D., Jao, W.-C., & Hambly, N. C. 2004, AJ, 128, 2460 * Høg et al. (2000) Høg, E., et al. 2000, A&A, 355, L27 * Lépine (2005) Lépine, S. 2005, AJ, 130, 1247 * Lépine (2008) Lépine, S. 2008, AJ, 135, 2177 * Luyten (1979) Luyten, W. J. 1979, LHS Catalogue (Minneapolis: Univ. of Minnesota Press) * Luyten (1980) Luyten, W. J. 1980, Proper Motion Survey with the 48-inch Telescope, Univ. Minnesota, 55, 1 (1980), 55, 1 * Oppenheimer et al. (2001) Oppenheimer, B. R., Hambly, N. C., Digby, A. P., Hodgkin, S. T., & Saumon, D. 2001, Science, 292, 698 * Pokorny et al. (2003) Pokorny, R. S., Jones, H. R. A., & Hambly, N. C. 2003, A&A, 397, 575 * Roeser et al. (2010) Roeser, S., Demleitner, M., & Schilbach, E. 2010, AJ, 139, 2440 * Salim & Gould (2003) Salim, S., & Gould, A. 2003, ApJ, 582, 1011 * Scholz et al. (2000) Scholz, R.-D., Irwin, M., Ibata, R., Jahreiß, H., & Malkov, O. Y. 2000, A&A, 353, 958 * Scholz et al. (2002) Scholz, R.-D., Szokoly, G. P., Andersen, M., Ibata, R., & Irwin, M. J. 2002, ApJ, 565, 539 * (28) Subasavage, J. P., Henry, T. J., Hambly, N. C., Brown, M. A., & Jao, W. C. 2005, AJ, 129, 413 * (29) Subasavage, J. P., Henry, T. J., Hambly, N. C., Brown, M. A., Jao, W. C., & Finch, C. T. 2005, AJ, 130, 1658 * Wroblewski & Torres (1994) Wroblewski, H., & Torres, C. 1994, A&AS, 105, 179 * Wroblewski & Costa (1999) Wroblewski, H., & Costa, E. 1999, A&AS, 139, 25 * Zacharias et al. (2000) Zacharias, N., et al. 2000, AJ, 120, 2131 * Zacharias et al. (2004) Zacharias, N., Urban, S. E., Zacharias, M. I., Wycoff, G. L., Hall, D. M., Monet, D. G., & Rafferty, T. J. 2004, AJ, 127, 3043 (UCAC2 paper) * Zacharias et al. (2008) Zacharias, N., Winter, L, Holdenried, E.R., De Cuyper,J.-P., Rafferty, T.J., Wycoff, G.L., 2008, PASP, 120, 644 (astro-ph 0806.0256) * Zacharias et al. (2010) Zacharias, N., et al. 2010, AJ, 139, 218 * Zacharias (2010) Zacharias, N. 2010, AJ, 139, 2208 * Zacharias et al. (2011) Zacharias, N. et al. in preparation (UCAC4 release paper) Figure 1: Color-apparent magnitude diagram for all proper motion systems in the sample having an $R_{59F}-J$ color. New proper motion objects are represented by solid circles while known objects (CPM companions to new objects) are represented with open triangles. Data below $R_{59F}=$ 17 are CPM candidates noticed during visual inspection. Figure 2: RPM diagram for all proper motion systems in this sample having an $R_{59F}-J$ color. New proper motion objects are represented by solid circles while known objects (CPM companions to new objects) are represented with open triangles. The empirical line separates the subdwarfs from where white dwarf candidates would be found. No white dwarf candidates were found in the current search. Figure 3: Comparisons of proper motions in each coordinate, $\mu_{\alpha}\cos\delta$ (top) and $\mu_{\delta}$ (bottom), for components in CPM systems. Proper motions from the UCAC3 catalog are represented by solid circles while proper motions manually obtained through other means are denoted by open circles. The solid line indicates perfect agreement. Information on the outliers can be found in $\S$4.4 Figure 4: Comparisons of UCAC3, Hipparcos and Tycho-2 proper motions per coordinate, $\Delta\mu_{\alpha}\cos\delta$ (left column) and $\Delta\mu_{\delta}$ (right column). Figure 5: Comparisons of UCAC3, SuperCOSMOS and SPM4 proper motions per coordinate, $\Delta\mu_{\alpha}\cos\delta$ (left column) and $\Delta\mu_{\delta}$ (right column). Figure 6: Sky distribution of all UCAC3 proper motion survey objects reported in U3PM1 (plus signs) and this paper (solid circles), i.e. those between declinations $-$90$\arcdeg$ and 0$\arcdeg$ having 0$\farcs$40 yr-1 $>$ $\mu$ $\geq$ 0$\farcs$18 yr-1. The curve represents the Galactic plane. Figure 7: Histogram showing the number of proper motion objects in 0$\farcs$01 yr-1 bins for the entire UCAC3 proper motion sample (empty bars) and the number of those objects having distance estimates within 50 pc (filled bars). Table 1: New Proper Motion Systems from the UCAC3 and SCR proper motion surveys Paper \| New Systems \| New Systems \| References ---\|---\|---\|--- \| total \| $\leq$ 25 pc \| U3PM1 \| 442 \| 15 \| (Finch et al. 2010b, ) U3PM2 \| 474 \| 4 \| this paper TSN08 \| 5 \| 2 \| (Hambly et al., 2004) TSN10 \| 4 \| 4 \| (Henry et al., 2004) TSN12 \| 141 \| 12 \| (Subasavage et al. 2005a, ) TSN15 \| 152 \| 25 \| (Subasavage et al. 2005b, ) TSN18 \| 1605 \| 30 \| (Finch et al., 2007) TSN25 \| 2817 \| 79 \| (Boyd et al., 2011) totals \| 5640 \| 171 \| Table 2: New UCAC3 High Proper Motion Systems between Declinations $-$47$\arcdeg$ and 0$\arcdeg$ with 0$\farcs$40 yr-1 $>$ $\mu$ $\geq$ 0$\farcs$18 yr-1 Name \| RA J2000.0 \| DEC J2000.0 \| $\mu_{\alpha}\cos\delta$ \| $\mu_{\delta}$ \| sig$\mu_{\alpha}$ \| sig$\mu_{\delta}$ \| $B_{J}$ \| $R_{59F}$ \| $I_{IVN}$ \| $J$ \| $H$ \| $K_{s}$ \| $R_{59F}$ $-$ $J$ \| Est Dist \| Notes ---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|--- \| (deg) \| (deg) \| (mas/yr) \| (mas/yr) \| (mas/yr) \| (mas/yr) \| \| \| \| \| \| \| \| (pc) \| UPM 0004-0833 \| 1.1472383 \| -8.5664958 \| 182.2 \| -26.3 \| 8.6 \| 8.6 \| 14.680 \| 12.649 \| 11.643 \| 10.874 \| 10.255 \| 10.072 \| 1.775 \| 63.8 \| UPM 0004-1258 \| 1.2061531 \| -12.9791222 \| 102.0 \| -156.8 \| 10.2 \| 6.6 \| 14.504 \| 12.478 \| 11.385 \| 10.774 \| 10.160 \| 9.973 \| 1.704 \| 62.3 \| UPM 0009-1539 \| 2.4914975 \| -15.6590597 \| 184.0 \| -15.7 \| 8.7 \| 8.7 \| 17.100 \| 15.050 \| 13.144 \| 12.387 \| 11.793 \| 11.562 \| 2.663 \| 93.0 \| UPM 0011-1448 \| 2.9555050 \| -14.8079581 \| 179.8 \| -34.9 \| 9.0 \| 9.0 \| 14.785 \| 13.131 \| 12.454 \| 12.005 \| 11.353 \| 11.246 \| 1.126 \| 115.6 \| UPM 0014-0029 \| 3.6589169 \| -0.4939803 \| 161.0 \| -104.7 \| 13.7 \| 14.0 \| 17.819 \| 15.829 \| 13.947 \| 12.292 \| 11.708 \| 11.457 \| 3.537 \| 58.7 \| UPM 0014-1219 \| 3.7410256 \| -12.3317842 \| 183.9 \| -21.6 \| 7.0 \| 3.9 \| 17.189 \| 15.068 \| 13.388 \| 12.808 \| 12.230 \| 11.974 \| 2.260 \| 130.3 \| UPM 0025-2547 \| 6.3606617 \| -25.7849942 \| 170.4 \| 65.2 \| 9.4 \| 9.3 \| 20.968 \| 18.826 \| 16.962 \| 15.167 \| 14.514 \| 14.163 \| 3.659 \| 179.8 \| aaProper motions suspect UPM 0044-1647 \| 11.1954375 \| -16.7984897 \| 187.8 \| 29.7 \| 14.6 \| 14.1 \| $\cdots$ \| $\cdots$ \| 13.280 \| 12.405 \| 11.777 \| 11.549 \| $\cdots$ \| 91.5 \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 0045-3602 \| 11.2618550 \| -36.0381975 \| 118.0 \| -136.9 \| 3.2 \| 2.4 \| 15.862 \| 13.863 \| 11.982 \| 11.398 \| 10.888 \| 10.610 \| 2.465 \| 67.0 \| UPM 0048-0217 \| 12.1068206 \| -2.2840133 \| 163.9 \| -75.0 \| 8.9 \| 8.3 \| 16.715 \| 14.628 \| 12.563 \| 11.042 \| 10.488 \| 10.194 \| 3.586 \| 31.8 \| UPM 0058-0158 \| 14.6628417 \| -1.9776981 \| -48.2 \| -181.1 \| 11.2 \| 11.2 \| 18.275 \| 16.265 \| 14.745 \| 13.405 \| 12.870 \| 12.662 \| 2.860 \| 141.3 \| UPM 0106-1342 \| 16.7276508 \| -13.7017203 \| 173.8 \| 95.7 \| 7.4 \| 7.4 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 12.372 \| 11.775 \| 11.689 \| $\cdots$ \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 0122-0003 \| 20.6797608 \| -0.0530356 \| 151.7 \| 162.8 \| 9.6 \| 9.1 \| $\cdots$ \| $\cdots$ \| 13.816 \| 12.612 \| 11.930 \| 11.729 \| $\cdots$ \| 76.4 \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 0126-0000 \| 21.7387347 \| -0.0141164 \| 188.1 \| 43.8 \| 12.2 \| 12.1 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 12.074 \| 11.374 \| 11.219 \| $\cdots$ \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 0137-0537 \| 24.3814961 \| -5.6331831 \| 236.7 \| -75.4 \| 8.4 \| 8.7 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 12.573 \| 12.028 \| 11.810 \| $\cdots$ \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 0155-4344 \| 28.9240619 \| -43.7373506 \| 148.1 \| 106.2 \| 2.0 \| 3.5 \| 16.195 \| 13.912 \| 12.128 \| 11.368 \| 10.829 \| 10.575 \| 2.544 \| 59.8 \| UPM 0202-4056 \| 30.6998675 \| -40.9468992 \| 180.1 \| -67.9 \| 2.6 \| 2.6 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 10.454 \| 9.754 \| 9.572 \| $\cdots$ \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable ccSuperCOSMOS plate magnitudes suspect UPM 0209-3339A \| 32.4333397 \| -33.6586869 \| -86.1 \| -166.9 \| 4.2 \| 2.0 \| 16.526 \| 14.533 \| 12.628 \| 11.437 \| 10.885 \| 10.605 \| 3.096 \| 49.5 \| ddCommon proper motion companion; see Table 4 UPM 0209-3339B \| 32.4371381 \| -33.6580217 \| -112.9 \| -170.2 \| 7.6 \| 8.3 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 12.025 \| 11.489 \| 11.131 \| $\cdots$ \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable ddCommon proper motion companion; see Table 4 eeNot detected during automated search but noticed by eye during the blinking process UPM 0218-0120 \| 34.5873781 \| -1.3488447 \| 166.5 \| -88.5 \| 9.9 \| 10.5 \| 13.640 \| 12.620 \| 12.263 \| 12.468 \| 12.109 \| 12.034 \| 0.152 \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 0238-1348 \| 39.6759683 \| -13.8016906 \| 173.3 \| -114.7 \| 9.9 \| 9.6 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 11.998 \| 11.375 \| 11.263 \| $\cdots$ \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 0241-1647 \| 40.2660111 \| -16.7848081 \| 16.5 \| -182.4 \| 11.2 \| 11.0 \| 15.606 \| 14.528 \| 14.078 \| 13.429 \| 12.940 \| 12.857 \| 1.099 \| [216.7] \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable ffSubdwarf candidate selected from RPM diagram; plate distance [in bracket] is incorrect UPM 0302-1307 \| 45.5532964 \| -13.1308544 \| 139.0 \| -117.1 \| 16.5 \| 14.2 \| 17.954 \| 16.054 \| 14.408 \| 13.364 \| 12.816 \| 12.536 \| 2.690 \| 146.8 \| UPM 0308-0532 \| 47.0978864 \| -5.5470081 \| 82.1 \| -175.0 \| 3.4 \| 4.6 \| 11.813 \| 10.766 \| 10.286 \| 10.222 \| 9.783 \| 9.698 \| 0.544 \| 53.7 \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 0314-0150 \| 48.5090014 \| -1.8370214 \| 114.2 \| -186.2 \| 8.8 \| 8.6 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 11.672 \| 11.088 \| 10.843 \| $\cdots$ \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 0314-0415 \| 48.6207794 \| -4.2554964 \| 145.6 \| -111.8 \| 9.0 \| 10.1 \| 14.625 \| 12.900 \| 12.119 \| 11.176 \| 10.525 \| 10.401 \| 1.724 \| 71.5 \| UPM 0330-0047 \| 52.7083053 \| -0.7902283 \| 193.7 \| -15.3 \| 3.3 \| 6.2 \| 15.120 \| 14.071 \| 13.434 \| 13.066 \| 12.604 \| 12.561 \| 1.005 \| [190.6] \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable ffSubdwarf candidate selected from RPM diagram; plate distance [in bracket] is incorrect UPM 0356-0828 \| 59.0545181 \| -8.4810458 \| 3.7 \| -184.8 \| 10.8 \| 12.5 \| $\cdots$ \| 11.147 \| 10.560 \| 10.892 \| 10.507 \| 10.400 \| 0.255 \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 0358-1617 \| 59.5358836 \| -16.2881436 \| 184.6 \| -37.0 \| 12.3 \| 12.5 \| 17.926 \| 15.806 \| 13.993 \| 12.933 \| 12.442 \| 12.212 \| 2.873 \| 115.0 \| UPM 0359-2449 \| 59.8231094 \| -24.8298217 \| -105.0 \| -149.1 \| 4.6 \| 4.6 \| 17.990 \| 15.911 \| 13.990 \| 12.757 \| 12.211 \| 11.956 \| 3.154 \| 88.6 \| UPM 0403-0635 \| 60.7777225 \| -6.5860825 \| 205.2 \| 13.7 \| 11.2 \| 10.6 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 12.140 \| 11.563 \| 11.276 \| $\cdots$ \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 0405-1951 \| 61.2543614 \| -19.8521608 \| 34.8 \| -179.9 \| 2.4 \| 2.4 \| $\cdots$ \| 13.753 \| 13.046 \| 12.967 \| 12.410 \| 12.355 \| 0.786 \| 199.6 \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable ccSuperCOSMOS plate magnitudes suspect UPM 0410-0742 \| 62.6567331 \| -7.7134942 \| 139.1 \| -175.2 \| 11.8 \| 10.8 \| 15.766 \| 13.474 \| 11.071 \| 10.642 \| 10.054 \| 9.770 \| 2.832 \| 37.0 \| UPM 0413-4212 \| 63.2980333 \| -42.2011461 \| 145.8 \| 124.2 \| 3.3 \| 3.1 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 12.229 \| 11.632 \| 11.368 \| $\cdots$ \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 0417-0431 \| 64.3450542 \| -4.5310789 \| 244.9 \| 78.0 \| 10.3 \| 10.6 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 11.392 \| 10.819 \| 10.545 \| $\cdots$ \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 0424-0307 \| 66.1372464 \| -3.1186994 \| 133.7 \| -176.5 \| 12.7 \| 15.8 \| 17.311 \| 14.694 \| 12.752 \| 12.087 \| 11.504 \| 11.214 \| 2.607 \| 72.7 \| UPM 0443-4129A \| 70.7825969 \| -41.4844111 \| 186.1 \| 4.3 \| 3.6 \| 3.8 \| 14.948 \| 12.775 \| 10.462 \| 10.422 \| 9.846 \| 9.594 \| 2.353 \| 39.3 \| aaProper motions suspect ddCommon proper motion companion; see Table 4 UPM 0446-4337 \| 71.5897783 \| -43.6211606 \| 59.2 \| 171.3 \| 3.6 \| 1.7 \| 12.496 \| 11.113 \| 10.479 \| 11.096 \| 10.602 \| 10.500 \| 0.017 \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 0454-4217 \| 73.5640600 \| -42.2886506 \| 159.7 \| 83.2 \| 10.9 \| 10.5 \| 16.433 \| 14.437 \| 12.943 \| 12.095 \| 11.431 \| 11.275 \| 2.342 \| 91.8 \| UPM 0456-1138 \| 74.1655053 \| -11.6350681 \| 30.5 \| -250.8 \| 11.5 \| 11.3 \| 17.224 \| 15.471 \| 13.518 \| 12.269 \| 11.716 \| 11.471 \| 3.202 \| 74.7 \| UPM 0502-1938 \| 75.6838831 \| -19.6478917 \| 54.4 \| 172.2 \| 4.0 \| 2.8 \| 16.562 \| 14.733 \| 12.874 \| 11.574 \| 10.917 \| 10.641 \| 3.159 \| 48.8 \| UPM 0507-1302 \| 76.8394261 \| -13.0453781 \| 184.1 \| -119.4 \| 9.8 \| 9.7 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 12.601 \| 12.004 \| 11.786 \| $\cdots$ \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 0508-0617 \| 77.0132517 \| -6.2872236 \| 170.4 \| -67.8 \| 7.5 \| 7.6 \| $\cdots$ \| 14.170 \| 12.642 \| 11.679 \| 11.078 \| 10.854 \| 2.491 \| 66.8 \| UPM 0521-0448 \| 80.2826481 \| -4.8134942 \| -20.5 \| -181.1 \| 4.5 \| 4.1 \| 14.255 \| 13.177 \| 12.882 \| 12.238 \| 11.770 \| 11.736 \| 0.939 \| 122.1 \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 0527-2006 \| 81.8761928 \| -20.1036386 \| 25.6 \| -183.7 \| 3.4 \| 3.4 \| $\cdots$ \| 10.973 \| 9.523 \| 11.168 \| 10.591 \| 10.335 \| -0.195 \| 60.1 \| aaProper motions suspect bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 0528-4313A \| 82.0375869 \| -43.2255300 \| -75.6 \| 164.7 \| 4.0 \| 4.0 \| 16.795 \| 14.566 \| 12.807 \| 11.889 \| 11.312 \| 11.067 \| 2.677 \| 70.4 \| ddCommon proper motion companion; see Table 4 UPM 0528-4313B \| 82.0298050 \| -43.2357586 \| -86.3 \| 163.2 \| 4.8 \| 5.5 \| 19.746 \| 17.653 \| 15.348 \| 13.943 \| 13.296 \| 13.015 \| 3.710 \| 109.1 \| ddCommon proper motion companion; see Table 4 eeNot detected during automated search but noticed by eye during the blinking process UPM 0532-1458 \| 83.2114242 \| -14.9684436 \| 183.3 \| -28.1 \| 10.2 \| 10.2 \| 16.626 \| 14.662 \| 12.946 \| 12.145 \| 11.623 \| 11.410 \| 2.517 \| 95.2 \| UPM 0534-1510 \| 83.6021194 \| -15.1696369 \| 177.5 \| -39.2 \| 8.5 \| 8.5 \| 14.461 \| 12.575 \| 11.551 \| 11.040 \| 10.429 \| 10.249 \| 1.535 \| 70.9 \| UPM 0540-2757 \| 85.0146869 \| -27.9660519 \| 33.1 \| -120.7 \| 3.6 \| 3.7 \| 18.317 \| 16.303 \| 14.451 \| 13.494 \| 12.946 \| 12.753 \| 2.809 \| 154.1 \| aaProper motions suspect eeNot detected during automated search but noticed by eye during the blinking process UPM 0542-4544 \| 85.6736511 \| -45.7491522 \| 64.1 \| 183.2 \| 4.3 \| 7.9 \| 15.664 \| 13.488 \| 11.376 \| 10.453 \| 9.854 \| 9.586 \| 3.035 \| 30.9 \| UPM 0545-0222 \| 86.3577581 \| -2.3675183 \| -54.4 \| -189.2 \| 10.0 \| 10.1 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 12.424 \| 11.861 \| 11.617 \| $\cdots$ \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 0556-3937 \| 89.0922544 \| -39.6272753 \| 84.1 \| -206.8 \| 4.7 \| 5.6 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 13.607 \| 13.063 \| 12.885 \| $\cdots$ \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 0557-1351 \| 89.2654756 \| -13.8644606 \| 25.1 \| -181.4 \| 5.6 \| 3.7 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 11.038 \| 10.475 \| 10.205 \| $\cdots$ \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 0558-3745 \| 89.7238911 \| -37.7586756 \| 140.6 \| -115.6 \| 3.9 \| 5.8 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 11.662 \| 11.035 \| 10.785 \| $\cdots$ \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 0601-1512 \| 90.4218800 \| -15.2024047 \| 105.0 \| -157.1 \| 7.5 \| 5.2 \| 14.492 \| 12.684 \| 11.569 \| 11.484 \| 10.841 \| 10.682 \| 1.200 \| 91.1 \| UPM 0602-1917 \| 90.6749681 \| -19.2984797 \| -132.2 \| -172.6 \| 10.4 \| 10.4 \| 15.906 \| 13.961 \| 12.074 \| 11.323 \| 10.858 \| 10.519 \| 2.638 \| 60.7 \| UPM 0603-1417 \| 90.8013183 \| -14.2981833 \| -184.4 \| -12.4 \| 4.5 \| 10.7 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 11.711 \| 11.092 \| 10.805 \| $\cdots$ \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 0603-1433 \| 90.8642300 \| -14.5603703 \| 49.8 \| -195.9 \| 3.6 \| 3.6 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 12.090 \| 11.491 \| 11.327 \| $\cdots$ \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 0614-3350 \| 93.6930136 \| -33.8383233 \| 235.6 \| 92.1 \| 10.9 \| 10.9 \| $\cdots$ \| 15.462 \| 14.028 \| 13.052 \| 12.565 \| 12.354 \| 2.410 \| 141.9 \| UPM 0615-0735 \| 93.9365817 \| -7.5865625 \| 96.7 \| -156.9 \| 11.5 \| 13.3 \| $\cdots$ \| 16.025 \| 14.794 \| 13.709 \| 13.130 \| 12.892 \| 2.316 \| 178.1 \| UPM 0620-0312 \| 95.1532386 \| -3.2076081 \| -19.4 \| -204.3 \| 10.5 \| 10.9 \| 16.634 \| 14.530 \| $\cdots$ \| 12.965 \| 12.383 \| 12.177 \| 1.565 \| [189.4] \| ffSubdwarf candidate selected from RPM diagram; plate distance [in bracket] is incorrect UPM 0629-1737 \| 97.4615906 \| -17.6326997 \| 143.0 \| -130.9 \| 6.4 \| 9.5 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 12.895 \| 12.363 \| 12.198 \| $\cdots$ \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 0639-0451 \| 99.9698181 \| -4.8652011 \| -229.8 \| 121.5 \| 7.5 \| 7.6 \| 16.020 \| 13.918 \| 12.144 \| 11.205 \| 10.624 \| 10.385 \| 2.713 \| 52.0 \| UPM 0641-0255 \| 100.2841003 \| -2.9306503 \| -72.7 \| -167.6 \| 18.7 \| 11.4 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 11.551 \| 10.989 \| 10.716 \| $\cdots$ \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 0645-0045 \| 101.2847811 \| -0.7658139 \| -168.9 \| -95.2 \| 7.2 \| 6.6 \| 17.124 \| 15.014 \| 13.172 \| 12.186 \| 11.639 \| 11.379 \| 2.828 \| 78.6 \| UPM 0652-0150 \| 103.0572631 \| -1.8487686 \| 64.1 \| -174.4 \| 1.4 \| 2.2 \| 13.647 \| 11.537 \| 10.484 \| 11.121 \| 10.480 \| 10.334 \| 0.416 \| 76.3 \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 0652-2243 \| 103.2459989 \| -22.7292186 \| -83.2 \| 169.2 \| 2.8 \| 3.2 \| $\cdots$ \| 14.509 \| 12.303 \| 11.303 \| 10.719 \| 10.419 \| 3.206 \| 43.4 \| UPM 0655-0715 \| 103.8760131 \| -7.2643236 \| 61.7 \| -240.9 \| 7.8 \| 7.9 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 10.392 \| 9.779 \| 9.563 \| $\cdots$ \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 0659-0052A \| 104.8421303 \| -0.8801367 \| -58.3 \| -184.1 \| 2.7 \| 1.8 \| 15.062 \| 12.642 \| 10.985 \| 11.253 \| 10.672 \| 10.466 \| 1.389 \| 78.2 \| ddCommon proper motion companion; see Table 4 UPM 0659-0052B \| 104.8439419 \| -0.8835767 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| $\cdots$ \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 13.969 \| 13.436 \| 13.172 \| $\cdots$ \| $\cdots$ \| aaProper motions suspect bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable ddCommon proper motion companion; see Table 4 eeNot detected during automated search but noticed by eye during the blinking process UPM 0702-2053 \| 105.5694072 \| -20.8837058 \| 82.0 \| -169.4 \| 8.2 \| 9.3 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 12.524 \| 11.899 \| 11.691 \| $\cdots$ \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 0704-0602A \| 106.1883350 \| -6.0386081 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 12.488 \| 11.140 \| 12.557 \| 11.932 \| 11.758 \| -0.069 \| 123.4 \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable aaProper motions suspect ccSuperCOSMOS plate magnitudes suspect ddCommon proper motion companion; see Table 4 eeNot detected during automated search but noticed by eye during the blinking process UPM 0704-0602B \| 106.1882444 \| -6.0352122 \| 99.5 \| -153.0 \| 5.0 \| 5.0 \| $\cdots$ \| 14.488 \| 13.710 \| 11.284 \| 10.723 \| 10.499 \| 3.204 \| 37.8 \| ddCommon proper motion companion; see Table 4 UPM 0704-2033 \| 106.1582275 \| -20.5532164 \| 90.1 \| -158.2 \| 10.3 \| 10.1 \| 17.435 \| 15.252 \| 13.485 \| 12.546 \| 12.008 \| 11.769 \| 2.706 \| 98.2 \| UPM 0705-2830 \| 106.3123578 \| -28.5031700 \| -23.5 \| 182.2 \| 4.7 \| 19.1 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 11.831 \| 11.247 \| 10.980 \| $\cdots$ \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 0706-2301 \| 106.5726647 \| -23.0297381 \| -164.1 \| 105.6 \| 8.5 \| 7.7 \| 17.987 \| 15.928 \| 14.413 \| 13.449 \| 12.936 \| 12.646 \| 2.479 \| 164.7 \| UPM 0708-2539 \| 107.2233878 \| -25.6601586 \| 115.2 \| -144.9 \| 13.0 \| 17.3 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 12.311 \| 11.694 \| 11.461 \| $\cdots$ \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 0716-2342 \| 109.2296050 \| -23.7101622 \| 139.8 \| -113.7 \| 9.7 \| 9.5 \| $\cdots$ \| 15.025 \| 12.960 \| 12.396 \| 11.860 \| 11.611 \| 2.629 \| 98.8 \| UPM 0724-0949 \| 111.1639358 \| -9.8236900 \| -192.9 \| 15.5 \| 5.0 \| 5.4 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 11.127 \| 10.519 \| 10.353 \| $\cdots$ \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 0725-0207 \| 111.4631742 \| -2.1191361 \| -185.1 \| -14.9 \| 5.0 \| 5.0 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 11.251 \| 10.700 \| 10.483 \| $\cdots$ \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 0730-4042 \| 112.7328597 \| -40.7005344 \| -237.6 \| 113.1 \| 7.4 \| 7.0 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 10.452 \| 9.848 \| 9.609 \| $\cdots$ \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 0736-4256 \| 114.0765108 \| -42.9425744 \| 171.4 \| -62.3 \| 14.4 \| 12.4 \| $\cdots$ \| 15.719 \| 14.674 \| 13.476 \| 12.860 \| 12.698 \| 2.243 \| 164.2 \| UPM 0740-2114 \| 115.1509197 \| -21.2383561 \| -192.2 \| 8.5 \| 14.0 \| 13.3 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 12.420 \| 11.875 \| 11.621 \| $\cdots$ \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 0740-3055 \| 115.1015369 \| -30.9323542 \| -235.9 \| 96.7 \| 7.7 \| 7.8 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 12.146 \| 11.616 \| 11.364 \| $\cdots$ \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 0742-2501 \| 115.5580681 \| -25.0244169 \| 176.9 \| 49.8 \| 9.7 \| 9.7 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 11.848 \| 11.231 \| 10.983 \| $\cdots$ \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 0745-4149 \| 116.2558594 \| -41.8251314 \| -89.8 \| 159.3 \| 2.2 \| 2.4 \| $\cdots$ \| 14.674 \| 13.178 \| 12.175 \| 11.583 \| 11.358 \| 2.499 \| 83.7 \| UPM 0746-3729 \| 116.5312928 \| -37.4876147 \| 11.2 \| -182.4 \| 4.4 \| 4.2 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 12.527 \| 11.994 \| 11.710 \| $\cdots$ \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable ccSuperCOSMOS plate magnitudes suspect UPM 0747-0320 \| 116.8618994 \| -3.3466481 \| -158.2 \| 89.5 \| 7.0 \| 7.6 \| $\cdots$ \| 14.709 \| 12.737 \| 11.930 \| 11.345 \| 11.126 \| 2.779 \| 71.9 \| UPM 0747-2537A \| 116.9783592 \| -25.6193264 \| -148.5 \| 101.9 \| 4.2 \| 2.1 \| $\cdots$ \| 9.801 \| 8.849 \| 9.593 \| 8.958 \| 8.814 \| 0.208 \| 40.6 \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable ddCommon proper motion companion; see Table 4 UPM 0747-2537B \| 116.9752431 \| -25.6211242 \| -151.3 \| 102.3 \| 3.5 \| 3.5 \| $\cdots$ \| 13.947 \| 13.441 \| 11.500 \| 10.965 \| 10.741 \| 2.447 \| 47.3 \| ddCommon proper motion companion; see Table 4 UPM 0748-0619 \| 117.1063369 \| -6.3225081 \| 109.8 \| -155.9 \| 6.1 \| 6.2 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 10.356 \| 9.749 \| 9.579 \| $\cdots$ \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 0750-1807 \| 117.5875447 \| -18.1294489 \| -182.7 \| 45.5 \| 7.0 \| 6.3 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 12.248 \| 11.756 \| 11.471 \| $\cdots$ \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 0751-0214 \| 117.7670006 \| -2.2466175 \| 9.8 \| -211.2 \| 13.9 \| 11.7 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 12.186 \| 11.708 \| 11.441 \| $\cdots$ \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 0752-0751 \| 118.0258133 \| -7.8544356 \| 12.8 \| -195.6 \| 5.9 \| 5.9 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 10.377 \| 9.794 \| 9.516 \| $\cdots$ \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 0752-1602 \| 118.1642550 \| -16.0351689 \| -86.0 \| -193.7 \| 7.3 \| 7.5 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 11.770 \| 11.214 \| 10.983 \| $\cdots$ \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 0800-0617A \| 120.2021822 \| -6.2902914 \| 135.2 \| -233.8 \| 9.0 \| 9.4 \| $\cdots$ \| 14.669 \| $\cdots$ \| 12.944 \| 12.321 \| 12.176 \| 1.725 \| [175.5] \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable ddCommon proper motion companion; see Table 4 ffSubdwarf candidate selected from RPM diagram; plate distance [in bracket] is incorrect UPM 0800-0617B \| 120.2007742 \| -6.2896208 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| $\cdots$ \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 14.720 \| 14.193 \| 13.983 \| $\cdots$ \| $\cdots$ \| aaProper motions suspect bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable ddCommon proper motion companion; see Table 4 eeNot detected during automated search but noticed by eye during the blinking process UPM 0801-1005 \| 120.4445300 \| -10.0905344 \| 177.8 \| -90.1 \| 3.8 \| 6.2 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 9.969 \| 9.388 \| 9.154 \| $\cdots$ \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 0801-4347 \| 120.4321706 \| -43.7879044 \| -100.9 \| 158.0 \| 3.6 \| 3.4 \| 14.186 \| 11.870 \| 10.471 \| 10.146 \| 9.546 \| 9.327 \| 1.724 \| 44.4 \| UPM 0802-2010 \| 120.6659497 \| -20.1753642 \| 191.5 \| -210.6 \| 7.7 \| 7.7 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 11.925 \| 11.310 \| 11.141 \| $\cdots$ \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 0803-4518 \| 120.7579836 \| -45.3141208 \| 49.4 \| 199.4 \| 3.3 \| 3.3 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 11.645 \| 11.109 \| 10.813 \| $\cdots$ \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 0805-3827 \| 121.3922181 \| -38.4596086 \| 153.3 \| -151.8 \| 6.6 \| 6.2 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 12.108 \| 11.584 \| 11.251 \| $\cdots$ \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 0807-0121 \| 121.8875364 \| -1.3593131 \| -178.8 \| 94.3 \| 10.2 \| 10.5 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 11.071 \| 10.473 \| 10.277 \| $\cdots$ \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 0807-0930 \| 121.8525969 \| -9.5078494 \| 197.5 \| 32.8 \| 3.9 \| 12.3 \| 13.602 \| 11.228 \| 9.767 \| 9.758 \| 9.148 \| 8.921 \| 1.470 \| 38.5 \| UPM 0807-2025 \| 121.8779764 \| -20.4218667 \| 110.5 \| -144.7 \| 3.0 \| 3.0 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 10.948 \| 10.335 \| 10.137 \| $\cdots$ \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 0808-0943 \| 122.0687272 \| -9.7311883 \| 98.7 \| -173.6 \| 4.5 \| 3.6 \| 14.522 \| 12.217 \| 10.408 \| 10.195 \| 9.679 \| 9.442 \| 2.022 \| 43.2 \| UPM 0809-0943 \| 122.2784283 \| -9.7207692 \| 147.0 \| -157.5 \| 8.1 \| 7.0 \| 15.935 \| 13.769 \| 12.187 \| 11.895 \| 11.357 \| 11.154 \| 1.874 \| 101.9 \| UPM 0809-4519 \| 122.4626147 \| -45.3295575 \| 204.4 \| -10.1 \| 12.6 \| 12.6 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 10.236 \| 9.686 \| 9.446 \| $\cdots$ \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 0813-0429 \| 123.3408675 \| -4.4854875 \| -32.9 \| 225.8 \| 8.2 \| 9.0 \| $\cdots$ \| 15.922 \| $\cdots$ \| 13.271 \| 12.743 \| 12.499 \| 2.651 \| 137.2 \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 0813-1048 \| 123.3918000 \| -10.8166544 \| -147.7 \| -139.7 \| 10.7 \| 10.8 \| 17.723 \| 15.527 \| 13.202 \| 12.094 \| 11.610 \| 11.292 \| 3.433 \| 58.5 \| UPM 0813-4604 \| 123.2772747 \| -46.0709556 \| -76.9 \| 168.7 \| 3.0 \| 3.1 \| 15.131 \| 13.421 \| 11.180 \| 10.955 \| 10.281 \| 10.073 \| 2.466 \| 53.7 \| UPM 0814-0835 \| 123.6868333 \| -8.5917794 \| -221.5 \| 113.0 \| 7.4 \| 6.7 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 11.087 \| 10.481 \| 10.297 \| $\cdots$ \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 0814-3645 \| 123.7166631 \| -36.7522964 \| -80.9 \| -167.1 \| 10.2 \| 4.1 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 11.660 \| 11.150 \| 10.963 \| $\cdots$ \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 0815-3058 \| 123.7880314 \| -30.9746633 \| -170.6 \| 205.9 \| 6.1 \| 6.2 \| $\cdots$ \| $\cdots$ \| 11.245 \| 11.346 \| 10.782 \| 10.473 \| $\cdots$ \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 0821-1452 \| 125.3709208 \| -14.8695183 \| -22.6 \| -183.9 \| 3.6 \| 5.5 \| 12.697 \| $\cdots$ \| $\cdots$ \| 10.073 \| 9.481 \| 9.376 \| $\cdots$ \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 0821-4626 \| 125.4251800 \| -46.4334139 \| -73.3 \| 166.4 \| 3.3 \| 3.3 \| 15.017 \| 13.307 \| 11.553 \| 11.528 \| 10.980 \| 10.748 \| 1.779 \| 82.2 \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 0826-3942 \| 126.5266758 \| -39.7096897 \| -98.3 \| 196.1 \| 6.2 \| 6.3 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 10.745 \| 10.153 \| 9.911 \| $\cdots$ \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 0828-1438 \| 127.0004200 \| -14.6352944 \| -166.9 \| 87.9 \| 8.4 \| 5.0 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 12.204 \| 11.541 \| 11.285 \| $\cdots$ \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 0832-3942 \| 128.0769122 \| -39.7043753 \| 224.2 \| 4.0 \| 14.0 \| 13.9 \| 18.269 \| 16.870 \| 14.829 \| 13.047 \| 12.517 \| 12.256 \| 3.823 \| 87.7 \| UPM 0836-0059 \| 129.0747936 \| -0.9920019 \| 109.8 \| -147.3 \| 6.8 \| 6.7 \| 15.555 \| 13.318 \| 11.750 \| 10.932 \| 10.342 \| 10.051 \| 2.386 \| 49.2 \| UPM 0838-3247 \| 129.6943083 \| -32.7966136 \| -106.7 \| 147.5 \| 4.6 \| 3.9 \| $\cdots$ \| 14.149 \| 12.693 \| 12.312 \| 11.637 \| 11.492 \| 1.837 \| 115.5 \| UPM 0840-3641 \| 130.0511394 \| -36.6993433 \| -173.6 \| 49.8 \| 4.9 \| 3.8 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 11.154 \| 10.611 \| 10.320 \| $\cdots$ \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable ccSuperCOSMOS plate magnitudes suspect UPM 0840-4437 \| 130.1662658 \| -44.6211017 \| -84.9 \| 161.1 \| 2.2 \| 2.2 \| $\cdots$ \| 12.483 \| 11.812 \| 11.471 \| 11.056 \| 11.013 \| 1.012 \| 106.9 \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 0842-0302 \| 130.6869978 \| -3.0436728 \| 51.0 \| -176.8 \| 7.3 \| 7.3 \| $\cdots$ \| 15.299 \| 13.728 \| 12.879 \| 12.353 \| 12.126 \| 2.420 \| 128.1 \| UPM 0842-0907 \| 130.5108458 \| -9.1258817 \| 130.2 \| -126.5 \| 7.0 \| 7.2 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 10.276 \| 9.675 \| 9.407 \| $\cdots$ \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 0842-4532 \| 130.7227961 \| -45.5387481 \| -151.7 \| 98.1 \| 6.7 \| 4.4 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 12.610 \| 12.124 \| 11.903 \| $\cdots$ \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 0843-3209 \| 130.9895922 \| -32.1573656 \| -165.4 \| 76.0 \| 3.2 \| 2.6 \| 15.446 \| 13.483 \| 12.412 \| 11.449 \| 10.860 \| 10.650 \| 2.034 \| 77.5 \| UPM 0846-2639 \| 131.5501986 \| -26.6632953 \| -136.2 \| 122.8 \| 2.5 \| 2.6 \| $\cdots$ \| $\cdots$ \| 11.530 \| 11.086 \| 10.434 \| 10.236 \| $\cdots$ \| 62.3 \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 0848-2542 \| 132.1816267 \| -25.7036467 \| -181.8 \| 7.0 \| 1.8 \| 6.2 \| $\cdots$ \| 13.801 \| 12.021 \| 11.335 \| 10.765 \| 10.529 \| 2.466 \| 61.3 \| UPM 0850-3052 \| 132.6042867 \| -30.8716572 \| 73.5 \| -167.7 \| 5.9 \| 3.8 \| 17.121 \| 14.799 \| 13.286 \| 12.239 \| 11.726 \| 11.439 \| 2.560 \| 86.4 \| UPM 0856-1741 \| 134.0761817 \| -17.6836933 \| -124.7 \| 134.3 \| 1.7 \| 2.9 \| 17.414 \| 15.472 \| 13.932 \| 12.293 \| 11.719 \| 11.460 \| 3.179 \| 69.5 \| UPM 0856-2909 \| 134.0409614 \| -29.1542111 \| 255.3 \| 2.9 \| 9.1 \| 8.8 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 11.988 \| 11.453 \| 11.215 \| $\cdots$ \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 0900-4308 \| 135.2269369 \| -43.1379197 \| -177.1 \| 38.5 \| 3.9 \| 2.1 \| $\cdots$ \| 15.782 \| 13.872 \| 12.083 \| 11.549 \| 11.323 \| 3.699 \| 47.7 \| UPM 0913-4303 \| 138.2865189 \| -43.0630097 \| -167.0 \| 78.6 \| 3.8 \| 4.0 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 12.673 \| 12.118 \| 11.868 \| $\cdots$ \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 0919-3821 \| 139.7869492 \| -38.3544064 \| 181.0 \| -27.6 \| 10.7 \| 5.2 \| 18.201 \| 16.059 \| 13.594 \| 12.398 \| 11.839 \| 11.577 \| 3.661 \| 60.3 \| UPM 0923-2518 \| 140.8532544 \| -25.3069511 \| -174.6 \| 66.3 \| 5.1 \| 5.0 \| 18.212 \| 16.305 \| 14.672 \| 13.104 \| 12.527 \| 12.241 \| 3.201 \| 99.2 \| UPM 0934-4355 \| 143.5423842 \| -43.9186631 \| -178.6 \| -36.3 \| 6.1 \| 4.5 \| $\cdots$ \| 15.225 \| 12.743 \| 11.673 \| 11.138 \| 10.879 \| 3.552 \| 49.2 \| UPM 0936-4557 \| 144.2119747 \| -45.9659628 \| -175.8 \| 51.4 \| 3.4 \| 3.3 \| 12.959 \| 11.119 \| 9.990 \| 9.830 \| 9.194 \| 9.028 \| 1.289 \| 41.8 \| UPM 0937-0014 \| 144.4922225 \| -0.2380892 \| -186.6 \| -52.5 \| 6.9 \| 7.1 \| 17.167 \| 15.231 \| 14.180 \| 13.184 \| 12.723 \| 12.483 \| 2.047 \| 184.6 \| UPM 0937-3214 \| 144.4728728 \| -32.2365067 \| -92.6 \| 155.7 \| 3.6 \| 3.4 \| $\cdots$ \| 15.513 \| 14.370 \| 13.263 \| 12.703 \| 12.492 \| 2.250 \| 152.5 \| UPM 0940-3918 \| 145.0291622 \| -39.3121189 \| -75.3 \| 164.2 \| 6.9 \| 6.9 \| $\cdots$ \| 13.681 \| 11.832 \| 11.519 \| 10.957 \| 10.738 \| 2.162 \| 76.4 \| UPM 0941-4439 \| 145.3561000 \| -44.6615028 \| -203.9 \| -71.1 \| 11.3 \| 11.0 \| $\cdots$ \| 15.616 \| 13.837 \| 12.855 \| 12.327 \| 12.069 \| 2.761 \| 109.0 \| UPM 0941-4518 \| 145.3343672 \| -45.3073242 \| -174.8 \| 89.0 \| 10.9 \| 6.3 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 11.682 \| 11.132 \| 10.884 \| $\cdots$ \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 0948-4147 \| 147.2015628 \| -41.7863742 \| -245.0 \| -69.8 \| 8.4 \| 8.2 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 10.646 \| 10.069 \| 9.803 \| $\cdots$ \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 0951-4429 \| 147.7806564 \| -44.4948356 \| -143.8 \| 109.8 \| 3.5 \| 2.1 \| 16.883 \| 14.919 \| 12.584 \| 11.698 \| 11.158 \| 10.966 \| 3.221 \| 58.8 \| UPM 0956-0940 \| 149.2293078 \| -9.6741689 \| -192.7 \| -24.4 \| 2.5 \| 2.0 \| 14.329 \| $\cdots$ \| 10.369 \| 10.191 \| 9.584 \| 9.352 \| $\cdots$ \| 43.6 \| UPM 1003-2717 \| 150.9001517 \| -27.2959828 \| -274.6 \| -39.5 \| 10.7 \| 10.2 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 10.565 \| 10.007 \| 9.712 \| $\cdots$ \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 1006-1144 \| 151.5145706 \| -11.7484167 \| 66.9 \| -170.5 \| 2.0 \| 2.0 \| 13.619 \| 11.759 \| 10.923 \| 11.193 \| 10.586 \| 10.439 \| 0.566 \| 90.8 \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 1009-0501B \| 152.4402375 \| -5.0217161 \| -190.5 \| 92.1 \| 7.4 \| 7.2 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 12.645 \| 12.105 \| 11.889 \| $\cdots$ \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable ddCommon proper motion companion; see Table 4 UPM 1011-4235 \| 152.9185411 \| -42.5901381 \| -178.6 \| -25.9 \| 2.8 \| 2.8 \| 15.694 \| 13.413 \| $\cdots$ \| 11.073 \| 10.461 \| 10.227 \| 2.340 \| 54.9 \| UPM 1015-0859 \| 153.8675097 \| -8.9998406 \| -151.3 \| 101.4 \| 2.0 \| 2.0 \| 14.108 \| 12.279 \| 11.435 \| 10.688 \| 10.044 \| 9.917 \| 1.591 \| 59.7 \| UPM 1020-0633A \| 155.2034769 \| -6.5554489 \| -179.8 \| -27.8 \| 7.3 \| 7.2 \| 16.023 \| 13.625 \| 11.335 \| 10.670 \| 10.073 \| 9.809 \| 2.955 \| 34.8 \| ddCommon proper motion companion; see Table 4 UPM 1020-0642 \| 155.0026631 \| -6.7037208 \| -211.0 \| 70.6 \| 7.6 \| 7.6 \| 16.635 \| 14.687 \| 12.883 \| 11.920 \| 11.350 \| 11.148 \| 2.767 \| 75.2 \| UPM 1024-0317 \| 156.1723939 \| -3.2861519 \| -143.2 \| -126.3 \| 7.9 \| 7.9 \| 16.780 \| 14.686 \| 12.708 \| 11.846 \| 11.276 \| 11.025 \| 2.840 \| 67.0 \| UPM 1030-2400 \| 157.5368944 \| -24.0092022 \| -164.3 \| 76.2 \| 4.0 \| 2.0 \| 16.458 \| 14.603 \| 12.580 \| 11.291 \| 10.690 \| 10.449 \| 3.312 \| 42.7 \| UPM 1031-0024A \| 157.7905397 \| -0.4118389 \| -207.4 \| -105.6 \| 11.4 \| 10.3 \| 14.016 \| 11.944 \| 10.371 \| 10.561 \| 10.048 \| 9.747 \| 1.383 \| 55.1 \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable ddCommon proper motion companion; see Table 4 UPM 1031-0024B \| 157.7925844 \| -0.4118883 \| -142.5 \| -96.9 \| 6.3 \| 6.3 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| $\cdots$ \| $\cdots$ \| $\cdots$ \| $\cdots$ \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable ddCommon proper motion companion; see Table 4 eeNot detected during automated search but noticed by eye during the blinking process UPM 1041-3913 \| 160.2807536 \| -39.2238969 \| 38.1 \| -182.1 \| 1.6 \| 14.5 \| 14.487 \| 12.444 \| 11.094 \| 10.148 \| 9.600 \| 9.347 \| 2.296 \| 38.6 \| UPM 1044-2414 \| 161.2433072 \| -24.2419103 \| -178.9 \| 57.3 \| 11.1 \| 9.2 \| 16.647 \| 14.667 \| 13.149 \| 12.003 \| 11.441 \| 11.199 \| 2.664 \| 78.4 \| UPM 1046-3046 \| 161.5409683 \| -30.7693019 \| -84.2 \| -159.9 \| 3.6 \| 3.6 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 12.983 \| 12.405 \| 12.164 \| $\cdots$ \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 1048-1538 \| 162.1992581 \| -15.6490136 \| -177.7 \| 42.4 \| 3.3 \| 4.4 \| 17.396 \| 15.382 \| 13.938 \| 13.043 \| 12.412 \| 12.221 \| 2.339 \| 141.8 \| UPM 1056-0542A \| 164.1791192 \| -5.7066733 \| -98.1 \| -173.8 \| 8.2 \| 8.3 \| 15.970 \| 14.281 \| 12.816 \| 11.682 \| 11.141 \| 10.939 \| 2.599 \| 76.5 \| ddCommon proper motion companion; see Table 4 UPM 1056-0542B \| 164.1816494 \| -5.7064783 \| -63.9 \| -173.3 \| 4.4 \| 4.1 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 12.485 \| 11.963 \| 11.709 \| $\cdots$ \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable ddCommon proper motion companion; see Table 4 eeNot detected during automated search but noticed by eye during the blinking process UPM 1058-4441 \| 164.5727686 \| -44.6861050 \| -168.3 \| 66.0 \| 9.8 \| 4.2 \| $\cdots$ \| 16.249 \| 15.218 \| 13.872 \| 13.176 \| 13.037 \| 2.377 \| 175.5 \| UPM 1059-0020 \| 164.9792050 \| -0.3418581 \| 102.1 \| -152.1 \| 2.9 \| 5.0 \| $\cdots$ \| 11.771 \| $\cdots$ \| 10.356 \| 9.748 \| 9.586 \| 1.415 \| 57.3 \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 1059-3022 \| 164.7671950 \| -30.3801764 \| -174.6 \| 59.9 \| 2.1 \| 2.1 \| 12.886 \| 11.393 \| 10.988 \| 10.299 \| 9.826 \| 9.725 \| 1.094 \| 50.7 \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 1108-0644 \| 167.1859397 \| -6.7383533 \| -205.9 \| -1.5 \| 8.7 \| 8.5 \| 14.710 \| 12.648 \| 11.392 \| 10.927 \| 10.290 \| 10.088 \| 1.721 \| 65.0 \| UPM 1109-1032 \| 167.3681956 \| -10.5479519 \| -162.3 \| 89.0 \| 10.1 \| 10.1 \| 17.490 \| 15.334 \| 13.246 \| 12.484 \| 11.943 \| 11.663 \| 2.850 \| 89.5 \| UPM 1113-0113 \| 168.3352431 \| -1.2199069 \| -184.6 \| 23.4 \| 10.5 \| 10.0 \| 17.881 \| 15.813 \| 14.194 \| 12.982 \| 12.530 \| 12.253 \| 2.831 \| 119.8 \| UPM 1113-0148 \| 168.3518608 \| -1.8163381 \| -174.3 \| -64.9 \| 7.4 \| 11.6 \| 17.348 \| 15.208 \| 13.359 \| 12.395 \| 11.858 \| 11.551 \| 2.813 \| 84.9 \| UPM 1122-4530 \| 170.6087600 \| -45.5091333 \| -179.3 \| -29.6 \| 1.9 \| 1.9 \| 17.065 \| 15.177 \| 12.936 \| 11.666 \| 11.061 \| 10.816 \| 3.511 \| 46.4 \| UPM 1130-1622 \| 172.5409028 \| -16.3801233 \| -200.6 \| 57.3 \| 6.6 \| 7.0 \| 17.842 \| 15.713 \| 14.155 \| 13.372 \| 12.854 \| 12.562 \| 2.341 \| 166.1 \| aaProper motions suspect UPM 1131-0725 \| 172.9437592 \| -7.4266169 \| -195.6 \| -5.9 \| 9.9 \| 9.5 \| 14.975 \| 13.881 \| 13.264 \| 13.077 \| 12.616 \| 12.538 \| 0.804 \| 194.3 \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 1133-2302 \| 173.3741581 \| -23.0458669 \| -191.9 \| 0.6 \| 1.7 \| 1.7 \| 16.127 \| 14.100 \| 12.923 \| 12.523 \| 11.928 \| 11.745 \| 1.577 \| 145.4 \| UPM 1136-0525 \| 174.1784361 \| -5.4211494 \| 158.0 \| -140.2 \| 13.6 \| 13.5 \| 17.294 \| 15.429 \| 13.996 \| 12.813 \| 12.249 \| 11.989 \| 2.616 \| 117.1 \| UPM 1138-4553 \| 174.6189286 \| -45.8979317 \| -191.9 \| 1.9 \| 3.8 \| 2.5 \| 16.204 \| 14.087 \| 12.336 \| 11.305 \| 10.616 \| 10.411 \| 2.782 \| 49.2 \| UPM 1142-2055A \| 175.5808186 \| -20.9279653 \| -186.7 \| 44.2 \| 2.4 \| 2.5 \| 14.178 \| 11.408 \| 10.020 \| 10.028 \| 9.348 \| 9.172 \| 1.380 \| 41.2 \| ddCommon proper motion companion; see Table 4 UPM 1142-2055B \| 175.5814128 \| -20.9302367 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| $\cdots$ \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 12.223 \| 11.647 \| 11.370 \| $\cdots$ \| $\cdots$ \| aaProper motions suspect bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable ddCommon proper motion companion; see Table 4 eeNot detected during automated search but noticed by eye during the blinking process UPM 1143-4443 \| 175.8747928 \| -44.7271675 \| -191.0 \| 2.6 \| 5.1 \| 4.0 \| 18.119 \| 16.055 \| 14.341 \| 12.943 \| 12.399 \| 12.169 \| 3.112 \| 99.0 \| UPM 1149-0019B \| 177.2613256 \| -0.3233508 \| -201.5 \| 2.2 \| 6.8 \| 3.2 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 9.963 \| 9.345 \| 9.145 \| $\cdots$ \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable ddCommon proper motion companion; see Table 4 UPM 1153-1747 \| 178.3206078 \| -17.7875069 \| -186.5 \| -58.7 \| 1.3 \| 1.3 \| 15.381 \| 13.160 \| 11.557 \| 10.957 \| 10.352 \| 10.110 \| 2.203 \| 55.0 \| UPM 1158-0912 \| 179.6672408 \| -9.2024986 \| -189.9 \| 52.2 \| 9.2 \| 9.4 \| 16.060 \| 14.074 \| 12.342 \| 11.466 \| 10.893 \| 10.655 \| 2.608 \| 63.3 \| UPM 1159-0339 \| 179.9315281 \| -3.6635244 \| -36.4 \| -193.0 \| 12.2 \| 13.0 \| $\cdots$ \| 15.447 \| 13.581 \| 12.770 \| 12.177 \| 11.932 \| 2.677 \| 106.6 \| UPM 1159-3623A \| 179.9150078 \| -36.3841397 \| -182.1 \| -101.4 \| 12.7 \| 13.1 \| $\cdots$ \| 15.642 \| 14.060 \| 12.934 \| 12.402 \| 12.169 \| 2.708 \| 113.1 \| ddCommon proper motion companion; see Table 4 UPM 1159-3623B \| 179.9110964 \| -36.3820372 \| -172.6 \| -92.4 \| 8.4 \| 6.5 \| $\cdots$ \| 18.260 \| 16.344 \| 14.447 \| 14.007 \| 13.646 \| 3.813 \| 132.4 \| ddCommon proper motion companion; see Table 4 eeNot detected during automated search but noticed by eye during the blinking process UPM 1202-0625 \| 180.5348450 \| -6.4258914 \| 87.6 \| -161.6 \| 5.7 \| 7.2 \| 14.976 \| 12.697 \| 11.068 \| 10.489 \| 9.857 \| 9.595 \| 2.208 \| 42.5 \| UPM 1203-0053 \| 180.8805247 \| -0.8924633 \| -208.2 \| -4.3 \| 7.3 \| 3.9 \| $\cdots$ \| 14.348 \| 13.979 \| 13.036 \| 12.618 \| 12.547 \| 1.312 \| [183.7] \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable ffSubdwarf candidate selected from RPM diagram; plate distance [in bracket] is incorrect UPM 1206-1851 \| 181.5521064 \| -18.8579533 \| -190.4 \| -26.2 \| 5.6 \| 5.9 \| $\cdots$ \| 15.193 \| 14.577 \| 13.870 \| 13.261 \| 13.128 \| 1.323 \| [255.4] \| ffSubdwarf candidate selected from RPM diagram; plate distance [in bracket] is incorrect UPM 1208-0904 \| 182.0832711 \| -9.0831478 \| -202.5 \| 129.8 \| 9.2 \| 9.0 \| 16.076 \| 13.897 \| $\cdots$ \| 12.038 \| 11.475 \| 11.253 \| 1.859 \| 113.1 \| UPM 1209-0721 \| 182.4658233 \| -7.3607258 \| -185.8 \| 7.3 \| 9.5 \| 9.4 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 11.706 \| 11.107 \| 10.926 \| $\cdots$ \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 1214-4016 \| 183.6045086 \| -40.2807997 \| -180.5 \| 3.8 \| 14.4 \| 2.7 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 11.455 \| 10.789 \| 10.516 \| $\cdots$ \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 1217-1032 \| 184.2883256 \| -10.5362844 \| -141.8 \| -116.8 \| 6.0 \| 4.4 \| 17.627 \| 15.433 \| 13.371 \| 12.421 \| 11.834 \| 11.602 \| 3.012 \| 79.2 \| UPM 1219-0238 \| 184.8517200 \| -2.6367969 \| -63.6 \| -186.1 \| 5.0 \| 3.5 \| 15.593 \| 14.171 \| 13.591 \| 13.297 \| 12.765 \| 12.675 \| 0.874 \| 204.3 \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 1223-1757 \| 185.8103789 \| -17.9516678 \| -183.7 \| 56.5 \| 3.3 \| 2.4 \| $\cdots$ \| 14.157 \| 12.414 \| 11.911 \| 11.341 \| 11.106 \| 2.246 \| 87.1 \| UPM 1223-2947 \| 185.9150278 \| -29.7938564 \| -158.1 \| 97.2 \| 2.5 \| 2.5 \| 16.417 \| 14.610 \| 12.933 \| 11.803 \| 11.181 \| 10.935 \| 2.807 \| 66.8 \| UPM 1226-2020A \| 186.6684756 \| -20.3442347 \| -137.6 \| -119.8 \| 8.0 \| 14.1 \| 13.216 \| 11.819 \| 11.024 \| 10.896 \| 10.327 \| 10.198 \| 0.923 \| 72.3 \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable ddCommon proper motion companion; see Table 4 UPM 1226-2020B \| 186.6675500 \| -20.3425083 \| -146.3 \| -117.3 \| 5.6 \| 5.7 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 13.457 \| 12.901 \| 12.690 \| $\cdots$ \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable ddCommon proper motion companion; see Table 4 eeNot detected during automated search but noticed by eye during the blinking process UPM 1226-3516B \| 186.6989514 \| -35.2778583 \| -200.5 \| 38.3 \| 7.0 \| 5.2 \| 19.238 \| 17.467 \| 15.558 \| 13.823 \| 13.303 \| 13.082 \| 3.644 \| 127.5 \| ddCommon proper motion companion; see Table 4 eeNot detected during automated search but noticed by eye during the blinking process UPM 1226-3516C \| 186.7202739 \| -35.2868633 \| -115.2 \| 8.5 \| 7.7 \| 6.1 \| 19.891 \| 18.036 \| 16.387 \| 14.792 \| 14.321 \| 14.107 \| 3.244 \| 243.4 \| aaProper motions suspect ddCommon proper motion companion; see Table 4 eeNot detected during automated search but noticed by eye during the blinking process UPM 1229-3351 \| 187.4657011 \| -33.8588425 \| -276.0 \| -141.9 \| 15.2 \| 15.2 \| 17.955 \| 16.382 \| $\cdots$ \| 13.235 \| 12.747 \| 12.476 \| 3.147 \| 129.1 \| UPM 1230-0436 \| 187.7377119 \| -4.6000967 \| 56.9 \| -175.8 \| 3.4 \| 3.2 \| 15.575 \| 13.562 \| 11.944 \| 11.098 \| 10.568 \| 10.359 \| 2.464 \| 59.1 \| UPM 1230-0439 \| 187.5189581 \| -4.6604111 \| -155.7 \| 105.0 \| 4.8 \| 4.5 \| 16.484 \| 14.340 \| 12.884 \| 11.915 \| 11.353 \| 11.065 \| 2.425 \| 78.8 \| UPM 1230-1444 \| 187.6268586 \| -14.7465403 \| -192.9 \| 28.8 \| 2.1 \| 2.3 \| 15.721 \| 14.376 \| 13.613 \| 13.446 \| 12.910 \| 12.826 \| 0.930 \| 244.1 \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 1232-0322 \| 188.0907006 \| -3.3744908 \| -126.2 \| -141.1 \| 14.5 \| 7.9 \| 17.217 \| 14.989 \| 13.389 \| 12.279 \| 11.703 \| 11.451 \| 2.710 \| 81.8 \| UPM 1232-4612 \| 188.1934892 \| -46.2156881 \| -181.0 \| -67.6 \| 2.7 \| 2.7 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 11.123 \| 10.578 \| 10.303 \| $\cdots$ \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable ccSuperCOSMOS plate magnitudes suspect UPM 1234-0023 \| 188.5658922 \| -0.3942053 \| -159.8 \| -94.0 \| 1.9 \| 3.6 \| $\cdots$ \| $\cdots$ \| 12.901 \| 12.589 \| 12.062 \| 11.998 \| $\cdots$ \| 148.2 \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 1235-0331 \| 188.8826600 \| -3.5177714 \| -132.6 \| -124.9 \| 8.4 \| 8.1 \| $\cdots$ \| $\cdots$ \| 13.529 \| 13.101 \| 12.480 \| 12.423 \| $\cdots$ \| 174.8 \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 1237-2708 \| 189.4098558 \| -27.1349803 \| -165.6 \| -89.2 \| 18.4 \| 5.4 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 10.255 \| 9.641 \| 9.414 \| $\cdots$ \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 1238-0038 \| 189.7220656 \| -0.6494094 \| -50.5 \| -210.9 \| 7.5 \| 7.4 \| 16.055 \| 14.536 \| 13.893 \| 13.547 \| 13.009 \| 12.833 \| 0.989 \| 217.0 \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 1239-2521 \| 189.9172039 \| -25.3574425 \| -131.8 \| -125.4 \| 1.0 \| 0.9 \| 12.470 \| 11.395 \| 10.838 \| 10.671 \| 10.251 \| 10.189 \| 0.724 \| 66.7 \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 1240-1232 \| 190.1730197 \| -12.5345394 \| -172.3 \| -54.7 \| 3.4 \| 3.5 \| 17.309 \| 14.763 \| $\cdots$ \| 11.828 \| 11.189 \| 10.977 \| 2.935 \| 52.3 \| UPM 1243-0333 \| 190.8523322 \| -3.5658900 \| -55.5 \| -171.6 \| 13.7 \| 13.0 \| $\cdots$ \| 16.023 \| 14.247 \| 13.040 \| 12.509 \| 12.241 \| 2.983 \| 104.4 \| UPM 1246-3423 \| 191.7417556 \| -34.3920336 \| -182.1 \| 10.0 \| 6.5 \| 9.4 \| 17.930 \| 15.820 \| 14.086 \| 12.864 \| 12.270 \| 12.097 \| 2.956 \| 101.8 \| UPM 1250-3103 \| 192.7028144 \| -31.0625358 \| -182.0 \| -16.8 \| 1.5 \| 1.4 \| 16.080 \| 14.009 \| 12.308 \| 11.528 \| 10.821 \| 10.633 \| 2.481 \| 63.0 \| UPM 1254-0633 \| 193.6115931 \| -6.5503981 \| -183.5 \| 50.8 \| 7.4 \| 7.2 \| 16.847 \| 14.677 \| 13.142 \| 12.012 \| 11.460 \| 11.202 \| 2.665 \| 75.6 \| UPM 1255-0123 \| 193.9155961 \| -1.3989186 \| -162.8 \| 77.8 \| 7.5 \| 7.6 \| 15.737 \| 13.545 \| 12.313 \| 11.503 \| 10.919 \| 10.741 \| 2.042 \| 78.7 \| UPM 1255-0201 \| 193.9661833 \| -2.0191072 \| -178.6 \| -59.1 \| 7.7 \| 7.9 \| 14.210 \| 12.182 \| 11.223 \| 10.337 \| 9.752 \| 9.539 \| 1.845 \| 48.9 \| UPM 1301-2002 \| 195.2599392 \| -20.0495206 \| -186.6 \| -1.2 \| 2.0 \| 2.3 \| 16.066 \| 13.927 \| 12.307 \| 11.773 \| 11.085 \| 10.872 \| 2.154 \| 79.5 \| UPM 1302-1739 \| 195.5729917 \| -17.6554925 \| 66.6 \| -168.0 \| 4.3 \| 4.3 \| $\cdots$ \| 14.986 \| 13.042 \| 12.419 \| 11.829 \| 11.567 \| 2.567 \| 96.0 \| UPM 1303-0529 \| 195.7612992 \| -5.4846467 \| -182.4 \| 32.7 \| 13.2 \| 12.6 \| 17.727 \| 15.728 \| 13.900 \| 12.816 \| 12.321 \| 12.085 \| 2.912 \| 109.1 \| UPM 1305-0509 \| 196.4285619 \| -5.1588822 \| -184.2 \| -24.3 \| 5.9 \| 4.9 \| 14.155 \| 13.390 \| 12.892 \| 12.725 \| 12.392 \| 12.321 \| 0.665 \| 178.0 \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 1305-1023 \| 196.3293772 \| -10.3901519 \| -146.7 \| 104.8 \| 3.9 \| 4.8 \| 13.943 \| 11.997 \| 11.040 \| 10.876 \| 10.238 \| 10.101 \| 1.121 \| 68.8 \| UPM 1311-4557 \| 197.8996369 \| -45.9601217 \| -170.7 \| -71.8 \| 14.0 \| 7.3 \| $\cdots$ \| 15.818 \| 13.636 \| 12.716 \| 12.123 \| 11.900 \| 3.102 \| 91.1 \| UPM 1313-4112 \| 198.3192592 \| -41.2098522 \| -20.5 \| 208.9 \| 11.3 \| 11.2 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 11.363 \| 10.788 \| 10.487 \| $\cdots$ \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 1314-0453 \| 198.6887014 \| -4.8900147 \| 94.2 \| -153.5 \| 9.5 \| 9.2 \| 17.647 \| 15.646 \| 13.948 \| 12.364 \| 11.732 \| 11.523 \| 3.282 \| 67.0 \| UPM 1315-0157 \| 198.9190489 \| -1.9517944 \| -180.4 \| -21.2 \| 9.9 \| 9.9 \| 17.456 \| 15.511 \| 13.763 \| 12.417 \| 11.920 \| 11.672 \| 3.094 \| 82.8 \| UPM 1315-2904A \| 198.9894239 \| -29.0780553 \| -190.6 \| -27.5 \| 16.7 \| 23.5 \| 18.025 \| 15.871 \| 14.281 \| 12.758 \| 12.274 \| 11.988 \| 3.113 \| 89.6 \| ddCommon proper motion companion; see Table 4 UPM 1315-2904B \| 198.9885206 \| -29.0765469 \| -209.7 \| -1.5 \| 12.8 \| 11.1 \| 20.271 \| 17.824 \| 15.971 \| 14.325 \| 13.751 \| 13.613 \| 3.499 \| 149.3 \| ddCommon proper motion companion; see Table 4 eeNot detected during automated search but noticed by eye during the blinking process UPM 1318-3910 \| 199.5230214 \| -39.1766283 \| -148.1 \| -115.5 \| 4.5 \| 4.5 \| 18.164 \| 16.073 \| 14.089 \| 12.549 \| 11.947 \| 11.661 \| 3.524 \| 63.1 \| UPM 1323-3009 \| 200.8448642 \| -30.1521306 \| -134.2 \| -125.7 \| 3.5 \| 3.8 \| 17.704 \| 15.816 \| 14.100 \| 13.061 \| 12.557 \| 12.291 \| 2.755 \| 130.2 \| UPM 1323-4541 \| 200.7692875 \| -45.6888108 \| -179.7 \| 46.3 \| 6.1 \| 4.9 \| 17.679 \| 15.619 \| 13.966 \| 12.586 \| 11.936 \| 11.747 \| 3.033 \| 82.2 \| UPM 1324-2631 \| 201.2206433 \| -26.5186122 \| -120.6 \| -140.9 \| 4.0 \| 2.2 \| 16.723 \| 14.650 \| 13.048 \| 11.813 \| 11.228 \| 10.997 \| 2.837 \| 64.5 \| UPM 1327-4606 \| 201.8371664 \| -46.1093275 \| -179.2 \| -25.0 \| 6.3 \| 6.6 \| 18.233 \| 16.153 \| 14.141 \| 12.751 \| 12.173 \| 11.906 \| 3.402 \| 76.2 \| UPM 1329-3729 \| 202.4278942 \| -37.4868347 \| -174.5 \| -67.1 \| 2.8 \| 6.8 \| 17.777 \| 15.655 \| 13.751 \| 12.475 \| 11.882 \| 11.606 \| 3.180 \| 72.3 \| UPM 1331-1706 \| 202.8177619 \| -17.1053169 \| -152.4 \| -103.8 \| 8.8 \| 8.8 \| 13.616 \| 11.668 \| 10.400 \| 10.293 \| 9.600 \| 9.397 \| 1.375 \| 48.2 \| UPM 1335-1706 \| 203.9090931 \| -17.1108244 \| -170.3 \| 83.7 \| 3.9 \| 6.3 \| 17.345 \| 15.345 \| 13.960 \| 12.976 \| 12.380 \| 12.125 \| 2.369 \| 134.0 \| UPM 1337-0155 \| 204.2700961 \| -1.9167561 \| -204.1 \| -165.2 \| 11.0 \| 11.3 \| 16.263 \| 14.421 \| 12.192 \| 10.983 \| 10.497 \| 10.223 \| 3.438 \| 38.3 \| UPM 1338-1459 \| 204.7328261 \| -14.9929269 \| -180.8 \| 6.1 \| 8.5 \| 8.7 \| 14.357 \| 12.632 \| 11.951 \| 11.520 \| 10.912 \| 10.801 \| 1.112 \| 94.7 \| UPM 1343-0220 \| 205.9086011 \| -2.3335592 \| -144.7 \| 128.3 \| 5.1 \| 4.4 \| 14.009 \| 11.642 \| 10.754 \| 10.064 \| 9.431 \| 9.219 \| 1.578 \| 43.4 \| UPM 1343-3728 \| 205.9051978 \| -37.4830681 \| -132.3 \| -132.2 \| 3.7 \| 3.4 \| 18.097 \| 15.807 \| 14.010 \| 12.689 \| 12.118 \| 11.856 \| 3.118 \| 80.8 \| UPM 1344-0757 \| 206.1694717 \| -7.9526650 \| -21.2 \| -186.8 \| 13.0 \| 12.9 \| 17.328 \| 15.072 \| 12.915 \| 12.111 \| 11.585 \| 11.300 \| 2.961 \| 71.2 \| UPM 1346-2111B \| 206.7059456 \| -21.1849233 \| -112.1 \| -60.0 \| 6.6 \| 5.7 \| 18.343 \| 16.407 \| 14.513 \| 13.004 \| 12.414 \| 12.148 \| 3.403 \| 86.8 \| aaProper motions suspect ddCommon proper motion companion; see Table 4 eeNot detected during automated search but noticed by eye during the blinking process UPM 1347-0042 \| 206.8132836 \| -0.7141892 \| -209.1 \| 50.0 \| 11.3 \| 11.1 \| 16.141 \| 14.211 \| 12.183 \| 10.993 \| 10.425 \| 10.156 \| 3.218 \| 38.7 \| UPM 1349-4228 \| 207.2552625 \| -42.4784189 \| -161.9 \| -84.6 \| 1.3 \| 3.0 \| 14.468 \| 11.703 \| $\cdots$ \| 9.449 \| 8.863 \| 8.622 \| 2.254 \| 24.4 \| UPM 1349-4603 \| 207.4748908 \| -46.0662628 \| -171.7 \| -66.6 \| 2.4 \| 3.5 \| 16.545 \| 14.507 \| 12.621 \| 11.706 \| 11.099 \| 10.819 \| 2.801 \| 61.3 \| UPM 1350-2538 \| 207.5852117 \| -25.6338006 \| -126.3 \| -130.5 \| 2.3 \| 2.9 \| 17.261 \| 15.147 \| 13.177 \| 11.576 \| 11.033 \| 10.753 \| 3.571 \| 41.2 \| UPM 1355-2724 \| 208.7766217 \| -27.4151128 \| -180.6 \| -12.6 \| 1.4 \| 1.2 \| 15.733 \| 13.991 \| 13.089 \| 12.058 \| 11.411 \| 11.310 \| 1.933 \| 110.2 \| UPM 1355-3547 \| 208.9479931 \| -35.7875306 \| -148.0 \| -118.6 \| 2.6 \| 19.8 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 11.689 \| 11.076 \| 10.853 \| $\cdots$ \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 1412-3518 \| 213.1385153 \| -35.3079467 \| -177.6 \| -39.0 \| 2.1 \| 3.8 \| 17.023 \| 14.997 \| 13.834 \| 12.895 \| 12.301 \| 12.085 \| 2.102 \| 145.3 \| UPM 1413-0615 \| 213.4542447 \| -6.2657081 \| -205.7 \| -9.9 \| 18.1 \| 6.5 \| 14.141 \| 12.262 \| 11.387 \| 11.476 \| 10.845 \| 10.750 \| 0.786 \| 103.0 \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 1413-2727 \| 213.4797344 \| -27.4651342 \| -173.9 \| -51.0 \| 1.2 \| 1.5 \| $\cdots$ \| 14.874 \| 14.198 \| 13.268 \| 12.728 \| 12.647 \| 1.606 \| [205.3] \| ffSubdwarf candidate selected from RPM diagram; plate distance [in bracket] is incorrect UPM 1426-3807 \| 216.7439033 \| -38.1297039 \| -173.8 \| 59.9 \| 3.3 \| 3.3 \| 17.549 \| 15.605 \| 13.932 \| 12.591 \| 12.052 \| 11.855 \| 3.014 \| 92.7 \| UPM 1433-1006 \| 218.2923342 \| -10.1017839 \| -178.4 \| -34.4 \| 3.9 \| 3.7 \| 17.319 \| 15.500 \| 14.052 \| 13.008 \| 12.426 \| 12.204 \| 2.492 \| 138.0 \| UPM 1440-1216 \| 220.0069028 \| -12.2808661 \| -95.2 \| -154.7 \| 2.1 \| 2.1 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 10.770 \| 10.118 \| 9.951 \| $\cdots$ \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 1440-2346 \| 220.0484783 \| -23.7699506 \| -166.3 \| -81.9 \| 6.1 \| 5.5 \| 18.074 \| 15.948 \| 14.523 \| 12.990 \| 12.405 \| 12.171 \| 2.958 \| 102.3 \| UPM 1443-1350 \| 220.7918803 \| -13.8418344 \| -132.3 \| -123.7 \| 8.4 \| 8.2 \| 16.306 \| 14.202 \| 12.724 \| 12.002 \| 11.453 \| 11.244 \| 2.200 \| 96.2 \| UPM 1443-3318 \| 220.8361869 \| -33.3092458 \| -169.7 \| -65.4 \| 2.6 \| 1.8 \| $\cdots$ \| 13.706 \| 11.765 \| 11.365 \| 10.778 \| 10.538 \| 2.341 \| 65.6 \| UPM 1444-1414 \| 221.1479233 \| -14.2457728 \| -110.5 \| -155.5 \| 7.7 \| 5.7 \| $\cdots$ \| $\cdots$ \| 11.465 \| 10.983 \| 10.424 \| 10.194 \| $\cdots$ \| 61.4 \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 1445-4228 \| 221.4660325 \| -42.4748283 \| -160.6 \| -90.7 \| 7.4 \| 7.5 \| 16.627 \| 14.542 \| 13.149 \| 12.083 \| 11.525 \| 11.310 \| 2.459 \| 88.6 \| UPM 1447-2307 \| 221.8288436 \| -23.1260342 \| -169.5 \| 68.6 \| 3.9 \| 3.3 \| 17.332 \| 15.364 \| 13.932 \| 12.472 \| 11.945 \| 11.733 \| 2.892 \| 91.3 \| UPM 1449-2906 \| 222.4226139 \| -29.1078767 \| -184.3 \| -32.6 \| 1.9 \| 1.7 \| 16.131 \| 13.917 \| 12.176 \| 11.357 \| 10.700 \| 10.475 \| 2.560 \| 55.6 \| UPM 1451-2451 \| 222.9401378 \| -24.8589333 \| 126.6 \| -131.3 \| 1.6 \| 1.6 \| 16.739 \| 14.847 \| 13.100 \| 11.702 \| 11.110 \| 10.884 \| 3.145 \| 55.3 \| UPM 1453-4446 \| 223.4601097 \| -44.7760186 \| -204.1 \| -63.0 \| 19.9 \| 25.5 \| 14.117 \| 12.089 \| 10.891 \| 10.938 \| 10.262 \| 10.120 \| 1.151 \| 69.6 \| ggPossible NLTT star with a position difference $>$ 90$\arcsec$ when compared to UCAC3 position UPM 1454-3423 \| 223.5418183 \| -34.3933181 \| -227.7 \| -119.4 \| 14.0 \| 13.6 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 13.049 \| 12.497 \| 12.336 \| $\cdots$ \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 1456-3806 \| 224.1946236 \| -38.1156314 \| -174.1 \| 52.7 \| 3.1 \| 2.1 \| 13.065 \| 11.263 \| 10.578 \| 10.002 \| 9.379 \| 9.216 \| 1.261 \| 46.1 \| UPM 1456-4218 \| 224.1659378 \| -42.3046658 \| -220.1 \| -37.0 \| 7.1 \| 7.8 \| 15.332 \| 13.396 \| 11.931 \| 10.866 \| 10.258 \| 10.035 \| 2.530 \| 48.5 \| UPM 1457-0555 \| 224.4482581 \| -5.9233250 \| -115.2 \| -179.5 \| 9.0 \| 8.7 \| 17.059 \| 15.107 \| 13.937 \| 13.015 \| 12.418 \| 12.184 \| 2.092 \| 154.0 \| UPM 1504-0235 \| 226.2061903 \| -2.5983772 \| -22.5 \| -194.4 \| 9.3 \| 9.5 \| 16.692 \| 14.908 \| 14.139 \| 13.546 \| 12.938 \| 12.775 \| 1.362 \| [234.4] \| ffSubdwarf candidate selected from RPM diagram; plate distance [in bracket] is incorrect UPM 1506-4504 \| 226.5962444 \| -45.0685394 \| -151.5 \| -119.5 \| 8.6 \| 10.8 \| 16.934 \| 15.119 \| 13.136 \| 12.013 \| 11.368 \| 11.151 \| 3.106 \| 65.1 \| UPM 1509-1356 \| 227.3431372 \| -13.9350375 \| -200.2 \| 28.8 \| 9.2 \| 8.8 \| $\cdots$ \| 14.921 \| 12.843 \| 11.865 \| 11.280 \| 11.034 \| 3.056 \| 60.9 \| UPM 1514-0519 \| 228.6674336 \| -5.3324128 \| -148.8 \| -106.8 \| 9.7 \| 9.5 \| 13.534 \| 12.904 \| 12.538 \| 12.494 \| 12.211 \| 12.190 \| 0.410 \| 169.1 \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 1514-0712 \| 228.7084150 \| -7.2062508 \| -168.0 \| -75.5 \| 6.4 \| 7.2 \| 13.339 \| 11.338 \| 10.441 \| 9.950 \| 9.295 \| 9.178 \| 1.388 \| 44.3 \| UPM 1516-2731 \| 229.0559556 \| -27.5278219 \| -131.4 \| -129.1 \| 4.8 \| 4.7 \| 17.417 \| 15.647 \| 13.761 \| 12.139 \| 11.608 \| 11.364 \| 3.508 \| 61.2 \| UPM 1525-4622 \| 231.3542089 \| -46.3688022 \| -181.0 \| -151.9 \| 6.3 \| 5.8 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 12.271 \| 11.915 \| 11.824 \| $\cdots$ \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable ggPossible NLTT star with a position difference $>$ 90$\arcsec$ when compared to UCAC3 position UPM 1529-0707 \| 232.3161231 \| -7.1222567 \| -181.5 \| -61.1 \| 9.7 \| 9.5 \| 13.142 \| 12.257 \| 11.859 \| 11.709 \| 11.387 \| 11.321 \| 0.548 \| 112.6 \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 1532-2833 \| 233.0121631 \| -28.5547736 \| -20.5 \| -132.9 \| 7.1 \| 6.7 \| 19.359 \| 17.550 \| 15.606 \| 13.724 \| 13.168 \| 12.897 \| 3.826 \| 104.9 \| aaProper motions suspect eeNot detected during automated search but noticed by eye during the blinking process UPM 1532-2834 \| 233.0302039 \| -28.5679217 \| -120.7 \| -24.6 \| 7.7 \| 7.3 \| 19.707 \| 17.755 \| 15.689 \| 13.749 \| 13.193 \| 12.916 \| 4.006 \| 94.7 \| aaProper motions suspect eeNot detected during automated search but noticed by eye during the blinking process UPM 1533-0251 \| 233.4913458 \| -2.8522467 \| 56.7 \| -179.0 \| 10.2 \| 10.9 \| 13.565 \| 12.658 \| 11.989 \| 11.098 \| 10.676 \| 10.588 \| 1.560 \| 77.8 \| UPM 1533-2126 \| 233.3307392 \| -21.4486364 \| -171.6 \| -68.5 \| 2.2 \| 2.1 \| 14.692 \| 12.723 \| 11.510 \| 11.830 \| 11.200 \| 10.980 \| 0.893 \| 97.6 \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable ccSuperCOSMOS plate magnitudes suspect UPM 1534-3744 \| 233.7133039 \| -37.7359719 \| -31.5 \| -260.7 \| 16.1 \| 14.4 \| 17.894 \| 15.895 \| 14.857 \| 13.171 \| 12.605 \| 12.408 \| 2.724 \| 130.8 \| UPM 1535-1652 \| 233.8295736 \| -16.8676697 \| -5.0 \| -239.3 \| 14.7 \| 14.3 \| 17.677 \| 15.648 \| 13.758 \| 12.655 \| 12.135 \| 11.875 \| 2.993 \| 94.0 \| UPM 1536-2307 \| 234.0091364 \| -23.1279594 \| -162.8 \| 78.1 \| 9.0 \| 9.2 \| 17.168 \| 15.250 \| 13.539 \| 11.960 \| 11.442 \| 11.192 \| 3.290 \| 60.2 \| UPM 1542-4520 \| 235.5850419 \| -45.3483308 \| -130.8 \| -130.5 \| 5.0 \| 5.0 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 11.572 \| 10.999 \| 10.746 \| $\cdots$ \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 1546-2553 \| 236.6958903 \| -25.8843933 \| -159.9 \| -135.9 \| 8.8 \| 9.0 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 11.478 \| 10.909 \| 10.700 \| $\cdots$ \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 1548-2228 \| 237.1036078 \| -22.4715383 \| -103.0 \| -151.7 \| 14.2 \| 4.1 \| 16.985 \| 15.801 \| 14.246 \| 12.667 \| 12.144 \| 11.938 \| 3.134 \| 107.0 \| UPM 1551-0438 \| 237.8945833 \| -4.6492825 \| -226.5 \| -3.9 \| 8.7 \| 8.9 \| 16.552 \| 15.226 \| 13.429 \| 11.657 \| 11.074 \| 10.847 \| 3.569 \| 51.3 \| UPM 1551-1335 \| 237.9823981 \| -13.5938706 \| -269.4 \| -96.6 \| 12.3 \| 14.4 \| 17.386 \| 15.241 \| 13.040 \| 12.180 \| 11.646 \| 11.376 \| 3.061 \| 72.1 \| UPM 1552-1033 \| 238.0100178 \| -10.5601664 \| -13.0 \| -184.1 \| 8.8 \| 9.6 \| 16.537 \| 14.730 \| 13.454 \| 12.042 \| 11.431 \| 11.245 \| 2.688 \| 80.5 \| UPM 1552-1511 \| 238.1625247 \| -15.1866033 \| -272.3 \| -209.5 \| 11.8 \| 12.2 \| 14.647 \| 13.024 \| 13.305 \| 12.638 \| 11.919 \| 11.698 \| 0.386 \| 107.4 \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 1552-3825 \| 238.2234847 \| -38.4229861 \| -58.0 \| -177.4 \| 4.8 \| 4.8 \| 14.200 \| 12.733 \| 12.214 \| 11.750 \| 11.166 \| 11.067 \| 0.983 \| 92.9 \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 1553-0244 \| 238.2634961 \| -2.7434919 \| -36.1 \| -176.6 \| 5.8 \| 17.9 \| 15.437 \| 13.462 \| 11.798 \| 10.683 \| 10.149 \| 9.892 \| 2.779 \| 41.4 \| UPM 1600-0137 \| 240.0560244 \| -1.6199883 \| 67.4 \| -178.8 \| 19.8 \| 4.9 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 9.820 \| 9.247 \| 8.981 \| $\cdots$ \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 1605-0010 \| 241.3375458 \| -0.1716083 \| -197.4 \| -64.0 \| 8.3 \| 8.5 \| 17.541 \| 15.676 \| 14.006 \| 12.518 \| 11.979 \| 11.736 \| 3.158 \| 82.7 \| UPM 1606-3534 \| 241.5239356 \| -35.5674578 \| -175.1 \| -51.7 \| 2.4 \| 2.2 \| 15.487 \| 13.696 \| 12.562 \| 10.799 \| 10.175 \| 9.954 \| 2.897 \| 40.1 \| UPM 1607-2307 \| 241.8580489 \| -23.1169094 \| -145.2 \| -133.7 \| 20.8 \| 5.3 \| 16.383 \| 14.608 \| 12.686 \| 11.766 \| 11.127 \| 10.854 \| 2.842 \| 64.3 \| UPM 1609-4639 \| 242.2884686 \| -46.6565003 \| -140.9 \| -196.7 \| 6.6 \| 6.6 \| $\cdots$ \| 16.743 \| 15.550 \| 13.013 \| 12.434 \| 12.241 \| 3.730 \| 67.3 \| UPM 1610-0227 \| 242.7059622 \| -2.4608150 \| -9.3 \| -186.0 \| 8.9 \| 8.2 \| 17.288 \| 15.364 \| 14.344 \| 12.667 \| 12.074 \| 11.838 \| 2.697 \| 102.0 \| UPM 1614-4033 \| 243.6298703 \| -40.5574253 \| -128.0 \| -138.4 \| 5.2 \| 4.3 \| 17.591 \| 16.426 \| 14.743 \| 12.402 \| 11.821 \| 11.543 \| 4.024 \| 60.5 \| UPM 1619-2602 \| 244.8460106 \| -26.0425961 \| -133.9 \| -154.6 \| 4.3 \| 3.1 \| 15.425 \| 14.502 \| 14.568 \| 11.879 \| 11.310 \| 11.109 \| 2.623 \| 55.3 \| UPM 1621-0031 \| 245.3020250 \| -0.5209803 \| -115.7 \| -183.5 \| 7.6 \| 7.7 \| 14.421 \| 13.350 \| 12.756 \| 12.074 \| 11.596 \| 11.532 \| 1.276 \| [122.1] \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable ffSubdwarf candidate selected from RPM diagram; plate distance [in bracket] is incorrect UPM 1623-0641 \| 245.7802481 \| -6.6980106 \| -186.4 \| 40.7 \| 6.5 \| 6.5 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 11.051 \| 10.424 \| 10.149 \| $\cdots$ \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 1624-3133 \| 246.2206158 \| -31.5640650 \| -161.0 \| 87.9 \| 9.8 \| 9.8 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 12.711 \| 11.997 \| 11.845 \| $\cdots$ \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 1634-1540 \| 248.5736717 \| -15.6766797 \| -72.1 \| -197.0 \| 11.3 \| 10.9 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 12.908 \| 12.156 \| 11.902 \| $\cdots$ \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 1638-1033 \| 249.6061678 \| -10.5661297 \| -217.3 \| -41.9 \| 10.2 \| 9.8 \| 15.959 \| 14.378 \| 13.123 \| 11.491 \| 10.891 \| 10.645 \| 2.887 \| 58.2 \| UPM 1638-2541 \| 249.7123400 \| -25.6911056 \| -136.2 \| -139.7 \| 3.6 \| 3.6 \| 16.105 \| 13.846 \| 13.824 \| 12.311 \| 11.583 \| 11.458 \| 1.535 \| 105.3 \| UPM 1638-3439 \| 249.6381294 \| -34.6511661 \| -96.1 \| -156.2 \| 6.4 \| 14.4 \| $\cdots$ \| $\cdots$ \| 13.568 \| 12.165 \| 11.622 \| 11.370 \| $\cdots$ \| 58.3 \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 1639-2331 \| 249.7811856 \| -23.5252967 \| -131.3 \| -134.4 \| 9.2 \| 3.7 \| 16.818 \| 15.295 \| 13.685 \| 11.451 \| 10.828 \| 10.559 \| 3.844 \| 37.5 \| UPM 1642-2833 \| 250.6688403 \| -28.5619450 \| -166.5 \| -100.9 \| 6.4 \| 22.1 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 13.377 \| 12.849 \| 12.652 \| $\cdots$ \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 1646-3709 \| 251.7445328 \| -37.1657722 \| 193.7 \| -2.2 \| 7.7 \| 7.7 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 11.056 \| 10.530 \| 10.293 \| $\cdots$ \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 1647-3213 \| 251.8940731 \| -32.2208019 \| -188.2 \| -37.3 \| 7.8 \| 8.3 \| $\cdots$ \| 15.988 \| 15.028 \| 12.602 \| 12.081 \| 11.870 \| 3.386 \| 66.4 \| UPM 1648-3459 \| 252.1200667 \| -34.9967942 \| 178.6 \| 142.5 \| 4.4 \| 4.4 \| $\cdots$ \| 14.631 \| $\cdots$ \| 10.687 \| 10.161 \| 9.907 \| 3.994 \| 22.1 \| UPM 1648-3538 \| 252.0849597 \| -35.6427183 \| -102.0 \| -156.6 \| 7.6 \| 5.0 \| 15.970 \| 14.748 \| $\cdots$ \| 11.008 \| 10.503 \| 10.266 \| 3.740 \| 38.5 \| UPM 1648-3539 \| 252.0724997 \| -35.6630103 \| 76.1 \| -169.7 \| 2.3 \| 2.3 \| 15.961 \| 14.549 \| 13.571 \| 10.890 \| 10.316 \| 10.056 \| 3.659 \| 29.8 \| UPM 1650-2440 \| 252.7400719 \| -24.6745247 \| -37.8 \| -177.8 \| 35.2 \| 30.4 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 11.982 \| 11.355 \| 11.127 \| $\cdots$ \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 1654-3105 \| 253.6846164 \| -31.0961000 \| -32.4 \| -215.9 \| 7.4 \| 7.2 \| 15.122 \| 13.553 \| 11.977 \| 10.072 \| 9.482 \| 9.237 \| 3.481 \| 23.8 \| ggPossible NLTT star with a position difference $>$ 90$\arcsec$ when compared to UCAC3 position UPM 1658-2811 \| 254.7136367 \| -28.1977644 \| 190.7 \| 179.0 \| 10.3 \| 10.3 \| 17.979 \| 16.677 \| 14.129 \| 13.417 \| 12.904 \| 12.677 \| 3.260 \| 149.9 \| UPM 1658-3931 \| 254.5610944 \| -39.5308172 \| -140.3 \| -191.4 \| 8.8 \| 8.9 \| $\cdots$ \| 16.787 \| 14.502 \| 12.161 \| 11.556 \| 11.266 \| 4.626 \| 30.2 \| UPM 1700-0857 \| 255.0531956 \| -8.9576528 \| -19.6 \| -199.7 \| 13.2 \| 13.2 \| 17.447 \| 15.705 \| 14.262 \| 12.535 \| 11.870 \| 11.639 \| 3.170 \| 77.4 \| UPM 1700-2913 \| 255.1488828 \| -29.2189083 \| -158.8 \| 96.5 \| 3.5 \| 6.1 \| $\cdots$ \| 14.142 \| 12.927 \| 11.843 \| 11.326 \| 11.094 \| 2.299 \| 79.9 \| UPM 1701-0657 \| 255.3763442 \| -6.9526483 \| 105.6 \| -160.9 \| 13.9 \| 14.1 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 12.695 \| 12.064 \| 11.808 \| $\cdots$ \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 1701-2502 \| 255.4740292 \| -25.0337903 \| -78.8 \| -190.1 \| 4.6 \| 4.6 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 13.016 \| 12.491 \| 12.276 \| $\cdots$ \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 1701-3150 \| 255.3130186 \| -31.8452472 \| -165.1 \| -154.8 \| 9.2 \| 9.2 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 11.344 \| 10.841 \| 10.566 \| $\cdots$ \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 1701-3345 \| 255.2964308 \| -33.7500906 \| -157.2 \| -99.6 \| 5.6 \| 5.6 \| $\cdots$ \| 14.898 \| 13.449 \| 11.496 \| 10.928 \| 10.692 \| 3.402 \| 38.2 \| UPM 1704-1459 \| 256.2319803 \| -14.9893325 \| -114.1 \| -148.4 \| 4.4 \| 4.5 \| 17.336 \| $\cdots$ \| $\cdots$ \| 12.347 \| 11.791 \| 11.571 \| $\cdots$ \| 85.2 \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 1704-3141 \| 256.2325125 \| -31.6892650 \| -175.2 \| -169.6 \| 7.3 \| 7.1 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 12.655 \| 12.006 \| 11.834 \| $\cdots$ \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 1706-0254 \| 256.6927197 \| -2.9013867 \| -122.8 \| -153.3 \| 12.3 \| 11.9 \| $\cdots$ \| 16.016 \| 15.214 \| 13.721 \| 13.074 \| 12.860 \| 2.295 \| 164.7 \| UPM 1707-0345 \| 256.8042544 \| -3.7595239 \| 6.0 \| -191.4 \| 7.8 \| 7.8 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 10.865 \| 10.218 \| 10.073 \| $\cdots$ \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 1707-1438 \| 256.7945428 \| -14.6436947 \| -94.6 \| -164.4 \| 4.7 \| 4.8 \| 17.411 \| 15.843 \| 15.096 \| 13.158 \| 12.519 \| 12.288 \| 2.685 \| 123.9 \| UPM 1709-1715 \| 257.4969336 \| -17.2529044 \| -196.8 \| -4.0 \| 6.0 \| 6.1 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 12.313 \| 11.758 \| 11.530 \| $\cdots$ \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 1709-2858 \| 257.4467800 \| -28.9713886 \| -173.0 \| -126.4 \| 10.8 \| 10.3 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 11.234 \| 10.636 \| 10.270 \| $\cdots$ \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 1710-1055 \| 257.6343403 \| -10.9268356 \| -89.1 \| -246.0 \| 11.3 \| 11.3 \| 17.190 \| 15.900 \| 13.995 \| 12.060 \| 11.508 \| 11.226 \| 3.840 \| 54.9 \| UPM 1711-3942 \| 257.8640486 \| -39.7006067 \| -168.1 \| -100.2 \| 11.1 \| 8.4 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 11.759 \| 11.166 \| 10.948 \| $\cdots$ \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 1712-4432 \| 258.1053478 \| -44.5375533 \| -149.3 \| -158.6 \| 7.2 \| 6.7 \| 14.386 \| 13.041 \| 14.109 \| 12.028 \| 11.541 \| 11.404 \| 1.013 \| [ 33.9] \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable ccSuperCOSMOS plate magnitudes suspect ffSubdwarf candidate selected from RPM diagram; plate distance [in bracket] is incorrect UPM 1712-4657 \| 258.0571681 \| -46.9649431 \| -154.8 \| -109.7 \| 4.3 \| 4.3 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 11.992 \| 11.409 \| 11.147 \| $\cdots$ \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable ccSuperCOSMOS plate magnitudes suspect UPM 1713-3512 \| 258.3514606 \| -35.2062569 \| -182.8 \| -132.3 \| 7.5 \| 7.3 \| 17.994 \| 16.680 \| 16.283 \| 12.680 \| 11.851 \| 11.656 \| 4.000 \| 40.7 \| UPM 1714-2118 \| 258.5668300 \| -21.3145236 \| -183.9 \| -129.4 \| 3.1 \| 3.1 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 11.710 \| 11.032 \| 10.878 \| $\cdots$ \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 1718-2245A \| 259.6129147 \| -22.7616683 \| -160.7 \| -160.8 \| 7.2 \| 6.8 \| 15.469 \| 13.836 \| 13.155 \| 10.385 \| 9.806 \| 9.572 \| 3.451 \| 25.4 \| ddCommon proper motion companion; see Table 4 UPM 1718-2245B \| 259.6213031 \| -22.7746183 \| -161.1 \| -154.8 \| 10.9 \| 9.6 \| $\cdots$ \| 14.787 \| 13.289 \| 10.207 \| 9.608 \| 9.375 \| 4.580 \| 13.2 \| ddCommon proper motion companion; see Table 4 eeNot detected during automated search but noticed by eye during the blinking process UPM 1720-1252 \| 260.1161614 \| -12.8698767 \| -41.6 \| -184.2 \| 17.1 \| 12.3 \| $\cdots$ \| 13.214 \| $\cdots$ \| 11.034 \| 10.467 \| 10.254 \| 2.180 \| 61.0 \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable ggPossible NLTT star with a position difference $>$ 90$\arcsec$ when compared to UCAC3 position UPM 1721-4500 \| 260.3965597 \| -45.0118219 \| 111.2 \| -181.3 \| 7.0 \| 6.6 \| 14.978 \| 12.715 \| 11.539 \| 10.365 \| 9.776 \| 9.537 \| 2.350 \| 38.8 \| UPM 1722-4136 \| 260.5690697 \| -41.6059956 \| -170.4 \| -72.7 \| 5.3 \| 20.1 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 11.924 \| 11.366 \| 11.112 \| $\cdots$ \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 1724-0318A \| 261.1610800 \| -3.3011636 \| 143.7 \| -126.7 \| 8.7 \| 8.9 \| 15.536 \| 13.860 \| 13.019 \| 11.833 \| 11.201 \| 10.994 \| 2.027 \| 92.0 \| ddCommon proper motion companion; see Table 4 UPM 1724-0318B \| 261.1602147 \| -3.2999100 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 15.536 \| 13.860 \| 13.019 \| 12.925 \| 12.291 \| 12.042 \| 0.935 \| 169.5 \| aaProper motions suspect bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable ddCommon proper motion companion; see Table 4 eeNot detected during automated search but noticed by eye during the blinking process UPM 1724-2217 \| 261.2276967 \| -22.2935397 \| -251.6 \| -93.6 \| 7.0 \| 7.0 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 12.042 \| 11.492 \| 11.260 \| $\cdots$ \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 1725-1703 \| 261.4728428 \| -17.0606703 \| -80.5 \| -163.3 \| 2.8 \| 2.8 \| $\cdots$ \| 12.446 \| 12.710 \| 10.927 \| 10.559 \| 10.513 \| 1.519 \| 61.0 \| UPM 1725-1749 \| 261.4613711 \| -17.8228856 \| -163.8 \| -123.4 \| 6.4 \| 7.3 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 10.639 \| 10.250 \| 10.035 \| $\cdots$ \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 1731-1316 \| 262.9469089 \| -13.2742833 \| 73.1 \| -199.8 \| 19.3 \| 11.7 \| 17.463 \| 16.004 \| 14.528 \| 12.690 \| 12.191 \| 11.941 \| 3.314 \| 93.3 \| UPM 1732-1639 \| 263.0863078 \| -16.6663169 \| -170.8 \| -147.8 \| 11.0 \| 10.7 \| $\cdots$ \| 13.326 \| 12.859 \| 11.800 \| 11.231 \| 11.145 \| 1.526 \| 93.7 \| UPM 1733-2051 \| 263.2673667 \| -20.8637517 \| 41.8 \| -211.8 \| 14.5 \| 5.7 \| 14.941 \| 13.518 \| 13.304 \| 10.787 \| 10.182 \| 9.974 \| 2.731 \| 37.3 \| UPM 1737-2324 \| 264.4041736 \| -23.4064956 \| -244.7 \| -127.3 \| 8.3 \| 8.3 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 11.322 \| 10.711 \| 10.416 \| $\cdots$ \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 1738-1917 \| 264.7263142 \| -19.2951633 \| 21.8 \| -202.1 \| 4.6 \| 5.1 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 11.358 \| 10.832 \| 10.622 \| $\cdots$ \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 1740-4110 \| 265.2443694 \| -41.1801894 \| -68.5 \| -166.7 \| 6.8 \| 7.3 \| $\cdots$ \| 14.331 \| 12.963 \| 11.261 \| 10.678 \| 10.382 \| 3.070 \| 38.2 \| UPM 1741-4536 \| 265.4011836 \| -45.6043703 \| 168.4 \| 194.1 \| 10.4 \| 10.4 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 11.021 \| 10.512 \| 10.236 \| $\cdots$ \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 1743-3957 \| 265.9756108 \| -39.9611264 \| -191.0 \| 84.3 \| 8.3 \| 8.0 \| $\cdots$ \| 15.433 \| 13.811 \| 12.184 \| 11.692 \| 11.431 \| 3.249 \| 61.1 \| UPM 1745-1322 \| 266.4679789 \| -13.3765100 \| -183.1 \| -35.8 \| 6.2 \| 6.1 \| 17.039 \| 15.378 \| 14.462 \| 12.724 \| 12.077 \| 11.902 \| 2.654 \| 111.3 \| UPM 1745-4336 \| 266.3273608 \| -43.6112189 \| 23.4 \| -220.2 \| 8.6 \| 7.9 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 10.374 \| 9.790 \| 9.503 \| $\cdots$ \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 1746-1246 \| 266.5727764 \| -12.7717856 \| 44.4 \| -186.4 \| 6.8 \| 6.6 \| 17.622 \| 16.100 \| 14.686 \| 12.556 \| 12.011 \| 11.767 \| 3.544 \| 76.0 \| UPM 1749-3138 \| 267.4376153 \| -31.6363372 \| -139.1 \| -122.8 \| 2.0 \| 2.0 \| 14.024 \| 12.310 \| 11.420 \| 10.334 \| 9.724 \| 9.567 \| 1.976 \| 48.9 \| UPM 1749-4135 \| 267.4756842 \| -41.5986675 \| -107.8 \| -184.7 \| 4.3 \| 4.3 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 10.349 \| 9.736 \| 9.544 \| $\cdots$ \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable ccSuperCOSMOS plate magnitudes suspect UPM 1749-4313 \| 267.4106522 \| -43.2245928 \| 64.1 \| -172.6 \| 2.7 \| 2.7 \| $\cdots$ \| 13.347 \| 11.682 \| 11.380 \| 10.832 \| 10.596 \| 1.967 \| 75.3 \| UPM 1749-4404B \| 267.4638828 \| -44.0790978 \| 0.0 \| -204.9 \| 4.3 \| 3.9 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 11.688 \| 11.147 \| 10.870 \| $\cdots$ \| $\cdots$ \| aaProper motions suspect bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable ccSuperCOSMOS plate magnitudes suspect ddCommon proper motion companion; see Table 4 UPM 1750-0406 \| 267.6260231 \| -4.1000422 \| -108.5 \| -145.0 \| 7.6 \| 8.5 \| 16.733 \| 15.701 \| 14.428 \| 11.896 \| 11.381 \| 11.146 \| 3.805 \| 51.0 \| UPM 1750-1456 \| 267.7385731 \| -14.9423142 \| -61.0 \| -172.0 \| 22.7 \| 4.5 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 8.874 \| 8.281 \| 8.030 \| $\cdots$ \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 1751-2404 \| 267.7996450 \| -24.0717064 \| -114.2 \| -184.3 \| 15.7 \| 13.6 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 11.631 \| 10.728 \| 10.473 \| $\cdots$ \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 1754-3805 \| 268.6284406 \| -38.0912072 \| 57.7 \| -206.3 \| 7.1 \| 6.8 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 11.641 \| 11.077 \| 10.839 \| $\cdots$ \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 1756-2126 \| 269.2477247 \| -21.4437422 \| -161.3 \| 92.0 \| 5.7 \| 7.0 \| 17.869 \| 16.628 \| 16.074 \| 12.485 \| 11.797 \| 11.548 \| 4.143 \| 38.5 \| UPM 1756-4052 \| 269.1487281 \| -40.8780769 \| -77.5 \| -196.1 \| 19.7 \| 19.7 \| $\cdots$ \| 14.372 \| 13.688 \| 13.299 \| 12.667 \| 12.607 \| 1.073 \| [215.7] \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable ffSubdwarf candidate selected from RPM diagram; plate distance [in bracket] is incorrect UPM 1757-3936 \| 269.4195217 \| -39.6043317 \| -266.3 \| -158.6 \| 7.3 \| 7.0 \| $\cdots$ \| 13.759 \| $\cdots$ \| 11.284 \| 10.734 \| 10.545 \| 2.475 \| 61.1 \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 1757-4013 \| 269.2794617 \| -40.2270539 \| -75.0 \| -194.2 \| 6.5 \| 6.1 \| $\cdots$ \| 12.349 \| $\cdots$ \| 11.883 \| 11.302 \| 11.200 \| 0.466 \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 1757-4632B \| 269.3754317 \| -46.5427658 \| 59.7 \| -185.1 \| 10.6 \| 6.5 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 10.847 \| 10.264 \| 10.007 \| $\cdots$ \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable ddCommon proper motion companion; see Table 4 UPM 1758-4615 \| 269.5915203 \| -46.2631747 \| -171.7 \| -82.6 \| 25.0 \| 7.2 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 12.842 \| 12.323 \| 12.077 \| $\cdots$ \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 1759-4528 \| 269.7754394 \| -45.4816806 \| -42.4 \| -188.1 \| 10.4 \| 10.8 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 11.429 \| 10.891 \| 10.601 \| $\cdots$ \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 1800-4642 \| 270.1561303 \| -46.7136864 \| -44.7 \| -206.8 \| 25.0 \| 19.7 \| $\cdots$ \| $\cdots$ \| 11.532 \| 10.872 \| 10.267 \| 10.009 \| $\cdots$ \| 51.3 \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 1804-0902 \| 271.0980389 \| -9.0451769 \| 164.6 \| -158.9 \| 5.8 \| 6.4 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 12.081 \| 11.395 \| 11.162 \| $\cdots$ \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 1805-4112 \| 271.4910153 \| -41.2111136 \| 14.8 \| -185.5 \| 7.9 \| 7.9 \| $\cdots$ \| 14.013 \| 12.634 \| 12.757 \| 12.060 \| 11.870 \| 1.256 \| [155.9] \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable ffSubdwarf candidate selected from RPM diagram; plate distance [in bracket] is incorrect UPM 1806-1700 \| 271.7052058 \| -17.0047383 \| -23.7 \| -181.0 \| 4.8 \| 4.9 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 10.873 \| 10.214 \| 9.957 \| $\cdots$ \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable ccSuperCOSMOS plate magnitudes suspect UPM 1810-4412 \| 272.7016906 \| -44.2066736 \| 133.6 \| -139.9 \| 4.8 \| 3.9 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 11.616 \| 11.037 \| 10.793 \| $\cdots$ \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable ccSuperCOSMOS plate magnitudes suspect UPM 1811-0139 \| 272.9795842 \| -1.6625278 \| -13.3 \| -182.6 \| 6.5 \| 6.5 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 11.236 \| 10.637 \| 10.563 \| $\cdots$ \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 1811-3907 \| 272.9450564 \| -39.1204408 \| 198.7 \| 207.8 \| 5.3 \| 5.4 \| $\cdots$ \| 15.619 \| 15.210 \| 13.780 \| 13.230 \| 12.961 \| 1.839 \| [180.2] \| ccSuperCOSMOS plate magnitudes suspect ffSubdwarf candidate selected from RPM diagram; plate distance [in bracket] is incorrect UPM 1812-0445 \| 273.0286425 \| -4.7643792 \| 239.4 \| -61.1 \| 7.8 \| 7.8 \| 18.072 \| 16.984 \| 15.462 \| 12.316 \| 11.828 \| 11.509 \| 4.668 \| 50.4 \| UPM 1812-3958 \| 273.1333542 \| -39.9774706 \| -125.7 \| -150.8 \| 8.0 \| 8.2 \| $\cdots$ \| $\cdots$ \| 12.478 \| 12.721 \| 12.126 \| 11.969 \| $\cdots$ \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 1817-0833 \| 274.4449242 \| -8.5561069 \| -65.6 \| -189.9 \| 12.2 \| 9.1 \| 17.222 \| 16.383 \| 16.030 \| 12.504 \| 11.794 \| 11.599 \| 3.879 \| 51.0 \| UPM 1818-2854 \| 274.5728911 \| -28.9023606 \| -179.0 \| 56.8 \| 13.3 \| 12.7 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 12.613 \| 12.037 \| 11.831 \| $\cdots$ \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 1818-3727 \| 274.6185211 \| -37.4610811 \| -161.2 \| -187.0 \| 11.1 \| 15.9 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 11.970 \| 11.454 \| 11.210 \| $\cdots$ \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 1818-3931 \| 274.6280806 \| -39.5268339 \| 134.9 \| -221.7 \| 8.6 \| 8.6 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 11.548 \| 10.966 \| 10.776 \| $\cdots$ \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 1819-0734 \| 274.8175769 \| -7.5811236 \| -173.8 \| -209.9 \| 8.7 \| 8.8 \| 15.950 \| 15.187 \| 15.621 \| 11.121 \| 10.363 \| 10.143 \| 4.066 \| 33.4 \| UPM 1822-3206 \| 275.5753822 \| -32.1135317 \| -56.7 \| -174.1 \| 18.7 \| 16.7 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 14.030 \| 13.493 \| 13.280 \| $\cdots$ \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 1822-3810 \| 275.7248661 \| -38.1719125 \| -152.6 \| -111.2 \| 3.9 \| 4.0 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 11.978 \| 11.444 \| 11.237 \| $\cdots$ \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 1823-3237 \| 275.9087906 \| -32.6205678 \| -206.0 \| 37.0 \| 7.1 \| 6.9 \| $\cdots$ \| 12.757 \| 11.493 \| 10.897 \| 10.321 \| 10.060 \| 1.860 \| 58.9 \| UPM 1823-4055 \| 275.8996567 \| -40.9293583 \| -196.9 \| -178.2 \| 8.0 \| 8.0 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 11.567 \| 11.025 \| 10.817 \| $\cdots$ \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 1824-1843 \| 276.0411947 \| -18.7243439 \| -181.5 \| -26.4 \| 4.6 \| 11.4 \| $\cdots$ \| 11.434 \| $\cdots$ \| 10.554 \| 10.150 \| 10.057 \| 0.880 \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 1827-3955 \| 276.8721647 \| -39.9218242 \| -45.5 \| -184.0 \| 7.4 \| 7.4 \| $\cdots$ \| 12.337 \| 10.815 \| 10.803 \| 10.208 \| 9.942 \| 1.534 \| 61.2 \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 1828-3011 \| 277.1039697 \| -30.1852867 \| -227.6 \| 9.7 \| 34.3 \| 7.7 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 12.077 \| 11.530 \| 11.244 \| $\cdots$ \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 1828-3826 \| 277.1754025 \| -38.4387294 \| -107.1 \| -226.0 \| 7.2 \| 7.1 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 11.835 \| 11.212 \| 10.950 \| $\cdots$ \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 1828-3918 \| 277.1296564 \| -39.3128139 \| 39.5 \| -217.8 \| 7.2 \| 7.0 \| 16.517 \| 14.283 \| 12.758 \| 11.920 \| 11.401 \| 11.185 \| 2.363 \| 86.3 \| UPM 1831-1919 \| 277.8215872 \| -19.3332856 \| -121.3 \| -139.6 \| 4.1 \| 4.8 \| 16.419 \| 14.678 \| 13.446 \| 11.001 \| 10.419 \| 10.211 \| 3.677 \| 33.7 \| UPM 1831-2005 \| 277.8505681 \| -20.0885764 \| 2.2 \| -198.3 \| 22.2 \| 10.8 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 11.451 \| 10.748 \| 10.566 \| $\cdots$ \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 1831-3017 \| 277.9810053 \| -30.2859400 \| -95.5 \| -162.4 \| 7.4 \| 7.0 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 12.022 \| 11.433 \| 11.198 \| $\cdots$ \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 1832-3853 \| 278.1483619 \| -38.8879883 \| 24.3 \| -185.5 \| 4.8 \| 4.9 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 12.239 \| 11.703 \| 11.441 \| $\cdots$ \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 1834-2656 \| 278.5639625 \| -26.9406214 \| 169.4 \| 75.0 \| 11.8 \| 11.8 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 13.405 \| 12.820 \| 12.658 \| $\cdots$ \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 1834-3308 \| 278.5452358 \| -33.1453625 \| 46.8 \| -198.0 \| 44.6 \| 23.6 \| 15.886 \| 13.239 \| 11.394 \| 11.229 \| 10.667 \| 10.424 \| 2.010 \| 63.6 \| UPM 1836-0915 \| 279.0383847 \| -9.2573003 \| -154.3 \| 153.0 \| 6.0 \| 6.1 \| 17.858 \| 16.610 \| 15.380 \| 12.580 \| 12.055 \| 11.792 \| 4.030 \| 58.7 \| UPM 1839-1913 \| 279.9685653 \| -19.2177800 \| 151.3 \| -104.3 \| 8.4 \| 8.5 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 13.262 \| 12.550 \| 12.334 \| $\cdots$ \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 1840-1934 \| 280.1299600 \| -19.5831464 \| 201.2 \| -89.8 \| 8.8 \| 8.1 \| $\cdots$ \| 13.733 \| 12.400 \| 10.264 \| 9.706 \| 9.450 \| 3.469 \| 20.7 \| UPM 1840-2334 \| 280.0762169 \| -23.5667811 \| -128.2 \| -143.8 \| 6.8 \| 3.0 \| 12.605 \| 11.900 \| 11.719 \| 11.549 \| 11.300 \| 11.228 \| 0.351 \| 107.6 \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 1840-2614 \| 280.2257300 \| -26.2364583 \| -176.7 \| -77.1 \| 9.7 \| 22.6 \| 13.515 \| 12.155 \| 11.377 \| 10.890 \| 10.258 \| 10.105 \| 1.265 \| 68.3 \| UPM 1841-1841 \| 280.4988975 \| -18.6836078 \| 72.4 \| -210.7 \| 11.8 \| 32.7 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 11.705 \| 11.141 \| 10.844 \| $\cdots$ \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 1841-1902 \| 280.4059922 \| -19.0437531 \| -14.8 \| -252.1 \| 10.0 \| 10.1 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 11.847 \| 11.254 \| 10.958 \| $\cdots$ \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 1841-2852 \| 280.3630056 \| -28.8708058 \| -113.3 \| -182.5 \| 10.6 \| 10.6 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 12.277 \| 11.757 \| 11.445 \| $\cdots$ \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable ccSuperCOSMOS plate magnitudes suspect UPM 1841-4049 \| 280.3558444 \| -40.8271589 \| 110.8 \| -160.4 \| 6.5 \| 5.9 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 13.108 \| 12.509 \| 12.313 \| $\cdots$ \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 1842-2736 \| 280.7361064 \| -27.6091083 \| 77.9 \| -163.2 \| 9.3 \| 9.3 \| 13.426 \| $\cdots$ \| 10.951 \| 10.023 \| 9.356 \| 9.184 \| $\cdots$ \| 29.6 \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 1846-0947 \| 281.5706144 \| -9.7963978 \| -150.4 \| -115.9 \| 6.7 \| 7.3 \| 17.167 \| $\cdots$ \| $\cdots$ \| 12.344 \| 11.764 \| 11.497 \| $\cdots$ \| 88.7 \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 1848-0252 \| 282.0143783 \| -2.8768681 \| 32.0 \| -183.3 \| 6.0 \| 8.4 \| 17.550 \| 16.574 \| 15.273 \| 11.510 \| 10.973 \| 10.640 \| 5.064 \| 26.9 \| UPM 1850-1011 \| 282.7145042 \| -10.1934358 \| -102.2 \| -165.6 \| 7.5 \| 7.5 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 10.893 \| 10.296 \| 10.068 \| $\cdots$ \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 1851-1431 \| 282.9501331 \| -14.5258975 \| -164.4 \| -92.0 \| 4.5 \| 4.5 \| 15.938 \| 14.662 \| 13.257 \| 11.551 \| 11.005 \| 10.746 \| 3.111 \| 60.0 \| UPM 1851-2232 \| 282.9338092 \| -22.5417239 \| 209.9 \| 10.0 \| 7.3 \| 9.5 \| 16.115 \| 13.559 \| 12.603 \| 11.955 \| 11.379 \| 11.194 \| 1.604 \| 105.8 \| UPM 1851-3840 \| 282.8074922 \| -38.6741419 \| -161.2 \| 82.0 \| 6.2 \| 6.2 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 10.575 \| 9.983 \| 9.704 \| $\cdots$ \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 1852-2751 \| 283.0031728 \| -27.8602172 \| 82.6 \| -174.6 \| 7.9 \| 8.8 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 12.419 \| 11.857 \| 11.593 \| $\cdots$ \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 1858-3548 \| 284.6541453 \| -35.8062356 \| -102.9 \| 178.4 \| 6.7 \| 6.7 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 11.275 \| 10.871 \| 10.760 \| $\cdots$ \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable ccSuperCOSMOS plate magnitudes suspect UPM 1859-1701 \| 284.8129344 \| -17.0311458 \| -127.7 \| -164.7 \| 8.3 \| 8.3 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 10.923 \| 10.345 \| 10.034 \| $\cdots$ \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 1900-0511 \| 285.2324742 \| -5.1836367 \| -90.8 \| -168.3 \| 7.2 \| 7.0 \| 13.538 \| 11.997 \| 11.474 \| 10.426 \| 9.765 \| 9.570 \| 1.571 \| 43.5 \| UPM 1902-3731 \| 285.5237094 \| -37.5289453 \| 163.2 \| -82.3 \| 12.1 \| 12.2 \| 18.665 \| 16.574 \| 15.884 \| 13.833 \| 13.187 \| 12.935 \| 2.741 \| 158.4 \| UPM 1907-0221 \| 286.7786636 \| -2.3570300 \| -23.4 \| -192.6 \| 7.2 \| 7.1 \| 13.359 \| 12.590 \| 12.326 \| 11.287 \| 10.812 \| 10.740 \| 1.303 \| 76.8 \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 1917-2949 \| 289.4523744 \| -29.8228444 \| -70.6 \| -223.4 \| 10.8 \| 10.4 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 11.927 \| 11.374 \| 11.094 \| $\cdots$ \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 1920-1606 \| 290.0811403 \| -16.1009153 \| -175.4 \| 68.9 \| 10.9 \| 11.0 \| $\cdots$ \| $\cdots$ \| 17.198 \| 16.119 \| 15.579 \| 15.625 \| $\cdots$ \| 574.5 \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 1920-2206 \| 290.1098181 \| -22.1062475 \| 122.9 \| -161.4 \| 14.4 \| 9.4 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 12.085 \| 11.542 \| 11.280 \| $\cdots$ \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 1923-4026 \| 290.7523894 \| -40.4484822 \| -154.6 \| -96.7 \| 7.7 \| 9.5 \| $\cdots$ \| 16.086 \| 14.192 \| 12.847 \| 12.337 \| 12.100 \| 3.239 \| 88.5 \| UPM 1925-0916 \| 291.4896428 \| -9.2686953 \| -58.8 \| -197.4 \| 6.4 \| 6.2 \| $\cdots$ \| 14.429 \| $\cdots$ \| 12.428 \| 11.800 \| 11.617 \| 2.001 \| 121.9 \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 1925-3712 \| 291.4461181 \| -37.2007175 \| 138.7 \| -165.4 \| 15.2 \| 15.1 \| $\cdots$ \| $\cdots$ \| 13.683 \| 12.134 \| 11.536 \| 11.289 \| $\cdots$ \| 47.8 \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 1931-4001 \| 292.9411467 \| -40.0227744 \| -3.1 \| -185.0 \| 1.9 \| 1.9 \| 14.966 \| 12.936 \| 11.356 \| 10.714 \| 10.092 \| 9.861 \| 2.222 \| 50.2 \| UPM 1933-2916 \| 293.3358858 \| -29.2750011 \| -90.4 \| -159.5 \| 2.9 \| 5.1 \| 15.935 \| 13.822 \| 11.914 \| 10.767 \| 10.151 \| 9.882 \| 3.055 \| 34.6 \| UPM 1940-0508 \| 295.1044458 \| -5.1410269 \| 22.0 \| -181.3 \| 8.1 \| 8.0 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 11.602 \| 11.074 \| 10.825 \| $\cdots$ \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 1943-0035 \| 295.9131372 \| -0.5899383 \| -256.9 \| 22.2 \| 11.4 \| 10.0 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 12.195 \| 11.669 \| 11.372 \| $\cdots$ \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 1947-3542 \| 296.7613983 \| -35.7157425 \| 24.0 \| -180.4 \| 2.5 \| 2.6 \| 16.879 \| 14.954 \| 13.555 \| 12.269 \| 11.659 \| 11.436 \| 2.685 \| 86.0 \| UPM 1950-0135 \| 297.6302547 \| -1.5941806 \| 223.2 \| 2.3 \| 10.2 \| 10.1 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 12.355 \| 11.803 \| 11.575 \| $\cdots$ \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 1952-0813 \| 298.1828144 \| -8.2235433 \| 103.0 \| -220.6 \| 7.8 \| 9.1 \| 15.568 \| 13.655 \| 11.782 \| 10.585 \| 9.968 \| 9.683 \| 3.070 \| 32.6 \| UPM 1953-0508 \| 298.3124231 \| -5.1386344 \| -45.1 \| -196.0 \| 7.3 \| 7.1 \| $\cdots$ \| 14.484 \| 13.942 \| 12.870 \| 12.395 \| 12.296 \| 1.614 \| 160.4 \| UPM 1954-0139 \| 298.6503828 \| -1.6506567 \| -93.6 \| -164.6 \| 9.0 \| 9.1 \| 16.468 \| 15.236 \| 14.497 \| 13.541 \| 12.951 \| 12.815 \| 1.695 \| [213.2] \| ffSubdwarf candidate selected from RPM diagram; plate distance [in bracket] is incorrect UPM 1955-3426 \| 298.9125511 \| -34.4337736 \| -92.3 \| -161.9 \| 4.8 \| 4.7 \| 15.620 \| 15.545 \| $\cdots$ \| 12.908 \| 12.406 \| 12.128 \| 2.637 \| 116.8 \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable ccSuperCOSMOS plate magnitudes suspect UPM 1957-3801 \| 299.4516389 \| -38.0268108 \| 119.4 \| -160.3 \| 6.6 \| 17.2 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 12.317 \| 11.721 \| 11.521 \| $\cdots$ \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 1958-3348 \| 299.6281386 \| -33.8102706 \| 122.5 \| -132.4 \| 5.7 \| 5.0 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 12.145 \| 11.624 \| 11.377 \| $\cdots$ \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 2007-0132 \| 301.7682706 \| -1.5366978 \| 6.6 \| -184.1 \| 16.4 \| 16.1 \| $\cdots$ \| 15.506 \| $\cdots$ \| 12.448 \| 11.871 \| 11.591 \| 3.058 \| 71.5 \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 2007-0312 \| 301.9236269 \| -3.2085131 \| -159.3 \| -86.5 \| 7.3 \| 7.2 \| 16.388 \| 14.715 \| 12.952 \| 11.235 \| 10.702 \| 10.462 \| 3.480 \| 41.6 \| UPM 2009-0041 \| 302.4692369 \| -0.6924347 \| 115.0 \| -191.9 \| 14.0 \| 14.7 \| 17.493 \| 15.536 \| 13.355 \| 12.277 \| 11.749 \| 11.497 \| 3.259 \| 72.2 \| UPM 2009-3305 \| 302.2734844 \| -33.0964475 \| 60.3 \| -179.0 \| 22.0 \| 16.9 \| 17.666 \| 15.906 \| 14.344 \| 13.039 \| 12.533 \| 12.329 \| 2.867 \| 129.1 \| UPM 2011-0002 \| 302.9155206 \| -0.0494872 \| 82.3 \| -262.2 \| 10.4 \| 10.7 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 11.420 \| 10.819 \| 10.684 \| $\cdots$ \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 2011-0010 \| 302.7694014 \| -0.1825764 \| -141.8 \| -130.9 \| 11.1 \| 8.4 \| $\cdots$ \| $\cdots$ \| 12.726 \| 11.881 \| 11.339 \| 11.119 \| $\cdots$ \| 79.4 \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 2011-0139 \| 302.8264014 \| -1.6649031 \| 129.2 \| -127.5 \| 15.9 \| 16.0 \| 17.948 \| 15.746 \| 13.609 \| 12.433 \| 11.860 \| 11.584 \| 3.313 \| 67.8 \| UPM 2012-0133 \| 303.1161578 \| -1.5506017 \| 236.9 \| -65.7 \| 14.0 \| 13.5 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 11.962 \| 11.370 \| 11.068 \| $\cdots$ \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 2014-0124 \| 303.6403675 \| -1.4061056 \| 144.1 \| -231.7 \| 7.8 \| 7.9 \| $\cdots$ \| $\cdots$ \| 12.139 \| 11.284 \| 10.729 \| 10.508 \| $\cdots$ \| 59.2 \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 2014-0624 \| 303.5245169 \| -6.4119417 \| 3.3 \| -220.4 \| 17.6 \| 17.2 \| 17.444 \| 15.620 \| 14.056 \| 12.989 \| 12.373 \| 12.223 \| 2.631 \| 131.8 \| UPM 2014-1634 \| 303.6036331 \| -16.5819139 \| -167.7 \| -192.2 \| 12.1 \| 13.4 \| $\cdots$ \| 13.477 \| $\cdots$ \| 12.772 \| 12.137 \| 12.020 \| 0.705 \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 2019-0754 \| 304.8205911 \| -7.9011086 \| 127.5 \| -157.6 \| 6.9 \| 7.4 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 12.462 \| 12.101 \| 12.066 \| $\cdots$ \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 2020-0826 \| 305.0236797 \| -8.4449589 \| 174.3 \| -48.6 \| 11.2 \| 8.1 \| $\cdots$ \| 14.934 \| 12.955 \| 11.903 \| 11.347 \| 11.099 \| 3.031 \| 62.7 \| UPM 2024-0638 \| 306.0952239 \| -6.6395264 \| 75.9 \| -247.6 \| 10.9 \| 11.1 \| 16.931 \| 14.871 \| 13.039 \| 11.878 \| 11.357 \| 11.085 \| 2.993 \| 64.4 \| UPM 2045-0612 \| 311.4779600 \| -6.2101506 \| -143.8 \| -120.3 \| 10.0 \| 10.1 \| $\cdots$ \| 15.646 \| 13.667 \| 12.937 \| 12.409 \| 12.167 \| 2.709 \| 122.0 \| UPM 2047-0429 \| 311.8002206 \| -4.4950453 \| 143.0 \| -117.9 \| 6.4 \| 6.6 \| 12.644 \| 11.975 \| 11.578 \| 11.659 \| 11.348 \| 11.398 \| 0.316 \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 2047-2232 \| 311.9131269 \| -22.5408683 \| -136.3 \| -128.9 \| 9.8 \| 10.0 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 13.261 \| 12.733 \| 12.634 \| $\cdots$ \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 2047-3246 \| 311.9777267 \| -32.7689467 \| -242.1 \| 197.9 \| 7.2 \| 7.4 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 10.917 \| 10.373 \| 10.143 \| $\cdots$ \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 2049-0304A \| 312.4151003 \| -3.0785519 \| 83.1 \| -185.7 \| 7.0 \| 7.3 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 12.889 \| 12.288 \| 12.072 \| $\cdots$ \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable ddCommon proper motion companion; see Table 4 UPM 2049-0304B \| 312.4151678 \| -3.0771406 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| $\cdots$ \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 14.444 \| 13.943 \| 13.673 \| $\cdots$ \| $\cdots$ \| aaProper motions suspect bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable ddCommon proper motion companion; see Table 4 eeNot detected during automated search but noticed by eye during the blinking process UPM 2050-1541 \| 312.5709881 \| -15.6922022 \| 84.2 \| -178.4 \| 3.2 \| 4.7 \| 15.656 \| 13.702 \| 12.131 \| 11.456 \| 10.862 \| 10.580 \| 2.246 \| 70.2 \| UPM 2050-4535 \| 312.5937853 \| -45.5929939 \| 17.8 \| -207.4 \| 9.5 \| 9.6 \| 17.265 \| 15.170 \| 14.302 \| 13.405 \| 12.863 \| 12.688 \| 1.765 \| [214.1] \| ffSubdwarf candidate selected from RPM diagram; plate distance [in bracket] is incorrect UPM 2057-0911 \| 314.4559047 \| -9.1875750 \| 129.1 \| -168.5 \| 8.7 \| 8.4 \| $\cdots$ \| 12.472 \| 11.512 \| 11.501 \| 10.806 \| 10.656 \| 0.971 \| 87.5 \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 2058-0332 \| 314.6363908 \| -3.5394206 \| -38.0 \| -198.2 \| 6.6 \| 8.1 \| 13.785 \| 11.439 \| 10.095 \| 10.436 \| 9.823 \| 9.618 \| 1.003 \| 44.8 \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 2105-1703 \| 316.4376317 \| -17.0536781 \| 81.7 \| -161.3 \| 2.1 \| 2.3 \| 15.955 \| 14.016 \| 12.396 \| 11.385 \| 10.749 \| 10.506 \| 2.631 \| 57.4 \| UPM 2115-0631 \| 318.8872478 \| -6.5276206 \| 191.1 \| 28.7 \| 7.6 \| 8.4 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 12.633 \| 12.090 \| 11.891 \| $\cdots$ \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 2118-2101 \| 319.5260700 \| -21.0311750 \| 188.4 \| 30.4 \| 15.9 \| 17.3 \| 18.188 \| 15.914 \| 14.188 \| 13.103 \| 12.484 \| 12.212 \| 2.811 \| 108.6 \| UPM 2131-3027 \| 322.7657644 \| -30.4592144 \| 47.4 \| -173.7 \| 3.7 \| 3.7 \| 17.497 \| 15.514 \| 13.704 \| 12.264 \| 11.687 \| 11.415 \| 3.250 \| 65.9 \| UPM 2140-0613 \| 325.0265956 \| -6.2270500 \| -56.1 \| -210.2 \| 7.9 \| 8.2 \| 13.765 \| 12.664 \| 13.526 \| 12.762 \| 12.356 \| 12.341 \| -0.098 \| 156.2 \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 2145-4145 \| 326.4178564 \| -41.7660619 \| 137.1 \| -125.2 \| 1.7 \| 2.1 \| 16.569 \| 13.895 \| 12.271 \| 11.407 \| 10.885 \| 10.560 \| 2.488 \| 55.7 \| ccSuperCOSMOS plate magnitudes suspect UPM 2152-1147 \| 328.2452208 \| -11.7888700 \| 150.3 \| -123.1 \| 1.9 \| 4.1 \| 16.766 \| 14.875 \| 12.787 \| 11.466 \| 10.888 \| 10.632 \| 3.409 \| 44.4 \| UPM 2154-0143 \| 328.6042781 \| -1.7273786 \| 259.8 \| -7.5 \| 11.2 \| 11.7 \| 13.464 \| 11.889 \| 10.739 \| 11.170 \| 10.542 \| 10.353 \| 0.719 \| 79.4 \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 2157-0251 \| 329.4304583 \| -2.8516900 \| 139.8 \| -132.6 \| 18.1 \| 17.9 \| 18.051 \| $\cdots$ \| 14.875 \| 13.480 \| 12.902 \| 12.672 \| $\cdots$ \| 152.1 \| UPM 2222-3528 \| 335.5426575 \| -35.4820467 \| -116.0 \| -155.8 \| 5.9 \| 5.7 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 13.313 \| 12.672 \| 12.461 \| $\cdots$ \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 2229-0432B \| 337.4473828 \| -4.5360572 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| $\cdots$ \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 12.140 \| 11.556 \| 11.319 \| $\cdots$ \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliableeeNot detected during automated search but noticed by eye during the blinking process ddCommon proper motion companion; see Table 4 aaProper motions suspect UPM 2231-1642 \| 337.7917883 \| -16.7142497 \| -123.3 \| -139.2 \| 1.8 \| 1.9 \| 16.421 \| 14.446 \| 12.668 \| 11.850 \| 11.283 \| 11.036 \| 2.596 \| 76.2 \| UPM 2232-0225 \| 338.1181078 \| -2.4226789 \| 181.2 \| -31.2 \| 11.1 \| 11.5 \| 16.226 \| 14.169 \| 12.605 \| 11.514 \| 10.938 \| 10.684 \| 2.655 \| 60.9 \| UPM 2233-0003 \| 338.2930958 \| -0.0525181 \| 247.8 \| 99.7 \| 8.7 \| 8.8 \| $\cdots$ \| 15.656 \| $\cdots$ \| 12.763 \| 12.209 \| 11.978 \| 2.893 \| 94.6 \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 2237-0118 \| 339.3505139 \| -1.3070036 \| -147.7 \| -110.1 \| 7.0 \| 7.0 \| 14.926 \| 13.000 \| 11.919 \| 11.674 \| 11.029 \| 10.877 \| 1.326 \| 97.5 \| UPM 2244-4333 \| 341.0835264 \| -43.5574258 \| 183.0 \| -51.4 \| 14.8 \| 15.1 \| 17.708 \| 15.718 \| 14.089 \| 13.114 \| 12.529 \| 12.353 \| 2.604 \| 138.3 \| UPM 2246-0017 \| 341.6494342 \| -0.2863092 \| 193.6 \| -28.5 \| 10.3 \| 10.3 \| 17.545 \| 15.654 \| 13.935 \| 12.665 \| 12.130 \| 11.873 \| 2.989 \| 95.3 \| UPM 2248-3255 \| 342.1992344 \| -32.9243678 \| -126.6 \| -174.9 \| 3.8 \| 3.9 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 12.662 \| 12.120 \| 11.932 \| $\cdots$ \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 2250-2908A \| 342.5898331 \| -29.1365486 \| 186.9 \| -9.2 \| 5.0 \| 3.0 \| 15.217 \| 13.358 \| 11.795 \| 12.126 \| 11.517 \| 11.293 \| 1.232 \| 115.3 \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable ddCommon proper motion companion; see Table 4 UPM 2250-2908B \| 342.5883269 \| -29.1372669 \| 182.3 \| -17.0 \| 3.7 \| 3.9 \| 15.217 \| 13.358 \| 11.795 \| 13.184 \| 12.599 \| 12.327 \| 0.174 \| 155.7 \| aaProper motions suspect ddCommon proper motion companion; see Table 4 bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable eeNot detected during automated search but noticed by eye during the blinking process UPM 2251-0201 \| 342.7661222 \| -2.0260436 \| 50.1 \| -185.5 \| 7.6 \| 7.6 \| 15.605 \| 13.598 \| 12.627 \| 11.995 \| 11.372 \| 11.176 \| 1.603 \| 110.8 \| UPM 2305-4612 \| 346.2905092 \| -46.2043653 \| 143.5 \| -110.4 \| 7.8 \| 2.5 \| 16.914 \| 15.068 \| 13.649 \| 12.773 \| 12.240 \| 11.993 \| 2.295 \| 136.2 \| UPM 2306-0315 \| 346.6112989 \| -3.2637836 \| 181.1 \| -55.6 \| 10.2 \| 10.2 \| 17.564 \| 15.549 \| 13.841 \| 12.506 \| 11.947 \| 11.661 \| 3.043 \| 80.8 \| UPM 2308-1954 \| 347.1959358 \| -19.9085208 \| 183.6 \| -0.3 \| 13.0 \| 12.1 \| 13.693 \| 12.059 \| 11.387 \| 11.479 \| 11.092 \| 11.006 \| 0.580 \| 131.5 \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable aaProper motions suspect UPM 2310-0410 \| 347.5503997 \| -4.1822806 \| 155.6 \| -93.0 \| 11.0 \| 11.1 \| $\cdots$ \| 15.438 \| 14.685 \| 13.920 \| 13.313 \| 13.160 \| 1.518 \| [263.6] \| ffSubdwarf candidate selected from RPM diagram; plate distance [in bracket] is incorrect UPM 2320-0006 \| 350.0481022 \| -0.1140686 \| 78.7 \| -165.2 \| 9.3 \| 2.5 \| 13.636 \| 11.776 \| 10.884 \| 10.299 \| 9.637 \| 9.479 \| 1.477 \| 50.0 \| UPM 2325-1715 \| 351.4973083 \| -17.2589567 \| 180.0 \| 254.5 \| 11.5 \| 10.5 \| 16.160 \| 13.342 \| 11.790 \| 11.113 \| 10.486 \| 10.218 \| 2.229 \| 50.7 \| UPM 2328-0546 \| 352.0857586 \| -5.7781261 \| 183.3 \| 62.8 \| 6.6 \| 7.2 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 10.995 \| 10.415 \| 10.162 \| $\cdots$ \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 2329-0510 \| 352.4821203 \| -5.1685144 \| 159.2 \| 88.3 \| 12.7 \| 9.7 \| 17.693 \| 15.621 \| 13.802 \| 12.583 \| 12.002 \| 11.781 \| 3.038 \| 85.5 \| UPM 2331-0233 \| 352.8670611 \| -2.5567644 \| -189.0 \| -114.3 \| 8.6 \| 11.3 \| 17.038 \| 14.938 \| 13.029 \| 11.864 \| 11.303 \| 11.000 \| 3.074 \| 58.3 \| UPM 2331-0617 \| 352.7987228 \| -6.2871728 \| 180.6 \| 36.1 \| 10.3 \| 10.3 \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 12.722 \| 12.079 \| 11.904 \| $\cdots$ \| $\cdots$ \| bbNumber of relations used for distance estimate $<$ 6: plate distance less reliable UPM 2334-4145 \| 353.6053231 \| -41.7568239 \| -20.3 \| -187.7 \| 2.4 \| 3.2 \| 17.676 \| 15.973 \| 15.112 \| 14.059 \| 13.421 \| 13.279 \| 1.914 \| [273.8] \| ffSubdwarf candidate selected from RPM diagram; plate distance [in bracket] is incorrect UPM 2349-1023 \| 357.2962428 \| -10.3857086 \| 169.5 \| 71.8 \| 13.1 \| 14.1 \| 17.811 \| 15.720 \| 13.845 \| 12.954 \| 12.352 \| 12.065 \| 2.766 \| 109.4 \| UPM 2356-0453 \| 359.1178706 \| -4.8861358 \| 189.6 \| -44.4 \| 19.9 \| 18.6 \| 18.052 \| 16.087 \| 14.310 \| 12.885 \| 12.374 \| 12.113 \| 3.202 \| 95.4 \| Table 3: New UCAC3 High Proper Motion Systems estimated to be within 25 pc between Declinations $-$47$\arcdeg$ and 0$\arcdeg$ with 0$\farcs$40 yr-1 $>$ $\mu$ $\geq$ 0$\farcs$18 yr-1 Name \| RA J2000.0 \| DEC J2000.0 \| $\mu_{\alpha}\cos\delta$ \| $\mu_{\delta}$ \| sig $\mu_{\alpha}$ \| sig $\mu_{\delta}$ \| $B_{J}$ \| $R_{59F}$ \| $I_{IVN}$ \| $J$ \| $H$ \| $K_{s}$ \| $R_{59F}$ $-$ $J$ \| Est Dist \| Notes ---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|--- \| (deg) \| (deg) \| (mas/yr) \| (mas/yr) \| (mas/yr) \| (mas/yr) \| \| \| \| \| \| \| \| (pc) \| UPM 1349-4228 \| 207.2552625 \| -42.4784189 \| -161.9 \| -84.6 \| 1.3 \| 3.0 \| 14.468 \| 11.703 \| $\cdots$ \| 9.449 \| 8.863 \| 8.622 \| 2.254 \| 24.4 \| UPM 1648-3459 \| 252.1200667 \| -34.9967942 \| 178.6 \| 142.5 \| 4.4 \| 4.4 \| $\cdots$ \| 14.631 \| $\cdots$ \| 10.687 \| 10.161 \| 9.907 \| 3.944 \| 22.1 \| UPM 1654-3105 \| 253.6846164 \| -31.0961000 \| -32.4 \| -215.9 \| 7.4 \| 7.2 \| 15.122 \| 13.553 \| 11.977 \| 10.072 \| 9.482 \| 9.237 \| 3.481 \| 23.8 \| aaPossible NLTT star with a position difference $>$ 90$\arcsec$ when compared to UCAC3 position UPM 1718-2245A \| 259.6129147 \| -22.7616683 \| -160.7 \| -160.8 \| 7.2 \| 6.8 \| 15.469 \| 13.836 \| 13.155 \| 10.385 \| 9.806 \| 9.572 \| 3.451 \| 25.4 \| bbCommon proper motion companion; see Table 4 UPM 1718-2245B \| 259.6213031 \| -22.7746183 \| -161.1 \| -154.8 \| 10.9 \| 9.6 \| $\cdots$ \| 14.787 \| 13.289 \| 10.207 \| 9.608 \| 9.375 \| 4.580 \| 13.2 \| bbCommon proper motion companion; see Table 4 ccNot detected during automated search but noticed by eye during the blinking process UPM 1840-1934 \| 280.1299600 \| -19.5831464 \| 201.2 \| -89.8 \| 8.8 \| 8.1 \| $\cdots$ \| 13.733 \| 12.400 \| 10.264 \| 9.706 \| 9.450 \| 3.469 \| 20.7 \| Table 4: Common Proper Motion Candidate Systems Primary \| $\mu_{\alpha}\cos\delta$ \| $\mu_{\delta}$ \| Distance \| Secondary/Tertiary \| $\mu_{\alpha}\cos\delta$ \| $\mu_{\delta}$ \| Distance \| Separation \| $\theta$ \| notes ---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|--- \| (mas/yr) \| (mas/yr) \| (pc) \| \| (mas/yr) \| (mas/yr) \| (pc) \| ($\arcsec$) \| ($\arcdeg$) \| UPM 0209-3339A \| -86.1 \| -166.9 \| 49.5 \| UPM 0209-3339B \| -112.9 \| -170.2 \| $\cdots$ \| 11.6 \| 78.1 \| aaNot detected during automated search but noticed by eye during the blinking process bbNumber of relations used for distance estimate $<$ 6; plate distance less reliable UPM 0443-4129A \| 186.1 \| 4.3 \| 39.3 \| 2MASS J04430760-4128575B \| -107.5 \| -53.1 \| $\cdots$ \| 6.8 \| 339.2 \| aaNot detected during automated search but noticed by eye during the blinking process bbNumber of relations used for distance estimate $<$ 6; plate distance less reliable ccProper motions suspect UPM 0528-4313A \| -75.6 \| 164.7 \| 70.4 \| UPM 0528-4313B \| -86.3 \| 163.2 \| 109.1 \| 42.1 \| 209.0 \| aaNot detected during automated search but noticed by eye during the blinking process UPM 0659-0052A \| -58.3 \| -184.1 \| 78.2 \| UPM 0659-0052B \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 13.8 \| 151.6 \| aaNot detected during automated search but noticed by eye during the blinking process bbNumber of relations used for distance estimate $<$ 6; plate distance less reliable ccProper motions suspect UPM 0704-0602A \| $\cdots$ \| $\cdots$ \| 123.4 \| UPM 0704-0602B \| 99.5 \| -153.0 \| 37.8 \| 12.2 \| 359.1 \| aaNot detected during automated search but noticed by eye during the blinking process bbNumber of relations used for distance estimate $<$ 6; plate distance less reliable ccProper motions suspect ddSuperCOSMOS plate magnitudes suspect UPM 0747-2537A \| -148.5 \| 101.9 \| 40.6 \| UPM 0747-2537B \| -151.3 \| 102.3 \| 47.3 \| 12.0 \| 237.4 \| bbNumber of relations used for distance estimate $<$ 6; plate distance less reliable UPM 0800-0617A \| 135.2 \| -233.8 \| [175.5] \| UPM 0800-0617B \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 5.8 \| 297.2 \| aaNot detected during automated search but noticed by eye during the blinking process bbNumber of relations used for distance estimate $<$ 6; plate distance less reliable ccProper motions suspect eeSubdwarf candidate selected from RPM diagram; plate distance [in bracket] is incorrect BD-04 2807A \| -142.1 \| -37.1 \| 19.5 \| UPM 1009-0501B \| -190.5 \| 92.1 \| $\cdots$ \| 20.9 \| 338.5 \| aaNot detected during automated search but noticed by eye during the blinking process bbNumber of relations used for distance estimate $<$ 6; plate distance less reliable UPM 1020-0633A \| -179.8 \| -27.8 \| 34.8 \| SCR 1020-0634B \| -181.5 \| -24.2 \| 37.5 \| 87.3 \| 157.2 \| UPM 1031-0024A \| -207.4 \| -105.6 \| 55.1 \| UPM 1031-0024B \| -142.5 \| -96.9 \| $\cdots$ \| 7.4 \| 91.4 \| aaNot detected during automated search but noticed by eye during the blinking process bbNumber of relations used for distance estimate $<$ 6; plate distance less reliable ffSource not in 2MASS UPM 1056-0542A \| -98.1 \| -173.8 \| 76.5 \| UPM 1056-0542B \| -63.9 \| -173.3 \| $\cdots$ \| 9.0 \| 86.2 \| aaNot detected during automated search but noticed by eye during the blinking process bbNumber of relations used for distance estimate $<$ 6; plate distance less reliable UPM 1142-2055A \| -186.7 \| 44.2 \| 41.2 \| UPM 1142-2055B \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 8.3 \| 167.6 \| aaNot detected during automated search but noticed by eye during the blinking process bbNumber of relations used for distance estimate $<$ 6; plate distance less reliable ccProper motions suspect NLTT 28641A \| -201.8 \| 6.2 \| $\cdots$ \| UPM 1149-0019B \| -201.5 \| 2.2 \| $\cdots$ \| 27.3 \| 128.2 \| aaNot detected during automated search but noticed by eye during the blinking process bbNumber of relations used for distance estimate $<$ 6; plate distance less reliable UPM 1159-3623A \| -182.1 \| -101.4 \| 113.1 \| UPM 1159-3623B \| -172.6 \| -92.4 \| 132.4 \| 13.6 \| 303.7 \| aaNot detected during automated search but noticed by eye during the blinking process UPM 1226-2020A \| -137.6 \| -119.8 \| 72.3 \| UPM 1226-2020B \| -146.3 \| -117.3 \| $\cdots$ \| 7.0 \| 333.3 \| aaNot detected during automated search but noticed by eye during the blinking process bbNumber of relations used for distance estimate $<$ 6; plate distance less reliable SCR 1226-3515A \| -192.3 \| 41.0 \| 56.5 \| UPM 1226-3516B \| -200.5 \| 38.3 \| 127.5 \| 49.8 \| 191.3 \| aaNot detected during automated search but noticed by eye during the blinking process ggPossible NLTT star with a position difference $>$ 90$\arcsec$ when compared to UCAC3 position \| \| \| \| UPM 1226-3516C \| -115.2 \| 8.5 \| 243.3 \| 97.0 \| 146.9 \| aaNot detected during automated search but noticed by eye during the blinking process ccProper motions suspect ggPossible NLTT star with a position difference $>$ 90$\arcsec$ when compared to UCAC3 position UPM 1315-2904A \| -190.6 \| -27.5 \| 89.6 \| UPM 1315-2904B \| -209.7 \| -1.5 \| 149.3 \| 5.9 \| 332.6 \| aaNot detected during automated search but noticed by eye during the blinking process 2MASS J13465039-2112266A \| -186.8 \| -44.0 \| 46.8 \| UPM 1346-2111B \| -112.1 \| -60.0 \| 86.8 \| 82.2 \| 350.8 \| aaNot detected during automated search but noticed by eye during the blinking process ccProper motions suspect UPM 1718-2245A \| -160.7 \| -160.8 \| 25.4 \| UPM 1718-2245B \| -161.1 \| -154.8 \| 13.2 \| 54.3 \| 149.2 \| aaNot detected during automated search but noticed by eye during the blinking process UPM 1724-0318A \| 143.7 \| -126.7 \| 92.0 \| UPM 1724-0318B \| $\cdots$ \| $\cdots$ \| 169.5 \| 5.5 \| 325.4 \| aaNot detected during automated search but noticed by eye during the blinking process bbNumber of relations used for distance estimate $<$ 6; plate distance less reliable ccProper motions suspect TYC 7897 997 1A \| -10.4 \| -160.2 \| 40.3 \| UPM 1749-4404B \| 0.0 \| -204.9 \| $\cdots$ \| 19.8 \| 254.0 \| aaNot detected during automated search but noticed by eye during the blinking process bbNumber of relations used for distance estimate $<$ 6; plate distance less reliable ccProper motions suspect ddSuperCOSMOS plate magnitudes suspect TYC 8344 154 1A \| 51.6 \| -179.5 \| $\cdots$ \| UPM 1757-4632B \| 59.7 \| -185.1 \| $\cdots$ \| 30.7 \| 296.9 \| aaNot detected during automated search but noticed by eye during the blinking process bbNumber of relations used for distance estimate $<$ 6; plate distance less reliable UPM 2049-0304A \| 83.1 \| -185.7 \| $\cdots$ \| UPM 2049-0304B \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 5.3 \| 1.5 \| aaNot detected during automated search but noticed by eye during the blinking process bbNumber of relations used for distance estimate $<$ 6; plate distance less reliable ccProper motions suspect 2MASS J22294694-0432036A \| 160.9 \| -86.5 \| 50.8 \| UPM 2229-0432B \| $\cdots$ \| $\cdots$ \| $\cdots$ \| 8.7 \| 134.7 \| aaNot detected during automated search but noticed by eye during the blinking process bbNumber of relations used for distance estimate $<$ 6; plate distance less reliable ccProper motions suspect UPM 2250-2908A \| 186.9 \| -9.2 \| 115.3 \| UPM 2250-2908B \| 182.3 \| -17.0 \| 155.7 \| 5.4 \| 241.4 \| aaNot detected during automated search but noticed by eye during the blinking process bbNumber of relations used for distance estimate $<$ 6; plate distance less reliable ccProper motions suspect	arxiv-papers	2011-11-29T13:19:39	2024-09-04T02:49:24.729963	{ "license": "Public Domain", "authors": "C. T. Finch, N. Zacharias, M. R. Boyd, T. J. Henry, N. C. Hambly", "submitter": "Charlie Finch", "url": "https://arxiv.org/abs/1111.6805" }
1111.6829	# Heavy Flavour Results at the LHC P. Koppenburg Nikhef, Amsterdam, The Netherlands On behalf of the LHCb collaboration, including Atlas, CMS and Alice results. ###### Abstract We present a brief overview of the first flavour physics results at the LHC. Cross-section for charm and beauty production have been measured by several experiments and the first competitive results on $D$ and $B$ decays are presented. ## I Introduction Precise measurements of CP violation and searches for rare decays have a high potential for discovering effects of New Physics. They are a specific task of the LHCb experiment and are complementary to direct searches performed by general purpose experiments. CP violation and rare decays are sensitive to new particles and couplings in an indirect way via interferences with Standard Model (SM) processes. This not only probes a potentially higher mass scale than direct searches for new particles, but also gives access to amplitudes and phases of the new couplings. The $b$ and $c$ quark decays are the best laboratory for this programme. ## II The LHC as a the new Flavour Factory After a very successful decade dominated by the $B$ factories, Belle and Babar, the LHC is taking over as the new flavour factory. While PEP-2 and KEK-B have produced around $10^{9}$ $b\bar{b}$ pairs during their lifetime, the LHC has produced close to $4\cdot 10^{12}$ just in 2011 thanks to the very large $b$ cross-section in high energy proton collisions. Of course the major challenge at the LHC is to efficiently collect the most interesting among all these events. ### II.1 The LHCb Experiment The LHCb experiment [1] is dedicated to precision measurements of CP violation and rare decays in beauty and charm decays. Its forward geometry covering the range $2<\eta<5$ exploits the dominant heavy flavour production mechanism at the LHC and covers about 40% of the differential cross-section. Among the features unique to LHCb are its high precision vertex detector, which is retracted away from the beam at injection and moved as close as $8\>$mm from the beam during data taking. It is followed by a tracking system located in a dipole magnet of which the polarity can be reversed, allowing to cancel detector asymmetries in CP-violation measurements. A system of two Ring Imaging Cherenkov detectors (RICH) allows a very good pion/kaon/proton separation over the momentum range $1$–$100\>{\rm GeV}/c$. Due to the very large total cross section an effective on-line event selection is required where the rate is reduced by a first level hardware trigger and further by two levels of software triggers [2]. LHCb uses hadrons, muons, electrons and photons throughout the trigger chain, thus maximising the trigger efficiency on all heavy quark decays. In order not to saturate the trigger and keep the event multiplicity low, the LHCb experiment is not operating at the maximum LHC luminosity. During most of 2011 LHCb has kept their luminosity at the constant value of $3.5\cdot 10^{32}\>\rm cm^{-2}s^{-1}$ by displacing the proton beams laterally in real time. ### II.2 Flavour Physics at Atlas, CMS and Alice The Atlas [3] and CMS [4] detectors are multi-purpose central detectors optimised for searches of heavy objects. At high luminosity, their potential for flavour physics is limited by their triggering capabilities and focus mainly on $b$ and charmonium decays involving dimuons. But at the lower luminosities at which the LHC operated during 2010 and part of 2011, a more open trigger allowed an interesting favour physics programme that is complementary to LHCb’s in particular for cross-section measurements in the central region. The Alice [5] detector is optimised for heavy ion collisions and, while covering mostly the central region, has similar particle ID and tracking capabilities as LHCb. They are thus a key player for cross-section measurements (which are often a necessary normalisation or their QGP programme) but are not competitive for CP violation and rare decays searches due to the lower luminosity at which they operate. ### II.3 Data Samples In 2011, Atlas and CMS have been delivered around $5.7\>\rm fb^{-1}$ each, LHCb $1.2\>\rm fb^{-1}$ and Alice $5\>\rm pb^{-1}$, of which more than 90% have been recorded and are useful for physics. The measurements reported below use only a fraction of these data, in most cases the $1.1\>\rm fb^{-1}$ (Atlas, CMS) and $370\>\rm pb^{-1}$ (LHCb) collected until end of June 2011. Most cross-section measurements use the lower-luminosity data sample of about $40\>\rm pb^{-1}$ collected during 2010. ## III Heavy Flavour Production The four LHC experiments offer a vast coverage of rapidity: Atlas and CMS cover the central region up to $2.5$, Alice $0$–$1$ and $2.5$–$4$ and LHCb $2$–$5.5$. A combination of differential cross-section measurements would allow to cover the range $0<\|\eta\|<5.5$, but in most cases such combinations have not yet been performed. Yet, many measurements from the various experiments are available and give a good picture of heavy flavour production in $pp$ collisions at $\sqrt{s}=7\>\rm TeV$. ### III.1 Charmonium The prompt $J/\psi$ production has been measured by all four experiments [6, 7, 8, 9] with 2010 data in bins of rapidity and transverse momentum (See Fig. 1) and compared with theoretical models. No large discrepancies are seen with the present level of uncertainties. The unknown polarisation is the main uncertainty in all measurements, and more data is needed to be able to resolve it. LHCb also reports a cross-section for double $J/\psi$ production, which is very sensitive to the production mechanism [10]. Figure 1: Differential $J/\psi$ cross section measurement at CMS [6]. ### III.2 Open Charm The open charm cross-section has been measured using $D^{0}$, $D^{+}$, $D_{s}$ and $D^{\ast+}$ modes by Atlas and LHCb [11, 12] using the first few $\rm nb^{-1}$ delivered by the LHC in 2010. The low luminosity during this period allowed to profit from a very open trigger which helps keeping systematic errors low. The unexpected high total $c\bar{c}$ cross section of $6.10\pm 0.93\rm\>mb$, extrapolated from these measurements is very encouraging for charm physics in $7\>$TeV collisions. This is about 10% of the total inelastic cross-section. ### III.3 Beauty The inclusive production of beauty and charm hadrons in pp collisions has been measured by LHCb. In particular using semi-leptonic decays $b\to D^{0}(K\pi)\mu\bar{\nu}X$ [13] the cross section $\sigma(pp\to b\bar{b}X)=284\pm 20\pm 49\mu b$ is obtained [14], extrapolating to the full phase space. All species of beauty hadrons can be produced in pp collisions, including $b$ baryons [15, 16] and $B_{c}^{+}$ [17]. The knowledge of the relative fractions of the various $b$ hadron species is of crucial importance for all measurements of branching rations, most prominently for $B_{s}\to\mu\mu$. LHCb have measured the $B_{s}$ to $B_{d}$ production ratio using semileptonic $B$ meson decays [18] and $SU(3)$ partner decays $B_{d}\to DK$ and $B_{s}\to D_{s}\pi$ [19]. Both measurements are consistent and get an average of $f_{s}/f_{d}=0.267{\>}^{+{\>}0.021}_{-{\>}0.020}$ [20]. Figure 2: Invariant mass relative to threshold of $B^{0}\pi$ system ($m_{B^{0}\pi}-m_{B}-m_{\pi}$, top) and normalised fit residuals (bottom) [21]. LHCb also studies orbitally excited $B$ mesons, notably observing for the first time states decaying to $B^{0}\pi^{+}$ (Fig 2) [21]. ## IV Flavour Physics The LHCb experiment has been optimised for flavour physics at the LHC and therefore all the results presented in this Section have been obtained by this experiment, with the notable exception of the $B_{s}\to\mu\mu$ result from CMS. ### IV.1 Charm Mixing The most interesting topic of charm physics is the characterisation of neutral $D$ meson mixing and the hunt for $CP$ violation in $D$ meson decays. As for any other neutral long-lived meson (e.g. $K^{0}$, $B^{0}$, $B_{s}^{0}$), the neutral $D$ system can be described in terms of two flavour eigenstates $D^{0}$, $\overline{D}^{0}$ or two mass eigenstates: $D_{1,2}=p\left\|D^{0}\right>\pm q\left\|\overline{D}^{0}\right>$ of masses $m_{1,2}$ and decay widths $\Gamma_{1,2}$. This allows to define the quantities $x=\frac{m_{2}-m_{1}}{2\Gamma}\qquad\text{and}\qquad y=\frac{\Gamma_{2}-\Gamma_{1}}{2\Gamma}.$ The HFAG averages for these quantities [22] differ from the no mixing hypothesis $x=0,y=0$ by $10.2\sigma$ but no single measurement excludes this hypothesis at $5\sigma$. Using two-body $D$ decays selected in $26\>\rm pb^{-1}$ of 2010 data the LHCb experiment measures a linear combination of these quantities as [23] $\displaystyle y_{CP}$ $\displaystyle=$ $\displaystyle\frac{\hat{\Gamma}(D^{0}\to K^{-}K^{+})}{\hat{\Gamma}(D^{0}\to K^{-}\pi^{+})}-1$ $\displaystyle=$ $\displaystyle y\cos\phi-x\sin\phi\left(\frac{A_{m}}{2}+A_{\text{prod}}\right)$ $\displaystyle=$ $\displaystyle\left(-0.55\pm 0.63\pm 0.41\right)\%,$ where $1+A_{m}=\|q/p\|$ and the production asymmetry $A_{\text{prod}}$ is measured to be very small. In the limit of vanishing $CP$ violation $y_{CP}=y$. Using the same data sample, LHCb also measure the lifetime difference [24] $\displaystyle A_{\Gamma}$ $\displaystyle=$ $\displaystyle\frac{\tau(\overline{D}^{0}\to K^{+}{}K^{-})-\tau(D^{0}\to K^{+}{}K^{-})}{\tau(\overline{D}^{0}\to K^{+}{}K^{-})+\tau(D^{0}\to K^{+}{}K^{-})}$ $\displaystyle=$ $\displaystyle\left(-0.59\pm 0.59\pm 0.21\right).$ Figure 3: $D^{0}$ impact parameter (left) and proper time [24] The key to the lifetime measurement is a good separation of prompt and secondary charm (from $b$ decays), illustrated in Fig 3. ### IV.2 $CP$ Violation in Charm $CP$ violation is expected to be vanishingly small in the charm sector in the Standard Model. A non-zero $CP$ asymmetry above a few per-mille in $D^{0}\to h^{+}h^{-}$ ($h=\pi,K$) decays would be strong sign of new physics. Experimentally the flavour of the $D$ meson is tagged using the decay $D^{\ast}\to D^{0}\pi^{+}$. The raw $CP$ asymmetry of tagged $D^{0}\to f$ and $\overline{D}^{0}\to f$ can be factorised as $A_{\text{RAW}}(f)=A_{CP}(f)+A_{\text{D}}(f)+A_{\text{D}}(\pi_{s})+A_{\text{P}}(D^{\ast})$ where $A_{D}$ are detector asymmetries related to the final state $f$ and the bachelor pion $\pi_{s}$ and $A_{\text{P}}(D^{\ast})$ is the production asymmetry in $pp$ collisions. The detection asymmetry for $f$ vanishes when one uses decays to $CP$ eigenstates, e.g. $\pi^{+}\pi^{-}$ or $K^{+}K^{-}$. All other asymmetries can be cancelled at first order by measuring the difference of the two $CP$ asymmetries in these two channels. While writing these proceedings the following interesting result has become available. Using $580\>\rm pb^{-1}$ of 2011 data the LHCb collaboration gets very clean samples of $1.44\cdot 10^{6}$ tagged $D\to K^{+}K^{-}$ and $0.38\cdot 10^{6}$ $D\to\pi^{+}\pi^{-}$ (Fig 4) [25]. Figure 4: Fits to the mass difference spectrum in tagged $D\to K^{+}K^{-}$ (left) and $D\to\pi^{+}\pi^{-}$ (right) decays [25] Due to the different lifetime acceptance of the two channels a small contribution from mixing induced $CP$ violation does not cancel out in the measurement but its magnitude can be extracted from data: $\displaystyle\Delta A_{CP}$ $\displaystyle\equiv$ $\displaystyle A_{CP}^{\rm raw}(K^{+}K^{-})-A_{CP}^{\rm raw}(\pi^{+}\pi^{-})$ $\displaystyle=$ $\displaystyle A_{CP}^{\rm dir}(K^{+}K^{-})-A_{CP}^{\rm dir}(\pi^{+}\pi^{-})+0.098\>A_{CP}^{\rm ind}$ $\displaystyle=$ $\displaystyle\left(\\-0.82\pm 0.21\pm 0.11\right)\%.$ The measured difference of $CP$ asymmetries the first ($3.5\sigma$) evidence of $CP$ violation in the charm sector. ### IV.3 Rare $b$ Decays The SM prediction for the Branching Ratios (BR) of the decays $B_{q}\to\mu^{+}\mu^{-}$ have been computed to be ${\rm BR}(B_{s}\to\mu^{+}\mu^{-})=(3.2\pm 0.2)\cdot 10^{-9}$ and ${\rm BR}(B_{d}\to\mu^{+}\mu^{-})=(0.10\pm 0.01)\cdot 10^{-9}$ [26]. However, many extensions of the SM predict large enhancements to these BR. The first search for this at the LHC decay was reported by the LHCb collaboration with 2010 data [27]. Recently LHCb and CMS collaborations have presented new searches based on $0.3$ and $1.1\>\rm fb^{-1}$ samples collected in 2011, respectively [28, 29]. Figure 5: $B_{s}\to\mu\mu$ search windows at LHCb [28] (top) and CMS [29] (bottom). While CMS uses a cut-based approach, LHCb uses a boosted decision tree calibrated on $B\to hh$ ($h=\pi,K$) decays which have the same topology as the signal. The estimated yield is then normalised using $B_{d}\to K\pi$ (LHCb only), $B^{+}\to J/\psi K$ and $B_{s}\to J/\psi\phi$. No excess of signal is observed at neither of the two experiments (Fig. 5) and upper limits are set by LHCb as ${\rm BR}(B_{s}\to\mu^{+}\mu^{-})<1.6\cdot 10^{-8}$ (95% C.L.) and ${\rm BR}(B_{d}\to\mu^{+}\mu^{-})<5.1\cdot 10^{-9}$, and CMS as ${\rm BR}(B_{s}\to\mu^{+}\mu^{-})<1.8\cdot 10^{-8}$. The combined LHC result is ${\rm BR}(B_{s}\to\mu^{+}\mu^{-})<1.1\cdot 10^{-8}$ [30] thus not confirming the excess reported by CDF [31]. The limits set by the LHC strongly constrain the allowed SUSY parameter space, especially at large $\tan\beta$ [32]. The rare decay $B_{d}\to\mu\mu K^{\ast}$ is a $b\to s$ flavour changing neutral current decay which is in the SM mediated by electroweak box and penguin diagrams. It can be a highly sensitive probe for new right handed currents and new scalar and pseudoscalar couplings. These New Physics contributions can be probed by its contribution to the angular distributions of the $B^{0}$ daughter particles. The most prominent observable is the forward-backward asymmetry of the muon system ($A_{\rm FB}$). $A_{\rm FB}$ varies with the invariant mass-squared of the dimuon pair ($q^{2}$) and in the SM changes sign at a well defined point, where the leading hadronic uncertainties cancel. In many NP models the shape of $A_{\rm FB}$ as a function of $q^{2}$ can be dramatically altered. Figure 6: $B_{d}\to\mu\mu K^{\ast}$ forward-backward asymmetry $A_{\rm FB}$ (top) and $K^{\ast}$ polarisation fraction $F_{L}$ in bins of dimuon mass $q^{2}$ [33]. The latest LHCb analysis [33] uses $309\rm\>pb^{-1}$ of data collected during 2011 to measure $A_{\rm FB}$, the fraction of longitudinal polarisation of the $K^{\ast}$, $F_{L}$, and the differential branching fraction, $dB/dq^{2}$, as a function of the dimuon invariant mass squared, $q^{2}$. There is good agreement between recent Standard Model predictions and the LHCb measurement of $A_{\rm FB}$, $F_{L}$ and $dB=dq^{2}$ in the six $q^{2}$ bins (Fig. 6). In a $1<q^{2}<6\rm\>GeV^{2}$ bin, LHCb measures $A_{\rm FB}=0.10\pm 0.14\pm 0.05$, to be compared with theoretical predictions of $A_{\rm FB}=0.04\pm 0.03$. The experimental uncertainties are presently statistically dominated, and will improve with a larger data set. Such a data set would also enable LHCb to explore a wide range of new observables. Using a very similar selection, LHCb also searched for Majorana Neutrinos [34] giving raise to $B^{+}\to K^{-}\mu^{+}\mu^{+}$ and $B^{+}\to\pi^{-}\mu^{+}\mu^{+}$ decays. No excess was found and 95% C.L. limits have been set at $5.4\cdot 10^{-8}$ and $5.8\cdot 10^{-8}$, respectively. Figure 7: $B\to K^{\ast}\gamma$ and $B_{s}\to\phi\gamma$ mass peaks [35]. Other $b\to s$ transitions of interest are radiative decays $B\to K^{\ast}\gamma$ and $B_{s}\to\phi\gamma$. LHCb reports the first measurement of the ratio of branching fractions of these two decays as $1.52\pm 0.15\pm 0.10\pm 0.12$ where the last error comes from ($f_{d}/f_{s}$) [35]. The mass resolution (Fig 7) is dominated by the photon energy resolution. LHCb already has the largest sample of $B_{s}\to\phi\gamma$, which will become to measure or constrain non-standard right-handed currents. ### IV.4 CP violation in $B$ decays Decays of neutral $B$ mesons provide a unique laboratory to study CP-violation originating from a non-trivial complex phase in the CKM matrix. The relative phase between the direct decay amplitude and the amplitude of decay via mixing gives rise to time-dependent CP-violation, a difference in the proper decay time distribution of $B$-meson and anti-$B$-meson decays. The decay $B_{s}\to J/\psi\phi$ is considered the golden modes for measuring this type of CP- violation, In the Standard Model the CP-violating phase in this decay is predicted to be $\phi_{s}\simeq-2\beta_{s}$ where $\beta_{s}=\arg(-V_{ts}V_{tb}^{\ast}/V_{cs}V_{cb}^{\ast})$. The indirect determination via global fits to experimental data gives $2\beta_{s}=(0.0363{\>}^{+{\>}0.0016}_{-{\>}0.0015})\>\rm rad$. New Physics contributions could significantly alter this phase. The channel $B_{s}\to J/\psi f_{0}(980)$ is also sensitive to the same phase. It has been first observed by the LHCb collaboration [36] using 2010 data and quickly confirmed by Belle [37] and CDF [38]. LHCb report measurements of the phase $\phi_{s}$ for each of these channels using $338\>\rm fb^{-1}$, and also performing a simultaneous fit to both channels. In both cases, flavour-tagged and untagged events are used, and the tagging efficiency is calibrated to control channels. The trigger and selection bias, in particular with respect to lifetime, is also extracted from the data itself. Due to the vector nature of the $\phi$ meson, the $B_{s}\to J/\psi\phi$ needs an angular analysis to disentangle the CP-even and CP-odd final states (Figs. 8 and 9). This is not necessary in the $f_{0}(980)$ case. Figure 8: Definitions of the decay angles in $B_{s}\to J/\psi\phi$ decay. Figure 9: Fits projections to proper time (top left), and the three angles in the $B_{s}\to J/\psi\phi$ decay [39]. The results are [39, 40, 41] $\displaystyle\phi_{s}^{J/\psi f_{0}}$ $\displaystyle=$ $\displaystyle-0.44\pm 0.44\pm 0.02~{}\mathrm{rad}$ $\displaystyle\phi_{s}^{J/\psi\phi}$ $\displaystyle=$ $\displaystyle+0.13\pm 0.18\pm 0.07~{}\mathrm{rad}$ $\displaystyle\phi_{s}^{\rm Comb}$ $\displaystyle=$ $\displaystyle+0.03\pm 0.16\pm 0.07~{}\mathrm{rad}$ which are consistent with the SM prediction. A sign ambiguity remains under the sign reversal of ($\phi_{s}$) and ($\Delta\Gamma_{s}$). The allowed regions are shown in Fig 10. Figure 10: Constraints of LHCb’s measurement on the $\phi_{s}$–$\Delta\Gamma_{s}$ plane from $B_{s}\to J/\psi\phi$ [39]. With increasing precision on $CP$ violating phases in $b\to c\bar{c}s$ transitions, assumed to be dominated by tree-level topologies, it will become crucial in the future to understand contributions from penguin topologies [42, 43]. These can be studied using Cabibbo-suppressed decays that are related by $U$-spin symmetry. One example is the $B_{s}\to J/\psi K_{S}^{0}$ decay, which is the partner of the golden mode $B_{d}\to J/\psi K_{S}^{0}$. CDF [44] and LHCb [45] have recently reported on the branching ratio of this channel and more precision will become available when more data is collected. ## V Conclusions With its large $b$ and $c$ cross-sections, the LHC is the new flavour factory. Many flavour physics results, mostly from LHCb, are becoming available yielding an unexpected and interesting pattern of measurements. The long awaited $B_{s}\to\mu\mu$ decay has not yet been observed, thus excluding large regions of the SUSY parameter space. Similarly the $CP$ violating phase in $B_{s}$ decays is compatible with the SM expectation, as well as the angular distributions in $B\to K^{\ast}\mu\mu$. Yet all these measurements are still affected by statistical errors much larger than the theoretical errors, which leaves a lot of room for observations of new physics. The biggest surprise comes from the measurement of $\Delta A_{CP}$ which exhibits a $3.5\sigma$ evidence for new physics. This will all have to be followed up very closely with increasing statistics. As high-precision beauty and charm physics is sensitive to energy scales much beyond the LHC centre-of- mass energy it is likely that flavour physics is paving the way for direct observations of new particles by the general purpose detectors at the LHC. ## References * [1] LHCb Collaboration. The LHCb Detector at the LHC. JINST, 3:S08005, 2008. * [2] Eric van Herwijnen. The LHCb trigger. PoS, ICHEP2010:027, 2010. * [3] ATLAS Collaboration. The ATLAS Experiment at the CERN Large Hadron Collider. JINST, 3:S08003, 2008. * [4] CMS Collaboration. The CMS experiment at the CERN LHC. JINST, 3:S08004, 2008. * [5] ALICE Collaboration. The ALICE experiment at the CERN LHC. JINST, 3:S08002, 2008. * [6] CMS Collaboration. Prompt and non-prompt $J/\psi$ production in $pp$ collisions at $\sqrt{s}=7$ TeV. Eur. Phys. J., C71:1575, 2011. * [7] ATLAS Collaboration. Measurement of the differential cross-sections of inclusive, prompt and non-prompt $J/\psi$ production in proton-proton collisions at $\sqrt{s}=7$ TeV. Nucl. Phys., B850:387–444, 2011. * [8] LHCb Collaboration. Measurement of $J/\psi$ production in $pp$ collisions at $\sqrt{s}=7$ TeV. Eur. Phys. J., C71:1645, 2011. * [9] ALICE Collaboration. Rapidity and transverse momentum dependence of inclusive $J/\psi$ production in $pp$ collisions at $\sqrt{s}=7$ TeV. Phys. Lett., B704:442–455, 2011. * [10] LHCb Collaboration. Observation of $J/\psi$ pair production in $pp$ collisions at $\sqrt{s}=7$ TeV. 2011\. * [11] LHCb Collaboration. Prompt charm production in $pp$ collisions at $\sqrt{s}=7$ TeV. Dec 2010. * [12] ATLAS Collaboration. Measurement of $D^{()}$ meson production cross sections in pp collisions at $\sqrt{s}=7$ TeV with the ATLAS detector. Technical Report ATLAS-CONF-2011-017, Geneva, Mar 2011. [13] LHCb Collaboration. First observation of $B_{s}->D_{s2}^{+}X\mu\nu$ decays. Phys. Lett., B698:14–20, 2011. [14] LHCb Collaboration. Measurement of $\sigma(pp\to b\bar{b}X)$ at $\sqrt{s}=7$ TeV in the forward region. Phys.Lett., B694:209–216, 2010. * [15] LHCb Collaboration. $b$-hadron lifetime measurements with exclusive $b\to J/\psi X$ decays reconstructed in the 2010 data. Mar 2011. LHCb-CONF-2011-001. * [16] LHCb Collaboration. Studies of beauty baryons decaying to $D^{0}p\pi^{-}$ and $D^{0}pK^{-}$. Jul 2011. LHCb-CONF-2011-036. * [17] LHCb Collaboration. Measurement of the $B_{c}^{+}$ to $B^{+}$ production cross-section ratios at $\sqrt{s}$ = 7 TeV in LHCb. Apr 2011. LHCb-CONF-2011-017. * [18] LHCb Collaboration. Measurement of $b$-hadron production fractions in 7 TeV centre-of-mass energy $pp$ collisions. Jun 2011. LHCb-CONF-2011-028. * [19] LHCb Collaboration. Determination of $f_{s}/f_{d}$ for 7 TeV $pp$ collisions and a measurement of the branching fraction of the decay $B_{d}\to D^{-}K^{+}$. 2011\. * [20] LHCb Collaboration. Average $f_{s}/f_{d}$ b-hadron production fraction for 7 TeV $pp$ collisions. 2011\. * [21] LHCb Collaboration. Observations of Orbitally Excited $B_{(s)}^{*}$ Mesons. Oct 2011. LHCb-CONF-2011-053. [22] Heavy Flavor Averaging Group. Averages of $b$-hadron, $c$-hadron, and $\tau$-lepton Properties. 2010\. * [23] LHCb Collaboration. Measurement of the Charm Mixing Parameter $y_{CP}$ in Two-Body Charm Decays. Aug 2011. LHCb-CONF-2011-054. * [24] LHCb Collaboration. Measurement of the $CP$ Violation Parameter $\cal{A}_{\Gamma}$ in Two-Body Charm Decays. Jul 2011. LHCb-CONF-2011-046. * [25] LHCb Collaboration. A search for time-integrated $CP$ violation in $D^{0}\to h^{+}h^{-}$ decays. Nov 2011. LHCb-CONF-2011-061. * [26] Andrzej J. Buras. Minimal flavour violation and beyond: Towards a flavour code for short distance dynamics. Acta Phys. Polon., B41:2487–2561, 2010. * [27] LHCb Collaboration. Search for the rare decays $B^{0}_{(s)}\to\mu^{+}\mu^{-}$. Phys.Lett., B699:330–340, 2011. * [28] LHCb Collaboration. Search for the rare decays $B^{0}_{(s)}\to\mu^{+}\mu^{-}$ with 300 pb-1 at LHCb. Jul 2011. LHCb-CONF-2011-037. * [29] CMS collaboration. Search for $B^{0}_{s}\to\mu^{+}\mu^{-}$ and $B^{0}\to\mu^{+}\mu^{-}$ decays in $pp$ collisions at $\sqrt{s}=7$ TeV. Aug 2011. * [30] LHCb and CMS Collaborations. Search for the rare decay $B^{0}_{s}\to\mu^{+}\mu^{-}$ at the LHC with the CMS and LHCb experiments. Aug 2011. * [31] CDF Collaboration. Search for $B_{s}\to\mu^{+}\mu^{-}$ and $B_{d}\to\mu^{+}\mu-$ Decays with CDF II. 2011\. * [32] A. G. Akeroyd, F. Mahmoudi, and D. Martinez Santos. The decay $B_{s}\to\mu^{+}\mu^{-}$: updated SUSY constraints and prospects. 2011\. * [33] LHCb Collaboration. Angular analysis of $B^{0}\rightarrow K^{0}\mu^{+}\mu^{-}$. Aug 2011. [34] LHCb Collaboration. Search for the lepton number violating decays $B^{+}\rightarrow\pi^{-}\mu^{+}\mu^{+}$ and $B^{+}\rightarrow K^{-}\mu^{+}\mu^{+}$. 2011\. * [35] LHCb Collaboration. Measurement of the ratio of branching fractions $\mathcal{B}(B_{d}\to K^{0}\gamma)/\mathcal{B}(B_{s}\to\phi\gamma)$ with the LHCb experiment at $\sqrt{s}=7$ TeV. Aug 2011. LHCb-CONF-2011-055. [36] LHCb Collaboration. First observation of $B_{s}\to J/\psi f_{0}(980)$ decays. Phys.Lett., B698:115–122, 2011. * [37] Belle Collaboration. Observation of $B_{s}^{0}\to J/\psi f_{0}(980)$ and Evidence for $B_{s}^{0}\to J/\psi f_{0}(1370)$. Phys. Rev. Lett., 106:121802, 2011. * [38] CDF Collaboration. Measurement of branching ratio and $B_{s}^{0}$ lifetime in the decay $B_{s}^{0}\to J/\psi f^{0}(980)$ at CDF. 2011\. * [39] LHCb Collaboration. Tagged time-dependent angular analysis of $B_{s}\to J/\psi\phi$ decays with 337 pb-1 at LHCb. Aug 2011. LHCb-CONF-2011-049. * [40] LHCb Collaboration. Measurement of $\phi_{s}$ in $B_{s}\to J/\psi f_{0}(980)$. Sep 2011. LHCb-CONF-2011-051. * [41] LHCb Collaboration. Combination of $\phi_{s}$ measurements from $B_{s}^{0}\to J/\psi\phi$ and $B_{s}^{0}\to J/\psi f_{0}(980)$. Aug 2011. LHCb-CONF-2011-056. * [42] Kristof De Bruyn, Robert Fleischer, and Patrick Koppenburg. Extracting $\gamma$ and Penguin Topologies through CP Violation in $B_{s}^{0}\rightarrow J/\psi K_{\mathrm{S}}$. Eur. Phys. J. C, 70(4):1025–1035, December 2010. arXiv:1010.0089 [hep-ph]. * [43] Kristof De Bruyn, Robert Fleischer, and Patrick Koppenburg. Extracting $\gamma$ and Penguin Parameters from $B_{s}^{0}\rightarrow J/\psi K_{\mathrm{S}}$. Talk at CKM2010, Warwick, UK, September 6–10, 2010. arXiv:1012.0840 [hep-ph]. * [44] CDF Collaboration. Observation of $B_{s}\to J/\psi K^{\ast}$ and $B_{s}\to J/\psi K_{S}$ Decays. Phys. Rev., D83:052012, 2011. * [45] LHCb Collaboration. Measurement of the $B^{0}_{s}\to J/\psi K^{0}_{s}$ branching fraction. Sep 2011. LHCb-CONF-2011-048.	arxiv-papers	2011-11-29T14:50:12	2024-09-04T02:49:24.767495	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Patrick Koppenburg", "submitter": "Patrick Koppenburg", "url": "https://arxiv.org/abs/1111.6829" }
1111.7004	# Arcsecond resolution mapping of Sulfur Dioxide emission in the circumstellar envelope of VY Canis Majoris Roger Fu11affiliation: Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 22affiliation: Department of Earth, Atmospheric, and Planetary Sciences, Massachusetts Institute of Technology, Cambridge, MA , Arielle Moullet11affiliation: Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA , Nimesh A. Patel11affiliation: Harvard- Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 33affiliation: Author for correspondence: npatel@cfa.harvard.edu , John Biersteker11affiliation: Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA , Kimberly L. Derose11affiliation: Harvard- Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA , Kenneth H. Young11affiliation: Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA ###### Abstract We report Submillimeter Array observations of SO2 emission in the circumstellar envelope of the red supergiant VY CMa, with an angular resolution of $\approx 1"$. SO2 emission appears in three distinct outflow regions surrounding the central continuum peak emission that is spatially unresolved. No bipolar structure is noted in the sources. A fourth source of SO2 is identified as a spherical wind centered at the systemic velocity. We estimate the SO2 column density and rotational temperature assuming local thermal equilibrium (LTE) as well as perform non-LTE radiative transfer analysis using RADEX. Column densities of SO2 are found to be $\sim 10^{16}$ cm-2 in the outflows and in the spherical wind. Comparison with existing maps of the two parent species OH and SO shows the SO2 distribution to be consistent with that of OH. The abundance ratio $f_{SO_{2}}/f_{SO}$ is greater than unity for all radii greater than at least $3\times 10^{16}$ cm. SO2 is distributed in fragmented clumps compared to SO, PN, and SiS molecules. These observations lend support to specific models of circumstellar chemistry that predict $f_{SO_{2}}/f_{SO}>1$ and may suggest the role of localized effects such as shocks in the production of SO2 in the circumstellar envelope. stars: individual (VY CMa (catalog )) — stars: late-type — circumstellar matter — submillimeter — radio lines: stars ††slugcomment: Accepted on 22 November 2011 for publication in ApJ. ## 1 Introduction Massive stars ($M_{}\gtrsim 8\,M_{\odot}$), enter a red supergiant (RSG) phase during which the star experiences mass-loss at rates of $\dot{M}\sim\,10^{-5}-10^{-3}M_{\odot}yr^{-1}$ (van Loon et al., 2005). The time variation of this mass-loss rate is not well-constrained by theoretical studies (Yoon and Cantiello, 2010). As a result, the total amount of mass lost over the course of the RSG phase remains uncertain for a given initial mass (Smith et al., 2009). Observations of mass-loss events have shown them to be sporadic and spatially anisotropic (de Wit et al., 2008). VY Canis Majoris (VY CMa) is an oxygen-rich red supergiant with an estimated mass of $M_{}\approx 25\,M_{\odot}$ and a mass-loss rate estimated to be $\dot{M}\sim\,2-4\times 10^{-4}M_{\odot}yr^{-1}$ (Danchi et al., 1994; Smith et al., 2009). Optical images of this source show multiple discrete and asymmetric mass-loss events, ranging in age from 1700 to 157 years ago, that are distinct from the general flow of diffuse material (Humphreys et al., 2007). Detailed studies of mm/sub-mm molecular spectra of VY CMa have been carried out, revealing the chemical complexity in the envelope (Ziurys et al., 2007; Royer et al., 2010; Tenenbaum et al., 2010). Spatial structures in the visible and IR bands have also been obtained (Smith et al., 2001). High angular resolution Submillimeter Array (SMA)111The Submillimeter Array is a joint project between the Smithsonian Astrophysical Observatory and the Academia Sinica Institute of Astronomy and Astrophysics, and is funded by the Smithsonian Institution and the Academia Sinica. observations of VY CMa produced maps of the spatial distribution of CO and SO (Muller et al., 2007). Millimeter wavelength observations have long shown evidence of SO2 in the circumstellar envelopes (CSE) of oxygen-rich red supergiants (e.g., Guilloteau et al., 1986) and SO2 emission in VY CMa was first detected by Sahai & Wannier (1992). Later infrared observations have also shown the production of SO2 within several radii of O-rich AGB stars (in the ”inner wind”) Yamamura et al. (1999). From a theoretical perspective, the presence of SO2 in circumstellar envelopes has been predicted by isotropic, non-equilibrium models of stellar chemistry (Scalo & Slavsky, 1980; Nejad & Millar, 1988; Willacy & Millar, 1997). This class of models assume an isotropic geometry of the CSE with specified values for the mass loss rate and expansion velocity. The resulting model CSEs consist of an inner region where assumed parent species in the outflow are broken down, a intermediate region where chemical reactions lead to high abundances of daughter species, and an outer region where the daughter species are destroyed by photodissociation. SO2 in these models has been assigned as a daughter species, although it has since been shown to exist as a possible parent species created in the inner wind (Decin et al., 2010). Cherchneff (2006) studied the role of shocks in SO2 formation in the inner wind within a few stellar radii of the photosphere. Observational works (e.g., Jackson & Nguyen, 1988; Yamamura et al., 1999; Decin et al., 2010) have repeatedly shown SO2 abundance to be higher both in the extended CSE and the inner wind than expected from CSE and inner wind chemistry models (Willacy & Millar, 1997; Cherchneff, 2006). Local abundance of SO2 may be enhanced in the ISM by the passage of shocks (Hartquist et al., 1980), and shock chemistry has been proposed to explain the high observed SO2 relative abundances in CSEs (Jackson & Nguyen, 1988). Alternatively, low model abundances of SO2 presented in Nejad & Millar (1988) compared to that of Scalo & Slavsky (1980) may be due to the latter’s lower assumed value for the photodissociation of SO2, which is the main mode of destruction of this molecule at large radii (Sahai & Wannier, 1992). Interferometric mapping of SO2 distribution in the CSE of an evolved star may contribute to the improved understanding of sulfur chemistry by directly constraining the relative abundances of sulfur-bearing species as a function of radius. Furthermore, associations of SO2 enhancement with discrete outflow features may favor the idea of localized production and provide evidence for non-isotropic processes in CSEs (Jackson & Nguyen, 1988). In this paper we present results of high spatial resolution maps obtained from SMA observations of SO2 around VY CMa. We describe four spatially discrete sources of SO2 emission and use multiple transitions of SO2 and derive rotational temperatures and column densities (assuming local thermodynamical equilibrium). Furthermore, we perform non-LTE radiative transfer analysis using the RADEX package (van der Tak et al., 2007) to derive local kinetic temperatures, H2 densities, and SO2 abundances of the SO2 emitting regions. We then compare our maps to published visible, IR, and submillimeter observations to evaluate their consistency with the above cited models of circumstellar chemistry. ## 2 Observations VY CMa was observed with the SMA on 2009 February 18, with the array in extended configuration, offering baselines from 44.2 to 225.9 m. The projected baseline lengths were from 14 to 188 m. The frequency coverage was 234.36 to 236.34 GHz in the lower sideband and 244.36 to 246.34 GHz in the upper sideband. The phase center was at $\alpha(2000)=07^{h}22^{m}58^{s}.27,\delta(2000)=-25^{\circ}46^{\prime}03.4^{\prime\prime}$. The quasars 0730-116 and 0538-440 were observed every 20 minutes for gain calibration, and the spectral band-pass was calibrated using quasar 3c273. Flux calibration was done using recent SMA measurements of 0730-116 (2.75 Jy at 1 mm) and 0538-440 (3.96 Jy at 1 mm). Nominal flux calibration accuracy is 15 to 20%, depending on the phase stability. In our observations, the uncertainty appears to be better than 10%, based on the good agreement with the ARO spectra (see section 3.2 and Figure 3). The on-source integration time on VY CMa was 5.66 hours. Tsys (SSB) varied approximately from 200 to 400 K during the track with an atmospheric zenith optical depth of $\sim$0.1 at the standard reporting frequency of 225 GHz, measured at the nearby Caltech Submillimeter Observatory. The conversion factor between Kelvin and Jansky for our observations is $\sim 14$ K/Jy. The visibility data were calibrated using the MIR package in IDL and imaged with the MIRIAD software222http://www.cfa.harvard.edu/$\sim$cqi/mircook.html (Sault, Teuben & Wright, 1995). The field of view (FWHM primary beam) varies from 53′′ to 56′′ whereas the largest angular extent of the source is expected to be about 5′′. The synthesized beam size, representing the obtained spatial resolution, is $1.^{\prime\prime}48\times 1.03^{\prime\prime}$. The adopted weighting mode for imaging is “natural” weighting (specified in the Miriad task INVERT). The final rms noise level is $\sim$60 mJy/beam per channel in the spectral line images and 3.9 mJy/beam in the continuum map. Line emission was subtracted b yvisually examining the spectra in visibility amplitudes, and using the line-free channels specification to the Miriad task UVLIN. ## 3 Results ### 3.1 Continuum emission We detected continuum emission in both the upper and lower sidebands, which is well described by a point source centered at RA=$07^{h}22^{m}58.336^{s}$, Dec=$-25^{\circ}46^{\prime}03.063^{\prime\prime}$. A 2D Gaussian fit in the image plane yields an integrated flux of $335.0\pm 66$ mJy at $235.4$ GHz and $359.3\pm 72$ mJy at $245.4$ GHz (maximum errors are of $20\%$, mainly resulting from the uncertainty in flux calibration). Previous SMA observations of VY CMa at $215$ and $225$ GHz have yielded continuum fluxes of $270\pm 40$ mJy and $288\pm 25$ mJy respectively (Shinnaga et al., 2004; Muller et al., 2007). These four values are consistent under the Rayleigh-Jeans approximation with black-body radiation with a brightness temperature higher than 12 K. ### 3.2 Line emission In the two sidebands of $\sim$4 GHz bandwidth, we detected a total of 14 lines, which show emissions from CS, H2O, PN, SiS and SO2. In Figure 1 we present the spectra and maps of the SiS J=13-12 and the PN J=5-4 lines. These lines are representative of a relatively compact emission that is centered on the peak of the continuum emission, hence they allow us to estimate the systemic velocity of the star. The SiS spectrum shows the distinctive triple peak morphology as described by Ziurys et al. (2007); Tenenbaum et al. (2010) that is interpreted as the signature of a slowly expanding shell and a pair of faster outflows nearly collimated with the line of sight. We use the fitted velocity of the central peak of these lines to find the systemic velocity of the star $v_{LSR}=19.5$ km s-1. This value is consistent with other results from mm and sub-mm observations (Ziurys et al., 2007; Muller et al., 2007) and is adopted as the star rest-frame velocity in the following sections. For the remainder of this work we focus only on the SO2 lines, which are summarized in Table 1. The excitation energies of the four identified SO2 lines vary from 19 to 130 K. All four of these lines were detected in the spectral line survey of Tenenbaum et al. (2010). Our interferometric observations reveal the spatial distribution of the SO2 emission for the first time. The spectra and integrated maps of all four lines are shown in Figure 2. Figure 3 compares the spectra of two SO2 lines with single dish data from the Submillimeter Telescope of the Arizona Radio Observatory, while channel maps of all four lines are presented in Figures 4, 5, 6 and 7. Maps of SO2 lines maps show well resolved spatially extended emission over an area of $3^{\prime\prime}\times 5^{\prime\prime}$, in which we distinguish four distinct components, hereafter designated as sources A, B, C, and D. Figure 8 shows all four sources in a position-velocity space diagram of the 245563.4 MHz line. Source positions, orientations, and intensities were fitted to 2D Gaussians using the MIRIAD routine IMFIT. Source A is a very strong blue-shifted source with $v_{lsr}$ between $-18$ and $+10$ km s-1. It is spatially offset to the East of the stellar position by $\sim 0.7"$. Of similar intensity to source A is source B, an elongated source offset by $\sim 0.8"$ to the West. Source B has a broader spectral profile than source A and is heavily red-shifted with $v_{lsr}$ between $+26$ and $+66$ km s-1. Source C is a strong source with a similar radial velocity as source B. In contrast to sources A and B, however, it is heavily offset to the West by $\sim 2.5"$. Source D is significantly weaker than sources A, B, and C. In velocity space, it is centered on the star’s frame of motion ($12$ km s-1 $<v_{LSR}$$<26$ km s-1) with a distinctly narrower spectral profile and similar spatial dimensions as the other sources. It is also spatially centered at the star’s location. Assuming local thermal equilibrium, we can use the integrated intensities measured in the detected SO2 lines in each of the four identified SO2 sources to constrain their respective column density $N$ and rotational temperature $T_{rot}$, using the expression from Friedel et al. (2005): $\ln[\frac{3c^{2}I_{beam}}{16\pi^{3}B\theta_{a}\theta_{b}\nu^{3}S\mu^{2}}]=\ln[\frac{N}{Z}]-\frac{E_{u}}{T_{rot}}$ (1) where $I_{beam}$ is the integrated flux in Jy km s-1 beam-1, $\theta_{a}\theta_{b}$ is the beam size, and $B$ the beam filling factor varying from $0.8$ to $0.95$ for the four distinct sources, as derived from Snyder et al. (2005): $\Theta_{source}/(\Theta_{beam}+\Theta_{source})$. $Z$ is the rotational partition function (Müller et al., 2005), $S$ is the line strength, $\mu$ the dipole moment (1.63 Debye for SO2), $E_{u}$ the line’s upper rotational state energy in K taken from the JPL spectroscopic database (Pickett, 1991), and $\Theta$ the solid angle. By performing linear fits of the rotational temperature diagram shown in Figure 9, we find that the three faster outflows (sources A, B, and C) show variations in rotational temperature. Sources A, B, and C have rotational temperature of $61_{-17}^{+39}$ K (one $\sigma$ error), $110_{-29}^{+63}$ K, and $69_{-17}^{+34}$ K, respectively. The column densities (N) of all three sources are similar (A: $9.2_{-4.4}^{+7.8}\times 10^{15}$ cm-2; B: $1.9_{-0.6}^{+0.8}\times 10^{16}$ cm-2; C: $1.1_{-0.4}^{+0.7}\times 10^{16}$ cm-2). The rotational temperature of the close-in source D is much higher at $240_{-52}^{+91}$ K, while its column density is likely lower than that of the others at $7.6\pm 1.0\times 10^{15}$ cm-2. The derived parameters for source D are uncertain since a significant portion of flux in its velocity range was missed by our interferometric observations due to the lack of short spacing data (our shortest projected baseline 14 m). Comparison with single dish data from the Arizona Radio Observatory shows that features as large as 4” are fully recovered (Figure 3). Features much larger than 4” may be subject to an underestimation of its flux, although 4” is an highly conservative estimate. To address this deficiency, we repeated the rotational temperature diagram analysis for source D using a single dish observation made by the Arizona Radio Observatory 10 meter Submillimeter Telescope (Tenenbaum et al., 2010). Assuming a source size of 4” by 4” we derive a lower rotational temperature of $97_{-31}^{+84}$ and a column density of $1.9^{+2.9}_{-1.8}\times 10^{15}$ cm-2. Because this column density represents an average over our assumed source size, it cannot be directly compared to our derived column densities, which represent the values at the center of each source. This averaged column density also represents an upper limit due to our assumption of the smallest possible angular size. The validity of the LTE approximation can be assessed by checking the linearity of the data points in the rotational temperature diagram. Deviation from a linear trend suggests non-LTE conditions in the source or line misidentification. In the case of source D, deviation from linearity is very small compared to the error. We are therefore confident that LTE is a valid assumption for this source. For the outflow sources A through C, a systematic concave up shape is noted in the rotation temperature diagram, suggesting departure from LTE conditions. These findings justify the need to consider a non-LTE radiative transfer model to interpret the measured fluxes. We perform this analysis using the RADEX package (van der Tak et al., 2007), which calculates expected line intensities for a given column density of SO2, kinetic temperature (Tkin), and volume density of H2. For each source, we assume spatially homogeneous kinetic temperature and volume density. The SO2 column density is fixed for each source and its value is drawn from the LTE analysis, which gives $10^{16}$ cm-2 for sources A, B, and C and $10^{15}$ cm-2 for source D. Varying column density one order of magnitude in each direction does not significantly affect the results. We perform calculations for a wide range of values for the kinetic temperature and the H2 density. We then evaluated the resulting ratios between the flux of each line and that of the $235151.7$ MHz transition for the outflow sources A, B, and C. Due to the lack of emission from the $235151.7$ MHz line in the vicinity of source D, the 236216.7 MHz line was used instead to calculate line ratios. Fits to our data are presented as the $\chi^{2}$ statistic p-value for each assumed value of Tkin, and H2 density (Figure 10) and best fit results are tabulated in Table 3. We see that the non-LTE RADEX results for $T_{kin}$ show general agreement with $T_{rot}$ derived with LTE assumption. A reliable estimate of H2 density is elusive. Because the flux of each line becomes more similar at greater values of H2 density, we are able only to constrain a lower bound on this value. The constraints on the temperature and H2 density in source D are much weaker. However, the best fit values of Tkin span the 130 K to 330 K range, which brackets the value of T${}_{rot}=240$ K obtained from the LTE analysis. This agreement between Trot and Tkin is consistent with LTE conditions in source D. Our RADEX analysis also allows us to evaluate our assumption of optically thin SO2 lines necessary for the LTE analysis performed above. Although for lower H2 densities ($\apprle 10^{7}$ cm-3), the 235.151 GHz line comes close to being optically thick, for the inferred H2 densities, all four SO2 have optical thickness below $10^{-2}$. Assuming that the extent of the sources in the line of sight direction is similar to that in the plane of the sky, and that the density is homogenous, we estimate the fractional abundance of SO2, $f_{SO_{2}}$ in each source. These are tabulated in Table 3. ## 4 Discussion ### 4.1 Source properties We begin by discussing the positions of the four identified SO2 sources. Source C, offset $2.5"$ to the West, is clearly an isolated body with no antipodal companion to the East. Its existence was suspected in Ziurys et al. (2007), but was not treated as a distinct source. On the other hand sources A and B are found to be offset in opposite directions relative to the star position with similar blue and redshift velocities relative to the stellar frame. SMA observations of the CO and SO analogs of sources A and B were interpreted to be antipodal companions oriented $15^{\circ}$ from the line of sight by Muller et al. (2007). On the other hand, Ziurys et al. (2009) found that single dish observations of CO and other molecules are best explained by a blueshifted and a redshifted source at 20∘ and 45∘ from the line of sight. Similarly, visible and IR HST observations (Humphreys et al., 2007; Smith et al., 2001) have found no evidence of antipodal structure around VY CMa. A careful inspection of sources A and B in our observation shows that the faster (in radial velocity relative to the star’s reference frame) sections of both bodies are offset towards the southwest, while the slower sections are offset towards the northeast. This observation argues against a bipolar geometry, in which antipodal subsections of the two outflows should have similar velocities. We therefore treat sources A, B, and C as three distinct outflows probably unrelated to the symmetry axis of the star. These sources do not seem to correspond to any visible or IR features. Our maps show that all SO2 outflow sources are too close to the star to be identified as the ”curved nebulous tail” or the numbered arcs presented in Smith et al. (2001), which are located $\sim 3.5"$ from the star. We identify sources A and B with the blue and redshifted outflows of the previous authors (Ziurys et al., 2007; Muller et al., 2007; Ziurys et al., 2009). The P-V space morphology of these sources match closely with outflows of both CO and SO lines mapped by Muller et al. (2007) (Figure 8). These bodies are hypothesized to have originated in an episode of anomalously high mass loss at uncorrelated locations on the stellar surface (Smith et al., 2001). Assuming that the ages of the outflows are similar and adopting the $\sim 500$ year age found by Muller et al. (2007), we can attempt to derive their locations in three-dimensions. We find that the deprojected radii of sources A and B are $4.2\times 10^{16}$ and $4.8\times 10^{16}$ cm and that both are situated at $22^{\circ}$ from the line of sight. Their deprojected star-frame velocities of $28$ and $30$ km s-1 fall within the range of measured outflow velocities from the multi-epoch observations of Humphreys et al. (2007). If we assume the same age as for sources A and B, then source C is found at a similar radius from the star ($6.9\times 10^{16}$ cm), but it is much faster at $44$ km s-1 and is situated $54^{\circ}$ from the line of sight. SO2 abundance at large radii is expected to be controlled by a balance between the rates of production, expansion, and photodissociation (Scalo & Slavsky, 1980). Outflows with faster expansion velocity are expected to maintain high SO2 abundance out to greater radii, as in the case of our source C. Source D is elongated and centered at the stellar position, with a resolved minor and major radii of $1"$ and $1.6"$, corresponding to $2.2\times 10^{16}$ and $3.5\times 10^{16}$ cm. The minor axis radius corresponds to between $180$ and $540$ stellar radii, depending on the adopted stellar radius (Monnier et al., 1999; Massey et al., 2006). A significant proportion of source D flux is missing from our observation when compared to single dish results (Figure 3). The source we observe therefore appears to represent the warm core of a larger extended envelope found at the systemic velocity with lower average column density. This extended source is analogous to the spherical wind described in previous millimeter wavelength observations of VY CMa by Ziurys et al. (2007), Ziurys et al. (2009), Muller et al. (2007), and Tenenbaum et al. (2010); however, only the first and last of these works observed SO2 and did not identify it in the spherical wind. These previous works have found a relatively low expansion velocity of between $15$ and $20$ km s-1, which is high given the narrow velocity range of source in our channel maps (Figures 4 \- 7). However, this discrepancy may be due to missing source D flux in our observations. The rotational temperatures derived from SMA and ARO data indicate the existence of a thermalized compact region with elevated temperatures with diameter $\approx 3\times 10^{16}$ cm. In the sections below, we refer to this inner region as the core of source D. Our derived values of $T_{rot}$ and $T_{kin}$ distinguish between the lower temperatures of sources A, B, and C and a much hotter core of source D. In comparison to previous works, our temperatures for source A, B, and C bracket the range of temperatures derived by Muller et al. (2007) and Ziurys et al. (2009) ($57$K and $85$K, respectively). This may be expected, as the preceding authors adopted the same best fit power-law temperature profile for both outflows, which are assumed to be at the same radius. As such, these previous results may represent an average of the temperatures of the outflows. Our inferred H2 densities from RADEX radiative transfer analysis are higher than the values found in previous studies. Ziurys et al. (2009) adopted an isotropic H2 density profile based on an assumed mass-loss rate that gives a value of $\sim 1\times 10^{6}$ cm-3 at $10^{16}$ cm radius. Muller et al. (2007) use a similar procedure to arrive at a lower value of $4.5\times 10^{5}$ cm-3 at the same location. A higher density of $3\times 10^{6}$ cm-3 is assumed for the inner wind region in Tenenbaum et al. (2007). Part of this discrepancy between our values for H2 density and that of previous authors may be explained by the latter’s assumption of isotropic mass flow, which does not account for density concentration in the spatially confined outflow regions. However, this reason alone may not be able to account for the more than two orders of magnitude difference. More significantly, our derived densities of between $5\times 10^{6}$ and $2\times 10^{8}$ cm-3 may be due to the presence of SO2 in regions of local density enhancement above the expected values from an isotropic model. We speculate that such high density regions are the result of shocks, and their existence around VY CMa can be inferred from the observations of OH masers, which overlap with sources A and B of SO2 emission (Bowers et al., 1983). The activation of the observed 1612 MHz maser line requires H2 densities of between $1\times 10^{6}$ and $3\times 10^{7}$ cm-3 at 100 K (Pavlakis & Kylafis, 1996). The upper end of this range is similar to our inferred value of minimum H2 density for sources B and C, while that of source A is much higher. However, OH masers may still be active in our source A despite its high density since its temperature of $\sim 55$K is cooler than that assumed in Pavlakis & Kylafis (1996). ### 4.2 Sulfur chemistry in the CSE For the remainder of the Discussion, we address the implications of our observations for circumstellar chemistry by comparing SO2 distribution with those of the OH and SO molecules. SO2 in CSEs is formed via the following reaction (Scalo & Slavsky, 1980; Nejad & Millar, 1988; Willacy & Millar, 1997; Cherchneff, 2006): $SO+OH\longrightarrow SO_{2}+H$ The radial abundance of SO2 is therefore expected to reflect that of the two reactant molecules. A striking similarity between the spectral profiles of SO2 and OH masers has already been noted by Ziurys et al. (2007). Maps of the 1612 MHz OH maser line show that it coincides with SO2 in sources A and B, but it is weak or undetectable in the outflow source C or the spherical wind source D. The lack of OH maser detection in source C (or perhaps a very weak detection; see Bowers et al., 1983) may be explained by anisotropic nature of maser radiation. Assuming that the velocity of outflows is oriented radially away from the star, relative velocities between different clumps of gas along a photon’s line of travel are smallest when the path is parallel or antiparallel to the gas expansion velocity. The most efficient pumping of a masering state is then achieved when photons travel radially inward or outward from the star. Therefore, the strongest maser emissions are observed from sources along our line of sight (Elitzur et al., 1976). This condition is nearly met for sources A and B ($15^{\circ}$ to $22^{\circ}$ from the line of sight). On the other hand source C, found at a line of sight angle of $54^{\circ}$, may also produce the 1612 MHz OH maser, but its signal is weak along the line of sight. The weak OH maser emission in the stellar velocity frame may be attributed to the high kinetic temperature of gas in this region. For a given volume density of gas, temperatures above a certain threshold tend to induce thermal equilibrium in the emitting body, undoing the population inversions responsible for maser emissions (Pavlakis & Kylafis, 1996). For temperatures of $\sim 200$K, the maximum allowable H2 density for the 1612 MHz OH maser is $3\times 10^{6}$ cm-3, which is more than an order of magnitude lower than our inferred density for source D. The highly linear trend of Source D data in the rotational temperature diagram (Figure 9) corroborates the prevalence of LTE conditions in source D. We therefore find that OH and SO2 distributions are generally similar and that their differences are reconcilable. Finally, we address the discrepancies between the SO2 and SO distributions of Muller et al. (2007), who have mapped the distribution of SO around VY CMa at a similar resolution to ours using a single rotational line of SO (J = 65 \- 54; $E_{u}=35$K). SO was found in all four source regions described in this work, although the red-shifted SO emitter is not fragmented into two discrete sources as in the case of SO2. Because of the strongly contrasting values of H2 density adopted in Muller et al. (2007) and this work, direct comparisons between column densities are more instructive than comparisons between fractional abundances. For column density to act as a valid proxy for abundance, we must assume that the line-of-sight dimension of the corresponding SO and SO2 sources are similar and that the distribution of the each molecules within each source is homogeneous. While we cannot be certain that the line-of-sight thickness of the SO and SO2 sources are equal, their plane of sky dimensions are similar ($\sim$30% discrepancies). The combined column density of both SO outflows and the inner wind where they overlap along the line of sight was found to be $\sim 10^{16}$ cm-2. This value reflected the combined column density from both outflows and the spherical wind (corresponding to our sources A, B, and D). Given that the three regions contribute approximately equal amounts to the total SO column density, the value of $N_{SO}$ in each region is on the order of a few $10^{15}$ cm-2. In contrast, we find column density of SO2 to be $1$ to $2\times 10^{16}$ cm-2 in $\it{each}$ outflow source. The column density of SO2 in the outflow sources A and C is therefore $\apprge 3$ times greater than that of SO in the same regions. This statement likely applies as well to outflow source C, given its similar SO2 column density compared to the other outflows and the lack of a discrete SO source at its location. Furthermore, even with missing flux, the column density of SO2 in the compact core of source D is within error as that of the outflow sources, implying that $N_{SO_{2}}>N_{SO}$ in the core of the spherical wind $<3\times 10^{16}$ cm from the star. Non-equilibrium CSE chemistry models (Scalo & Slavsky, 1980; Nejad & Millar, 1988; Willacy & Millar, 1997) make differing predictions about ratio of SO2 to SO abundance ($f_{SO_{2}}/f_{SO}$). The Scalo & Slavsky (1980) (SS) model adopts a reaction rate for SO2 formation that is fast compared to the rate of SO2 destruction via photodissociation. Under these conditions, SO is quickly converted into SO2, which becomes the primary S-bearing species. SO2 therefore reaches maximum abundance at a greater radius than SO. SS has predicted the radius of this transition where $f_{SO_{2}}/f_{SO}>1$ to be around several times $10^{15}$ cm. Our observations suggest that $f_{SO_{2}}/f_{SO}>1$ for the outflow regions ($>4\times 10^{16}$ cm from star) and in the compact region within $3\times 10^{16}$ cm of the star. If there does exist a transitional radius at which $f_{SO_{2}}/f_{SO}=1$, it must be less than the latter radius, making our study consistent with the Scalo & Slavsky (1980) model. On the other hand, models that predict $f_{SO_{2}}/f_{SO}<1$ at all radii are inconsistent with our data (Nejad & Millar, 1988; Willacy & Millar, 1997). In addition, the SS model predicts a steady increase in the value of $f_{SO_{2}}/f_{SO}$ outward from the $f_{SO_{2}}/f_{SO}=1$ transition radius at $\sim 2\times 10^{15}$ cm. The value of $f_{SO_{2}}/f_{SO}$ increases by more than an order of magnitude by radius 1016 cm. Observations of the outflow sources A, B, and C do not support this view, as $f_{SO_{2}}/f_{SO}$ in on the order of 3 for all four sources. More precise comparisons between SO2 and SO abundance in each source is hindered by the lack of more detailed interpretations of SO maps (the Muller et al. (2007) observation included only one SO line). Therefore, quantitative comparison between model radial distribution of the two species and our results is elusive and we regard our support of the SS model as only qualitative. Other uncertainties remain in our understanding of sulfur chemistry in the CSE of VY CMa. Our observations cannot be used to constrain whether the SO2 originates in the CSE, as assumed by the cited models, or in shocks in an ”inner wind” within a few stellar radii of the photosphere as modeled by Cherchneff (2006) and observed by Decin et al. (2010) and Yamamura et al. (1999). Furthermore, as proposed earlier by Jackson & Nguyen (1988) and Willacy & Millar (1997), shock events similar to those that occur in the inner wind may be the source of SO2 enrichment in the outflow lobes. Indeed, comparison of SO2 and maps of SO, PN, and SiS (Muller et al., 2007; Figure 1, this work) shows that SO2 exhibits a more fragmented distribution with discrete sources in the red-shifted outflow. Unlike these other molecules, the red-shifted SO2 outflow is partitioned into two regions of high local abundance (sources B and C), suggesting that local effects may participate in the formation of SO2. However, CSE chemistry models that include the effect of shocks (similar to Cherchneff, 2006) remain to be done. A further uncertainty involves the very high mass loss rate of VY CMa, which is, for example, about two orders of magnitude faster than the asymptotic giant branch star IK Tau (Olofsson et al., 1998). Varying the mass loss rate in CSE chemistry models does not significantly affect the radial abundance profiles of chemicals species while it does result in globally larger envelopes for higher mass loss rates Willacy & Millar (1997). Therefore the discussions of the $f_{SO_{2}}/f_{SO}$ profile in this work are also relevant to stars with lower mass loss rates, while the actual measured radii of peak SO2 abundance around VY CMa are unique this object. ### 4.3 Summary 1. 1. Four rotational lines of the SO2 molecule are mapped with $\sim 1"$ resolution around VY CMa. SO2 is found in four discrete sources, three of which are fast ($28$ to $44$ km s-1) outflows far from the star and one is a slower spherical wind near the star. No symmetrical relationship among the faster outflows or visible and IR features are found. 2. 2. The three fast outflows are found at similar distances from the star and probably originated around 500 years ago. 3. 3. Comparison between our SO2 maps and those of the 1612 MHz OH maser line suggests that the two species are strongly correlated and that the OH maser detection may be limited by high temperature and density in the spherical wind. 4. 4. SO2 is more abundant than SO in all three outflow sources, supporting the non- equilibrium chemistry model of Scalo & Slavsky (1980). It is inconsistent with models that predict $f_{SO_{2}}/f_{SO}<1$ for all radii (e.g., Nejad & Millar, 1988; Willacy & Millar, 1997). The distribution of SO2 in discrete clumps when compared to other molecules may point to the role of localized effects, such as shocks, in the enhancement of SO2 abundance. We are grateful to Raymond Blundell, Thomas Dame and Patrick Thaddeus for the opportunity to carry out this research using the SMA as part of the Laboratory Astrophysics (Astro 191) course at Harvard University. We also thank Carl Gottlieb for helpful comments on the manuscript. ## References * Bowers et al. (1983) Bowers, P. F., Johnston, K. J., & Spencer, J. H. 1983, ApJ, 274, 733 * Cherchneff (2006) Cherchneff, I. 2006, A&A, 456, 1001 * Danchi et al. (1994) Danchi, W. C., Bester, M., Degiacomi, C. G., Greenhill, L. J., & Townes, C. H. 1994, AJ, 107, 1469 * Decin et al. (2010) Decin, L., et al. 2010, A&A, 516, A69 * de Wit et al. (2008) de Wit, W. J., et al. 2008, ApJ, 685, L75 * Elitzur et al. (1976) Elitzur, M., Goldreich, P., & Scoville, N. 1976, ApJ, 205, 384 * Friedel et al. (2005) Friedel, D. N., Snyder, L. E., Remijan, A. J., & Turner, B. E. 2005, ApJ, 632, L95 * Guilloteau et al. (1986) Guilloteau, S., Lucas, R., Omont, A., & Nguyen-Q-Rieu 1986, A&A, 165, L1 * Hartquist et al. (1980) Hartquist, T. W., Dalgarno, A., & Oppenheimer, M. 1980, ApJ, 236, 182 * Humphreys et al. (2007) Humphreys, R. M., Helton, L. A., & Jones, T. J. 2007, AJ, 133, 2716 * Jackson & Nguyen (1988) Jackson, J. M., & Nguyen, Q.-R. 1988, ApJ, 335, L83 * Massey et al. (2006) Massey, P., Levesque, E. M., & Plez, B. 2006, ApJ, 646, 1203 * Monnier et al. (1999) Monnier, J. D., Tuthill, P. G., Lopez, B., Cruzalebes, P., Danchi, W. C., & Haniff, C. A. 1999, ApJ, 512, 351 * Müller et al. (2005) Müller, H. S. P., Schlöder, F., Stutzki, J., & Winnewisser, G. 2005, Journal of Molecular Structure, 742, 215 * Muller et al. (2007) Muller, S., Dinh-V-Trung, Lim, J., Hirano, N., Muthu, C., & Kwok, S. 2007, ApJ, 656, 1109 * Olofsson et al. (1998) Olofsson, H., Lindqvist, M., Nyman, L.-A., & Winnberg, A. 1998, A&A, 329, 1059 * Nejad & Millar (1988) Nejad, L. A. M., & Millar, T. J. 1988, MNRAS, 230, 79 * Pavlakis & Kylafis (1996) Pavlakis, K. G., & Kylafis, N. D. 1996, ApJ, 467, 309 * Pickett (1991) Pickett, H. , 1991, Journal of Molecular Spectroscopy 148, 371 * Royer et al. (2010) Royer, P., et al. 2010, A&A, 518, L145 * Sahai & Wannier (1992) Sahai, R., & Wannier, P. G. 1992, ApJ, 394, 320 * Sault, Teuben & Wright (1995) Sault, R.J, Teuben, P.& Wright, M.C.H., 1995, A Retrospective View of Miriad, in: Astronomical Data Analysis Software and Systems IV, ed. R. A. Shaw, H. E. Payne and J.J.E. Hayes. PASP Conf. Series 77,433 * Scalo & Slavsky (1980) Scalo, J. M., & Slavsky, D. B. 1980, ApJ, 239, L73 * Shinnaga et al. (2004) Shinnaga, H., Moran, J. M., Young, K. H., & Ho, P. T. P. 2004, ApJ, 616, L47 * Smith et al. (2001) Smith, N., Humphreys, R. M., Davidson, K., Gehrz, R. D., Schuster, M. T., & Krautter, J. 2001, AJ, 121, 1111 * Smith et al. (2009) Smith, N., Hinkle, K. H., & Ryde, N. 2009, AJ, 137, 3558 * Snyder et al. (2005) Snyder, L. E., et al. 2005, ApJ, 619, 914 * Tenenbaum et al. (2007) Tenenbaum, E. D., Woolf, N. J., & Ziurys, L. M. 2007, ApJ, 666, L29 * Tenenbaum et al. (2010) Tenenbaum, E. D., Dodd, J. L., Milam, S. N., Woolf, N. J., & Ziurys, L. M. 2010, ApJS, 190, 348 * van der Tak et al. (2007) van der Tak, F. F. S., Black, J. H., Schöier, F. L., Jansen, D. J., & van Dishoeck, E. F. 2007, A&A, 468, 627 * van Loon et al. (2005) van Loon, J. T., Cioni, M.-R. L., Zijlstra, A. A., & Loup, C. 2005, A&A, 438, 273 * Willacy & Millar (1997) Willacy, K., & Millar, T. J. 1997, A&A, 324, 237 * Yamamura et al. (1999) Yamamura, I., de Jong, T., Onaka, T., Cami, J., & Waters, L. B. F. M. 1999, A&A, 341, L9 * Yoon and Cantiello (2010) Yoon, S.-C., & Cantiello, M. 2010, ApJ, 717, L62 * Ziurys et al. (2007) Ziurys, L. M., Milam, S. N., Apponi, A. J., & Woolf, N. J. 2007, Nature, 447, 1094 * Ziurys et al. (2009) Ziurys, L. M., Tenenbaum, E. D., Pulliam, R. L., Woolf, N. J., & Milam, S. N. 2009, ApJ, 695, 1604 Table 1: Summary of observed SO2 lines. SO2 \| Frequency \| Eu11Eu is the upper rotational state energy. \| $S$22$S$ is the line strength. ---\|---\|---\|--- transition \| (MHz) \| (K) \| $4_{2,2}-3_{1,3}$ \| 235151.7 \| 19.0 \| 1.71 $10_{3,7}-10_{2,8}$ \| 245563.4 \| 72.7 \| 5.44 $15_{2,14}-15_{1,15}$ \| 248057.4 \| 119.3 \| 5.27 $16_{1,15}-15_{2,14}$ \| 236216.7 \| 130.7 \| 6.05 Table 2: Description of SO2 line emission sourcesElliptical gaussian fits of the identified SO2 emission sources. Offsets are measured in arcseconds from the center of the continuum emission (RA=$07^{h}22^{m}58.336^{s}$, Dec=$-25^{\circ}46^{\prime}03.063^{\prime\prime}$). Source size represents the major and minor axis of the ellipse, and its orientation angle is that of the major axis. Angles are given as rotations from north through east. Fits were performed on maps of the 235151.7 MHz line. Source \| Offset \| Position \| Size \| Orientation ---\|---\|---\|---\|--- (vlsr range in km s-1) \| (“) \| Angle (∘) \| (“) \| A: ($-18,\,+10$) \| $0.66\pm 0.10$ \| $33^{\circ}$ \| $2.5\times 2.4$ \| $126^{\circ}$ B: ($+26,\,+66$) \| $0.83\pm 0.10$ \| $316^{\circ}$ \| $2.9\times 1.3$ \| $41^{\circ}$ C: ($+26,\,+66$) \| $2.45\pm 0.09$ \| $272^{\circ}$ \| $2.9\times 1.8$ \| $127^{\circ}$ D: ($+12,\,+26$) \| $\sim 0$ \| $N/A$ \| $3.2\times 1.9$ \| $\sim-40^{\circ}$ Table 3: Summary of LTE and RADEX modeling results Source \| $T_{rot}$ \| SO2 Column Density \| $T_{kin}$ \| H2 Density \| SO2 Abundance ( $f_{SO_{2}}$) ---\|---\|---\|---\|---\|--- \| (K) \| (cm-2) \| (K) \| (cm-3) \| A \| $61_{-17}^{+39}$ \| $9.2_{-4.4}^{+7.8}\times 10^{15}$ cm-2 \| 50 \| $\apprge 2$$\times 10^{8}$ \| $7\times 10^{-10}$ B \| $110_{-29}^{+63}$ \| $1.9_{-0.6}^{+0.8}\times 10^{16}$ \| $\sim$ 75 \| $\apprge 3$$\times 10^{7}$ \| $2\times 10^{-8}$ C \| $69_{-17}^{+34}$ \| $1.1_{-0.4}^{+0.7}\times 10^{16}$ \| $\sim$ 100 \| $\apprge 3$$\times 10^{7}$ \| $8\times 10^{-9}$ D \| $240_{-52}^{+91}$ \| $7.6\pm 1.0\times 10^{15}$Values for source D suffer from missing extended flux. Using single dish data, average $T_{rot}=97$ K and $N=1.8\times 10^{15}$ cm-2 assuming that the emitter is an uniform 4” by 4” region. \| $\sim$ 220 \| $\apprge 4$$\times 10^{7}$ \| $5\times 10^{-10}$ Figure 1: Left: PN J=5–4 and SiS J=13–12 spectra showing the peak emission at 19.49 km s-1 (from the line rest frequencies). We adopt this velocity as that of the stellar frame. ${\it Right:}$ Integrated intensity maps of PN (top) and SiS (bottom) emission, which show maxima close to the peak of the continuum emission. Figure 2: Left: SO2 spectra averaged in a region of 3”x3” around the continuum peak. Right: Integrated intensity maps corresponding SO2 lines over velocity intervals $-20$ and $10$ km s-1 are shown in blue contours. The red contours show the integrated intensity over velocity intervals 20 to 60 km s-1. The starting value and interval of contour levels are set to 35 mJy beam-1 km s-1. The continuum emission is shown in grey scale (repeated in all panels). The location of the three identified sources A, B and C is shown in the bottom two maps. Figure 3: Left: SO2 spectra from our (black) and 10 meter single dish observations by the Arizona Radio Observatory’s Submillimeter Telescope (red) for the two lowest temperature transitions (235151.7 and 245563.4 MHz). The SMA data was smoothed at the appropriate scale to approximate the field of view of the single dish telescope ( 30”). Figure 4: Channel maps of SO2 4(2,2)–3(1,3) emission at 235151.7 MHz toward VY CMa. In each panel, emission is integrated over a 10 km s-1 wide velocity range centered on the velocity indicated on the top right corner. Contour levels are same as in Fig. 2. Figure 5: Same as in Fig.4 for SO2 10(3,7)–10(2,8) emission at 245563.4 MHz. Figure 6: Same as in Fig.4 for SO2 15(2,14)–15(1,15) emission at 248057.4 MHz. Figure 7: Same as in Fig.4 for SO2 15(1,15)–15(2,14) emission at 236216.6 MHz. Figure 8: Left: A position velocity cut along east-west at Declination offset of 0′′ in the SO2 line emission at 245563.4 MHz, at position angle 105∘. The contours correspond to 10% of the peak emission. Sources A, B, C and D are clearly separated. Figure 9: Rotational temperature diagram of the four SO2 emission sources. Red, blue, and black curves and points indicate sources A, B, and C respectively. Orange represents the spherical wind (source D). Variable “L” on the vertical axis represents the left-hand side of Equation 1. The dashed orange line represents the spherical wind based on ARO single dish data Tenenbaum et al. (2010). The errors plotted are set to three sigma. Horizontal coordinates of the plotted points have been staggered for clarity. Figure 10: Fits between our measured line ratios and predictions with Radex radiative transfer. Solid, dashed, and dotted contours enclose regions of $\chi^{2}$ test p-values greater than 5%, 30%, and 95%. Owing to lower S/N and weak dependence of line ratios to these quantities, the derived temperature and density of source D are much less well-constrained than the other three.	arxiv-papers	2011-11-29T22:01:38	2024-09-04T02:49:24.778084	{ "license": "Public Domain", "authors": "Roger Fu, Arielle Moullet, Nimesh A. Patel, John Biersteker, Kimberly\n L. DeRose, Kenneth H. Young", "submitter": "Nimesh Patel", "url": "https://arxiv.org/abs/1111.7004" }
1111.7068	# Unveiling the super-orbital modulation of LS I $+$61∘303 in X-rays Jian Li11affiliation: Laboratory for Particle Astrophysics, Institute of High Energy Physics, Beijing 100049, China. Email: jianli@ihep.ac.cn , Diego F. Torres22affiliation: Institut de Ciències de l’Espai (IEEC-CSIC), Campus UAB, Torre C5, 2a planta, 08193 Barcelona, Spain 33affiliation: Institució Catalana de Recerca i Estudis Avançats (ICREA). , Shu Zhang11affiliation: Laboratory for Particle Astrophysics, Institute of High Energy Physics, Beijing 100049, China. Email: jianli@ihep.ac.cn , Daniela Hadasch22affiliation: Institut de Ciències de l’Espai (IEEC-CSIC), Campus UAB, Torre C5, 2a planta, 08193 Barcelona, Spain , Nanda Rea22affiliation: Institut de Ciències de l’Espai (IEEC-CSIC), Campus UAB, Torre C5, 2a planta, 08193 Barcelona, Spain , G. Andrea Caliandro22affiliation: Institut de Ciències de l’Espai (IEEC-CSIC), Campus UAB, Torre C5, 2a planta, 08193 Barcelona, Spain , Yupeng Chen11affiliation: Laboratory for Particle Astrophysics, Institute of High Energy Physics, Beijing 100049, China. Email: jianli@ihep.ac.cn , Jianmin Wang11affiliation: Laboratory for Particle Astrophysics, Institute of High Energy Physics, Beijing 100049, China. Email: jianli@ihep.ac.cn ###### Abstract From the longest monitoring of LS I $+$61∘303 done to date by the Rossi X-ray Timing Explorer (RXTE) we found evidence for the long-sought, years-long modulation in the X-ray emission of the source. The time evolution of the modulated fraction in the orbital lightcurves can be well fitted with a sinusoidal function having a super-orbital period of 1667 days, the same as the one reported in non-contemporaneous radio measurements. However, we have found a 281.8 $\pm$ 44.6 days shift between the super-orbital variability found at radio frequencies extrapolated to the observation time of our campaign and what we found in the super-orbital modulation of the modulated fraction of our X-ray data. We also find a super-orbital modulation in the maximum count rate of the orbital lightcurves, compatible with the former results, including the shift. X-rays: binaries, X-rays: individual (LS I $+$61∘303) ## 1 Introduction LS I $+$61∘303 is one of a handful high-mass X-ray binaries that have been detected at all frequencies, including TeV and GeV energies. Its nature is still under debate, with rotationally-powered pulsar-composed systems (see Maraschi & Treves 1981, Dubus 2006) and microquasar jets (see Bosch-Ramon & Khangulyan 2009 for a review) being discussed. Recently, evidence favoring LS I $+$61∘303 as the source of a very short X-ray burst led to the analysis of a third alternative, in which LS I $+$61∘303 is a magnetar binary (see Torres et al. 2011, also Bednarek 2009, and Dubus 2010). Long-term monitoring of the source, ideally across all wavelengths, is a key ingredient to disentangle differences in behavior which could point to the underlying source nature. Here, we report on the analysis of _RXTE_ -Proportional Counter Array (PCA) monitoring observations of the $\gamma$-ray binary system LS I $+$61∘303\. The dataset we consider covers the period between 2007 September and 2011 September (2007-08-28 – 2011-09-15), and it is the longest monitoring campaign done for this source. Smaller datasets included in this campaign have been analyzed and results have been presented in our previous papers I and II (Torres et al. 2010, and Li et al. 2011, respectively). In them, we focused on establishing the orbit-to-orbit X-ray variability and on studying the spectral properties and the flares. The current data enhances in more than one additional year the reported coverage on the source, and for the first time, it is sufficient to consider the possible super-orbital modulation of the X-ray emission from LS I $+$61∘303. ## 2 Observations, data analysis, and results Our current analysis includes 473 _RXTE_ -PCA pointed observations identified by proposal numbers 93100, 93101, 93102, 93017, 94102, 95102 and 96102. The analysis of PCA data was performed using HEASoft 6.9. We filtered the data using the standard _RXTE_ -PCA criteria. Only PCU2 (in the 0-4 numbering scheme) has been used for the analysis because it was the only Proportional Counter Unit (PCU) that was 100% on during the observations. We select time intervals where the source elevation is $>$10∘ and the pointing offset is $<0.02^{\circ}$. The background file used in the analysis of PCA data is the most recent available one from the HEASARC Web site for faint sources, and detector breakdown events have been removed.111The background file is pca_bkgd_cmfaintl7_eMv20051128.mdl. The data have been barycentered using the FTOOLS routine faxbary using the JPL DE405 solar system ephemeris. Our flux and count rate values are given for an energy range of 3–30 keV. At first, we consider the complete X-ray lightcurve of our campaign, and fitted it with a constant to obtain an average flux, resulting in (1.616$\pm$0.006) counts s-1. In order to remove the influence of the several ks-long flares detected from the source, we cut all observations that presented a count rate larger than three times this average. The remaining on- source time amounts to 684.3 ks (460 observations), and it is uniformly distributed in the system’s orbital (between 60 and 80 ks per each 0.1 of orbital phase bin) as well as in the system’s super-orbital phase (around 60 ks per each 0.1 of super-orbital phase bin, except at super-orbital phase 0.8, where an intensive campaign increased the exposure to 120 ks) as defined by the radio observations (Gregory 2002). Given a 6-months time bin (or approximately 6.8 orbits of the system), we take the peak X-ray flux in that period and compute the modulated flux fraction. The latter is defined as ($c_{max}-c_{min})/(c_{max}+c_{min}$ ), where $c_{max}$ and $c_{min}$ are the maximum and minimum count rates in the 3–30 keV orbital profile of that period, after background subtraction. The minimum count rates are roughly constant around 1 count s-1 all along the observation time. Results are shown in Figure 1. Table 1 presents the values of the reduced $\chi^{2}$ for fitting different models to the modulation fraction and the peak flux in X-rays. It compares the results of fitting a horizontal line, a linear fit, and two sinusoidal functions. One of the latter has the same period and phase of the radio modulation (from Gregory 2002, labelled as ‘Radio’ in Table 1, dotted line in Figure 1). The other sine function has the same period as in radio but allowing for a phase shift from it (a solid line in Figure 1, labelled as ‘Shifted’ in Table 1). It is clear that there is variability in the data (and thus that a constant fit is unacceptable), as well as that the sinusoidal description with a phase-shift is better than the linear one, which is obvious by visual inspection of Figure 1. The phase shift derived by fitting the modulated fraction is 281.8 $\pm$ 44.6 days, corresponding in phase to $\sim 0.2$ of the 1667$\pm$8 days super-orbital period. The phase shift derived by fitting instead the maximum flux is 300.1 $\pm$ 39.1 days, and results compatible with the former. Table 1: Reduced $\chi^{2}$ for fitting different models to the modulation fraction and the peak flux in X-rays. \| Constant \| Linear \| Radio \| Shifted ---\|---\|---\|---\|--- Modulation Fraction \| 88.2 / 7 \| 38.0 / 6 \| 42.1 / 6 \| 1.1 / 5 Peak Flux \| 212.8 / 7 \| 114.8 / 6 \| 91.8 / 6 \| 4.9 / 5 Figure 2 analyzes whether there is anything special concerning the 6 months time bin just chosen to present the former results. It presents the peak flux and modulated fraction in different time bins, from 4 to 12 months. The black lines in Figure 2 shows the sine fitting with a 1667 days period fixed. The black lines in all left (right) panels are the same as the one depicted in the left (right) panel of Figure 1. It is interesting to note that the smaller the time bin the larger it is the scattering around the sinusoidal fit, which can be understood as an effect of the increasing similarity between the time bin itself and the orbital period of the system (of 26.4960$\pm$0.0028, Gregory 2002). Indeed, orbit-to-orbit variability is known to exist in our data, and can be similar or larger than the super-orbital induced variability at times (e.g., see the variations found for the same phase-bin in contiguous orbits in Figure 4 of Torres et al. 2010). Thus, the shorter the time bin, the less likely it is that the super-orbital induced variation could be detected, which may be sub-dominant to the local-in-time changes. On the contrary, as soon as the integrated time bin is large enough in comparison with the orbital period of LS I $+$61∘303, the super-orbital variability is consistently observable. This same fact makes a direct comparison of the sinusoidal trend of the super- orbital X-ray modulation of Figure 1 with the flux obtained from LS I $+$61∘303 under short and isolated observations untrustable. For example the observations at soft X-rays conducted by XMM-Newton (Neronov & Chernyakova 2006; Chernyakova et al. 2006, Sidoli et al. 2006), Chandra (Paredes et al. 2007, Rea et al. 2010), ASCA (Leahy et al. 1997), ROSAT (Goldoni & Mereghetti 1995; Taylor et al. 1996), and Einstein (Bignami et al. 1981) were all too short to cover even a single full orbit. Similarly, earlier campaigns with PCA during the month of March 1996 (Harrison et al. 2000, Greiner and Rau 2001) are also too short. Comparing their obtained flux values with the super- orbital trend would be meaningless since there is not enough time coverage to average out the possible orbit-to-orbit variations. Finally, we note that we have also analyzed 15 years of _RXTE_ -ASM data on LS I $+$61∘303, but the larger error bars on the count rates preclude to obtain any conclusion from that dataset. Figure 1: Left: Peak count rate of the X-ray emission from LS I $+$61∘303 as a function of time and the super-orbital phase. Right: modulated fraction, see text for details. The dotted line shows the sine fitting to the modulated flux fraction and peak flux with a period and phase fixed at the radio parameters (from Gregory 2002). The solid curve stands for sinusoidal fit obtained by fixing the period at the 1667 days value, but letting the phase vary. The time bin corresponds to six months. The colored boxes represent the times of the TeV observations that covered the broadly-defined apastron region. The boxes in green denote the times when TeV observations are in low state while boxes in yellow are TeV observations in high state. Figure 2: Peak flux (left) and modulation fraction (right) at different time scales. From top to bottom we plot the results obtained by binning the data in 4, 6, 8, and 12 months bins. The black line shows the sine fitting with a 1667 days period fixed, as in the corresponding panels of Figure 1 ## 3 Discussion We notice that the X-ray long-term monitoring (2007–2011) on LS I $+$61∘303 started about 7 years later from the end of the campaign used to determine the super-orbital period in radio (1977–2000, see Gregory 2002). We will assume then that the radio-determined super-orbital modulation of the source is, although possibly variable, active today with similar features as the ones claimed a decade earlier. This appears possible given that recent reports of variation in the orbital radio maxima are of only $\sim$0.1 in phase (see Trushkin & Nizhelskij 2010). Under such assumptions, we showed that there is a phase shift between the radio and the X-ray super orbital modulation. Interestingly, this shift is the same as the one hinted at between the radio and the H$\alpha$ line (Zamanov et al. 1999, Zamanov & Martí 2000). Indeed, the optical observations that covered the period 1989–1999 were fitted with a period of $\sim$1584 days (Zamanov et al. 1999), a value reported prior to the work by Gregory (2002), where the super-orbital period was refined to 1667$\pm$8 days. To investigate further the long-term modulation of LS I $+$61∘303 at optical wavelengths and how it compares with the current findings in X-rays, we took the H$\alpha$ data from Table 4 of Paredes et al. (1994), Table 1 of Zamanov et al. (1999), and Figure 1 of Zamanov et al. (2000), and as stated in Zamanov et al. (1999), considered an error of 10% for all the equivalent widths. We performed an analysis similar to the one done for X-rays in the previous section, and derived and optical phase lag of 290.1$\pm$16.7 days with respect to the radio phase. Thus, the optical phase lag is coincident with the one derived at X-rays, although the observations at the two bands are about 8 years apart. The stellar disk of Be stars are well known to grow larger as the equivalent width of the H$\alpha$ emission line increases (e.g., Hanushik et al. 1988; Grundstrom & Gies 2006). The optical variability has been most likely attributed to the cyclic variation of Be circumstellar disk. Thus, the possible coincidence with the X-ray phase lag suggests that the stellar disk may play an important role also for the X-ray emission, and probably for the higher-energy non-thermal emission of LS I $+$61∘303 too. The coincidence between the X-ray and optical shift with respect to the 1667 days radio modulation has to be taken with the necessary prudence prompted by it being based on non-simultaneous observations. In particular, it seems that the optical observations present the largest degree of variation in time. We checked that in addition of the H$\alpha$ measurements mentioned above, there are more recent ones in the works of e.g., Liu et al. (2005), Grundstrom et al. (2007), Zamanov et al. (2007), and McSwain et al. (2010). However, the latter span 0.51 (at best, being usually much shorter) of the super-orbital period, and as such we can not directly use them for a comparison in long- terms. Nevertheless, they seem to hint at that the H$\alpha$ variability is not strictly periodic or at least at a changing amplitude. Based (among other reasons discussed in Torres et al. 2011) on the analysis of a the _Swift_ -BAT detection of a short, magnetar-like burst from the direction of LS I $+$61∘303, we have proposed that the system’s compact object is a high magnetic field, slow period pulsar. In that case, we proved that the LS I $+$61∘303-system would most likely be subject to a flip-flop behavior, from a rotationally powered regime (in apastron, also known as ejector), to a propeller regime (in periastron) along each of the system’s eccentric orbits. The multi-wavelength phenomenology can be put in the context of the former model, and in particular, also the highest energy TeV emission, which has also shown low and high states which are apparently modulated by the same super- orbital period as well. Within this model, we notice that an increase in the accreted mass onto the compact object (unavoidably linked to the mass-loss rate of the star) by a factor of a few222Estimations of the cyclical variations in the mass loss-rate from the Be star in LS I $+$61∘303 are given as the ratio between maximal and minimal values obtained either from radio emission (a factor of 4 was determined by Gregory & Neish 2002) or from H$\alpha$ measurements, which span from a factor of 5.6 (Gregory et al. 1989) to 1.5 (Zamanov et al. 1999). can put the system in a permanent propeller stage along the orbit, including at the apastron region. This change of behavior for such an small change in mass loss rate can be the reason behind the evolution of the modulated fraction. Indeed, using the formulae in Torres et al. (2011), and considering to simplify that the condition $R_{m}=R_{lc}$, where $R_{m}$ stands for the magnetic radius and $R_{lc}$ for the light cylinder, establishes both the out-of-ejector and into-ejector condition, one sees that the period–mass-loss–magnetic field relation for the apastron of LS I $+$61∘303 is $\left({P}/{\rm 1\;s}\right)\sim a\\!\times\\!15({B}/{10^{14}\;{\rm G}})^{4/7}({\dot{M}_{}}/{10^{18}{\rm\;g\;s^{-1}}})^{-2/7}$ where $a$ represents a constant of order 1, and we have assumed an eccentricity of 0.6 and a semi-major axis of $6\times 10^{12}$ cm. For periods shorter than the former, the system is in an ejector phase. For larger periods, it is in a propeller stage (see Torres et al. 2011 for details). High values of magnetic field and slow periods would make the transition possible: a cyclical change by a factor of a few in $\dot{M}_{}$ can make the system to abandon the ejector phase in apastron. For instance, under a variation by a factor of 4 in $\dot{M}_{}$, a case that leads to a super-orbital induced transition is given by a magnetic field of $5\times 10^{13}$ G, and a typical period of magnetars ($\sim 7$ s). This may also happen for smaller values of the magnetic field but only in the case of a relatively long period. For instance, again under a variation by a factor of 4 in $\dot{M}_{}$ and for $B=10^{12}$ G, the period should be between 700 ms and 1s in order for the system to flip- flop in the super-orbital evolution, although no known pulsar in these parameter ranges has a rotational energy in excess of $10^{36}$ erg s-1 (ATNF Catalogue version: 1.43), which would be needed to account for the multi- wavelength output of the system. Note in particular that the behavior of the LS I $+$61∘303 system containing a pulsar with $B\sim 10^{12}$ G and $P<700$ ms would be unaffected by the cyclical variation of the mass-loss rate: it would act as an ejector in apastron along the whole super-orbital period. The flip-flop mechanism can then be used to qualitatively explain why LS I $+$61∘303 has entered in a low TeV state (see, e.g., Acciari et al. 2010) when at the maximum of the radio super-orbital variability, but perhaps also to explain why the modulated X-ray flux fraction varies as we found in Figure 1. When the mass-loss rate is low, the inter-wind shock formed at the collision region between the pulsar and the stellar wind would be present at the broad apastron region (and so will the TeV emission there), disappearing at periastron. In this situation, there are two contributors to the X-ray emission along the orbit, expected to be roughly at the same level (e.g., Zamanov et al. 1999); the shock at apastron and the propeller at periastron, and the modulated fraction is consistently low. When at the maximum of the mass-loss rate, the inter-wind shock may not form, and abundant TeV particles would not be produced since shocks at the magnetic radius are unable to reach TeV energies. Thus there is only one process generating the X-ray emission along the system’s orbit, the propeller, and the modulated fraction is then maximum. The exact position of the X-ray maximum along each of the orbits would depend on the local-in-time conditions of the accreted mass onto the compact object, which established the relative weight of the two X-ray contributors, and it is thus expected to vary beyond the super-orbital trend in short timescales, and not always be located at periastron. However, given that the H$\alpha$ cycle represents the cyclical modulation of the mass loss rate, it would be natural to expect that the X-ray emission be correlated with it in long timescales (i.e., with how much mass is falling towards the compact object, e.g., see Bednarek 2009 or Bednarek & Pabich 2011). Zamanov et al. (2001) already discussed when the radio emission is expected to peak in each of the system orbits: The switch on of the ejector phase will activate the pulsar wind, creating a cavern around the neutron star which will start to expand. This means that the radio outburst will peak with some delay after the change of regimes, which is supposed to happen somewhere after the periastron, when the accretion rate onto the compact object diminishes enough. In a cyclical variability of the mass loss rate of the star, the ejector- propeller transition moves in phase: at lower mass loss rates, the ejector will switch on earlier, and the radio outburst will peak at earlier orbital phases than at higher mass loss rates. A generic TeV and radio anti- correlation is thus expected since the more mass fuels the propeller phase the more violent the radio outburst will be, and the less effectively the inter- wind shock will generate TeV particles. Figure 3: Peak flux per orbit in TeV shown in red (all of them happening in the 0.6–1.0 orbital phase range) as a function of superorbital phase, together with radio (top panel) and H$\alpha$ (bottom panel) data(black) as described in the text. The upper gray dashed line stands for the TeV flux level at discovery of the source in 2006, whereas the lower dashed line stands for 1/3 of this flux value. Two super-orbital phases are shown for clarity. Whenever there are both MAGIC and VERITAS compatible observations for the same orbit, they are averaged. The colored boxes in Figure 1 represent the times of the TeV observations that covered the broadly-defined apastron region (from Albert et al. 2006, 2008, Anderhub et al. 2009, Aleksic et al. 2011; Acciari et al. 2008, 2009, 2010). Those boxes colored in green denote the times for which the observations led only to imposing an upper-limit or to a detection with a flux that is about 3 times less than the one obtained at the discovery observations of 2006 (Albert et al. 2006), which defines the low state. The yellow boxes stand for those observations for which the level of the TeV emission was roughly compatible with the original discovery. There is a trend for finding a low TeV state towards the maximum of the super-orbital low-frequency cycles. This is perhaps more clearly seen in Figure 3, where we plot the peak flux per orbit in TeV (all of them happening in the 0.6–1.0 orbital phase range) as a function of superorbital phase, together with radio and H$\alpha$ data. However, the scarcity (and non-simultaneity) of the TeV coverage precludes reaching a definite conclusion on whether there is an anti-correlation of the TeV emission with the radio or with the H$\alpha$ curves. It would seem, however, that the TeV emission is rather anti-correlated with the radio flux and not with H$\alpha$, but this could not be quantitatively proven with the data at hand, especially given the caveats of dealing with non-contemporaneous observations. A simultaneous optical-TeV campaign is needed to establish the nature of the anti-correlation. The latter would be particularly useful for the forthcoming extrapolated radio maximum around October-November 2012. We acknowledge support from the grants AYA2009-07391 and SGR2009-811, as well as the Formosa Program TW2010005, by the National Natural Science Foundation of China via NSFC-10325313, 10521001, 10733010,11073021, and 10821061, the CAS key Project KJCX2-YW-T03, and 973 program 2009CB824800. YPC thanks the Natural Science Foundation of China for support via NSFC-11103020 and 11133002. NR is supported by a Ramon y Cajal Fellowship. We also acknowledge the use of the High Energy Astrophysics Science Archive Research Center (HEASARC), provided by NASA’s Goddard Space Flight Center. JL acknowledges the hospitality of IEEC-CSIC, where this research was conducted. ## References * Albert J., et al. (2006) Albert J., et al. 2006, Science, 312, 1771 * Albert J. et al. (2008) Albert J. et al. 2008, ApJ 684, 1351 * Anderhub., et al. (2009) Anderhub., et al. 2009, ApJ 693, 303 * Aleksic J., et al. (2011) Aleksic J., et al. 2011, submitted. * Acciari V. A. et al. (2008) Acciari V. A. et al. 2008, ApJ 679, 1427 * Acciari V. A. et al. (2009) Acciari V. A. et al. 2009, ApJ 700, 1034 * Acciari V. A. et al. (2010) Acciari V. A. et al. 2010, ApJ in press, arXiv:1105.0449 * Bednarek, W. (2009) Bednarek, W. 2009, MNRAS 397, 1420 * Bednarek, W. & Pabich, J. (2011) Bednarek, W. & Pabich, J. 2011, MNRAS 411, 1701 * Bignami et al. (1981) Bignami G.F.B., et al. 1981, ApJ 247, L85 * Bosch-Ramon V., & Khangulyan D. (2009) Bosch-Ramon V., & Khangulyan D. 2009, International Journal of Modern Physics D18, 347 * Chernyakova M., Neronov A., & Walter R. (2006) Chernyakova M., Neronov A., & Walter R. 2006, MNRAS 372, 1585 * Dubus, G. (2006) Dubus, G. 2006, A&A 456, 801 * Dubus, G. (2010) Dubus, G. 2010, ASP Conference Series 422, 23; Proceedings of the workshop on High Energy Phenomena in Massive Stars. * Gregory P.C. et al. (1989) Gregory P.C., Huang-Jian Xu, Bachhouse C.J., & Reid A., 1989, ApJ 339, 1054 * Gregory (2002) Gregory P. C. 2002, ApJ 575, 427 * Gregory (2002) Gregory, P. C., & Neish, C., 2002, ApJ, 580, 1133 * Greiner (2001) Greiner, J. & Rau, A., 2001, A&A, 375, 145 * Grundstrom (2006) Grundstrom, E. D. & Gies, D. R., 2006, ApJ Letters, 651, 53 * Grundstrom (2007) Grundstrom, E. D. et al., 2007, ApJ, 656, 437 * Goldoni et al. (1995) Goldoni P., & Mereghetti S. 1995, A&A 299, 751 * Harrison (2000) Harrison F. A. 2000, ApJ 528, 454 * Hanushik (1988) Hanushik R. W., Kozok J. R., & Kaizer D., A&A 189, 147 (1988) * Leahy et al. (1997) Leahy D. A., et al. 1997, ApJ 475, 823 * Li (2011) Li J., et al. 2011, ApJ 733, 89 (paper II) * Liu (2005) Liu Q. Z. & Yan J. Z. 2005, New Astronomy 11, 130 * Maraschi (1981) Maraschi L. & Treves A. 1981, MNRAS, 194, 1 * McSwain (2010) McSwain M. V., Grundstrom, E. D., Gies, D. R., & Ray, P. S., 2010, ApJ 724, 379 * Neronov et al. (2006) Neronov A., & Chernyakova M. 2006, MNRAS, 372, 1585 * Paredes (1994) Paredes J. M., et al., Astron. Astrophys. 288, 519 (1994). * Paredes et al. (2007) Paredes J. M., et al. 2007, ApJ 664, L39 * Rea (2010) Rea, N., et al. 2010, MNRAS 405, 2206 * Sidoli et al. (2006) Sidoli L., et al. 2006, A&A 459, 901 * Taylor (1996) Taylor A.R., et al. 1996, A&A 305, 817 * Torres (2010) Torres D. F., et al. 2010, ApJ Letters 719, 104 (paper I) * Torres (2011) Torres D. F., et al. 2011, ApJ in press, astro-ph (arXiv:1109.5008) * Trushkin (2010) Trushkin, S. & Nizhelskij, N. 2010, Poster at the 38th COSPAR Scientific Assembly 2010, Accretion on Compact Objects and Fast Phenomena in Multiwavelength Era (E13), Symposium E, session 13, paper number E13-0075-10 (Poster, Nr. Wed-152) * Zamanov (1999) Zamanov R. K., Martí, J., Paredes J. M. et al. 1999, A&A 351, 543 * Zamanov (2000) Zamanov R. K., & Martí J. M., 2000, A&A 358, 55 * Zamanov (2001) Zamanov R. K., Martí J. M., & Marziani P. 2001, arXiV 0110114 * Zamanov (2007) Zamanov R. K., Stoyanov K. A., & Tomov N. A. 2007, Commission 27 and 42 of the IAU, Information Bulletin on Variable Stars, number 5776	arxiv-papers	2011-11-30T07:10:06	2024-09-04T02:49:24.789950	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Jian Li, Diego F. Torres, Shu Zhang, Daniela Hadasch, Nanda Rea, G.\n Andrea Caliandro, Yupeng Chen, Jianmin Wang", "submitter": "Jian Li", "url": "https://arxiv.org/abs/1111.7068" }
1111.7084	# Failed Gamma-Ray Bursts: Thermal UV/Soft X-ray Emission Accompanied by Peculiar Afterglows M. Xu11affiliation: Department of Astronomy, Nanjing University, Nanjing 210093, China; hyf@nju.edu.cn 22affiliation: Yukawa Institute for Theoretical Physics, Oiwake-cho, Kitashirakawa, Sakyo-ku, Kyoto 606-8502, Japan 33affiliation: Department of Physics, Yunnan University, Kunming 650091, China , S. Nagataki22affiliation: Yukawa Institute for Theoretical Physics, Oiwake- cho, Kitashirakawa, Sakyo-ku, Kyoto 606-8502, Japan , Y. F. Huang11affiliation: Department of Astronomy, Nanjing University, Nanjing 210093, China; hyf@nju.edu.cn , and S.-H. Lee22affiliation: Yukawa Institute for Theoretical Physics, Oiwake-cho, Kitashirakawa, Sakyo-ku, Kyoto 606-8502, Japan ###### Abstract We show that the photospheres of “failed” Gamma-Ray Bursts (GRBs), whose bulk Lorentz factors are much lower than 100, can be outside of internal shocks. The resulting radiation from the photospheres is thermal and bright in UV/Soft X-ray band. The photospheric emission lasts for about one thousand seconds with luminosity about several times $10^{46}$ erg/s. These events can be observed by current and future satellites. It is also shown that the afterglows of failed GRBs are peculiar at the early stage, which makes it possible to distinguish failed GRBs from ordinary GRBs and beaming-induced orphan afterglows. gamma-ray bursts: general — radiation mechanisms: thermal ## 1 Intronduction Gamma-Ray Bursts (GRBs) are the most powerful explosion in the universe. The origin of prompt emission remains unresolved, owing to the fact that the prompt emission has a large explosion energy showing non-thermal spectrum with rapid time-variabilities. It is widely accepted that the prompt emission is coming from a highly- relativistic flow, because it can reduce the optical depth of the flow which makes the radiation spectrum non-thermal (Rees & Mészáros 1994). In fact, in the internal shock scenario, which is one of the most promising scenarios, the relativistic shells collide with each other after the system becomes optically thin (e.g. Piran 1994 for a review). However, it was pointed out by Mészáros & Rees (2000) that even such a relativistic flow should have a photosphere inevitably and thermal radiation should be coming from there. Since then, there have been many theoretical (Daigne & Mochkovitch 2002; Pe′er et al. 2006; Pe′er 2008; Pe′er & Ryde 2011) and observational (Ghirlanda et al. 2003; Ryde 2004, 2005; Ryde et al. 2010; Guiriec et al. 2010; Ryde et al. 2011) studies on how the thermal component contributes to the prompt emission (or the precursor). Nowadays, the internal shock model with a photosphere is frequently discussed (e.g. Toma et al. 2011; Wu & Zhang 2011). In this picture, the radius of the photosphere, $R_{\rm PS}$, is usually smaller than the radius $R_{\rm IS}$ where internal shocks are happening. The above scenario is based on the assumption that the bulk Lorentz factor of the jet is as large as 100-1000. But what happens if the bulk Lorentz factor is not so high? Theoretically, it is natural to consider such a case, because it is very hard to realize such a clean, highly-relativistic flow. Especially, in case of long GRBs, some bursts are at least coming from the death of massive stars where a lot of baryons should be surrounding the central engine (MacFadyen & Woosley 1999; Proga et al. 2003; Nagataki et al. 2007; Nagataki 2009, 2010). Thus we can expect there are a lot of “failed GRBs” that have dirty, not so highly-relativistic flows in the universe (e.g. Dermer et al. 1999; Huang et al. 2002; Paragi et al. 2010; Xu et al. 2011). Recently, Bromberg et al. (2011a, 2011b) suggested the existence of a large population of failed GRBs if the jets failed to break out of the progenitor stars in collapsar model. Qualitatively, $R_{\rm IS}$ becomes smaller if the bulk Lorentz factor of the flow is smaller, while $R_{\rm PS}$ increases with the decreasing of bulk Lorentz factor. Thus we can expect that the photosphere will become outside of the internal shock region for some lower Lorentz factors (see Fig. 1a). In such a case, $\gamma$-rays from the internal shocks cannot escape. Instead, softer thermal radiation from the photosphere followed by an afterglow will be seen. We note that such a situation was considered in case of a “successful GRB” (Rees & Mészáros 2005; Lazzati et al. 2009; Mizuta et al. 2011; Nagakura et al. 2011; Ryde & Pe′er 2009; Ryde et al. 2010, 2011). In the photospheric model, it is considered that the photospheric emission itself is the origin of the prompt emission: its spectrum is modified to non-thermal due to the heating by relativistic electrons that are produced at the internal shocks inside the photosphere (Beloborodov 2010; Vurm et al. 2011). Thus the photospheric model can explain normal successful GRBs. Here, in this study, we consider the case of “failed GRBs”. We show that the direct emission from the photosphere of a failed-GRB will be a UV/soft X-ray burst, followed by an afterglow with a peculiar spectrum at early stage. It is also shown that the afterglows of failed GRBs can be distinguished from ordinary GRB afterglows and beaming-induced orphan afterglows. Figure 1: (a) A sketch illustrating the emission regions of a failed-GRB. Note that the internal shock radius is smaller than the photosphere radius. (b) Schematic diagram showing the path of a photospheric photon escaping from the ejecta in the stellar frame. A photon emitted at point E will escape from the ejecta at point G. ## 2 Photospheric emission In this study, we consider axisymmetric jet and assume that the observer is on the axis $\overline{OD}$ in Fig. 1b. This is the extension of the 1-D formulation derived by Daigne & Mochkovitch (2002). For a baryon-rich ejecta in the stellar frame (i.e., burst source frame, and from now on all variables are defined in this frame), we assume the ejecta has been accelerated at a distance $r_{\rm acc}$ from the central engine. The mass flux of the ejecta is written as $\dot{M}=\dot{E}/\Gamma c^{2}$, where $\Gamma$ is the Lorentz factor of the ejecta and $\dot{E}$ is the energy injection rate. The energy injection begins at $t_{\rm inj}$=0, and stops at $t_{\rm inj}=t_{\rm w}$, i.e., the central engine activity lasts for a period of time $t_{\rm w}$. The ejecta can be subdivided into a series of concentric layers, where each layer has been injected at $r_{\rm acc}$ at a certain injection time $t_{\rm j}$. Each ejecta layer becomes transparent when it has expanded to a distance $r(t_{\rm inj}=t_{\rm j},t)$ at a specific time $t$, where $r(t_{\rm inj}=t_{\rm j},t)=r_{\rm acc}+\beta c(t-t_{\rm j})$ and $\beta=\sqrt{1-1/\Gamma^{2}}$. The photons emitted at $r(t_{\rm inj}=t_{\rm j},t)$ will escape from the ejecta at a time $t_{\rm esc}$ at a distance $r_{\rm esc}(t_{\rm inj}=t_{\rm j},t)$. Here $r_{\rm esc}$ and $t_{\rm esc}$ are defined where a photon emitted at time t by the shell ejected at time $t_{\rm inj}=t_{\rm j}$ escapes the outflow, i.e. reaches the first shell emitted at time $t_{\rm inj}=0$. In other words, the optical depth from $r(t_{\rm inj}=t_{\rm j},t)$ to $r_{\rm esc}(t_{\rm inj}=t_{\rm j},t)$ is unity. A geometric sketch illustrating the escape path of photons inside the ejecta is shown in Fig. 1b. A photon emitted at point E (at a distance $r(t_{\rm inj}=t_{\rm j},t)$ and propagation angle $\varphi$) will escape from the ejecta at point G, such that the optical depth from E to G is $\tau(t_{\rm j},\varphi)=\int^{r_{\rm esc}(t_{\rm inj}=t_{\rm j},t)}_{r(t_{\rm inj}=t_{\rm j},t)}d\tau(r),$ (1) where $d\tau(r)$ can be estimated as (Abramowicz et al. 1991; Daigne & Mochkovitch 2002; Pe′er 2008) $d\tau(r)=\frac{\kappa\dot{M}(1-\beta{\rm cos}\varphi)}{4\pi r^{2}}dr.$ (2) We define the photospheric radius $R_{\rm PS}(t_{\rm j})=r(t_{\rm j},t)$ at which $\tau$ is equal to unity. In light of Fig. 1b, we choose a cylindrical coordinate system (Pe′er 2008) with the central point $O$ being the stellar center, and the observer located along the +z-direction (defined by the direction of $\overline{OD}$). Photons are emitted at a perpendicular distance $r_{\rm min}=r(t_{\rm j},t){\rm sin}\varphi$ from the z-axis and a distance $z_{\rm min}=r(t_{\rm j},t){\rm cos}\varphi$ along the z-axis from point $O$. The escape radius can be estimated from the triangle $OEG$, i.e., $r_{\rm esc}(t_{\rm inj}=t_{\rm j},t)=r(t_{\rm j},t)+\beta c(t_{\rm j}-0)+\beta c(t_{\rm esc}-t)=\\{r(t_{\rm j},t)^{2}+[c(t_{\rm esc}-t)]^{2}-2r(t_{\rm j},t)c(t_{\rm esc}-t){\rm cos}(\pi-\varphi)\\}^{\frac{1}{2}}$, The integration for the optical depth can be conveniently rewritten in cylindrical coordinates as the following $\tau(t_{\rm j},\varphi)=\int^{z_{\rm max}}_{z_{\rm min}}\frac{\kappa\dot{M}(1-\beta{\rm cos}\varphi)}{4\pi r^{2}}\frac{dr}{dz}dz,$ (3) where $r=\sqrt{z^{2}+r_{\rm min}^{2}}$, $dz/dr=z/\sqrt{z^{2}+r_{\rm min}^{2}}$ and $z_{\rm max}=\sqrt{r_{\rm esc}^{2}-r_{\rm min}^{2}}$. The photospheric radius ($R_{\rm PS}$), which depends on the propagation angle ($\varphi$), can be found readily by defining $\tau=1$. For a relativistic ejecta with Lorentz factor $\Gamma$, the arrival time of photons emitted at the photosphere in the observer frame are delayed relative to that measured in the stellar frame $t_{\rm obs}=t-R_{\rm PS}{\rm cos}\varphi/c=t_{\rm j}+(1-\beta{\rm cos}\varphi)R_{\rm PS}/\beta c,$ (4) i.e., the observer time is a function of injection time and propagation angle. In this equation, we have neglected the effect of the acceleration radius ($r_{\rm acc}$) because it is much smaller than the photosphere radius. The evolution of photospheric radius with propagation angle is shown in Fig. 2b. The parameters are taken as $\Gamma=10$, $\dot{E}=10^{51}$erg/s and $t_{\rm w}=2000$s. The solid curve shows its evolution at observer time $t_{\rm obs}=100$ s and the dashed curve at $t_{\rm obs}=2000$ s. From this panel, we can see that in the observer frame, the photospheric radius decreases with propagation angle. Figure 2: Evolution of the photospheric radius and temperature with respect to the observer time and photon propagation angle. The parameters are taken as $\Gamma=10$, $\dot{E}=10^{51}$erg/s and $t_{\rm w}=2000$s. (a) The photospheric radius vs. the observer time. The solid curve is plot for photons propagating along the expansion direction of the ejecta ($\varphi=0$). The dashed curve corresponds to $\varphi=0.1$ rad. (b) The photospheric radius vs. the photon propagation angle. The solid and dashed curves correspond to the observer time of 100 s and 2000 s respectively. (c) The photospheric temperature vs. observer time. The solid and dashed curves correspond to photon propagation angles of 0 and 0.1 rad, respectively. (d) The photospheric temperature vs. the photon propagation angle. The solid and dashed curves are for $t_{\rm obs}$=100 s and 2000 s, respectively. We also show the evolution of photospheric radius with observer time in Fig. 2a. The solid, dashed curves present the cases for $\varphi=0$ and $\varphi=0.1$ rad respectively. As is shown in this panel, the photospheric radius increases with time, and the duration of the photospheric emission is prolonged at larger propagation angle. The end points of the two curves indicate the observer time when the last layer of the ejecta becomes transparent. According to the fireball model, the temperature of a layer at its photospheric radius is given by (Piran 1999) $kT_{\rm PS}=\frac{D}{\Gamma}kT^{0}\big{(}\frac{R_{\rm PS}}{r_{\rm acc}}\big{)}^{-2/3},$ (5) where $D=[\Gamma(1-\beta{\rm cos}\varphi)]^{-1}$ is the Doppler factor, $r_{\rm acc}$ is the saturation radius and $T^{0}$ is its blackbody temperature. In Fig. 2c and Fig. 2d, we show the evolution of $T_{\rm PS}$ with respect to the propagation angle and observer time respectively. From the two panels, we find that the photosperic temperature decreases with the propagation angle and time in the observer frame. The evolution of the injection time with respect to the observer time and the propagation angle is shown in Fig. 3a and 3b, respectively. For the “standard” parameter set ($\Gamma=10$, $\dot{E}=10^{51}$ erg/s, $t_{\rm w}$=2000 s), $t_{\rm inj}(t_{\rm obs})$ is mildly smaller than $t_{\rm obs}$ for different $\varphi$, and $t_{\rm inj}(\varphi)$ is almost independent of $\varphi$ for different $t_{\rm obs}$. In Fig. 3c and Fig. 3d, we also show the evolution of the escaping radius $r_{\rm esc}$ with respect to the observer time and the propagation angle, respectively. The relations of $t_{\rm obs}$ and $\varphi$ with respect to $r_{\rm esc}$ are similar to that of the photospheric radius $R_{\rm PS}$. Figure 3: Evolution of the injection time and escaping radius with respect to the observer time and photon propagation angle. The parameters are the same as those in Fig. 2. (a) The injection time vs. the observer time. The solid curve is plot for photons propagating along the expansion direction of the ejecta ($\varphi=0$). The dashed curve is the evolution of the injection time for $\varphi=0.1$ rad. (b) The injection time vs. the photon propagation angle. The solid and dashed curves are observed at 100s and 2000s respectively. (c) The escaping radius vs. the observer time. The solid and dashed curves are plot with a propagation angle of 0 and 0.1 rad, respectively. (d) The escaping radius vs. the photon propagation angle. The solid and dashed curves are for $t_{\rm obs}$=100 s and 2000 s, respectively. As for a jet with half-opening angle $\theta=0.1$ rad, constant Lorentz factor $\Gamma=10$, and energy injection from $t_{\rm inj}=0$ to $t_{\rm inj}=t_{\rm w}=2000$ s with energy injection rate per solid angle $\dot{E}/4\pi=10^{51}/4\pi$ erg/s, we can estimate that $r_{\rm acc}\simeq 9\times 10^{7}$cm and $kT^{0}\simeq 0.41$ MeV for a fireball model (Piran 1999; $\rm M\acute{e}sz\acute{a}ros~{}\&~{}Rees$ 2000; Daigne & Mochkovitch 2002). If the line-of-sight is along the jet central axis, we can find that the photospheric radius is about $1.1\times 10^{14}$ cm when the last layer becomes transparent, the observer’s time can be calculated from Eq. 4, which is found to be about 2020 s and corresponds to the end point of the solid curve in Fig. 2a. The observed luminosity of photospheric emission can be determined by integrating over the surface of the photosphere $L=\int_{0}^{\theta}\sigma T_{\rm PS}^{4}dS{\rm cos}\vartheta$ (6) where $\vartheta$ is the angle between the tangential direction of the photosphere surface and the line-of-sight when the propagation angle is $\varphi$. $dS{\rm cos}\vartheta=2\pi R_{\rm PS}(t_{\rm obs},\varphi){\rm sin}\varphi[R_{\rm PS}(t_{\rm obs},\varphi+d\varphi){\rm sin}(\varphi+d\varphi)-R_{\rm PS}(t_{\rm obs},\varphi){\rm sin}\varphi]$ is the photospheric surface area from propagation angle $\varphi$ to $\varphi+d\varphi$. The evolution of the photospheric luminosity with observer time is shown by the solid curve in Fig. 4. There is a break in the light curve at about $t_{\rm obs}\simeq$ 2020 s, which is attributed to the stop of energy injection by the central engine and when the last layer of the ejecta became transparent as the photons propagate along the line-of-sight. Afterwards, only photospheric emission at high latitude (large propagation angles) contributes to the observed luminosity. The photospheric emission ceases when the last layer with propagation angle $\varphi=\theta=0.1$ become transparent, which is about 2050 s in the observer frame. Figure 4: Evolution of the photospheric luminosity (solid curve) and effective temperature (dashed curve) with observer time for a jet with parameters of $\theta=0.1$ rad, $\Gamma=10$, $\dot{E}=10^{51}$erg/s and $t_{\rm w}$=2000 s. We can also define an effective temperature for the photosphere $T_{\rm eff}=\frac{\int_{0}^{\theta}TdL}{\int_{0}^{\theta}dL}.$ (7) This effective temperature is shown as a dashed curve in Fig. 4. We can find that the photospheric emission of a failed GRB is presented as a short soft X-ray burst and then becomes a UV burst which lasts for about several thousand seconds. We also investigated the parameter effect on the photospheric emission, which are shown in Fig. 5 and Fig. 6. All the curves are derived when the last layer of ejecta along the line-of-sight became transparent , i.e., $\varphi=0$ and $t_{\rm j}=t_{\rm w}$. As is shown in Fig. 5a, the photospheric radii are decreasing with the increase of Lorentz factor for different sets of parameters. The solid curve corresponds to the standard parameters ($\dot{E}=10^{51}$erg/s, $t_{\rm w}=2000$ s), while the parameters for the dashed curve and the dotted curve are $\dot{E}=10^{49}$erg/s, $t_{\rm w}=2000$ s and $\dot{E}=10^{51}$erg/s, $t_{\rm w}=200$ s, respectively. A lower energy injection rate and shorter injection time will decease the radius of the photosphere. Fig. 5b shows the evolution of the observer time with Lorentz factor. The parameters for each curve are the same as Fig. 5a. This time period can be interpreted as the duration of the photospheric emission. From this panel, we can find that the duration of photospheric emission decreases with an increase of Lorentz factor. Lower energy injection rate and shorter injection time will decease the duration of the photospheric emission. Figure 5: Parameter dependence of the photospheric emission. All curves are obtained when the last layer of the ejecta became transparent along the line- of-sight, i.e., $\varphi=0$ and $t_{\rm j}=t_{\rm w}$. (a) Parameter dependence of the photospheric radius. The solid curve is derived using the standard parameters, i.e., $\dot{E}=10^{51}$erg/s and $t_{\rm w}$=2000 s; the dashed curve is for $\dot{E}=10^{49}$erg/s and $t_{\rm w}$=2000 s; and the dotted curve is for $\dot{E}=10^{51}$erg/s and $t_{\rm w}$=200 s. The evolution of the internal shock radii $R_{\rm IS}$ for a variability timescale of $\delta t$=1 s, 0.33 s and 0.1 s are shown and marked correspondingly. (b) Parameter dependence of the observer time. The identities of the curves are the same as in (a). In Fig. 6, we show the parameter effect on the photospheric luminosity and effective temprature. The parameters of each curve are the same as Fig. 5. From Fig. 6a, we can find that the luminosity of the photospheric emission are low in both high and low Lorentz factor. Lower energy injection rate results in lower luminosity. As is shown in Fig. 6b, the effective temperatures are increasing with the increase of Lorentz factor. Lower energy injection rate and shorter injection time will decease the radii of the photosphere and hence results in a higher effective temperature. Figure 6: Parameter dependence of the photospheric luminosity (a) and the effective temperature (b) in the Lorentz factor space. All curves are obtained when the last layer of ejecta became transparent along the line-of-sight, i.e., $\varphi=0$ and $t_{\rm j}=t_{\rm w}$. The identities of the curves are the same as in Figure 5. As for a jet with half-opening angle of about 0.1 rad, Lorentz factor $\Gamma=2-20$, energy injection rate $\dot{E}=10^{49-51}$erg/s and injection time $t_{\rm w}$=200-2000s, from Fig. 5 and 6, we can conclude that the prompt emission for a failed GRB is thermal soft X-ray or UV photospheric emission, there will be no significant non-thermal gamma-ray emission. The photospheric luminosity is about $10^{46}$erg/s and last for about one thousand seconds. Note that the photospheric luminosity is far lower than the energy injection power, most of of the energy is re-converted into the ejecta’s kinetic energy. From Fig. 5a, we find that the radius of the photosphere is about $10^{14}$ cm, which is larger then the prediction for the internal shock’s radius, i.e., $R_{\rm IS}\simeq\Gamma^{2}c\delta t\simeq 10^{12}$ cm (Mészáros 2006), where $\delta t\sim 0.33$ s here is the variability timescale of the prompt emission. The evolution of $R_{\rm IS}$ with respect to $\Gamma$ for $\delta t$=1 s, 0.33 s and 0.1 s are shown and marked in Fig. 5a correspondingly. This radius is consistent with Fig. 1a. This thermal radiation will be in the UV or soft X-ray band. The lower band of Swift-XRT ($0.2-10$ keV) may cover this energy range and it is sensitive enough to detect such a photospheric emission component (Gehrels et al. 2004). MAXI-SSC monitors all-sky in the energy range of $0.5-10$ keV and also has a chance to detect such events (Matsuoka et al. 1997). Future UV satellites may be also have the capability to detect these events, such as TAUVEX (wavelength range 120nm-350nm) (Safonova et al. 2008). ## 3 Afterglow emission As the outflow expands outward, it will collide with the surrounding medium and afterglow will be produced. The dynamical evolution of a relativistic jet in interstellar medium has been studied by Huang et al. (1999). Their codes can be used in both ultra-relativistic and non-relativistic phases. In our model, we consider a jet with the bulk Lorentz factor $\Gamma=10$, the half-opening angle $\theta=0.1$ and the isotropic energy $E=10^{50}$ erg. The jet expands laterally at the comoving sound speed and collides with a medium whose number density is $n_{\rm ISM}=1$ cm-3. We also assume typical values for some other parameters of the jet, i.e., the electron energy fraction $\epsilon_{e}=0.1$, the magnetic energy fraction $\epsilon_{B}=0.01$ and the power-law index of the energy distribution function of electrons $p=2.5$. Multiband afterglow emission is expected from synchrotron radiation of relativistic electrons. Using this exquisite model, we numerically calculated the afterglow light curves and spectra with line of sight parallel to the jet axis. We assume a redshift $z=1$ and a standard cosmology with $\Omega_{M}=0.27$, $\Omega_{\Lambda}=0.73$ and with the Hubble constant of $H_{0}=71$ km s-1 Mpc-1. Our results for the afterglow spectra of failed GRBs are shown in Fig. 7. The thick and thin solid curves are the spectra observed at $10^{3}$ s and $10^{6}$ s respectively. In this figure, we can find that the spectrum becomes softer with the elapse of the observational time. At the early stage ($10^{3}$ s), the peak flux appears at about $5\times 10^{12}$ Hz, i.e., in the IR band. Both the peak flux and peak frequency decrease with time. At late time ($10^{6}$ s), the peak flux is more than one magnitude less than that in the early stage. The peak frequency decreases to about $10^{9}$ Hz at late stage. It is in the radio band and more than three magnitudes less than the peak frequency at the early stage. Figure 7: Evolution of the afterglow spectra for the three types of GRBs. The solid, dashed and dotted curves are spectra of a failed GRB afterglow, an ordinary GRB afterglow and a beaming-induced orphan afterglow respectively. The thick and thin curves are the spectra observed at $10^{3}$ s and $10^{6}$ s respectively. For comparison, we also show the afterglow spectra of an ordinary GRB in Fig. 7. Here we choose the same parameters as the failed GRB except for a much larger bulk Lorentz factor ($\Gamma=300$). The dashed curves show the afterglow spectra of this GRB with observing angle $\theta_{\rm obs}=0$ (the line of sight is parallel to the jet axis). As is shown in Fig. 7, the spectra of the failed GRB and the ordinary GRB are similar at the late stage (thin solid and thin dashed curves) because their energies and Lorentz factors are both similar at this moment. But at early stage, they are very different (thick solid and thick dashed curves) due to their very different initial Lorentz factors and the corresponding minimum Lorentz factors of electrons (Sari et al. 1998). The peak frequency of the GRB afterglow is much larger than that of the failed GRB afterglow. In Fig. 7, we also show the spectra of a beaming-induced orphan afterglow (afterglow from an ordinary highly collimated GRB outflow, but with the observing angle larger than the jet half-opening angle so that no prompt gamma-rays can be observed in the main burst phase, Rhoads 1997; Huang et al. 2002). Here we assume the same parameters as the ordinary GRB except $\theta_{\rm obs}=0.125$. The early and late spectra of this orphan afterglow are shown in Fig. 7 with thick dotted curve and thin dotted curve. From this figure, we can find the spectra of beaming-induced orphan afterglow are similar to that of the ordinary GRB. Although it is hard to distinguish a beaming-induced orphan afterglow from a failed GRB afterglow through their afterglow light curves (Huang et al. 2002), they can be potentially distinguished from their spectra at the early stages. Their spectra of early afterglows are very different: the peak frequency of a failed GRB afterglow is far lower than that of a beaming-induced orphan afterglow. Another way to distinguish them is through their early light curves. Early afterglow of a beaming-induced orphan afterglow will show a rebrightening while failed GRB will not (Huang et al. 1999, 2002; Xu & Huang 2010). ## 4 Conclusion and Discussions The analysis in this paper shows that the emission of ejecta with low Lorentz factors is very different from that expected from ejecta with high Lorentz factors. Prompt emission of a GRB is non-thermal and bright in the gamma-ray band. For a failed GRB, however, the emission originates from the photosphere with a thermal spectrum, and is bright in the UV or soft X-ray band instead of gamma-rays. This photospheric emission lasts for about a thousand seconds with a luminosity about several times $10^{46}$ erg/s. Since the photospheric emission manifests as a UV or soft X-ray transient, it can be detected by some current and future satellites, such as Swift-XRT, MAXI-SSC and TAUVEX etc. On 2008 January 9, Swift-XRT discovered a peculiar X-ray transient 080109 in NGC 2770 (Berger & Soderberg 2008; Page et al. 2008). No gamma-ray emission was detected. This X-ray transient reached its peak at about $60s$ and lasted for about $600$ s. Its spectrum can be fitted with an absorbed double blackbody model with temperatures about $0.36$ keV and $1.24$ keV respectively (Li 2008). This transient may be a candidate of photospheric emission from a failed GRB. Meanwhile, some unidentified X-ray transients have been detected by MAXI during its one-year monitoring (Nakajima et al. 2009; Suzuki et al. 2010). These transients generally showed an absorbed blackbody spectrum and lasted for tens of seconds. It is possible that some of them are photospheric emission from failed GRBs. If we extend the injection time to about $10^{5}$ s and the jet half-opening to about 0.4 rad in our model, we find that the photospheric radius is about $10^{15}$ cm and the effective temperature is deceased to lower than 1 eV, i.e., there will be an optical burst. This kind of optical burst will last for about several thousand seconds with a luminosity about $10^{42}$ erg/s, which may be detected by the Hyper-Suprime Camera of the $Subaru$ telescope in the future. In this work, we have assumed that the prompt emission is thermal radiation coming from the photosphere where the optical depth is unity. Due to the low density of GRB jets, however, it has been pointed out that the last-scattering positions of the observed photons may not simply coincide with the photosphere, but instead possess a finite distribution around it (e.g. Pe′er et al. 2006; Pe′er 2008; Beloborodov 2010; Pe′er & Ryde 2011). This stochastic effect can lead to differentiation of the observed spectrum from a thermal one of purely photospheric origin. Such mechanism can work even in failed GRBs, and it is our future work to study how the spectrum will be reshaped using Monte-Carlo calculations. We are planning to investigate this effect in the context of failed GRBs as a next step of our study. From the comparison of afterglow emissions from failed and ordinary GRBs, while we find it challenging to distinguish them at their late stage of evolution, their spectra at the early stage are profoundly different. We conclude that it is possible to identify failed GRBs by observing their afterglow emission in the early stage. The typical frequency at peak flux in the afterglow phase for failed GRBs is much lower than that for ordinary GRBs (or beaming-induced orphan GRBs). We can thus define a hardness ratio, for instance, as the flux contrast between $10^{12}$ Hz and $10^{14}$ Hz at an observed time of $1000$ s, i.e., $f_{\rm 1ks}\equiv F_{10^{12}{\rm Hz}}/F_{10^{14}{\rm Hz}}$. If $f_{\rm 1ks}>1$, then it is quite likely that the emission is coming from a failed GRB. If $f_{\rm 1ks}<1$, then it would be more likely to come from an ordinary GRB afterglow or a beaming-induced orphan afterglow. In addition, at the early afterglow stage, a rebrightening phase will be present in the case of a beaming-induced orphan GRB, while it is not expected for ordinary or failed GRBs. Therefore, the afterglows of failed GRBs can be distinguished from both ordinary GRB afterglows and beaming-induced orphan afterglows through observations at the early stages. We thank the anonymous referee for many of the useful suggestions and comments. We also would like to thank P. Mészáros, T. Piran and J. Aoi for helpful discussions. This work was supported by the National Natural Science Foundation of China (Grant No. 11033002), the National Basic Research Program of China (973 Program, Grant No. 2009CB824800) and the Grant-in-Aid for the ’Global COE Bilateral International Exchange Program’ of Japan, Grant-in-Aid for Scientific Research on Priority Areas No. 19047004 and Scientific Research on Innovative Areas No. 21105509 by Ministry of Education, Culture, Sports, Science and Technology (MEXT), Grant-in-Aid for Scientific Research (S) No. 19104006 and Scientific Research (C) No. 21540404 by Japan Society for the Promotion of Science (JSPS). ## References * (1) Abramowicz, M. A., Novikov, I. D., & Paczyński, B. 1991, ApJ, 369, 175 * (2) Amati, L., et al. 2002, A&A, 390, 81 * (3) Berger, E., & Soderberg, A. M. 2008, GCN Circ. 7159 * (4) Beloborodov, A. M. 2010, MNRAS, 407, 1033 * (5) Bromberg, O., Nakar, E., & Piran, T. 2011a, ApJ, 739, L55 * (6) Bromberg, O., Nakar, E., Piran, T., & Sari, R. 2011b, arXiv:1111.2990 * (7) Daigne, F., & Mochkovitch, R. 2002, MNRAS, 336, 1271 * (8) Dermer, C. D., Böttcher, M., & Chiang, J. 1999, ApJ, 515, L49 * (9) Gehrels, N., et al. 2004, ApJ, 611, 1005 * (10) Ghirlanda, G., Celotti, A., & Ghisellini, G. 2003, A&A, 406, 879 * (11) Guiriec, S., et al. 2011, ApJ, 727, L33 * (12) Huang, Y. F., Dai, Z. G., & Lu, T. 1999, MNRAS, 309, 513 * (13) Huang, Y. F., Dai, Z. G., & Lu, T. 2002, MNRAS, 332, 735 * (14) Huang, Y. F., Gou, L. J., Dai, Z. G., & Lu, T. 2000, ApJ, 543, 90 * (15) Lazzati, D., Morsony, B.J., Begelman, M.C. 2009, ApJ, 700, L47 * (16) Li, L. X. 2008, MNRAS, 388, 603 * (17) Liang, E. W., et al. 2010, ApJ, 725, 2209 * (18) MacFadyen, A. I., & Woosley, S. E. 1999, ApJ, 524, 262 * (19) Matsuoka, M., et al. 1997, SPIE, 3114, 414 * (20) Mészáros, P. 2006, Rep. Prog. Phys., 69, 2259 * (21) Mészáros, P., Ramirez-Ruiz, E., Rees, M. J., & Zhang, B, 2002, ApJ, 578, 812 * (22) Mészáros, P., & Rees, M. J. 2000, ApJ, 530, 292 * (23) Mizuta, A., Nagataki, S. Aoi, J. 2011, ApJ, 732,26 * (24) Nagakura, H., Ito, H., Kiuchi, K., Yamada, S. ApJ, 731,80 * (25) Nagataki, S. 2009, ApJ, 704, 937 * (26) Nagataki, S. 2010, arXiv:1010.4964 * (27) Nagataki, S., et al. 2007, ApJ, 659, 512 * (28) Nakajima, M., et al. 2009, ATel 2321 * (29) Paczyński, B. 1990, ApJ, 363, 218 * (30) Page, K. L., et al. 2008, GCN Report 110.1 * (31) Paragi, Z., et al. 2010, Nature, 463, 516 * (32) Pe′er, A., Mészáros, P., & Rees, M. J. 2006, ApJ, 642, 995 * (33) Pe′er, A. 2008, ApJ, 682, 463 * (34) Pe′er, A., Ryde, F. 2011, ApJ, 732, 49 * (35) Piran, T., 1999, Physics Reports, 314, 575 * (36) Piran, T. 2004, Rev. Mod. Phys., 76, 1143 * (37) Proga, D., et al. 2003, ApJ, 599, L5 * (38) Rees, M. J., & Mészáros, P., 1994, ApJ, 430, L93 * (39) Rees, M. J., & Mészáros, P. 2005, ApJ, 628, 847 * (40) Rhoads, J. 1997, ApJ, 487, L1 * (41) Rybicki, G., & Lightman, A., 1979, Radiative Processes in Astrophysics (Wiley: New York) * (42) Ryde, F. 2004, ApJ, 614, 827 * (43) Ryde, F. 2005, ApJ, 625, L95 * (44) Ryde, F., & Pe′er, A. 2009, ApJ, 702, 1211 * (45) Ryde, F., et al. 2010, ApJ, 709, L172 * (46) Ryde, F., et al. 2011, MNRAS, 415, 3693 * (47) Safonova, M., Sivaram, C., & Murthy, J. 2008, Astrophys. Space Sci., 318, 1 * (48) Sari, R., Piran, T., & Narayan, R. 1998, ApJ, 497, L17 * (49) Suzuki, M., et al. 2010, ATel 2415 * (50) Toma, K., Wu, X-F., & Mészáros, P. 2011, MNRAS, 415, 1663 * (51) Vurm, I., Beloborodov, A. & Poutanen, J. 2011, ApJ, 738, 77 * (52) Wu, X. F., & Zhang, B. 2011, to be submitted * (53) Xu, M., & Huang, Y. F. 2010, A&A, 523, 5 * (54) Xu, M., Nagataki, S., & Huang, Y. F. 2011, ApJ, 735, 3	arxiv-papers	2011-11-30T08:54:53	2024-09-04T02:49:24.797762	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "M. Xu, S. Nagataki, Y.-F. Huang, and S.-H. Lee", "submitter": "Ming Xu", "url": "https://arxiv.org/abs/1111.7084" }
1111.7114	# Novel Fermi Liquid of 2D Polar Molecules Zhen-Kai Lu1,2,3 and G. V. Shlyapnikov2,4 1Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-Straße 1, 85748 Garching, Germany 2Laboratoire de Physique Théorique et Modèles Statistiques, CNRS and Université Paris Sud, UMR8626, 91405 Orsay, France 3 Département de Physique, École Normale Supérieure, 75005, Paris, France 4Van der Waals-Zeeman Institute, University of Amsterdam, Science Park 904, 1098 XH Amsterdam, The Netherlands ###### Abstract We study Fermi liquid properties of a weakly interacting 2D gas of single- component fermionic polar molecules with dipole moments $d$ oriented perpendicularly to the plane of their translational motion. This geometry allows the minimization of inelastic losses due to chemical reactions for reactive molecules and, at the same time, provides a possibility of a clear description of many-body (beyond mean field) effects. The long-range character of the dipole-dipole repulsive interaction between the molecules, which scales as $1/r^{3}$ at large distances $r$, makes the problem drastically different from the well-known problem of the two-species Fermi gas with repulsive contact interspecies interaction. We solve the low-energy scattering problem and develop a many-body perturbation theory beyond the mean field. The theory relies on the presence of a small parameter $k_{F}r_{}$, where $k_{F}$ is the Fermi momentum, and $r_{}=md^{2}/\hbar^{2}$ is the dipole-dipole length, with $m$ being the molecule mass. We obtain thermodynamic quantities as a series of expansion up to the second order in $k_{F}r_{}$ and argue that many-body corrections to the ground-state energy can be identified in experiments with ultracold molecules, like it has been recently done for ultracold fermionic atoms. Moreover, we show that only many-body effects provide the existence of zero sound and calculate the sound velocity. ## I Introduction The recent breakthrough in creating ultracold diatomic polar molecules in the ground ro-vibrational state Ni ; Deiglmayr2008 ; Inouye ; Nagerl and cooling them towards quantum degeneracy Ni has opened fascinating prospects for the observation of novel quantum phases Baranov2008 ; Lahaye2009 ; Pupillo2008 ; Wang2006 ; Buchler2007 ; Taylor ; Cooper2009 ; Pikovski2010 ; Lutchyn ; Potter ; Barbara ; Sun ; Miyakawa ; Gora ; Ronen ; Parish ; Babadi ; Baranov . A serious problem in this direction is related to ultracold chemical reactions, such as KRb+KRb$\Rightarrow$K2+Rb2 observed in the JILA experiments with KRb molecules Jin ; Jin2 , which places severe limitations on the achievable density in three-dimensional samples. In order to suppress chemical reactions and perform evaporative cooling, it has been proposed to induce a strong dipole-dipole repulsion between the molecules by confining them to a (quasi)two-dimensional (2D) geometry and orienting their dipole moments (by a strong electric field) perpendicularly to the plane of the 2D translational motion Bohn1 ; Baranov1 . The suppression of chemical reactions by nearly two orders of magnitude in the quasi2D geometry has been demonstrated in the recent JILA experiment Ye . At the same time, not all polar molecules of alkali atoms, on which experimental efforts are presently focused, may undergo these chemical reactions Jeremy . In particular, they are energetically unfavorable for RbCs bosonic molecules obtained in Innsbruck Nagerl , or for NaK and KCs molecules which are now being actively studied by several experimental groups (see, e.g. MITZ ). It is thus expected that future experimental studies of many-body physics will deal with non-reactive polar molecules or with molecules strongly confined to the 2D regime. Therefore, the 2D system of fermionic polar molecules attracts a great deal of interest, in particular when they are in the same internal state. Various aspects have been discussed regarding this system in literature, in particular the emergence and beyond mean field description of the topological $p_{x}+ip_{y}$ phase for microwave-dressed polar molecules Cooper2009 ; Gora , interlayer superfluids in bilayer and multilayer systems Pikovski2010 ; Ronen ; Potter ; Zinner , the emergence of density-wave phases for tilted dipoles Sun ; Miyakawa ; Parish ; Babadi ; Baranov . The case of superfluid pairing for tilted dipoles in the quasi2D geometry beyond the simple BCS approach has been discussed in Ref. Baranov . The Fermi liquid behavior of this system has been addressed by using the Fourier transform of the dipole-dipole interaction potential Das1 ; Das2 ; Miyakawa ; Taylor ; Pu ; Das3 ; Baranov and then employing various types of mean field approaches, such as the Hartree-Fock approximation Miyakawa or variational approaches Taylor ; Pu . It should be noted, however, that the short-range physics can become important for the interaction between such polar molecules, since in combination with the long- range behavior it introduces a peculiar momentum dependence of the scattering amplitude Gora . On the other hand, there is a subtle question of many-body (beyond mean field) effects in the Fermi liquid behavior of 2D polar molecules, and it can be examined in ultracold molecule experiments. For the case of atomic fermions, a milestone in this direction is the recent result at ENS, where the experiment demonstrated the many-body correction to the ground state energy of a short- range interacting two-species fermionic dilute gas Salomon1 ; Salomon2 . This correction was originally calculated by Huang, Lee, and Yang Huang ; Lee by using a rather tedious procedure. Later, it was found by Abrikosov and Khalatnikov Abr in an elegant way based on the Landau Fermi liquid theory Landau . In this paper, we study a weakly interacting 2D gas of fermionic polar molecules which are all in the same internal state. It is assumed that each molecule has an average dipole moment $d$ which is perpendicular to the plane of the translational motion, so that the molecule-molecule interaction at large separations $r$ is $U(r)=\frac{d^{2}}{r^{3}}=\frac{\hbar^{2}r_{}}{mr^{3}},$ (1) where $r_{}=md^{2}/\hbar^{2}$ is the characteristic dipole-dipole distance, and $m$ is the molecule mass. The value of $d$ depends on the external electric field. At ultralow temperatures that are much smaller than the Fermi energy, characteristic momenta of particles are of the order of the Fermi momentum $k_{F}$, and the criterion of the weakly interacting regime is: $k_{F}r_{}\ll 1.$ (2) As a consequence, the Fermi liquid properties of this system, such as the ground state energy, compressibility, effective mass, can be written as a series of expansion in the small parameter $k_{F}r_{}$. We obtain explicit expressions of these quantities up to the second order in $k_{F}r_{}$, which requires us to reveal the role of the short-range physics in the scattering properties and develop a theory beyond the mean field. Our analysis shows that only many-body (beyond mean field) effects provide the existence of undamped zero sound in the collisionless regime. The paper is organized as follows. In Section II we analyze the low-energy 2D scattering of the polar molecules due to the dipole-dipole interaction. We obtain the scattering amplitude for all scattering channels with odd orbital angular momenta. The leading part of the amplitude comes from the so-called anomalous scattering, that is the scattering related to the interaction between particles at distances of the order of their de Broglie wavelength. This part of the amplitude corresponds to the first Born approximation and, due to the long-range $1/r^{3}$ character of the dipole-dipole interaction, it is proportional to the relative momentum $k$ of colliding particles for any orbital angular momentum $l$. We then take into account the second Born correction, which gives a contribution proportional to $k^{2}$. For the $p$-wave scattering channel it is necessary to include the short-range contribution, which together with the second Born correction leads to the term behaving as $k^{2}\ln{k}$. In Section III, after reviewing the Landau Fermi liquid theory for 2D systems, we specify two-body (mean field) and many-body (beyond mean field) contributions to the ground state energy for 2D fermionic polar molecules in the weakly interacting regime. We then calculate the interaction function of quasiparticles on the Fermi surface and, following the idea of Abrikosov-Khalatnikov Abr , obtain the compressibility, ground state energy, and effective mass of quasiparticles. In Section IV we calculate the zero sound velocity and stress that the many-body contribution to the interaction function of quasiparticles is necessary for finding the undamped zero sound. We conclude in Section V, emphasizing that the 2D gas of fermionic polar molecules represents a novel Fermi liquid, which is promising for revealing many-body effects. Moreover, we show that with present facilities it is feasible to obtain this system in both collisionless and hydrodynamic regimes. ## II Low-energy scattering of fermionic polar molecules in 2D ### II.1 General relations We first discuss low-energy two-body scattering of identical fermionic polar molecules undergoing the 2D translational motion and interacting with each other at large separations via the potential $U(r)$ (1). The term low-energy means that their momenta satisfy the inequality $kr_{}\ll 1$. In order to develop many-body theory for a weakly interacting gas of such molecules, we need to know the off-shell scattering amplitude defined as $f(\mathbf{k}^{\prime},\mathbf{k})=\int\exp(-i\mathbf{k}^{\prime}\mathbf{r})U(r)\tilde{\psi}_{\mathbf{k}}(\mathbf{r})d^{2}\mathbf{r},$ (3) where $\tilde{\psi}_{\mathbf{k}}(\mathbf{r})$ is the true wavefunction of the relative motion with momentum $\mathbf{k}$. It is governed by the Schrödinger equation $\left(-\frac{\hbar^{2}}{m}\Delta+U(r)\right)\tilde{\psi}_{\mathbf{k}}(\mathbf{r})=\frac{\hbar^{2}k^{2}}{m}\tilde{\psi}_{\mathbf{k}}(\mathbf{r}).$ (4) For $\|{\bf k}^{\prime}\|=\|{\bf k}\|$ we have the on-shell amplitude which enters an asymptotic expression for $\psi_{\mathbf{k}}(\mathbf{r})$ at $r\rightarrow\infty$ Lan2 ; Gora : $\tilde{\psi}_{\mathbf{k}}(\mathbf{r})=\exp(i{\bf kr})-\frac{m}{\hbar^{2}}\sqrt{\frac{i}{8\pi kr}}f(k,\varphi)\exp(ikr),$ (5) with $\varphi$ being the scattering angle, i.e. the angle between the vectors ${\bf k}^{\prime}$ and ${\bf k}$. The wavefunction $\tilde{\psi}_{{\bf k}}({\bf r})$ can be represented as a sum of partial waves $\tilde{\psi}_{l}(k,r)$ corresponding to the motion with a given value of the orbital angular momentum $l$: $\tilde{\psi}_{{\bf k}}({\bf r})=\sum_{l=-\infty}^{\infty}\tilde{\psi}_{l}(k,r)i^{l}\exp(il\varphi).$ (6) Using the relation $\exp(i{\bf kr})=\sum_{l=-\infty}^{\infty}i^{l}J_{l}(kr)\exp[il(\varphi_{k}-\varphi_{r})],$ (7) where $J_{l}$ is the Bessel function, and $\varphi_{k}$ and $\varphi_{r}$ are the angles of the vectors ${\bf k}$ and ${\bf r}$ with respect to the quantization axis. Eqs. (6) and (7) allow one to express the scattering amplitude as a sum of partial-wave contributions: $f(\mathbf{k}^{\prime},\mathbf{k})=\sum_{l=-\infty}^{\infty}\exp(il\varphi)f_{l}(k^{\prime},k),$ (8) with the off-shell $l$-wave amplitude given by $f_{l}(k^{\prime},k)=\int_{0}^{\infty}J_{l}(k^{\prime}r)U(r)\tilde{\psi}_{l}(k,r)2\pi rdr.$ (9) Similar relations can be written for the on-shell scattering amplitude: $\displaystyle f(k,\varphi)=\sum_{l=-\infty}^{\infty}\exp(il\varphi)f_{l}(k),$ (10) $\displaystyle f_{l}(k)=\int_{0}^{\infty}J_{l}(k^{\prime}r)U(r)\tilde{\psi}_{l}(k,r)2\pi rdr.$ (11) The asymptotic form of the wavefunction of the $l$-wave relative motion at $r\rightarrow\infty$ may be represented as $\tilde{\psi}_{l}(k,r)\propto\frac{\cos(kr-\pi/4+\delta_{l}(k))}{\sqrt{kr}},$ (12) where $\delta_{l}(k)$ is the scattering phase shift. This is obvious because in the absence of scattering the $l$-wave part of the plane wave $\exp(i{\bf kr})$ at $r\rightarrow\infty$ is $(kr)^{-1/2}\cos(kr-\pi/4)$. Comparing Eq. (12) with the $l$-wave part of Eq. (5) we obtain a relation between the partial on-shell amplitude and the phase shift: $f_{l}(k)=-\frac{4\hbar^{2}}{m}\frac{\tan\delta_{l}(k)}{1-i\tan\delta_{l}(k)}.$ (13) Note that away from resonances the scattering phase shift is small in the low- momentum limit $kr_{}\ll 1$. For the solution of the scattering problem it is more convenient to normalize the wavefunction of the radial relative motion with orbital angular momentum $l$ in such a way that it is real and for $r\rightarrow\infty$ one has: $\displaystyle\psi_{l}(k,r)$ $\displaystyle=\left[J_{l}(kr)-\tan\delta_{l}(k)N_{l}(kr)\right]$ (14) $\displaystyle\propto\cos(kr-l\pi/2-\pi/4+\delta_{l}(k)),$ where $N_{l}$ is the Neumann function. One checks straightforwardly that $\tilde{\psi}_{l}(k,r)=\frac{\psi_{l}(k,r)}{1-i\tan\delta_{l}(k)}.$ Using this relation the off-shell scattering amplitude (9) can be represented as $f_{l}(k^{\prime},k)=\frac{{\bar{f}}_{l}(k^{\prime},k)}{1-i\tan\delta_{l}(k)},$ (15) where ${\bar{f}}_{l}(k^{\prime},k)$ is real and follows from Eq. (9) with $\tilde{\psi}_{l}(k,r)$ replaced by $\psi_{l}(k,r)$. Setting $k^{\prime}=k$ we then obtain the related on-shell scattering amplitude: ${\bar{f}}_{l}(k,k)\equiv{\bar{f}}_{l}(k)=-\frac{4\hbar^{2}}{m}\tan\delta_{l}(k).$ (16) ### II.2 Low-energy $p$-wave scattering As we will see, the slow $1/r^{3}$ decay of the potential $U(r)$ at sufficiently large distances makes the scattering drastically different from that of short-range interacting atoms. For identical fermionic polar molecules, only the scattering with odd orbital angular momenta $l$ is possible. For finding the amplitude of the $p$-wave scattering in the ultracold limit, $kr_{}\ll 1$, we employ the method developed in Ref. Gora and used there for the scattering potential containing an attractive $1/r^{3}$ dipole-dipole tail. We divide the range of distances into two parts: $r<r_{0}$ and $r>r_{0}$, where $r_{0}$ is in the interval $r_{}\ll r_{0}\ll k^{-1}$. In region I where $r<r_{0}$, the $p$-wave relative motion of two particles is governed by the Schrödinger equation with zero kinetic energy: $-\frac{\hbar^{2}}{m}\left(\frac{d^{2}\psi_{I}}{dr^{2}}+\frac{1}{r}\frac{d\psi_{I}}{dr}-\frac{\psi_{I}}{r^{2}}\right)+U(r)\psi_{I}=0.$ (17) At distances where the potential $U(r)$ already acquires the form (1), the solution of Eq. (17) can be expressed in terms of growing and decaying Bessel functions: $\psi_{I}(r)\propto\left[AI_{2}\left(2\sqrt{\frac{r_{}}{r}}\right)+K_{2}\left(2\sqrt{\frac{r_{}}{r}}\right)\right].$ (18) The constant $A$ is determined by the behavior of $U(r)$ at short distances where Eq. (1) is no longer valid. If the interaction potential $U(r)$ has the form (1) up to very short distances, then $A=0$, so that for $r\rightarrow 0$ equation (18) gives an exponentially decaying wavefunction. It should be noted here that for the quasi2D regime obtained by a tight confinement of the translational motion in one direction, we can encounter the situation where $r_{}\lesssim l_{0}$, with $l_{0}$ being the confinement length. However, we may always select $r_{0}\gg l_{0}$ if the condition $kl_{0}\ll 1$ is satisfied. Therefore, our results for the 2D $p$-wave scattering obtained below in this section remain applicable for the quasi2D regime. The character of the relative motion of particles at distances $r\lesssim l_{0}$ is only contained in the value of the coefficient $A$, and the extra requirement is the inequality $kl_{0}\ll 1$. At large distances, $r>r_{0}$, the relative motion is practically free and the potential $U(r)$ can be considered as perturbation. To zero order, the relative wavefunction is given by $\psi_{II}^{(0)}(r)=J_{1}(kr)-\tan\delta_{I}(k)N_{1}(kr),$ (19) where the phase shift $\delta_{I}(k)$ is due to the interaction between particles in region I. Equalizing the logarithmic derivatives of $\psi_{I}(r)$ and $\psi_{II}^{(0)}$ at $r=r_{0}$ we obtain: $\\!\\!\tan\delta_{I}\\!=\\!-\frac{\pi k^{2}r_{0}r_{}}{8}\left[1\\!+\\!\frac{r_{}}{r_{0}}\left(2C\\!-\\!\frac{1}{2}\\!-\\!2A\\!+\\!\ln\frac{r_{}}{r_{0}}\right)\right],\\!\\!$ (20) with $C=0.5772$ being the Euler constant. We now include perturbatively the contribution to the $p$-wave scattering phase shift from distance $r>r_{0}$. In this region, to first order in $U(r)$, the relative wavefunction is given by $\\!\psi^{(1)}_{II}(r)=\psi^{(0)}_{II}(r)\\!-\\!\int_{r_{0}}^{\infty}G(r,r^{\prime})U(r^{\prime})\psi^{(0)}_{II}(r^{\prime})2\pi r^{\prime}dr^{\prime}\\!,$ (21) where the Green function for the free $p$-wave motion obeys the radial equation: $\displaystyle-\frac{\hbar^{2}}{m}\left(\frac{d^{2}}{dr^{2}}+\frac{1}{r}\frac{d}{dr}-\frac{1}{r^{2}}+k^{2}\right)G(r,r^{\prime})=\frac{\delta(r-r^{\prime})}{2\pi r}.$ For the normalization of the relative wavefunction chosen in Eq. (14), we have: $G(r,r^{\prime})=-\frac{m}{4\hbar^{2}}\begin{cases}\psi^{(0)}_{II}(r^{\prime})N_{1}(kr),&r>r^{\prime}\\\ \\\ \psi^{(0)}_{II}(r)N_{1}(kr^{\prime}).&r<r^{\prime}\end{cases}$ (22) Substituting this Green function into Eq. (21) and taking the limit $r\rightarrow\infty$, for the first order contribution to the phase shift we have: $\\!\\!\tan\delta_{1}^{(1)}(k)\\!=\\!\tan\delta_{I}(k)\\!-\\!\frac{m}{4\hbar^{2}}\int_{r_{0}}^{\infty}\\!\\![\psi^{(0)}_{II}(r)]^{2}U(r)2\pi rdr.$ (23) Using Eqs. (19) and (20) we then obtain: $\\!\\!\\!\tan\delta_{1}^{(1)}(k)\\!=\\!-\frac{2kr_{}}{3}\\!-\\!\frac{\pi k^{2}r_{}^{2}}{8}\left(\\!\\!-2A\\!+\\!2C\\!+\\!\ln\frac{r_{}}{r_{0}}\\!-\\!\frac{3}{2}\\!\right).\\!$ (24) To second order in $U(r)$, we have the relative wavefunction: $\displaystyle\psi^{(2)}_{II}(r)$ $\displaystyle=\psi^{(1)}_{II}(r)+\int_{r_{0}}^{\infty}G(r,r^{\prime})U(r^{\prime})2\pi r^{\prime}dr^{\prime}$ (25) $\displaystyle\times\int_{r_{0}}^{\infty}G(r^{\prime},r^{\prime\prime})U(r^{\prime\prime})\psi^{(0)}_{II}(r^{\prime\prime})2\pi r^{\prime\prime}dr^{\prime\prime}.$ Taking the limit $r\rightarrow\infty$ in this equation we see that including the second order contribution, the scattering phase shift becomes: $\displaystyle\tan\delta_{1}(k)$ $\displaystyle=\tan\delta^{(1)}(k)-\frac{m^{2}}{8\hbar^{4}}\int_{r_{0}}^{\infty}\psi^{(0)}_{II}(r)^{2}U(r)2\pi rdr$ (26) $\displaystyle\times\int_{r}^{\infty}N_{1}(kr^{\prime})U(r^{\prime})\psi^{(0)}_{II}(r^{\prime})2\pi r^{\prime}dr^{\prime}.$ As we are not interested in terms that are proportional to $k^{3}$ or higher powers of $k$, we may omit the term $\tan\delta_{I}(k)N_{1}(kr)$ in the expression for $\psi^{(0)}_{II}(r)$. Then the integration over $dr^{\prime}$ leads to: $\displaystyle\tan\delta_{1}(k)=\tan\delta_{1}^{(1)}(k)-\frac{(\pi kr_{})^{2}}{2}\int_{kr_{0}}^{\infty}\frac{J^{2}_{1}(x)}{x^{2}}dx$ $\displaystyle\times\Big{[}\frac{2}{3}x\left(N_{0}(x)J_{2}(x)-N_{1}(x)J_{1}(x)\right)$ $\displaystyle\;\;\;\;-\frac{1}{2}N_{0}(x)J_{1}(x)+\frac{1}{6}N_{1}(x)J_{2}(x)-\frac{1}{\pi x}\Big{]}.$ (27) For the first four terms in the square brackets, we may put the lower limit of integration equal to zero and use the following relations: $\displaystyle\int_{0}^{\infty}J^{3}_{1}(x)N_{1}(x)\frac{dx}{x}=-\frac{1}{4\pi},$ $\displaystyle\int_{0}^{\infty}J^{2}_{1}(x)J_{2}(x)N_{0}(x)\frac{dx}{x}=\frac{1}{8\pi},$ $\displaystyle\int_{0}^{\infty}J^{3}_{1}(x)N_{0}(x)\frac{dx}{x^{2}}=\frac{1}{16\pi},$ $\displaystyle\int_{0}^{\infty}J^{2}_{1}(x)J_{2}(x)N_{1}(x)\frac{dx}{x^{2}}=-\frac{1}{16\pi}.$ For the last term in the square brackets we have: $\int_{kr_{0}}^{\infty}J^{2}_{1}(x)\frac{dx}{x^{3}}\approx\frac{1}{16}-\frac{C}{4}+\frac{\ln 2}{4}-\frac{1}{4}\ln kr_{0}.$ (28) We then obtain: $\displaystyle\tan\delta_{1}(k)$ $\displaystyle=\tan\delta_{1}^{(1)}(k)-\frac{\pi(kr_{})^{2}}{8}\left[\frac{7}{12}+C-\ln 2+\ln kr_{0}\right]$ $\displaystyle=-\frac{2kr_{}}{3}-\frac{\pi k^{2}r^{2}_{}}{8}\ln\xi kr_{},$ (29) where: $\xi=\exp\left(3C-\ln 2-\frac{11}{12}-2A\right).$ (30) Using Eqs. (16) and (29) we represent the on-shell $p$-wave scattering amplitude ${\bar{f}}_{1}(k)$ in the form: ${\bar{f}}_{1}(k)={\bar{f}}^{(1)}_{1}(k)+{\bar{f}}^{(2)}_{1}(k),$ (31) with ${\bar{f}}^{(1)}_{1}(k)=\frac{8\hbar^{2}}{3m}kr_{}$ (32) and ${\bar{f}}^{(2)}_{1}(k)=\frac{\pi\hbar^{2}}{2m}(kr_{})^{2}\ln\xi kr_{}.$ (33) The leading term is ${\bar{f}}^{(1)}_{1}(k)\propto k$. It appears to first order in $U(r)$ and comes from the scattering at distances $r\sim 1/k$. This term can be called “anomalous scattering” term (see Lan2 ). The term $f^{(2)}_{1}(k)\propto k^{2}\ln\xi kr_{}$ comes from both large distances $\sim 1/k$ and short distances. Note that the behavior of the wavefunction at short distances where $U(r)$ is no longer given by Eq. (1), is contained in Eq. (29) only through the coefficient $\xi$ under logarithm. ### II.3 Scattering with $\|l\|>1$ The presence of strong anomalous $p$-wave scattering, i.e. the scattering from interparticle distances $\sim 1/k$, originates from the slow $1/r^{3}$ decay of the potential $U(r)$ at large $r$. The strong anomalous scattering is also present for partial waves with higher $l$. In this section we follow the same method as in the case of the $p$-wave scattering and calculate the amplitude of the $l$-wave scattering with $\|l\|>1$. For simplicity we consider positive $l$, having in mind that the scattering amplitude and phase shift depend only on $\|l\|$. To zero order in $U(r)$, the wavefunction of the $l$-wave relative motion at large distances $r>r_{0}$ is written as: $\psi^{(0)}_{l(II)}(k,r)=\left[J_{l}(kr)-\tan\delta_{l(I)}(k)N_{l}(kr)\right],$ (34) where $\delta_{l(I)}(k)$ is the $l$-wave scattering phase shift coming from the interaction at distances $r<r_{0}$. We then match $\psi^{(0)}_{l(II)}(k,r)$ at $r=r_{0}$ with the short-distance wavefunction $\psi_{l(I)}(r)$ which follows from the Schrödinger equation for the $l$-wave relative motion in the potential $U(r)$ at $k=0$. This immediately gives a relation: $\tan\delta_{l(I)}(k)=\frac{kJ^{\prime}_{l}(kr_{0})-w_{l}J_{l}(kr_{0})}{kN^{\prime}_{l}(kr_{0})-w_{l}N_{l}(kr_{0})},$ (35) where the momentum-independent quantity $w_{l}$ is the logarithmic derivative of $\psi_{l(I)}(r)$ at $r=r_{0}$. Since we have the inequality $kr_{0}\ll 1$, the arguments of the Bessel functions in Eq. (35) are small and they reduce to $J_{l}(x)\sim x^{l}\;,\;N_{l}(x)\sim x^{-l}$. This leads to $\tan\delta_{l(I)}(k)\sim(kr_{0})^{2l}$. Thus, the phase shift coming from the interaction at short distances is of the order of $(kr_{0})^{2l}$. As we confine ourselves to second order in $k$, we may put $\tan\delta_{l(I)}(k)=0$ for the scattering with $\|l\|>1$. Then, like for the $p$-wave scattering, we calculate the contribution to the phase shift from distances $r>r_{0}$ by considering the potential $U(r)$ as perturbation. To first and second order in $U(r)$, at $r>r_{0}$ we have similar expressions as Eq. (23), (25) for the relative wavefunction of the $l$-wave motion. Following the same method as in the case of the $p$-wave scattering and retaining only the terms up to $k^{2}$, for the first order phase shift we have: $\displaystyle\tan\delta_{l}^{(1)}(k)=-\frac{m}{4\hbar^{2}}\int_{r_{0}}^{\infty}[\psi^{(0)}_{l(II)}(r)]^{2}U(r)2\pi rdr$ $\displaystyle\simeq-\frac{\pi kr_{}}{2}\int_{kr_{0}}^{\infty}J_{l}^{2}(x)\frac{1}{x^{2}}dx=-\frac{2kr_{}}{4l^{2}-1}.$ (36) The second order phase shift is: $\displaystyle\tan\delta^{(2)}_{l}(k)=-\frac{m^{2}}{8\hbar^{4}}\int_{r_{0}}^{\infty}\psi^{(0)}_{l(II)}(r)^{2}U(r)2\pi rdr$ $\displaystyle\;\;\times\int_{r}^{\infty}N_{l}(kr^{\prime})U(r^{\prime})\psi^{(0)}_{l(II)}(r^{\prime})2\pi r^{\prime}dr^{\prime}$ $\displaystyle\simeq-\frac{(\pi kr_{})^{2}}{2}\int_{kr_{0}}^{\infty}\frac{J^{2}_{l}(x)}{x^{2}}dx\int_{x}^{\infty}\frac{N_{l}(y)J_{l}(y)}{y^{2}}dy,$ (37) and we may put the lower limit of integration equal to zero. For the integral over $dy$, we obtain : $\displaystyle\int_{x}^{\infty}\frac{N_{l}(y)J_{l}(y)}{y^{2}}dy$ $\displaystyle=\frac{1}{2l(2l-1)}J_{l}(x)N_{l-1}(x)+\frac{1}{2l(2l+1)}J_{l+1}(x)N_{l}(x)$ $\displaystyle\\!\\!+\frac{2x}{4l^{2}-1}\big{[}N_{l-1}(x)J_{l+1}(x)\\!-\\!J_{l}(x)N_{l}(x)\big{]}-\frac{1}{\pi lx}.$ (38) Then, using the relations: $\int_{0}^{\infty}\frac{J_{l}^{2}(x)}{x^{3}}dx=\frac{1}{4l(l^{2}-1)},$ $\int_{0}^{\infty}\frac{J_{l}^{2}(x)}{x}N_{l-1}(x)J_{l+1}(x)dx=\frac{1}{4l(l+1)\pi},$ $\int_{0}^{\infty}\frac{J_{l}^{3}(x)}{x}N_{l}(x)dx=-\frac{1}{4l^{2}\pi},$ $\int_{0}^{\infty}\frac{J_{l}^{2}(x)}{x^{2}}J_{l}(x)N_{l-1}(x)dx=\frac{1}{8l^{2}(l+1)\pi},$ $\int_{0}^{\infty}\frac{J_{l}^{2}(x)}{x^{2}}J_{l+1}(x)N_{l}(x)dx=-\frac{1}{8l^{2}(l+1)\pi},$ we find the following result for the second order phase shift: $\displaystyle\tan\delta^{(2)}_{l}(k)=\frac{3\pi(kr_{})^{2}}{8}\frac{1}{l(l^{2}-1)(4l^{2}-1)}.$ (39) So, the total phase shift is given by $\displaystyle\tan\delta_{l}(k)$ $\displaystyle=\tan\delta_{l}^{(1)}(k)+\tan\delta_{l}^{(2)}(k)$ $\displaystyle=-\frac{2kr_{}}{4l^{2}-1}+\frac{3\pi(kr_{})^{2}}{8l(l^{2}-1)(4l^{2}-1)}.$ (40) Then, according to Eq. (16) the on-shell scattering amplitude ${\bar{f}}_{l}(k)$ is ${\bar{f}}_{l}(k)={\bar{f}}^{(1)}_{l}(k)+{\bar{f}}^{(2)}_{l}(k),$ (41) where ${\bar{f}}^{(1)}_{l}(k)=\frac{8\hbar^{2}kr_{}}{m}\frac{1}{4l^{2}-1},$ (42) ${\bar{f}}^{(2)}_{l}(k)=-\frac{3\pi\hbar^{2}}{2m}(kr_{})^{2}\frac{1}{\|l\|(l^{2}-1)(4l^{2}-1)}.$ (43) Note that Eqs. (42) and (43) do not contain short-range contributions as those are proportional to $k^{2\|l\|}$ and can be omitted for $\|l\|>1$. ### II.4 First order Born approximation and the leading part of the scattering amplitude As we already said above, in the low-momentum limit for both $\|l\|=1$ and $\|l\|>1$ the leading part of the on-shell scattering amplitude ${\bar{f}}_{l}(k)$ is ${\bar{f}}_{l}^{(1)}(k)$ and it is contained in the first order contribution from distances $r>r_{0}$. For $\|l\|>1$ it is given by Eq. (42) and follows from Eq. (II.3) with $\psi^{(0)}_{l(II)}=J_{l}(kr)$. In the case of $\|l\|=1$ this leading part is given by Eq. (32) and follows from the integral term of Eq. (23) in which one keeps only $J_{1}(kr)$ in the expression for $\psi^{(0)}_{II}(r)$. This means that ${\bar{f}}_{l}^{(1)}(k)$ actually follows from the first order Born approximation. The off-shell scattering amplitude can also be represented as ${\bar{f}}_{l}(k^{\prime},k)={\bar{f}}_{l}^{(1)}(k^{\prime},k)+{\bar{f}}_{l}^{(2)}(k^{\prime},k)$, and the leading contribution ${\bar{f}}_{l}^{(1)}(k^{\prime},k)$ follows from the first Born approximation. It is given by Eq. (9) in which one should replace $\tilde{\psi}_{l}(k,r)$ by $J_{l}(kr)$: ${\bar{f}}_{l}^{(1)}(k^{\prime},k)=\int_{0}^{\infty}J_{l}(kr)J_{l}(kr^{\prime})U(r)2\pi rdr.$ (44) Note that it is not important that we put zero for the lower limit of the integration, since this can only give a correction which behaves as $k^{2}$ or a higher power of $k$. Then, putting $U(r)=\hbar^{2}r_{}/mr^{3}$ in Eq. (44), we obtain: $\displaystyle{\bar{f}}^{(1)}_{l}(k^{\prime},k)=$ $\displaystyle\frac{\pi\hbar^{2}}{m}\frac{\Gamma(l-1/2)}{\sqrt{\pi}}\frac{k^{l}r_{}}{(k^{\prime})^{l-1}}$ $\displaystyle\times F\left(-\frac{1}{2},-\frac{1}{2}+l,1+l,\frac{k^{2}}{k^{\prime 2}}\right),$ (45) where $F$ is the hypergeometric function. The result of Eq. (II.4) corresponds to $k<k^{\prime}$, and for $k>k^{\prime}$ one should interchange $k$ and $k^{\prime}$. For identical fermions the full scattering amplitude contains only partial amplitudes with odd $l$. Since the scattered waves with relative momenta ${\bf k}^{\prime}$ and $-{\bf k}^{\prime}$ correspond to interchanging the identical fermions, the scattering amplitude can be written as (see, e.g. Lan2 ): $\tilde{f}({\bf k}^{\prime},{\bf k})=f({\bf k}^{\prime},{\bf k})-f(-{\bf k}^{\prime},{\bf k}).$ (46) Then, according to equation (10) one can write: $\tilde{f}({\bf k}^{\prime},{\bf k})=2\sum_{l\,odd}f_{l}(k^{\prime},k)\exp(il\varphi).$ (47) In the first Born approximation there is no difference between $f_{l}(k^{\prime},k)$ and ${\bar{f}}_{l}(k^{\prime},k)$ because $\tan\delta_{l}(k)$ in the denominator of Eq. (15) is proportional to $k$ and can be disregarded. Therefore, one may use ${\bar{f}}_{l}^{(1)}(k^{\prime},k)$ of Eq.(II.4) for $f_{l}(k^{\prime},k)$ in Eq. (47). One can represent $\tilde{f}({\bf k}^{\prime},{\bf k})$ in a different form recalling that in the first Born approximation we have: $f({\bf k}^{\prime},{\bf k})=\int U(r)\exp[i({\bf k}-{\bf k}^{\prime}){\bf r}]d^{2}r.$ (48) Performing the integration in this equation, with $U(r)$ given by Eq. (1), and using Eq. (46) we obtain: $\tilde{f}({\bf k}^{\prime},{\bf k})=\frac{2\pi\hbar^{2}r_{}}{m}\\{\|{\bf k}+{\bf k}^{\prime}\|-\|{\bf k}-{\bf k}^{\prime}\|\\}.$ (49) Equation (49) is also obtained by a direct summation over odd $l$ in Eq. (47), with $f_{l}(k^{\prime},k)$ following from Eq. (II.4). ## III Thermodynamics of a weakly interacting 2D gas of fermionic polar molecules at $T=0$ ### III.1 General relations of Fermi liquid theory Identical fermionic polar molecules undergoing a two-dimensional translational motion and repulsively interacting with each other via the potential (1) represent a 2D Fermi liquid. General relations of the Landau Fermi liquid theory remain similar to those in 3D (see, e.g. Landau ). The number of “dressed” particles, or quasiparticles, is the same as the total number of particles $N$, and the (quasi)particle Fermi momentum is $k_{F}=\sqrt{\frac{4\pi N}{S}},$ (50) where $S$ is the surface area. At $T=0$ the momentum distribution of free quasiparticles is the step function $n({\bf k})=\theta(k_{F}-k),$ (51) i.e. $n({\bf k})=1$ for $k<k_{F}$ and zero otherwise.The chemical potential is equal to the boundary energy at the Fermi circle, $\mu=\epsilon_{F}\equiv\epsilon(k_{F})$. The quasiparticle energy $\epsilon({\bf k})$ is a variational derivative of the total energy with respect to the distribution function $n({\bf k})$. Due to the interaction between quasiparticles, the deviation $\delta n$ of this distribution from the step function (51) results in the change of the quasiparticle energy: $\delta\epsilon(\mathbf{k})=\int F(\mathbf{k},\mathbf{k}^{\prime})\delta n(\mathbf{k}^{\prime})\frac{d^{2}k^{\prime}}{(2\pi)^{2}}.$ (52) The interaction function of quasiparticles $F(\mathbf{k},\mathbf{k}^{\prime})$ is thus the second variational derivative of the total energy with regard to $n({\bf k})$. The quantity $\delta n({\bf k})$ is significantly different from zero only near the Fermi surface, so that one may put ${\bf k}=k_{F}{\bf n}$ and ${\bf k}^{\prime}=k_{F}{\bf n}^{\prime}$ in the arguments of $F$ in Eq. (52), where ${\bf n}$ and ${\bf n}^{\prime}$ are unit vectors in the directions of ${\bf k}$ and ${\bf k}^{\prime}$. The quasiparticle energy near the Fermi surface can be written as: $\epsilon(\mathbf{k})=\epsilon_{F}+\hbar v_{F}(k-k_{F})+\int F(\mathbf{k},\mathbf{k}^{\prime})\delta n(\mathbf{k}^{\prime})\frac{d^{2}k^{\prime}}{(2\pi)^{2}}.$ (53) The quantity $v_{F}=\partial\epsilon({\bf k})/\hbar\partial k\|_{k=k_{F}}$ is the Fermi velocity, and the effective mass of a quasiparticle is defined as $m^{}=\hbar k_{F}/v_{F}$. It can be obtained from the relation (see Landau ): $\displaystyle\frac{1}{m}=\frac{1}{m^{}}+\frac{1}{(2\pi\hbar)^{2}}\int_{0}^{2\pi}F(\theta)\cos\theta d\theta,$ (54) where $\theta$ is the angle between the vectors ${\bf n}$ and ${\bf n}^{\prime}$, and $F(\theta)=F(k_{F}{\bf n},k_{F}{\bf n}^{\prime})$. The compressibility $\kappa$ at $T=0$ is given by Landau : $\kappa^{-1}=\frac{N^{2}}{S}\frac{\partial\mu}{\partial N}.$ (55) The chemical potential is $\mu=\epsilon_{F}$, and the variation of $\mu$ due to a change in the number of particles can be expressed as $\delta\mu=\int F(k_{F}{\bf n},\mathbf{k}^{\prime})\delta n(\mathbf{k^{\prime}})\frac{d^{2}k^{\prime}}{(2\pi)^{2}}+\frac{\partial\epsilon_{F}}{\partial k_{F}}\delta k_{F}.$ (56) The quantity $\delta n(\mathbf{k^{\prime}})$ is appreciably different from zero only when $\mathbf{k^{\prime}}$ is near the Fermi surface, so that we can replace the interaction function $F$ by its value on the Fermi surface. Then the first term of Eq. (56) becomes $\displaystyle\int F(\theta)\frac{d\theta}{2\pi}\int\delta n(\mathbf{k^{\prime}})\frac{d^{2}k^{\prime}}{(2\pi)^{2}}=\frac{\delta N}{2\pi S}\int F(\theta)d\theta.$ The second term of Eq. (56) reduces to $\displaystyle\frac{\partial\epsilon_{F}}{\partial k_{F}}\delta k_{F}=\frac{\hbar^{2}k_{F}}{m^{}}\delta k_{F}=\frac{2\pi\hbar^{2}}{m^{}}\frac{\delta N}{S}.$ (57) We thus have (see Landau ): $\displaystyle\frac{\partial\mu}{\partial N}$ $\displaystyle=\frac{1}{2\pi S}\int_{0}^{2\pi}F(\theta)d\theta+\frac{2\pi\hbar^{2}}{m^{}S}$ $\displaystyle=\frac{2\pi\hbar^{2}}{mS}+\frac{1}{2\pi S}\int_{0}^{2\pi}(1-\cos\theta)F(\theta)d\theta.$ (58) Equation (III.1) shows that the knowledge of the interaction function of quasiparticles on the Fermi surface, $F(\theta)$, allows one to calculate $\partial\mu/\partial N$ and, hence, the chemical potential $\mu=\partial E/\partial N$ and the ground state energy $E$. This elegant way of finding the ground state energy has been proposed by Abrikosov and Khalatnikov Abr . It was implemented in Ref. Abr for a two-component 3D Fermi gas with a weak repulsive contact (short-range) interspecies interaction. We develop a theory beyond the mean field for calculating the interaction function of quasiparticles for a single-component 2D gas of fermionic polar molecules in the weakly interacting regime. We obtain the ground state energy as a series of expansion in the small parameter $k_{F}r_{}$ and confine ourselves to the second order. In this sense our work represents a sort of Lee-Huang-Yang Huang ; Lee and Abrikosov-Khalatnikov Abr calculation for this dipolar system. As we will see, the long-range character of the dipole- dipole interaction makes the result quite different from that in the case of short-range interactions. ### III.2 Two-body and many-body contributions to the ground state energy We first write down the expression for the kinetic energy and specify two-body (mean field) and many-body (beyond mean field) contributions to the interaction energy. The Hamiltonian of the system reads: $\\!\\!\hat{\cal H}\\!=\\!\sum_{\mathbf{k}}\frac{\hbar^{2}k^{2}}{2m}\hat{a}_{\mathbf{k}}^{{\dagger}}\hat{a}_{\mathbf{k}}\\!+\\!\frac{1}{2S}\\!\\!\\!\sum_{\mathbf{k_{1}},\mathbf{k_{2}},\mathbf{q}}\\!\\!\\!U(\mathbf{q})\hat{a}_{\mathbf{k_{1}}+\mathbf{q}}^{{\dagger}}\hat{a}_{\mathbf{k_{2}}-\mathbf{q}}^{{\dagger}}\hat{a}_{\mathbf{k_{2}}}\hat{a}_{\mathbf{k_{1}}},\\!\\!$ (59) where $\hat{a}^{\dagger}_{{\bf k}}$ and $\hat{a}_{{\bf k}}$ are creation and annihilation operators of fermionic polar molecules, and $U(\mathbf{q})$ is the Fourier transform of the interaction potential $U(r)$: $U(\mathbf{q})=\int d^{2}\mathbf{r}U(r)e^{-i\mathbf{q}\cdot\mathbf{r}},$ (60) The first term of Eq. (59) represents the kinetic energy and it gives the main contribution to the total energy $E$ of the system. This term has only diagonal matrix elements, and using the momentum distribution (51) at $T=0$ we have: $\displaystyle\frac{E_{kin}}{S}=\int_{0}^{k_{F}}\frac{\hbar^{2}k^{2}}{2m}\frac{2\pi kdk}{(2\pi)^{2}}=\frac{\hbar^{2}k_{F}^{4}}{16m}.$ (61) The interaction between the fermionic molecules is described by the second term in Eq. (59) and compared to the kinetic energy it provides a correction to the total energy $E$. The first order correction is given by the diagonal matrix element of the interaction term of the Hamiltonian: $\displaystyle E^{(1)}$ $\displaystyle=\frac{1}{2S}\sum_{{\bf{k_{1}}},{\bf{k_{2}}},{\bf q}}U(\mathbf{q})\langle\hat{a}_{\mathbf{k_{1}}+\mathbf{q}}^{{\dagger}}\hat{a}_{\mathbf{k_{2}}-\mathbf{q}}^{{\dagger}}\hat{a}_{\mathbf{k_{2}}}\hat{a}_{\mathbf{k_{1}}}\rangle$ $\displaystyle=\frac{1}{2S}\sum_{\mathbf{k_{1}},\mathbf{k_{2}}}\left[U(0)-U(\mathbf{k_{2}}-\mathbf{k_{1}})\right]n_{\mathbf{k_{1}}}n_{\mathbf{k_{2}}}.$ (62) The second order correction to the energy of the state $\left\|{j}\right>$ of a non-interacting system can be expressed as: $\displaystyle E_{j}^{(2)}=\sum_{m\neq j}\frac{V_{jm}V_{mj}}{E_{j}-E_{m}},$ (63) where the summation is over eigenstates $\left\|{m}\right>$ of the non- interacting system, and $V_{jm}$ is the non-diagonal matrix element. In our case the symbol $j$ corresponds to the ground state and the symbol $m$ to excited states. The non-diagonal matrix element is $\\!\\!V_{jm}\\!=\\!\frac{1}{2S}\left<\\!m\left\|\sum_{\mathbf{k_{1}},\mathbf{k_{2}},\mathbf{q}}U(\mathbf{q})\hat{a}_{\mathbf{k_{1}}+\mathbf{q}}^{{\dagger}}\hat{a}_{\mathbf{k_{2}}-\mathbf{q}}^{{\dagger}}\hat{a}_{\mathbf{k_{2}}}\hat{a}_{\mathbf{k_{1}}}\right\|j\\!\right>.\\!$ (64) This matrix element corresponds to the scattering of two particles from the initial state $\mathbf{k_{1}}$, $\mathbf{k_{2}}$ to an intermediate state $\mathbf{k^{\prime}_{1}}$, $\mathbf{k^{\prime}_{2}}$, and the matrix element $V_{mj}$ describes the reversed process in which the two particles return from the intermediate to initial state. Taking into account the momentum conservation law ${\bf k}_{1}+{\bf k}_{2}={\bf k}^{\prime}_{1}+{\bf k}^{\prime}_{2}$ the quantity $V_{jm}V_{mj}=\|V_{jm}\|^{2}$ is given by $\displaystyle\|V_{jm}\|^{2}$ $\displaystyle=\frac{1}{(2S)^{2}}n_{\mathbf{k_{1}}}n_{\mathbf{k_{2}}}(1-n_{\mathbf{k_{1}^{\prime}}})(1-n_{\mathbf{k_{2}^{\prime}}})$ $\displaystyle\times\left\|U(\mathbf{k^{\prime}_{1}}-\mathbf{k_{1}})-U(\mathbf{k^{\prime}_{2}}-\mathbf{k_{1}})\right\|^{2},$ (65) and the second order correction to the ground state energy takes the form: $\displaystyle E^{(2)}=$ $\displaystyle\frac{1}{(2S)^{2}}\sum_{\mathbf{k_{1}},\mathbf{k_{2}},\mathbf{k_{1}^{\prime}}}\Bigg{[}\left\|U({\bf k}^{\prime}_{1}-{\bf k}_{1})-U({\bf k}^{\prime}_{2}-{\bf k}_{1})\right\|^{2}$ $\displaystyle\times\frac{n_{\mathbf{k_{1}}}n_{\mathbf{k_{2}}}(1-n_{\mathbf{k_{1}^{\prime}}})(1-n_{\mathbf{k_{2}^{\prime}}})}{\hbar^{2}(\mathbf{k^{2}_{1}}+\mathbf{k^{2}_{2}}-\mathbf{k^{\prime 2}_{1}}-\mathbf{k^{\prime 2}_{2}})/2m}\Bigg{]}.$ (66) From Eq. (III.2) we see that the second order correction diverges because of the term proportional to $n_{\mathbf{k_{1}}}n_{\mathbf{k_{2}}}$, which is divergent at large $k^{\prime}_{1}$. This artificial divergence is eliminated by expressing the energy correction in terms of a real physical quantity, the scattering amplitude. The relation between the Fourier component of the interaction potential and the off-shell scattering amplitude is given by Lan2 : $f(\mathbf{k^{\prime}},\mathbf{k})=U(\mathbf{k}^{\prime}-\mathbf{k})+\frac{1}{S}\sum_{\mathbf{k^{\prime\prime}}}\frac{U(\mathbf{k}^{\prime}-\mathbf{k^{\prime\prime}})f(\mathbf{k^{\prime\prime}},\mathbf{k})}{(E_{\mathbf{k}}-E_{\mathbf{k^{\prime\prime}}}-i0)},$ (67) where $E_{\mathbf{k}}=\hbar^{2}{\bf k}^{2}/m$ and $E_{\mathbf{k^{\prime\prime}}}=\hbar^{2}{\bf k}^{\prime\prime 2}/m$ are relative collision energies. Obviously, we have: $E_{\mathbf{k}}-E_{\mathbf{k^{\prime\prime}}}=\hbar^{2}(\mathbf{k_{1}}^{2}+\mathbf{k_{2}}^{2}-\mathbf{k^{\prime\prime}_{1}}^{2}-\mathbf{k^{\prime\prime}_{2}}^{2})/2m$, with ${\bf k}_{1}$, ${\bf k}_{2}$ (${\bf k}^{\prime\prime}_{1}$, ${\bf k}^{\prime\prime}_{2}$) being the momenta of colliding particles in the initial (intermediate) state, as the relative momenta are given by $\mathbf{k}=(\mathbf{k}_{1}-\mathbf{k}_{2})/2$, $\mathbf{k^{\prime\prime}}=(\mathbf{k^{\prime\prime}_{1}}-\mathbf{k^{\prime\prime}_{2}})/2$. We thus can write: $\\!\\!\\!U(\mathbf{k}^{\prime}\\!-\mathbf{k})\\!=\\!f(\mathbf{k}^{\prime}\\!,\\!\mathbf{k})\\!-\\!\frac{2m}{\hbar^{2}S}\sum_{\mathbf{k_{1}^{\prime\prime}}}\frac{U(\mathbf{k}^{\prime}\\!-\\!\mathbf{k}^{\prime\prime})f(\mathbf{k^{\prime\prime}}\\!,\\!\mathbf{k})}{\mathbf{k^{2}_{1}}\\!+\\!\mathbf{k^{2}_{2}}\\!-\\!\mathbf{k^{\prime\prime 2}_{1}}\\!-\\!\mathbf{k^{\prime\prime 2}_{2}}\\!-\\!i0}.\\!\\!$ (68) Then, putting ${\bf k}^{\prime}={\bf k}$ we have $\displaystyle U(0)=f({\bf k},{\bf k})-\frac{2m}{\hbar^{2}S}\sum_{{\bf k}^{\prime\prime}_{1}}\frac{U({\bf k}-{\bf k}^{\prime\prime})f({\bf k}^{\prime\prime},{\bf k})}{\mathbf{k^{2}_{1}}+\mathbf{k^{2}_{2}}-\mathbf{k^{\prime\prime 2}_{1}}-\mathbf{k^{\prime\prime 2}_{2}}-i0},$ and setting ${\bf k}^{\prime}=-{\bf k}$ we obtain $\displaystyle\\!\\!U({\bf k}_{2}\\!-\\!{\bf k}_{1})=f(-{\bf k},{\bf k})\\!-\\!\frac{2m}{\hbar^{2}S}\sum_{{\bf k}^{\prime\prime}_{1}}\frac{U(-{\bf k}\\!-\\!{\bf k}^{\prime\prime})f({\bf k}^{\prime\prime},{\bf k})}{\mathbf{k^{2}_{1}}\\!+\\!\mathbf{k^{2}_{2}}\\!-\\!\mathbf{k^{\prime\prime 2}_{1}}\\!-\\!\mathbf{k^{\prime\prime 2}_{2}}\\!-\\!i0},$ Using these relations the first order correction (III.2) takes the form: $\displaystyle E^{(1)}=\frac{1}{2S}\sum_{\mathbf{k_{1}},\mathbf{k_{2}}}\left[f(\mathbf{k},\mathbf{k})-f(\mathbf{-k},\mathbf{k})\right]n_{\mathbf{k_{1}}}n_{\mathbf{k_{2}}}$ $\displaystyle\\!\\!-\frac{1}{2S^{2}}\\!\\!\\!\\!\\!\sum_{\mathbf{k_{1}},\mathbf{k_{2}},\mathbf{k^{\prime}_{1}}}\\!\\!\frac{[U(\mathbf{k}\\!-\\!\mathbf{k^{\prime}})\\!-\\!U({\\!-\bf k}\\!-\\!{\bf k}^{\prime})]f(\mathbf{k^{\prime}},\mathbf{k})}{\hbar^{2}(\mathbf{k^{2}_{1}}\\!+\\!\mathbf{k^{2}_{2}}\\!-\\!\mathbf{k^{\prime 2}_{1}}\\!-\\!\mathbf{k^{\prime 2}_{2}}\\!-\\!i0)/2m}n_{{\bf k}_{1}}n_{{\bf k}_{2}}.\\!\\!$ (69) The quantity $[U({\bf k}-{\bf k}^{\prime})-U(-{\bf k}-{\bf k}^{\prime})]$ in the second term of Eq. (III.2), being expanded in circular harmonics $\exp(il\varphi)$ contains terms with odd $l$. Therefore, partial amplitudes with even $l$ in the expansion of the multiple $f({\bf k}^{\prime},{\bf k})$ vanish after the integration over $d^{2}k^{\prime}$. Hence, this amplitude can be replaced by $[f({\bf k}^{\prime},{\bf k})-f({\bf k}^{\prime},-{\bf k})]/2$. As we are interested only in the terms that behave themselves as $\sim k$ or $\sim k^{2}$, the amplitudes in the second term of Eq. (III.2) are the ones that follow from the first Born approximation and are proportional to $k$. Therefore, we may put $[U({\bf k}-{\bf k}^{\prime})-U(-{\bf k}-{\bf k}^{\prime})]=[f({\bf k},{\bf k}^{\prime})-f(-{\bf k},{\bf k}^{\prime})]$ and $f({\bf k}^{\prime},{\bf k})=f^{}({\bf k},{\bf k}^{\prime})$. Then the first order correction takes the form: $\displaystyle E^{(1)}=\frac{1}{2S}\sum_{\mathbf{k_{1}},\mathbf{k_{2}}}\left[f(\mathbf{k},\mathbf{k})-f(\mathbf{-k},\mathbf{k})\right]n_{\mathbf{k_{1}}}n_{\mathbf{k_{2}}}-\frac{1}{(2S)^{2}}$ $\displaystyle\times\\!\\!\sum_{\mathbf{k_{1}},\mathbf{k_{2}},\mathbf{k^{\prime}_{1}}}\frac{\|f(\mathbf{k^{\prime}},\mathbf{k})\\!-\\!f({\bf k}^{\prime},-{\bf k})\|^{2}}{\hbar^{2}(\mathbf{k^{2}_{1}}\\!+\\!\mathbf{k^{2}_{2}}\\!-\\!\mathbf{k^{\prime 2}_{1}}\\!-\\!\mathbf{k^{\prime 2}_{2}}\\!-\\!i0)/2m}n_{{\bf k}_{1}}n_{{\bf k}_{2}}.$ (70) Using the expansion of the full scattering amplitude in terms of partial amplitudes as given by Eq. (47) we represent the first order correction as $\displaystyle E^{(1)}=\frac{1}{S}\sum_{{\bf k}_{1},{\bf k}_{2}}\sum_{l\,odd}f_{l}(k)n_{{\bf k}_{1}}n_{{\bf k}_{2}}-\frac{1}{S^{2}}\sum_{{\bf k}_{1},{\bf k}_{2}}\sum_{l\,odd}$ $\displaystyle\\!\times\\!\\!\int\\!\frac{d^{2}k^{\prime}}{(2\pi)^{2}}\frac{f_{l}^{2}(k)}{\hbar^{2}(\mathbf{k^{2}_{1}}\\!+\\!\mathbf{k^{2}_{2}}\\!-\\!\mathbf{k^{\prime 2}_{1}}\\!-\\!\mathbf{k^{\prime 2}_{2}}\\!-\\!i0)/2m}n_{{\bf k}_{1}}n_{{\bf k}_{2}}.\\!\\!$ (71) The contribution of the pole in the integration over $d^{2}k^{\prime}$ in the second term of Eq. (III.2) gives $imf_{l}^{2}(k)/4\hbar^{2}$ for each term in the sum over ${\bf k}_{1}$, ${\bf k}_{2}$, and $l$, and we may use here the amplitude ${\bar{f}}^{(1)}_{l}(k)$. In the first term of Eq. (III.2) we should use $f_{l}(k)=f_{l}^{(1)}(k)+f_{l}^{(2)}(k)$. However, we may replace $f_{l}^{(2)}$ by ${\bar{f}}_{l}^{(2)}$ because the account of $\tan\delta(k)$ in the denominator of Eq. (15) leads to $k^{3}$ terms and terms containing higher powers of $k$. For the amplitude $f_{l}^{(1)}(k)$, we use the expression: $f_{l}^{(1)}(k)={\bar{f}}^{(1)}_{l}+i\tan\delta(k){\bar{f}}^{(1)}_{l}={\bar{f}}^{(1)}_{l}-im[{\bar{f}}^{(1)}_{l}]^{2}/4\hbar^{2},$ which assumes a small scattering phase shift. The second term of this expression, being substituted into the first line of Eq. (III.2), exactly cancels the contribution of the pole in the second term of (III.2). Thus, we may use the amplitude ${\bar{f}}_{l}$ in the first term of equation (III.2) and take the principal value of the integral in the second term. The resulting expression for the first order correction reads: $\displaystyle E^{(1)}$ $\displaystyle=\frac{1}{S}\sum_{{\bf k}_{1},{\bf k}_{2}}{\bar{f}}({\bf k})n_{{\bf k}_{1}}n_{{\bf k}_{2}}-\frac{1}{(2S)^{2}}$ $\displaystyle\times\sum_{\mathbf{k_{1}},\mathbf{k_{2}},\mathbf{k^{\prime}_{1}}}\frac{2m\|f(\mathbf{k},\mathbf{k^{\prime}})-f({-\bf k},{\bf k}^{\prime})\|^{2}}{\hbar^{2}(\mathbf{k^{2}_{1}}+\mathbf{k^{2}_{2}}-\mathbf{k^{\prime 2}_{1}}-\mathbf{k^{\prime 2}_{2}})}n_{{\bf k}_{1}}n_{{\bf k}_{2}},$ (72) where ${\bar{f}}({\bf k})=\sum_{l\,\,odd}{\bar{f}}_{l}(k)$. The second order correction (III.2) can also be expressed in terms of the scattering amplitude by using Eq.(67). Replacing $U({\bf k}_{1}-{\bf k}^{\prime}_{1})=U({\bf k}-{\bf k}^{\prime})$ and $U({\bf k}^{\prime}_{2}-{\bf k}_{1})=U(-{\bf k}-{\bf k}^{\prime})$ by $f({\bf k}^{\prime},{\bf k})$ and $f(-{\bf k},{\bf k}^{\prime})$, respectively, we have: $\displaystyle E^{(2)}=\frac{1}{(2S)^{2}}$ $\displaystyle\sum_{\mathbf{k_{1}},\mathbf{k_{2}},\mathbf{k_{1}^{\prime}}}\Big{[}\frac{\left\|f(\mathbf{k^{\prime}},\mathbf{k})-f(\mathbf{k^{\prime}},-\mathbf{k})\right\|^{2}}{\hbar^{2}(\mathbf{k^{2}_{1}}+\mathbf{k^{2}_{2}}-\mathbf{k^{\prime 2}_{1}}-\mathbf{k^{\prime 2}_{2}})/2m}$ $\displaystyle\times n_{\mathbf{k_{1}}}n_{\mathbf{k_{2}}}(1-n_{\mathbf{k_{1}^{\prime}}})(1-n_{\mathbf{k_{2}^{\prime}}})\Big{]}.$ (73) Note that the divergent term proportional to $n_{{\bf k}_{1}}n_{{\bf k}_{2}}$ in Eq. (III.2) and the (divergent) second term of Eq. (III.2) exactly cancel each other, and the sum of the first and second order corrections can be represented as $E^{(1)}+E^{(2)}=\tilde{E}^{(1)}+\tilde{E}^{(2)},$ where $\tilde{E}^{(1)}=\frac{1}{S}\sum_{{\bf k}_{1},{\bf k}_{2}}{\bar{f}}({\bf k})n_{{\bf k}_{1}}n_{{\bf k}_{2}},$ (74) and $\displaystyle\tilde{E}^{(2)}=\frac{1}{(2S)^{2}}$ $\displaystyle\sum_{\mathbf{k_{1}},\mathbf{k_{2}},\mathbf{k^{\prime}_{1}}}\Big{\\{}\frac{\|f(\mathbf{k^{\prime}},\mathbf{k})-f({\bf k^{\prime}},-{\bf k})\|^{2}}{\hbar^{2}(\mathbf{k^{2}_{1}}+\mathbf{k^{2}_{2}}-\mathbf{k^{\prime 2}_{1}}-\mathbf{k^{\prime 2}_{2}})/2m}$ $\displaystyle\times n_{{\bf k}_{1}}n_{{\bf k}_{2}}[(1-n_{{\bf k}^{\prime}_{1}})(1-n_{{\bf k}^{\prime}_{2}})-1]\Big{\\}}.$ (75) The term $\tilde{E}^{(1)}$ originates from the two-body contributions to the interaction energy and can be quoted as the mean field term. The term $\tilde{E}^{(2)}$ is the many-body contribution, which is beyond mean field. It is worth noting that the term proportional to the product of four occupation numbers vanishes because its numerator is symmetrical and the denominator is antisymmetrical with respect to an interchange of ${\bf k}_{1},{\bf k}_{2}$ and ${\bf k}^{\prime}_{1},{\bf k}^{\prime}_{2}$. The terms containing a product of three occupation numbers, $n_{{\bf k}_{1}}n_{{\bf k}_{2}}n_{{\bf k}^{\prime}_{1}}$ and $n_{{\bf k}_{1}}n_{{\bf k}_{2}}n_{{\bf k}^{\prime}_{2}}$ are equal to each other because the denominator is symmetrical with respect to an interchange of ${\bf k}^{\prime}_{1}$ and ${\bf k}^{\prime}_{2}$. We thus reduce Eq. (III.2) to $\\!\\!\\!\\!\tilde{E}^{(2)}\\!\\!=\\!\\!-\frac{1}{2S^{2}}\\!\\!\\!\\!\sum_{\mathbf{k_{1}},\mathbf{k_{2}},\mathbf{k^{\prime}_{1}}}\\!\\!\\!\frac{2m\|f(\mathbf{k^{\prime}},\mathbf{k})\\!-\\!f({\bf k^{\prime}},\\!-{\bf k})\|^{2}}{\hbar^{2}(\mathbf{k^{2}_{1}}\\!+\\!\mathbf{k^{2}_{2}}\\!-\\!\mathbf{k^{\prime 2}_{1}}\\!\\!-\\!\mathbf{k^{\prime 2}_{2}})}n_{{\bf k}_{1}}\\!n_{{\bf k}_{2}}\\!n_{{\bf k}^{\prime}_{1}}.\\!\\!\\!\\!$ (76) Equations (74) and (76) allow a direct calculation of the ground state energy. With respect to the mean field term $\tilde{E}^{(1)}$ this is done in Appendix A. However, a direct calculation of the many-body correction $\tilde{E}^{(2)}$ is even a more tedious task than in the case of two-component fermions with a contact interaction. We therefore turn to the Abrikosov-Khalatnikov idea of calculating the ground state energy (and other thermodynamic quantities) through the interaction function of quasiparticles on the Fermi surface. ### III.3 Interaction function of quasiparticles The interaction function of quasiparticles $F({\bf k},{\bf k}^{\prime})$ is the second variational derivative of the total energy with respect to the distribution $n_{{\bf k}}$. The kinetic energy of our system is linear in $n_{{\bf k}}$ (see Eq. (59)), and the second variational derivative is related to the variation of the interaction energy $\tilde{E}$. We have Landau : $\delta\tilde{E}=\frac{1}{2S}\sum_{{\bf k},{\bf k}^{\prime}}F({\bf k},{\bf k}^{\prime})\delta n_{\mathbf{k}}\delta n_{\mathbf{k^{\prime}}},$ (77) where $\tilde{E}=\tilde{E}^{(1)}+\tilde{E}^{(2)}$, and the quantities $\tilde{E}^{(1)}$ and $\tilde{E}^{(2)}$ are given by equations (74) and (76). On the Fermi surface we should put $\|{\bf k}\|=\|{\bf k}^{\prime}\|=k_{F}$, so that the interaction function will depend only on the angle $\theta$ between ${\bf k}$ and $\mathbf{k^{\prime}}$. Hereinafter it will be denoted as $\tilde{F}(\theta)$. The contribution $\tilde{F}^{(1)}(\theta)=2S\delta\tilde{E}^{(1)}/\delta n_{{\bf k}}\delta n_{{\bf k}^{\prime}}$ is given by $\tilde{F}^{(1)}(\theta)=2f\left(\frac{\|{\bf k}-{\bf k}^{\prime}\|}{2}\right)=2\sum_{l\,odd}{\bar{f}}_{l}\left(k_{F}\|\sin{\frac{\theta}{2}}\|\right),$ (78) where ${\bar{f}}_{l}={\bar{f}}_{l}^{(1)}+{\bar{f}}_{l}^{(2)}$, and the amplitudes ${\bar{f}}_{l}^{(1)}$ and ${\bar{f}}_{l}^{(2)}$ follow from Eqs. (32) and (33) at $\|l\|=1$, and from Eqs. (42), (43) at $\|l\|>1$. We thus may write equation (42), ${\bar{f}}^{(1)}_{l}(k)=\frac{8\hbar^{2}}{m}\frac{1}{4l^{2}-1}kr_{},$ for any odd $l$, and $\displaystyle{\bar{f}}^{(2)}_{l}(k)=\frac{\pi\hbar^{2}}{2m}(kr_{})^{2}\times\begin{cases}\ln(\xi kr_{});&\text{$\|l\|=1$}\\\ -\frac{3}{\|l\|(l^{2}-1)(4l^{2}-1)};&\text{$\|l\|>1$}\end{cases}$ with $\xi$ from Eq. (30). Making a summation over all odd $l$ we obtain: ${\bar{f}}^{(1)}(k)=\sum_{l\,odd}f^{(1)}_{l}(k)=\frac{2\pi\hbar^{2}}{m}kr_{},\\\ $ (79) $\\!\\!\\!{\bar{f}}^{(2)}(k)\\!=\\!\\!\\!\sum_{l\,odd}f^{(2)}_{l}(k)\\!\\!=\\!\\!\frac{\pi\hbar^{2}}{m}(kr_{}\\!)^{2}\\!\\!\left[\ln(\xi kr_{}\\!)\\!-\\!\frac{25}{12}\\!+\\!3\\!\ln 2\\!\right]\\!\\!.\\!\\!\\!$ (80) Putting $k=k_{F}\|\sin(\theta/2)\|$ and substituting the results of equations (79) and (80) into Eq. (78) we find: $\displaystyle\tilde{F}^{(1)}(\theta)$ $\displaystyle=\frac{4\pi\hbar^{2}k_{F}r_{}}{m}\|\sin\frac{\theta}{2}\|+\frac{2\hbar^{2}}{m}(k_{F}r_{})^{2}$ $\displaystyle\times\pi\sin^{2}\frac{\theta}{2}\left[\ln\|\xi r_{}k_{F}\sin\frac{\theta}{2}\|-\frac{25}{12}+3\ln 2\right].$ (81) The many-body correction (76) we represent as $\tilde{E}^{(2)}=\tilde{E}_{1}^{(2)}+\tilde{E}_{2}^{(2)}$, where $\displaystyle\tilde{E}_{1}^{(2)}\\!\\!\\!=\\!\\!-\frac{8(\pi\hbar r_{})^{2}}{mS^{2}}\\!\\!\\!\\!\\!\sum_{\mathbf{k_{1}},\mathbf{k_{2}},\mathbf{k_{1}^{\prime}}}\\!\\!\\!\frac{\|\mathbf{k^{\prime}_{1}}\\!-\\!\mathbf{k_{1}}\|^{2}}{\mathbf{k^{2}_{1}}\\!+\\!\mathbf{k^{2}_{2}}\\!-\\!\mathbf{k^{\prime 2}_{1}}\\!-\\!\mathbf{k^{\prime 2}_{2}}}n_{\mathbf{k_{1}}}n_{\mathbf{k_{2}}}n_{\mathbf{k_{1}^{\prime}}}\\!,\\!\\!\\!$ (82) $\displaystyle\tilde{E}_{2}^{(2)}\\!\\!\\!=\\!\frac{8(\pi\hbar r_{})^{2}}{mS^{2}}\\!\\!\\!\\!\\!\sum_{\mathbf{k_{1}},\mathbf{k_{2}},\mathbf{k_{1}^{\prime}}}\\!\\!\\!\frac{\|\mathbf{k_{1}}\\!-\\!\mathbf{k^{\prime}_{1}}\|\\!\cdot\\!\|\mathbf{k_{2}}\\!-\\!\mathbf{k^{\prime}_{1}}\|}{\mathbf{k^{2}_{1}}\\!+\\!\mathbf{k^{2}_{2}}\\!-\mathbf{k^{\prime 2}_{1}}\\!-\\!\mathbf{k^{\prime 2}_{2}}}n_{\mathbf{k_{1}}}n_{\mathbf{k_{2}}}n_{\mathbf{k_{1}^{\prime}}}\\!,\\!\\!$ (83) and we used Eqs (46) and (49) for the scattering amplitudes. The contribution to the interaction function from $\tilde{E}_{1}^{(2)}$ is calculated in Appendix B and it reads: $\\!\\!\\!\tilde{F}_{1}^{(2)}\\!(\theta)\\!\\!=\\!\\!\frac{2\hbar^{2}(k_{F}r_{}\\!)^{2}}{m}\\!\left[\\!3\pi\\!+\\!2\pi\sin^{2}\frac{\theta}{2}\left(\\!\frac{4}{3}\\!-\\!\ln\\!\|\tan\frac{\theta}{2}\|\right)\right]\\!.\\!\\!$ (84) The contribution from $\tilde{E}_{2}^{(2)}$ is calculated in Appendix C. It is given by $\displaystyle\tilde{F}^{(2)}_{2}(\theta)=$ $\displaystyle\frac{2\hbar^{2}k^{2}_{F}r^{2}_{}}{m}\Big{\\{}-\sin^{2}\frac{\theta}{2}\left(\pi\ln 2+\frac{\pi}{2}-\pi\ln\|\sin\frac{\theta}{2}\|+4\ln\|\cos\frac{\theta}{2}\|-4\ln(1+\|\sin\frac{\theta}{2}\|)+\mathcal{G}(\theta)+\frac{4\arcsin\|\sin\frac{\theta}{2}\|-2\pi}{\|\cos\frac{\theta}{2}\|}\right)$ $\displaystyle-\frac{1}{\|\cos\frac{\theta}{2}\|}\left(\pi-2\arcsin\|\sin\frac{\theta}{2}\|+\|\sin\theta\|\right)-4\left[\cos^{2}\frac{\theta}{2}\ln\frac{1+\|\sin(\theta/2)\|}{1-\|\sin(\theta/2)\|}+2\|\sin\frac{\theta}{2}\|\right]\Big{\\}},$ (85) where $\mathcal{G}(\theta)=\int_{0}^{\pi}2\sin^{2}\varphi\ln\left(\sin\varphi+\sqrt{\sin^{2}\frac{\theta}{2}+\cos^{2}\frac{\theta}{2}\sin^{2}\varphi}\right)d\varphi,$ (86) so that $\frac{d\mathcal{G}(\theta)}{d\theta}=\frac{\pi}{2}\cot\frac{\theta}{2}-\frac{\|\sin(\theta/2)\|}{\sin(\theta/2)}\frac{1}{\cos(\theta/2)}+\frac{\arcsin\|\cos(\theta/2)\|}{\|\cos(\theta/2)\|}\left(\tan\frac{\theta}{2}-\cot\frac{\theta}{2}\right).$ (87) We thus have $\tilde{F}(\theta)=\tilde{F}^{(1)}(\theta)+\tilde{F}^{(2)}_{1}(\theta)+\tilde{F}^{(2)}_{2}(\theta)$, where $\tilde{F}^{(1)}$, $\tilde{F}^{(2)}_{1}$, $\tilde{F}^{(2)}_{2}$ follow from Eqs. (81), (84), and (85). This allows us to proceed with the calculation of thermodynamic quantities. ### III.4 Compressibility, ground state energy, and effective mass We first calculate the compressibility at $T=0$. On the basis of Eq. (III.1) we obtain: $\displaystyle\frac{\partial\mu}{\partial N}=\frac{2\pi\hbar^{2}}{mS}+\frac{1}{2\pi S}\int(1-\cos\theta)\left[\tilde{F}^{(1)}(\theta)+\tilde{F}_{1}^{(2)}(\theta)+\tilde{F}_{2}^{(2)}(\theta)\right]d\theta$ $\displaystyle=\frac{2\pi\hbar^{2}}{mS}+\frac{32\hbar^{2}}{3mS}k_{F}r_{}+\frac{3\pi\hbar^{2}}{2mS}(k_{F}r_{})^{2}\left(\ln[4\xi k_{F}r_{}]-\frac{3}{2}\right)+\frac{6\pi\hbar^{2}}{mS}(k_{F}r_{})^{2}-\frac{\hbar^{2}}{\pi mS}(k_{F}r_{})^{2}(30-8G+21\zeta(3)),$ (88) where $G=0.915966$ is the Catalan constant, and $\zeta(3)=1.20206$ is the Riemann zeta function. Calculating coefficients and recalling that $k_{F}=\sqrt{4\pi N/S}$ we represent the inverse compressibility following from Eq. (55) in a compact form: $\\!\\!\kappa^{-1}\\!=\\!\frac{\hbar^{2}k_{F}^{2}}{2m}\frac{N}{S}\\!\left(\\!1\\!+\\!\frac{16}{3\pi}k_{F}r_{}\\!+\\!\frac{3}{4}(k_{F}r_{})^{2}\ln(\zeta_{1}k_{F}r_{})\right)\\!,\\!\\!\\!\\!$ (89) where we obtain the coefficient $\zeta_{1}=2.16\exp(-2A)$ by using Eq. (30) for the coefficient $\xi$ which depends on the short-range behavior through the constant $A$ (see Eq. (18)). For the chemical potential and ground state energy we obtain: $\displaystyle\mu$ $\displaystyle=$ $\displaystyle\frac{2\pi\hbar^{2}N}{mS}+\frac{64\hbar^{2}N}{9mS}k_{F}r_{}+\frac{3\pi\hbar^{2}N}{4mS}(k_{F}r_{})^{2}\left(\ln[4\xi k_{F}r_{}]-\frac{7}{4}\right)+\frac{3\pi\hbar^{2}N}{mS}(k_{F}r_{})^{2}-\frac{\hbar^{2}N}{2\pi mS}(k_{F}r_{})^{2}(30-8G+21\zeta(3))$ (90) $\displaystyle=$ $\displaystyle\frac{\hbar^{2}k_{F}^{2}}{2m}\left(1+\frac{32}{9\pi}k_{F}r_{}+\frac{3}{8}(k_{F}r_{})^{2}\ln(\zeta_{2}k_{F}r_{})\right).$ $\displaystyle\frac{E}{N}$ $\displaystyle=$ $\displaystyle\frac{\pi\hbar^{2}N}{mS}+\frac{128\hbar^{2}N}{45mS}k_{F}r_{}+\frac{\pi\hbar^{2}N}{4mS}(k_{F}r_{})^{2}\left(\ln[4\xi k_{F}r_{}]-\frac{23}{12}\right)+\frac{\pi\hbar^{2}N}{mS}-\frac{\hbar^{2}N}{6\pi mS}(30-8G+21\zeta(3))$ (91) $\displaystyle=$ $\displaystyle\frac{\hbar^{2}k_{F}^{2}}{4m}\left(1+\frac{128}{45\pi}k_{F}r_{}+\frac{1}{4}(k_{F}r_{})^{2}\ln(\zeta_{3}k_{F}r_{})\right),$ with numerical coefficients $\zeta_{2}=1.68\exp(-2A)$ and $\zeta_{3}=1.43\exp(-2A)$. Note that the first term in the second line of Eq. (III.4) and the first terms in the first lines of Eqs. (90) and (91) represent the contributions of the kinetic energy, the second and third terms correspond to the contributions of the mean field part of the interaction energy, and the last two terms are the contributions of the many-body effects. The effective mass is calculated in a similar way by using Eq. (54): $\displaystyle\frac{1}{m^{}}\\!=\\!\frac{1}{m}\\!-\\!\frac{1}{(2\pi\hbar)^{2}}\int_{0}^{2\pi}\\!(F^{(1)}(\theta)\\!+\\!F^{(2)}_{1}(\theta)\\!+\\!F^{(2)}_{2}(\theta))\cos\theta d\theta\\!\\!$ $\displaystyle\\!=\\!\\!\frac{1}{m}\\!\\!\left[\\!1\\!\\!+\\!\frac{4k_{F}r_{\\!}}{3\pi}\\!+\\!\frac{(k_{F}r_{\\!})^{\\!2}}{4}\\!\\!\left(\\!\ln{\\![4k_{F}r_{\\!}\xi]}\\!\\!-\\!\frac{8}{3}\\!+\\!\frac{48G\\!\\!-\\!\\!20\\!-\\!\\!14\zeta(3)}{\pi^{2}}\\!\right)\\!\right]\\!\\!$ $\displaystyle\\!=\frac{1}{m}\left[1+\frac{4}{3\pi}k_{F}r_{}+\frac{1}{4}(k_{F}r_{})^{2}\ln(\zeta_{4}k_{F}r_{})\right],$ (92) where the numerical coefficient $\zeta_{4}=0.65\exp(-2A)$. Note that if the potential $U(r)$ has the dipole-dipole form (1) up to very short distances, we have to put $A=0$ in the expressions for the coefficients $\zeta_{1},\,\zeta_{2},\,\zeta_{3},\,\zeta_{4}$. Considering the quasi2D regime, this will be the case for $r_{}$ greatly exceeding the length of the sample in the tightly confined direction, $l_{0}$. Then, as one can see from equations (89), (90), (91), and (92), the terms proportional to $(k_{F}r_{})^{2}$ are always negative in the considered limit $k_{F}r_{}\ll 1$. These terms may become significant for $k_{F}r_{}>0.3$. ## IV Zero sound In the collisionless regime of the Fermi liquid at very low temperatures, where the frequency of variations of the momentum distribution function greatly exceeds the relaxation rate of quasiparticles, one has zero sound waves. For these waves, variations $\delta n({\bf q},{\bf r},t)$ of the momentum distribution are related to deformations of the Fermi surface, which remains a sharp boundary between filled and empty quasiparticle states. At $T\rightarrow 0$ the equilibrium distribution $n_{\bf q}$ is the step function (51), so that $\partial n_{\bf q}/\partial{\bf q}=-{\bf n}\delta(q-k_{F})=-\hbar{\bf v}\delta(\epsilon_{q}-\epsilon_{F})$, where ${\bf v}=v_{F}{\bf n}$, with ${\bf n}$ being a unit vector in the direction of ${\bf q}$. Then, searching for the variations $\delta n$ in the form: $\delta n({\bf q},{\bf r},t)=\delta(\epsilon_{q}-\epsilon_{F})\nu({\bf n})\exp{i({\bf kr}-\omega t)}$ and using Eq. (52), from the kinetic equation in the collisionless regime: $\displaystyle\frac{\partial\delta n}{\partial t}+\mathbf{v}\cdot\frac{\partial\delta n}{\partial\mathbf{r}}-\frac{\partial n_{\bf q}}{\partial\mathbf{q}}\cdot\frac{\partial\delta\epsilon_{q}}{\hbar\partial\mathbf{r}}=0,$ one obtains an integral equation for the function $\nu({\bf n})$ representing displacements of the Fermi surface in the direction of ${\bf n}$ Landau : $\displaystyle(\omega- v_{F}\mathbf{n}\cdot\mathbf{k})\nu(\mathbf{n})=\frac{k_{F}}{(2\pi)^{2}\hbar}\mathbf{n}\cdot\mathbf{k}\int F(k_{F}\mathbf{n},k_{F}\mathbf{n^{\prime}})\nu(\mathbf{n^{\prime}})d{\bf n}^{\prime}.$ Introducing the velocity of zero sound $u_{0}=\omega/k$ and dividing both sides of this equation by $v_{F}k$ we have: $(s-\cos\theta)\nu(\theta)=\frac{m^{}\cos\theta}{(2\pi\hbar)^{2}}\int_{0}^{2\pi}\tilde{F}(\theta-\theta^{\prime})\nu(\theta^{\prime})d\theta^{\prime},$ (93) where $s=u_{0}/v_{F}$, and $\theta,\,\theta^{\prime}$ are the angles between ${\bf k}$ and ${\bf n},\,{\bf n}^{\prime}$, so that $\theta-\theta^{\prime}$ is the angle between ${\bf n}$ and ${\bf n}^{\prime}$. The dependence of the interaction function of quasiparticles $\tilde{F}=\tilde{F}^{(1)}+\tilde{F}^{(2)}_{1}+\tilde{F}^{(2)}_{2}$ on $(\theta-\theta^{\prime})$ follows from Eqs. (81), (84), and (85) in which one has to replace $\theta$ by $(\theta-\theta^{\prime})$. The solution of equation (93) gives the function $\nu(\theta)$ and the velocity of zero sound $u_{0}$, and in principle one may obtain several types of solutions. It is important to emphasize that undamped zero sound requires the condition $s>1$, i.e. the sound velocity should exceed the Fermi velocity Landau . We will discuss this issue below. For solving Eq. (93) we represent the interaction function $\tilde{F}$ as a sum of the part proportional to $k_{F}r_{}$ and the part proportional to $(k_{F}r_{})^{2}$. As follows from Eqs. (81), (84), and (85), we have: $\\!\\!\\!\\!\tilde{F}(\theta\\!-\\!\theta^{\prime})\\!=\\!\frac{4\pi\hbar^{2}}{m}k_{F}r_{}\\!\left\|\sin\frac{\theta\\!-\\!\theta^{\prime}}{2}\right\|+\frac{2\hbar^{2}}{m}(k_{F}r_{}\\!)^{2}\Phi(\theta\\!-\\!\theta^{\prime}),\\!\\!\\!\\!$ (94) where the function $\Phi(\theta-\theta^{\prime})$ is given by the sum of three terms. The first one is the term in the second line of Eq. (81), the second term is the expression in the square brackets in Eq. (84), the third term is the one in curly brackets in Eq. (85), and we should replace $\theta$ by $(\theta-\theta^{\prime})$ in all these terms. It is important that the function $\Phi(\theta-\theta^{\prime})$ does not have singularities and $\Phi(0)=\Phi(\pm 2\pi)=2\pi$. Using Eq. (94) the integral equation (93) is reduced to the form: $\displaystyle(s-\cos\theta)\nu(\theta)$ $\displaystyle=\beta\cos\theta\int_{0}^{2\pi}\nu(\theta^{\prime})\left\|\sin\frac{\theta-\theta^{\prime}}{2}\right\|d\theta^{\prime}$ $\displaystyle+\frac{\beta^{2}m}{2m^{}}\cos\theta\int_{0}^{2\pi}\nu(\theta^{\prime})\Phi(\theta-\theta^{\prime})d\theta^{\prime},$ (95) where $\beta=(m^{}/\pi m)k_{F}r_{}\ll 1$. We now represent the function $\nu(\theta)$ as $\nu(\theta)=\sum_{p=0}^{\infty}C_{p}\cos p\theta.$ (96) Then, integrating over $d\theta^{\prime}$ in Eq. (95), multiplying both sides of this equation by $\cos{j\theta}$ and integrating over $d\theta$, we obtain a system of linear equations for the coefficients $C_{j}$. We write this system for the coefficients $\eta_{j}=C_{j}(1-\beta/(j^{2}-1/4))$, so that $C_{j}=\eta_{j}(1+\beta U_{j})$, where $U_{j}=(j^{2}-1/4-\beta)^{-1}$. The system reads: $\displaystyle\\!\\!\\!\\!(s-1)(1+\beta U_{0})\eta_{0}+[\eta_{0}-\frac{1}{2}\eta_{1}]+\beta U_{0}\eta_{0}=\frac{\beta^{2}}{2}{\bar{\Phi}}_{0};\\!$ (97) $\displaystyle\\!\\!\\!\\!(s\\!-\\!1)(1\\!+\\!\beta U_{1})\eta_{1}\\!+\\![\eta_{1}-\eta_{0}-\frac{1}{2}\eta_{2}]+\beta U_{1}\eta_{1}\\!\\!=\\!\frac{\beta^{2}}{2}{\bar{\Phi}}_{1};$ (98) $\displaystyle\\!\\!\\!\\!(\\!s\\!\\!-\\!\\!1\\!)(\\!1\\!\\!+\\!\\!\beta U_{j}\\!)\eta_{j}\\!\\!+\\!\\![\eta_{j}\\!\\!-\\!\\!\frac{1}{2}\\!(\\!\eta_{j\\!-\\!1}\\!\\!+\\!\eta_{j\\!+\\!1)\\!}\\!]\\!\\!+\\!\\!\beta U_{j}\eta_{j}\\!\\!=\\!\\!\frac{\beta^{2}}{2}\\!{\bar{\Phi}}_{j}\\!;\,j\\!\\!\geq\\!\\!2,\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!$ (99) where $\\!\\!{\bar{\Phi}}_{j}\\!=\\!\frac{\tilde{C}_{j}}{\pi}\\!\\!\int_{0}^{2\pi}\\!\\!\\!\\!\\!\\!\\!\\!\cos\theta\,\cos j\theta d\theta\\!\\!\int_{0}^{2\pi}\\!\\!\sum_{p=0}^{\infty}C_{p}\cos p\theta^{\prime}\Phi(\theta-\theta^{\prime})d\theta^{\prime}\\!,\\!\\!\\!\\!\\!\\!\\!$ (100) with $\tilde{C}_{j}=1$ for $j\geq 1$ and $\tilde{C}_{0}=1/2$, and we put $m^{}=m$ in the terms proportional to $\beta^{2}$. In the weakly interacting regime the velocity of zero sound is close to the Fermi velocity and, hence, we have $(s-1)\ll 1$ (see, e.g. Landau ). Since $\beta\ll 1$, we first find coefficients $\eta_{j}$ omitting the terms proportional to $\beta$ and $\beta^{2}$ in Eqs. (97)-(99). For $j\gg 1$ equation (99) then becomes: $(s-1)\eta_{j}-\frac{1}{2}\frac{d^{2}\eta_{j}}{dj^{2}}=0,$ and searching for $s>1$ we may write $\eta_{j}\simeq\exp\\{-\sqrt{2(s-1)}j\\};\,\,\,j\gg 1.$ (101) If $j\ll 1/\sqrt{s-1}$, then we may also omit the terms proportional to $(s-1)$ in the system of linear equations for $\eta_{j}$ (97)-(99). The system then takes the form: $\displaystyle\eta_{0}-\frac{1}{2}\eta_{1}=0;$ $\displaystyle\eta_{1}-\eta_{0}-\frac{1}{2}\eta_{2}=0;$ $\displaystyle\eta_{j}-\frac{1}{2}[\eta_{j-1}+\eta_{j+1}]=0;\,\,\,\,j\geq 2.$ Without loss of generality we may put $\eta_{0}=1/2$. This immediately gives $\eta_{j}=1$ for $j\geq 1$, which is consistent with Eq. (101) at $j\ll 1/\sqrt{s-1}$. We thus have the zero order solution: $\begin{cases}\eta_{0}=1/2;\\\ \eta_{j}=1;\,\,\,1\leq j\ll 1/\sqrt{s-1}.\end{cases}$ (102) In order to find the coefficients $\eta_{j}$ taking into account the terms linear in $\beta$, we consider $j$ such that $\beta U_{j}\sim\beta/j^{2}\gg(s-1)$, i.e. $j\ll\sqrt{\beta/(s-1)}$. Then we may omit the terms proportional to $(s-1)$ in equations (97)-(99). Omitting also the terms proportional to $\beta^{2}$ this system of equations becomes: $\displaystyle\eta_{0}-\frac{1}{2}\eta_{1}+\beta U_{0}\eta_{0}=0;$ (103) $\displaystyle\eta_{1}-\eta_{0}-\frac{1}{2}\eta_{2}+\beta U_{1}\eta_{1}=0;$ (104) $\displaystyle\eta_{j}-\frac{1}{2}[\eta_{j-1}+\eta_{j+1}]+\beta U_{j}\eta_{j}=0;\,\,j\geq 2.$ (105) Putting again $\eta_{0}=1/2$ the solution of these equations reads: $\displaystyle\eta_{1}=1+\beta U_{0};$ $\displaystyle\eta_{j}=1+\beta jU_{0}+2\beta\sum_{p=1}^{j-1}(j-p)U_{p};\,\,\,j\geq 2.$ Confining ourselves to terms linear in $\beta$ we put $U_{p}=1/(p^{2}-1/4)$ and, hence, $U_{0}=-4$. Then, using the relation $\sum_{p=1}^{j-1}\frac{1}{p^{2}-1/4}=\frac{4(j-1)}{2j-1},$ which is valid for $j\geq 2$, we obtain: $\displaystyle\begin{cases}&\eta_{0}=\frac{1}{2};\\\ \\\ &\eta_{1}=1-4\beta;\\\ \\\ &\eta_{j}=1-2\beta\left\\{\frac{2j}{2j-1}+\sum_{p=1}^{j-1}\frac{p}{p^{2}-1/4}\right\\};\,\,\,j\geq 2.\end{cases}$ (106) For $j\gtrsim\sqrt{\beta/(s-1)}$ we should include the terms proportional to $(s-1)$ in Eq. (105). This leads to the solution in the form of the decaying Bessel function: $\eta_{j}\simeq\sqrt{2(s-1)/\pi}K_{\sqrt{1/4+\beta}}(\sqrt{s-1}j)$, which for small $\beta$ is practically equivalent to Eq. (101). We now make a summation of equations (97)-(99) from $j=0$ to $j=j_{}\ll 1/\sqrt{s-1}$. The summation of the second terms of these equations gives $\sqrt{(s-1)/2}$, whereas the contribution of the terms proportional to $(s-1)$ is much smaller and will be omitted. The sums $\sum_{j=0}^{j_{}}U_{j}\eta_{j}$ and $\sum_{j=0}^{j_{}}{\bar{\Phi}}_{j}$ converge at $j\ll 1/\sqrt{s-1}$, and the upper limit of summation in these terms can be formally replaced by infinity. We thus obtain a relation: $\sqrt{\frac{s-1}{2}}+\sum_{j=0}^{\infty}\beta\eta_{j}U_{j}-\frac{\beta^{2}}{2}\sum_{j=0}^{\infty}{\bar{\Phi}}_{j}=0.$ (107) Confining ourselves to contributions up to $\beta^{2}$, in the second term on the left hand side of Eq. (107) we use coefficients $\eta_{j}$ given by Eqs. (106), and write $U_{j}=1/(j^{2}-1/4)+\beta/(j^{2}-1/4)^{2}$. In the expressions for ${\bar{\Phi}}_{j}$ we use $C_{0}=1/2$ and $C_{p}=1$ for $p\geq 1$. We then have: $\\!\\!\sum_{j=0}^{\infty}\beta\eta_{j}U_{j}\\!=\\!-2\beta\\!+\\!\sum_{j=1}^{\infty}\frac{\beta}{\\!j^{2}\\!-\\!1/4}\\!+\\!\beta^{2}\\{\\!8\\!+\\!S_{1}\\!-\\!2S_{2}\\!-2S_{3}\\!\\}.\\!\\!\\!\\!$ (108) The contribution linear in $\beta$ vanishes because $\sum_{j=1}^{\infty}1/(j^{2}-1/4)=2$. The quantities $S_{1},\,S_{2}$, and $S_{3}$ are given by $\displaystyle\\!\\!\\!S_{1}=\sum_{j=1}^{\infty}\frac{1}{(j^{2}-1/4)^{2}}=\pi^{2}-8;$ $\displaystyle\\!\\!\\!S_{2}\\!=\\!\sum_{j=2}^{\infty}\frac{1}{j^{2}\\!-\\!1/4}\\!\sum_{p=1}^{j-1}\frac{p}{p^{2}\\!-\\!1/4}\\!=\\!\sum_{j=1}\frac{j}{(j^{2}\\!-\\!1/4)(j\\!+\\!1/2)};$ $\displaystyle\\!\\!\\!S_{3}=\sum_{j=1}^{\infty}\frac{j}{(j-1/2)(j^{2}-1/4)},$ so that $S_{2}+S_{3}=\sum_{j=1}^{\infty}\frac{2j^{2}}{(j^{2}-1/4)^{2}}=\frac{\pi^{2}}{2}.$ We thus see that the contribution quadratic in $\beta$ also vanishes because the term in the curly brackets in Eq. (108) is exactly equal to zero. Hence, we have $\sum_{j=0}^{\infty}\beta\eta_{j}U_{j}=0$ up to terms proportional to $\beta^{2}$. The sum in the third term on the left hand side of Eq. (107), after putting $C_{0}=1/2$ and $C_{p}=1$ for $p\geq 1$ in the relations for ${\bar{\Phi}}_{j}$, reduces to $\displaystyle\sum_{j=0}^{\infty}{\bar{\Phi}}_{j}$ $\displaystyle=\frac{1}{4\pi}\\!\sum_{j=-\infty}^{\infty}\\!\int_{0}^{2\pi}\\!\\!\\!\\!\cos\theta\,\cos{j\theta}d\theta$ $\displaystyle\times\sum_{p=-\infty}^{\infty}\\!\int_{0}^{2\pi}\\!\\!\\!\\!\cos{p\theta^{\prime}}\Phi(\theta\\!-\\!\theta^{\prime})d\theta^{\prime}.\\!\\!\\!\\!$ (109) For $\theta$ in the interval $0\leq\theta\leq 2\pi$ we have a relation: $\sum_{j=-\infty}^{\infty}\cos{j\theta}=\pi[\delta(\theta)+\delta(\theta-2\pi)],$ which transforms Eq. (IV) to $\sum_{j=0}^{\infty}{\bar{\Phi}}_{j}=\frac{\pi}{4}[2\Phi(0)+\Phi(2\pi)+\Phi(-2\pi)]=2\pi^{2},$ (110) and equation (107) becomes: $\displaystyle\sqrt{\frac{s-1}{2}}-\beta^{2}\pi^{2}=0.$ This gives $s=1+2(\beta\pi)^{4}$, and recalling that $\beta=k_{F}r_{}/\pi$ (we put $m^{}=m$) we obtain for the velocity of zero sound: $u_{0}=v_{F}[1+2(k_{F}r_{})^{4}].$ (111) Note that in contrast to the 3D two-species Fermi gas with a weak repulsive contact interaction (scattering length $a$), where the correction $(u_{0}-v_{F})$ exponentially depends on $k_{F}a$, for our 2D dipolar gas we obtained a power law dependence. This is a consequence of dimensionality of the system. It is important that confining ourselves to only the leading part of the interaction function $\tilde{F}$, which is proportional to $k_{F}r_{}$ and is given by the first term of Eq. (81), we do not obtain undamped zero sound ($s>1$) comment . This corresponds to omitting the terms $\beta^{2}{\bar{\Phi}}_{j}/2$ in equations (97)-(99) and is consistent with numerical calculations Baranov . Only the many-body corrections to the interaction function of quasiparticles, given by equations (84) and (85), provide non-zero positive values of $\Phi(0)$ and $\Phi(\pm 2\pi)$, thus leading to a positive value of $(u_{0}-v_{F})$. One then sees that many-body effects are crucial for the propagation of zero sound. In principle, we could obtain the result of Eq. (111) in a simpler way, similar to that used for the two-species Fermi gas with a weak repulsive interaction (see, e.g. Landau ). Representing the function $\nu(\theta)$ as $\nu(\theta)=\cos{\theta}\tilde{\nu}(\theta)/(s-\cos{\theta})$ we transform Eq. (93) to the form: $\tilde{\nu}(\theta)=\frac{m^{}}{(2\pi\hbar)^{2}}\int_{0}^{2\pi}\frac{\tilde{F}(\theta-\theta^{\prime})\tilde{\nu}(\theta^{\prime})\cos{\theta^{\prime}}}{s-\cos{\theta^{\prime}}}d\theta^{\prime}.$ (112) Since $s$ is close to unity, it looks reasonable to assume that the main contribution to the integral in Eq. (112) comes from $\theta^{\prime}$ close to zero and to $2\pi$. Using the fact that $\tilde{F}(\theta)=\tilde{F}(2\pi-\theta)$ we then obtain: $\tilde{\nu}(\theta)=\frac{m^{}\tilde{F}(\theta)\tilde{\nu}(0)}{4\pi\hbar^{2}}\sqrt{\frac{2}{s-1}}.$ (113) We now take the limit $\theta\rightarrow 0$ and substitute $\tilde{F}(0)=(4\pi\hbar^{2}/m)(k_{F}r_{})^{2}$ as follows froms Eqs. (81), (84), and (85). Putting $m^{}=m$ we then obtain $s=1+2(k_{F}r_{})^{4}$ and arrive at Eq. (111). Note, however, that for very small $\theta$ or $\theta$ very close to $2\pi$ the dependence $\tilde{F}(\theta)$ is very steep. For $\theta\rightarrow 0$ the leading part of the interaction function, which is linear in $k_{F}r_{}$, vanishes, and only the quadratic part contributes to $\tilde{F}(0)$. Therefore, strictly speaking the employed procedure of calculating the integral in Eq. (112) is questionable for very small $\theta$. This prompted us to make the analysis based on representing $\nu(\theta)$ in the form (96) and on solving the system of linear equations (97)-(99). Equation (112) is useful for understanding why undamped zero sound requires the condition $s>1$ so that $u_{0}>v_{F}$. For $s<1$ there is a pole in the integrand of Eq. (112), which introduces an imaginary part of the integral. As a result, the zero sound frequency $\omega$ will also have an imaginary part at real momenta $k$, which means the presence of damping (see, e.g. Landau ). We could also consider an odd function $\nu(\theta)$, namely such that $\nu(2\pi-\theta)=-\nu(\theta)$ and $\nu(0)=\nu(2\pi)=0$. In this case, however, we do not obtain an undamped zero sound. ## V Concluding remarks We have shown that (single-component) fermionic polar molecules in two dimensions constitute a novel Fermi liquid, where many-body effects play an important role. For dipoles oriented perpendicularly to the plane of translational motion, the many-body effects provide significant corrections to thermodynamic functions. Revealing these effects is one of the interesting goals of up-coming experimental studies. The investigation of the full thermodynamics of 2D polar molecules, including many-body effects, can rely on the in-situ imaging technique as it has been done for two-component atomic Fermi gases Salomon1 ; Salomon2 . This method can also be extended to 2D systems for studying thermodynamic quantities Zwierlein ; Dalibard2 . Direct imaging of a 3D pancake-shaped dipolar molecular system has been recently demonstrated at JILA Ye2 . For 2D polar molecules discussed in our paper, according to equations (89)-(91), the contribution of many-body corrections proportional to $(k_{F}r_{})^{2}$ can be on the level of $10\%$ or $20\%$ for $k_{F}r_{}$ close to $0.5$. Thus, finding many-body effects in their thermodynamic properties looks feasible. It is even more important that the many-body effects are responsible for the propagation of zero sound waves in the collisionless regime of the 2D Fermi liquid of polar molecules with dipoles perpendicular to the plane of translational motion. This is shown in Section IV of our paper, whereas mean- field calculations do not find undamped zero sound Baranov . Both collisionless and hydrodynamic regimes are achievable in on-going experiments. This is seen from the dimensional estimate of the relaxation rate of quasiparticles. At temperatures $T\ll\epsilon_{F}$ the relaxation of a non- equilibrium distribution of quasiparticles occurs due to binary collisions of quasiparticles with energies in a narrow interval near the Fermi surface. The width of this interval is $\sim T$ and, hence, the relaxation rate contains a small factor $(T/\epsilon_{F})^{2}$ (see, e.g. Landau ). Then, using the Fermi Golden rule we may write the inverse relaxation time as $\tau^{-1}\sim(g_{eff}^{2}/\hbar)(m/\hbar^{2})n(T/\epsilon)^{2}$, where $n$ is the 2D particle density, the quantity $\sim m/\hbar^{2}$ represents the density of states on the Fermi surface, and the quantity $g_{eff}$ is the effective interaction strength. Confining ourselves to the leading part of this quantity, from Eqs. (III.2) and (79) we have $g_{eff}\sim\hbar^{2}k_{F}r_{}/m$. We thus obtain: $\frac{1}{\tau}\sim\frac{\hbar n}{m}(k_{F}r_{})^{2}\left(\frac{T}{\epsilon_{F}}\right)^{2}.$ (114) Note that as $\epsilon_{F}\approx\hbar^{2}k_{F}^{2}/2m\approx 2\pi\hbar^{2}n/m$, for considered temperatures $T\ll\epsilon_{F}$ the relaxation time $\tau$ is density independent. Excitations with frequencies $\omega\ll 1/\tau$ are in the hydrodynamic regime, where on the length scale smaller than the excitation wavelength and on the time scale smaller than $1/\omega$ the system reaches a local equilibrium. On the other hand, excitations with frequencies $\omega\gg 1/\tau$ are in the collisionless regime. Assuming $T\sim 10$nK, for KRb molecules characterized by the dipole moment $d\simeq 0.25$ D in the electric field of $5$kV/cm as obtaind in the JILA experiments, we find $\tau$ on the level of tens of milliseconds. The required condition $T\ll\epsilon_{F}$ is satisfied for $\epsilon_{F}\gtrsim 70$ nK, which corresponds to $n\gtrsim 2\cdot 10^{8}$ cm-2. In such conditions excitations with frequencies of the order of a few Hertz or lower will be in the hydrodynamic regime, and excitations with larger frequencies in the collisionless regime. The velocity of zero sound is practically equal to the Fermi velocity $v_{F}=\hbar k_{F}/m^{}$. This is clearly seen from Eq. (111) omitting a small correction proportional to $(k_{F}r_{})^{4}$. Then, using Eq. (92) for the effective mass and retaining only corrections up to the first order in $k_{F}r_{}$, we have: $u_{0}\simeq\frac{\hbar k_{F}}{m}\left(1+\frac{4}{3\pi}k_{F}r_{}\right).$ (115) In the hydrodynamic regime the sound velocity is: $u=\sqrt{\frac{N}{m}\frac{\partial\mu}{\partial N}}\simeq\frac{\hbar k_{F}}{m}\left(1+\frac{8}{3\pi}k_{F}r_{}\right),$ (116) where we used Eq. (III.4) for $\partial\mu/\partial N$ and retained corrections up to the first order in $k_{F}r_{}$. The hydrodynamic velocity $u$ is slightly larger than the velocity of zero sound $u_{0}$, and the difference is proportional to the interaction strength. This is in sharp contrast with the 3D two-component Fermi gas, where $u_{0}\approx v_{F}>u\approx v_{F}/\sqrt{3}$. We thus see that it is not easy to distinguish between the hydrodynamic and collisionless regimes from the measurement of the sound velocity. A promising way to do so can be the observation of damping of driven excitations, which in the hydrodynamic regime is expected to be slower. Another way is to achieve the values of $k_{F}r_{}$ approaching unity and still discriminate between $u_{0}$ and $u$ in the measurement of the sound velocity. For example, in the case of dipoles perpendicular to the plane of their translational motion the two velocities are different from each other by about $20\%$ at $k_{F}r_{}\simeq 0.5$. These values of $k_{F}r_{}$ are possible if the 2D gas of dipoles still satisfies the Pomeranchuk criteria of stability. These criteria require that the energy of the ground state corresponding to the occupation of all quasiparticle states inside the Fermi sphere, remains the minimum energy under an arbitrarily small deformation of the Fermi sphere. The generalization of the Pomeranchuk stability criteria to the case of the 2D single-component Fermi liquid with dipoles perpendicular to the plane of their translational motion reads: $1+\frac{m^{}}{(2\pi\hbar)^{2}}\int_{0}^{2\pi}\tilde{F}(\theta)\cos{j\theta}\,d\theta>0,$ (117) and this inequality should be satisfied for any integer $j$. As has been found in Ref. Baranov , the Pomeranchuk stability criteria (117) are satisfied for $k_{F}r_{}$ approaching unity from below if the interaction function of quasiparticles contains only the first term of Eq. (81), which is the leading mean field term. We have checked that the situation with the Pomeranchuk stability does not change when we include the full expression for the interaction function, $\tilde{F}(\theta)=\tilde{F}^{(1)}(\theta)+\tilde{F}^{(2)}_{1}(\theta)+\tilde{F}^{(2)}_{2}(\theta)$, following from Eqs. (81), (84), and (85). Thus, achieving $k_{F}r_{}$ approaching unity looks feasible. For KRb molecules with the (oriented) dipole moment of $0.25$ D the value $k_{F}r_{}\approx 0.5$ requires densities $n\approx 2\cdot 10^{8}$ cm-2. Finally, we would like to emphasize once more that our results are applicable equally well for the quasi2D regime, where the dipole-dipole length $r_{}$ is of the order of or smaller than the confinement length $l_{0}=(\hbar/m\omega_{0})^{1/2}$, with $\omega_{0}$ being the frequency of the tight confinement. The behavior at distances $r\lesssim l_{0}$ is contained in the coefficient $A$ defined in Eq. (18). Therefore, the results for the velocity of zero sound which is independent of $A$, are universal in the sense that they remain unchanged when going from $r_{}\gg l_{0}$ to $r_{}\lesssim l_{0}$. The only requirement is the inequality $k_{F}l_{0}\ll 1$. It is, however, instructive to examine the ratio $r_{}/l_{0}$ that can be obtained in experiments with ultracold polar molecules. Already in the JILA experiments using the tight confinement of KRb molecules with frequency $\omega_{0}\approx 30$ kHz and achieving the average dipole moment $d\simeq 0.25$ D in electric fields of $5$ kV/cm, we have $r_{}\simeq 100$ nm and $l_{0}\simeq 50$ nm so that $r_{}/l_{0}\simeq 2$. A decrease of the confinement frequency to $5$ kHz and a simultaneous decrease of the dipole moment by a factor of 2 leads to $r_{}/l_{0}\sim 0.2$. On the other hand, for $d$ close to $0.5$ (which is feasible to obtain for other molecules) one can make the ratio $r_{}/l_{0}$ close to $10$ at the same confinement length. ## Acknowledgements We are grateful to M.A. Baranov and S.I. Matveenko for fruitful discussions. We acknowledge support from EPSRC Grant No. EP/F032773/1, from the IFRAF Institute, and from the Dutch Foundation FOM. This research has been supported in part by the National Science Foundation under Grant No. NSF PHYS05-51164. LPTMS is a mixed research unit No. 8626 of CNRS and Université Paris Sud. ## Appendix A Direct calculation of the first order contribution to the interaction energy For directly calculating the first order (mean field) contribution to the interaction energy $\tilde{E}^{(1)}$ (74), we represent it as $\tilde{E}^{(1)}=\tilde{E}^{(1)}_{1}+\tilde{E}^{(1)}_{2}$ where $\displaystyle\tilde{E}^{(1)}_{1}=\int{\bar{f}}^{(1)}\left(\frac{\|{\bf k}_{1}-{\bf k}_{2}\|}{2}\right)n_{{\bf k}_{1}}n_{{\bf k}_{2}}\frac{d^{2}k_{1}d^{2}k_{2}}{(2\pi)^{4}},$ (118) $\displaystyle\tilde{E}^{(1)}_{2}=\int{\bar{f}}^{(2)}\left(\frac{\|{\bf k}_{1}-{\bf k}_{2}\|}{2}\right)n_{{\bf k}_{1}}n_{{\bf k}_{2}}\frac{d^{2}k_{1}d^{2}k_{2}}{(2\pi)^{4}},$ (119) and the amplitudes ${\bar{f}}^{(1)}$ and ${\bar{f}}^{(2)}$ are given by Eqs. (79) and (80), respectively. In the calculation of the integrals for $\tilde{E}^{(1)}_{1}$ and $\tilde{E}^{(1)}_{2}$ we turn to the variables ${\bf x}=({\bf k}_{1}-{\bf k}_{2})/2k_{F}$ and ${\bf y}=({\bf k}_{1}+{\bf k}_{2})/2k_{F}$, so that $d^{2}k_{1}d^{2}k_{2}=8\pi k_{F}^{4}d^{2}xd^{2}yd\varphi$, where $\varphi$ is the angle between the vectors ${\bf x}$ and ${\bf y}$, and the integration over $d\varphi$ should be performed from $0$ to $2\pi$. The distribution functions $n_{{\bf k}_{1}}$ and $n_{{\bf k}_{2}}$ are the step functions (51). The integration over $dk_{1}$ and $dk_{2}$ from $0$ to $k_{F}$ corresponds to the integration over $dy$ from $0$ to $y_{0}(x,\varphi)=-x\|\cos\varphi\|+\sqrt{1-x^{2}\sin 2\varphi}$ and over $dx$ from $0$ to $1$. Using Eq. (79) we reduce Eq. (118) to $\tilde{E}^{(1)}_{1}=\frac{S\hbar^{2}k_{F}^{4}}{\pi^{2}m}k_{F}r_{}I_{1},$ (120) where $\displaystyle I_{1}=\int_{0}^{2\pi}d\varphi\int_{0}^{1}x^{2}dx\int_{0}^{y_{0}(x,\varphi)}ydy=\frac{1}{2}\int_{0}^{2\pi}d\varphi\int_{0}^{1}x^{2}dx$ $\displaystyle\times[1-2\|\cos\varphi\|\sqrt{1-x^{2}\sin^{2}\varphi}+x^{2}(\cos^{2}\varphi-\sin^{2}\varphi)].$ The last term of the second line vanishes, and the integration of the first two terms over $d\varphi$ and $dx$ gives: $I_{1}=\int_{0}^{1}x^{2}\left(\pi-2x\sqrt{1-x^{2}}-2\arcsin{x}\right)=\frac{8}{45}.$ Then Eq. (120) yields: $\tilde{E}^{(1)}_{1}=\frac{8S}{45\pi^{2}}\frac{\hbar^{2}k_{F}^{4}}{m}k_{F}r_{}=\frac{N^{2}}{S}\frac{128}{45}\frac{\hbar^{2}k_{F}^{2}}{m}k_{F}r_{},$ (121) which exactly coincides with the second term of the first line of Eq. (91). Using Eq. (80) the contribution $\tilde{E}^{(1)}_{2}$ takes the form: $\\!\\!\\!\\!\tilde{E}^{(\\!1\\!)}_{2}\\!\\!\\!=\\!\frac{S\hbar^{2}\\!k_{F}^{4}}{2\pi^{2}m}(k_{F}r_{}\\!)^{2}\\!\left\\{\\!\left[\ln(\xi k_{F}r_{}\\!)\\!-\\!\frac{25}{12}\\!+\\!3\ln{2}\right]\\!I_{2}\\!+\\!I_{3}\\!\right\\}\\!\\!,\\!\\!\\!\\!\\!\\!\\!\\!$ (122) where the integrals $I_{2}$ and $I_{3}$ are given by $\displaystyle I_{2}$ $\displaystyle=\int_{0}^{2\pi}d\varphi\int_{0}^{1}x^{3}dx\int_{0}^{y_{0}(x,\varphi)}ydy=\frac{1}{2}\int_{0}^{2\pi}d\varphi\int_{0}^{1}x^{3}dx$ $\displaystyle\times\big{[}1-2\|\cos\varphi\|x\sqrt{1-x^{2}\sin^{2}\varphi}+x^{2}(\cos^{2}\varphi-\sin^{2}\varphi)\big{]}$ $\displaystyle=\frac{1}{2}\int_{0}^{1}x^{3}[2\pi-4x\sqrt{1-x^{2}}-4\arcsin{x}]dx=\frac{\pi}{32},$ and $\displaystyle I_{3}=\int_{0}^{2\pi}d\varphi\int_{0}^{1}x^{3}\ln{x}dx\int_{0}^{y_{0}(x,\varphi)}ydy=\frac{1}{2}\int_{0}^{2\pi}d\varphi\int_{0}^{1}dx$ $\displaystyle\times x^{3}\ln{x}\big{[}1-2\|\cos\varphi\|x\sqrt{1-x^{2}\sin^{2}\varphi}+x^{2}(\cos^{2}\varphi-\sin^{2}\varphi)\big{]}$ $\displaystyle=\frac{1}{2}\int_{0}^{1}x^{3}\ln{x}[2\pi-4x\sqrt{1-x^{2}}-4\arcsin{x}]dx$ $\displaystyle=\frac{\pi}{32}\left(\frac{1}{6}-\ln{2}\right).$ Substituting the calculated $I_{2}$ and $I_{3}$ into Eq. (122) we obtain: $\displaystyle\tilde{E}^{(1)}_{2}=\frac{S\hbar^{2}k_{F}^{4}}{64\pi m}(k_{F}r_{})^{2}\left[\ln(4\xi k_{F}r_{})-\frac{23}{12}\right]$ $\displaystyle=\frac{N^{2}}{S}\frac{\pi\hbar^{2}}{4m}(k_{F}r_{})^{2}\left[\ln(4\xi k_{F}r_{})-\frac{23}{12}\right].$ (123) This exactly reproduces the third term of the first line of Eq. (91). ## Appendix B Calculation of the interaction function $\tilde{F}_{1}^{(2)}$ The interaction function $\tilde{F}_{1}^{(2)}$ is the second variational derivative of the many-body contribution to the interaction energy, $\tilde{E}^{(2)}_{1}$ (82), with respect to the momentum distribution function. It can be expressed as $\tilde{F}_{1}^{(2)}(\mathbf{k},\mathbf{k^{\prime}})=-\frac{2\hbar^{2}}{m}(k_{F}r_{})^{2}(\tilde{I}_{1}+\tilde{I}_{2}+\tilde{I}_{3}),$ (124) where $\displaystyle\tilde{I}_{1}=2\\!\int_{\|\mathbf{k_{1}}\|<k_{F}}\\!\frac{d^{2}k_{1}}{k_{F}^{2}}\frac{\|\mathbf{k}-\mathbf{k_{1}}\|^{2}}{\mathbf{k^{2}}\\!+\\!\mathbf{k^{\prime 2}}\\!-\\!\mathbf{k^{2}_{1}}\\!-\\!\mathbf{k^{2}_{2}}}\delta_{\mathbf{k}\\!+\\!\mathbf{k^{\prime}}\\!-\\!\mathbf{k_{1}}\\!-\\!\mathbf{k_{2}}},$ (125) $\displaystyle\tilde{I}_{2}=2\\!\int_{\|\mathbf{k_{1}}\|<k_{F}}\\!\frac{d^{2}k_{1}}{k_{F}^{2}}\frac{\|\mathbf{k}-\mathbf{k^{\prime}}\|^{2}}{\mathbf{k^{2}}\\!+\\!\mathbf{k^{2}_{1}}\\!-\\!\mathbf{k^{\prime 2}}\\!-\\!\mathbf{k^{2}_{2}}}\delta_{\mathbf{k}\\!+\\!\mathbf{k_{1}}\\!-\\!\mathbf{k^{\prime}}\\!-\\!\mathbf{k_{2}}},$ (126) $\displaystyle\tilde{I}_{3}=2\\!\int_{\|\mathbf{k_{1}}\|<k_{F}}\\!\frac{d^{2}k_{1}}{k_{F}^{2}}\frac{\|\mathbf{k_{1}}-\mathbf{k^{\prime}}\|^{2}}{\\!\mathbf{k^{2}_{1}}\\!+\\!\mathbf{k^{2}}\\!-\\!\mathbf{k^{\prime 2}}\\!-\\!\mathbf{k^{2}_{2}}}\delta_{\mathbf{k_{1}}\\!+\\!\mathbf{k}\\!-\\!\mathbf{k^{\prime}}\\!-\\!\mathbf{k_{2}}},$ (127) and the presence of the Kronecker symbols $\delta_{\bf q}$ reflects the momentum conservation law. On the Fermi surface we put $\|{\bf k}\|=\|{\bf k}^{\prime}\|=k_{F}$ and denote the angle between ${\bf k}$ and ${\bf k}^{\prime}$ as $\theta$. Due to the symmetry property: $F(\mathbf{k},\mathbf{k^{\prime}})=F(\mathbf{k^{\prime}},\mathbf{k})$ we have $F(\theta)=F(2\pi-\theta)$ and may consider $\theta$ in the interval from $0$ to $\pi$. In order to calculate the integral $\tilde{I}_{1}$, we use the quantities $\mathbf{s}=(\mathbf{k}+\mathbf{k^{\prime}})/2k_{F}$ and $\mathbf{m}=(\mathbf{k}-\mathbf{k^{\prime}})/2k_{F}$ and turn to the variable $\mathbf{x}=(\mathbf{k_{1}}-\mathbf{k_{2}})/2k_{F}=(2\mathbf{k_{1}}-\mathbf{s})/2k_{F}$. For given vectors $\mathbf{k}$ and $\mathbf{k^{\prime}}$, the vectors $\mathbf{s}$ and $\mathbf{m}$ are fixed and $\|\mathbf{s}\|=\cos(\theta/2)$, $\|\mathbf{m}\|=\sin(\theta/2)$. The integral can then be rewritten as: $\tilde{I}_{1}=\int\frac{m^{2}+x^{2}}{m^{2}-x^{2}}d^{2}x.$ The integration region is shown in Fig.1, where the distance between the points $O_{1}$ and $O_{2}$ is ${\bf R}_{O_{1}O_{2}}=\mathbf{s}$. The distance between the points $O_{1}$ and $N$ is ${\bf R}_{O_{1}N}=\mathbf{k_{1}}/2k_{F}$, and ${\bf R}_{NO_{2}}=\mathbf{k_{2}}/2k_{F}$, so that ${\bf R}_{ON}=\mathbf{x}/2$. The quantity $\|{\bf x}\|$ changes from $0$ to $l_{1}(\varphi)$ where $l_{1}^{2}(\varphi)+\cos^{2}\frac{\theta}{2}-2l_{1}(\varphi)\cos\frac{\theta}{2}\cos\varphi=1,$ and $l_{1}(\varphi)\cdot l_{1}(\varphi+\pi)=\sin^{2}(\theta/2)$, with $\varphi$ being an angle between ${\bf m}$ and ${\bf x}$. In the polar coordinates the integral $\tilde{I}_{1}$ takes the form: $\tilde{I}_{1}=\int_{0}^{2\pi}d\varphi\int_{0}^{l_{1}(\varphi)}\left(-1+2\sin^{2}\frac{\theta}{2}\frac{1}{\sin^{2}(\theta/2)-x^{2}}\right)xdx,$ and after a straightforward integration we obtain: $\tilde{I}_{1}=\pi\left(2\sin^{2}\frac{\theta}{2}\ln\|\tan\frac{\theta}{2}\|-1\right).$ (128) Figure 1: (color online). Left: The integration area for $\tilde{I}_{1}$ (in blue). The distance between the points $O_{1}$ and $P$ is ${\bf R}_{O_{1}P}=\mathbf{k}/2k_{F}$, ${\bf R}_{PO_{2}}=\mathbf{k^{\prime}}/2k_{F},$ and ${\bf R}_{O_{1}N}=\mathbf{k_{1}}/2k_{F}$. Right: The integration area for $\tilde{I}_{2}$ and $\tilde{I}_{3}$ (in red). The distance between the points $O_{2}$ and $N$ is ${\bf R}_{O_{2}N}=\mathbf{k_{1}}/2k_{F}$, ${\bf R}_{O_{1}P}={\bf k}/2k_{F}$, and ${\bf R}_{O_{2}P}={\bf k}^{\prime}/2k_{F}$. In the integral $\tilde{I}_{2}$, using the variable $\mathbf{y}=(\mathbf{k_{1}}+\mathbf{k_{2}})/2k_{F}$ we observe that it changes from $0$ to $l_{2}(\tilde{\varphi})$ where $l_{2}^{2}(\tilde{\varphi})+\sin^{2}\frac{\theta}{2}-2l_{2}(\tilde{\varphi})\sin\frac{\theta}{2}\cos(\tilde{\varphi})=1$ and $l_{2}(\tilde{\varphi})-l_{2}(\tilde{\varphi}+\pi)=2\sin\frac{\theta}{2}\cos\tilde{\varphi}$, with $\tilde{\varphi}$ being an angle between ${\bf y}$ and ${\bf m}$. We then have: $\displaystyle\tilde{I}_{2}$ $\displaystyle=-2\int\frac{m^{2}}{\mathbf{m}\cdot\mathbf{y}}d^{2}y=-2\sin\frac{\theta}{2}\int_{0}^{2\pi}d\tilde{\varphi}\int_{0}^{l_{2}(\tilde{\varphi})}\frac{dy}{\cos\tilde{\varphi}}$ $\displaystyle=-4\pi\sin^{2}\frac{\theta}{2}.$ (129) For the integral $\tilde{I}_{3}$ we have: $\displaystyle\tilde{I}_{3}=-\frac{1}{2}\int\frac{s^{2}+y^{2}-2\mathbf{s}\cdot\mathbf{y}}{\mathbf{m}\cdot\mathbf{y}}$ $\displaystyle=-\frac{1}{2\sin\frac{\theta}{2}}\int_{0}^{2\pi}\\!\\!\frac{d\tilde{\varphi}}{\cos\tilde{\varphi}}\\!\int_{0}^{l_{2}(\tilde{\varphi})}\\!\\!\\!dy\left[y^{2}+\cos^{2}\frac{\theta}{2}-2y\cos\frac{\theta}{2}\sin\tilde{\varphi}\right]$ $\displaystyle=-2\pi\left(\cos^{2}\frac{\theta}{2}+\frac{1}{3}\sin^{2}\frac{\theta}{2}\right),$ (130) where we used the relation $l_{2}(\tilde{\varphi})=l_{2}(-\tilde{\varphi})$. Using integrals $\tilde{I}_{1}$ (128), $\tilde{I}_{2}$ (129), and $\tilde{I}_{3}$ (130) in Eq. (124), we obtain equation (84): $\\!\\!\\!\tilde{F}_{1}^{(2)}(\theta)\\!\\!=\\!\\!\frac{2\hbar^{2}r^{2}_{}k^{2}_{F}}{m}\\!\\!\left[\\!3\pi\\!+\\!2\pi\sin^{2}\frac{\theta}{2}\left(\frac{4}{3}\\!-\\!\ln\|\tan\frac{\theta}{2}\|\right)\\!\right].$ . ## Appendix C Calculation of the interaction function $\tilde{F}_{2}^{(2)}$ The interaction function $\tilde{F}_{2}^{(2)}$ is the second variational derivative of the many-body contribution to the interaction energy, $\tilde{E}^{(2)}_{2}$ (83), with respect to the momentum distribution. It reads: $\tilde{F}_{2}^{(2)}(\mathbf{k},\mathbf{k^{\prime}})=\frac{2\hbar^{2}}{m}(k_{F}r_{})^{2}(I^{\prime}_{1}+I^{\prime}_{2}),$ (131) where $\displaystyle I^{\prime}_{1}=2\\!\int_{\|\mathbf{k_{1}}\|<k_{F}}\\!\frac{d^{2}k_{1}}{k_{F}^{2}}\frac{\|\mathbf{k}\\!-\\!\mathbf{k_{1}}\|\cdot\|\mathbf{k^{\prime}}\\!-\\!\mathbf{k_{1}}\|}{\mathbf{k^{2}}\\!+\\!\mathbf{k^{\prime 2}}\\!-\\!\mathbf{k^{2}_{1}}\\!-\\!\mathbf{k^{2}_{2}}}\delta_{\mathbf{k}\\!+\\!\mathbf{k^{\prime}}\\!-\\!\mathbf{k_{1}}\\!-\\!\mathbf{k_{2}}},$ (132) $\displaystyle I^{\prime}_{2}=4\\!\int_{\|\mathbf{k_{1}}\|<k_{F}}\\!\frac{d^{2}k_{1}}{k_{F}^{2}}\frac{\|\mathbf{k}\\!-\\!\mathbf{k^{\prime}}\|\cdot\|\mathbf{k_{1}}\\!-\\!\mathbf{k^{\prime}}\|}{\mathbf{k^{2}}\\!+\\!\mathbf{k^{2}_{1}}\\!-\\!\mathbf{k^{\prime 2}}\\!-\\!\mathbf{k^{2}_{2}}}\delta_{\mathbf{k}\\!+\\!\mathbf{k_{1}}\\!-\\!\mathbf{k^{\prime}}\\!-\\!\mathbf{k_{2}}}$ (133) The integration area for $I^{\prime}_{1}$ is shown in Fig. 2, where the distance between the points $O_{1}$ and $P$ is ${\bf R}_{O_{1}P}={\bf k}/2k_{F}$, ${\bf R}_{PO_{2}}={\bf k}^{\prime}/2k_{F}$, ${\bf R}_{O_{1}N}={\bf k}_{1}/2k_{F}$, and ${\bf R}_{ON}={\bf x}/2$. We thus have ${\bf R}_{NP}=({\bf k}-{\bf k}_{1})/2k_{F}$ and ${\bf R}_{NP^{\prime}}=({\bf k}^{\prime}-{\bf k}_{1})/2k_{F}$. In the region of integration we should have $\|{\bf R}_{O_{1}N}\|=k_{1}/2k_{F}\leq 1/2$. This leads to $\displaystyle I^{\prime}_{1}$ $\displaystyle=4\int\frac{\|{\bf R}_{NP}\|\cdot\|{\bf R}_{NP^{\prime}}\|}{m^{2}-x^{2}}d^{2}n=-\int_{0}^{2\pi}d\varphi\int_{0}^{l_{3}(\varphi)}xdx$ $\displaystyle\times\frac{\sqrt{[x^{2}+\sin^{2}(\theta/2)]^{2}-4x^{2}\sin^{2}(\theta/2)\cos^{2}\varphi}}{x^{2}-\sin^{2}(\theta/2)},$ (134) where $\varphi$ is the angle between ${\bf x}$ and ${\bf m}$ (see Fig. 2), and the quantity $l_{3}(\varphi)$ obeys the equation $l_{3}^{2}(\varphi)-2\cos\frac{\theta}{2}\sin\varphi\cdot l_{3}(\varphi)+\cos^{2}\frac{\theta}{2}=1.$ Turning to the variable $z=r^{2}-\sin^{2}(\theta/2)$ the integral $I^{\prime}_{1}$ is reduced to $\displaystyle I^{\prime}_{1}=-\frac{1}{2}\int_{0}^{2\pi}d\varphi\int_{-\sin^{2}(\theta/2)}^{l_{3}^{2}(\varphi)-\sin^{2}(\theta/2)}\frac{\sqrt{R}}{z}dz,$ (135) with $R=z^{2}+4z\sin^{2}\frac{\theta}{2}\sin^{2}\varphi+4\sin^{4}\frac{\theta}{2}\sin^{2}\varphi.$ Figure 2: (color online). Left: The integration area for $I^{\prime}_{1}$ (in blue): ${\bf R}_{O_{1}P}=\mathbf{k}/2k_{F}$, ${\bf R}_{PO_{2}}=\mathbf{k^{\prime}}/2k_{F}$, ${\bf R}_{O_{1}N}=\mathbf{k_{1}}/2k_{F}$, and $\varphi$ is the angle between the vectors ${\bf R}_{OP}$ and ${\bf R}_{ON}$, which is the same as the angle between ${\bf m}$ and ${\bf x}$. Right: The integration area for $I^{\prime}_{2}$ (in red): ${\bf R}_{O_{1}P}=\mathbf{k}/2k_{F}$, ${\bf R}_{O_{2}P}=\mathbf{k^{\prime}}/2k_{F}$, ${\bf R}_{O_{2}N}=\mathbf{k_{1}}/2k_{F}$, $\alpha$ is the angle between ${\bf R}_{PM}$ and ${\bf R}_{PN}$, and $\phi$ is the angle between ${\bf R}_{PN}$ and ${\bf R}_{OO_{2}}$. It is easy to see that: $\displaystyle I_{r}=\int_{-\sin^{2}(\theta/2)}^{l_{3}^{2}(\varphi)-\sin^{2}(\theta/2)}\frac{\sqrt{R}}{z}dz$ $\displaystyle=\Big{\\{}\sqrt{R}-\sqrt{a}\ln\left(2a+bz+2\sqrt{aR}\right)$ $\displaystyle+\frac{b}{2}\ln\left(2\sqrt{R}+2z+b\right)\Big{\\}}\Big{\|}^{l_{3}^{2}(\varphi)-\sin^{2}(\theta/2)}_{-\sin^{2}(\theta/2)}$ $\displaystyle+\sqrt{a}\cdot P\int_{-\sin^{2}(\theta/2)}^{l_{3}^{2}(\varphi)-\sin^{2}(\theta/2)}\frac{dz}{z}=I_{r\uparrow}-I_{r\downarrow},$ where $a=4\sin^{4}(\theta/2)\sin^{2}\varphi$, $b=4\sin^{2}(\theta/2)\sin^{2}\varphi$, and the symbol $P$ stands for the principal value of the integral. The quantities $I_{r\uparrow}$ and $I_{r\downarrow}$ denote the values of the integral at the upper and lower bounds, respectively (in the last line we have to take the principal value of the integral and, hence, if the upper bound of the integral is positive we have to replace the lower bound with $\sin^{2}(\theta/2)$). Then $I_{r\uparrow}$ and $I_{r\downarrow}$ are given by: $\displaystyle I_{r\uparrow}=2\|\sin\varphi\|\cdot l(\varphi)-2\sin^{2}\frac{\theta}{2}\|\sin\varphi\|\cdot\left[\ln\left(8\sin^{2}\frac{\theta}{2}\sin^{2}\varphi\right)+\ln\left(\sin^{2}\frac{\theta}{2}+\cos\frac{\theta}{2}\sin\varphi\cdot l(\varphi)+l(\varphi)\right)\right]$ $\displaystyle+2\sin^{2}\frac{\theta}{2}\|\sin\varphi\|\cdot\ln\|2\cos\frac{\theta}{2}\sin\varphi\cdot l(\varphi)\|+2\sin^{2}\frac{\theta}{2}\sin^{2}\varphi\left[\ln 4+\ln\left(\|\sin\varphi\|\cdot l(\varphi)+\cos\frac{\theta}{2}\sin\varphi\cdot l(\varphi)+\sin^{2}\frac{\theta}{2}\sin^{2}\varphi\right)\right],$ $\displaystyle I_{r\downarrow}=\sin^{2}\frac{\theta}{2}-2\sin^{2}\frac{\theta}{2}\|\sin\varphi\|\cdot\left[\ln\left(4\sin^{4}\frac{\theta}{2}\right)+\ln\left(\sin^{2}\varphi+\|\sin\varphi\|\right)\right]+2\sin^{2}\frac{\theta}{2}\|\sin\varphi\|\cdot\ln\left(\sin^{2}\frac{\theta}{2}\right)$ $\displaystyle+2\sin^{2}\frac{\theta}{2}\sin^{2}\varphi\cdot\ln\left(4\sin^{2}\frac{\theta}{2}\sin^{2}\varphi\right).$ The integral $I^{\prime}_{1}$ can be expressed as: $I^{\prime}_{1}=-\frac{1}{2}\int_{0}^{2\pi}[I_{r\uparrow}-I_{r\downarrow}]d\varphi,$ and for performing the calculations we notice that $l_{3}(\varphi)\cdot l_{3}(\varphi+\pi)=\sin^{2}\frac{\theta}{2}$, $l_{3}(\varphi)-l_{3}(\varphi+\pi)=2\cos\frac{\theta}{2}\sin\varphi$, and $l_{3}(\varphi)+l_{3}(\varphi+\pi)=2\sqrt{\cos^{2}\frac{\theta}{2}\sin^{2}\varphi+\sin^{2}\frac{\theta}{2}}$. We then obtain: $\displaystyle I^{\prime}_{1}=$ $\displaystyle-\sin^{2}\frac{\theta}{2}\left(\pi\ln 2+\pi/2-\pi\ln\sin\frac{\theta}{2}+4\ln\|\cos\frac{\theta}{2}\|-4\ln(1+\sin\frac{\theta}{2})+\mathcal{G}(\theta)-\frac{2\pi}{\|\cos\frac{\theta}{2}\|}-\frac{4\arcsin(\sin\frac{\theta}{2})}{\|\cos\frac{\theta}{2}\|}\right)$ $\displaystyle-\frac{k^{2}_{F}}{\|\cos\frac{\theta}{2}\|}\left(\pi-2\arcsin(\sin\frac{\theta}{2})+\|\sin\theta\|\right),$ (136) with $\mathcal{G}(\theta)=\int_{0}^{\pi}2\sin^{2}\varphi\ln\left(\sin\varphi+\sqrt{\sin^{2}\frac{\theta}{2}+\cos^{2}\frac{\theta}{2}\sin^{2}\varphi}\right)d\varphi.$ (137) The integration area for $I^{\prime}_{2}$ is shown in Fig. 2, and we get: $I^{\prime}_{2}=-4\int\frac{\|\mathbf{m}\|\cdot\|{\bf R}_{PN}\|}{\mathbf{m}\cdot\mathbf{y}}d^{2}y=-8\int\frac{d^{2}\rho}{\cos\phi},$ (138) where we denote ${\bf R}_{PN}=$ $\bm{\rho}$, and $\phi=\alpha-\theta/2$ is the angle between the vectors $\mathbf{m}$ and $\bm{\rho}$, with $\alpha$ being the angle between the vectors ${\bf R}_{PM}$ and ${\bf R}_{PN}$ (see Fig. 2). We then have: $\displaystyle I^{\prime}_{2}$ $\displaystyle=-8\int_{0}^{\pi}d\alpha\int_{0}^{\sin\alpha}\frac{\rho d\rho}{\cos(\alpha-\theta/2)}$ $\displaystyle=-4\left[\cos^{2}\frac{\theta}{2}\ln\frac{1+\sin(\theta/2)}{1-\sin(\theta/2)}+2\sin\frac{\theta}{2}\right].$ (139) Using $I^{\prime}_{1}$ (C) and $I^{\prime}_{2}$ (C) in Eq. (131) we get equation (85) for the interaction function $\tilde{F}^{(2)}_{2}(\theta)$. ## References (1) K.-K. Ni, S.Ospelkaus, M.H.G. de Miranda, A.Pe’er, B.Neyenhuis, J.J.Zirbel, S.Kotochigova, P.S.Julienne, D.S.Jin and J.Ye, Science 322, 231(2008). * (2) J. Deiglmayr, A. Grocola, M. Repp, K. Mörtlbauer, C. Glück, J. Lange, O. Dulieu, R. Wester, and M. Weidemüller, Phys. Rev. Lett. 101, 133004 (2008). * (3) K. Aikawa, D. Akamatsu, M. Hayashi, K. Oasa, J. Kobayashi, P. Naidon, T. Kishimoto, M. Ueda, and S. Inouye, Phys. Rev. Lett. 105, 203001 (2010). * (4) M. Deatin, T. Takekoshi, R.Rameshan, L. Reichsollner, F. Felaino, R. Grimm, R. Vexiau, N. Bouloufa, O. Dulieu, and H.-C. Naegerl, arXiv:1106.0129 and an up-coming preprint. * (5) M.A. Baranov, Physics Reports 464, 71 (2008). * (6) T. Lahaye, C. Menotti, L. Santos, M. Lewenstein, and T. Pfau, Rep. Prog. Phys. 72, 126401 (2009). * (7) Cold Molecules: Theory, Experiment, Applications, edited by R.V. Krems, B. Friedrich, and W.C. Stwalley, (CRC Press, Taylor & Francis Group 2009). * (8) D.-W. Wang, M.D. Lukin, and E. Demler, Phys. Rev. Lett. 97, 180413 (2006). * (9) H.P. Büchler, E. Demler, M. Lukin, A. Micheli, N. Prokof’ev, G. Pupillo, and P. Zoller, Phys. Rev. Lett. 98, 060404 (2007). * (10) G. M. Bruun and E. Taylor, Phys. Rev. Lett. 101, 245301 (2008). * (11) N.R. Cooper and G.V. Shlyapnikov, Phys. Rev. Lett. 103, 155302 (2009). * (12) A. Pikovski, M. Klawunn, G.V. Shlyapnikov, and L. Santos, Phys. Rev. Lett. 105, 215302 (2010). * (13) R.M. Lutchyn, E. Rossi, and S. Das Sarma, Phys. Rev. A 82, 061604 (2010). * (14) A.C. Potter, E. Berg, D.-W. Wang, B.I. Halpern, and E. Demler, Phys. Rev. Lett. 105, 220406 (2010). * (15) B. Capagrosso-Snasone, C. Trefzger, M. Lewenstein, P. Zoller, and G. Pupillo, Phys. Rev. Lett. 104, 125301 (2010). * (16) K. Sun, C. Wu, and S. Das Sarma, Phys. Rev. B 82, 075105 (2010). * (17) Y. Yamaguchi, T. Sogo, T. Ito, and T. Miyakawa, Phys. Rev. A 82, 013643 (2010). * (18) J. Levinsen, N. R. Cooper, and G. V. Shlyapnikov, Phys. Rev. A 84, 013603 (2011) * (19) M.A. Baranov, A. Micheli, S. Ronen, and P. Zoller, Phys. Rev. A 83, 043602 (2011). * (20) M.M. Parish and F.M. Marchetti, arXiv:1109.2464. * (21) M. Babadi and E. Demler, arXiv:1109.3755. * (22) L. M. Sieberer, M. A. Baranov, arXiv:1110.3679. * (23) S. Ospelkaus, K.-K. Ni, D. Wang, M.H.G. de Miranda, B. Neyenhuis, G. Quemener, P. S. Julienne, J. L. Bohn, D. S. Jin, and J. Ye, Science 327, 853 (2010). * (24) K.-K. Ni, S. Ospelkaus, D. Wang, G. Quemener, B. Neyenhuis, M.H.G. de Miranda, J.L. Bohn, J. Ye, and D.S. Jin, Nature 464, 1324 (2010). * (25) G. Quéméner and J. L. Bohn, Phys. Rev. A 81, 060701(2010). * (26) A. Micheli, Z. Idziaszek, G. Pupillo, M. A. Baranov, P. Zoller, and P. S. Julienne, Phys. Rev. Lett. 105, 073202(2010) * (27) M.H.G. de Miranda, A. Chotia, B. Neyenhuis, D. Wang, G. Quemener, S. Ospelkaus, J.L. Bohn, J. Ye, and D.S. Jin, Nature Phys. 7, 502 (2011). * (28) P.S. Zuchowski and J.M. Hutson, Phys. Rev. A 81, 060703(R) (2010). * (29) J.W. Park, C.-H. Wu, I. Santiago, T.G. Tiecke, P. Ahmadi, and M.W. Zwierlein, arXiv:1110.4552. * (30) N.T. Zinner, B. Wunsch, D. Pekker, and D.-W. Wang, arXiv:1009.2030. * (31) J. P. Kestner and S. Das Sarma, Phys. Rev. A 82, 033608 (2010). * (32) C-K Chan, C Wu, W-C Lee and S. Das Sarma, Phys. Rev. A 81, 023602 (2010). * (33) T.Miyakawa, T. Sogo, and H. Pu, Phys. Rev. A 77, 061603 (2008). * (34) Q. Li, E. H. Hwang, and S. Das Sarma, Phys. Rev. B 82, 235126 (2010). * (35) N. Navon, S. Nascimbene, F. Chevy and C. Salomon, Science 328, 729 (2010). * (36) S. Nascimbéne, N. Navon, K. J. Jiang, F. Chevy, and C. Salomon, Nature 463, 1057 (2010). * (37) K. Huang and C.N. Yang, Phys. Rev. 105, 767 (1957). * (38) T. D. Lee and C. N. Yang, Phys. Rev. 105, 1119 (1957). * (39) A.A.Abrikosov and I.M.Khalatnikov, Sov. Phys. JETP 6, 888 (1958) * (40) E.M.Lifshitz and L.P.Pitaevskii, Statistical Physics, Part 2 (Pergamon Press, Oxford, 1980). * (41) L.D. Landau and E.M. Lifshitz, Quantum Mechanics, Non-Relativistic Theory (Butterworth-Heinemann, Oxford, 1999). * (42) In fact, we arrived at this conclusion confining ourselves to the contributions up to $\beta^{2}$. In stead of considering higher order contributions we checked numerically that the solution with $s>1$ is absent if we omit the $(k_{F}r_{})^{2}$-terms in the interaction function of quasiparticles. (43) M. J. H. Ku, A. T. Sommer, L. W. Cheuk, and M. W. Zwierlein, arXiv:1110.3309. * (44) T. Yefsah, R. Desbuquois, L. Chomaz, K. J. Gunter, and J. Dalibard, Phys. Rev. Lett. 107, 130401 (2011). * (45) D. Wang, B. Neyenhuis, M. H. G. de Miranda, K.-K. Ni, S. Ospelkaus, D. S. Jin, and J. Ye, Phys. Rev. A 81, 061404(R) (2010).	arxiv-papers	2011-11-30T10:34:54	2024-09-04T02:49:24.806793	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Zhen-Kai Lu and G. V. Shlyapnikov", "submitter": "Zhenkai Lu Lu Zhenkai", "url": "https://arxiv.org/abs/1111.7114" }
1111.7188	¡html¿¡head¿ ¡meta http-equiv=”content-type” content=”text/html; charset=ISO-8859-1”¿ ¡title¿CERN-2010-002¡/title¿ ¡/head¿ ¡body¿ ¡h1¿¡a href=”http://indico.cern.ch/conferenceDisplay.py?confId=97971”¿EuCARD- AccNet-EuroLumi Workshop: The High-Energy Large Hadron Collider¡/a¿¡/h1¿ ¡h2¿Villa Bighi, Malta, 14 - 16 Oct 2010¡/h2¿ ¡h2¿Proceedings - CERN Yellow Report ¡a href=”http://cdsweb.cern.ch/record/1344820”¿CERN-2011-003¡/a¿¡/h2¿ ¡h3¿editor: E. Todesco and F. Zimmermann ¡/h3¿ ¡p¿This report contains the proceedings of the EuCARD-AccNet-EuroLumi Workshop on a High-Energy Large Hadron Collider ‘HE-LHC10’ which was held on Malta from 14 to 16 October 2010. This is the first workshop where the possibility of building a 33 TeV centre- of-mass energy proton–proton accelerator in the LHC tunnel is discussed. The key element of such a machine will be the 20 T magnets needed to bend the particle beams: therefore much space was given to discussions about magnet technologies for high fields. The workshop also discussed possible parameter sets, issues related to beam dynamics and synchrotron radiation handling, and the need for new injectors, possibly with 1 TeV energy. The workshop searched for synergies with other projects and studies around the world facing similar challenges or pushing related technologies, revisited past experience, and explored a possible re-use of existing superconducting magnets. Last not least, it reinforced the inter-laboratory collaborations within EuCARD, especially between CERN and its European, US, and Japanese partners.¡/p¿ ¡h2¿Lectures¡/h2¿ ¡p¿ ¡!– Elements of a physics case for a High-Energy LHC –¿ Title: ¡a href=”http://cdsweb.cern.ch/record/1403000”¿Elements of a physics case for a High-Energy LHC¡/a¿ ¡br¿ Author: Wells, James D¡br¿ Journal-ref: CERN Yellow Report CERN-2011-003, pp. 1-5¡br¿ ¡br¿ ¡!–CERN Accelerator Strategy –¿ LIST:arXiv:1108.1115¡br¿ ¡!– HE-LHC beam-parameters, optics and beam-dynamics issues –¿ LIST:arXiv:1108.1617¡br¿ ¡!– Conceptual design of 20 T dipoles for high-energy LHC –¿ LIST:arXiv:1108.1619¡br¿ ¡br¿ ¡!–What can the SSC and the VLHC studies tell us for the HE-LHC? –¿ Title: ¡a href=”http://cdsweb.cern.ch/record/1403041”¿What can the SSC and the VLHC studies tell us for the HE-LHC?¡/a¿ ¡br¿ Author: Wienands, U¡br¿ Journal-ref: CERN Yellow Report CERN-2011-003, pp. 20-26¡br¿ ¡br¿ ¡!–A high energy LHC machine: experiments ’first’ impressions –¿ LIST:arXiv:1108.1621¡br¿ ¡!– Progress in high field accelerator magnet development by the US LHC Accelerator Research Program–¿ LIST:arXiv:1108.1625¡br¿ ¡!– LBNL high field core program –¿ LIST:arXiv:1108.1868¡br¿ ¡!–KEK effort for high field magnets –¿ LIST:arXiv:1108.1626¡br¿ ¡!– EUCARD magnet development –¿ LIST:arXiv:1108.1627¡br¿ ¡!– Status of Nb3Sn accelerator magnet R&D at Fermilab –¿ LIST:arXiv:1108.1869¡br¿ ¡!– Status of high temperature superconductor based magnets and the conductors they depend upon –¿ LIST:arXiv:1108.1634¡br¿ ¡!– 20 T dipoles and Bi-2212: the path to LHC energy upgrade –¿ LIST:arXiv:1108.1640¡br¿ ¡!– Heat loads and cryogenics for HE-LHC –¿ LIST:arXiv:1108.1870¡br¿ ¡!– HE-LHC: requirements from beam vacuum –¿ LIST:arXiv:1108.1642¡br¿ ¡!– Beam screen issues –¿ LIST:arXiv:1108.1643¡br¿ ¡!– Synchrotron radiation damping, intrabeam scattering and beam-beam simulations for HE-LHC –¿ LIST:arXiv:1108.1644¡br¿ ¡!– Beam-beam studies for the High-Energy LHC –¿ LIST:arXiv:1108.1871¡br¿ ¡!– Preliminary considerations about the injectors of the HE-LHC –¿ LIST:arXiv:1108.1652¡br¿ ¡!– Using tevatron magnets for HE-LHC or new ring in LHC tunnel –¿ LIST:arXiv:1108.1653¡br¿ ¡!– Magnet design issues and concepts for the new injector –¿ LIST:arXiv:1108.1654¡br¿ ¡!– Using LHC as injector and possible uses of HERA magnets/coils –¿ LIST:arXiv:1108.1655¡br¿ ¡!– Intensity issues and machine protection of the HE-LHC –¿ LIST:arXiv:1108.1663¡br¿ ¡!– Injection and dump considerations for a 16.5 TeV HE-LHC –¿ LIST:arXiv:1108.1664¡br¿ ¡!– Radiation protection issues after 20 years of LHC operation –¿ LIST:arXiv:1108.1669¡br¿ ¡!– 20 T dipoles and Bi-2212: the path to LHC energy upgrade –¿ ¡br¿ Title: ¡a href=”http://cdsweb.cern.ch/record/1373679”¿Summary of session 1: introduction and overview¡/a¿ ¡br¿ Author: J.P. Koutchouk and R. Bailey ¡br¿ Journal-ref: CERN Yellow Report CERN-2011-003, pp. 137-139 ¡br¿ ¡br¿ Title: ¡a href=”http://cdsweb.cern.ch/record/1373681”¿Summary of session 2: magnets for the HE-LHC¡/a¿ ¡br¿ Author: L. Rossi and E. Todesco ¡br¿ Journal-ref: CERN Yellow Report CERN-2011-003, pp. 140-142 ¡br¿ ¡br¿ LIST:arXiv:1202.3811¡br¿ ¡br¿ Title: ¡a href=”http://cdsweb.cern.ch/record/1373683”¿Summary of session 4: HE-LHC injectors and infrastructure¡/a¿ ¡br¿ Author: E. Prebys and L. Bottura ¡br¿ Journal-ref: CERN Yellow Report CERN-2011-003, pp. 145-148 ¡br¿ ¡/p¿ ¡/body¿¡/html¿	arxiv-papers	2011-11-30T14:44:06	2024-09-04T02:49:24.822151	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "E. Todesco and F. Zimmermann", "submitter": "Scientific Information Service CERN", "url": "https://arxiv.org/abs/1111.7188" }
1111.7211	# Disconnecting Open Solar Magnetic Flux C.E. DeForest∗, T.A. Howard∗, and D.J. McComas∗† ∗Southwest Research Institute, 1050 Walnut Street Suite 300, Boulder CO 80302 †University of Texas at San Antonio, San Antonio TX 78249 Disconnection of open magnetic flux by reconnection is required to balance the injection of open flux by CMEs and other eruptive events. Making use of recent advances in heliospheric background subtraction, we have imaged many abrupt disconnection events. These events produce dense plasma clouds whose distinctive shape can now be traced from the corona across the inner solar system via heliospheric imaging. The morphology of each initial event is characteristic of magnetic reconnection across a current sheet, and the newly- disconnected flux takes the form of a “U”-shaped loop that moves outward, accreting coronal and solar wind material. We analyzed one such event on 2008 December 18 as it formed and accelerated at 20 m s-2 to 320 km s-1, expanding self-similarly until it exited our field of view 1.2 AU from the Sun. From acceleration and photometric mass estimates we derive the coronal magnetic field strength to be $8\,\mu T$, 6$R_{\odot}$ above the photosphere, and the entrained flux to be $1.6\times 10^{11}Wb$ ($1.6\times 10^{19}Mx$). We model the feature’s propagation by balancing inferred magnetic tension force against accretion drag. This model is consistent with the feature’s behavior and accepted solar wind parameters. By counting events over a 36 day window, we estimate a global event rate of $1\,d^{-1}$ and a global solar minimum unsigned flux disconnection rate of $6\times 10^{13}Wb\,y^{-1}$ ($6\times 10^{21}Mx\,y^{-1}$) by this mechanism. That rate corresponds to $\sim-0.2\,nT\,y^{-1}$ change in the radial heliospheric field at 1 AU, indicating that the mechanism is important to the heliospheric flux balance. ## 1 Introduction The solar dynamo generates magnetic flux inside the Sun, whch is transported outward and emerges through the Sun’s surface into the corona. Magnetic loops build up “closed” magnetic flux (connected to the Sun at both ends) in the corona. Some of these closed loops subsequently “open” into interplanetary space – that is, they are connected to the Sun at only one end with the other extending to great distances in the heliosphere or beyond. Owing to the very high electrical conductivity, open magnetic flux is frozen into the solar wind and carried out with it. The magnetized solar wind expands continuously outward from the Sun in all directions, filling and inflating our heliosphere and protecting the inner solar system from the vast majority of galactic cosmic rays. The balance between the opening and closing of magnetic flux from the Sun is thus critical and fundamental both to the solar wind and to the radiation environment of our solar system. Magnetic flux opens when coronal mass ejections (CMEs) erupt through the corona, carrying previously closed magnetic loops beyond the critical point where the solar wind exceeds the Alfvén speed (typically <$20\,R_{\odot}$) and can no longer return to the Sun. CMEs were first studied in OSO-7 and Skylab observations of the corona (e.g.Tousey 1973; Gosling et al. 1974; Hundhausen 1993), and since then continued work has provided an increasingly detailed picture of these transient magnetic structures both during their formation and ejection, and as they continue to evolve and interact with the solar wind. Long lasting, radial "legs" are often observed along the flanks of a CME and persisting behind it. These legs are generally interpreted as evidence for at least some continued magnetic connection of CMEs back to the Sun and hence the opening of new magnetic flux with CME ejections. That picture is further supported by observatation, _in situ_ , of beamed suprathermal halo electrons streaming in both directions along the local interplanetary magnetic field (IMF) during the passage of an interplanetary CME (ICME) cloud (e.g. Gosling 1990, 1993 & references therein), which are commonly interpreted as signatures of direct connection of the ICME magnetic field to the solar corona in both directions, and hence of newly opening magnetic flux. However, less is known about the equally necessary process of disconnection that must be present to remove newly opened flux and prevent the IMF from growing without limit. Because of the continual opening of magnetic flux through CMEs, McComas and coworkers in the early 1990s pursued a series of studies to determine how magnetic flux could be closed back off and avoid a so-called magnetic flux “catastrophe” of ever increasing magnetic field strength in the interplanetary magnetic field (McComas 1995& references therein). The amount of open magnetic flux in interplanetary space can be approximated with the "total flux integral" which removes the effects of variations in the solar wind speed in determining the amount of magnetic flux crossing 1 AU (McComas et al. 1992a). Using this integral, McComas et al. (1992a; 1995) showed that if all counterstreaming electron events represent simply connected opening magnetic loops, then for solar maximum CME rates, the amount of flux crossing 1 AU would double over only ~9 months. For flux rope CMEs, significantly more magnetic flux may be observed in the loops crossing 1 AU than what remains attached to the Sun along the CMEs’ legs; however, it must be stressed that if CMEs retain any solar attachment whatsoever, the flux catastrophe will ultimately occur in the absence of some other process to close off previously open fields. Of course a magnetic flux catastrophe is not observed in the solar wind and, in fact, the overall magnitude of the IMF and amount of open flux seems to vary over the solar cycle. For cycle 21, the average magnitude varied by ~50% (Slavin et al. 1986) while the total flux integral varied by ~60% (McComas et al. 1992a; 1992b), with maxima shortly after solar maximum and minima shortly after solar minimum McComas 1994. Since these studies, the solar wind has gone through a prolonged (multi-cycle) reduction in both solar wind power (the dynamic pressure of the solar wind that ultimately inflates the heliosphere) (McComas et al. 2008) and magnetic field magnitude (Smith & Balogh 2008). The lack of a flux catastrophe, solar cycle variation and now long-term reduction in the open magnetic flux from the Sun all show that there must be some process for closing off previously open field regions and returning magnetic flux to the Sun. Magnetic reconnection plays an important role in regulating the topology of solar magnetic flux, however, once the top of a loop passes the critical point, its magnetic flux remains open until some other process occurs to close it off below the critical point. That is, reconnection above the critical point can only rearrange the topology of open magnetic flux in the heliosphere – only reconnection between two oppositely directed (inward and outward field) regions of open flux close to the Sun can close off previously open magnetic flux. The most obvious method of reducing the amount of magnetic flux open to interplanetary space is via reconnection between oppositely directed, previously open field lines (McComas et al. 1989), which creates closed field loops that can return to the Sun and the release of disconnected U-shaped field structures into interplanetary space. An example of such a coronal disconnection event was shown by McComas et al. 1991 using SMM coronagraph images from 1 June 1989\. An even older example of a likely coronal disconnection event can be found as far back as the 16 April 1893 solar eclipse (e.g., Cliver 1989), where sketches (data in 1893) made in time ordered sequence from Chile, Brazil, and Senegal, indicate the outward motion of a large U-shaped structure (McComas 1994). For the opening and closing of the solar magnetic flux to maintain some sort of equilibrium, there must be some type of feedback between these two processes. McComas et al. (1989; 1991) suggested that this feedback occurs through transverse magnetic pressure in the corona, where the expansion of newly opened field regions must enhance transverse pressure and compress already open flux elsewhere around the Sun. When enough pressure builds up, reconnection between oppositely direct open flux would reduce the pressure and amount of open flux. The sequence of images from the 27 June 1988 coronal disconnection event, in fact showed just such a compression, indicated by the deflection of the streamers in the corona, just prior to and appearing to precipitate the coronal disconnection event. Another line of supporting evidence was provided by numerical simulations (Linker et al. 1992), which indicated that increased magnetic pressure could lead to reconnection across a helmet streamer and the release of disconnected flux. Schwadron et al. (2010) recently reexamined the flux balance issue in light of the anomalously long solar minimum between cycles 23 and 24 and modeled the level of magnetic flux in the inner heliosphere as a balance of that flux injected by CMEs, lost through disconnection, and closed flux lost through interchange reconnection near the Sun. McComas et al. (1992c)conducted a statistical study of three months of SMM coronagraph observations (Hundhausen (1993)) to assess the frequency of coronal disconnection events. These authors found that while the initial survey (St. Cyr & Burkepile 1990) found no obvious disconnections, six of the 53 transient events during this interval (11%) showed some evidence of disconnection in more than one frame and 13 (23%) showed a single frame with an outward "U" or "V" structure. Given the imaging and analysis technology of the day, McComas et al. (1992c) concluded that magnetic disconnection events on previously open field lines may be far more common than previously appreciated. With today’s imaging and exceptional analysis capabilities, the question of coronal disconnection events should finally be resolvable. For this study we used image sequences, collected by the SECCHI (Howard et al. 2008) instrument suite on board NASA’s _STEREO-A_ spacecraft, of Thomson- scattered sunlight from free electrons in the interplanetary plasma. The observations span from the deep solar corona to beyond 1 AU at elongation angles of up to $70^{\circ}$ from the solar disk; this continuous observation is enabled by recently developed background subtraction techniques (DeForest et al. 2011) operating on the _STEREO_ data. The signature U-shaped loops of disconnected plasma are far clearer in the processed heliospheric data far from the Sun than in the coronagraph data close to the Sun, and we detect 12 characteristic departing “V” or “U” events in 36 days – far more than the expected number based on scaling the results of McComas et al. (1992c). For this initial report, we focus on quantitative analysis of a single event. In Sections 2.1-2.7, we describe the observations and calculate the geometry, the mass evolution, and (by assuming the U-loop is accelerated by the tension force) the coronal magnetic field sand entrained flux in the disconnecting structure. As a plausibility check, we explore the tension force scenario and its consequences for the long-term evolution of the feature, and find that the scenario is consistent with accepted values for the solar wind density and speed. In Section 3, we discuss broader consequences of the observation, including estimating the disconnection rate based on the number of similar events in our data set, and discuss implications for the global magnetic flux balance. ## 2 Observations The SECCHI suite on STEREO was intended to be used as a single integrated imaging instrument (e.g. Howard et al. 2008). It consists of an EUV imager (EUVI) observing the disk of the Sun, and four visible light imagers (COR-1, COR-2, HI-1, and HI-2) with progressively wider overlapping fields of view, to cover the entire range of angles between the solar disk and the Earth. The visible light imagers view sunlight that has been Thomson scattered off of free electrons in the corona and interplanetary space; the theory of Thomson scattering observations has been recently reviewed by Howard & Tappin (2009a). We set out to view coronal and heliospheric events in the weeks around 2008 December, using newly developed background subtraction techniques to observe solar wind features in the HI-1 and HI-2 fields of view (DeForest et al. (2011); Howard & DeForest (2011)). In the initial 36 day data set we prepared, we observed 12 disconnection events identified by a clear “V” or “U” shaped bright structure propagating outward in the heliosphere. We chose a particuarly clearly presented one, which was easily traceable to its origin in the low corona on 2008-Dec-12 at 04:00, for further detailed study. ### 2.1 Image Preparation Data preparation followed standard and published techniques. For COR-1 and COR-2, we downloaded Level 1 (photometrically calibrated) data, and further processed them by fixed background subtraction: we acquired images for an 11-day period, and found the 10 percentile value of each pixel across the entire 11-day dataset. This image was median filtered over a 5x5 pixel window to generate a background image that included the F corona, any instrumental stray light, and the smooth, steady portion of the K corona. Subtracting this background from each image yielded familiar coronal images of excess feature brightness compared to the smooth, steady background. We performed one additional step: motion filtration to suppress stationary image components. This step matches the motion filtration step used for HI-1 and HI-2 (below), and suppresses the stationary streamer belt while not greatly affecting the moving features under study. The heliospheric imagers required further processing to remove the starfield, which is quite bright compared to the faint Thomson scattering signal far from the Sun in the image plane. We processed the STEREO-A HI-2 data as described by DeForest et al. (2011). The HI-1 data used a similar process adapted to the higher background gradients in that field of view and described by Howard & DeForest (2011). All the imagers yielded calibrated brightness data in physical units of the mean solar surface brightness ( $B_{\odot}=2.3\times 10^{7}W\,m^{-2}\,SR^{-1}$). Because of the wide field of view, as a subsequent processing step we distorted the images into azimuthal coordinates, in which one coordinate is azimuth (solar position angle) in the image plane and the other is either elongation angle $\epsilon$ (“radius” on the celestial sphere) or its logarithm. The latter projection, if scaled properly, is conformal: it preserves the shape of features that are small compared to their distance from the Sun. To equalize brightness, we applied radial filters to the images for presentation, with either a $\epsilon^{3.5}$ scaling (for coronal images) or a $\epsilon^{3}$ scaling (for heliospheric images). Figure 1 shows snapshots of the disconnected plasma and associated cusp, as observed by four separate instruments over the course of four days as it propagated outward. Shortly after 2008 Dec 18 04:00, the streamer belt at $160^{\circ}$ ecliptic azimuth ($20^{\circ}$ CCW of the Sun-Earth line) pinched and separated, forming a “U” loop that retracted outward, with a trailing cusp, over the course of the following three days. The feature remained visible in Thomson scattered light because of plasma scooped up during the early acceleration period in the lower corona: this plasma remained denser than the surrounding medium, yielding a bright feature throughout the data set. The disconnected plasma completely missed the ecliptic plane and was therefore not observed _in situ_ by any of the near-Earth or STEREO probes. Figure 1: The disconnection event of 2008 Dec 18 in context \- 8 still images showing formation and evolution of the U-loop: left to right, top to bottom. ### 2.2 Observing Geometry and 3-D structure The observing geometry for the 2008 Dec 18 event is shown in Figure 2, from an overhead (northward out-of-ecliptic) point of view co-rotating with the STEREO-A orbit. The event departure angle was measured both using direct triangulation between the coronagraphs in STEREO-A and STEREO-B. We used the triangulation method described by Howard & Tappin (2008). Although the disconnection event is small compared to most CMEs, subtending just a few degrees in latitude, it is still large enough to cast doubt on the simple triangulation results, so we also used “TH model” semi empirical transient event reconstruction tool (Tappin & Howard 2009) to extract the departure angle. TH was developed to reconstruct CME leading edge (“sheath”) overall envelope and propagation speed, but is also applicable to smaller transient events such as this one. Details, applications, and limitations of the TH model are further described by Tappin & Howard 2009 and by Howard & Tappin (2009b; 2010). Departure longitude was measured to be $-10^{\circ}\text{\textpm}5^{\circ}$ in heliographic coordinates, with an estimated event width of under $5^{\circ}.$ We took the disconnected feature’s trajectory to have constant radial motion (the “Fixed-$\Phi$ aproximation”) in the solar inertial frame – this leads to the slightly curved aspect to the trajectory in the co-rotating heliographic ecliptic frame, which maintains the prime meridian at the Earth-Sun line. Figure 2 shows an out-of-ecliptic projected view of the observing geometry, including construction angles and distances used in Section 2.4 for trajectory calculations. Figure 2: Observing geometry in the ecliptic plane on 2008 Dec 18 - 2008 Dec 22 ### 2.3 Feature Evolution To analyze the feature’s evolution across a two-order-of-magnitude shift in scale over its observed lifetime, we transformed the processed _STEREO-A_ source images into local heliographic radial coordinates – i.e. zero azimuth is due solar West from the viewpoint of _STEREO-A_ , with azimuthal coordinate increasing clockwise around the image plane; this follows early work by DeForest et al. (2001) in imaging polar plumes. Distances from Sun center are recorded as elongation angle $\epsilon$ from the center of the Sun, as a reminder of the angular nature of the wide-field observations. To avoid aliasing in the resampling process, we resampled the images using the optimized resampling package described by DeForest (2004). Figures 3 and 4 show the liftoff and propagation of the feature across 65 degrees of elongation from its origin in the solar streamer belt. Both figures have a radial gain filter applied to equalize the feature’s brightness, which varies by over seven orders of magnitude: from $1.5\times 10^{-9}B_{\odot}$ in the low streamer belt at 2008 Dec 18 04:30 to $5.7\times 10^{-17}B_{\odot}$ six days later, at $\epsilon=65^{\circ}$. The bright feature takes the classic wishbone shape of reconnecting field lines emerging from a current sheet (e.g. Chapter 4 of Priest et al. 2000). The aspect ratio of the wishbone may be estimated by dividing the vertical height from cusp to the top of the visible horns, by the width between the horns. This aspect ratio varies from ~10:1 when the horns are first clearly resolved near 2008 Dec 18 08:00, to approximately 2:1 some four hours later and 1:1 by 2008 Dec 19 04:00 – one full day after the first pinch is observed in the streamer belt. After 2008 Dec 19, the feature expands approximately self-similarly as it propagates, subtending approximately $16^{\circ}$ of azimuth and not changing its aspect ratio throughout the rest of its trajectory. Note that aspect ratio is _not_ preserved by the linear azimuthal mapping used in Figure 3, which was selected to show the early acceleration clearly; aspect ratio is preserved by the (conformal) logarithmic mapping used in Figure 4, which shows nearly self-similar expansion in the image plane despite perspective effects that come into play above about $\epsilon=30^{\circ}$ The scaling of brightness is reassuring because, in a uniformly propagating wind with no acceleration, density must decrease as $r^{-2}$ and feature column density must thus decrease as $r^{-1}$, while illumination decreases as $r^{-2}$, so feature brightness is expected to decrease as $r^{-3}$. The fact that brightness levels do not change much across Figure 4, which is scaled by $\epsilon^{3}$, suggests that the disconnected flux and material entrained in it are indeed propagating approximately uniformly. The fact that they _do_ change slightly, with brighter images to the right, indicates that the feature is gaining intrinsic brightnsss by accumulating material as it propagates. The horizontal positions and error bars in Figures 3 and 4 are the results of manual feature location of the cusp, with a point-and-click interface. The white error bars are based on the sharpness of the feature. In the excess brightness plot, the feature is easy to see but blurs near the top of Figure 3 due to the higher levels of both photon noise and motion blur as the feature accelerates to the top of the coronagraph field of view. The running difference plot highlights fine scale feature and helps identify the cusp location near the top of the COR-2 field of view. Figure 3: Formation and early acceleration of the 2008 Dec 18 disconnection event through the STEREO-A COR-1 and COR-2 fields of view The trailing edge of the event is marked, with error bars based on feature identification. TOP: direct excess-brightness images show feature formation and overall structure. BOTTOM: running-difference images show detail. These stack plots include a small image of the feature at each sampled time to show evolution. Intensities are scaled with $\epsilon^{3.5}$ to equalize brightness vs. height. The individual images have been resampled into linear azimuthal (radial) coordinates, and the horizontal range is $160^{\circ}-174^{\circ}$ of azimuth. Note that this projection does not preserve aspect ratio: despite appearances, the event widens as it rises. Figure 4: Propagation and evolution of the 2008 Dec 18 disconnection event through the STEREO-A HI-1 and HI-2 fields of view. The trailing edge of the event is marked, with error bars based on feature identification. These stack plots include a small image of the feature at each sampled time to show evolution. The individual images have been resampled into logarithmic azimuthal (radial) coordinates, and the horizontal range is $162^{\circ}-178\text{\textdegree}$of azimuth. This projection is conformal, so the shape of the feature is preserved in each image. Intensities are scaled with $\epsilon^{3}$ to equalize brightness vs. height. Note self-similar expansion: the angular width and shape of the feature are preserved. ### 2.4 Acceleration profile Converting angular observed coordinates to examine the inertial behavior of the plasma requires triangulation using the Law of Sines. Using the “fixed $\Phi"$ approximation (assuming the feature’s cusp is small and that it propagates in a radial line from the Sun), the feature’s radius from the Sun is easily calculated: $r_{ev}=r_{A}\frac{{sin\left(\epsilon^{\prime}\right)}}{sin\left(\epsilon\right)},$ (1) where the variables take the meanings in Figure 2: $\epsilon$ is the solar elongation of the feature as seen from _STEREO-A_ , $\epsilon^{\prime}$ is the solar elongation of _STEREO-A_ as seen from the feature), and the _STEREO-A_ solar distance $r_{A}$ is found by spacecraft tracking and is supplied by the mission. Although no camera was present at the event itself, $\epsilon^{\prime}$ is calculated by noting that $\epsilon^{\prime}=180^{\circ}-\epsilon-\left(L-L_{ev}\right)$, as the feature, _STEREO-A,_ and the Sun form a triangle. Figure 5 shows the results of the tracking from Figures 3 and 4, propagated through Equation 1. As expected, the event rapidly accelerates during the early phase, reaching a peak acceleration of $20\,m\,s^{-2}$ as the aspect ratio changes in the initial hours. The acceleration peaks 4-5 hours after the initial pinch in the streamer belt, or 2-3 hours after the first observation of a well-formed cusp. The feature reaches its final speed of $\sim 320\pm 15\,km\,s^{-1}$ within just 8 hours of the initial pinch at 04:00 and within 6 hours of the first observation of the well formed cusp at 06:00, and undergoes no further significant acceleration nor deceleration during its obsered passage to beyond 1 AU over the next five days. Figure 5: Inferred position, speed, and acceleration of the disconnected plasma from the 2008-Dec-18 event, during onset (LEFT) and over the full observation period (RIGHT). Error bars are derived by propagating _a priori_ location error and geometric error in the longitude of the event. The shaded region indicates the full time range of the left-side plots. ### 2.5 Mass profile We extracted photometric densities using the feature brightness in suitable frames. The feature brightness is determined from the density via the Thomson scattering equation (see, e.g., Howard & Tappin 2009a for a clear exposition). Compact features can be treated as nearly point sources, and the line-of-sight integral for the optically thin medium reduces to: $B=B_{\odot}\Omega_{\odot}(r)\sigma_{e}\left(1+cos^{2}\chi\right)\rho\mu_{av}^{-1}d$ (2) where $B$ is the measured feature brightness (in units of emissivity: $Wm^{-2}SR^{-1}$), $B_{\odot}$ is (still) the solar surface brightness; $\Omega_{\odot}(r)$ is the solid angle subtended by the Sun at the point of scatter, well approximated by $\pi r_{\odot}^{2}r_{ev}^{-2}$ everywhere above about 4 $r_{\odot}$; $\sigma_{e}$ is the differential Thomson scattering cross section, given by half of the square of the classical electron radius $r_{e}^{2}/2=4.0\times 10^{-30}m^{2}$; $\chi$ is the scattering angle (equal to $\epsilon^{\prime}$ in Figure 2); $\rho$ is the mass density; $\mu_{av}$ is the average mass per electron in the coronal plasma; and $d$ is the depth of the feature. $mu_{av}$ may be calculated from the spectroscopically measured 5% He/H number ratio in the corona (Laming & Feldman 2000) and the assumption that the helium is fully ionized (yielding two electrons per ion). This yields $mu_{av}=1.1m_{p}=1.84\times 10^{-27}kg$. Solving for the line-of-sight integrated mass surface density $\rho d$ gives: $\rho d=\mu_{av}\frac{B}{B_{\odot}}\Omega_{\odot}^{-1}(r)\sigma_{e}^{-1}\left(1+cos^{2}\epsilon^{\prime}\right)^{-1}$ (3) and therefore $m_{ev}=\left(\rho d\right)wh=\mu_{av}\frac{B}{B_{\odot}}\Omega_{\odot}^{-1}(r)\sigma_{e}^{-1}\left(1+cos^{2}\epsilon^{\prime}\right)^{-1}\Omega_{ev}^{\left(1+cos^{2}\epsilon^{\prime}\right)}S^{2}$ (4) where $w$ and $h$ are the dimensions shown in Figure 7; $\Omega_{ev}$ is the solid angle subtended by the feature in the images; and $S$ is the calculated spacecraft-feature distance, calculated by the law of sines as for $r_{ev}.$ To extract the mass profile from the data, we generated an image sequence containing the feature, and marked the locus of the feature visually using a pixel paint program. Using the generated masks, we summed masked pixels in the feature for each photometric image, thereby integrating the feature brightness over the solid angle represented by the corresponding pixels, to obtain an intensity and an average brightness within the feature. To account for errors in visual masking, we assigned error bars based on one-pixel dilation and one- pixel contraction of the masked locus. We omitted frames with excessive noise, encroachment of an image boundary, or a star or cosmic ray in or near the feature. The results of the calculation are given in Figure 6, which shows steady accretion of material through most of the journey through the heliosphere. Because our photometric analysis is based on subtraction of a calculated background derived from the data set itself, we measure only excess feature brightness (not absolute brightness) from Thomson scattering; thus our brightness measurements and mass estimates are biased low, because we cannot measure the absolute density of the background. The initial derived mass of 20-25 Tg translates to an electron number density of $2\times 10^{7}\,cm^{-3}$ in the lower corona, which is comparable to the density in bright coronal features – so the total mass may be up to a factor of order two higher. The final “feature excess” mass is $8\pm 2\times 10^{10}kg$, and the final subtended solid angle is 0.028 SR, for a presented cross-section of $4.3\pm 0.1\times 10^{20}m^{2}$. Taking the depth to be the square root of the observed cross section yields an estimated volume at 1 AU of $8.9\pm 0.3\times 10^{31}m^{3}$, for a total estimated excess electron density of $5\pm 1.4\,cm^{-3}$ at 1 AU, which is in good agreement with slow solar wind densities ($3-10\,cm^{-3}$ when scaled to 1 AU) that were observed by _Ulysses_ in situ in the same heliographic latitude range (e.g. McComas et al. 2000). Approximately 2/3 of this excess density appears to have been accumulated enroute from the surroundign solar wind; this is further described in Section 2.7. Figure 6: Photometrically determined excess mass profile of the retracting disconnected feature of 2008 Dec 18. Error bars are based on identification of the feature boundary in the images. The trendline is extracted from regression of the HI-1 and HI-2 data. The mass shown is excess mass in the feature compared to the background solar wind (see text). ### 2.6 Entrained magnetic flux Figure 7: Cartoon of the initial acceleration process of a disconnection event. Tension force along newly released field lines is balanced by mass entrained on the field lines. By measuring the acceleration and mass we infer the amount of magnetic flux that was disconnected. From the mass of the feature, and its acceleration, it is possible to extract the entrained magnetic field by measuring the rate of change of momentum and inferring a magnetic tension force via $f=ma$. The system is sketched in Figure 7. The magnetic tension force is conserved along the open field lines, so we can calculate it at any convenient cut plane including the one shown. Tension force is frequently referred to as a “curvature force” and calculated locally; here we integrate around the “U”, and notice that the integrated force is just the unbalanced tension on the field lines contained in the “U” shape. It is therefore given by $f=m_{ev}a_{ev}=f_{B}=\frac{B^{2}A}{2\mu_{0}}=\frac{{\Phi^{2}}}{2\mu_{0}A}$ (5) where, here, $B$ is the magnetic field strength (not brightness, as before). Solving for $\Phi$, $\Phi=\sqrt{2\mu_{0}dwm_{ev}a_{ev}}$ (6) taking $m_{ev}$ to be $25Tg$ ($2.5\times 10^{10}kg$) during the peak of the acceleration, and taking $w=d=0.2R_{\odot}$ (based on the measured width of the feature’s fork during maximum acceleration, at $6R_{s}$ from the surface ($7R_{s}$ from Sun center) gives $\Phi=1.6\times 10^{11}Wb$ ($1.6\times 10^{19}Mx$ ), corresponding to an average field strength of 8$\mu T$ ( 0.08 Gauss) at that altitude, or an equivalent $r^{2}$-scaled field of 400$\mu T$ (4 Gauss) at the surface; this is comparable to accepted values of the open flux density at the solar surface at solar minimum. Because of the way $m$ was calculated (section 2.5 above) this figure is probably low by a factor of order $\sqrt{2}$. ### 2.7 Accretion and force balance As the disconnectioned structure travels outward, it accretes new material. This effect is dramatic: as seen in Figure 6, the mass increases by a factor of 3 from the corona to 1 AU. We conjecture that the material is accreted by “snowplow” effects from the plasma ahead of the disconnected cusp as it propagates. For the observed mass growth in the feature, new material must be compressed to become visible in our Thomson scattering images, and the most plausible way for it to be compressed is via ram effects. This scenario also neatly explains the constant speed of the feature, by balancing the continued tension force from the cusp with accretion momentum transfer. Here we explore the concept of force balance between accretion and the tension force, to identify whether some other model is required in addition to this simple one. Extending Newton’s law to include momentum transfer by accretion, and neglecting all but the tension force, $\frac{\Phi^{2}}{2\mu_{0}A_{\Phi}}=m_{ev}a_{ev}+\frac{dm_{ev}}{dt}\Delta v,$ (7) where the LHS is just the tension force from Equation 5, with the modification that the cross section of the exiting field lines is written $A_{\Phi}$; $a_{ev}=0$ after the initial acceleration; and the second term represents momentum transfer into accreted material, with $\Delta v$ being the difference between the feature speed and surrounding wind speed. The feature is thus in equilibrium between accretion drag and continued acceleration by the tension force. This accretion drag is important to the observed increase in feature mass, because ram pressure against the surrounding wind material is what compresses incoming material and renders it visible in the data against the subtracted background. Applying conservation of mass, we can relate $\Delta v$ and the average density of the background solar wind through which the feature is propagating: $\rho_{sw}=\frac{dm_{ev}/dt}{A_{ev}\Delta v}.$ (8) where $A_{ev}$ is the geometrical area presented by the feature to the slow wind ahead of it. Solving Equations 7 and 8 to eliminate $\Delta v$ gives $\rho_{sw}=\left(\frac{dm_{ev}}{dt}\right)^{2}\left(\frac{2\mu_{0}}{\Phi^{2}}\right)\left(\frac{A_{\Phi}}{A_{ev}}\right),$ (9) which gives the background solar wind density in terms of the accumulation rate of mass in the observed feature, assuming constant outflow for both the wind and the feature, and acceleration by the tension force. Given the conservation of mass and the approximately constant speed of the solar wind, $\rho_{sw}$ falls as approximately $r^{-2}$. Further, we observe nearly self- similar expansion throughout most of the heliospheric range, so $A_{\Phi}/A_{ev}$ is constant in that part of the trajectory – hence $dm_{ev}/dt$ must also fall as $r^{-1}$ during the approximately constant speed portion of the feature’s lifetime. Using this functional form, we can extract an analytic expresson for the feature mass versus radius. We introduce the $r^{-1}$ dependence by switching from the linear regression used in Figure 6, to a semi-log regression that assumes $dm_{ev}/d(log_{e}(r)$ to be constant. Figure 8 shows such a regression, with the result that $dm_{ev}/dr=34\pm 3\times 10^{9}\left(R_{\odot}/r\right)kgR_{\odot}^{-1}$. Including the measured outflow speed of $315\pm 15\,km\,s^{-1}$, we find that $dm_{ev}/dt=\left(1AU/r\right)\left(7.1\pm 1\times 10^{4}kg\,s^{-1}\right)$. Including all of these values into Equation 9, together with the average particle mass from Section 2.5, yields a background wind numeric density of $n_{sw}\left(1\,AU\right)$ of $30\pm 6\,cm^{-3}\left(A_{\Phi}/A_{ev}\right)$. From the morphology of the feature in Figure 4, we conservatively estimate $A_{\Phi}/A_{ev}<0.25$, i.e. the forward cross section of the “horns” of the vee appears to be well under 1/4 of the cross section of the vee itself. This value yields a derived background solar wind density of $n_{sw}<8\,cm^{-3}$ at 1 AU to maintain the force balance in Equation 7. That figure is again in line with the wind measurements from _Ulysses_ at $15^{\circ}$ heliographic latitude (McComas et al. 2000), adding to the plausibility of the accretion force balance picture. The corresponding mass density limit is $\rho_{sw}<8\times 10^{-19}\,kg\,m^{-3}$ at 1 AU As a sanity check, we can use this $\rho_{sw}$ limit and Equation 8 to find that $\Delta v$ must then be a few tens of $km\,s^{-1},$ i.e. the background wind speed must be close to $300\,km\,s^{-1}$. We conclude that the picture of force balance between snowplow accretion and the tension force is at least broadly consistent with the observed feature, though further study of more events (preferably with corresponding _in situ_ measurements of the feature itself) is necessary. Figure 8: Semilog regression fit of $m_{ev}(r)$ ## 3 Discussion Using data from STEREO/SECCHI, we have identified and measured the characteristics of a single flux disconnection event and associated cusp feature, similar to that discovered by McComas et al. (1992), from initial detection in the lower corona to distances beyond 1 AU. The cusp feature is formed in the classic X-point geometry and rapidly accelerates under the tension force to approximately $320\,km\,s^{-1}$, which it reaches in under 4 hours at an altitude of approximately 10$R_{\odot}$. Thereafter the feature continues to accumulate mass but maintains approximately constant speed until it is lost to sight 1.2 AU from the Sun. Based on photometry, we are able to estimate the onset mass of the event as $25\,Tg$ and the entrained flux as $160\,GWb,$ corresponding to a coronal field strength of $0.08\,G$ and an $r^{2}$-normalized surface open field of $4\,G$ over the projected surface footprint of the feature. These estimates are likely low by a factor of order $\sqrt{2}$, because they make use of feature excess brightness rather than absolute Thomson-scattered brightness in the coronagraph images; using polarized-brightness imagery could improve the measurement by separating the non-transient component of the Thomson scattering signal from the unwanted F coronal background. Because our measurements are all based on morphology and photometry, we have performed several consistency checks to build confidence in the calculated parameters of the feature as it propagates. In particular, a model of simple force balance between the tension force and mass accretion is consistent with both the inferred magnetic field and accepted values for background slow solar wind density and speed. Simple accretion models such as we developed here demonstrate clearly why ejected features such as U-loops or CMEs seem frequently to propagate at near constant speed: under continuous weak driving, an equilibrium forms rapidly between the driving force and momentum transfer by mass accretion. The equilibrium outflow speed is the sum of a large, fixed (or at least driver- independent) speed – that of the surrounding wind – with a smaller offset speed that drives mass accretion. Thus the feature speed is quite insensitive to the driver. In our case, doubling the tension force would only increase the outflow rate by $\sim 10\%$. The event under study is well presented, but is not unusual at all; such events are easy to identify in heliospheric image sequences, because of their distinctive “U” and cusp shape; they are readily traced back to the corona. This technique represents a new, very effective way of finding these disconnection events, which are small and hard to identify __ in the coronagraph sequences alone, but are strongly and easily visible in the processed heliospheric images. In an initial reduced data set of 36 days near the deepest part of the recent extended solar minimum (2008 Dec – 2009 Jan), we identified 12 such events; all of them were identified by tracking “V” or “U” shapes back from the heliospheric images to the corona. Assuming the present feature to be typical, and considering that the single viewpoint affords clear coverage of about 1/4 of the circumference of the Sun, we estimate the global disconnection feature rate at that time to be over $1\,event\,d^{-1}$, and the flux disconnection rate to thus be at least of order $60\,TWb\,y^{-1}$. Expanded to a 1 AU sphere, this amounts to a rate of change of the open field of order $0.2\,nT\,y^{-1}$, which is a significant fraction of the observed cycle- dependent rate of change of the open heliospheric field (e.g. Schwadron, Connick, & Smith 2010). These figures are based on a single calculated flux and an event rate obtained by initial visual inspection of a single 36-day data set, and hence are merely rough estimates – but they indicate that flux disconnections of this type are important to the global balance of open flux. Further study, in the form of a systematic survey, is needed to determine whether they are the primary mechanism of flux disconnection from the Sun. The authors thank the STEREO instrument teams for making their data available. Our image processing made heavy use of the freeware Perl Data Language (http://pdl.perl.org). The work was enhanced by enlightening conversations with J. Burkepile, C. Eyles, and N. Schwadron, to whom we are indebted. This work was supported by NASA’s SHP-GI program, under grant NNG05GK14G. ## 4 References ## References * Cliver (1989) Cliver, E. W. 1989, Sol. Phys., 122, 319 * DeForest (2004) DeForest, C. E. 2004, Sol. Phys., 219, 3 * DeForest et al. (2011) DeForest, C. E., Howard, T. A., & Tappin, S. J. 2011, ApJ, 738, 103 * DeForest et al. (2001) DeForest, C. E., Plunkett, S. P., & Andrews, M. D. 2001, ApJ, 546, 569 * Gosling (1990) Gosling, J. T. 1990, Washington DC American Geophysical Union Geophysical Monograph Series, 58, 343 * Gosling (1993) —. 1993, J. Geophys. Res., 981, 18937 * Gosling et al. (1974) Gosling, J. T., Hildner, E., MacQueen, R. M., Munro, R. H., Poland, A. I., & Ross, C. L. 1974, J. Geophys. Res., 79, 4581 * Howard et al. (2008) Howard, R. A., et al. 2008, Space Sci. Rev., 136, 67 * Howard & DeForest (2011) Howard, T. A., & DeForest, C. E. 2011, ApJ, (submitted) * Howard & Tappin (2008) Howard, T. A., & Tappin, S. J. 2008, Sol. Phys., 252, 373 * Howard & Tappin (2009a) —. 2009a, Space Sci. Rev., 147, 31 * Howard & Tappin (2009b) —. 2009b, Space Sci. Rev., 147, 89 * Howard & Tappin (2010) —. 2010, Space Weather, 80, S07004 * Hundhausen (1993) Hundhausen, A. J. 1993, J. Geophys. Res., 981, 13177 * Linker et al. (1992) Linker, J. A., van Hoven, G., & McComas, D. J. 1992, J. Geophys. Res., 971, 13733 * McComas (1994) McComas, D. J. 1994, NASA STI/Recon Technical Report N, 951, 10770 * McComas (1995) —. 1995, Tongues, bottles, and disconnected loops: The opening and closing of the interplanetary magnetic field (95RG00124), Tech. rep. * McComas et al. (2008) McComas, D. J., Ebert, R. W., Elliott, H. A., Goldstein, B. E., Gosling, J. T., Schwadron, N. A., & Skoug, R. M. 2008, Geophys. Res. Lett., 351, L18103 * McComas et al. (1992a) McComas, D. J., Gosling, J. T., & Phillips, J. L. 1992a, J. Geophys. Res., 97, 171 * McComas et al. (1992b) McComas, D. J., Gosling, J. T., & Phillips, J. L. 1992b, in Solar Wind Seven Colloquium, ed. E. Marsch & R. Schwenn, 643–646 * McComas et al. (1989) McComas, D. J., Gosling, J. T., Phillips, J. L., Bame, S. J., Luhmann, J. G., & Smith, E. J. 1989, J. Geophys. Res., 94, 6907 * McComas et al. (1991) McComas, D. J., Phillips, J. L., Hundhausen, A. J., & Burkepile, J. T. 1991, Geophys. Res. Lett., 18, 73 * McComas et al. (1992c) McComas, D. J., Phillips, J. L., Hundhausen, A. J., & Burkepile, J. T. 1992c, in Solar Wind Seven Colloquium, ed. E. Marsch & R. Schwenn, 225–228 * McComas et al. (2000) McComas, D. J., et al. 2000, J. Geophys. Res., 105, 10419 * Priest et al. (2000) Priest, E., Forbes, T., & Murdin, P. 2000, Magnetohydrodynamics, ed. Murdin, P. * Schwadron et al. (2010) Schwadron, N. A., Connick, D. E., & Smith, C. 2010, ApJ, 722, L132 * Slavin et al. (1986) Slavin, J. A., Jungman, G., & Smith, E. J. 1986, Geophys. Res. Lett., 13, 513 * Smith & Balogh (2008) Smith, E. J., & Balogh, A. 2008, Geophys. Res. Lett., 352, L22103 * St. Cyr & Burkepile (1990) St. Cyr, O. C., & Burkepile, J. T. 1990, NASA STI/Recon Technical Report N, 911, 17996 * Tappin & Howard (2009) Tappin, S. J., & Howard, T. A. 2009, Space Sci. Rev., 147, 55 * Tousey (1973) Tousey, R. 1973, in Space Research XIII, Vol. 2, p. 713 - 730, ed. M. J. Rycroft & S. K. Runcorn, 713–730	arxiv-papers	2011-11-30T15:37:46	2024-09-04T02:49:24.827543	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "C. E. DeForest, T. A. Howard, D. J. McComas", "submitter": "Craig DeForest", "url": "https://arxiv.org/abs/1111.7211" }
1111.7243	Quantifying the dynamics of coupled networks of switches and oscillators Matthew R. Francis1, Elana J. Fertig2,∗, 1 Physics Department, Randolph-Macon College, Ashland, VA, USA 2 Department of Oncology and Division of Oncology Biostatistics and Bioinformatics, Sidney Kimmel Comprehensive Cancer Center, School of Medicine, Johns Hopkins University, Baltimore, MD, USA $\ast$ E-mail: Corresponding ejfertig@jhmi.edu ## Abstract Complex network dynamics have been analyzed with models of systems of coupled switches or systems of coupled oscillators. However, many complex systems are composed of components with diverse dynamics whose interactions drive the system’s evolution. We, therefore, introduce a new modeling framework that describes the dynamics of networks composed of both oscillators and switches. Both oscillator synchronization and switch stability are preserved in these heterogeneous, coupled networks. Furthermore, this model recapitulates the qualitative dynamics for the yeast cell cycle consistent with the hypothesized dynamics resulting from decomposition of the regulatory network into dynamic motifs. Introducing feedback into the cell-cycle network induces qualitative dynamics analogous to limitless replicative potential that is a hallmark of cancer. As a result, the proposed model of switch and oscillator coupling provides the ability to incorporate mechanisms that underlie the synchronized stimulus response ubiquitous in biochemical systems. ## Introduction The dynamics in systems ranging from intercellular gene regulation to organogenesis are driven by complex interactions (represented as edges) in subcomponents (represented as nodes) in networks. If the structure of these networks is known, network-wide models of coupled systems have been applied to predict their qualitative dynamics. For example, models of coupled switches based upon Glass networks [1] have been applied to model systems such as neuronal synapses [2] and gene regulatory networks [3]. Similarly, models of coupled oscillators along networks based upon the Kuramoto model [4] have been used to model synchronization of oscillators in diverse systems reviewed in [5]. In biochemical systems, in vivo oscillator synchronization has been observed in synthetic oscillatory fluorescent bacteria [6, 7], yeast gene regulatory networks [8, 9], and human cell fate decisions [10]. Such spontaneous synchronization has also been attributed to the development of the mammalian cardiac pacemaker cells (reviewed in [11]) and cortical systems (reviewed in [12]) including notably the circadian pacemaker (e.g., [13]). More recently, these network models have been found to be insufficient to model more complex dynamics in neuronal information transfer [14, 15, 16, 12, 17] and cardiac arrhythmias [18, 19, 20, 21]. These limitations extend to physical systems, such as the coupled lasers studied in [22]. Therefore, numerous studies have modified these network models to account for evolving networks [15, 23, 24, 25, 26, 27, 28], dynamic frequencies [15, 29, 30], or phase delays [16, 31, 32, 33]. However, these mathematical modifications typically do not encode the mechanism underlying the limitations in the Kuramoto and Glass network models. We hypothesize that the observed limitations in the standard Kuramoto and Glass models arise from their exclusion of coupling components with qualitatively different dynamics. Several studies have inferred that biochemical systems contain “network motifs” with both oscillatory and switch- like dynamics [34, 35]. The dynamics of these motifs are inferred from the topology of subgraphs in the networks of these systems. Their structures are statistically overrepresented in biochemical networks [36, 37] such as intracellular regulatory networks [38], implicating evolutionary preservation (and thus utility) of these network motifs [39]. The dynamics of these motifs have been used to model yeast cell cycle regulation [40] and have been further confirmed in synthetic, designed biochemical circuits (reviewed in [41]). Because these heterogeneous network motifs are all identified as components within a single biochemical network, their interactions must drive the global dynamics of the network [42]. Previously, [43] have shown that coupling small sets of heterogeneous network motifs ensures the robustness of motif dynamics and [42] have shown that coupling networks changes their dynamics in isolation. However, the network-level dynamics that result from coupling oscillatory and switch-like components have not been studied comprehensively. In this paper, we develop a theoretical framework to quantify the network-wide dynamics resulting from coupling switches and oscillators. This model is based upon introducing cross-coupling between the Kuramoto and Glass models, due to their wide success in modeling the dynamics in networks of oscillators and networks of switches, respectively. Simulations with the proposed model across state-space in an all-to-all network yields four operational states: (1) switches remain “on” and oscillators synchronize, (2) switches are “off” and oscillators freeze, (3) switches fluctuate in sync with oscillators, and (4) switches fluctuate transitionally until oscillators freeze. Further application of our model to the network motifs identified in yeast in [44] recapitulates the qualitative dynamics of the system observed in that study. However, a simple rewiring of this cell-cycle network that introduces feedback causes a cancer-like sustained re-activation of the cell cycle machinery without regard for external signal growth signals. These dynamics suggest that modeling cross-motif coupling may predict critical processes in the dynamics of biochemical networks with minimal parameterization. ### The Kuramoto model of coupled oscillators Quantitative studies of coupled oscillators often apply the Kuramoto model of $M$ oscillators coupled in an all-to-all network. In this model, the change in time $\dot{\theta}_{i}$ of the phase of the $i^{\mbox{th}}$ oscillator, $\theta_{i}$, is governed by $\dot{\theta_{i}}=\hat{\omega}_{i}+\frac{\kappa_{\theta,\theta}}{M}\sum_{j=1}^{M}\sin\left(\theta_{j}-\theta_{i}\right),$ (1) where $\hat{\omega}_{i}$ is the natural frequency of the $i^{\mbox{th}}$ oscillator and $\kappa_{\theta,\theta}\geq 0$ is the coupling strength of the oscillators [4]. Typically, the $\hat{\omega}_{i}$ values are drawn from a normal distribution centered at 0 with variance $\sigma_{\omega}$. In the Kuramoto model, the phases of the oscillators will synchronize if $\kappa_{\theta,\theta}$ is above a threshold coupling strength $\hat{\kappa}_{\theta,\theta}$. Such synchronization is quantified with the mean field of the oscillators as $r_{\theta}e^{i\psi}=\frac{1}{M}\sum_{j=1}^{M}e^{i\theta_{j}}.$ (2) Here $\psi$ is the average phase of the oscillators and the coherence $r_{\theta}$ represents the spread of the oscillators from that average phase. Based upon eq. (2), $r_{\theta}=1$ if each $\theta_{i}=\psi$ and $r_{\theta}=0$ if the values of $\theta_{i}$ are distributed uniformly between $[0,2\pi)$ [45]. ### Glass networks of coupled switches Coupled sets of $N$ switches, which adopt one of a set of binary states, are modeled with Glass networks [1]. These models describe the evolution of the $i^{\mbox{th}}$ switch ($\tilde{x}_{i}$) as follows $\dot{x}_{i}=-x_{i}+F_{i}\left(\tilde{x}_{1},\tilde{x}_{2},\ldots,\tilde{x}_{n}\right)\mbox{, and}$ (3) $\tilde{x}_{i}=0\mbox{ if }x_{i}<0\mbox{; }1\mbox{ otherwise},$ (4) where $\dot{x}_{i}$ represents the change in time of the value of each $x_{i}$, which are unobservable continuous variables that control the time of switching between observable, discrete states in $\tilde{x}_{i}$. In this model, $F_{i}$ describes the change in state of the $i^{\mbox{th}}$ switch due to the coupling with the other $N$ switches in the network [1]. In specified network structures and functions $F_{i}$, such Glass networks can exhibit complex dynamics, including periodic and aperiodic orbits (e.g., [46]). One type of Glass network, called a Hopfield network [2], has dynamics applicable to the smooth-decay of signal in biochemical switches [2]. The Hopfield model lets $F_{i}=\kappa_{x,x}\sum_{j=1}^{N}w_{ij}\tilde{x}_{j}-\tau_{i},$ (5) where $w_{ij}$ takes values between $-1$ and $1$ representing the relative strength of the connection between switches $i$ and $j$, $\kappa_{x,x}$ is the magnitude of coupling strengths, and $\tau_{i}$ the threshold for switch activation. Similar to the Kuramoto model, sets of the switches will synchronize for $\kappa_{x,x}$ above a threshold $\hat{\kappa}_{x,x}$ in appropriate network topologies. ## Results ### Network model of coupled oscillators and switches By combining the established models for switches and oscillators, we model the dynamics of the heterogeneous system of coupled switches and oscillators in systems including biochemical networks with the following set of equations: $\displaystyle\dot{x}_{i}$ $\displaystyle=$ $\displaystyle- x_{i}+G_{i}\left(\tilde{x}_{1},\tilde{x}_{2},\ldots,\tilde{x}_{N},\theta_{1},\theta_{2},\ldots,\theta_{M}\right)$ (6) $\displaystyle\dot{\theta}_{l}$ $\displaystyle=$ $\displaystyle\omega_{l}\left(\tilde{x}_{1},\tilde{x}_{2},\ldots,\tilde{x}_{N}\right)$ $\displaystyle+$ $\displaystyle H_{l}\left(\tilde{x}_{1},\tilde{x}_{2},\ldots,\tilde{x}_{N},\theta_{1},\theta_{2},\ldots,\theta_{M}\right).$ Here, eq. (6) is analogous to the Glass network in eq. (3) and $\tilde{x}_{i}$ is defined according to eq. (4). In this study, we explore a case of the switch-oscillator model in eqs. (6) and (Network model of coupled oscillators and switches) which contains an all- to-all network that couples the Kuramoto model, eq. (1), and Hopfield network, eqs. (3) - (5), as follows $\displaystyle\dot{x}_{i}$ $\displaystyle=$ $\displaystyle- x_{i}+\frac{\kappa_{x,x}}{N}\sum_{j\neq.~{}i}^{N}\tilde{x}_{j}+\frac{\kappa_{x,\theta}}{M}\sum_{k=1}^{M}\tilde{\theta}_{k}-\tau_{i},$ (8) $\displaystyle\dot{\theta}_{l}$ $\displaystyle=$ $\displaystyle\omega_{l}+\frac{\kappa_{\theta,\theta}}{M}\sum_{k=1}^{M}\sin\left(\theta_{k}-\theta_{l}\right),$ (9) $\displaystyle\dot{\omega}_{l}$ $\displaystyle=$ $\displaystyle\frac{\kappa_{\theta,x}}{N}\sum_{j=1}^{N}(\tilde{x}_{j}\hat{\omega}_{l}-\omega_{l}),$ (10) where $\kappa_{x,\theta}$ and $\kappa_{\theta,x}$ are cross-component coupling strengths. In eq. (10), $\omega_{l}$ is the time-varying frequency of the $l^{\mbox{th}}$ oscillator resulting from switch coupling, with initial values $\omega_{l}(t=0)=\hat{\omega}_{l}$, and $\tilde{\theta}_{l}=\left\\{\begin{array}[]{ll}1&\mbox{ if }0\leq\theta_{l}<\pi\\\ 0&\mbox{ otherwise}\\\ \end{array}\right..$ (11) In this system, zero values of the cross-coupling parameters $\kappa_{x,\theta}$ and $\kappa_{\theta,x}$ cause the model to reduce to the standard uncoupled Kuramoto and Hopfield models. Similar decoupling of the models occurs if the switch and oscillator systems are at vastly different timescales, determined by the $\tau_{i}$ and $\hat{\omega}_{l}$ parameters, respectively. The transformation in eq. (11) facilitates comparable switch- like dynamics in the oscillators when they interact with switches in eq. (8). Nonzero switch-oscillator ($\kappa_{x,\theta}$) interactions will cause an oscillator in the “up” part ($\tilde{\theta}_{l}=1$) of its cycle to feed energy into the switch in question, nudging it towards the “on” state if off or delaying its decay if already on. Similarly, an “on” switch with a nonzero oscillator-switch interaction ($\kappa_{\theta,x}$) will feed energy into the oscillators causing them to cycle at their natural frequency if coupled to that switch. By thus incorporating coupling between switches and oscillators within the framework established by the standard Kuramoto and Hopfield models, the dynamics of our model in eqs. (8) - (10) can be analyzed within the framework of these well established models. Similar to analysis of the Kuramoto model and Glass network, we summarize the dynamics of our system using order parameters. For oscillators, we utilize the order parameter defined in eq. (2). We introduce a new order parameter $r_{\omega}(t)=\frac{1}{M}\sum_{\mu}\frac{\omega_{\mu}(t)}{\hat{\omega}_{\mu}}$ (12) that tracks how closely each individual oscillator’s frequency $\omega_{\mu}$ corresponds to the natural frequency $\hat{\omega}_{\mu}$. Analogously, we measure the fraction of switches that are in the “on” position at a given time using a switch-switch order parameter defined by $r_{x}(t)=\frac{1}{N}\sum_{j}\tilde{x}_{j}(t).$ (13) Both of these functions will have a maximum of 1 when all switches are on, and minimum 0 if all switches are off. ### Simulation results in all-to-all networks We first explore the qualitative dynamics of the heterogeneous system through numerical simulations in all-to-all networks. We limited these simulations to all-to-all networks, because of the ability of this network topology to describe the qualitative dynamics from the Kuramoto model. These simulations explore the majority of parameter space defined by $\kappa_{x,x}$, $\kappa_{x,\theta}$, $\kappa_{\theta,\theta}$, $\kappa_{\theta,x}$, and $\sigma_{\omega}$. Specifically, we select $\kappa_{x,x}=1<\hat{\kappa}_{x,x}$ to ensure that switches are able to turn off without appropriate stimulation from the oscillators. We consider the effects of switches on oscillators for values of $\kappa_{\theta,\theta}$ both above and below the Kuramoto threshold $\hat{\kappa}_{\theta,\theta}$. Figure 1 plots the time-dependent order parameters observed in the four qualitative states observed in simulations of the coupled model eqs. (8) - (10) that are reflective of the qualitative dynamics observed in simulations with these parameter values. Supplemental videos S1-S4 further summarize the results of these simulations. We note that these four states were the only qualitative states observed for our coupled model in all-to-all networks simulated according to the description in the Methods section. Because $\tau$, $\kappa_{x,\theta}$, and $\sigma_{\omega}$ all control the relative timing of switches and oscillators, their values were selected in these simulations to optimize visualization in the supplemental videos. When exploring the effect of timing on the system dynamics, we hold $\tau$ and $\kappa_{x,\theta}$ fixed while varying $\sigma_{\omega}$. Figure 2 shows the probability of observing the states in Figures 1-1 in 100 simulations of all-to-all systems containing 100 switches and oscillators as a function of $\kappa_{\theta,x}$ and $\sigma_{\omega}$. Because of their common control of system timing, we would obtain comparable distributions when varying either $\tau$ or $\kappa_{x,\theta}$ instead of $\sigma_{\omega}$. #### The coupled system preserves synchronization in both oscillators and switches. Figure 1 shows a state of the model in which the switches are all in the “on” state and oscillators are synchronized ($r_{x}$ near 1, $r_{\theta}$ near 1, and $\psi$ oscillating between $[0,2\pi)$ periodically). While such synchronization is observed in the uncoupled Hopfield and Kuramoto models, the oscillator-switch cross coupling extends the region of parameter space over which this synchronization occurs. Specifically, modest values of $\kappa_{x,\theta}$ can induce sustained switch activity for parameter values of $\kappa_{x,x}$ in which switches would decay in the uncoupled system. Furthermore, this switch synchronization will occur for all values of $\kappa_{x,x}$ in which synchronization occurs in the uncoupled Hopfield model (i.e., all $\kappa_{x,x}$ larger than a threshold value $\hat{\kappa}_{x,x}$) because the oscillators only contribute positively to the derivative in eq. (8) in our model. On the other hand, no value of $\kappa_{\theta,x}$ will cause oscillator phases to synchronize if $\kappa_{\theta,\theta}$ is below the critical coupling parameter for the pure Kuramoto model ($\hat{\kappa}_{\theta,\theta}$). However, there are parameter regimes in which this synchronization occurs stochastically, depending on the initial values selected for $x_{i}$, $\theta_{j}$, and $\hat{\omega}_{j}$ (Figure 2). In these cases, the average decrease in oscillator natural frequencies caused by decreasing $\kappa_{\theta,x}$ or $\sigma_{\omega}$ will increase the effective period of oscillators, thereby increasing the probability of switches being locked in the “on” state and oscillator synchronization in the heterogeneous system. #### Coupling switches to unsynchronized oscillators can freeze network-wide dynamics. Figure 1 depicts a model state in which switches are all “off” ($r_{x}$ near zero) and oscillators “freeze”: each $\theta_{j}\left(t\right)=\psi\left(t\right)=\Psi$ for some constant values $\Psi$ for all $t$ beyond the preliminary freezing time $t_{f}$. While the decaying switches are observed in an uncoupled Hopfield model, the freezing oscillators cannot be simulated in the uncoupled Kuramoto model. Such oscillator freezing will occur whenever the oscillators decay to the “off” state by virtue of the coupling of the oscillators to switches through the $\omega_{j}$ in eq. (10). Specifically, this frozen state can occur whenever $\kappa_{x,x}<\hat{\kappa}_{x,x}$ depending on the values of $\tilde{x}_{i}$, $\theta_{j}$, and $\hat{\omega}_{j}$. However, the probability of selecting these initial states is decreased when the heterogeneity of the oscillators increases through incomplete synchronization ($r_{\theta}(t)<1$) or increased $\sigma_{\omega}$ (Figure 2). In these cases, a single oscillator in the “up” phase ($\tilde{\theta}=1$) can contribute positively to the switch states, forcing the system out of this frozen state. The probability of obtaining this frozen model state further depends on the relative timing of switch decay and oscillator freezing. Specifically, the probability of obtaining the frozen state decreases with the average oscillator frequency, determined predominantly by the parameter $\kappa_{\theta,x}$ (Figure 2). #### Coupling switches to synchronized oscillators can induce synchronized oscillations in switches. An additional consequence of coupling switches and oscillators in a state in which switches vacillate between all “on” and all “off” along with the synchronized oscillator frequency (Figure 1). This oscillatory synchronization occurs when the pure Hopfield model would turn switches “off” ($\kappa_{x,x}<\hat{\kappa}_{x,x}$), the pure Kuramoto model would induce oscillator synchronization ($\kappa_{\theta,\theta}>\hat{\kappa}_{\theta,\theta}$), and the timing between the oscillators and switches are balanced such that the average period of the coupled oscillators is slightly less than the average decay time of the system of switches. Figure 2 shows that this balance in switch-oscillator timescales increases with decreasing $\kappa_{\theta,x}$ and depends non- monotonically on $\sigma_{\omega}$. As we see in the plot of $r_{\omega}\left(t\right)$ in Figure 1, the average oscillator natural frequencies will decrease towards the end of the “down” phase in response to switches turning off, and then increase to their full natural values in the “up” phase as switches turn back “on”. Therefore, if synchronized oscillator period is too slow (i.e., $\sigma_{\omega}$ is too large), the system will tend to be locked in the “on” state (Figure 2); if too fast (i.e., $\sigma_{\omega}$ too small) the system will tend to be locked in the “off” state (Figure 2). #### Synchronization of network-wide oscillations may be transitory. Oscillatory behavior in the switches is also observed for unsynchronized oscillators ($\kappa_{\theta,\theta}<\hat{\kappa}_{\theta,\theta}$) as depicted in Figure 1. In this case, the value of $\kappa_{\theta,x}$ must be large enough to enable switches to freeze the oscillators’ phases. However, because the oscillators are uncoupled, a small subset of oscillators in the “up” phase can drive the switches to turn on for large-enough values of $\kappa_{x,\theta}$. These switch oscillations are transitory, ending when at last the switch coupling dominates the system and induces all of the oscillators to freeze. For unsynchronized oscillators in the parameter range of Figure 1, the transitional oscillations in the switch state occurs regularly in 21 of 100 simulations. In 8 of these 21 simulations, the switch state turns “on” after decaying at least twice. More rarely, transitory changes in switch state may be induced by a similar mechanism in simulations for which $\kappa_{\theta,\theta}>\hat{\kappa}_{\theta,\theta}$ and switches ultimately settle on the all “on” or all “off” states. #### System size affects the distribution of qualitative dynamics We also explored the dynamics of the coupled system for networks of sizes ranging from $N=M=10$ to $N=M=500$ nodes, described in the methods. For networks of all sizes, we observe that the dynamics of the system was limited to the four qualitative behaviors observed for networks of size $N=M=100$ depicted in Figure 1. However, the system size does have a notable effect on the frequency with which each of these behaviors occurs. Supplemental Figures S5-S7 plot the observed frequencies for each of the network sizes as a function of the $\kappa_{\theta,x}$ and $\sigma_{\omega}$ values considered in Figure 2. When $\kappa_{\theta,x}=0.01$, the observed frequencies of the system states depend most strongly on network size in simulations using the smallest value of $\sigma_{\omega}=1$ is also small (Supplemental Figure S5). In this case, the probability of observing the system with synchronized oscillatory dynamics in both switches and oscillators decays as the network grows. Both the state in which the switches are on and oscillators are synchronized and the state in which the switches are off and oscillators are frozen have with compensatory increases in probability (Figure 3). The relative probability of obtaining the frozen state increases, with notable decay in the probability of obtaining the state in which switches are “on” and oscillators are synchronized in large networks. On the other hand, when $\kappa_{\theta,x}=1$, the system size has the greatest influence on the resulting dynamics for large values of $\sigma_{\omega}$ (Supplemental Figure S7). In this case, the system changes from containing mostly switches in the on state and synchronized oscillators to switches that are entirely in the “frozen” state for large network sizes (Figure 4). We hypothesize that the system is forced into the frozen state in larger networks because of increased oscillator synchronization in large networks. Therefore, small networks would have a higher probability of having few oscillators that are unsynchronized and in the “up” phase ($\tilde{\theta}=1$), causing the switches to turn “on” ($\tilde{x}=1$) due to the structure of eq. (8) as was discussed previously. Furthermore, the rare oscillations observed in both switches and oscillators when $\kappa_{\theta,x}=1$ occur only when the network is small. Intermediate values of $\kappa_{\theta,x}=0.1$ show similar changes to those described for $\kappa_{\theta,x}=0.01$ when $\sigma_{\omega}=1$ and to those described for $\kappa_{\theta,x}=1$ when $\sigma_{\omega}=10$ (Supplemental Figure S6). ### The heterogeneous network models qualitative dynamics of the yeast cell cycle derived from network motifs. Previous work by [47] make the cell cycle processes controlling mitotic division of fission yeast Schizosaccharomyces pombe cells provides an optimal system in which to apply our model. The biochemical reactions responsible for driving the cell cycle are well understood and the resulting dynamics in each of the stages of the cell cycle have been characterized extensively in [47, 44, 40]. The cell cycle machinery in mitosis is divided into four, sequential stages: phase 1 is a gap or rest phase (G1); phase 2 is a DNA synthesis stage (S); phase 3 is an additional gap stage (G2); and phase 4 is the mitotic division stage (M). Previously, [47] observed that the dynamics of the yeast cell cycle can be divided into three sequentially interacting modules, triggered by a signal based upon cell size: (1) G1/S transitions with a toggle-switch, (2) S/G2 transitions with a toggle-switch, and (3) G2/M transitions with an oscillator. Although the specific timing differs from [47], we observe similar qualitative dynamics to that observed in [47] when applying our heterogeneous model to evolve the state of these cell cycle stages (Figure 5) as described in the Methods section. We note that the response in this system is consistent with the transitory oscillations observed in Figure 1 in the case of all-to-all coupling. We also modeled this cell-cycle system in a rewired-network, in which the G2/M transitions feedback into G1/S (Figure 6). In this case, we observe sustained reactivation of the cell cycle regardless of the external signal. These dynamics are analogous to the synchronized dynamics in Figure 2 and consistent with cell growth arising from re-wiring biochemical reactions in cancer cells [48]. ## Discussion Our model of coupled switches and oscillators in all-to-all networks demonstrates that networks with components having heterogeneous dynamics can exhibit synchronization similar to that observed in homogeneous systems. As is the case in homogeneous models (e.g., [49, 50, 51, 52]), we expect analogous synchronization to hold in small-world, biochemical network topologies (e.g., [53]). However, these alternative topologies would likely change the probability of observing each of the qualitative model behaviors similar to the observed dependence of probabilities in network size. In this alternative network topologies, the qualitative states of the network model may have greater variability in small network sizes in accordance with the findings of [54]. Finally, in these topologies the heterogeneous model could yield additional, complex qualitative dynamic states, resulting from the complex dynamics that they cause in models of coupled switches alone [46]. While uncoupled network motifs may adopt switch-like or oscillatory dynamics, coupling between these components can induce switch-like behavior in oscillators and oscillatory behavior in switches. These qualitative changes in component dynamics occur stochastically, depending on the distribution of frequencies and switch states. They are more likely to occur in simulations with an imbalance in relative timescales, in which the dynamics of the faster network motif will dominate the system. Similarly, when $\kappa_{\theta,x}$ and $\sigma_{\omega}$ are both small, the coordinated oscillations in the switches and oscillators that occur in frequently small networks are largely eliminated in larger networks. We hypothesize that this larger network effectively increases the range of natural frequencies and phases, making the simulation less likely to have the constrained distribution required to obtain such synchronized oscillations. We can expect that biological systems have evolved components according to these distributions to ensure the robustness of the dynamics in the system. For example, multiple proteins can often serve similar functions in cell signaling pathways, which would increase the system size and decrease the probability of transient behaviors in our model. This robustness will be further ensured through the sheer size of most biochemical systems. For example, in humans yeast two-hybrid maps and metabolic network maps both contain on the order of thousands of interactions between thousands of species [53]. Furthermore, we have also observed that the heterogeneous network model will freeze the oscillator dynamics in the presence of inactive switches and then subsequently activate in synchrony in the presence of active switches. As a result, our model provides a natural mechanism for the coordination of complex machinery such as the initiation of cell-cycle dynamics. For example, when we apply our model to the yeast cell cycle motifs in [47], we recapitulate the qualitative dynamics of delayed initiation of stages of the cell cycle observed in simulations with differential equations of the regulatory dynamics in [47]. Additional tuning of the model parameters or incorporation of additional cell cycle checkpoints would facilitate a precise match of the timing of [47]. Because parameters are defined for modules and their interaction, our model requires far fewer rate parameters than any differential equation model of sets of biochemical reactions of the yeast cell cycle. Generally, the oscillator in the final G2/M step of the cell cycle is active only when the series of switches in the previous steps of the cell cycle are activated, consistent with the transient dynamics observed in our network model. However, rewiring the network to introduce feedback from the G2/M stage to the G1/S stage of the cell cycle will cause the modeled cell cycle machinery to engage continually without regard to the external growth signals, consistent with the malignant rewring in cancer cells [48]. Similar to the oscillatory behavior induced in switches in simulations in all-to-all networks, this small modification to the topology of cell cycle interactions altered the resulting dynamics of the network motifs for the G1/S and S/G2 motifs. We, therefore, hypothesize that motif dynamics predicted by the structure of subgraphs may not accurately describe their in vivo dynamics if considered in isolation, consistent with the hypothesis in [55] and findings of [42]. We observed that the switches in the cell cycle block activation of the yeast cell cycle when no external signal is present. Similarly, when part of the larger but sparse networks that compose biochemical systems [53], inactive switches would effectively destroy links between nodes on the network. As a result, the proposed heterogeneous model provides a potential mechanism for Kuramoto-based models with evolving network topologies such as [15, 23, 24, 25, 26, 27, 28]. Similarly, we observed that the intermediate switches delay the oscillations in the final G2/M motif in the simulated yeast cell cycle. As a result, we hypothesize that coupling switches to oscillators through their frequencies in this model also provides a natural mechanism for extensions of the Kuramoto model with dynamic frequencies [15, 29, 30] or phase delays [16, 31, 32, 33]. The heterogeneous network model described in this paper facilitates characterization of the dynamics of complex, biochemical systems by abstracting the dynamics of their composite motifs such as the yeast cell cycle based upon [47]. We note that the proposed heterogeneous network model is deterministic once the initial values of all the switches and oscillator frequencies have been specified. However, many intracellular reactions (e.g., [56]) and neuronal systems (reviewed in [57, 58]) evolve stochastically. In these cases, the Hopfield networks used to model the switches could be replaced with probabilistic Boolean networks [3] and the oscillators evolved with stochastic solvers such as the stochastic simulation algorithm (reviewed in [59]), integrated with the methodology developed in [60]. Similar modifications could also extend the heterogeneous model to incorporate coupling with components of additional dynamics pertinent to biochemical systems, such as those of the network motifs enumerated in [44, 34, 35]. These studies would also ideally consider the dynamics of the heterogeneous network model in additional small-world and random network topologies, as well as the topologies defined by neuronal systems and gene regulatory networks. ## Materials and Methods ### Numerical simulations in the all-to-all network In this study, we analyze the range of possible dynamics of the coupled, heterogeneous networks by applying this model to all-to-all networks. Analyses were performed for networks with equal number of switches and oscillators ($N=M$) of sizes 10, 50, 100, 200, and 500. All simulations are run one hundred times from random initial conditions for the state of switches ($x_{i},i=1,\ldots,N$) and oscillators ($\theta_{j},j=1,\ldots,M$), drawn from a Gaussian distribution and a uniform distribution on $[0,2\pi)$, respectively. Similarly, oscillator natural frequencies are drawn randomly from a Gaussian distribution of mean zero and standard deviation parameter $\sigma_{\omega}$. Simulations of 100 seconds (in the arbitrary units of the model), with a time step of 0.01 seconds were found sufficient to reflect the range of possible model behaviors and verify consistency across initial conditions. The behavior of each simulation is summarized based on the time- dependent order parameters $r_{\theta}\left(t\right)$ and $\psi\left(t\right)$, $r_{\omega}\left(t\right)$, and $r_{x}\left(t\right)$. ### Numerical simulations of the yeast cell cycle Based upon [47], we model the yeast cell cycle as an initiating external signal (namely the cell size), coupled to a toggle switch representing the transition between G1/S, a toggle switch representing the transition between S/G2, and an oscillator representing the transition from G2/M. While the external signal is incorporated into the model with coupling to the other switches in eq. (6), its state is not updated by the model. The duration of this external signal is set at 10 simulated minutes, based upon [47]. Similarly, the initial values of the hidden variable $x$ for the switches in the G1/S and S/G2 modules are set at -0.5, $\tau$ to 1, and $\kappa_{x,x}$ to 2 to reproduce the approximate 10 minute duration of these switches in [47]. The natural frequency is for the G2/M module set to $\frac{2\pi}{60}\mbox{min}^{-1}$ to likewise reflect the timescale reported in [47], while the remainder of the coupling parameters are left untuned, set to $\kappa_{\theta,x}=\kappa_{x,\theta}=\kappa_{\theta,\theta}=2$ because we sought only to reproduce the qualitative dynamics of the [47] model. The rewiring in the system with enduring cell cycle activation is achieved by adding an edge from the module for G2/M to the switch in the G1/S module. ## Acknowledgments This work was sponsored by NCI (CA141053). We thank LV Danilova, B Fertig, AV Favorov, BR Hunt, MJ Stern, MF Ochs, E Ott, E Webster, and LM Weiner for advice. Code available upon request. ## References * 1. Glass L, Kauffman S (1973) The logical analysis of continuous, non-linear biochemical control networks. Journal of Theoretical Biology 39: 103-129. * 2. Hopfield JJ (1982) Neural networks and physical systems with emergent collective computational abilities. Proc Natl Acad Sci U S A 79: 2554-8. * 3. Shmulevich I, Dougherty ER, Zhang W (2002) From boolean to probabilistic boolean networks as models of genetic regulatory networks. Proceedings of the IEEE 90: 1778-1792. * 4. Kuramoto Y (1975) International Symposium on Mathematical Problems in Theoretical Physics, volume 39 of _Lecture Notes in Physics_. Springer, 420 pp. * 5. Strogatz SH (2003) Sync: the emerging science of spontaneous order. New York: Hyperion, 1st ed edition. * 6. Garcia-Ojalvo J, Elowitz MB, Strogatz SH (2004) Modeling a synthetic multicellular clock: repressilators coupled by quorum sensing. Proc Natl Acad Sci U S A 101: 10955-60. * 7. Danino T, Mondragón-Palomino O, Tsimring L, Hasty J (2010) A synchronized quorum of genetic clocks. Nature 463: 326-30. * 8. Tu BP, Kudlicki A, Rowicka M, McKnight SL (2005) Logic of the yeast metabolic cycle: temporal compartmentalization of cellular processes. Science 310: 1152-8. * 9. Klevecz RR, Li CM, Marcus I, Frankel PH (2008) Collective behavior in gene regulation: the cell is an oscillator, the cell cycle a developmental process. FEBS J 275: 2372-84. * 10. Huang S, Eichler G, Bar-Yam Y, Ingber D (2005) Cell fates as high-dimensional attractor states of a complex gene regulatory network. Phys Rev Lett 94: 128701. * 11. Peskin C (1975) Mathematical aspects of heart physiology. Courant institute lecture notes, Courant Institute of Mathematical Sciences. * 12. Breakspear M, Heitmann S, Daffertshofer A (2010) Generative models of cortical oscillations: neurobiological implications of the kuramoto model. Front Hum Neurosci 4: 190. * 13. Strogatz SH (1987) Human sleep and circadian rhythms: a simple model based on two coupled oscillators. J Math Biol 25: 327-47. * 14. Fox JJ, Jayaprakash C, Wang D, Campbell SR (2001) Synchronization in relaxation oscillator networks with conduction delays. Neural Comput 13: 1003-21. * 15. Cumin D, Unsworth C (2007) Generalizing the Kuramoto model for the study of neuronal synchronization in the brain. Physica D 226: 181-196. * 16. Wang Q, Duan Z, Perc M, Chen G (2008) Synchronization transitions on small-world neuronal networks: Effects of information transmission delay and rewiring probability. EPL 83: 50008. * 17. Pérez T, Garcia GC, Eguíluz VM, Vicente R, Pipa G, et al. (2011) Effect of the topology and delayed interactions in neuronal networks synchronization. PLoS One 6: e19900. * 18. Kunysz A, Shrier A, Glass L (1997) The challenge of complex dynamics for theoretical models of cardiac activity. Journal of Theoretical Medicine 1: 79–90. * 19. Chang HS, Staras K, Gilbey MP (2000) Multiple oscillators provide metastability in rhythm generation. J Neurosci 20: 5135-43. * 20. Sato D, Xie LH, Sovari AA, Tran DX, Morita N, et al. (2009) Synchronization of chaotic early afterdepolarizations in the genesis of cardiac arrhythmias. Proc Natl Acad Sci U S A 106: 2983-8. * 21. Zahanich I, Sirenko SG, Maltseva LA, Tarasova YS, Spurgeon HA, et al. (2011) Rhythmic beating of stem cell-derived cardiac cells requires dynamic coupling of electrophysiology and ca cycling. J Mol Cell Cardiol 50: 66-76. * 22. Tiana-Alsina J, Garcia-Lopez JH, Torrent M, Garcia-Ojalvo J (2011) Dual-lag synchronization between coupled chaotic lasers due to path-delay interference. Chaos 21: 043102. * 23. Mondal A, Sinha S, Kurths J (2008) Rapidly switched random links enhance spatiotemporal regularity. Phys Rev E Stat Nonlin Soft Matter Phys 78: 066209. * 24. Sorrentino F, Ott E (2008) Adaptive synchronization of dynamics on evolving complex networks. Phys Rev Lett 100: 114101. * 25. Jiang M, Ma P (2009) Coherence resonance induced by rewiring in complex networks. Chaos 19: 013115. * 26. Volman V, Perc M (2010) Fast random rewiring and strong connectivity impair fast random rewiring and strong connectivity impair subthreshold signal detection in excitable networks. New Journal of Physics 12: 043013. * 27. Adhikari BM, Prasad A, Dhamala M (2011) Time-delay-induced phase-transition to synchrony in coupled bursting neurons. Chaos 21: 023116. * 28. Gómez-Gardeñes J, Gómez S, Arenas A, Moreno Y (2011) Explosive synchronization transitions in scale-free networks. Phys Rev Lett 106: 128701. * 29. Martens E, Barreto E, Strogatz S, Ott E, So P, et al. (2009) Exact results for the kuramoto model with a bimodal frequency distribution. Phys Rev E 79: 026204. * 30. Taylor D, Ott E, Restrepo JG (2010) Spontaneous synchronization of coupled oscillator systems with frequency adaptation. Phys Rev E Stat Nonlin Soft Matter Phys 81: 046214. * 31. Ermentrout B, Ko TW (2009) Delays and weakly coupled neuronal oscillators. Philos Transact A Math Phys Eng Sci 367: 1097-115. * 32. Wang Q, Perc M, Duan Z, Chen G (2009) Synchronization transitions on scale-free neuronal networks due to finite information transmission delays. Phys Rev E Stat Nonlin Soft Matter Phys 80: 026206. * 33. Wang Q, Chen G, Perc M (2011) Synchronous bursts on scale-free neuronal networks with attractive and repulsive coupling. PLoS One 6: e15851. * 34. Alon U (2007) Network motifs: theory and experimental approaches. Nat Rev Genet 8: 450–461. * 35. Novak B, Tyson JJ (2008) Design principles of biochemical oscillators. Nat Rev Mol Cell Biol 9: 981–991. * 36. Milo R, Shen-Orr S, Itzkovitz S, Kashtan N, Chklovskii D, et al. (2002) Network motifs: simple building blocks of complex networks. Science 298: 824-7. * 37. Shen-Orr SS, Milo R, Mangan S, Alon U (2002) Network motifs in the transcriptional regulation network of escherichia coli. Nat Genet 31: 64-8. * 38. Rao CV, Arkin AP (2001) Control motifs for intracellular regulatory networks. Annu Rev Biomed Eng 3: 391-419. * 39. Prill RJ, Iglesias PA, Levchenko A (2005) Dynamic properties of network motifs contribute to biological network organization. PLoS Biol 3: e343. * 40. Csikász-Nagy A, Novák B, Tyson JJ (2008) Reverse engineering models of cell cycle regulation. Adv Exp Med Biol 641: 88-97. * 41. Purcell O, Savery NJ, Grierson CS, di Bernardo M (2010) A comparative analysis of synthetic genetic oscillators. J R Soc Interface 7: 1503-24. * 42. Taylor D, Restrepo JG (2011) Network connectivity during mergers and growth: optimizing the addition of a module. Phys Rev E Stat Nonlin Soft Matter Phys 83: 066112. * 43. Kim JR, Yoon Y, Cho KH (2008) Coupled feedback loops form dynamic motifs of cellular networks. Biophys J 94: 359-65. * 44. Tyson JJ, Chen KC, Novak B (2003) Sniffers, buzzers, toggles and blinkers: dynamics of regulatory and signaling pathways in the cell. Curr Opin Cell Biol 15: 221–231. * 45. Strogatz SH (2000) From Kuramoto to Crawford: exploring the onset of synchronization in populations of coupled oscillators. Physica D 143: 1–20. * 46. Edwards R (2000) Analysis of continuous-time switching networks. Physica D 146: 165–199. * 47. Tyson JJ, Chen K, Novak B (2001) Network dynamics and cell physiology. Nat Rev Mol Cell Biol 2: 908. * 48. Hanahan D, Weinberg R (2000) The hallmarks of cancer. Cell 100: 57–70. * 49. Restrepo JG, Ott E, Hunt B (2005) Onset of synchronization in large networks of coupled oscillators. Phys Rev E 71: 036151. * 50. Restrepo JG, Ott E, Hunt B (2005) Synchronization in large directed networks of coupled phase oscillators. Chaos 16: 015107. * 51. Restrepo JG, Ott E, Hunt B (2006) Emergence of coherence in complex networks of heterogeneous dynamical systems. Phys Rev Lett 96: 254103. * 52. Restrepo JG, Ott E, Hunt B (2006) Emergence of synchronization in complex networks of interacting dynamical systems. Physica D 224: 114-122. * 53. Barabási AL, Oltvai ZN (2004) Network biology: understanding the cell’s functional organization. Nat Rev Genet 5: 101-13. * 54. Pikovsky A, Zaikin A, de la Casa MA (2002) System size resonance in coupled noisy systems and in the ising model. Phys Rev Lett 88: 050601. * 55. Dorogovtsev S (2010) Lectures on Complex Networks. Oxford University Press. * 56. Kaern M, Elston TC, Blake WJ, Collins JJ (2005) Stochasticity in gene expression: from theories to phenotypes. Nat Rev Genet 6: 451-464. * 57. Stein RB, Gossen ER, Jones KE (2005) Neuronal variability: noise or part of the signal? Nat Rev Neurosci 6: 389-97. * 58. Faisal AA, Selen LPJ, Wolpert DM (2008) Noise in the nervous system. Nat Rev Neurosci 9: 292-303. * 59. Gillespie DT (2007) Stochastic simulation of chemical kinetics. Annu Rev Phys Chem 58: 35-55. * 60. Fertig EJ, Danilova LV, Favorov AV, Ochs MF (2011) Hybrid modeling of cell signaling and transcriptional reprogramming and its application in c. elegans development. Frontiers in Bioinformatics and Computational Biology 2: 77. ## Figure Legends ###### List of Figures 1. 1 Summary of the qualitative dynamics of the heterogeneous network model of eqs. (8) - (10). In all figures, top-panel shows temporal evolution of the mean field statistics ($r_{\theta}$ black, solid; $r_{x}$ green, dashed; and $r_{\omega}$ blue, dash-dotted) and the bottom-panel shows the evolution of the mean phase $\psi$ (red, solid). (a) Oscillators synchronize and switches stay “on” ($\kappa_{x,x}=11$, $\kappa_{x,\theta}=1.5$, $\kappa_{\theta,x}=1$, $\kappa_{\theta,\theta}=40$, and $\sigma_{\omega}=10$), (b) oscillators freeze (as evidenced by unchanging $\psi$) and switches stay “off” ($\kappa_{x,x}=1$, $\kappa_{x,\theta}=1.5$, $\kappa_{\theta,x}=1$, $\kappa_{\theta,\theta}=40$, and $\sigma_{\omega}=10$), (c) oscillators synchronize and switches oscillate ($\kappa_{x,x}=1$, $\kappa_{x,\theta}=160$, $\kappa_{\theta,x}=0.2$, $\kappa_{\theta,\theta}=42$, and $\sigma_{\omega}=3$), and (d) transitory oscillations in oscillators and switches ($\kappa_{x,x}=0.1$, $\kappa_{x,\theta}=1.4$, $\kappa_{\theta,x}=2$, $\kappa_{\theta,\theta}=1.8$, and $\sigma_{\omega}=10$). 2. 2 Percentage of simulations in which the qualitative dynamics in Figure 1 occur. In (a) oscillators synchronize and switches are “on”, in (b) oscillators freeze and switches are “off”, and in (c) switches vary with oscillators vs $\sigma_{\omega}$ for $\kappa_{\theta,x}=0.01$ (solid), $\kappa_{\theta,x}=0.1$ (dotted) and $\kappa_{\theta,x}=1$ (dashed). $\kappa_{x,x}=1<\hat{\kappa}_{x,x}$, $\kappa_{x,\theta}=1.5$, and $\kappa_{\theta,\theta}=40>\hat{\kappa}_{\theta,\theta}$. 3. 3 Dependence on network size for qualitative states for $\kappa_{\theta,x}=0.01$ and $\sigma_{\omega}=1$. Percentage of simulations in which the qualitative dynamics have switches off and oscillators frozen (blue, solid), switches on and oscillators synchronized (green, dashed), and oscillatory switches and synchronized oscillators (red, dotted). 4. 4 Dependence on network size for qualitative states for $\kappa_{\theta,x}=1$ and $\sigma_{\omega}=10$. Percentage of simulations with qualitative dynamics plotted as described in Figure 3. 5. 5 Simulated dynamics of the yeast cell cycle Evolution of the states of the cell cycle modules (G1/S top, green dashed; S/G2 top, red dotted; G2/M bottom, black) in response to an external stimulus to initiate the cell cycle (top, blue solid) 6. 6 Simulated dynamics of an aberrant cell cycle network. As for Figure 5 with a network topology linking the G2/M module to the G1/S module. Figure 1: Summary of the qualitative dynamics of the heterogeneous network model of eqs. (8) - (10). In all figures, top-panel shows temporal evolution of the mean field statistics ($r_{\theta}$ black, solid; $r_{x}$ green, dashed; and $r_{\omega}$ blue, dash-dotted) and the bottom-panel shows the evolution of the mean phase $\psi$ (red, solid). (a) Oscillators synchronize and switches stay “on” ($\kappa_{x,x}=11$, $\kappa_{x,\theta}=1.5$, $\kappa_{\theta,x}=1$, $\kappa_{\theta,\theta}=40$, and $\sigma_{\omega}=10$), (b) oscillators freeze (as evidenced by unchanging $\psi$) and switches stay “off” ($\kappa_{x,x}=1$, $\kappa_{x,\theta}=1.5$, $\kappa_{\theta,x}=1$, $\kappa_{\theta,\theta}=40$, and $\sigma_{\omega}=10$), (c) oscillators synchronize and switches oscillate ($\kappa_{x,x}=1$, $\kappa_{x,\theta}=160$, $\kappa_{\theta,x}=0.2$, $\kappa_{\theta,\theta}=42$, and $\sigma_{\omega}=3$), and (d) transitory oscillations in oscillators and switches ($\kappa_{x,x}=0.1$, $\kappa_{x,\theta}=1.4$, $\kappa_{\theta,x}=2$, $\kappa_{\theta,\theta}=1.8$, and $\sigma_{\omega}=10$). Figure 2: Percentage of simulations in which the qualitative dynamics in Figure 1 occur. In (a) oscillators synchronize and switches are “on”, in (b) oscillators freeze and switches are “off”, and in (c) switches vary with oscillators vs $\sigma_{\omega}$ for $\kappa_{\theta,x}=0.01$ (solid), $\kappa_{\theta,x}=0.1$ (dotted) and $\kappa_{\theta,x}=1$ (dashed). $\kappa_{x,x}=1<\hat{\kappa}_{x,x}$, $\kappa_{x,\theta}=1.5$, and $\kappa_{\theta,\theta}=40>\hat{\kappa}_{\theta,\theta}$. Figure 3: Dependence on network size for qualitative states for $\kappa_{\theta,x}=0.01$ and $\sigma_{\omega}=1$. Percentage of simulations in which the qualitative dynamics have switches off and oscillators frozen (blue, solid), switches on and oscillators synchronized (green, dashed), and oscillatory switches and synchronized oscillators (red, dotted). Figure 4: Dependence on network size for qualitative states for $\kappa_{\theta,x}=1$ and $\sigma_{\omega}=10$. Percentage of simulations with qualitative dynamics plotted as described in Figure 3. Figure 5: Simulated dynamics of the yeast cell cycle Evolution of the states of the cell cycle modules (G1/S top, green dashed; S/G2 top, red dotted; G2/M bottom, black) in response to an external stimulus to initiate the cell cycle (top, blue solid) Figure 6: Simulated dynamics of an aberrant cell cycle network. As for Figure 5 with a network topology linking the G2/M module to the G1/S module. ### Supplemental Figure Captions Figure S5. Dependence of dynamics on network size for $\kappa_{\theta,x}=0.01$. Number of simulations (of 100) for which the switches are off and oscillators are frozen (left panel), the switches are on and the oscillators are synchronized (center panel), and both the oscillators and switches have synchronized oscillations (right). Figure S6. Dependence of dynamics on network size for $\kappa_{\theta,x}=0.1$. As for Supplemental Figure S5. Figure S7. Dependence of dynamics on network size for $\kappa_{\theta,x}=1$. As for Supplemental Figure S5.	arxiv-papers	2011-11-30T17:33:02	2024-09-04T02:49:24.837198	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Matthew R. Francis and Elana J. Fertig", "submitter": "Elana Fertig", "url": "https://arxiv.org/abs/1111.7243" }
1111.7302	# Self-Similar Solutions for Viscous and Resistive ADAF Kazem Faghei School of Physics e-mail: kfaghei@du.ac.ir Damghan University Damghan Iran (Received 2011 January; accepted 2011 April) ###### Abstract In this paper, the self-similar solution of resistive advection dominated accretion flows (ADAF) in the presence of a pure azimuthal magnetic field is investigated. The mechanism of energy dissipation is assumed to be the viscosity and the magnetic diffusivity due to turbulence in the accretion flow. It is assumed that the magnetic diffusivity and the kinematic viscosity are not constant and vary by position and $\alpha$-prescription is used for them. In order to solve the integrated equations that govern the behavior of the accretion flow, a self-similar method is used. The solutions show that the structure of accretion flow depends on the magnetic field and the magnetic diffusivity. As, the radial infall velocity and the temperature of the flow increase, and the rotational velocity decreases. Also, the rotational velocity for all selected values of magnetic diffusivity and magnetic field is sub- Keplerian. The solutions show that there is a certain amount of magnetic field that the rotational velocity of the flow becomes zero. This amount of the magnetic field depends on the gas properties of the disc, such as adiabatic index and viscosity, magnetic diffusivity, and advection parameters. The solutions show the mass accretion rate increases by adding the magnetic diffusivity and in high magnetic pressure case, the ratio of the mass accretion rate to the Bondi accretion rate decreases as magnetic field increases. Also, the study of Lundquist and magnetic Reynolds numbers based on resistivity indicates that the linear growth of magnetorotational instability (MRI) of the flow decreases by resistivity. This property is qualitatively consistent with resistive magnetohydrodynamics (MHD) simulations. ###### keywords: accretion, accretion disks, magnetohydrodynamics: MHD ## 1 Introduction The standard geometrically thin, optically thick accretion disc model can successfully explain most of the observational features in active galactic nuclei (AGN) and X-ray binaries (Shakura & Sunyaev 1973). In the standard thin model, the motion of the matter in the accretion disc is nearly Keplerian, and the gravitational energy released in the disc is radiated away locally. During recent years, another type of accretion flow has been studied in which it is assumed that energy released through dissipation processes in the disc may be trapped within accreting gas, and only the small fraction of the energy released in the accretion flow is radiated away due to inefficient cooling, and most of the energy is stored in the accretion flow and advected to the central object. This kind of accretion flow is known as advection-dominated accretion flow (ADAF). The basis ideas of ADAF models have been developed by a number of researchers (e.g. Ichimaru 1977; Rees et al. 1982; Narayan & Yi 1994, 1995; Abramowicz et al. 1995; Ogilvie 1999; Blandford & Begelman 1999). It seems that accretion discs, whether in star-forming regions, in X-ray binaries, in cataclysmic variables, or in the centers of active galactic nuclei, are likely to be threaded by magnetic fields. Consequently, the role of the magnetic fields on ADAFs has been analyzed in detail by a number of investigators (Bisnovatyi-kogan & Lovelace 2001; Akizuki & Fukue 2006, hereafter AF06; Shadmehri 2004, hereafter Sh04; Ghanbari et al., 2007; Bu, Yung, & Xie 2009; Khesali & Faghei 2009, hereafter KF09). The existence of the toroidal magnetic field has been proven in the outer regions of the discs of young stellar objects (YSOs; Aitken et al. 1993; Wright et al. 1993) and in the Galactic center (Novak et al. 2003; Chuss et al. 2003). Thus, considering the accretion discs with a toroidal magnetic field have been studied by several authors (AF06; Begelman & Pringle 2007; Khesali & Faghei 2008, hereafter KF08; Bu, Yung, & Xie 2009; KF09). The resistive diffusion of magnetic field is important in some systems, such as the protostellar discs (Stone et al. 2000; Fleming & Stone 2003), discs in dwarf nova systems (Gammie & Menou 1998), the discs around black holes (Kudoh & Kaburaki 1996), and Galactic center (Melia & Kowalenkov 2001; Kaburaki et al. 2010). Also, two and three dimensional simulations of local shearing box have shown that resistive dissipation is one of the crucial processes that determines the saturation amplitude of the magnetorotational instability (MRI). As, linear growth rate of MRI can be reduced significantly because of the suppression by ohmic dissipation (Fleming et al. 2000; Masada & Sano 2008). AF06 proposed a self-similar advection-dominated accretion flow that the disc plasma is highly ionized, so they assumed that resistivity of the plasma is zero, and only viscosity is due to turbulence and dissipation in the disc. However, recent works represent importance of magnetic diffusivity in accretion discs (e.g. Kuwabara et al. 2000; Kaburaki 2000 ;Kaburaki et al. 2002; Sh04; Ghanbari et al. 2007; Krasnopolsky et al. 2010; Kaburaki et al. 2010). Sh04 studied a quasi-spherical accretion flow that dominant mechanism of energy dissipation was assumed to be the magnetic diffusivity due to turbulence in the accretion flow and the viscosity of the fluid was completely neglected. The main focus of Sh04 was nonrotating accretion flow and ignored from toroidal magnetic field. Also, Sh04 studied induction equation of magnetic field in a steady state that is not according to anti-dynamo theorem (e.g. Cowling 1981) and is usefull only in particular systems where the magnetic dissipation time is very long, much longer than the age of the system. Ghanbari, Salehi, & Abbassi (2007) considered an axisymmetric, rotating, isothermal steady accretion flow, which contains a poloidal magnetic field of the central star and from toroidal magnetic field of the flow neglected. They assumed that the mechanisms of energy dissipation are the turbulence viscosity and magnetic diffusivity due to the magnetic field of the central star. They explored the effect of viscosity on a rotating disc in the presence of constant magnetic diffusivity. Similar to Sh04 they considered the flow in balance between escape and creation of the magnetic field, and ignored from toroidal component of magnetic field. As mentioned the observational evidences and the MHD simulations express that the toroidal component of magnetic field and the magnetic diffusivity are important in accretion discs. Thus in this paper by using AF06 technique we will study the influence of presence of toroidal component of magnetic field in a viscous and resistive accreting gas, and investigate the role of non- constant magnetic diffusivity in systems that escaping and creating of magnetic field are not balanced. We will show that the present model from some aspects is qualitatively consistent with the observational evidences and the resistive MHD simulation results. The paper is organized as follows. In section 2, the basic equations of constructing a model for quasi-spherical magnetized advection dominated accretion flow will be defined. In section 3, self-similar method for solving equations which govern the behavior of the accreting gas was utilized. The summary of the model will appear in section 4. ## 2 Basic Equations We use spherical coordinate $(r,\theta,\varphi)$ centered on the accreting object and make the following standard assumptions: * (i) The flow is assumed to be steady and axisymmetric $\partial_{t}=\partial_{\varphi}=0$, so all flow variables are a function of only $r$ ; * (ii) The magnetic field has only an azimuthal component; * (iii) The gravitational force on a fluid element is characterized by the Newtonian potential of a point mass, $\Psi=-{GM_{}}/{r}$, with $G$ representing the gravitational constant and $M_{}$ standing for the mass of the central star; * (iv) The equations written in spherical coordinates are considered in the equatorial plane $\theta=\pi/2$ and terms with any $\theta$ and $\varphi$ dependence are neglected (Ogilvie 1999; KF09). * (v) For the sake of simplicity, the self-gravity and general relativistic effects have been neglected; The behavior of such system can be analyzed by magnetohydrodynamics (MHD) equations. The general MHD equations are written as follows: $\frac{\partial\rho}{\partial t}+\nabla\cdot(\rho{\bf v})=0,$ (1) $\rho\left[\frac{\partial{\bf v}}{\partial t}+({\bf v}\cdot\nabla){\bf v}\right]=-\nabla p_{gas}-\rho\nabla\Psi+\frac{1}{4\pi}{\bf J}\times\bf{B}+{\bf{F}}_{vis},$ (2) ${\bf J}=\nabla\times{\bf B}$ (3) $\frac{\partial{\bf B}}{\partial t}=\nabla\times\left({\bf v}\times{\bf B}-\eta{\bf J}\right),$ (4) $\rho\left[\frac{1}{\gamma-1}\frac{d}{dt}\left(\frac{p_{gas}}{\rho}\right)+\left(\frac{p_{gas}}{\rho}\right)(\nabla\cdot{\bf v})\right]=Q_{\rm diss}-Q_{\rm cool}\equiv fQ_{\rm diss},$ (5) $\nabla\cdot{\bf B}=0,$ (6) where $\rho$, $\mathbf{v}$, and $p_{gas}$ are the density, the velocity field, and the gas pressure, respectively; $\mathbf{F}_{vis}$ is the viscous force per unit volume; $\mathbf{B}$ is the magnetic field; ${\bf J}$ is the current density and $\eta$ represents the magnetic diffusivity; $\gamma$ is the adiabatic index; The term on the right hand side of the energy equation, $Q_{\rm diss}$, is the rate of heating of the gas by the dissipation and $Q_{\rm cool}$ represents the energy loss through radiative cooling. The advection factor, $f$ ($0\leq f\leq 1$), describes the fraction of the dissipation energy which is stored in the accretion flow and advected into the central object rather than radiated away. The advection factor of $f$ in general depends on the details of the heating and cooling mechanism and will vary with postion (e.g. Watari 2006, 2007; Sinha et al. 2009). However, we assume a constant $f$ for simplicity. Clearly, the case $f=1$ corresponds to the extreme limit of no radiative cooling and in the limit of efficient cooling, we have $f=0$. Under the assumptions (i)-(v), the approximation of quasi-spherical symmetry, the equations (1)-(6) become $\frac{1}{r^{2}}\frac{d}{dr}(r^{2}\rho v_{r})=\dot{\rho},$ (7) $v_{r}\frac{dv_{r}}{dr}+\frac{1}{\rho}\frac{dp}{dr}+\frac{GM_{}}{r^{2}}=r\Omega^{2}-\frac{B_{\varphi}}{4\pi r\rho}\frac{d}{dr}(rB_{\varphi}),$ (8) $\rho v_{r}\frac{d}{dr}(r^{2}\Omega)=\frac{1}{r^{2}}\frac{d}{dr}\left[\nu\rho r^{4}\frac{d\Omega}{\partial r}\right],$ (9) $\frac{1}{\gamma-1}\left[v_{r}\frac{dp}{dr}+\frac{\gamma p}{r^{2}}\frac{d}{dr}\left(r^{2}v_{r}\right)\right]=fQ_{diss},$ $\frac{1}{r}\frac{d}{dr}\left[rv_{r}B_{\varphi}-\eta\frac{d}{dr}(rB_{\varphi})\right]=\dot{B}_{\varphi}.$ (10) Here $v_{r}$ the radial velocity, $\dot{\rho}$ the mass-loss rate per unit volume, $\Omega$ the angular velocity, $B_{\varphi}$ the toroidal component of magnetic field, $\dot{B}_{\varphi}$ is the field escaping/creating rate due to a magnetic instability or dynamo effect, $\nu$ the kinematic viscosity coefficient. We assume both of the kinematic coefficient of viscosity and the magnetic diffusivity due to turbulence in the accretion flow, it is reasonable to use these parameters in analogy to the $\alpha$-prescription of Shakura & Sunyaev (1973) for the turbulent, $\nu=\alpha\frac{p_{gas}}{\rho\Omega_{K}}(1+\beta)^{1-\mu},$ (11) $\eta=\eta_{0}\frac{p_{gas}}{\rho\Omega_{K}}(1+\beta)^{1-\mu}$ (12) where $\Omega_{K}=({GM_{}}/{r^{3}})^{1/2}$ is the Keplerian angular velocity. Narayan & Yi (1995) used a similar form for kinematic coefficient of viscosity, i.e. $\nu=\alpha(p_{gas}/\rho\Omega_{K})$, also Sh04 applied a similar form for magnetic diffusivity, i.e. $\eta=\eta_{0}(p_{gas}/\rho\Omega_{K})$. Thus in comparison to Narayan & Yi (1995) and Sh04 prescriptions, we are using the above equations for the kinematic coefficient of viscosity $\nu$ and the magnetic diffusivity $\eta$. The parameters of $\alpha$ and $\eta_{0}$ are assumed to be positive constants, and we assume that $\alpha$, $\eta_{0}\leq 1$ (Campbell 1999; Kuwabara et al. 2000; Sh04; King et al. 2007). The parameter of $\beta[=p_{mag}/p_{gas}]$ is the degree of magnetic pressure, $p_{mag}=B^{2}_{\varphi}/8\pi$, to the gas pressure. Since, we will apply steady self-similar method for solving system equations, this parameter will be constant throughout the disc, but really this parameter varies by position (see, KF08, KF09). The parameter of $\mu$ is a constant and states importance degree of total pressure in the kinematic viscosity and the magnetic diffusivity. Clearly, the case of $\mu=1$ corresponds to traditional $\alpha$-model and in case of $\mu=0$ total pressure is used. Note the kinematic coefficient of viscosity and the magnetic diffusivity are not costant and depend on the physical quantities of the flow. we will show the quantities of $\nu$ and $\eta$ in our self-similar solution vary with radius $r^{1/2}$. The ratio of the kinematic coefficient of viscosity to the magnetic diffusivity is defined by the magnetic Prandtl number, $P_{m}=\nu/\eta$. By using equations (12) and (13) the magnetic Prandtl number in present model is $P_{m}=\alpha/\eta_{0}$. We will consider conditions that $P_{m}=\infty$ (in case that magnetic diffusivity is zero), $P_{m}\geq 1$, and $P_{m}<1$. For the heating term in equation (10), $Q_{diss}$, we use two sources of dissipation: the viscous and resistive dissipations. Thus, for $Q_{diss}$ we can write $Q_{diss}=\nu\rho r^{2}\left(\frac{d\Omega}{dr}\right)^{2}+\frac{\eta}{4\pi}J^{2}$ (13) where first term is related to viscous dissipation and second term is related to resistive dissipation. By converting the gas pressure and the magnetic pressure in terms of sound speed ($c^{2}_{s}=p_{gas}/\rho$) and Alfvén speed ($c^{2}_{A}=B^{2}_{\varphi}/4\pi\rho$), and by using equations (12)-(14) for the equations (7)-(10), we can write $\frac{1}{r^{2}}\frac{d}{dr}(r^{2}\rho v_{r})=\dot{\rho},$ (14) $v_{r}\frac{dv_{r}}{dr}+\frac{1}{\rho}\frac{d}{dr}(\rho c^{2}_{s})+\frac{GM_{}}{r^{2}}=r\Omega^{2}-\frac{c^{2}_{A}}{r}-\frac{1}{2\rho}\frac{d}{dr}(\rho c^{2}_{A}),$ (15) $\rho v_{r}\frac{d}{dr}(r^{2}\Omega)=\frac{\alpha}{r^{2}}\frac{d}{dr}\left[\frac{\rho c^{2}_{s}}{\Omega_{K}}(1+\beta)^{1-\mu}r^{4}\left(\frac{d\Omega}{dr}\right)\right],$ (16) $\frac{1}{\gamma-1}\left[v_{r}\frac{d}{dr}(\rho c^{2}_{s})+\frac{\gamma\rho c^{2}_{s}}{r^{2}}\frac{d}{dr}\left(r^{2}v_{r}\right)\right]=fQ_{diss},$ $\frac{1}{r}\frac{d}{dr}\left[\sqrt{4\pi\rho c^{2}_{A}}\left(rv_{r}-\frac{\eta_{0}(1+\beta)^{1-\mu}}{4\rho\beta\Omega_{K}}\frac{d}{dr}(r^{2}\rho c^{2}_{A})\right)\right]=\dot{B}_{\varphi},$ (17) where $Q_{diss}=\frac{(1+\beta)^{1-\mu}}{\Omega_{K}}\left[\alpha r^{2}\rho c^{2}_{s}\left(\frac{d\Omega}{dr}\right)^{2}+\frac{\eta_{0}}{8r^{4}\rho\beta}\left(\frac{d}{dr}(r^{2}\rho c^{2}_{A})\right)^{2}\right].$ (18) ## 3 Self-Similar Solutions We seek self-similar solutions in the following forms: $v_{r}=-c_{1}\alpha\sqrt{\frac{GM_{}}{r}}$ (19) $\Omega=c_{2}\sqrt{\frac{GM_{}}{r^{3}}}$ (20) $c^{2}_{s}=c_{3}\frac{GM_{}}{r}$ (21) $c^{2}_{A}=\frac{B^{2}_{\varphi}}{4\pi\rho}=2\beta c_{3}\frac{GM_{}}{r}$ (22) where coefficients $c_{1}$, $c_{2}$, and $c_{3}$ are determined later. we assume a power-law relation for density $\rho(r)=\rho_{0}r^{s}$ (23) where $\rho_{0}$ and $s$ are constant. By using above self-similar quantities, the mass-loss rate and the field escaping/creating rate must have the following form: $\dot{\rho}=\dot{\rho}_{0}r^{s-3/2}$ (24) $\dot{B}_{\varphi}=\dot{B}_{0}r^{\frac{s-4}{2}}$ (25) where $\dot{\rho}_{0}$ and $\dot{B}_{0}$ are constant. Substituting the above solutions in the continuity, momentum, angular momentum, energy, and induction equations [(15)-(20)], we can obtain the following relations: $\dot{\rho}_{0}=-\left(s+\frac{3}{2}\right)\alpha\rho_{0}c_{1}\sqrt{GM_{}},$ (26) $-\frac{1}{2}c^{2}_{1}\alpha^{2}+1-c^{2}_{2}+c_{3}\left(s-1+\beta(1+s)\right)=0,$ (27) $c_{1}=3(s+2)(1+\beta)^{1-\mu}c_{3},$ (28) $-\frac{1}{\gamma-1}\alpha c_{1}(2s-2+3\gamma)=\frac{1}{2}f(1+\beta)^{1-\mu}\left(9\alpha c^{2}_{2}+2\eta_{0}\beta c_{3}(1+s)^{2}\right),$ (29) $\dot{B}_{0}=-\frac{s}{2}GM_{}\sqrt{2\pi\rho_{0}\beta c_{3}}\left(2c_{1}\alpha+\eta_{0}c_{3}(1+s)(1+\beta)^{1-\mu}\right).$ (30) Above equations express for $s=-3/2$, there is no mass loss, while for $s>-3/2$ mass loss (wind) exists. The escape and creation of magnetic fields are balanced in $s=-2+(3\eta_{0})/(6\alpha+\eta_{0})$ that $\dot{B}_{0}$ becomes zero, solving it in $s=-3/2$ (no wind) implies that $\eta_{0}=(6/5)\alpha$. This amount of $\eta_{0}$ corresponds to the magnetic Prandtl number $5/6$. Thus, when the flow has $s=-3/2$ and $\eta_{0}=(6/5)\alpha$, we expect the escape and creation of magnetic field are balanced and there is no mass loss. In $\eta_{0}=0$ for balance of escape and creation of magnetic field, we have $s=-2$. In this paper, only case of no wind ($s=-3/2$) is considered that $\dot{\rho}_{0}=0$ and $\dot{B}_{\varphi}\propto r^{-11/4}$ , we note $\dot{B}_{\varphi}$ in $P_{m}=5/6$ will be zero, too. The equations (29)-(31) imply that for given $\alpha$, $\eta_{0}$, $\beta$, $s$, and $f$ form a closed set of equations of $c_{1}$, $c_{2}$, and $c_{3}$ which will determine behavior of the accretion flow. ### 3.1 Analysis By using the equations (29)-(31), the coefficients of $c_{i}$ have the following forms $c_{1}=\frac{1}{2\alpha^{2}}\left(D_{4}+\sqrt{D^{2}_{4}+8\alpha^{2}}\right),$ (31) $c^{2}_{2}=-\frac{2}{9}c_{1}D_{1}D_{2},$ (32) $c_{3}=\frac{1}{3(s+2)}c_{1}D_{1},$ (33) where $D_{1}=\frac{1}{(1+\beta)^{1-\mu}},$ (34) $D_{2}=\frac{(2s-2+3\gamma)}{(\gamma-1)f}+\frac{\eta_{0}\beta}{3\alpha}\frac{(1+s)^{2}}{(s+2)},$ (35) $D_{3}=\frac{s-1+\beta(1+s)}{s+2},$ (36) $D_{4}=\frac{2}{3}D_{1}\left(\frac{2}{3}D_{2}+D_{3}\right).$ (37) The obtained results imply that the model is parametrized by the ratio of specific heat, $\gamma$, the standard viscous parameter, $\alpha$, the magnetic diffusivity parameter, $\eta_{0}$, the degree of magnetic pressure to gas pressure, $\beta$, the advection parameter, $f$, and the mass-loss rate parameter, $s$. The equation (37) implies that the value of $D_{2}$ for a value of $\beta$ will be zero. From equations (22) and (34) we can write $\Omega\propto c_{2}^{2}\propto D_{2}$. Thus we conclude for a value of $\beta$ that we define it $\beta_{b}$ (braking $\beta$), the angular velocity will be zero. Over of $\beta_{b}$, $c_{2}^{2}$ becomes negative that it is not physical. By solving $D_{2}=0$ for the $\beta$ parameter, we can write $\beta_{b}=-\frac{3\alpha}{f\eta_{0}}\frac{(s+2)}{(s+1)^{2}}\frac{(2s-2+3\gamma)}{(\gamma-1)}.$ (38) In the case of no mass-loss, $s=-3/2$, we can write $\beta_{b}=\frac{18\alpha}{f\eta_{0}}\left(\frac{5/3-\gamma}{\gamma-1}\right).$ (39) The above equation in terms of the magnetic Prandtl number becomes $\beta_{b}=18(\frac{P_{m}}{f})\left(\frac{5/3-\gamma}{\gamma-1}\right).$ (40) The equations (40)-(42) express that the amount of the $\beta_{b}$ parameter depends on $f$, $P_{m}$, and $\gamma$. As $\beta_{b}$ decreases by adding the advection degree, while increases by adding the magnetic Prandtl number. In a flow with high conductivity, that $P_{m}$ is very large, $\beta_{b}$ becomes very large. Also in the flow with high resistivity and low viscosity, $\beta_{b}$ will be small. Since the magnetic pressure fraction always is positive or equal to zero ($\beta\geq 0$), so in presence model $\gamma\leq 5/3$. Examples of the coefficients $c_{i}$ in two cases of $\mu=0$ and $1.0$ are shown in figures (1) and (2) as a function of the degree of magnetic pressure to the gas pressure, ($\beta$), for different value of the magnetic diffusivity, $\eta_{0}$. In figures (1) and (2), $c_{4}$ is the viscous torque that is obtained from right hand side of equation (30), $c_{5}$ is total energy dissipation by the viscosity and the resistivity, and is calculated by right hand side of equation (31), and $c_{6}$ is the ratio of the resistive dissipation to the viscous dissipation. #### 3.1.1 First Case: $\mu=0$ The parameter of $\mu$ has appeared in the equations of the kinematic coefficient of viscosity and the magnetic diffusivity to indicate important degree of total pressure in them. In the case of $\mu=0$, the kinematic coefficient of viscosity and the magnetic diffusivity become $\displaystyle\nu=\alpha\frac{c^{2}_{s}}{\Omega_{K}}(1+\beta)$ $\displaystyle=\alpha c_{3}(1+\beta)\sqrt{GM_{}}~{}r^{1/2}$ (41) $\displaystyle\eta=\eta_{0}\frac{c^{2}_{s}}{\Omega_{K}}(1+\beta)$ $\displaystyle=\eta_{0}c_{3}(1+\beta)\sqrt{GM_{}}~{}r^{1/2}.$ (42) Figure 1: Physical quantities of the flow as a function of the degree of magnetic pressure to the gas pressure, for several values of $\eta_{0}=0$, $0.12$, $0.15$, and $0.2$ that corresponding to $P_{m}=\infty$, $5/6$, $2/3$, and $5/10$. The disc density profile is set to be $s=-3/2$ (no wind), the ratio of the specific heats is set to be $\gamma=1.3$, the viscous parameter is $\alpha=0.1$, and the advection parameter is $f=1.0$. In this case the function forms of the kinematic coefficient of viscosity and the magnetic diffusivity deviate by factor $(1+\beta)$ from the function forms of used by NY95 and Sh04. Also the profiles of non-resistive and non-magnetic case is shown to compare the present model with canonical ADAF solutions (e.g. NY95). The existence of $(1+\beta)$ in $\nu$ and $\eta$ causes the viscosity and resistivity increase by adding the toroidal magnetic field. The $\nu$ and $\eta$ have direct effects on the viscous torque ($c_{4}$) and the energy dissipation ($c_{5}$). Thus, we expect increase of $c_{4}$ and $c_{5}$ by adding the toroidal magnetic field that the profiles of $c_{4}$ and $c_{5}$ confirm it. Also, by adding the magnetic diffusivity ($\eta_{0}$) of the flow, $c_{4}$ and $c_{5}$ increase. Because the ohmic dissipation increases due to resistivity that it also increases the flow temperature (the profiles of $c_{3}$ confirm it), and since the viscous torque is proportional with temperature (sound speed), thus the viscous torque, $c_{4}$, increases by adding the magnetic diffusivity. The profiles of $c_{4}$ and $c_{5}$ imply that for all value of the $\beta$ and $\eta_{0}$, the total energy dissipation and the viscous torque are larger than the canonical ADAF solutions. The profiles of $c_{3}$ show that the temperature of the flow by adding the toroidal component of magnetic field decreases. This property is qualitatively consistent with the results of Bu et al. (2009) and KF09. The profiles of $c_{6}$ show that the ratio of the resistive dissipation to the viscous dissipation increases by adding the toroidal component of magnetic field ($\beta$) and the magnetic diffusivity ($\eta_{0}$). As in small amounts of magnetic field and the magnetic diffusivity, the dominant heat generated is the viscous dissipation, while in large values of magnetic field and the magnetic diffusivity, the dominant heat generated will be the resistive dissipation. Also, figure (1) shows by adding the $\beta$ and $\eta_{0}$ parameters, the radial infall velocity, $c_{1}$, increases, and the angular velocity, $c_{2}$, decreases. It can be due to increase of the viscous torque ($c_{4}$) by parameters of $\beta$ and $\eta_{0}$. The raise of the viscous torque by adding $\eta_{0}$ and $\beta$ parameters generates a larger negative torque in angular momentum equation and causes the angular velocity of the flow decreases, and the matter accretes with larger speed. In the present model, the flow rotates slower than canonical ADAF and accretes speeder than it. The increase of the radial infall velocity by adding the parameter of $\beta$ is qualitatively consistent with the results by Bu et al. (2009) and KF09. #### 3.1.2 Second Case: $\mu=1$ In the case of $\mu=1$, for the kinematic coefficient of viscosity and the magnetic diffusvity we can write $\displaystyle\nu=\alpha\frac{c^{2}_{s}}{\Omega_{K}}$ $\displaystyle=\alpha c_{3}\sqrt{GM_{}}~{}r^{1/2}.$ (43) $\displaystyle\eta=\eta_{0}\frac{c^{2}_{s}}{\Omega_{K}}$ $\displaystyle=\eta_{0}c_{3}\sqrt{GM_{}}~{}r^{1/2}.$ (44) In this case the function forms of the kinematic coefficient of viscosity and the magnetic diffusivity are the same as Sh04 and NY95 used. The behavior of the physical quantities of the flow in this case are shown in figure (2). The absence of $(1+\beta)$ in this case for $\nu$ and $\eta$ in comparison to previous case causes the value of them decrease by factor $(1+\beta)$. The effects of absence of this factor increases by adding parameter of $\beta$. The profiles of the angular momentum transport ($c_{4}$) and total energy dissipation ($c_{5}$) imply that the amounts of them by factor $(1+\beta)$ compared with previous case decrease. As their behavior in terms of the toroidal component of magnetic field have changed and they decrease by adding the $\beta$ parameter. However, the magnetic diffusivity has the previous effects and increase these two quantities. Also, the solutions show that the viscous torque and the total dissipation in this case are smaller than canonical ADAF. The weakening of $c_{4}$ and $c_{5}$ by adding the $\beta$ parameter reduces the radial infall velocity. Here, the radial infall velocity increases by adding $\eta_{0}$ that is due to increase of $c_{4}$ and $c_{5}$. The behavior of other physical quantities in terms of the $\beta$ and $\eta_{0}$ parameters does not change, and represent small variations. To compare the radial infall velocity profiles with canonical ADAF solutions implies that the flow accretes slower than canonical ADAF that is different with previous case. The decrease of the temperature and the viscous torque by adding the parameter of $\beta$ is qualitatively consistent with the results of Bu et al. (2009). Figure 2: Physical quantities of the flow as a function of the degree of magnetic pressure to the gas pressure, for several values of $\eta_{0}=0$, $0.12$, $0.15$, and $0.2$ that corresponding to $P_{m}=\infty$, $5/6$, $2/3$, and $5/10$. The disc density profile is set to be $s=-3/2$ (no wind), the ratio of the specific heats is set to be $\gamma=1.3$, the viscous parameter is $\alpha=0.1$, and the advection parameter is $f=1.0$. ### 3.2 Mass Accretion Rate In according to assumptions of section (2) the mass accretion rate defines as $\dot{M}=-4\pi r^{2}\rho v_{r}.$ (45) The mass accretion rate under self-similar transformations of equations (21) and (25) becomes $\dot{M}=4\pi\alpha\rho_{0}c_{1}\sqrt{GM_{}}~{}r^{s+3/2}.$ (46) In our interesting case, $s=-3/2$ (no wind), for the mass accretion rate we can write $\dot{M}=4\pi\alpha\rho_{0}c_{1}\sqrt{GM_{}}$ (47) that implies the mass accretion rate does not vary by position. This result is qualitatively consistent with the results by Sh04, Ghanbari et al (2007), and AF06. Although the present model of accretion flow is different from Bondi (1952) accretion in various aspect, we can define Bondi accretion rate as $\dot{M}_{Bondi}=\pi G^{2}M_{}^{2}\left(\frac{\rho(\infty)}{c_{s}^{3}(\infty)}\right)\left[\frac{2}{5-3\gamma}\right]^{(5-3\gamma)/2(\gamma-1)}$ (48) where $\rho(\infty)$ and $c_{s}(\infty)$ are the density and the sound speed in the gas far away from the star (Frank et al. 2002). Bondi accretion rate in terms of our self-similar transformations becomes $\dot{M}_{Bondi}=\pi\sqrt{GM_{}}\left(\frac{\rho_{0}}{c_{3}^{3/2}}\right)\left[\frac{2}{5-3\gamma}\right]^{(5-3\gamma)/2(\gamma-1)}.$ (49) Thus, we can write the mass accretion rate in term of Bondi accretion rate as follows $\dot{M}/\dot{M}_{Bondi}=4\alpha c_{1}c_{3}^{3/2}\left[\frac{2}{5-3\gamma}\right]^{(3\gamma-5)/2(\gamma-1)}r^{s+3/2}.$ (50) In our interesting case, $s=-3/2$ (no wind), we can write $c_{7}=\dot{M}/\dot{M}_{Bondi}=4\alpha c_{1}c_{3}^{3/2}\left[\frac{2}{5-3\gamma}\right]^{(3\gamma-5)/2(\gamma-1)}.$ (51) Here, we defined new parameter of $c_{7}$ that indicates the ratio of the mass accretion rate to Bondi accretion rate. Examples of the coefficient of $c_{7}$ in two cases of $\mu=0$ and $1.0$ are shown in figures (3) as a function of the degree of magnetic pressure to the gas pressure, ($\beta$), for different value of the magnetic diffusivity, $\eta_{0}$. The profiles of $c_{7}$ show that in the case $\mu=0$, the mass accretion rate increases by adding the toroidal magnetic field and the magnetic diffusivity. While the solutions for $\mu=1$ imply that the mass accretion rate decreases by adding the toroidal magnetic field and increases by adding the magnetic diffusivity. On the other hand, the magnetic diffusivity in two cases causes the mass accretion rate increase. In high magnetic pressure, the $c_{7}$ profiles for both cases show that the ratio of the mass accretion rate to the Bondi accretion rate is decreased with an increase in magnetic pressure. This property is qualitatively consistent with results of Kaburaki (2007). Comparision of the present model with canonical ADAF solutions implies that in the case of $\mu=0$ the flow accretes quicker than canonical ADAF, however in the case of $\mu=1$ the flow accretes slower than canonical ADAF. Also, the profiles of $c_{7}$ in both of cases show the mass accretion rate in our model is smaller than Bondi accretion rate that is in adapting with observational evidences from Sgr A∗, M87, and NGC 4261 (Kaburaki 2007). Figure 3: The ratio of mass accretion rate to Bondi accretion rate ($c_{7}=\dot{M}/\dot{M}_{Bondi}$) as a function of the degree of magnetic pressure to the gas pressure, for several values of $\eta_{0}=0$, $0.12$, $0.15$, and $0.2$ that corresponding to $P_{m}=\infty$, $5/6$, $2/3$, and $5/10$. The disc density profile is set to be $s=-3/2$ (no wind), the ratio of the specific heats is set to be $\gamma=1.3$, the viscous parameter is $\alpha=0.1$, and the advection parameter is $f=1.0$. In section 3.1, the upper limit of the magnetic field obtained and mentioned with $\beta_{b}$. By substituting $\beta_{b}$ and equations (33)-(39) in equation (53) and assume of $s=-3/2$, the mass accretion rate to the Bondi accretion rate ($c_{7}$) approximately is $\displaystyle c_{7}=\dot{M}/\dot{M}_{Bondi}\approx 24\sqrt{2}~{}\alpha~{}g_{1}~{}\frac{\left(1+\beta_{b}\right)^{1-\mu}}{\left(5+\beta_{b}\right)^{5/2}}$ $\displaystyle=24\sqrt{2}~{}\alpha~{}g_{1}~{}\frac{\left(1+\frac{18~{}g_{2}P_{m}}{f}\right)^{1-\mu}}{\left(5+\frac{18~{}g_{2}P_{m}}{f}\right)^{5/2}}$ (52) where $\displaystyle g_{1}=\left[\frac{2}{5-3\gamma}\right]^{\frac{(3\gamma-5)}{2(\gamma-1)}}$ $\displaystyle g_{2}=\left[\frac{5/3-\gamma}{\gamma-1}\right].$ To obtain the root of derivative of $c_{7}$ in terms of $f$, we can write $\displaystyle\frac{dc_{7}}{df}=0~{}~{}\Rightarrow~{}~{}f_{max}=\left(\frac{18}{5}\right)\left(\frac{3+2\mu}{1-2\mu}\right)g_{2}P_{m},$ The above equation states the mass accretion rate in $f_{max}$ becomes maximum and for $f>f_{max}$ the mass accretion rate decreases by advection degree. The $f_{max}$ for $\mu>1/2$ becomes negative, while $0\leq f\leq 1$. Thus, the relation of $f_{max}$ is valid only for the case of $\mu=0$. In the case of $\mu=0$, $f_{max}=\left(54/5\right)g_{2}P_{m}$ and $\beta_{b}=5/3$. Thus, we expect in dominant magnetic case ($\beta>1$), the accretion efficiency decreases by advection degree parameter. In the case of $\mu=1$, due to the lack of any extremum, the mass accretion rate only increases by advection degree and does not show any decrease. In the case of high magnetic pressure ($\beta\gg 1$), equation (54) becomes $\displaystyle c_{7}=\dot{M}/\dot{M}_{Bondi}\approx 24\sqrt{2}~{}\alpha~{}g_{1}~{}\left(\frac{f}{18~{}g_{2}P_{m}}\right)^{3/2+\mu}.$ The above equation implies that the mass accretion rate to Bondi accretion is strongly depends on viscosity parameter, Prandtl number and the advection degree. The mass accretion rate increases by $f$ and decreases by $P_{m}$. Thus, accretion efficiency increases by the advection degree parameter in high magnetic pressure. ### 3.3 Timescales To estimate the effect of viscosity and resistivity on the accretion discs, we compare the viscous and resistive timescales with accretion timescale. The accretion timescale, $t_{acc}$, and the viscous timescale, $t_{visc}$, are given by $t_{acc}=\frac{r}{-v_{r}},$ $t_{visc}=\frac{r^{2}}{\nu}.$ We are using a similar functional form of $t_{visc}$ for the resistive timescale, $t_{resis}$, that is given by $t_{resis}=\frac{r^{2}}{\eta}.$ By using self-similar forms of physical quantities, we can write $\displaystyle\frac{t_{resis}}{t_{acc}}=\frac{\alpha}{\eta_{0}}\frac{c_{1}}{c_{3}}(1+\beta)^{\mu-1}$ $\displaystyle=3(\frac{\alpha}{\eta_{0}})(s+2).$ (53) The equation (30) is used for fraction of $c_{1}/c_{3}$. As, we said in previous section, in present model $\alpha/\eta_{0}$ is the magnetic Prandtl number, $P_{m}$, so above equation becomes $\frac{t_{resis}}{t_{acc}}=3P_{m}(s+2).$ For our interesting case, $s=-3/2$ (no wind), we can write $\frac{t_{resis}}{t_{acc}}=\frac{3}{2}P_{m}.$ The above equation implies that for $P_{m}\leq 2/3$, the magnetic diffusivity timescale is shorter than or equal to accretion timescale, while for $P_{m}>2/3$ the accretion timescale is shorter. Similar calculations for the viscous timescale express $\frac{t_{visc}}{t_{acc}}=3(s+2),$ where in no wind case ($s=-3/2$) becomes $\frac{t_{visc}}{t_{acc}}=(3/2).$ Thus, the viscosity timescale will be longer than the accretion timescale. To compare the magnetic diffusivity with the viscous timescales, we can write $\displaystyle\frac{t_{resis}}{t_{visc}}=\frac{r^{2}/\eta}{r^{2}/\nu}$ $\displaystyle=\frac{\nu}{\eta}~{}~{}$ $\displaystyle=P_{m}.$ (54) Thus, the magnetic Prandtl number specifies which one is shorter. For example in flow with high conductivity (e.g. AF06; KF09), $\eta\rightarrow 0$, the magnetic Prandtl number limits to infinity, and so the magnetic diffusivity timescale will be very longer than the viscous timescale. On the other hand, for a flow with finite resistivity and tiny viscosity (e.g. Sh04), the magnetic Prandtl number limits to zero, and so the magnetic diffusivity timescale is very shorter than viscous timescale. When the resistivity and the viscosity are approximately equal, $P_{m}\sim 1$, we expect $t_{resis}\sim t_{visc}$. Also, in special case of $P_{m}=5/6$ and $s=-3/2$ that escape and creation of magnetic field are balanced and there is no mass-loss, $t_{resis}=(5/6)t_{visc}$. ## 4 Summary and Discussion In this paper, the influences of the resistivity on the structure of the advection-dominated accretion flow is investigated. It is used only azimuthal component of magnetic field that is consistent with observational evidence of Galactic center (Novak et al 2003; Chuss et al. 2003; Yuan 2006). The $\alpha$-prescription is used for the kinematic coefficient of viscosity and the magnetic diffusivity. The equations of the model are solved by a semi- analytical self-similar method in comparison with the self-similar solution by AF06. The physical quantities of disc are sensitive to the amounts of the magnetic pressure fraction ($\beta$) and the magnetic diffusivity ($\eta_{0}$) parameters. As, the angular velocity of the flow by adding the $\beta$ and $\eta_{0}$ parameters decreases. For a value of the magnetic pressure fraction, the angular velocity of disc becomes zero. This amount of the magnetic pressure fraction strongly depends on the properties of the accreting gas, such as the viscosity, resistivity, adiabatic index, and advection degree. The solutions represent the radial infall velocity increases by adding the magnetic diffusivity. Also the solutions show that the temperature of the flow decrease by adding the toroidal component of magnetic field. This result qualitatively is consistent with the results of Bu et al. (2009) and KF09. The profiles of the temperature of the flow show that it increases by adding the magnetic diffusivity that is due to the raise of the resistive dissipation. Comparison of the present model with Bondi accretion implies that for all values of the $\beta$ and $\eta_{0}$ parameters, the Bondi accretion rate is larger than the mass accretion rate that is in accord with observational evidences of Sgr A∗, M87, and NGC 4261 (Kaburaki 2007). Also, the mass accretion rate profiles at high magnetic field express that the magnetic field reduces the mass accretion rate that is similar to results of Kaburaki (2007). We found that in the small magnetic field, the more heat generated in the flow is due to the viscous dissipation, while the ohmic dissipation will be dominant in large amounts of magnetic field and resistivity. As noted in the introduction, the MHD simulations show that linear growth of MRI decreases significantly by ohmic dissipation. linear growth of the MRI in the resistive fluid can be characterized by the Lundquist number ($S_{MRI}=c_{A}^{2}/\eta\Omega$) and magnetic Reynolds number ($Re_{M}=c_{s}^{2}/\eta\Omega$), where $c_{A}$, $c_{s}$, $\eta$, and $\Omega$ have usual meaning. In terms of our self-similar transformations, the Lundquist number and magnetic Reynolds number become $S_{MRI}=2\beta/\eta_{0}c_{2}(1+\beta)^{1-\mu}$ and $Re_{M}=1/\eta_{0}c_{2}(1+\beta)^{1-\mu}$. The solutions of present model show that $S_{MRI}$ and $Re_{M}$ decrease by resistivity. This property is qualitatively consistent with MHD simulation results (Fleming et al. 2000; Masada & Sano 2008). Here, latitudinal dependence of physical quantities is ignored, while some authors showed that latitudinal dependence is important in structure consideration of a disc (Narayan & Yi 1995; Sh04; Ghanbari et al. 2007). One can investigate latitudinal behavior of such discs. Furthermore, in a realistic model the advection parameter $f$ is a function of position, one can consider such discs. ## Acknowledgements I would like to thank the referee for very useful comments that helped me to improve the initial version of the paper. I would also like to thank Markus Flaig for helpful discussion. ## References [1] [] Abramowicz, M., Chen, X., Kato, S., Lasota, J. P., Regev, O., 1995, ApJ, 438, L37 * [2] [] Aitken D. K., Wright C. M., Smith C. H., Roche P. F., 1993, MNRAS, 262, 456 * [3] [] Akizuki, C., Fukue, J., 2006, PASJ, 58, 469 (AF06) * [4] [] Balbus S. A., Hawley J. F., 1992, ApJ, 392, 662 * [5] [] Begelman, M. C., Pringle, J.E., 2007, MNRAS, 375, 1070 * [6] * [7] [] Bisnovatyi-Kogan,G. S., Lovelace, R. V. E., 2001, New Astron. Rev. 45, 663 * [8] * [9] [] Blandford, R. D., Begelman, M. C. 1999, MNRAS, 303, L1 * [10] [] Bondi, H., 1952. MNRAS, 112, 195 * [11] [] Bu D., Yuan F., Xie F., 2009, MNRAS, 392, 325 * [12] [] Campbell C. G., 1999, Geophys. Astrophys. Fluid Dynamics, 90, 113 * [13] [] Chuss, D. T., Davidson, J. A., Dotson, J. L., Dowell, C. D., Hildebrand, R. H., Novak, G., Vaillancourt, J. E., 2003, ApJ, 599, 1116 * [14] [] Cowling, T. G., 1981, ARA&A, 19, 115 * [15] [] Fleming T. P., Stone J. M., Hawley J. F., 2000, ApJ, 530, 464 * [16] [] Fleming T. P., Stone J. M., 2003, ApJ, 585, 908 * [17] [] Frank J., King A. R., Raine D., 2002, Accretion Power in Astrophysics. Cambridge Univ. Press, Cambridge * [18] [] Gammie C. F., Menou K., 1998, ApJ, 332, 659 * [19] [] Ghanbari J., Salehi F., Abbassi S., 2007, MNRAS, 381, 159 * [20] * [21] [] Ichimaru, S. 1977, ApJ, 214, 840 * [22] * [23] [] Kaburaki, O., 2000, ApJ, 531, 210 * [24] [] Kaburaki O., 2007, ApJ, 662, 102 * [25] [] Kaburaki, O., Yamazaki,N., Okuyama, Y. 2002, NewA, 7, 283 * [26] * [27] [] Kaburaki O., Nankou T., Tamura N., Wajima K., 2010, PASJ, 62, 1177 * [28] * [29] [] Khesali, A., Faghei, K., 2008, MNRAS, 389, 1218 (KF08) * [30] [] Khesali, A., Faghei, K., 2009, MNRAS, 398, 1361 (KF09) * [31] * [32] [] King A. R., Pringle J. E., Livio M., 2007, MNRAS, 376, 1790 * [33] * [34] [] Krasnopolsky R., Li Z. Y., Shang H., 2010, ApJ, 716, 1541 * [35] [] Kudoh T., Kaburaki O., 1996, ApJ, 460, 199 * [36] [] Kuwabara T., Shibata K., Kudoh T., Matsumoto R., 2000, PASJ, 52, 1109 * [37] [] Masada Y., Sano T, 2008, ApJ, 689, 1234 * [38] [] Narayan, R., Yi, I., 1994, ApJ, 428, L13 * [39] [] Narayan, R., Yi, I., 1995, ApJ, 453, 710 * [40] [] Novak, G., Chuss, D. T., Renbarger, T., et al. 2003, ApJL, 583, L83 * [41] * [42] [] Ogilvie, G. I., 1999, MNRAS, 306, L9O * [43] * [44] [] Rees, M. J., Begelman, M. C., Blandford, R. D., Phinney, E. S. 1982, Nature, 295, 17 * [45] * [46] [] Shadmehri, M., 2004, A&A, 424, 379 (Sh04) * [47] [] Shakura, N.I., Sunyaev, R.A., 1973, A&A, 24, 337 * [48] [] Sinha M., Rajesh S. R., Mukhopadhyay B., 2009, RAA, 9, 1331 * [49] [] Stone J. M., Gammie C. F., Balbus S. B., Hawley J. F., 2000, in Protostars and Planets IV, ed, V. Manning, A. Boss, S. Russell, Tuncsun: Uni. Arizona Press * [50] [] Watarai, K. 2006, ApJ, 648, 523 * [51] [] Watarai, K. Y. 2007, PASJ, 59, 443 * [52] [] Wright C. M., Aitken D. K., Smith C. H., Roche P. F., 1993, PASA, 10, 247 * [53] [] Yuan F., 2006, JPhCS, 54, 427 * [54]	arxiv-papers	2011-11-30T20:29:09	2024-09-04T02:49:24.847327	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Kazem Faghei", "submitter": "Kazem Faghei", "url": "https://arxiv.org/abs/1111.7302" }
1112.0045	CytoITMprobe: a network information flow plugin for Cytoscape Aleksandar Stojmirović , Alexander Bliskovsky and Yi-Kuo Yu**to whom correspondence should be addressed National Center for Biotechnology Information National Library of Medicine National Institutes of Health Bethesda, MD 20894 United States #### Background: Cytoscape is a well-developed flexible platform for visualization, integration and analysis of network data. Apart from the sophisticated graph layout and visualization routines, it hosts numerous user-developed plugins that significantly extend its core functionality. Earlier, we developed a network information flow framework and implemented it as a web application, called ITM Probe. Given a context consisting of one or more user-selected nodes, ITM Probe retrieves other network nodes most related to that context. It requires neither user restriction to subnetwork of interest nor additional and possibly noisy information. However, plugins for Cytoscape with these features do not yet exist. To provide the Cytoscape users the possibility of integrating ITM Probe into their workflows, we developed CytoITMprobe, a new Cytoscape plugin. #### Findings: CytoITMprobe maintains all the desirable features of ITM Probe and adds additional flexibility not achievable through its web service version. It provides access to ITM Probe either through a web server or locally. The input, consisting of a Cytoscape network, together with the desired origins and/or destinations of information and a dissipation coefficient, is specified through a query form. The results are shown as a subnetwork of significant nodes and several summary tables. Users can control the composition and appearance of the subnetwork and interchange their ITM Probe results with other software tools through tab-delimited files. #### Conclusions: The main strength of CytoITMprobe is its flexibility. It allows the user to specify as input any Cytoscape network, rather than being restricted to the pre-compiled protein-protein interaction networks available through the ITM Probe web service. Users may supply their own edge weights and directionalities. Consequently, as opposed to ITM Probe web service, CytoITMprobe can be applied to many other domains of network-based research beyond protein-networks. It also enables seamless integration of ITM Probe results with other Cytoscape plugins having complementary functionality for data analysis. ## Background Cytoscape [1, 2, 3] is a popular and flexible platform for visualization, integration and analysis of network data. Apart from the sophisticated graph layout and visualization routines, its main strength is in providing an API that allows developers other than its core authors to produce extension plugins. Over the last decade, a large number of plugins have been released, supporting the features such as import and export of data, network analysis, scripting and functional enrichment analysis. In this paper, we describe CytoITMprobe a plugin that brings to Cytoscape new functionality founded on information flow. Numerous approaches for analyzing biological networks based on information flow [4, 5, 6, 7, 8, 9, 10] have emerged in recent years. The main assumption of all such methods is information transitivity: information can flow through or can be exchanged via paths of biological interactions. Our contribution to this area [11, 12] is a context-specific framework based on discrete-time random walks (or equivalently, diffusion) over weighted directed graphs. In contrast to most other approaches, our framework explicitly accommodates directed networks as well as the information loss and leakage that generally occurs in all networks. Apart from the network itself and a user-specified context, it requires no prior restriction to the sub-network of interest nor additional and possibly noisy information. We implemented our framework as an application called ITM Probe [13] and made it available as a web service [14], where users can query protein-protein interaction (PPI) networks from several model organisms and visualize the results. In addition to implementing network flow algorithms, the ITM Probe web service possesses a number of useful features. Using the Graphviz [15] suite for layout and visualization of graphs, it displays in the user’s web browser the images of subnetworks consisting of nodes identified as significant by the information flow models and offers a choice of multiple coloring schemes. The entire query results can be retrieved in the CSV format or forwarded to a functional enrichment analysis tool to facilitate their interpretation. However, lacking a mechanism to decouple the algorithmic part from the interaction graph, the ITM Probe web service restricts users to querying only the few compiled PPI networks available on the website. Using a canned suite for graph layout, ITM Probe limits the users’ ability to manipulate network images. For example, the only way to change the layout of significant subnetworks is to choose a different seed and re-compute the layout. Most importantly, not having an adequate interface to a well-designed platform such as Cytoscape, it is difficult to use the results of the ITM Probe service within the workflows involving additional data and algorithms from other sources. We thus developed CytoITMprobe to meet these challenges by (1) providing an explicit decoupling between the algorithmic part and the interaction graph, (2) utilizing the core graph manipulation functionality of Cytoscape for a broader visualization choices, and (3) adding an appropriate input/output interface for seamless integration with other resources available in Cytoscape. Figure 1: ITM Probe is based on discrete-time random walks with boundary nodes and damping. As an example, consider the weighted directed network shown, containing 19 nodes and 44 links. Single-directional links are assigned weight 2 and are indicated using arrows while bi-directional edges are assigned weight 1 and are shown as lines. The first five graphs show the time progress of a random walk in the presence of damping and two absorbing boundary nodes (indicated by octagons). At $t=0$, 1000 random walkers start at a single point in the network. At $t=1$, they have progressed one step from their origin to the nodes adjacent to it, being distributed randomly in proportion to the weights of the edges leading from the origin. Only 900 walkers remain in the network at $t=1$ due to damping: the damping factor $\mu=0.9$ (dissipation $0.1$) means that $10\%$ of walkers are dissipated at each step. At $t=60$, most of the walks have terminated, either by dissipation, or by reaching one of the two boundary nodes. The number of walkers terminating at each boundary node depends on their starting location. The final graph shows the probability $F_{ik}$ for a random walk starting at any transient node in the network (indicated by circular shape) to terminate at the boundary node on the right- hand side (scaled by 1000). Note that the value indicated in the final graph for the starting node at $t=0$ (190) is the same as the final number of walks shown at $t=60$ as terminating at the right boundary node. ## Information Flow Framework ITM Probe extracts _context-specific_ information from networks. We elaborated on the information flow framework underlying ITM Probe in our previous publications [11, 12] and here we provide a non-technical explanation. Given a context consisting of one or more user-selected network nodes, the aim is to retrieve a set of other network nodes most related to that context. We model networks as weighted directed graphs, where nodes are linked by directional edges and each edge is assigned a positive weight. One can consider a random walker that wanders among network nodes in discrete steps. The rule of the walk is that the walker starts at a certain node and in each step moves randomly to some adjacent node with probability proportional to the weight of the edge linking these nodes (Fig. 1). If the graph is connected, that is, if there is a directed path linking any two nodes, such a walk never terminates and the walker will eventually visit every node in the graph. Our main idea is to set termination or _boundary_ nodes for the walkers while using random walks to explore the neighborhoods of the context nodes. Provided there is a directed path linking any node to a boundary node, every random walk here will eventually terminate. Furthermore, the nodes visited by a walker before termination will vary depending on the origin of the walk. Since a random walk is a stochastic process, and each walk is different, we are interested in the cumulative behavior of infinitely many walkers following the same rules. On average, we expect that the nodes more relevant to the context will be more visited than those that are less relevant. Thus, the main quantity of interest is the average number of visits to a node given the selected origins and destinations of the walk. A problem with the above approach is that random walkers may spend too much time in the graph if the origins and destinations of the walk are far apart. This could mean that the entire graph is visited so that the most significant nodes are just those with the largest degree. To ensure that the significant nodes are relatively close to the context nodes, our framework contains an additional ingredient, _damping_ : at each step of a walk, we assign a certain probability for the walker to dissipate, that is, to leave the network. We still evaluate the average number of visits to each node, but now only count the visits prior to the walker leaving the network. Evidently, the nodes that are close to the walker’s origin will be significantly visited. In addition to forcing locality, damping is also natural in physical or biological contexts. If we treat random walkers as information propagating through the network, it is natural to assume that some information is lost during transmission. For protein-protein interaction networks, where nodes are proteins and links are physical bindings between proteins, damping could be associated with protein degradation by proteases, which would diminish the strength of information propagation. ITM Probe framework contains three models: _emitting_ , _absorbing_ and _channel_. In the absorbing model (Fig. 1), the context nodes are interpreted as destinations or _sinks_ of random walks, while every non-boundary or _transient_ node is considered as a potential origin. For each transient node $i$ and each sink $k$, the model computes $F_{ik}$, the average number of visits to the terminating node $k$ by random walks originating at the node $i$. Since a walk can either terminate at one sink or the other, $F_{ik}$ can also be interpreted as the probability that a random walk from $i$ reaches $k$. In the absence of damping, the sum of $F_{ik}$ over all sinks will be exactly $1$ for any transient node $i$. However, in the presence of damping, the sum of $F_{ik}$ over all sinks may be much less than $1$ (Fig. 1). The emitting model (Fig. 2), offers a dual point of view. Here, the context nodes are interpreted as origins or _sources_ of random walks. The walks terminate by dissipating or by returning to the sources – the sources form an emitting boundary. Since the origins of the walks are fixed, the quantity of interest is the visits to the transient nodes. Specifically, for each source $s$ and each transient node $i$, the emitting model returns $H_{si}$, the average number of visits to $i$ by walkers originating at $s$. Figure 2: The emitting model counts visits from sources. Using the example network from Fig. 1 with the same damping factor, consider the case where 1000 random walkers start at the source node indicated by a hexagon. At each time step, some random walkers leave the network due to damping or by moving back to the source. In the first five graphs, the number in each node documents the total number of visits to that node from all random walkers, dissipated or not, up to the indicated time. The value of $H_{si}$ returned by the ITM Probe emitting mode ($s$ here denotes the source node) yields the expected number of visits to node $i$ per random walker that starts at $s$ over infinitely many time steps. The final graph shows the values of $H_{si}$ for this context, scaled by 1000. Note that the magnitude shown for one transient node is greater than 1000 because a walker may visit the same node multiple times. The values of $F_{ik}$ and $H_{si}$ can be efficiently computed by solving (sparse) systems of linear equations. Let $W_{ij}$ denote the weight of the directed link $i\to j$ and let $0<\mu<1$ denote the damping factor. For all pairs of nodes $i,j$, construct the random walk evolution operator $\mathbf{P}$, where $P_{ij}=\frac{\mu W_{ij}}{\sum_{j^{\prime}}W_{ij}}$. The operator $\mathbf{P}$ includes damping and hence $\sum_{j}P_{ij}<1$. Let $\mathbf{P}_{TT}$ denote the sub-operator of $\mathbf{P}$ with domain and range restricted only to transient nodes and let $\mathbf{G}=(\mathbb{I}-\mathbf{P}_{TT})^{-1}$, where $\mathbb{I}$ stands for the identity matrix. Then, it can be shown [11], that $\displaystyle F_{ik}$ $\displaystyle=\sum_{j}G_{ij}P_{jk},\qquad\text{and}$ $\displaystyle H_{si}$ $\displaystyle=\sum_{j}P_{sj}G_{ji}.$ More details, including the cases where $\mu=0$, $\mu=1$ or non-uniform damping are covered in [11, 12]. Figure 3: The channel model highlights the directed flow from origins to destinations. Consider once again the example network from Figs. 1 and 2, now with a single source (hexagon) and two sinks (octagons). In common with the case from Fig. 2, the walkers start at the source, but in this case can terminate only by reaching the sinks. The damping factor is implicit: it determines how far the walkers are allowed to deviate from the shortest path towards one of the sinks. In the first five graphs, the number in each transient node documents the total number of visits to that node from all random walkers up to the indicated time. However, the value in each sink node represents the likelihood to reach that sink from the source at the indicated time. The value of $\hat{\varPhi}_{i,K}^{s}$ returned by the ITM Probe normalized channel mode yields the expected number of visits to node $i$ per random walker that starts at $s$ over infinitely many time steps. Note that the sink nodes split the flow from the source depending on their location. In this example, over infinitely many time steps, the node closer to the source captures 970 walkers, while the further sink gets only the remaining 30. The channel model combines the emitting and the absorbing model, with both sources and sinks on the boundary. It illuminates the most likely paths from sources to sinks. For each source node $s$, transient node $i$ and sink node $k$, it computes $\varPhi_{i,k}^{s}=H_{si}F_{ik}$, the average number of visits to $i$ by a random walker that originates at $s$ and terminates at $k$. ITM Probe does not report $\varPhi_{i,k}^{s}$ directly, but instead shows a simpler, _normalized_ quantity $\hat{\varPhi}_{i,K}^{s}$ (Fig. 3), which is defined for each source $s$ and transient node $i$ by $\hat{\varPhi}_{i,K}^{s}=\frac{\sum_{k}H_{si}F_{ik}}{\sum_{k^{\prime}}F_{sk^{\prime}}}.$ (1) Here, the numerator $\sum_{k}H_{si}F_{ik}=\sum_{k}\varPhi_{i,k}^{s}$ gives the average number of visits, in the presence of damping, to $i$ by a random walker starting at $s$ and terminating at any sink. The denominator gives the total probability of a walker starting at $s$ to terminate at any sink. Hence, with the denominator off-setting the effect of damping, the value of $\hat{\varPhi}_{i,K}^{s}$ counts the average number of visits to $i$ by walkers that start at $s$ and terminate at any of the sinks as if no dissipation is present. Generally, damping in the emitting or the absorbing model determines how far the flow can reach away from its origins. In contrast, the damping parameter for the normalized channel model plays a different role (Fig. 4): it effectively determines the ‘width’ of the channel from sources to sinks. When damping is very strong, only the nodes on the shortest path from a source to its nearest sink will be visited. Given the close relationship between random walks and diffusion, it is also possible to interpret ITM Probe models through information diffusion (or information flow). Within that paradigm, a fixed amount information is constantly replenished at the source nodes while leaving the network at all boundary nodes and through dissipation. At equilibrium, when the rate of flow entering equals the rate of leaving, the amount of information occupying each transient node is equivalent to the average number of visits to that node (using the aforementioned non-replenishing random walk interpretation [11]). We call the set of nodes most influenced by the flow an _Information Transduction Module_ or ITM. Figure 4: An example of the results of running different ITM Probe models. Here we see the results of running the emitting (a,d,g), absorbing (b,e,h) and channel (c,f,i) model of ITM Probe with the same sources and sinks but different dissipation coefficients. The underlying undirected graph is derived from a square lattice by removing random nodes and edges. Sources are shown as hexagons, sinks as octagons, and transient nodes as squares. The top row (a,b,c) shows the runs with damping factor $\mu=0.95$ (dissipation $0.05$), the middle (d,e,f) with $\mu=0.75$ and the bottom with $\mu=0.25$. For the emitting and channel model, each basic cyan, magenta or yellow color is associated with a source. The coloring of each node arises by mixing the basic color in proportion to the strength of information flow from their respective sources. For the absorbing model, the nodes are shaded according to the total probability of absorption at any sink on a logarithmic scale. ## Software architecture CytoITMprobe architecture consists of two parts: the user interface front end and computational back end. The user interface, written in Java [16] using Cytoscape API, is accessed as a Cytoscape plugin. It consists of the query form, the results viewer and the ITM subnetwork (Fig. 5). The back end is the standalone ITM Probe program, written in Python, which can be installed locally or accessed through a web service. In either configuration, CytoITMprobe takes the user input through the graphical user interface, validates it, and passes a query to the back end. Upon receiving from the back end the entire query results, CytoITMprobe stores them as the node and network attributes of the original network. Consequently, the query output can be edited or manipulated within Cytoscape, as well as saved for later use. Figure 5: CytoITMprobe interface. At startup from the Plugins menu, CytoITMprobe embeds its query form into the Control Panel (left). After performing a query or loading previously obtained search results, it creates an ITM subnetwork showing significant nodes and a viewer embedded into Results Panel (right). The viewer allows closer examination of the results and manipulation of the contents and the look of the ITM subnetwork. The overall visual styling of CytoITMprobe components closely resembles that of the ITM Probe web version. Standalone ITM Probe is a part of the qmbpmn-tools Python package, which also contains the code supporting the ITM Probe and SaddleSum web services, as well as the scripts for constructing the underlying datasets. The ITM Probe part depends on Numpy and Scipy [17] packages for numerical computations. The performance of ITM Probe critically depends on the routines for computing direct solutions of large, sparse, nonsymmetric systems of linear equations. Scipy supports two sparse direct solver libraries (both written in C): SuperLU [18] as default and UMFPACK [19] as an optional add on through SciKits collection [20]. In our experience, UMFPACK runs faster than SuperLU and Scipy always uses it if available. However, for optimal performance, UMFPACK requires well-tuned Basic Linear Algebra Subroutines (BLAS) libraries and may not be easy to install. To support users who are inclined not to install UMFPACK or Scipy, CytoITMprobe supports remote queries by default. ## Input CytoITMprobe requires as input a weighted directed graph and the ITM Probe model parameters that include a selection of boundary nodes and a dissipation probability. ### Step one: defining a query graph In CytoITMprobe graph connectivity is specified by selecting a Cytoscape network. In addition, each link must be assigned a weight and a direction through the query form. Edge weights are set using the _Weight attribute_ dropdown box, which lists all available floating-point edge attributes of the selected network and the default option (_NONE_). If the default option is selected, CytoITMprobe assumes a weight $2$ for any self-pointing edge and $1$ for all other edges. If an attribute is selected, the weight of an edge is set to the value of the selected attribute for that edge. Null attribute values are treated as zero weights. Since Cytoscape edges are always internally treated as directed, the user must also indicate the directedness of each edge type through the query form. Whenever a new Cytoscape network is selected, CytoITMprobe updates the query form and places all of the network’s edge types into the _undirected_ category. The user can use arrow buttons to move some edge types to the _directed_ or _ignored_ category. Undirected edges are treated as bidirectional, with the same weight in both directions. Directed edges have a specified weight assigned only in the forward direction, with the backward direction receiving the zero weight. Ignored edges have zero weight in both directions. Since Cytoscape allows multiple edges of different types between the same nodes, CytoITMprobe collapses multiple edges in each direction into a single edge by appropriately summing their weights (Fig. 6). Figure 6: Edge weights example Consider the following example: Suppose A and B are nodes in a Cytoscape network linked by three edges of two types with shown edge weights. Assume two type I edges (lighter gray), $A\to B$ and $B\to A$ are directed, while a single type II edge (darker gray) $A\to B$ is undirected. At query time, CytoITMprobe creates two directed edges, $A\to B$ and $B\to A$, with weights $3$ and $6$, respectively. ### Step two: selecting a model and boundary nodes In addition to a weighted directed graph, ITM Probe requires an information flow model (emitting, absorbing or normalized channel), a selection of sources and/or sinks, and dissipation probability. The choice of the model determines the types of boundary nodes that need to be specified, as well as the ways in which the damping factor can be set (see ‘Step three: specifying dissipation probability’ below). The query form also allows users to specify _excluded nodes_. Any flow reaching excluded nodes is fully dissipated. This is a way to remove those nodes that do not participate in information propagation in the desired context or that introduce undesirable shortcuts. ### Step three: specifying dissipation probability The values of $H$, $F$, and $\hat{\varPhi}_{,}$ all implicitly depend on the dissipation probability. In ITM Probe the user can set the dissipation probability directly or specify a related quantity that can, using Newton’s method, determine the dissipation probability. The choice of the alternative quantity depends on the selected model. For the emitting model, this quantity is the average path length before termination, which we denote by $\bar{t}$. For example, the user can require a random walker to make on average three steps before terminating. The formula for $\bar{t}$ is $\bar{t}=1+\frac{1}{n_{S}}\sum_{s}\sum_{j}H_{sj},$ (2) where $n_{S}$ denotes the number of sources. For the normalized channel model, the path length before termination is given by $\bar{t}=1+\frac{1}{n_{S}}\sum_{s}\sum_{j}\hat{\varPhi}_{j,K}^{s}.$ (3) Since the normalized channel model counts only the random walkers actually terminating at sinks, $\bar{t}$ is in this case bounded below by the length of the shortest path from any source to any sink. Hence, ITM Probe accepts the desired value of $\bar{t}$ in terms of length deviation from the shortest path. There are two ways to set the average path-length deviation: in absolute units (steps) or as a proportion of the length of the shortest path. The absorbing model allows users to obtain the dissipation probability by setting the average absorption probability, denoted $\bar{r}$. The formula for $\bar{r}$ is $\bar{r}=\frac{1}{n_{T}}\sum_{i}\sum_{k}F_{ik},$ (4) where $k$ ranges over all sinks, $i$ ranges over all transient nodes _that are connected to at least one sink_ , and $n_{T}$ is the total number of such nodes. The value of $\bar{r}$ represents the likelihood of a random walk starting at a randomly selected point in the network to reach a sink. The dissipation probability obtained in this way is larger if the sinks are well- connected hubs near the center of the network, in contrast to the case when the chosen sinks are not as well connected. ### Step four: submitting a query After specifying all necessary input, the user submits a query by pressing the _QUERY_ button on the query form. The time required for a run depends on whether the query is local or remote, as well as on the size of the submitted graph and the number of selected sources and/or sinks. ## Output For every completed query, CytoITMprobe displays its results in a viewer embedded in Cytoscape Results Panel and a new Cytoscape network consisting of significant nodes (ITM subnetwork). The results viewer has five tabs: _Top Scoring Nodes_ , _Summary_ , _Input Parameters_ , _Excluded Nodes_ , and _Display Options_. The first four tabs contain information about the query and the results, while the last one contains a form that allows users to manipulate the ITM subnetwork. The form controls two aspects of the subnetwork: composition (what nodes are selected and how many) and node coloring. ### Displaying significant nodes Subnetwork nodes are selected through a _ranking attribute_ , which assigns a numerical value from ITM Probe results to each node. The nodes are listed in descending order of the ranking attribute and top nodes are displayed as the ITM subnetwork. The number of top nodes is determined by specifying a _selection criterion_ , which can be simply a number of nodes to show, a cutoff value or the ‘participation ratio’. Specifying a cutoff value $x$ selects the nodes with their ranking attribute greater than $x$. Participation ratio estimates the number of ‘significant’ nodes by considering all values of the ranking attribute in a scale-independent manner [11]. The available choices for the ranking attribute depend on the ITM Probe model and the number of boundary points. For the emitting and normalized channel model, the user can select visits to a node from each source or the sum of visits from all sources. It is also possible to use _interference_ [11], which denotes the minimum number of node visits, taken over all sources. For the absorbing model, the available attributes are absorbing probabilities to each sink and the total probability of termination at a sink. The values of all attributes for the subnetwork nodes are displayed in the _Top Scoring Nodes_ tab. The colors of the subnetwork nodes are determined by selecting _coloring attributes_ , a _scaling function_ and a _color map_. The list of coloring attributes is the same as the list of ranking attributes but the user can select up to three coloring attributes. If a single attribute is selected, node colors are determined by the selected eight-category ColorBrewer [21] color map. Otherwise, they are resolved by color mixing: each coloring attribute is assigned a single basic color (cyan, magenta or yellow), and the final node color is obtained by mixing the three basic colors in proportion to the values of their associated attributes at that node. The scaling function serves to scale and discretize the coloring attributes to the ranges appropriate for color maps. Figure 4 shows examples of mixed color scheme with three boundary points (left and right columns) and of a coloring using a single attribute (center column). ### Manipulating node attributes Since the ITM Probe query results are saved as Cytoscape attributes of the original network, they can be arbitrarily modified through Cytoscape. Any changes made are reflected in the results viewer and the corresponding ITM subnetwork after pressing the _RESET_ button on the Display Options form. Using the CytoITMprobe attribute nomenclature, users can create additional attributes to be used for ranking or coloring. Consider the following usage example. A user has run an emitting model query with three sources, S1, S2, and S3, and obtained the results in a viewer labeled ITME243. At the end of the run, CytoITMprobe created the attributes ITME243[S1], ITME243[S2] and ITME243[S3] for the nodes of the input network and saved the results as their values. The user creates a new floating-point node attribute with a label ITME243[avgS1S2] and fills it with an average of ITME243[S1] and ITME243[S2]. After resetting the Display Options form, an item ‘Custom [avgS1S2]’ is available for selection as a ranking or coloring attribute. This gives the user the flexibility to reinterpret S1 and S2 as if they were a single source of equal weight as S3. Another possibility is to combine the results of queries with different boundaries and display them together on the same subnetwork. ### Saving and restoring results The query network together with its attributes containing ITM Probe results can be saved as a Cytoscape session and later retrieved. After reloading the session, the user can regenerate the results viewer and the corresponding subnetwork for a stored ITM by pressing the _LOAD_ button on the CytoITMprobe query form and selecting the desired ITM from a list. Alternatively, the ITM Probe results can be exported to tab-delimited text files through the Cytoscape _Export_ menu. Each exported tab-delimited file contains all the information necessary to restore the results except the query network and can be easily manipulated both by humans and by external programs or scripts. The results from tab-delimited files can be imported into any selected Cytoscape network through the _Import_ menu. Since the selected network may be different from the original query network, only the results for the nodes in the selected network whose IDs match the IDs from the imported file will be loaded. After importing the results, CytoITMprobe generates a new results viewer and a subnetwork, as if the results originated from a direct ITM Probe query. ## Discussion The main function of ITM Probe, also applicable to domains other than PPI networks, is to retrieve information from large and complex networks by discovering the possible interface between network nodes that are hypothesized to be related. This paradigm works best with large networks, where such information cannot be easily accessed by other means. For examples of applications of the ITM Probe frameworks to protein-protein interaction networks, consult our earlier papers [11, 13, 12]. With a network as an _encyclopedia_ of domain-specific knowledge, ITM Probe enables a direct access to its specific portions related to a specified context. The user can learn about the objects representing individual nodes by setting them as sources and/or sinks and retrieving information about the most significant objects in the resulting ITM. This approach not only extracts a relevant subnetwork but also produces context-specific weights for each node. With their interpretation as average numbers of node visits, or equivalently, as average numbers of paths passing through a node, the ITM weights signify the relative importance of network nodes in the context of the query and thus can be used to refine its interpretation as a whole. For example, a single node with a large weight in an ITM resulting from a normalized channel model query represents a choke point _in the particular context of the query_. The same node need not have a high global centrality. Containing both sources and sinks, the normalized channel model offers the users the ability to formulate and evaluate network based hypotheses in silico. Since information flow that reaches one sink cannot subsequently terminate at any other, sink nodes can be associated with alternative hypotheses, such as different biological functions if the network is PPI. The information flow from each source will then, depending on the dissipation coefficient used, mainly trace the path towards the sink most likely to be reached first from that source (see Fig. 4, right column). The ITM Probe framework considers all weighted paths from sources to sinks and hence produces more robust results than approaches involving only the shortest paths. The path weights are tunable using the dissipation probability. Compared to the previously described web interface to ITM Probe [13], CytoITMprobe significantly benefits from being a part of the Cytoscape platform. Although the _Display Options_ form is very similar to the web version, the sophisticated network visualization functionality provided by Cytoscape allows significantly more versatility in displaying ITMs. For example, Cytoscape GUI allows users to manually alter node placements, rotate network views, or arbitrarily change the look of a network. In addition, Cytoscape interface enables users to directly manipulate node attributes representing ITM Probe results and possibly create new node summary variables appropriate to their problem. The newly created variables can be immediately reflected in the graphical representation of an ITM, which is not possible in the web setting. Most importantly, the results of ITM Probe can be integrated into workflows involving other Cytoscape plugins that provide complementary functionality. For instance, output ITMs can be related to terms from controlled vocabularies such as Gene Ontology [22] using functional enrichment analysis plugins such as PinGO [23] or our own recently released CytoSaddleSum [24]. The graph-theoretic structure of ITM subnetworks can be analyzed using a variety of algorithms such as MCODE [25] or GraphletCounter [26, 27]. The architecture of CytoITMprobe with a Cytoscape front end and an ITM Probe back end offers flexibility for a variety of usage scenarios. In contrast to the web version, it allows users to use ITM Probe with arbitrary networks and edge weights, rather than being limited to compiled PPIs from few model organisms. Most users will be content with accessing ITM Probe through the web server. However, the option to download and install the qmbpmn-tools package provides not only faster running times for queries but also the ability to use the command line interface for ITM Probe to perform batch queries and to locally reproduce its web service. The separation of the presentation layers (web or Cytoscape) from the ‘business’ layer (standalone ITM Probe) facilitates easy future updates to any components. ## Conclusion CytoITMprobe is a plugin that brings the previously unavailable network flow algorithms of ITM Probe to the Cytoscape platform. It enables users to extract context-specific subnetworks from large networks by specifying the origins and/or destinations of information flow. CytoITMprobe significantly extends the features of the previously released web version of ITM Probe. The main novelty of CytoITMprobe is that it allows the user to specify as input any Cytoscape network, rather than being restricted to the PPI networks available through the ITM Probe web service. Using Cytoscape attributes to hold their desired values, users may easily supply their own edge weights and denote edge directionalities. Additionally, the ability to manipulate and add new node attributes through Cytoscape reduces the workload required for visualizing various combinations of ITM components. In the context of biological cellular networks, this additional flexibility may lead to constructions of new node attributes that can better reflect biological significance, hence facilitating more educated hypothesis forming. By bringing ITM Probe to Cytoscape, CytoITMprobe enables seamless integration of ITM Probe results with other Cytoscape plugins having complementary functionality for data analysis. By decoupling the query network from the information flow algorithm, the newly developed CytoITMprobe can be applied to many other domains of network-based research beyond protein-networks. ## Availability and requirements ### CytoITMprobe plugin Project name: CytoITMprobe Project home page: http://www.ncbi.nlm.nih.gov/CBBresearch/Yu/downloads/itmprobe.html Documentation: http://www.ncbi.nlm.nih.gov/CBBresearch/Yu/mn/itm_probe/doc/cytoitmprobe.html Video tutorial: http://www.youtube.com/watch?v=4Cdf-mSKtWo Operating system(s): Platform independent Programming language: Java Other requirements: Java SE 6 or higher and Cytoscape 2.7 or higher License: All components written by the authors at the NCBI are released into Public Domain. Components included from elsewhere are available under their own open source licenses and attributed in the source code. ### Standalone ITM Probe (optional for CytoITMprobe) Project name: qmbpmn-tools Project home page: http://www.ncbi.nlm.nih.gov/CBBresearch/Yu/downloads/itmprobe.html Documentation: http://www.ncbi.nlm.nih.gov/CBBresearch/Yu/mn/itm_probe/doc/ Operating system(s): Platform independent Programming language: Python Other requirements: Python 2.6 or 2.7, Numpy 1.3 or higher and Scipy 0.7 or higher. UMFPACK Scikit is recommended for good performance. License: All components written by the authors at the NCBI are released into Public Domain. Components included from elsewhere are available under their own open source licenses and attributed in the source code. ## Acknowledgments This work was supported by the Intramural Research Program of the National Library of Medicine at the National Institutes of Health. ## References [1] Cline MS, Smoot M, Cerami E, Kuchinsky A, Landys N, Workman C, Christmas R, Avila-Campilo I, Creech M, Gross B, Hanspers K, Isserlin R, Kelley R, Killcoyne S, Lotia S, Maere S, Morris J, Ono K, Pavlovic V, Pico AR, Vailaya A, Wang PL, Adler A, Conklin BR, Hood L, Kuiper M, Sander C, Schmulevich I, Schwikowski B, Warner GJ, Ideker T, Bader GD: Integration of biological networks and gene expression data using Cytoscape. _Nat Protoc_ 2007, 2(10):2366–82. * [2] Shannon P, Markiel A, Ozier O, Baliga NS, Wang JT, Ramage D, Amin N, Schwikowski B, Ideker T: Cytoscape: a software environment for integrated models of biomolecular interaction networks. _Genome Res_ 2003, 13(11):2498–504. * [3] Smoot ME, Ono K, Ruscheinski J, Wang PL, Ideker T: Cytoscape 2.8: new features for data integration and network visualization. _Bioinformatics_ 2011, 27(3):431–2. * [4] Nabieva E, Jim K, Agarwal A, Chazelle B, Singh M: Whole-proteome prediction of protein function via graph-theoretic analysis of interaction maps. _Bioinformatics_ 2005, 21 Suppl 1:302–310. * [5] Tu Z, Wang L, Arbeitman M, Chen T, Sun F: An integrative approach for causal gene identification and gene regulatory pathway inference. _Bioinformatics_ 2006, 22:e489–496. * [6] Suthram S, Beyer A, Karp R, Eldar Y, Ideker T: eQED: an efficient method for interpreting eQTL associations using protein networks. _Mol. Syst. Biol._ 2008, 4:162. * [7] Zotenko E, Mestre J, O’Leary DP, Przytycka TM: Why do hubs in the yeast protein interaction network tend to be essential: reexamining the connection between the network topology and essentiality. _PLoS Comput Biol_ 2008, 4(8):e1000140. * [8] Missiuro P, Liu K, Zou L, Ross B, Zhao G, Liu J, Ge H: Information flow analysis of interactome networks. _PLoS Comput Biol_ 2009, 5(4):e1000350. * [9] Voevodski K, Teng S, Xia Y: Spectral affinity in protein networks. _BMC Syst Biol_ 2009, 3:112. * [10] Kim YA, Przytycki JH, Wuchty S, Przytycka TM: Modeling information flow in biological networks. _Phys Biol_ 2011, 8(3):035012. * [11] Stojmirović A, Yu YK: Information flow in interaction networks. _J Comput Biol_ 2007, 14(8):1115–43. * [12] Stojmirović A, Yu YK: Information flow in interaction networks II: channels, path lengths and potentials. _J Comput Biol_ 2012\. in press. * [13] Stojmirović A, Yu YK: ITM Probe: analyzing information flow in protein networks. _Bioinformatics_ 2009, 25(18):2447–9. * [14] ITM Probe Web Service [http://www.ncbi.nlm.nih.gov/CBBresearch/Yu/mn/itm_probe]. * [15] Gansner ER, North SC: An open graph visualization system and its applications to software engineering. _Software — Practice and Experience_ 2000, 30(11):1203–1233. * [16] Java [http://www.java.com]. * [17] Jones E, Oliphant T, Peterson P, et al.: SciPy: Open source scientific tools for Python 2001–. [http://www.scipy.org/]. * [18] Demmel JW, Eisenstat SC, Gilbert JR, Li XS, Liu JWH: A supernodal approach to sparse partial pivoting. _SIAM J. Matrix Analysis and Applications_ 1999, 20(3):720–755. * [19] Davis TA: Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. _ACM Trans. Math. Softw._ 2004, 30(2). * [20] SciKits [http://scikits.appspot.com/]. * [21] Harrower M, Brewer C: ColorBrewer.org: An Online Tool for Selecting Colour Schemes for Maps. _Cartogr J_ 2003, 40:27–37. * [22] Ashburner M, Ball CA, Blake JA, Botstein D, Butler H, Cherry JM, Davis AP, Dolinski K, Dwight SS, Eppig JT, Harris MA, Hill DP, Issel-Tarver L, Kasarskis A, Lewis S, Matese JC, Richardson JE, Ringwald M, Rubin GM, Sherlock G: Gene ontology: tool for the unification of biology. The Gene Ontology Consortium. _Nat Genet_ 2000, 25:25–29. * [23] Smoot M, Ono K, Ideker T, Maere S: PiNGO: a Cytoscape plugin to find candidate genes in biological networks. _Bioinformatics_ 2011, 27(7):1030–1. * [24] Stojmirovic A, Bliskovsky A, Yu YK: CytoSaddleSum: a functional enrichment analysis plugin for Cytoscape based on sum-of-weights scores. _Bioinformatics_ 2012\. [Doi://10.1093/bioinformatics/bts041]. * [25] Bader GD, Hogue CWV: An automated method for finding molecular complexes in large protein interaction networks. _BMC Bioinformatics_ 2003, 4:2. * [26] Whelan C, Sönmez K: Computing graphlet signatures of network nodes and motifs in Cytoscape with GraphletCounter. _Bioinformatics_ 2012, 28(2):290–1. * [27] Milenković T, Przulj N: Uncovering biological network function via graphlet degree signatures. _Cancer Inform_ 2008, 6:257–73.	arxiv-papers	2011-11-30T22:10:50	2024-09-04T02:49:24.861303	{ "license": "Public Domain", "authors": "Aleksandar Stojmirovi\\'c, Alexander Bliskovsky and Yi-Kuo Yu", "submitter": "Aleksandar Stojmirovi\\'c", "url": "https://arxiv.org/abs/1112.0045" }
1112.0051	# QCD measurements at the Tevatron Dmitry Bandurin (for the D0 and CDF Collaborations) Florida State University Department of Physics Tallahassee FL 32306 USA ###### Abstract Selected quantum chromodynamics (QCD) measurements performed at the Fermilab Run II Tevatron $p\bar{p}$ collider running at $\sqrt{s}=1.96$ TeV by CDF and D0 Collaborations are presented. The inclusive jet, dijet production and three-jet cross section measurements are used to test perturbative QCD calculations, constrain parton distribution function (PDF) determinations, and extract a precise value of the strong coupling constant, $\alpha_{s}(m_{Z})=0.1161^{+0.0041}_{-0.0048}$. Inclusive photon production cross-section measurements reveal an inability of next-to-leading-order (NLO) perturbative QCD (pQCD) calculations to describe low-energy photons arising directly in the hard scatter. The diphoton production cross-sections check the validity of the NLO pQCD predictions, soft-gluon resummation methods implemented in theoretical calculations, and contributions from the parton-to- photon fragmentation diagrams. Events with $W/Z$+jets productions are used to measure many kinematic distributions allowing extensive tests and tunes of predictions from pQCD NLO and Monte-Carlo (MC) event generators. The charged- particle transverse momenta ($p_{T}$) and multiplicity distributions in the inclusive minimum bias events are used to tune non-perturbative QCD models, including those describing the multiple parton interactions (MPI). Events with inclusive production of $\gamma$ and 2 or 3 jets are used to study increasingly important MPI phenomenon at high $p_{T}$, measure an effective interaction cross section, $\sigma_{\rm eff}=16.4\pm 2.3$ mb, and limit existing MPI models. ## 1 Introduction QCD, the theory of the strong interaction between quarks and gluons, is heavily tested in experimental studies at hadron colliders. QCD results from the CDF and D0 collaborations obtained with integrated luminosity up to 8 fb-1 are reviewed in this paper. These results provide a crucial tests for pQCD, PDFs, the strong coupling constant, non-perturbative models describing parton fragmentation, and MPI phenomena. At the same time, the results are used to search for new phenomena and impose limits on the corresponding models. The performed extensive studies prepare a solid base for the LHC era of $pp$ collisions. ## 2 Jet production Thorough testing of pQCD at short distances is provided through measurements of differential inclusive jet, dijet and three-jet cross sections. The measurements of the inclusive jet cross sections done by the D0 [1] and CDF [2] collaborations are in agreement with pQCD predictions in a few jet rapidity regions. However, data with uncertainties lower than theoretical (mostly PDF) ones, favor a smaller gluon content at high Feynman $x$ ($>$0.2). The D0 inclusive jet data has also been used to extract values of the strong coupling constant $\alpha_{s}$ in the interval of $50<p_{T}^{\rm jet}<145$ GeV [3]. The best fit over 22 data points leads to $\alpha_{s}(m_{Z})=0.1161^{+0.0041}_{-0.0048}$ with improved accuracy from the Run I CDF result [4] and also in agreement with result from HERA jet data [5]. The inclusive dijet cross sections have been measured in the D0 [6] and CDF [7] experiments. Both measurements cover the mass range up to about 1.2 TeV with good agreement with pQCD, and no indication of any new physics. CDF imposed limits on some models with exotic particles decaying into two jets [7]. D0 results are compared to mstw2008 [9] and cteq6.6m [8] PDF sets. They are in a better agreement with mstw2008 and are systematically lower than the central pQCD prediction at high rapidities ($1.2<\|y\|<2.4$). The D0 collaboration has also measured the three-jet mass cross sections using jets with leading (in $p_{T}$) jet $p_{T}>150$ GeV, and considering three regions with different lower cut on the 3rd jet $p_{T}$ ($40,70,100$ GeV) and three different jet rapidity regions ($\|y\|<2.4,1.6,0.8$) [10]. Results are shown in Fig. 2 and in agreement with NLO pQCD predictions, which use mstw2008, ct10 [11], nnpdf2.1 [12], hera1.0 [13] and abkm09 [14] PDF sets. Results favor more mstw2008 and nnpdf2.1 PDFs. D0 has also presented a measurement of the ratio of the inclusive 3-jet to 2-jet production cross sections [15]. The shape of the ratio is well described by NLO QCD and is practically independent of the PDF set. The results can potentially be used to test the running of $\alpha_{s}$ up to a $p_{T}$ scale of 500 GeV. The CDF collaboration studied structure of high $p_{T}$ jets by selecting only events with at least one jet having $p_{T}>400$ GeV, $0.1<\|y\|<0.7$ and considering jets with cone sizes $R=0.4,0.7$ and $1.0$ [16]. Such studies can be used to tune parton showering and search for heavy resonances decaying hadronically. The jet mass is calculated using 4-vectors of calorimeter towers in a jet. Fig. 3 shows the jet mass distribution for $R=0.7$ at high masses. The data are in agreement pythia predictions and interpolate between the QCD LLA predictions [17] for quark and gluon jets, and confirm that the high mass jets are mostly caused by quark fragmentation. Figure 1: Ratios of data/theory for the dijet mass cross sections measured in D0 and CDF are shown on the left and right plots. Figure 2: The 3-jet mass differential cross section in the three jet rapidity intervals are compared to NLO pQCD with different PDF sets. Figure 3: Distribution of jet mass for jets with $R=0.7$; black crosses are data, red dashed is QCD MC, theoretical “all quarks” and “all gluons” curves are presented as well; the inset plot compares the results with Midpoint/SC and Anti-kT jet algorithms. ## 3 $W/Z$ \+ jets production Both collaborations have extensively studied the $W/Z$ \+ jet productions since these events are the main background to top-quark, Higgs boson, SUSY and many other new physics production channels. In this section we review some of the latest results. Fig. 4 shows the inclusive cross section for $Z/\gamma^{\ast}$+jets production measured by CDF [18] as a function of dijet mass and jet multiplicity. The measurements are compared to LO and NLO pQCD predictions obtained with MCFM [19] and are in good agreement with the NLO theory predictions. D0 measured jet $p_{T}$ inclusive cross sections of $W+n$-jet production for jet multiplicities $n=1-4$ [20]. The measurements are compared to the NLO predictions for $n=1-3$ and to LO predictions for $n=4$. The measured cross sections are generally found to agree with the NLO calculation although certain regions of phase space are identified where the calculations could be improved. D0 recently published the measured cross section ratio $\sigma(Z+b)/\sigma(Z+{\rm jet})=0.0193\pm 0.0022({\rm stat})\pm 0.0015({\rm syst})$ for events with jet $p_{T}>20$ GeV and $\|\eta\|<2.5$ [21]. This most precise measurement of the $Z+b$ fraction is consistent with the NLO theory prediction, $0.0192\pm 0.0022$, (done with MCFM, renormalization and factorization scales set at $m_{Z}$) and the CDF result [22]. the CDF collaboration measured the cross section of $W+b$-jet production $\sigma(W+b)\cdot Br(W\to l\nu)=2.74\pm 0.27({\rm stat})\pm 0.42({\rm syst})$ pb with jet $p_{T}>20$ GeV, $\|\eta\|<2.0$ and $l=e,\mu$. The measurement significantly exceeds the NLO prediction $1.2\pm 0.14$ pb. Figure 4: Left: measured inclusive cross section for $Z/\gamma^{}$+jets production as a function of dijet mass compared to NLO pQCD predictions as determined using MCFM. Right: measured cross section as a function of inclusive jet multiplicity compared to LO and NLO pQCD predictions as determined using MCFM. Figure 5: Left: measured $W+n$ jet differential cross section as a function of jet $p_{T}$ for $n=1-4$, normalized to the inclusive $W\to e\nu$ cross section. The $W+1$ jet inclusive spectra are shown by the top curve, the $W+4$ jet inclusive spectra by the bottom curve. The measurements are compared to the fixed-order NLO predictions for $n=1-3$ and to LO predictions for $n=4$. Right: (a) total inclusive $n$-jet cross sections $\sigma_{n}$ as a function of $n$, (b) the ratio of the theory predictions to the measurements, and (c) $\sigma_{n}/\sigma_{n-1}$ ratios for data, Blackhat+Sherpa and Rocket+MCFM. The hashed areas represent the theoretical uncertainty arising from the choice of renormalization and factorization scale. Figure 6: Left (D0): the distributions of the $b$, $c$, light jets and data over the $b-$jet discriminant; MC templates are weighted by the fractions found from the fit to data. Right (CDF): the secondary vertex mass fit for the tagged jets in the selected sample. ## 4 Photon production Since high $p_{T}$ photons emerge directly from $p\bar{p}$ collisions and provide a direct probe of the parton hard scattering dynamics, they are of permanent interest in high energy physics. The inclusive photon production cross sections measured by D0 and CDF in the central rapidity region [24, 25] are in agreement within experimental uncertainties and both indicate a difficulty for NLO pQCD to describe the low $p_{T}$ behavior. In light of the Higgs boson search and other possible resonances decaying to a photon pair, both the collaborations performed a thorough study of the diphoton production. D0 measured the diphoton cross sections (see Fig. 7) as a function of the diphoton mass $M_{\gamma\gamma}$, the transverse momentum of the diphoton system $p_{T}^{\gamma\gamma}$, the azimuthal angle between the photons $\Delta\phi_{\gamma\gamma}$, and the polar scattering angle of the photons. The latter three cross sections are also measured in the three $M_{\gamma\gamma}$ bins, $30-50,50-80$ and $80-350$ GeV. The measurements are compared to NLO QCD and pythia [32] predictions. The results show that the largest discrepancies between data and NLO predictions for each of the kinematic variables originate from the lowest $M_{\gamma\gamma}$ region ($M_{\gamma\gamma}$ $<50$ GeV), where the contribution from $gg\to\gamma\gamma$ is expected to be largest [26]. The discrepancies between data and the theory predictions are reduced in the intermediate $M_{\gamma\gamma}$ region, and a quite satisfactory description of all kinematic variables is achieved for the $M_{\gamma\gamma}$$>80$ GeV region, the relevant region for the Higgs boson and new phenomena searches. The CDF collaboration measured the diphoton production cross sections functions of $M_{\gamma\gamma}$, $p_{T}^{\gamma\gamma}$and $\Delta\phi_{\gamma\gamma}$. They are shown in Fig. 8. None of the models describe the data well in all kinematic regions, in particular at low diphoton mass ($M_{\gamma\gamma}$$<60$ GeV), low $\Delta\phi_{\gamma\gamma}$($<1.7$ rad) and moderate $p_{T}^{\gamma\gamma}$($20-50$ GeV). Figure 7: The measured differential diphoton production cross sections as functions of (a) $M_{\gamma\gamma}$, (b) $p_{T}^{\gamma\gamma}$and (c) $\Delta\phi_{\gamma\gamma}$ in D0 experiment. The data are compared to the theoretical predictions from resbos, diphox, and pythia. The ratio of differential cross sections between data and resbos are displayed as black points with uncertainties in the bottom plots. The solid (dashed) line shows the ratio of the predictions from diphox (pythia) to those from resbos. In the bottom plots, the scale uncertainties are shown by dash-dotted lines and the PDF uncertainties by shaded regions. Figure 8: The measured differential diphoton production cross sections as functions of (from left to right) $M_{\gamma\gamma}$, $p_{T}^{\gamma\gamma}$and $\Delta\phi_{\gamma\gamma}$ in CDF experiment. ## 5 Multiple parton interactions The CDF and D0 collaborations comprehensively studied the phenomenon of MPI events in a few Run II measurements. In this section we mention some of the recently published results. CDF studied charged-particle $p_{T}$ sum densities and multiplicities in Drell-Yan and jet events [28]. Specifically, both distributions have been analyzed in the three regions: towards the total lepton pair (Z-boson) $\vec{p}_{T}$, opposite to this direction (“away” region) and in the region transverse to the lepton pair/jet $\vec{p}_{T}$. The charged-particle $p_{T}$ sum density in the three regions in the Drell-Yan events is shown on the left plot of Fig. 9 as a function of the lepton pair $p_{T}$. The same quantity is also plotted for the transverse region on the right plot. One can see a similar trend in both Drell-Yan and jet events which can be considered as MPI universality. The tuned pythia describes the data very well. Figure 9: Left: the comparison of the total charged-particle $p_{T}$ sum density $dp_{T}/d\eta d\phi$ in the three regions in the Drell-Yan events: “transverse”, “away” and “toward”. Right: the $p_{T}$ sum density in the transverse region in Drell-Yan and jet production as a function of the lepton pair or leading jet $p_{T}$. D0 has studied events with double parton (DP) scattering in $\gamma+3$ jet events [29], in which two pairs of partons undergo two hard interactions in a single $p\bar{p}$ collision. The DP events can be a background to many rare processes but they also provide insight into the spatial distribution of partons in the colliding hadrons. D0 measured the so-called effective cross section, that characterizes rates of the DP events, $\sigma^{\gamma j,jj}_{\rm DP}=\sigma^{\gamma j}\sigma^{jj}/\sigma_{\rm eff}$. The measurement was done in the three bins of the 2nd (ordered in $p_{T}$) jet $p_{T}$. The results are shown in Fig. 11. Using these three (almost uncorrelated0 points, the obtained average effective cross section is $\sigma_{\rm eff}=16.4\pm 0.3({\rm stat})\pm 2.3({\rm syst})$. It is in agreement with the previous CDF result [30]. To tune MPI models, D0 also measured cross sections for the azimuthal angle defined between the $p_{T}$ vectors of the $\gamma+$jet and dijet systems in the three $p_{T}$ bins of the 2nd jet $p_{T}$ [31]. Comparison of data with a few MPI and two “no MPI” models are shown in Fig. 11. One can see that data clearly contain DP events and favor more Perugia MPI tunes. Figure 10: The measured effective cross section vs 2nd jet $p_{T}$. Figure 11: Left: Normalized differential cross section in the $\gamma+3$-jet + X events, $(1/\sigma_{\gamma 3j})\sigma_{\gamma 3j}/d\Delta S$, in data compared to MC models and the ratio of data over theory, only for models including MPI, in the range $15<p_{T}^{jet2}<30$ GeV. Right: Normalized differential cross section in $\gamma+2$-jet + X events, $(1/\sigma_{\gamma 2j})\sigma_{\gamma 2j}/d\Delta\phi$, in data compared to MC models and the ratio of data over theory, only for models including MPI, in the range $15<p_{T}^{jet2}<20$ GeV. ## 6 Summary The Tevatron experiments provide precision QCD measurements of many fundamental observables. In most cases, the results are mutually consistent and/or complementary to each other. Jet measurements show good agreement with pQCD, sensitivity to PDF sets, the strongest constraint on high-$x$ gluon PDF, provide detailed studies of different jet algorithms, are used to extract $\alpha_{s}$, study jet substructure, and provide limits on many new phenomena models. The $W/Z+$jets results provide extensive tests of pQCD and tune existing MC models. The photon results test fixed order NLO pQCD predictions accounting for resummation and fragmentation effects and show that the theory should be better understood. Measurements of underlying/MPI events impose strong constraints and improve phenomenological MPI models at low and high $p_{T}$ regimes. ## References [1] V. M. Abazov et al. (D0), Phys. Rev. Lett. 101, 062001 (2008). * [2] T. Aaltonen et al. (CDF), Phys.Rev. D 78, 052006 (2008); Phys.Rev. D 75, 092006 (2007). * [3] V. M. Abazov et al. (D0), Phys. Rev. D 80, 111108 (2009). * [4] T. Affolder et al. (CDF), Phys. Rev. Lett. 88, 042001 (2002). * [5] S. Bethke, Eur. Phys. J. C 64, 689 (2009). * [6] V. M. Abazov et al. (D0), Phys. Lett. B 693, 531 (2010). * [7] T. Aaltonen et al. (CDF), Phys.Rev. D 79, 112002 (2009). * [8] P. M. Nadolsky et al., Phys. Rev. D 78 013004 (2008). * [9] A. D. Martin, W.J. Stirling, R.S. Thorne, G. Watt, Eur. Phys. J. C 63, 189 (2009). * [10] V. M. Abazov et al. (D0), Phys. Lett. B 704, 434 (2011), arXiv:1104.1986. * [11] H. L. Lai et al., Phys. Rev. D 82, 074024 (2010). * [12] R. D. Ball et al., arXiv:1101.1300 [hep-ph]. * [13] F. D. Aaron et al., (H1 and ZEUS), JHEP 01, 109 (2010). * [14] S. Alekhin, J. Blumlein, S. Klein and S. Moch, Phys. Rev. D 81 014032 (2010), arXiv:0908.2766. * [15] V. M. Abazov et al. (D0), D0 Note 6032-CONF. * [16] T. Aaltonen et al. (CDF), CDF Note CDF-10199 (2011). * [17] L. G. Almeida et al., Phys. Rev. D 79 074017 (2009). * [18] $http://www-cdf.fnal.gov/physics/new/qcd/QCD.html$ * [19] R. K. Ellis et al., J. High Energy Phys. 01, 012 (2009). * [20] V. M. Abazov et al. (D0), Phys. Lett. B 705, 200 (2011), arXiv:1106.1457. * [21] V. M. Abazov et al. (D0), Phys. Lett. B 682, 370 (2010). * [22] T. Aaltonen et al. (CDF), Phys.Rev. D 79, 052008 (2009). * [23] T. Aaltonen et al. (CDF), Phys.Rev.Lett. 10, 131801 (2010). * [24] V.M. Abazov et al. (D0), Phys.Lett. B 639, 151 (2006); ibid. 658, 285 (2008). * [25] T. Aaltonen et al. (CDF), Phys. Rev. D 80, 111106 (2009). * [26] V. M. Abazov et al. (D0), Phys. Lett. B 690, 108 (2010) and references therein. * [27] T. Aaltonen et al. (CDF), CDF Note CDF-10160 (2011). * [28] T. Aaltonen et al. (CDF), Phys.Rev. D 82, 034001 (2010). * [29] V. M. Abazov et al. (D0), Phys.Rev. D 81, 052012 (2010). * [30] F. Abe et al. (CDF), Phys.Rev. D 56, 3811 (1997). * [31] V. M. Abazov et al. (D0), Phys.Rev. D 83, 052008 (2011). * [32] T. Sjostrand, S. Mrenna, P. Z. Skands, JHEP 0605, 026 (2006).	arxiv-papers	2011-11-30T23:12:38	2024-09-04T02:49:24.872809	{ "license": "Public Domain", "authors": "Dmitry Bandurin (for the D0 and CDF Collaborations)", "submitter": "Dmitry Bandurin V", "url": "https://arxiv.org/abs/1112.0051" }
1112.0343	# Ontological Queries: Rewriting and Optimization (Extended Version)111This is an extended and revised version of the paper [1]. Georg Gottlob1,2, Giorgio Orsi1,3, Andreas Pieris1 $~{}^{1}$Department of Computer Science, University of Oxford, UK $~{}^{2}$Oxford-Man Institute of Quantitative Finance, University of Oxford, UK $~{}^{3}$Institute for the Future of Computing, University of Oxford, UK {georg.gottlob,giorgio.orsi,andreas.pieris}@cs.ox.ac.uk ###### Abstract Ontological queries are evaluated against an ontology rather than directly on a database. The evaluation and optimization of such queries is an intriguing new problem for database research. In this paper we discuss two important aspects of this problem: query rewriting and query optimization. Query rewriting consists of the compilation of an ontological query into an equivalent query against the underlying relational database. The focus here is on soundness and completeness. We review previous results and present a new rewriting algorithm for rather general types of ontological constraints. In particular, we show how a conjunctive query against an ontology can be compiled into a union of conjunctive queries against the underlying database. Ontological query optimization, in this context, attempts to improve this process so to produce possibly small and cost-effective UCQ rewritings for an input query. We review existing optimization methods, and propose an effective new method that works for linear Datalog±, a class of Datalog-based rules that encompasses well-known description logics of the _DL-Lite_ family. ## 1 Introduction This paper is about ontological query processing, an important new challenge to database research. We will review existing methods and propose new algorithms for compiling an ontological query, that is, a query against an ontology on top of a relational database, into a direct query against this database, and we will deal with optimization issues related to this process so as to obtain possibly small and efficient compiled queries. In this section, we first discuss a number of relevant concepts, and then illustrate query rewriting and optimization processes in the context of a small but non-trivial example. #### Ontologies. The use of ontologies and ontological reasoning in companies, governmental organizations, and other enterprises has become widespread in recent years. An ontology is an explicit specification of a conceptualization of an area of interest [2], and consists of a formal representation of knowledge as a set of concepts within a domain, and the relationships between those concepts [3]. To distinguish an enterprise ontology from a data dictionary, Dave McComb explicitly refers to the formal semantics of ontologies that enables automated processing and inferencing, while the interpretation of a data dictionary is strictly done by humans [4]. Moreover, ontologies have been adopted as high- level conceptual descriptions of the data contained in data repositories that are sometimes distributed and heterogeneous in the data models. Due to their high expressive power, ontologies are also substituting more traditional conceptual models such as UML class-diagrams and E/R schemata. #### Description Logics. Description logics (DLs) are logical languages for expressing and modelling ontologies. The best known DLs are those underlying the _OWL_ language222http://www.w3.org/TR/owl2-overview/. The main ontological reasoning and query answering tasks in the complete OWL language, called _OWL Full_ , are undecidable. For the most well-known decidable fragments of OWL, ontological reasoning and query answering is still computationally very hard, typically 2exptime-complete. In description logics, the ontological axioms are usually divided into two sets: The ABox (assertional box), which essentially contains factual knowledge such as “IBM is a company”, denoted by $\mathit{company}(\mathit{ibm})$, or “IBM is listed on the NASDAQ”, which could be represented as a fact of the form $\mathit{list\\_comp}(\mathit{ibm},\mathit{nasdaq})$, and a TBox (terminological box) which contains axioms and constraints that allow us, on the one hand, to infer new facts from those given in the ABox, and, on the other hand, to express restrictions such as keys. For example, a TBox may contain an axiom stating that for each fact $\mathit{list\\_comp}(X,Y)$, $Y$ must be a financial index, which in DL is expressed as $\exists\mathit{list\\_comp}^{-}\sqsubseteq\mathit{fin\\_idx}$. If the fact $\mathit{fin\\_idx}(\mathit{nasdaq})$ is not already present in the ABox, it can be derived via the above axiom from $\mathit{list\\_comp}(\mathit{ibm},\mathit{nasdaq})$. Thus, the atomic query “$q(X)\leftarrow\mathit{fin\\_idx}(X)$” would return $\mathit{nasdaq}$ as one of the answers. Note that the axiom $\exists\mathit{list\\_comp}^{-}\sqsubseteq\mathit{fin\\_idx}$, which corresponds to an inclusion dependency, is enforced by adding new tuples, rather than just being checked. This is one main difference between ontological constraints and classical database dependencies. In database terms, the above axiom is to be interpreted more like a trigger than a classical constraint. #### Ontology Based Data Access (OBDA). We are currently witnessing the marriage of ontological reasoning and database technology. In fact, this amalgamation consists in the realization of the obvious idea that ABoxes shall be implemented in form of a relational database, or even stored in classical RDBMSs. Moreover, very large existing databases are semantically enriched with ontological constraints. There are a number of recent commercial systems and experimental prototypes that extend RDBMSs with the possibility of querying an ontology that is rooted in a database (for examples, see Section 2). The main problem here is how to couple these two different types of technology smoothly and efficiently, and this is also the main theme of the present paper. One severe obstacle to efficient OBDA is the already mentioned high computational complexity of query answering with description logics. The situation clearly worsens when the ABoxes of enterprise ontologies are very large databases. To tackle this problem, new, lightweight DLs have been designed, that guarantee polynomial-time data complexity for conjunctive query answering. This means that based on a fixed TBox, a fixed query can be answered in polynomial time over variable databases. The best-known and best- studied examples of such lightweight DLs are the _DL-Lite_ [5] and $\mathcal{E}\mathcal{L}$ (see, e.g., [6]) families. These languages can be considered tractable subclasses of OWL. It was convincingly argued that simple DLs such as DL-Lite or $\mathcal{E}\mathcal{L}$ are sufficient for modelling an overwhelming number of applications. More recently, the Datalog± family of description logics was introduced [7, 8, 9, 10]. Its syntax is based on classical first-order logic, more specifically, on variants of the well-known Datalog language [11]. The basic Datalog± rules are known as _tuple-generating dependencies_ (TGDs) in the database literature [12]. Tractable DLs in this framework are guarded Datalog±, which is noticeably more general than both DL-Lite and $\mathcal{E}\mathcal{L}$, and the DLs linear Datalog± and sticky-join Datalog±, which both encompass DL- Lite. Besides being more expressive than DL-Lite, suitable Datalog± languages offer a more compact representation of the attributes of concepts and roles, since description logics are usually restricted to unary and binary predicates only. Consider, as an example, a relation $\mathit{stock}(\underline{{\sf id}},{\sf name},{\sf unit}$-${\sf price})$. Representing this relation in DL would require the introduction of a concept symbol $\mathit{stock}$, and of three attribute symbols $\mathit{id}$, $\mathit{name}$ and $\mathit{unit}$-$\mathit{price}$. These entities must be then weaved together by the TBox formula $\mathit{stock}\sqsubseteq\exists id\sqcap\exists\mathit{name}\sqcap\exists\mathit{unit}$-$\mathit{price}$. Datalog± represents the relation in a natural way by means of a ternary predicate $\mathit{stock}$. In the same way, Datalog± provides a more natural syntax for many other DL formulae; for example, an inverse role assertion $r\sqsubseteq s^{-}$ is represented as a (full) TGD $r(X,Y)\rightarrow s(Y,X)$, while an existential restriction $p\sqsubseteq\exists r.q$ is represented as a (partial) TGD $p(X)\rightarrow\exists Y\,r(X,Y),q(Y)$. #### First-Order Rewritability. Polynomial-time tractability is often considered not to be good enough for efficient query processing. Ideally, one would like to achieve the same complexity as for processing SQL queries, or, equivalently, first-order (FO) queries. An ontology language $\mathcal{L}$ is _first-order rewritable_ if, for every TBox $\Sigma$ expressed in $\mathcal{L}$ and a query $q$, a first- order query $q_{\Sigma}$ (called the perfect rewriting) can be constructed such that, given a database $D$, $q_{\Sigma}$ evaluated over $D$ yields exactly the same result as $q$ evaluated against $D$ and $\Sigma$. Since answering first-order queries is in the class ac0 in data complexity [13], it immediately follows that under FO-rewritable TGDs, query answering is also in ac0 in data complexity This notion was first introduced by Calvanese et al. [5] in the concept of description logics. If a DL guarantees the FO-rewritability of each query under every TBox, we simply say that the logic is FO-rewritable. FO- rewritability is a most desirable property since it ensures that the reasoning process can be largely decoupled from data access. In fact, to answer query $q$, a separate software can compile $q$ into $q_{\Sigma}$, and then just submit $q_{\Sigma}$ as a standard SQL query to the DBMS holding $D$, where it is evaluated and optimized in the usual way. Excitingly, it was shown that the members of the DL-Lite family, as well as the slightly more expressive language linear Datalog± are FO-rewritable. Moreover, even the much more expressive language of sticky-join Datalog± is FO-rewritable. For these languages, a pair $\langle\Sigma,q\rangle$, where $q$ is a CQ, is rewritten as an SQL expression equivalent to a UCQ $q_{\Sigma}$. The research challenge we address in this paper is precisely the question of how to rewrite $\langle\Sigma,q\rangle$ to $q_{\Sigma}$ correctly and efficiently. Let us illustrate this process by a small, but comprehensive example. Consider the following relational schema $\mathcal{R}$ representing financial information about companies and their stocks: $\begin{array}[]{rcl}&&\mathit{stock}({\sf id},{\sf name},{\sf unit}$-${\sf price})\\\ &&\mathit{company}({\sf name},{\sf country},{\sf segment})\\\ &&\mathit{list\\_comp}({\sf stock},{\sf list})\\\ &&\mathit{fin\\_idx}({\sf name},{\sf type},{\sf ref}$-${\sf mkt})\\\ &&\mathit{stock\\_portf}({\sf company},{\sf stock},{\sf qty}).\end{array}$ The $\mathit{stock}$ relation contains information about stocks such as the name, and the price per unit. The relation $\mathit{company}$ contains information about companies; in particular, the name, the country, and the market segment of a company. The relation $\mathit{list\\_comp}$ relates a stock to a financial index (i.e., NASDAQ, FTSE, NIKKEI) represented by the relation $fin\\_idx$ which, in turn, contains information about the types of stocks in the index, and the reference market (e.g., London Stock Exchange). Finally, $\mathit{stock\\_portf}$ relates companies to their stocks along with an indication of the amount of the investment. Datalog± provides the necessary expressive power to extend $\mathcal{R}$ with ontological constraints in an easy and intuitive way. Examples of such constraints follow: $\begin{array}[]{rcl}\sigma_{1}&:&\mathit{stock\\_portf(X,Y,Z)}\rightarrow\exists V\exists W\ \mathit{company(X,V,W)}\\\ \sigma_{2}&:&\mathit{stock\\_portf(X,Y,Z)}\rightarrow\exists V\exists W\ \mathit{stock(Y,V,W)}\\\ \sigma_{3}&:&\mathit{list\\_comp(X,Y)}\rightarrow\exists Z\exists W\ \mathit{fin\\_idx(Y,Z,W)}\\\ \sigma_{4}&:&\mathit{list\\_comp(X,Y)}\rightarrow\exists Z\exists W\ \mathit{stock(X,Z,W)}\\\ \sigma_{5}&:&\mathit{stock\\_portf(X,Y,Z)}\rightarrow\mathit{has\\_stock(Y,X)}\\\ \sigma_{6}&:&\mathit{has\\_stock(X,Y)}\rightarrow\exists Z\ \mathit{stock\\_portf(Y,X,Z)}\\\ \sigma_{7}&:&\mathit{stock(X,Y,Z)}\rightarrow\exists V\exists W\ \mathit{stock\\_portf(V,X,W)}\\\ \sigma_{8}&:&\mathit{stock(X,Y,Z)}\rightarrow\mathit{fin\\_ins(X)}\\\ \sigma_{9}&:&\mathit{company(X,Y,Z)}\rightarrow\mathit{legal\\_person(X)}\\\ \delta_{1}&:&\mathit{legal\\_person(X,Y,Z),fin\\_ins(X,V,W)\rightarrow\bot}.\end{array}$ Figure 1: A (partial) rewriting for the Stock Exchange example. $q^{[0]}(A,B,C)\leftarrow\mathit{fin\\_ins}(A),\mathit{stock\\_portf}(B,A,D),\mathit{company}(B,E,F),\mathit{list\\_comp}(A,C),\mathit{fin\\_idx}(C,G,H)$ --- $q^{[1]}(A,B,C)\leftarrow\mathit{fin\\_ins}(A),\underline{\mathit{has\\_stock}(A,B)},\mathit{company}(B,E,F),\mathit{list\\_comp}(A,C),\mathit{fin\\_idx}(C,G,H)$ $q^{[2]}(A,B,C)\leftarrow\mathit{fin\\_ins}(A),\mathit{has\\_stock}(A,B),\underline{\mathit{stock\\_portf}(B,E,F)},\mathit{list\\_comp}(A,C),\mathit{fin\\_idx}(C,G,H)$ $q^{[3]}(A,B,C)\leftarrow\underline{\mathit{stock}(A,J,K)},\mathit{has\\_stock}(A,B),\mathit{stock\\_portf}(B,E,F),\mathit{list\\_comp}(A,C),\mathit{fin\\_idx}(C,G,H)$ $\ldots$ The first four TGDs set the “domain” and the “range” of the $\mathit{stock\\_portf}$ and $\mathit{list\\_comp}$ relations, respectively. TGDs $\sigma_{5}$ and $\sigma_{6}$ assert that $\mathit{stock\\_portf}$ and $\mathit{has\\_stock}$ are “inverse relations”, while $\sigma_{7}$ expresses that each stock must belong to some stock portfolio. TGDs $\sigma_{8}$ and $\sigma_{9}$ model taxonomic relationships such as the facts that each stock is a financial instrument, and each company is a legal person. Finally, the negative constraint $\delta_{1}$ (where $\bot$ denotes the truth constant $\mathit{false}$) states that legal persons and financial instruments are disjoint sets. Consider now the following conjunctive query $q$ asking for all the triples $\langle a,b,c\rangle$, where $a$ is a financial instrument owned by the company $b$ and listed on $c$: $\begin{array}[]{rcl}\mathit{q(A,B,C)}&\leftarrow&\mathit{fin\\_ins}(A),\mathit{stock\\_portf}(B,A,D),\mathit{company}(B,E,F),\\\ &&\mathit{list\\_comp}(A,C),\mathit{fin\\_idx}(C,G,H).\end{array}$ Since $\Sigma=\\{\sigma_{1},\ldots,\sigma_{9}\\}$ is a set of linear TGDs, i.e., TGDs with single body-atom, query answering under $\Sigma$ is FO- rewritable. Thus, it is possible to reformulate $\langle\Sigma,q\rangle$ to a first-order query $q_{\Sigma}$ such that, for every database $D$, $D\cup\Sigma\models q$ iff $D\models q_{\Sigma}$. A naive rewriting procedure would use the TGDs of $\Sigma$ as rewriting rules for the atoms in $q$ to generate all the CQs of the perfect rewriting. Figure 1 shows a (partial) rewriting for $q$, where the query obtained at the $i$-th step is denoted as $q^{[i]}$, and the newly introduced atoms are underlined. In particular, $q^{[0]}$ is the given query $q$, $q^{[1]}$ is obtained from $q^{[0]}$ by using $\sigma_{6}$, $q^{[2]}$ is obtained from $q^{[1]}$ by applying $\sigma_{1}$, and $q^{[3]}$ is obtained from $q^{[2]}$ by using $\sigma_{8}$. The complete perfect rewriting contains more than 200 queries executing more than 1000 joins. However, by exploiting the set of constraints, it is possible to eliminate redundant atoms in the generated queries, and thus reduce the number of the CQs in the rewritten query. For example, in the given query $q$ above it is possible to eliminate the atom $\mathit{fin\\_ins(A)}$ since, due to the existence of the TGDs $\sigma_{2}$ and $\sigma_{8}$ in $\Sigma$, if the atom $\mathit{stock\\_portf}(B,A,D)$ is satisfied, then immediately the atom $\mathit{fin\\_ins(A)}$ is also satisfied. Notice that by eliminating a redundant atom from a query, we also eliminate all the atoms that are generated starting from it during the rewriting process. Moreover, due to the TGD $\sigma_{3}$, if the atom $\mathit{list\\_comp(A,C)}$ in $q$ is satisfied, then the atom $\mathit{fin\\_idx(C,G,H)}$ is also satisfied, and therefore can be eliminated. Finally, due to the TGD $\sigma_{1}$, if the atom $\mathit{stock\\_portf}(B,A,D)$ is satisfied, then the atom $\mathit{company(B,E,F)}$ is also satisfied, and hence is redundant. The query that has to be considered as input of the rewriting process is therefore $q(A,B,C)\leftarrow\mathit{stock\\_portf}(B,A,D),\mathit{list\\_comp}(A,C)$ that produces a perfect rewriting containing the following two queries executing only two joins: $\begin{array}[]{rcl}q(A,B,C)&\leftarrow&\mathit{list\\_comp}(A,C),\mathit{stock\\_portf}(B,A,D)\\\ q(A,B,C)&\leftarrow&\mathit{list\\_comp}(A,C),\mathit{has\\_stock}(A,B).\end{array}$ #### Contributions and Roadmap. After a review of previous work on ontology based data access in the next section, and some formal definitions and preliminaries in Section 3, we present a short overview of the Datalog± family in Section 4. We then proceed with new research results. In Section 5, we propose a new rewriting algorithm that improves the one stated in [14] by substantially reducing the number of redundant queries in the perfect rewriting. In Section 6, we present a polynomial-time optimization strategy based on the early-pruning of redundant atoms produced during the rewriting process. An implementation and experimental evaluation of the new method is discussed in Section 7. We also discuss the relationship between our optimization technique and optimal query minimization algorithms such as the _chase & back-chase_ algorithm [15]. We conclude with a brief outlook on further research. ## 2 Ontology Based Data Access Answering queries under constraints and the related optimization techniques are important topics in data management beyond the obvious research interest. In particular, they are profitable opportunities for companies that need to deliver efficient and effective data management solutions to their customers. This trend is becoming even more evident as a plethora of robust systems and APIs for Semantic Web data management proposed in the recent years. These systems span from open-source solutions such as Virtuoso333http://virtuoso.openlinksw.com/, Sesame444http://www.openrdf.org/, RDFSuite [16], KAON555http://kaon.semanticweb.org/ and Jena666http://jena.sourceforge.net/, to commercial implementations such as the semantic extensions implemented in Oracle Database 11g R2 [17] and BigOWLLim777http://www.ontotext.com/owlim/. In this Section we briefly analyze the systems providing rewriting-based access to databases under ontological constraints, and we highlight some crucial points that we want to address in this work. We first present the class of constraints identified by the members of the DL- Lite family [5], namely, DL-LiteA, DL-LiteF, and DL-LiteR, underlying the W3C OWL-QL profile of the OWL language. These constraints correspond to unary and binary _inclusion dependencies_ combined with a restricted form of _key constraints_. In order to perform query answering under this class of constraints, a rewriting algorithm, introduced in [5] and implemented in the QuOnto system, reformulates the given query into unions of conjunctive queries. The size of the reformulated query is unnecessarily large due to a number of reasons. In the first place, _(i)_ basic optimization techniques such as the identification of the connected components in the body of the input query, or the computation of any form of query decomposition [18], are not applied. Moreover, _(ii)_ the fact that the given set of constraints can be used to identify existential joins in the reformulated query which can be eliminated is not exploited. Finally, _(iii)_ the factorization step (which is needed to guarantee completeness) is applied exhaustively, and as a result many superfluous queries are generated. Peréz-Urbina et al. [19] proposed an alternative resolution-based rewriting algorithm, implemented in the Requiem system, that addressed the issue of the useless factorizations (and therefore of the redundant queries generated due to this weakness) by directly handling existential quantification through proper functional terms. The algorithm has then been extended to more expressive DL languages [19]. In this case the output of the rewriting is a Datalog program. Rosati et al. [20] recently proposed a very sophisticated rewriting technique, implemented in the Presto system, that addresses some of the issues described above. In particular, _(i)_ the unnecessary existential joins are eliminated by resorting to the concept of _most-general subsumees_ , which also avoids the unnecessary factorizations, and _(ii)_ the connectivity of the given query is checked before executing the algorithm; in case the query is not connected, Presto splits the query in connected components and rewrites them separately. Notice that Presto produces a non-recursive Datalog program, and not a union of conjunctive queries. This allows the “hiding” of the exponential blow-up inside the rules instead of generating explicitly the disjunctive normal form. The final rewriting is exponential only in the number of non-eliminable existential joins, but not in the size of the input query. The approaches presented above have been proven very effective when applied to very particular classes of description logic constraints. Following a more general approach for ontological query answering, Calì et al. [14] presented a backward-chaining rewriting algorithm which is able to deal with arbitrary sets of TGDs, providing that the class of TGDs under consideration satisfies suitable syntactic restrictions that guarantee the termination of the algorithm. However, this algorithm is inspired by the original QuOnto algorithm and inherits all its drawbacks. Despite the possibly exponential number of queries to be constructed, we know that all such queries are independent from each other, and thus they can be easily executed in parallel threads and distributed on multiple processors. Notice that a non-recursive Datalog program is not equally easy to distribute. Moreover, the optimizations implemented in current DBMS systems for (unions of) conjunctive queries are much more advanced than those implemented for the positive existential first-order queries resulting from the translation of a non-recursive Datalog program into a concrete query language such as SQL. It is clear that a trade-off between these two approaches must be found in order to exploit as much as possible the current optimization techniques, while keeping the size of the rewriting reasonably small in order to make the execution of it feasible in practice. A related research field is that of query minimization [21], in particular, in presence of views and constraints [22, 15]. Given a conjunctive query $q$, and a set of constraints $\Sigma$, the goal is to find all the minimal equivalent reformulations of $q$ under the constraints of $\Sigma$. The most interesting approach in this respect is the chase & back-chase algorithm (C&B) [15], implemented in the MARS system [23]. The algorithm freezes the atoms of $\mathit{body}(q)$ and, by considering them as a database $D_{q}$, applies the following two steps. During the chase-step, the chase of $D_{q}$ w.r.t. $\Sigma$ is constructed, and then the atoms of $\mathit{chase}(D_{q},\Sigma)$ are considered as the body-atoms of a query $q_{u}$, called the universal plan. The back-chase step considers all the possible subsets of the atoms of $\mathit{body}(q_{u})$, starting from those with a single-atom, which are then considered as the body of a query $q^{\prime}$. Whenever there exists a containment mapping from $\mathit{body}(q_{u})$ to $\mathit{chase}(D_{q^{\prime}},\Sigma)$, where $D_{q^{\prime}}$ is the database obtained by freezing $\mathit{body}(q^{\prime})$, then $q^{\prime}$ is an equivalent reformulation of $q$. Moreover, every time an equivalent reformulation $q^{\prime}$ is found, the back-chase does not consider any of the supersets of the atoms of $\mathit{body}(q^{\prime})$ because they will be automatically implied by the atoms of $q^{\prime}$, and therefore the produced query would be redundant. This particular exploration strategy guarantees the minimality of the reformulations. A non-negligible drawback of this approach is the fact that we need to compute the chase of $D_{q}$ w.r.t. $\Sigma$, and also the chase for the (exponentially many) databases $D_{q^{\prime}}$ w.r.t $\Sigma$. Clearly, this makes the procedure computationally expensive. ## 3 Preliminaries In this section we recall some basics on relational databases, conjunctive queries, tuple-generating dependencies, and the chase procedure. ### 3.1 Relational Databases and Conjunctive Queries Consider two pairwise disjoint (infinite) sets of symbols $\Delta_{c}$ and $\Delta_{z}$ such that: $\Delta_{c}$ is a set of _constants_ (which constitutes the domain of a database), and $\Delta_{z}$ is a set of _labeled nulls_ (used as placeholders for unknown values). Different constants represent different values (_unique name assumption_), while different nulls may represent the same value. Throughout the paper, we denote by $\mathbf{X}$ sequences of variables $X_{1},\ldots,X_{k}$, where $k\geq 0$, and by $[n]$ the set $\\{1,\ldots,n\\}$, for any $n\geq 1$. A _relational schema_ $\mathcal{R}$ (or simply _schema_) is a set of _relational symbols_ (or _predicate symbols_), each with its associated arity. A _position_ $r[i]$ (or $\langle r,i\rangle$) is identified by a predicate $r\in\mathcal{R}$ and its $i$-th argument. A _term_ $t$ is a constant, labeled null, or variable. An _atomic formula_ (or simply _atom_) has the form $r(t_{1},\ldots,t_{n})$, where $r\in\mathcal{R}$ has arity $n$, and $t_{1},\ldots,t_{n}$ are terms. Conjunctions of atoms are often identified with the sets of their atoms. A _substitution_ from one set of symbols $S_{1}$ to another set of symbols $S_{2}$ is a function $h:S_{1}\rightarrow S_{2}$. A _homomorphism_ from a set of atoms $A_{1}$ to a set of atoms $A_{2}$, both over the same schema $\mathcal{R}$, is a substitution $h$ from the set of terms of $A_{1}$ to the set of terms of $A_{2}$ such that: (i) if $t\in\Delta_{c}$, then $h(t)=t$, and (ii) if $r(t_{1},\ldots,t_{n})$ is in $A_{1}$, then $h(r(t_{1},\ldots,t_{n}))=r(h(t_{1}),\ldots,h(t_{n}))$ is in $A_{2}$. The notion of homomorphism naturally extends to conjunctions of atoms. A _relational instance_ (or simply _instance_) $I$ for a schema $\mathcal{R}$ is a (possibly infinite) set of atoms of the form $r({\bf t})$, where $r\in\mathcal{R}$ has arity $n$ and ${\bf t}\in(\Delta_{c}\cup\Delta_{z})^{n}$. A _database_ is a finite relational instance. A _conjunctive query_ (CQ) $q$ of arity $n$ over a schema $\mathcal{R}$ is a formula of the form $q(\mathbf{X})\leftarrow\phi(\mathbf{X},\mathbf{Y})$, where $\phi(\mathbf{X},\mathbf{Y})$ is a conjunction of atoms over $\mathcal{R}$, and $q$ is an $n$-ary predicate. $\phi(\mathbf{X},\mathbf{Y})$ is called the body of $q$, denoted as $body(q)$, and $q(\mathbf{X})$ is the head of $q$, denoted as $head(q)$. A Boolean conjunctive query (BCQ) is a CQ of arity zero. The answer to a CQ $q$ of arity $n$ over an instance $I$, denoted as $q(I)$, is the set of all $n$-tuples $\mathbf{t}\in(\Delta_{c})^{n}$ for which there exists a homomorphism $h:\mathbf{X}\cup\mathbf{Y}\rightarrow\Delta_{c}\cup\Delta_{z}$ such that $h(\phi(\mathbf{X},\mathbf{Y}))\subseteq I$ and $h(\mathbf{X})=\mathbf{t}$. A BCQ has only the empty tuple $\langle\rangle$ as possible answer, in which case it is said that has positive answer. Formally, a BCQ has _positive_ answer over $I$, denoted as $I\models q$, iff $\langle\rangle\in q(I)$. A _union of CQs_ (UCQ) $Q$ of arity $n$ is a set of CQs, where each $q\in Q$ has the same arity $n$ and uses the same predicate symbol in the head. The answer to $Q$ over an instance $I$, denoted as $Q(I)$, is defined as the set of tuples $\\{\mathbf{t}~{}\|~{}\textrm{there~{}exists~{}}q\in Q\textrm{~{}such~{}that~{}}\mathbf{t}\in q(I)\\}$. ### 3.2 Tuple-Generating Dependencies A tuple-generating dependency (TGD) $\sigma$ over a schema $\mathcal{R}$ is a first-order formula $\forall{\bf X}\forall{\bf Y}\phi({\bf X},{\bf Y})\rightarrow\exists\mathbf{Z}\,\psi({\bf X},{\bf Z})$, where $\phi(\mathbf{X},\mathbf{Y})$ and $\psi(\mathbf{X},\mathbf{Z})$ are conjunctions of atoms over $\mathcal{R}$, called the body and the head of $\sigma$, denoted as $\mathit{body}(\sigma)$ and $\mathit{head}(\sigma)$, respectively. Henceforth, to avoid notational clutter, we will omit the universal quantifiers in TGDs. Such $\sigma$ is satisfied by an instance $I$ for $\mathcal{R}$ iff, whenever there exists a homomorphism $h$ such that $h(\phi(\mathbf{X},\mathbf{Y}))\subseteq I$, there exists an extension $h^{\prime}$ of $h$ (i.e., $h^{\prime}\supseteq h$) such that $h^{\prime}(\psi(\mathbf{X},\mathbf{Z}))\subseteq I$. We now define the notion of _query answering_ under TGDs. Given a database $D$ for $\mathcal{R}$, and a set $\Sigma$ of TGDs over $\mathcal{R}$, the _models_ of $D$ w.r.t. $\Sigma$, denoted as $\mathit{mods}(D,\Sigma)$, is the set of all instances $I$ such that $I\models D\cup\Sigma$, which means that $I\supseteq D$ and $I$ satisfies $\Sigma$. The _answer_ to a CQ $q$ w.r.t. $D$ and $\Sigma$, denoted as $\mathit{ans}(q,D,\Sigma)$, is the set $\\{\mathbf{t}~{}\|~{}\mathbf{t}\in q(I){\rm~{}for~{}each~{}}I\in\mathit{mods}(D,\Sigma)\\}$. The _answer_ to a BCQ $q$ w.r.t. $D$ and $\Sigma$ is _positive_ , denoted as $D\cup\Sigma\models q$, iff $\mathit{ans}(q,D,\Sigma)\neq\varnothing$. Note that query answering under general TGDs is undecidable [24], even when the schema and the set of TGDs are fixed [25]. We recall that the two problems of answering CQs and BCQs under TGDs are equivalent [21, 26]. Roughly speaking, we can enumerate the polynomially many tuples of constants which are possible answers to $q$, and then, instead of answering the given query $q$, we answer the polynomially many BCQs that we obtain by replacing the variables in the body of $q$ with the appropriate constants. A certain tuple $\mathbf{t}$ of constants is in the answer of $q$ iff the answer to the BCQ that we obtain from $\mathbf{t}$ is positive. Henceforth, we thus focus only on the BCQ answering problem. ### 3.3 The TGD Chase The _chase procedure_ (or simply _chase_) is a fundamental algorithmic tool introduced for checking implication of dependencies [27], and later for checking query containment [28]. Informally, the chase is a process of repairing a database w.r.t. a set of dependencies so that the resulted database satisfies the dependencies. We shall use the term chase interchangeably for both the procedure and its result. The chase works on an instance through the so-called TGD _chase rule_. TGD Chase Rule: Consider a database $D$ for a schema $\mathcal{R}$, and a TGD $\sigma:\phi(\mathbf{X},\mathbf{Y})\rightarrow\exists\mathbf{Z}\,\psi(\mathbf{X},\mathbf{Z})$ over $\mathcal{R}$. If $\sigma$ is applicable to $D$, i.e., there exists a homomorphism $h$ such that $h(\phi(\mathbf{X},\mathbf{Y}))\subseteq D$ then: _(i)_ define $h^{\prime}\supseteq h$ such that $h^{\prime}(Z_{i})=z_{i}$, for each $Z_{i}\in\mathbf{Z}$, where $z_{i}\in\Delta_{z}$ is a “fresh” labeled null not introduced before, and _(ii)_ add to $D$ the set of atoms in $h^{\prime}(\psi(\mathbf{X},\mathbf{Z}))$, if not already in $D$. Given a database $D$ and a set of TGDs $\Sigma$, the chase algorithm for $D$ and $\Sigma$ consists of an exhaustive application of the TGD chase rule in a breadth-first fashion, which leads as result to a (possibly infinite) chase for $D$ and $\Sigma$, denoted as $\mathit{chase}(D,\Sigma)$. For the formal definition of the chase algorithm we refer the reader to [8]. The (possibly infinite) chase for $D$ and $\Sigma$ is a _universal model_ of $D$ w.r.t. $\Sigma$, i.e., for each instance $I\in\mathit{mods}(D,\Sigma)$, there exists a homomorphism from $\mathit{chase}(D,\Sigma)$ to $I$ [26, 29]. Using this fact it can be shown that $D\cup\Sigma\models q$ iff $\mathit{chase}(D,\Sigma)\models q$, for every BCQ $q$. ## 4 The Datalog± Family In this section we present the main Datalog± languages under which query answering is decidable, and (almost in all cases) also tractable in data complexity. ### 4.1 Decidability Paradigms We first discuss the three main paradigms for ensuring decidability of query answering, namely, chase termination, guardedness and stickiness. #### Chase Termination. In this case the chase always terminates and produces a finite universal model $U$. Thus, given a query we just need to evaluate it over the finite database $U$. The most notable syntactic restriction of TGDs guaranteeing chase termination is _weak-acyclicity_ , which is defined by means of a graph-based condition, for which we refer the reader to the landmark paper [29]. Roughly speaking, in the chase constructed under a weakly-acyclic set of TGDs over a schema $\mathcal{R}$, only a finite number of distinct values can appear at any position of $\mathcal{R}$, and thus after finitely many steps the chase procedure terminates. It is known that query answering under a weakly-acyclic set of TGDs is ptime-complete [29] and 2exptime-complete [10] in data and combined complexity, respectively. More general syntactic restrictions that guarantee chase termination were proposed in [26] and [30]. #### Guardedness. _Guarded_ TGDs, introduced and studied in [25], have an atom in their body, called the _guard_ , that contains all the universally quantified variables. For example, the TGD $r(X,Y),s(X,Y,Z)\rightarrow\exists Ws(Z,X,W)$ is guarded via the guard atom $s(X,Y,Z)$, while the TGD $r(X,Y),r(Y,Z)\rightarrow r(X,Z)$ is not. Decidability of query answering follows from the fact that the chase constructed under a set of guarded TGDs has the bounded treewidth property, i.e., is a “tree-like” structure. The data and combined complexity of query answering under a set of guarded TGDs is ptime-complete [7] and 2exptime- complete [25], respectively. _Linear_ TGDs, proposed in [7], is a FO-rewritable variant of guarded TGDs. A TGD is linear iff it contains only one atom in its body. Obviously a linear TGD is trivially guarded since the singleton body-atom is automatically a guard. Linear TGDs are more expressive than the well-known class of inclusion dependencies. Query answering under linear TGDs is in the highly tractable class ac0 in data complexity [7]. The same problem is pspace-complete in combined complexity; this result is immediately implied by results in [28]. An expressive class, which forms a generalization of guarded TGDs, is the class of _weakly-guarded_ sets of TGDs introduced in [25]. Intuitively speaking, a set $\Sigma$ of TGDs is weakly-guarded iff in the body of each TGD of $\Sigma$ there exists an atom, called the _weak-guard_ , that contains all the universally quantified variables that appear only at positions where a “fresh” null of $\Delta_{z}$ can appear during the construction of the chase. Query answering under a weakly-guarded set of TGDs is exptime-complete [25] and 2exptime-complete [25] in data and combined complexity, respectively. #### Stickiness. In this paragraph we present a Datalog± language (and its extensions), which hinges on a paradigm that is very different from guardedness. _Sticky_ sets of TGDs are defined formally by an efficiently testable condition involving variable-marking [9]. In what follows we just give an intuitive definition of this class. For every database $D$, assume that during the construction of chase of $D$ under a sticky set of TGDs, we apply a TGD $\sigma\in\Sigma$ that has a variable $V$ appearing more than once in its body; assume also that $V$ maps (via homomorphism) on the symbol $z$, and that by virtue of this application the atom $\underline{a}$ is introduced. In this case, for each atom $\underline{b}$ in $\mathit{body}(\sigma)$, we say that $\underline{a}$ is _derived_ from $\underline{b}$. Then, we have that $z$ appears in $\underline{a}$ and in all atoms resulting from some chase derivation sequence starting from $\underline{a}$, “sticking” to them (hence the name “sticky” sets of TGDs). Interestingly, sticky sets of TGDs are FO-rewritable, and thus query answering is feasible in ac0 in data complexity [9]. Combined complexity of query answering is known to be exptime-complete [9]. In [10] the FO-rewritable class of _sticky-join_ sets of TGDs, that captures both linear TGDs and sticky sets of TGDs, is introduced. Similarly to sticky sets of TGDs, sticky-join sets are defined formally by a testable condition based on variable-marking. However, this variable-marking procedure is more sophisticated than the one used for sticky sets, and due to this fact the problem of identifying sticky-join sets of TGDs is harder than the one of identifying sticky sets. In particular, given a set $\Sigma$ of TGDs, we can decide in ptime whether $\Sigma$ is sticky, while the problem whether $\Sigma$ is sticky-join is pspace-complete. Note that the data and combined complexity of query answering under sticky and sticky-join sets of TGDs coincide. ### 4.2 Additional Features In this subsection we briefly discuss how the languages presented above can be combined with negative constraints and key dependencies, without altering the complexity of query answering. #### Negative Constraints. A _negative constraint_ (NC) $\nu$ over a schema $\mathcal{R}$ is a first- order formula $\forall{\bf X}\,\phi({\bf X})\rightarrow\bot$, where $\bot$ denotes the truth constant false. NCs are vital when representing ontologies (see, e.g., [7, 9]), as well as conceptual schemas such as Entity-Relationship diagrams (see, e.g., [31, 32]). With NCs we can assert, for example, that students and professors are disjoint sets: $\forall X\mathit{student}(X),\mathit{professor}(X)\rightarrow\bot$. Also, we can state that a student cannot be the leader of a research group: $\forall X\forall Y\mathit{student}(X),\mathit{leads}(X,Y)\rightarrow\bot$. It is known that checking NCs is tantamount to query answering [7]. In particular, given an instance $I$, a set $\Sigma_{\bot}$ of NCs, and a set $\Sigma$ of TGDs, for each NC $\nu$ of the form $\forall\mathbf{X}\,\phi(\mathbf{X})\rightarrow\bot$, we answer the BCQ $q_{\nu}()\leftarrow\phi(\mathbf{X})$. If at least one of such queries answers positively, then $I\cup\Sigma\cup\Sigma_{\bot}\models\bot$ (i.e., the theory is inconsistent), and therefore $I\cup\Sigma\cup\Sigma_{\bot}\models q$, for every BCQ $q$; otherwise, given a BCQ $q$, we have $I\cup\Sigma\cup\Sigma_{\bot}\models q$ iff $I\cup\Sigma\models q$, i.e., we can answer $q$ by ignoring the set of NCs. #### Key Dependencies. It is well-known that the interaction of general TGDs and key dependencies (KDs) leads to undecidability of query answering [33]; we assume that the reader is familiar with the notion of KD (see, e.g., [34]). Thus, the classes of TGDs presented above cannot be combined arbitrarily with KDs. Suitable syntactic restrictions are needed in order to ensure decidability of query answering. A crucial concept towards this direction is separability [35], which formulates a controlled interaction of TGDs and KDs. Formally speaking, a set $\Sigma=\Sigma_{T}\cup\Sigma_{K}$ over a schema $\mathcal{R}$, where $\Sigma_{T}$ and $\Sigma_{K}$ are sets of TGDs and KDs, respectively, is _separable_ iff for every instance $I$ for $\mathcal{R}$, either $I$ violates $\Sigma_{K}$, or for every BCQ $q$ over $\mathcal{R}$, $I\cup\Sigma\models q$ iff $I\cup\Sigma_{T}\models q$. Notice that separability is a semantic notion. A sufficient syntactic criterion for separability of TGDs and KDs is given in [7]; TGDs and KDs satisfying the criterion are called _non-conflicting_. Obviously, in case of non-conflicting sets of TGDs and KDs, we just need to perform a preliminary check whether the given instance satisfies the KDs, and if this is the case, then we eliminate them, and proceed by considering only the set of TGDs. This preliminary check can be done using negative constraints. For example, to check whether the KD $\mathit{key}(r)=\\{1\\}$, stating that the first attribute of the binary relation $r$ is a key attribute, is satisfied by the database $D$, we just need to check whether the database $D_{\neq}$ obtained by adding to $D$ the set of atoms $\\{\mathit{neq}(a,b)~{}\|~{}a\neq b,\textrm{~{}and~{}}a,b\textrm{~{}are~{}constants~{}occurring~{}in~{}}D\\}$, where $\mathit{neq}$ is an auxiliary predicate, satisfies the negative constraint $r(X,Y),r(X,Z),\mathit{neq}(Y,Z)\rightarrow\bot$. The atom $\mathit{neq}(a,b)$ implies that $a$ and $b$ are different constants. Since, as already mentioned, checking NCs is tantamount to query answering, we immediately get that the complexity of query answering under non-conflicting sets of TGDs and KDs is the same as in the case of TGDs only. Interestingly, by combining non-conflicting linear (or sticky) sets of TGDs and KDs with NCs, we get strictly more expressive formalisms than the most widely-adopted tractable ontology languages, in particular DL-LiteA, DL-LiteF and DL-LiteR, without loosing FO-rewritability, and consequently high tractability of query answering in data complexity. For more details, we refer the interested reader to [7, 9]. ## 5 Datalog± for OBDA In this section we consider the problem of BCQ answering under the FO- rewritable members of the Datalog± family, namely, linear, sticky and sticky- join sets of TGDs. Given a BCQ $q$ and a set $\Sigma$ of TGDs, the actual computation of the rewriting is done by applying a backward-chaining resolution procedure using the rules of $\Sigma$ as rewriting rules. Our algorithm optimizes the algorithm presented in [14] by greatly reducing the number of BCQs in the rewriting, and therefore improves the overall performance of query answering. Before going into the details of the rewriting algorithm, we first give some useful notions. A set of atoms $A=\\{\underline{a}_{1},\ldots,\underline{a}_{n}\\}$, where $n\geqslant 2$, _unifies_ if there exists a substitution $\gamma$, called _unifier_ for $A$, such that $\gamma(\underline{a}_{1})=\ldots=\gamma(\underline{a}_{n})$. A _most general unifier (MGU)_ for $A$ is a unifier for $A$, denoted as $\gamma_{A}$, such that for each other unifier $\gamma$ for $A$, there exists a substitution $\gamma^{\prime}$ such that $\gamma=\gamma^{\prime}\circ\gamma_{A}$. Notice that if a set of atoms unify, then there exists a MGU. Furthermore, the MGU for a set of atoms is unique (modulo variable renaming). The MGU for a singleton set $\\{\underline{a}\\}$ is defined as the identity substitution on the set of terms that occur in $\underline{a}$. Let us now give some auxiliary results which will allow us to simplify our later technical definitions and proofs. The first such lemma states that we can restrict our attention on TGDs that have only one head-atom. ###### Lemma 1 BCQ answering under (general) TGDs and BCQ answering under TGDs with just one head-atom are logspace-equivalent problems. Proof. It suffices to show that BCQ answering under (general) TGDs can be reduced in logspace to BCQ answering under TGDs with just one head-atom. Consider a BCQ $q$ over a schema $\mathcal{R}$, a database $D$ for $\mathcal{R}$, and a set $\Sigma$ of TGDs over $\mathcal{R}$. We construct $\Sigma^{\prime}$ from $\Sigma$ by applying the following procedure. For each TGD $\sigma\in\Sigma$, where $\mathit{head}(\sigma)=\\{\underline{a}_{1},\ldots,\underline{a}_{k}\\}$ and $\mathbf{X}$ is the set of variables that occur in $\mathit{head}(\sigma)$, replace $\sigma$ with the following set of TGDs: $\begin{array}[]{rcl}\mathit{body}(\sigma)&\rightarrow&r_{\sigma}(\mathbf{X}),\\\ r_{\sigma}(\mathbf{X})&\rightarrow&\underline{a}_{1},\\\ r_{\sigma}(\mathbf{X})&\rightarrow&\underline{a}_{2},\\\ &\vdots&\\\ r_{\sigma}(\mathbf{X})&\rightarrow&\underline{a}_{k},\end{array}$ where $r_{\sigma}$ is an auxiliary predicate not occurring in $\mathcal{R}$ having the same arity as the number of variables in $\mathbf{X}$. It is not difficult to see that the above construction is feasible in logspace. By construction, except for the atoms with an auxiliary predicate, $\mathit{chase}(D,\Sigma)$ and $\mathit{chase}(D,\Sigma^{\prime})$ coincide. The auxiliary predicates, being introduced only during the above transformation, do not match any predicate symbol in $q$, and hence $\mathit{chase}(D,\Sigma)\models q$ iff $\mathit{chase}(D,\Sigma^{\prime})\models q$, or, equivalently, $D\cup\Sigma\models q$ iff $D\cup\Sigma^{\prime}\models q^{\prime}$. The next lemma implies that we can restrict our attention on TGDs that have only one existentially quantified variable which occurs only once. ###### Lemma 2 BCQ answering under (general) TGDs and BCQ answering under TGDs with at most one existentially quantified variable that occurs only once are logspace- equivalent problems. Proof. It suffices to show that BCQ answering under (general) TGDs can be reduced in logspace to BCQ answering under TGDs that have at most one existentially quantified variable which occurs only once. Consider a BCQ $q$ over a schema $\mathcal{R}$, a database $D$ for $\mathcal{R}$, and a set $\Sigma$ of TGDs over $\mathcal{R}$. We construct $\Sigma^{\prime}$ from $\Sigma$ by applying the following procedure. For each TGD $\sigma\in\Sigma$, where $\\{X_{1},\ldots,X_{n}\\}$, for $n\geqslant 1$, is the set of variables that occur both in $\mathit{body}(\sigma)$ and $\mathit{head}(\sigma)$, and $\\{Z_{1},\ldots,Z_{m}\\}$, for $m>1$, is the set of the existentially quantified variables of $\sigma$, replace $\sigma$ with the following set of TGDs: $\begin{array}[]{rcl}\mathit{body}(\sigma)&\rightarrow&\exists Z_{1}\,r_{\sigma}^{1}(X_{1},\ldots,X_{n},Z_{1}),\\\ r_{\sigma}^{1}(X_{1},\ldots,X_{n},Z_{1})&\rightarrow&\exists Z_{2}\,r_{\sigma}^{2}(X_{1},\ldots,X_{n},Z_{1},Z_{2}),\\\ &\vdots&\\\ r_{\sigma}^{m-1}(X_{1},\ldots,X_{n},Z_{1},\ldots,Z_{m-1})&\rightarrow&\exists Z_{m}\,r_{\sigma}^{m}(X_{1},\ldots,X_{n},Z_{1},\ldots,Z_{m}),\\\ r_{\sigma}^{m}(X_{1},\ldots,X_{n},Z_{1},\ldots,Z_{m})&\rightarrow&\mathit{head}(\sigma),\end{array}$ where $r_{\sigma}^{i}$ is an auxiliary predicate of arity $n+i$, for each $i\in[m]$. It is easy to see that the above procedure can be carried out in logspace. By construction, except for the atoms with an auxiliary predicate, $\mathit{chase}(D,\Sigma)$ and $\mathit{chase}(D,\Sigma^{\prime})$ are the same (modulo bijective variable renaming). The auxiliary predicates, being introduced only during the above construction, do not match any predicate symbol in $q$, and hence $\mathit{chase}(D,\Sigma)\models q$ iff $\mathit{chase}(D,\Sigma^{\prime})\models q$, or, equivalently, $D\cup\Sigma\models q$ iff $D\cup\Sigma^{\prime}\models q$. Since the transformations given above preserve the syntactic condition of linear, sticky and sticky-join sets of TGDs, henceforth we assume w.l.o.g. that every TGD has just one atom in its head which contains only one existentially quantified variable that occurs only once. In the rest of the paper, for notational convenience, given a TGD $\sigma$, we denote by $\pi_{\sigma}$ the position in $\mathit{head}(\sigma)$ at which the existentially quantified variable occurs. We now give the notion of _applicability_ of a TGD to a set of body-atoms of a query. Let us assume w.l.o.g that the variables that appear in the query, and those that appear in the TGD, constitute two disjoint sets. Given a BCQ $q$, a variable is called _shared_ in $q$ if it occurs more than once in $\mathit{body}(q)$. Notice that in the case of (non-Boolean) CQs, a variable is shared in $q$ if it occurs more than once in $q$ (considering also the head of $q$ and not just its body). ###### Definition 1 (Applicability) Consider a BCQ $q$ over a schema $\mathcal{R}$, and a TGD $\sigma$ over $\mathcal{R}$. Given a set of atoms $A\subseteq\mathit{body}(q)$ that unifies, we say that $\sigma$ is _applicable_ to $A$ if the following conditions are satisfied: (i) the set $A\cup\\{\mathit{head}(\sigma)\\}$ unifies, and (ii) for each $\underline{a}\in A$, if the term at position $\pi$ in $\underline{a}$ is either a constant or a shared variable in $q$, then $\pi\neq\pi_{\sigma}$. Let us now introduce the notion of _factorizability_ which, as we explain below, makes one of the main differences between our algorithm and the one presented in [14], due to which a perfect rewriting with less BCQs is obtained. ###### Definition 2 (Factorizability) Consider a BCQ $q$ over a schema $\mathcal{R}$, and a TGD $\sigma$ over $\mathcal{R}$ which contains an existentially quantified variable. A set of atoms $A\subseteq\mathit{body}(q)$, where $\|A\|\geqslant 2$, that unifies is _factorizable_ w.r.t. $\sigma$ if there exists a variable $V$ that occurs in every atom of $S$ only at position $\pi_{\sigma}$, and also $V$ does not occur in $\mathit{body}(q)\setminus S$. It is important to clarify that in the case of (non-Boolean) CQs, the notion of factorizability is defined as above, except that the variable $V$ does not occur in $(\\{\mathit{head}(\sigma)\\}\cup\mathit{body}(\sigma))\setminus S$. ###### Example 1 (Factorization) Consider the BCQs $\begin{array}[]{rcl}q_{1}&:&q()\,\leftarrow\,\underbrace{t(A,B,C),t(A,E,C)}_{S_{1}}\\\ q_{2}&:&q()\,\leftarrow\,s(C),\underbrace{t(A,B,C),t(A,E,C)}_{S_{2}}\\\ q_{3}&:&q()\,\leftarrow\,\underbrace{t(A,B,C),t(A,C,C)}_{S_{3}}\end{array}$ and the TGD $\sigma:s(X),r(X,Y)\,\rightarrow\,\exists Z\,t(X,Y,Z).$ Clearly, $S_{1}$ is factorizable w.r.t. $\sigma$ since the substitution $\\{E\rightarrow B\\}$ is a unifier for $S_{1}$, and also $C$ appears in both atoms of $S_{1}$ only at position $\pi_{\sigma}$. The factorization results in the query $q()\leftarrow t(A,B,C)$; notice that $\sigma$ is not applicable to $S_{1}$, but it is applicable to $\\{t(A,B,C)\\}$. On the contrary, despite the fact that $S_{2}$ unifies, it is not factorizable w.r.t. $\sigma$ since $C$ occurs also in $\mathit{body}(q_{2})\setminus S_{2}$. Finally, even if $S_{3}$ unifies, it is not factorizable w.r.t. $\sigma$ since $C$ appears in $S_{3}$, not only at position $\pi_{\sigma}$, but also at position $t[2]$. We are now ready to describe the algorithm TGD-rewrite, depicted in Algorithm 1, which is based on the rewriting algorithm presented in [14]. The perfect rewriting of a BCQ $q$ w.r.t. a set of TGDs $\Sigma$ is computed by exhaustively applying (i.e., until a fixpoint is reached) two steps: _factorization_ and _rewriting_. Input: a BCQ $q$ over a schema $\mathcal{R}$, a set $\Sigma$ of TGDs over $\mathcal{R}$ Output: the FO-rewriting $Q_{\textsc{fin}}$ of $q$ w.r.t. $\Sigma$ $Q_{\textsc{rew}}:=\\{\langle q,1\rangle\\}$; repeat $Q_{\textsc{temp}}:=Q_{\textsc{rew}}$; foreach _$\\{\langle q,x\rangle\\}\in Q_{\textsc{temp}}$ , where $x\in\\{0,1\\}$,_ do /* factorization step / foreach _$\sigma\in\Sigma$_ do $q^{\prime}:=\mathit{factorize}(q,\sigma)$; if _$\mathit{notExists}(\langle q^{\prime},y\rangle,Q_{\textsc{rew}})$ , where $y\in\\{0,1\\}$,_ then $Q_{\textsc{rew}}:=Q_{\textsc{rew}}\cup\\{\langle q^{\prime},0\rangle\\}$; / rewriting step / foreach _$A\subseteq\mathit{body}(q)$_ do foreach _$\sigma\in\Sigma$_ do if _$\mathit{isApplicable}(\sigma,A,q)$_ then $q^{\prime}:=\gamma_{A\cup\\{\mathit{head}(\sigma)\\}}(q[A/\mathit{body}(\sigma)])$; if _$\mathit{notExists}(\langle q^{\prime},1\rangle,Q_{\textsc{rew}})$_ then $Q_{\textsc{rew}}:=Q_{\textsc{rew}}\cup\\{\langle q^{\prime},1\rangle\\}$; until _$Q_{\textsc{temp}}=Q_{\textsc{rew}}$_ ; $Q_{\textsc{fin}}:=\\{q~{}\|~{}\langle q,x\rangle\in Q_{\textsc{rew}}\textrm{~{}and~{}}x=1\\}$; return _$Q_{\textsc{fin}}$_ Algorithm 1 The algorithm TGD-rewrite Factorization Step. The function $\mathit{factorize}(q,\sigma)$, providing that there exists a subset of $\mathit{body}(q)$ which is factorizable w.r.t. $\sigma$ (otherwise, the query $q$ is returned), first selects such a set $S\subseteq\mathit{body}(q)$. Then, the query $q^{\prime}$ is constructed by applying the MGU $\gamma_{S}$ for $S$ on $q$. Providing that there is no pair $\langle q^{\prime\prime},y\rangle$, where $y\in\\{0,1\\}$, in $Q_{\textsc{rew}}$ such that $q^{\prime}$ and $q^{\prime\prime}$ are the same (modulo bijective variable renaming), the pair $\langle q^{\prime},0\rangle$ is added to $Q_{\textsc{rew}}$; the label $0$ keeps track of the queries generated by the factorization step that must be excluded from the final rewriting. This is carried out by the $\mathit{notExists}$ function. Rewriting Step. If there exists a pair $\langle q,y\rangle$ and a TGD $\sigma\in\Sigma$ which is applicable to a set of atoms $A\subseteq\mathit{body}(q)$, then the algorithm constructs a new query $q^{\prime}=\gamma_{A\cup\\{\mathit{head}(\sigma)\\}}(q[A/\mathit{body}(\sigma)])$, that is, the BCQ obtained from $q$ by replacing $A$ with $\mathit{body}(\sigma)$ and then applying the MGU for the set $A\cup\\{\mathit{head}(\sigma)\\}$. Providing that there is no pair $\langle q^{\prime\prime},1\rangle$ in $Q_{\textsc{rew}}$ such that $q^{\prime}$ and $q^{\prime\prime}$ are the same (modulo bijective variable renaming), the pair $\langle q^{\prime},1\rangle$ is added to $Q_{\textsc{rew}}$; the label $1$ keeps track of the queries generated by the rewriting step which will be the final rewriting. ###### Example 2 (Rewriting) Consider the set $\Sigma$ of TGDs $\begin{array}[]{rcl}\sigma_{1}&:&s(X)\,\rightarrow\exists Z\ \,t(X,X,Z)\\\ \sigma_{2}&:&t(X,Y,Z)\,\rightarrow\,r(Y,Z)\end{array}$ and the query $q()\leftarrow t(A,B,C),r(B,C).$ TGD-rewrite first applies $\sigma_{2}$ to $\\{r(B,C)\\}$ since $\sigma_{1}$ is not applicable. The query $q_{1}:q()\leftarrow t(A,B,C),t(V^{1},B,C)$ is produced. Clearly, $\mathit{body}(q_{1})$ is factorizable w.r.t. $\sigma_{1}$ and the query $q_{2}:q()\leftarrow t(A,B,C)$ is obtained. Now, $\sigma_{1}$ is applicable to $\\{t(A,B,C)\\}$ and the query $q_{3}:q()\leftarrow s(A)$ is obtained. The perfect rewriting constructed by the algorithm is the set $\\{q,q_{1},q_{3}\\}$. The next example shows that dropping the applicability condition, then TGD- rewrite may produce unsound rewritings. ###### Example 3 (Loss of soundness) Suppose that we ignore the applicability condition during the rewriting process. Consider the set $\Sigma$ of TGDs given in Example 2, and also the BCQ $q_{1}:q()\leftarrow t(A,B,c)$, where $c$ is a constant of $\Delta_{c}$. A BCQ $q^{\prime}$ of the form $q()\leftarrow s(V)$ is obtained, where the information about the constant $c$ is lost. Consider now the database $D=\\{s(b),t(a,b,d)\\}$ for $\mathcal{R}$. The query $q^{\prime}$ maps to the atom $s(b)$ which implies that $D\models q^{\prime}$. However, the original query $q$ does not map to $\mathit{chase}(D,\Sigma)$, and thus $D\cup\Sigma\not\models q$. Therefore, any rewriting containing $q^{\prime}$ is not a sound rewriting of $q$ given $\Sigma$. Consider now the query $q^{\prime\prime}:q()\leftarrow t(A,B,B)$. The same query $q^{\prime}$ mapping to the atom $s(b)$ of $D$ is obtained. However, during the construction of $\mathit{chase}(D,\Sigma)$ it is not possible to get an atom of the form $t(X,Y,Y)$, where at positions $t[2]$ and $t[3]$ the same value occurs. This implies that there is no homomorphism that maps $q$ to $\mathit{chase}(D,\Sigma)$, and hence $D\cup\Sigma\not\models q$. Therefore, any rewriting containing $q^{\prime}$ is again unsound. The applicability condition may prevent the generation of queries that are vital to guarantee completeness of the rewritten query, as shown by the following example. This is exactly the reason why the factorization step is also needed. ###### Example 4 (Loss of completeness) Consider the set $\Sigma$ of TGDs $\begin{array}[]{rcl}\sigma_{1}&:&p(X)\,\rightarrow\,\exists Y\,t(X,Y)\\\ \sigma_{2}&:&t(X,Y)\,\rightarrow\,s(Y)\end{array}$ and the query $q:q()\leftarrow t(A,B),s(B).$ The only viable strategy in this case is to apply $\sigma_{2}$ to $\\{s(B)\\}$, since $\sigma_{1}$ is not applicable to $\\{t(A,B)\\}$ due to the shared variable $B$. The query that we obtain is $q^{\prime}:q()\leftarrow t(A,B),t(V^{1},B)$, where $V^{1}$ is a fresh variable. Notice that in $q^{\prime}$ the variable $B$ remains shared thus it is not possible to apply $\sigma_{1}$. It is obvious that without the factorization step there is no way to obtain the query $q^{\prime\prime}:q()\leftarrow p(A)$ during the rewriting process. Now, consider the database $D=\\{p(a)\\}$. Clearly, $\mathit{chase}(D,\Sigma)=\\{p(a),t(a,z_{1}),s(z_{1})\\}$, and therefore $\mathit{chase}(D,\Sigma)\models q$, or, equivalently, $D\cup\Sigma\models q$. However, the rewritten query is not entailed by the given database $D$, since $q^{\prime\prime}$ does not belong to it, which implies that it is not complete. We proceed now to establish soundness and completeness of the proposed algorithm. Towards this aim we need two auxiliary technical lemmas. The first one, which is needed for soundness, states that once the chase entails the rewritten query constructed by the rewriting algorithm, then the chase entails also the given query. In the sequel, for brevity, given a BCQ $q$ over a schema $\mathcal{R}$ and a set $\Sigma$ of TGDs over $\mathcal{R}$, we denote by $q_{\Sigma}$ the rewritten query $\textsf{TGD-rewrite}(q,\Sigma)$. ###### Lemma 3 Consider a BCQ $q$ over a schema $\mathcal{R}$, a database $D$ for $\mathcal{R}$, and a set $\Sigma$ of TGDs over $\mathcal{R}$. If $\mathit{chase}(D,\Sigma)\models q_{\Sigma}$, then $\mathit{chase}(D,\Sigma)\models q$. Proof. The proof is by induction on the number of applications of the rewriting step. We denote by $q_{\Sigma}^{[i]}$ the part of the rewritten query $q_{\Sigma}$ obtained by applying $i$ times the rewriting step. Base Step. Clearly, $q_{\Sigma}^{0}=q_{\Sigma}$, and the claim holds trivially. Inductive Step. Suppose now that $\mathit{chase}(D,\Sigma)\models q_{\Sigma}^{[i]}$, for $i\geq 0$. This implies that there exists $p\in q_{\Sigma}^{[i]}$ such that $\mathit{chase}(D,\Sigma)\models p$, and thus there exists a homomorphism $h$ such that $h(\mathit{body}(p))\subseteq\mathit{chase}(D,\Sigma)$. If $p\in q_{\Sigma}^{[i-1]}$, then the claim follows by induction hypothesis. The interesting case is when $p$ was obtained during the $i$-th application of the rewriting step from a BCQ $p^{\prime}\in q_{\Sigma}^{[i-1]}$, i.e., $q_{\Sigma}^{[i]}=q_{\Sigma}^{[i-1]}\cup\\{p\\}$. By induction hypothesis, it suffices to show that $\mathit{chase}(D,\Sigma)\models q_{\Sigma}^{[i-1]}$. Clearly, there exists a TGD $\sigma\in\Sigma$ of the form $\phi(\mathbf{X},\mathbf{Y})\rightarrow\exists Z\,r(\mathbf{X},Z)$ which is applicable to a set $A\subseteq\mathit{body}(p^{\prime})$, and $p$ is such that $\mathit{body}(p)=\gamma(p^{\prime}[A/\phi(\mathbf{X},\mathbf{Y})])$, where $\gamma$ is the MGU for the set $A\cup\\{\mathit{head}(\sigma)\\}$. Observe that $h(\gamma(\phi(\mathbf{X},\mathbf{Y})))\subseteq\mathit{chase}(D,\Sigma)$, and hence $\sigma$ is applicable to $\mathit{chase}(D,\Sigma)$; let $\mu=h\circ\gamma$. Thus, $\mu^{\prime}(r(\mathbf{X},Z))\in\mathit{chase}(D,\Sigma)$, where $\mu^{\prime}\supset\mu$. We define the substitution $h^{\prime}=h\cup\\{\gamma(Z)\rightarrow\mu^{\prime}(Z)\\}$. Let us first show that $h^{\prime}$ is a well-defined substitution. It suffices to show that $\gamma(Z)$ is not a constant, and also that $\gamma(Z)$ does not appear in the left-hand side of an assertion of $h$. Towards a contradiction, suppose that $\gamma(Z)$ is either a constant or appears in the left-hand side of an assertion of $h$. It is easy to verify that in this case there exists an atom $\underline{a}\in A$ such that at position $\pi_{\sigma}$ in $\underline{a}$ occurs either a constant or a variable which is shared in $p^{\prime}$. But this contradicts the fact that $\sigma$ is applicable to $A$. Consequently, $h^{\prime}$ is well-defined. It remains to show that the substitution $h^{\prime}\circ\gamma$ maps $\mathit{body}(p^{\prime})$ to $\mathit{chase}(D,\Sigma)$, and thus $\mathit{chase}(D,\Sigma)\models q_{\Sigma}^{[i-1]}$. Clearly, $\gamma(\mathit{body}(p^{\prime})\setminus A)\subseteq\mathit{body}(p)$. Since $h(\mathit{body}(p))\subseteq\mathit{chase}(D,\Sigma)$, we get that $h^{\prime}(\gamma(\mathit{body}(p^{\prime})\setminus A))\subseteq\mathit{chase}(D,\Sigma)$. Moreover, $\begin{array}[]{rcl}h^{\prime}(\gamma(A))&=&h^{\prime}(\gamma(r(\mathbf{X},Z)))\\\ &=&r(h^{\prime}(\gamma(\mathbf{X})),h^{\prime}(\gamma(Z)))\\\ &=&r(\mu(\mathbf{X}),\mu^{\prime}(Z))\\\ &=&\mu^{\prime}(r(\mathbf{X},Z))\\\ &\in&\mathit{chase}(D,\Sigma).\end{array}$ The proof is now complete. The second auxiliary lemma, which is needed for completeness, asserts that once the chase entails the rewritten query constructed by the rewriting algorithm, then the given database also entails the rewritten query. ###### Lemma 4 Consider a BCQ $q$ over a schema $\mathcal{R}$, a database $D$ for $\mathcal{R}$, and a set $\Sigma$ of TGDs over $\mathcal{R}$. If $\mathit{chase}(D,\Sigma)\models q_{\Sigma}$, then $D\models q_{\Sigma}$. Proof. We proceed by induction on the number of applications of the chase step. Base Step. Clearly, $\mathit{chase}^{[0]}(D,\Sigma)=D$, and the claim holds trivially. Inductive Step. Suppose now that $\mathit{chase}^{[i]}(D,\Sigma)\models q_{\Sigma}$, for $i\geq 0$. This implies that there exists $p\in q_{\Sigma}$ such that $\mathit{chase}^{[i]}(D,\Sigma)\models p$, and thus there exists a homomorphism $h$ such that $h(\mathit{body}(p))\subseteq\mathit{chase}^{[i]}(D,\Sigma)$. If $h(\mathit{body}(p))\subseteq\mathit{chase}^{[i-1]}(D,\Sigma)$, then the claim follows by induction hypothesis. The non-trivial case is when the atom $\underline{a}$, obtained during the $i$-th application of the chase step due to a TGD $\sigma\in\Sigma$ of the form $\phi(\mathbf{X},\mathbf{Y})\rightarrow\exists Z\,r(\mathbf{X},Z)$, belongs to $h(\mathit{body}(p))$. Clearly, there exists a homomorphism $\mu$ such that $\mu(\phi(\mathbf{X},\mathbf{Y}))\subseteq\mathit{chase}^{[i-1]}(D,\Sigma)$ and $\underline{a}=\mu^{\prime}(r(\mathbf{X},\mathbf{Y}))$, where $\mu^{\prime}\supseteq\mu$. By induction hypothesis, it suffices to show that $\mathit{chase}^{[i-1]}(D,\Sigma)\models q_{\Sigma}$. Before we proceed further, we need to establish an auxiliary technical claim. ###### Claim 5 There exists a BCQ $p^{\prime}\in q_{\Sigma}$ and a set of atoms $A\subseteq\mathit{body}(p^{\prime})$ such that $\sigma$ is applicable to $A$, and also there exists a homomorphism $\lambda$ such that $\lambda(\mathit{body}(p^{\prime})\setminus A)\subseteq\mathit{chase}^{[i-1]}(D,\Sigma)$ and $\lambda(A)=\underline{a}$. Proof. Clearly, there exists a set of atoms $B$ such that $h(\mathit{body}(p)\setminus B)\subseteq\mathit{chase}^{[i-1]}(D,\Sigma)$ and $h(B)=\underline{a}$. Observe that the null value that occurs in $\underline{a}$ at position $\pi_{\sigma}$ does not occur in $\mathit{chase}^{[i-1]}(D,\Sigma)$ or in $\underline{a}$ at some position other than $\pi_{\sigma}$. Therefore, the variables that occur in the atoms of $B$ at $\pi_{\sigma}$ do not appear at some other position. Consequently, $B$ can be partitioned into the sets $B_{1},\ldots,B_{m}$, where $m\geq 1$, and the following holds: for each $i\in[m]$, in the atoms of $B_{i}$ at position $\pi_{\sigma}$ the same variable $V_{i}$ occurs, and also $V_{i}$ does not occur in some other set $B\in\\{B_{1},\ldots,B_{m}\\}\setminus\\{B_{i}\\}$ or in $B_{i}$ at some position other than $\pi_{\sigma}$. It is easy to verify that each set $B_{i}$ is factorizable w.r.t. $\sigma$. Suppose that we factorize $B_{1}$. Then, the query $p_{1}=\gamma_{1}(p)$, where $\gamma_{1}$ is the MGU for $B_{1}$, is obtained. Observe that $h$ is a unifier for $B_{1}$. By definition of the MGU, there exists a substitution $\theta_{1}$ such that $h=\theta_{1}\circ\gamma_{1}$. Clearly, $\begin{array}[]{rcl}\theta_{1}(\mathit{body}(p_{1})\setminus\gamma_{1}(B))&=&\theta_{1}(\gamma_{1}(\mathit{body}(p))\setminus\gamma_{1}(B))\\\ &=&h(\mathit{body}(p)\setminus B)\\\ &\subseteq&\mathit{chase}^{[i-1]}(D,\Sigma),\end{array}$ and $\theta_{1}(\gamma_{1}(B))=h(B)=\underline{a}$. Now, observe that the set $\gamma_{1}(B_{2})\subseteq\mathit{body}(p_{1})$ is factorizable w.r.t. $\sigma$. By applying factorization we get the query $p_{2}=\gamma_{2}(p_{1})$, where $\gamma_{2}$ is the MGU for $\gamma_{1}(B_{2})$. Since $\theta_{1}$ is a unifier for $\gamma_{1}(B_{2})$, there exists a substitution $\theta_{2}$ such that $\theta_{1}=\theta_{2}\circ\gamma_{2}$. Clearly, $\begin{array}[]{rcl}\theta_{2}(\mathit{body}(p_{2})\setminus\gamma_{2}(\gamma_{1}(B)))&=&\theta_{2}(\gamma_{2}(\mathit{body}(p_{1}))\setminus\gamma_{2}(\gamma_{1}(B)))\\\ &=&\theta_{1}(\gamma_{1}(\mathit{body}(p))\setminus\gamma_{1}(B))\\\ &=&h(\mathit{body}(p)\setminus B)\\\ &\subseteq&\mathit{chase}^{[i-1]}(D,\Sigma),\end{array}$ and $\theta_{2}(\gamma_{2}(\gamma_{1}(B)))=\theta_{1}(\gamma_{1}(B))=h(B)=\underline{a}$. Eventually, by applying the factorization step as above, we will get the BCQ $p_{m}\ =\ \gamma_{m}\circ\ldots\circ\gamma_{1}(p),$ where $\gamma_{j}$ is the MGU for the set $\gamma_{j-1}\circ\ldots\circ\gamma_{1}(B_{j})$, for $j\in\\{2,\ldots,m\\}$ (recall that $\gamma_{1}$ is the MGU for $B_{1}$), such that $\theta_{m}(\mathit{body}(p_{m})\setminus\gamma_{m}\circ\ldots\circ\gamma_{1}(B))\subseteq\mathit{chase}^{[i-1]}(D,\Sigma)$ and $\theta_{m}(\gamma_{m}\circ\ldots\circ\gamma_{1}(B))=\underline{a}$. It is easy to verify that $\sigma$ is applicable to $A$. The claim follows with $p^{\prime}=p_{m}$, $A=\gamma_{m}\circ\ldots\circ\gamma_{1}(B)$ and $\lambda=\theta_{m}$. The above claim implies that during the rewriting process eventually we will get a BCQ $p^{\prime\prime}$ such that $\mathit{body}(p^{\prime\prime})=\gamma(\mathit{body}(p^{\prime})\setminus A)\cup\gamma(\phi(\mathbf{X},\mathbf{Y}))$, where $\gamma$ is the MGU for the set $A\cup\\{\mathit{head}(\sigma)\\}$. It remains to show that there exists a homomorphism that maps $\mathit{body}(p^{\prime\prime})$ to $\mathit{chase}^{[i-1]}(D,\Sigma)$. Since $\lambda\cup\mu^{\prime}$ is a well- defined substitution, we get that $\lambda\cup\mu^{\prime}$ is a unifier for $A\cup\\{\mathit{head}(\sigma)\\}$. By definition of the MGU, there exists a substitution $\theta$ such that $\lambda\cup\mu^{\prime}=\theta\circ\gamma$. Observe that $\begin{array}[]{rcl}\theta(\mathit{body}(p^{\prime\prime}))&=&\theta(\gamma(\mathit{body}(p^{\prime})\setminus A)\cup\gamma(\phi(\mathbf{X},\mathbf{Y})))\\\ &=&(\lambda\cup\mu^{\prime})(\mathit{body}(p^{\prime})\setminus A)\cup(\lambda\cup\mu^{\prime})(\phi(\mathbf{X},\mathbf{Y}))\\\ &=&\lambda(\mathit{body}(p^{\prime})\setminus A)\cup\mu^{\prime}(\phi(\mathbf{X},\mathbf{Y}))\\\ &\subseteq&\mathit{chase}^{[i-1]}(D,\Sigma).\end{array}$ Consequently, the desired homomorphism is $\theta$, and the claim follows. We are now ready to establish soundness and completeness of the algorithm TGD- rewrite. ###### Theorem 6 Consider a BCQ $q$ over a schema $\mathcal{R}$, a database $D$ for $\mathcal{R}$, and a set $\Sigma$ of TGDs over $\mathcal{R}$. It holds that, $D\models q_{\Sigma}$ iff $D\cup\Sigma\models q$. Proof. Suppose first that $D\models q_{\Sigma}$. Since $D\subseteq\mathit{chase}(D,\Sigma)$, we get that $\mathit{chase}(D,\Sigma)\models q_{\Sigma}$, and the claim follows by Lemma 3. Suppose now that $D\cup\Sigma\models q_{\Sigma}$. Since $q\in q_{\Sigma}$, we get that $\mathit{chase}(D,\Sigma)\models q_{\Sigma}$, and the claim follows by Lemma 4. Notice that the above result holds for arbitrary TGDs. However, termination of TGD-rewrite is guaranteed if we consider linear, sticky or sticky-join sets of TGDs since, during the rewriting process, only finitely many queries (modulo bijective variable renaming) are generated. ###### Theorem 7 The algorithm TGD-rewrite terminates under linear, sticky or sticky-join sets of TGDs. Approaches such as those of [5] and [14] resort to exhaustive factorizations of the atoms in the queries generated by the rewriting algorithm. By factorizing a query $q$ we obtain a subquery $q^{\prime}$, that is, $q$ implies $q^{\prime}$ (w.r.t. the given set of TGDs). Observe that by computing the factorized query $q^{\prime}$ we eliminate unnecessary shared variables, in the body of $q$, due to which the applicability condition is violated. Consider for example the query $q^{\prime}$ of Example 4. By factorizing the body of $q^{\prime}$ we obtain the query $q()\leftarrow t(A,B)$ which is a subquery (w.r.t. to the given set $\Sigma$ of TGDs) of $q^{\prime}$ (in this case equivalent to $q^{\prime}$), where the variable $B$ is no longer shared. Thus, the rewriting step can now apply $\sigma_{1}$ to $\\{t(A,B)\\}$ and produce the query $q()\leftarrow p(A)$ which is needed to ensure completeness. The exhaustive factorization produces a non-negligible number of redundant queries as demonstrated by the simple example above. It is thus necessary to apply a restricted form of factorization that generates a possibly small number of BCQs that are necessary to guarantee completeness of the rewritten query. This corresponds to the identification of all the atoms in the query whose shared existential variables come from the same atom in the chase, and they can be thus unified with no loss of information. The key principle behind our factorization process is that, in order to be applied, there must exist a TGD that can be applied to the output of the factorization. ### 5.1 Exploiting Negative Constraints It is well-known that negative constraints (NCs) of the form $\forall\mathbf{X}\,\phi(\mathbf{X})\rightarrow\bot$ are vital for representing ontologies. As already explained in Subsection 4.2, given a database $D$ for a schema $\mathcal{R}$, a set $\Sigma$ of TGDs over $\mathcal{R}$, and a set $\Sigma_{\bot}$ of NCs over $\mathcal{R}$, once the theory $D\cup\Sigma\cup\Sigma_{\bot}$ is consistent, then we are allowed to ignore the NCs since, for every BCQ $q$, $D\cup\Sigma\cup\Sigma_{\bot}\models q$ iff $D\cup\Sigma\models q$. However, as shown in the following example, by exploiting the given set of NCs it is possible to further reduce the size of the final rewriting. ###### Example 5 Consider the TGD $\sigma:t(X),s(Y)\rightarrow\exists Z\,p(Y,Z)$, the NC $\nu:r(X,Y),s(Y)\rightarrow\bot$, and the BCQ $q()\leftarrow r(A,B),p(B,C)$. Clearly, due to the rewriting step, the query $p:q()\leftarrow r(A,B),t(V^{1}),s(B)$ is obtained during the rewriting process. However, this query is not really needed since, for any database $D$ for $\mathcal{R}$, $D\not\models p$; otherwise, $D$ violates the NC $\nu$ which is a contradiction since we always assume that the theory $D\cup\\{\sigma,\nu\\}$ is consistent. It is not difficult to show that, given a BCQ $q$, and a set $\Sigma$ of TGDs, if a query $p\in q_{\Sigma}$ is not entailed by $\mathit{chase}(D,\Sigma)$, for an arbitrary database $D$, then any query $p^{\prime}\in q_{\Sigma}$ obtained during the rewriting process starting from $p$, also it is not entailed by $\mathit{chase}(D,\Sigma)$. Assume now that the set $\Sigma_{\bot}$ of NCs is part of the input. If we obtain a query $p\in q_{\Sigma}$ such that there exists a homomorphism that maps $\mathit{body}(\nu)$, for some NC $\nu\in\Sigma_{\bot}$, to $\mathit{body}(p)$, then we can safely ignore $p$ since $\mathit{chase}(D,\Sigma)$ does not entail $p$. From the above informal discussion, we conclude that we can further reduce the size of the final rewriting by modifying our algorithm as follows. During the execution of the rewriting algorithm TGD-rewrite (see Algorithm 1), after the factorization step (resp., rewriting step) we check whether there exists a homomorphism that maps $\mathit{body}(\nu)$, for some NC $\nu$ of the given set of NCs, to the body of the generated query $q^{\prime}$. If there exists such a homomorphism, then the pair $\langle q^{\prime},0\rangle$ (resp., $\langle q^{\prime},1\rangle$) is not added to the set $Q_{\textsc{rew}}$. Furthermore, the pair $\langle q,1\rangle$ is added to $Q_{\textsc{rew}}$ (see the first line of the algorithm) only if there is no homomorphism that maps $\mathit{body}(\nu)$, for some NC $\nu$ of the given set of NCs, to $\mathit{body}(q)$. If there exists such a homomorphism, then the algorithm terminates and returns the emptyset, which means that $\mathit{chase}(D,\Sigma)\not\models q$, for every database $D$ for $\mathcal{R}$. ## 6 Rewriting Optimization It is common knowledge that the perfect rewriting obtained by applying a backward-chaining rewriting algorithm (like TGD-rewrite) is, in general, not very well-suited for execution by a DB engine due to the large number of queries to be evaluated. In this section we propose a technique, called _query elimination_ , aiming at optimizing the obtained rewritten query under the class of linear TGDs. As we shall see, query elimination (which is an additional step during the execution of the algorithm TGD-rewrite) reduces _(i)_ the number of BCQs of the perfect rewriting, _(ii)_ the number of atoms in each query of the rewriting as well as _(iii)_ the number of joins. Note that in the rest of the paper we restrict our attention on linear TGDs. Recall that linear TGDs are TGDs with just one atom in their body. Since we also assume, as explained in the previous section, TGDs with just one atom in their head, henceforth, when using the term TGD, we shall refer to TGDs with just one body-atom and one head-atom. By exploiting the given set of TGDs, it is possible to identify atoms in the body of a certain query that are logically implied (w.r.t. the given set of TGDs) by other atoms in the same query. In particular, for each BCQ $q$ obtained by applying the rewriting step of TGD-rewrite, the atoms of $\mathit{body}(q)$ that are logically implied (w.r.t. the given set of TGDs) by some other atom of $\mathit{body}(q)$ are eliminated. Roughly speaking, the elimination of an atom from the body of a query implies the avoidance of the construction of redundant queries during the rewriting process. Thus, this step greatly reduces the number of BCQs in the perfect rewriting. Before going into the details, let us first introduce some necessary technical notions. ###### Definition 3 (Dependency Graph) Consider a set $\Sigma$ of TGDs over a schema $\mathcal{R}$. The _dependency graph_ of $\Sigma$ is a labeled directed multigraph $\langle N,E,\lambda\rangle$, where $N$ is the node set, $E$ is the edge set, and $\lambda$ is a labeling function $E\rightarrow\Sigma$. The node set $N$ is the set of positions of $\mathcal{R}$. If there is a TGD $\sigma\in\Sigma$ such that the same variable appears at position $\pi_{b}$ in $\mathit{body}(\sigma)$ and at position $\pi_{h}$ in $\mathit{head}(\sigma)$, then in $E$ there is an edge $e=(\pi_{b},\pi_{h})$ with $\lambda(e)=\sigma$. Intuitively speaking, the dependency graph of a set $\Sigma$ of TGDs describes all the possible ways of propagating a term from a position to some other position during the construction of the chase under $\Sigma$. More precisely, the existence of a path $P$ from $\pi_{1}$ to $\pi_{2}$ implies that it is possible (but not always) to propagate a term from $\pi_{1}$ to $\pi_{2}$. The existence of $P$ guarantees the propagation of a term from $\pi_{1}$ to $\pi_{2}$ if, for each pair of consecutive edges $e=(\pi,\pi^{\prime})$ and $e^{\prime}=(\pi^{\prime},\pi^{\prime\prime})$ of $P$, where $e$ and $e^{\prime}$ are labeled by the TGDs $\sigma$ and $\sigma^{\prime}$, respectively, the atom obtained during the chase by applying $\sigma$ triggers $\sigma^{\prime}$. To verify whether this holds we need an additional piece of information, the so-called _equality type_ , about the body-atom and the head- atom of each TGD that occurs in $P$. ###### Definition 4 (Equality Type) Consider an atom $\underline{a}$ of the form $r(t_{1},\ldots,t_{n})$, where $n\geq 1$. The _equality type_ of $\underline{a}$ is the set of equalities $\displaystyle\left\\{r[i]=r[j]~{}\|~{}t_{i},t_{j}\not\in\Delta_{c}\textrm{~{}and~{}}t_{i}=t_{j}\right\\}$ $\displaystyle\bigcup$ $\displaystyle\left\\{r[i]=c~{}\|~{}c\in\Delta_{c}\textrm{~{}and~{}}t_{i}=c\right\\}.$ We denote the above set as $\mathit{eq}(\underline{a})$. It is straightforward to see that, given a pair of TGDs $\sigma$ and $\sigma^{\prime}$, if $\mathit{eq}(\mathit{body}(\sigma^{\prime}))\subseteq\mathit{eq}(\mathit{head}(\sigma))$, then there exists a substitution $\mu$ such that $\mu(\mathit{body}(\sigma^{\prime}))=\mathit{head}(\sigma)$. This allows us to show that the atom obtained by applying $\sigma$ during the construction of the chase triggers $\sigma^{\prime}$. Consequently, the existence of a path $P$ (as above) guarantees the propagation of a term from $\pi_{1}$ to $\pi_{2}$ if, for each pair of consecutive edges $e$ and $e^{\prime}$ of $P$ which are labeled by $\sigma$ and $\sigma^{\prime}$, respectively, $\mathit{eq}(\mathit{body}(\sigma^{\prime}))\subseteq\mathit{eq}(\mathit{head}(\sigma))$. ###### Example 6 (Dependency Graph) Consider the set $\Sigma$ of TGDs $\begin{array}[]{rcl}\sigma_{1}&:&p(X,Y)\rightarrow\exists Zr(X,Y,Z)\\\ \sigma_{2}&:&r(X,Y,c)\rightarrow s(X,Y,Y)\\\ \sigma_{3}&:&s(X,X,Y)\rightarrow p(X,Y).\end{array}$ The equality type of the body-atoms and head-atoms of the TGDs of $\Sigma$ are as follows: $\begin{array}[]{rcl}\mathit{eq}(\mathit{body}(\sigma_{1}))&=&\varnothing\\\ \mathit{eq}(\mathit{head}(\sigma_{1}))&=&\varnothing\\\ \mathit{eq}(\mathit{body}(\sigma_{2}))&=&\\{r[3]=c\\}\\\ \mathit{eq}(\mathit{head}(\sigma_{2}))&=&\\{s[2]=s[3]\\}\\\ \mathit{eq}(\mathit{body}(\sigma_{3}))&=&\\{s[1]=s[2]\\}\\\ \mathit{eq}(\mathit{head}(\sigma_{3}))&=&\varnothing.\end{array}$ The dependency graph of $\Sigma$ is shown in Figure 2. We are now ready, by exploiting the dependency graph of a set of TGDs, and the equality type of an atom, to introduce _atom coverage_. Figure 2: Dependency graph for Example 6. ###### Definition 5 (Atom Coverage) Consider a BCQ $q$ over a schema $\mathcal{R}$, and a set $\Sigma$ of TGDs over $\mathcal{R}$. Let $\underline{a}$ and $\underline{b}$ be atoms of $\mathit{body}(q)$, where $\\{t_{1},\ldots,t_{n}\\}$, for $n\geq 0$, is the set of shared variables and constants that occur in $\underline{b}$. Also, let $G_{\Sigma}$ be the dependency graph of $\Sigma$. We say that $\underline{a}$ _covers_ $\underline{b}$ w.r.t. $q$ and $\Sigma$, written as $\underline{a}\prec_{\Sigma}^{q}\underline{b}$, if for each $i\in[n]$: (i) the term $t_{i}$ occurs also in $\underline{a}$, and (ii) if $t_{i}$ occurs in $\underline{a}$ and $\underline{b}$ at positions $\Pi_{\underline{a},i}$ and $\Pi_{\underline{b},i}$, respectively, then, there exists an integer $k\geq 2$ and a set of TGDs $\\{\sigma_{1},\ldots,\sigma_{k-1}\\}\subseteq\Sigma$, where $\mathit{eq}(\mathit{body}(\sigma_{1}))\subseteq\mathit{eq}(\underline{a})$ and, for each $j\in[k-2]$, $\mathit{eq}(\mathit{body}(\sigma_{j+1}))\subseteq\mathit{eq}(\mathit{head}(\sigma_{j}))$, such that, for each $\pi\in\Pi_{\underline{b},i}$, in $G_{\Sigma}$ there exists a path $\pi_{i_{1}}\pi_{i_{2}}\ldots\pi_{i_{k}}$, where $\pi_{i_{1}}\in\Pi_{\underline{a},i}$, $\pi_{i_{k}}=\pi$, and $\lambda((\pi_{i_{j}},\pi_{i_{j+1}}))=\sigma_{j}$, for each $j\in[k-1]$. Condition _(i)_ ensures that by removing $\underline{b}$ from $q$ we do not loose any constant, and also all the joins between $\underline{b}$ and the other atoms of $\mathit{body}(q)$, except $\underline{a}$, are preserved. Condition _(ii)_ guarantees that the atom $\underline{b}$ is logically implied (w.r.t. $\Sigma$) by the atom $\underline{a}$, and therefore can be eliminated. ###### Lemma 8 Consider a BCQ $q$ over a schema $\mathcal{R}$, and a set $\Sigma$ of linear TGDs over $\mathcal{R}$. Suppose that $\underline{a}\prec_{\Sigma}^{q}\underline{b}$, where $\underline{a},\underline{b}\in\mathit{body}(q)$, and $q^{\prime}$ is the BCQ obtained from $q$ by eliminating the atom $\underline{b}$. Then, $I\models q$ iff $I\models q^{\prime}$, for each instance $I$ that satisfies $\Sigma$. Proof (Sketch). ($\Rightarrow$) By hypothesis, there exists a homomorphism $h$ such that $h(\mathit{body}(q))\subseteq I$. Since, by definition of $q^{\prime}$, $\mathit{body}(q^{\prime})\subset\mathit{body}(q)$, we immediately get that $h(\mathit{body}(q^{\prime}))\subseteq I$, which implies that $I\models q^{\prime}$. ($\Leftarrow$) Conversely, there exists a homomorphism $h$ such that $h(\mathit{body}(q^{\prime}))\subseteq I$, and thus $h(\mathit{body}(q)\setminus\\{\underline{b}\\})\subseteq I$. It suffices to show that there exists an extension of $h$ which maps $\underline{b}$ to $I$. Since $\underline{a}\prec_{\Sigma}^{q}\underline{b}$, it is not difficult to verify that there exists an atom $\underline{c}\in I$ such that $\mathit{eq}(\underline{b})=\mathit{eq}(\underline{c})$, which implies that there exists a substitution $\mu$ such that $\mu(\underline{b})=\underline{c}$, and also $\mu$ is compatible with $h$. Consequently, $(h\cup\mu)(\mathit{body}(q))\subseteq I$, and thus $I\models q$. An _atom elimination strategy_ for a BCQ is a permutation of its body-atoms. Given a BCQ $q$ and a set $\Sigma$ of linear TGDs, the set of atoms of $\mathit{body}(q)$ that cover $\underline{a}\in\mathit{body}(q)$ w.r.t. $\Sigma$, denoted as $\mathit{cover}(\underline{a},q,\Sigma)$, is the set $\\{\underline{b}~{}\|~{}\underline{b}\in\mathit{body}(q)\textrm{~{}and~{}}\underline{b}\prec_{\Sigma}^{q}\underline{a}\\}$; when $q$ and $\Sigma$ are obvious from the context, we shall denote the above set as $\mathit{cover}(\underline{a})$. By exploiting the cover set of the atoms of $\mathit{body}(q)$, we associate to each atom elimination strategy $S$ for $q$ a subset of $\mathit{body}(q)$, denoted $\mathit{eliminate}(q,S,\Sigma)$, which is the set of atoms of $\mathit{body}(q)$ that can be safely eliminated (according to $S$) in order to obtain a logically equivalent query (w.r.t. $\Sigma$) with less atoms in its body. Formally, $\mathit{eliminate}(q,S,\Sigma)$ is computed by applying the following procedure; in the sequel, let $S=[\underline{a}_{1},\ldots,\underline{a}_{n}]$, where $\\{\underline{a}_{1},\ldots,\underline{a}_{n}\\}=\mathit{body}(q)$: $A:=\varnothing$; * foreach $i:=1$ to $n$ do * $\underline{a}:=S[i]$; * if $\mathit{cover}(\underline{a})\neq\varnothing$ then * $A:=A\cup\\{\underline{a}\\}$; * foreach $\underline{b}\in\mathit{body}(q)\setminus A$ do * $\mathit{cover}(\underline{b}):=\mathit{cover}(\underline{b})\setminus\\{\underline{a}\\}$; * return $A$. By exploiting the fact that the binary relation $\prec_{\Sigma}^{q}$ is transitive, it is possible to establish the uniqueness (w.r.t. the number of the eliminated atoms) of the atom elimination strategy for a BCQ. In particular, the following lemma can be shown. ###### Lemma 9 Consider a BCQ $q$ over a schema $\mathcal{R}$, and a set $\Sigma$ of linear TGDs over $\mathcal{R}$. Let $S_{1}$ and $S_{2}$ be arbitrary elimination strategies for $q$. It holds that, $\|\mathit{eliminate}(q,S_{1},\Sigma)\|=\|\mathit{eliminate}(q,S_{2},\Sigma)\|$. Since the elimination strategy for a query is unique (w.r.t. the number of the eliminated atoms), in the rest of this section we refer to the set of atoms that can be safely eliminated from a query $q$ (w.r.t. a set $\Sigma$ of linear TGDs) by $\mathit{eliminate}(q,\Sigma)$. We are now ready to describe how query elimination works. During the execution of the rewriting algorithm TGD-rewrite (see Algorithm 1), after the factorization step and the rewriting step the so-called _elimination_ step is applied. In particular, the factorized query $q^{\prime}$ obtained during the factorization step is the query $\mathit{eliminate}(\mathit{factorize}(q,\sigma),\Sigma)$, while the rewritten query obtained during the rewriting step is the query $\mathit{eliminate}(\gamma_{A\cup\\{\mathit{head}(\sigma)\\}}(q[A/\mathit{body}(\sigma)]),\Sigma)$. Moreover, instead of adding the given query $q$ in $Q_{\textsc{rew}}$, we add the eliminated query. In particular, the first line of the algorithm is replaced by $Q_{\textsc{rew}}:=\langle\mathit{eliminate}(q),1\rangle$. An example of query elimination follows. ###### Example 7 (Query Elimination) Consider the set $\Sigma$ of TGDs of Example 6, and the BCQ $\begin{array}[]{rcl}q()&\leftarrow&\underbrace{p(A,B)}_{\underline{a}},\underbrace{r(A,B,C)}_{\underline{b}},\underbrace{s(A,A,D)}_{\underline{c}}.\end{array}$ Based on the Definition 5, it is an easy task to verify that $\mathit{cover}(\underline{a})=\varnothing$, $\mathit{cover}(\underline{b})=\\{\underline{a}\\}$ and $\mathit{cover}(\underline{c})=\varnothing$. Therefore, the output of the function $\mathit{eliminate}(q,\Sigma)$ is the singleton set $\\{\underline{b}\\}$. Consequently, by applying the elimination step we get the BCQ $q()\leftarrow p(A,B),s(A,A,D)$. As already mentioned, the fact that an atom $\underline{a}$ covers some atom $\underline{b}$, means that $\underline{b}$ is logically implied (w.r.t. the given set of TGDs) by $\underline{a}$. However, as shown by the following example, this fact is not also necessary for the implication of $\underline{b}$ by $\underline{a}$. ###### Example 8 (Atom Implication) Consider the set $\Sigma$ of TGDs of Example 6, and the BCQ $q$ $\begin{array}[]{rcl}q()&\leftarrow&\underbrace{r(A,A,c)}_{\underline{a}},\underbrace{p(A,A)}_{\underline{b}},\end{array}$ where $c$ is a constant of $\Delta_{c}$. Observe that $\underline{a}$ does not cover $\underline{b}$ since, despite the existence of the paths $r[1]s[1]p[1]$ and $r[2]s[3]p[2]$ in the dependency graph of $\Sigma$, $\mathit{eq}(\mathit{body}(\sigma_{3}))\not\subseteq\mathit{eq}(\mathit{head}(\sigma_{2}))$. However, $\underline{b}$ is logically implied (w.r.t. $\Sigma$) by $\underline{a}$. In particular, for every instance $I$ that satisfies $\Sigma$, if $I\models\underline{a}$, which implies that an atom of the from $r(V,V,c)$ exists in $I$, then due to the TGDs $\sigma_{2}$ and $\sigma_{3}$ there exists also an atom $p(V,V)$, and thus $I\models\underline{b}$. Note that such cases are identified by the C&B algorithm [15]. Nevertheless, as already criticized in Section 2, this requires to pay a price in the number of queries in the rewritten query. It is not difficult to see that the function eliminate runs in quadratic time in the number of atoms of $\mathit{body}(q)$ (by considering the given set of TGDs as fixed). In particular, to compute the cover set of each body-atom of $q$ we need to consider all the pairs of atoms of $\mathit{body}(q)$. Note that the problem whether a certain atom $\underline{a}$ covers some other atom $\underline{b}$ is feasible in constant time since the given set of TGDs (and thus its dependency graph) is fixed. The following result implies that the rewriting algorithm $\textsf{TGD- rewrite}^{\star}$, obtained from TGD-rewrite by applying the additional step of elimination, is still sound and complete. ###### Theorem 10 Consider a BCQ $q$ over a schema $\mathcal{R}$, a database $D$ for $\mathcal{R}$, and a set $\Sigma$ of linear TGDs over $\mathcal{R}$. Then, $D\models\textsf{TGD-rewrite}^{\star}(\mathcal{R},\Sigma,q)$ iff $D\cup\Sigma\models q$. Proof (Sketch). This result follows from the fact that the algorithm TGD- rewrite is sound and complete under linear TGDs (see Theorem 6) and Lemma 8. It is important to clarify that the above result does not hold if we consider arbitrary TGDs (as in Theorem 6). This is because the proof of Lemma 8, which states that atom coverage implies logical implication (w.r.t. the given set of TGDs), is based heavily on the linearity of TGDs. Termination of $\textsf{TGD- rewrite}^{\star}$ follows immediately from the fact that TGD-rewrite terminates under linear TGDs (see Theorem 7). ## 7 Implementation and Experimental Setting TGD-rewrite (without the additional check described in Subsection 5.1) and the query elimination technique presented in Section 6 have been implemented in the prototype system Nyaya [36] available at http://mais.dia.uniroma3.it/Nyaya. The reasoning and query answering engine is based on the IRIS Datalog engine888http://www.iris-reasoner.org/. extended to support the FO-rewritable fragments of the Datalog± family. In particular, we extended IRIS to natively support existential variables in the head without introducing function symbols and to support the constant $\mathit{false}$ as head of a rule (used to represent negative constraints). Both IRIS and our extension are implemented in Java. Since TGD-rewrite is designed for reasoning over ontologies with large ABoxes, we put ourselves in a similar experimental setting such that of [19]. Thus, we use DL-LiteR ontologies with a varying number of axioms. The queries under consideration are based on canonical examples used in the research projects where these ontologies have been developed. VICODI (V) is an ontology of European history, and developed in the EU-funded VICODI project999http://www.vicodi.org.. STOCKEXCHANGE (S) is an ontology for representing the domain of financial institutions of the European Union. UNIVERSITY (U) is a DL-LiteR version of the LUBM Benchmark101010http://swat.cse.lehigh.edu/projects/lubm/., developed at Lehigh University, and describes the organizational structure of universities. ADOLENA (A) (Abilities and Disabilities OntoLogy for ENhancing Accessibility) is an ontology developed for the South African National Accessibility Portal, and describes abilities, disabilities and devices. The Path5 (P5) ontology is a synthetic ontology encoding graph structures and used to generate an exponential-blowup of the size of the rewritten queries. Recall that the transformation of a set of TGDs into an equivalent set of single-head TGDs with a single existential variable can introduce auxiliary predicates and rules (see Lemmas 1 and 2). The ontologies UX, AX and P5X are equivalent ontologies to U, A and P5 where the auxiliary predicates are considered part of the schema. These ontologies allow to study the impact of such transformations on the size of the rewriting. We compared our implementation with two other rewriting-based query answering systems for FO-rewritable ontologies: QuOnto111111http://www.dis.uniroma1.it/quonto/., based on [5] and developed by the University of Rome La Sapienza, and Requiem121212http://www.comlab.ox.ac.uk/projects/requiem/home.html., based on [19] and developed by the Knowledge Representation group of the University of Oxford. Table 1: Evaluation of Nyaya System. \| \| Size \| Length \| Width ---\|---\|---\|---\|--- \| \| QO \| RQ \| NY \| NY⋆ \| QO \| RQ \| NY \| NY⋆ \| QO \| RQ \| NY \| NY⋆ V \| $q_{1}$ \| 15 \| 15 \| 15 \| 15 \| 15 \| 15 \| 15 \| 15 \| 0 \| 0 \| 0 \| 0 $q_{2}$ \| 11 \| 10 \| 10 \| 10 \| 32 \| 30 \| 30 \| 30 \| 31 \| 30 \| 30 \| 30 $q_{3}$ \| 72 \| 72 \| 72 \| 72 \| 216 \| 216 \| 216 \| 216 \| 144 \| 144 \| 144 \| 144 $q_{4}$ \| 185 \| 185 \| 185 \| 185 \| 555 \| 555 \| 555 \| 555 \| 370 \| 370 \| 370 \| 370 $q_{5}$ \| 150 \| 30 \| 30 \| 30 \| 900 \| 210 \| 210 \| 210 \| 1,110 \| 270 \| 270 \| 270 S \| $q_{1}$ \| 6 \| 6 \| 6 \| 6 \| 6 \| 6 \| 6 \| 6 \| 0 \| 0 \| 0 \| 0 $q_{2}$ \| 204 \| 160 \| 160 \| 2 \| 566 \| 480 \| 480 \| 2 \| 362 \| 320 \| 320 \| 0 $q_{3}$ \| 1,194 \| 480 \| 480 \| 4 \| 5,026 \| 2,400 \| 2,400 \| 8 \| 4,778 \| 2,400 \| 2,400 \| 4 $q_{4}$ \| 1,632 \| 960 \| 960 \| 4 \| 7,384 \| 4,800 \| 4,800 \| 8 \| 7,112 \| 4,800 \| 4,800 \| 4 $q_{5}$ \| 11,487 \| 2,880 \| 2,880 \| 8 \| 67,664 \| 20,160 \| 20,160 \| 24 \| 84,064 \| 25,920 \| 25,920 \| 24 U \| $q_{1}$ \| 5 \| 2 \| 2 \| 2 \| 10 \| 4 \| 4 \| 4 \| 5 \| 2 \| 2 \| 2 $q_{2}$ \| 287 \| 148 \| 148 \| 1 \| 813 \| 444 \| 444 \| 1 \| 526 \| 296 \| 296 \| 0 $q_{3}$ \| 1,260 \| 224 \| 224 \| 4 \| 7,296 \| 1,344 \| 1,344 \| 16 \| 10,812 \| 2,016 \| 2,016 \| 20 $q_{4}$ \| 5,364 \| 1,628 \| 1,628 \| 2 \| 15,723 \| 4,884 \| 4,884 \| 2 \| 10,393 \| 3,256 \| 3,256 \| 0 $q_{5}$ \| 9,245 \| 2,960 \| 2,960 \| 10 \| 35,710 \| 11,840 \| 11,840 \| 20 \| 52,970 \| 17,760 \| 17,760 \| 20 A \| $q_{1}$ \| 783 \| 402 \| 402 \| 247 \| 1,540 \| 779 \| 779 \| 197 \| 757 \| 377 \| 377 \| 86 $q_{2}$ \| 1,812 \| 103 \| 103 \| 92 \| 5,350 \| 256 \| 256 \| 234 \| 3,538 \| 153 \| 153 \| 142 $q_{3}$ \| 4,763 \| 104 \| 104 \| 104 \| 23,804 \| 520 \| 520 \| 520 \| 23,804 \| 520 \| 520 \| 520 $q_{4}$ \| 7,251 \| 492 \| 492 \| 454 \| 21,406 \| 1,288 \| 1,288 \| 1,212 \| 14,155 \| 796 \| 796 \| 758 $q_{5}$ \| 66,068 \| 624 \| 624 \| 624 \| 195,042 \| 3,120 \| 3,120 \| 3,120 \| 128,974 \| 3,120 \| 3,120 \| 3,120 P5 \| $q_{1}$ \| 14 \| 6 \| 6 \| 6 \| 14 \| 6 \| 6 \| 6 \| 0 \| 0 \| 0 \| 0 $q_{2}$ \| 86 \| 10 \| 10 \| 10 \| 156 \| 16 \| 16 \| 16 \| 70 \| 6 \| 6 \| 6 $q_{3}$ \| 538 \| 13 \| 13 \| 13 \| 1,413 \| 29 \| 29 \| 29 \| 900 \| 16 \| 16 \| 16 $q_{4}$ \| 3,620 \| 15 \| 15 \| 15 \| 14,430 \| 44 \| 44 \| 44 \| 10,260 \| 29 \| 29 \| 29 $q_{5}$ \| 25,256 \| 16 \| 16 \| 16 \| 107,484 \| 60 \| 60 \| 60 \| 103,361 \| 44 \| 44 \| 44 UX \| $q_{1}$ \| 5 \| 5 \| 5 \| 5 \| 10 \| 10 \| 10 \| 10 \| 5 \| 5 \| 5 \| 5 $q_{2}$ \| 286 \| 240 \| 240 \| 1 \| 156 \| 147 \| 147 \| 1 \| 70 \| 70 \| 70 \| 0 $q_{3}$ \| 1,248 \| 1,008 \| 1,008 \| 12 \| 1,397 \| 1,125 \| 1,125 \| 48 \| 892 \| 735 \| 735 \| 60 $q_{4}$ \| 5,358 \| 5,000 \| 5,000 \| 5 \| 12,006 \| 7,578 \| 7,578 \| 5 \| 9,828 \| 5,625 \| 5,625 \| 0 $q_{5}$ \| 9,220 \| 8,000 \| 8,000 \| 25 \| 101,652 \| 47,656 \| 47,656 \| 50 \| 96,677 \| 37,890 \| 37,890 \| 50 AX \| $q_{1}$ \| 783 \| 782 \| 782 \| 555 \| 1,543 \| 1,541 \| 1,541 \| 1,084 \| 763 \| 761 \| 761 \| 529 $q_{2}$ \| 1,812 \| 1,781 \| 1,781 \| 1,737 \| 3,589 \| 3,528 \| 3,528 \| 3,514 \| 3,576 \| 3,516 \| 3,516 \| 3,401 $q_{3}$ \| 4,763 \| 4,752 \| 4,752 \| 4,741 \| 27,705 \| 23,760 \| 23,760 \| 23,760 \| 23,824 \| 23,815 \| 23,815 \| 23,694 $q_{4}$ \| 7,251 \| 7,100 \| 7,100 \| 6,564 \| 7,739 \| 7,578 \| 7,578 \| 6,178 \| 5,744 \| 5,625 \| 5,625 \| 5,201 $q_{5}$ \| - \| - \| 76,032 \| 76,032 \| - \| - \| 81,173 \| 81,173 \| - \| - \| 95,942 \| 95,942 P5X \| $q_{1}$ \| 14 \| 14 \| 14 \| 14 \| 14 \| 14 \| 14 \| 14 \| 0 \| 0 \| 0 \| 0 $q_{2}$ \| 86 \| 77 \| 77 \| 66 \| 156 \| 147 \| 147 \| 121 \| 70 \| 70 \| 70 \| 55 $q_{3}$ \| 530 \| 390 \| 390 \| 329 \| 1,397 \| 1,125 \| 1,125 \| 925 \| 892 \| 735 \| 735 \| 596 $q_{4}$ \| 3,476 \| 1,953 \| 1,953 \| 1,644 \| 12,006 \| 7,578 \| 7,578 \| 6,263 \| 9,828 \| 5,625 \| 5,625 \| 4,619 $q_{5}$ \| 23,744 \| 9,766 \| 9,766 \| 8,219 \| 101,652 \| 47,656 \| 47,656 \| 39,531 \| 96,677 \| 37,890 \| 37,890 \| 31,312 Since TGD-rewrite, as well as the algorithms presented in [5] and [19], are proven to be sound and complete, the most relevant way of judging the quality of the rewriting is the _size_ of the perfect rewriting, i.e., the number of CQs in the perfect UCQ rewriting. In addition, we use two additional metrics, namely, the _length_ of the rewriting, i.e., the number of atoms in the perfect rewriting, and the _width_ , i.e., the number of joins to be performed when the rewritten query is executed. We believe these metrics to be more appropriate than the number of symbols in the rewritten query used, for example, in [19], since they allow to establish in a more precise way the cost of executing the rewriting on a database system. Table 1 reports the results of our experiments131313Additional data can be found on the Nyaya’s Web site. while Table 2 shows the queries used in the experiments. We use the symbol “-” to denote those cases where the tool did not complete the rewriting within 15 minutes. By QO and RQ we refer to the QuOnto and Requiem systems, respectively, while NY and NY⋆ refer to Nyaya with factorisation and Nyaya with both factorisation and query elimination, respectively. All the tests have been performed on an Intel Core 2 Duo Processor at 2.50 GHz and 4GB of RAM. The OS is Ubuntu Linux 9.10 carrying a Sun JVM Standard Edition with maximum heap size set at 2GB of RAM. Table 2: Test Queries TBox \| Queries ---\|--- V \| $q_{1}(A)\leftarrow\mathit{Location(A).}$ $q_{2}(A,B)\leftarrow\mathit{Military\\_Person(A),hasRole(B,A),related(A,C).}$ $q_{3}(A,B)\leftarrow\mathit{Time\\_Dependant\\_Relation(A),hasRelationMember(A,B),Event(B).}$ $q_{4}(A,B)\leftarrow\mathit{Object(A),hasRole(A,B),Symbol(B).}$ $q_{5}(A)\leftarrow\mathit{Individual(A),hasRole(A,B),Scientist(B),hasRole(A,C),Discoverer(C),hasRole(A,D),Inventor(D).}$ S \| $q_{1}(A)\leftarrow\mathit{StockExchangeMember(A).}$ $q_{2}(A,B)\leftarrow\mathit{Person(A),hasStock(A,B),Stock(B).}$ $q_{3}(A,B,C)\leftarrow\mathit{FinantialInstrument(A),belongsToCompany(A,B),Company(B),hasStock(B,C),Stock(C).}$ $q_{4}(A,B,C)\leftarrow\mathit{Person(A),hasStock(A,B),Stock(B),isListedIn(B,C),StockExchangeList(C).}$ $q_{5}(A,B,C,D)\leftarrow\mathit{FinantialInstrument(A),belongsToCompany(A,B),Company(B),hasStock(B,C),Stock(C),}$ \| $\mathit{isListedIn(B,D),StockExchangeList(D).}$ U(X) \| $q_{1}(A)\leftarrow\mathit{worksFor(A,B),affiliatedOrganizationOf(B,C).}$ $q_{2}(A,B)\leftarrow\mathit{Person(A),teacherOf(A,B),Course(B).}$ $q_{3}(A,B,C)\leftarrow\mathit{Student(A),advisor(A,B),FacultyStaff(B),takesCourse(A,C),teacherOf(B,C),Course(C).}$ $q_{4}(A,B)\leftarrow\mathit{Person(A),worksFor(A,B),Organization(B).}$ $q_{5}(A)\leftarrow\mathit{Person(A),worksFor(A,B),University(B),hasAlumnus(B,A).}$ A(X) \| $q_{1}(A)\leftarrow\mathit{Device(A),assistsWith(A,B).}$ $q_{2}(A)\leftarrow\mathit{Device(A),assistsWith(A,B),UpperLimbMobility(B).}$ $q_{3}(A)\leftarrow\mathit{Device(A),assistsWith(A,B),Hear(B),affects(C,B),Autism(C).}$ $q_{4}(A)\leftarrow\mathit{Device(A),assistsWith(A,B),PhysicalAbility(B).}$ $q_{5}(A)\leftarrow\mathit{Device(A),assistsWith(A,B),PhysicalAbility(B),affects(C,B),Quadriplegia(C).}$ P5(X) \| $q_{1}(A)\leftarrow\mathit{edge(A,B).}$ $q_{2}(A)\leftarrow\mathit{edge(A,B),edge(B,C).}$ $q_{3}(A)\leftarrow\mathit{edge(A,B),edge(B,C),edge(C,D).}$ $q_{4}(A)\leftarrow\mathit{edge(A,B),edge(B,C),edge(C,D),edge(D,E).}$ $q_{4}(A)\leftarrow\mathit{edge(A,B),edge(B,C),edge(C,D),edge(D,E),edge(E,F).}$ As it can be seen, query elimination provides a substantial advantage in terms of the size of the perfect rewriting for the real-world ontologies A, U and S. In particular, for the queries denoted as Q2 in U and S, our procedure eliminates all the redundant atoms in the input query, and drastically reduces the number of queries in the final rewriting. On the other side, query elimination is not particularly effective in the synthetic test case P5 and P5X, since these cases have been intentionally created in order to generate perfect rewritings of exponential size. ## 8 Future Work We plan to investigate rewriting and optimization techniques for sticky-join sets of TGDs, and alternative forms of rewriting such as positive-existential queries. We also plan to develop improved techniques for rewriting an ontological query into a non-recursive Datalog program, rather than into a union of conjunctive queries (recall the discussion in Section 2). While the current approaches yield exponentially large non-recursive Datalog programs, it is possible to rewrite queries and TBoxes into non-recursive Datalog programs whose size is simultaneously polynomial in the query and the TBox. This will be dealt in a forthcoming paper. ## Acknowledgments G. Gottlob’s work was funded by the European Research Council under the European Community’s Seventh Framework Programme (FP7/2007-2013)/ERC grant no. 246858 – DIADEM. Gottlob gratefully acknowledges a Royal Society Wolfson Research Merit Award. G. Orsi and G. Gottlob also acknowledge the Oxford Martin School - Institute for the Future of Computing. A. Pieris’ work was funded by the EPSRC project “Schema Mappings and Automated Services for Data Integration and Exchange” (EP/E010865/1). We thank Michaël Thomazo for his useful and constructive comments on the conference version of this paper. ## References * [1] G. Gottlob, G. Orsi, and A. Pieris, “Ontological queries: Rewriting and optimization,” in _Proc. of the 27th Intl Conf. on Data Engineering (ICDE)_ , 2011, pp. 2–13. * [2] T. R. Gruber, “A translation approach to portable ontology specifications,” _Knowledge Acquisition_ , vol. 5, no. 2, pp. 199–220, 1993. * [3] Wikipedia, “Ontology (information science),” 2010. [Online]. Available: –http://en.wikipedia.org/wiki/Ontology“˙(information“˙science)˝. * [4] D. McComb, “The enterprise ontology,” 2006. [Online]. Available: –http://www.tdan.com/view-articles/5016˝. * [5] D. Calvanese, G. de Giacomo, D. Lembo, M. Lenzerini, and R. Rosati, “Tractable reasoning and efficient query answering in description logics: The DL-Lite Family,” _Journal of Automated Reasoning_ , vol. 39, no. 3, pp. 385–429, 2007. * [6] F. Baader, “Terminological cycles in a description logic with existential restrictions,” in _Proc. of 18th Intl Joint Conf. on Artificial Intelligence (IJCAI)_ , 2003, pp. 325–330. * [7] A. Calì, G. Gottlob, and T. Lukasiewicz, “A general Datalog-based framework for tractable query answering over ontologies,” in _Proc. of the 28th Symp. on Principles of Database Systems (PODS)_ , 2009, pp. 77–86. * [8] ——, “Datalog${}^{\mbox{$\pm$}}$: A unified approach to ontologies and integrity constraints,” in _Proc. of the 12th Intl Conf. on Database Theory (ICDT)_ , 2009, pp. 14–30. * [9] A. Calì, G. Gottlob, and A. Pieris, “Advanced processing for ontological queries,” in _Proc. of the 36th Intl Conf. on Very Large Databases (VLDB)_ , 2010, pp. 554–565. * [10] ——, “Query answering under non-guarded rules in Datalog±,” in _Proc. of the 4th Intl conf. on Web Reasoning and Rule Systems (RR)_ , 2010, pp. 1–17. * [11] S. Ceri, G. Gottlob, and L. Tanca, “What you always wanted to know about Datalog (and never dared to ask),” _IEEE Transactions on Knowledge and Data Engineering_ , vol. 1, no. 1, pp. 146–166, 1989. * [12] C. Beeri and M. Y. Vardi, “A proof procedure for data dependencies,” _Journal of the ACM_ , vol. 31, no. 4, pp. 718–741, 1984. * [13] M. Y. Vardi, “On the complexity of bounded-variable queries,” in _Proc. of the 14th Symp. on Principles of Database Systems (PODS)_ , 1995, pp. 266–276. * [14] A. Calì, G. Gottlob, and A. Pieris, “Query rewriting under non-guarded rules,” in _Proc. of the 4th Alberto Mendelzon Intl Work. on Foundations of Data Management (AMW)_ , 2010. * [15] A. Deutsch, L. Popa, and V. Tannen, “Query reformulation with constraints,” _SIGMOD Record_ , vol. 35, pp. 65–73, 2006. * [16] S. Alexaki, V. Christophides, G. Karvounarakis, D. Plexousakis, and K. Tolle, “The ICS-FORTH RDFSuite: Managing voluminous RDF description bases,” in _Proc. of the 2nd Intl Workshop on the Semantic Web (SemWeb)_ , 2001, pp. 109–113. * [17] E. Chong, S. Das, G. Eadon, and J. Srinivasan, “An efficient SQL-based RDF querying scheme,” in _Proc. of the 31th Intl Conf. on Very Large Data Bases (VLDB)_ , 2005, pp. 1216–1227. * [18] G. Gottlob, N. Leone, and F. Scarcello, “Hypertree decompositions and tractable queries,” in _In Proc. of the 18th Symp. on Principles of database systems (PODS)_ , 1999, pp. 21–32. * [19] H. Pérez-Urbina, B. Motik, and I. Horrocks, “Efficient query answering for OWL 2,” in _Proc. of the 8th Intl Semantic Web Conf. (ESWC)_ , 2009, pp. 489–504. * [20] R. Rosati and A. Almatelli, “Improving query answering over DL-Lite ontologies,” in _In Proc. of the 20th Intl Conf. on Principles of Knowledge Representation (KR)_ , 2010. * [21] A. K. Chandra and P. M. Merlin, “Optimal implementation of conjunctive queries in relational data bases,” in _Proc. of the 9th ACM Symp. on Theory of Computing (STOC)_ , 1977, pp. 77–90. * [22] A. Halevy, “Answering queries using views: A survey,” _The VLDB Journal_ , vol. 10, pp. 270–294, 2001. * [23] A. Deutsch and V. Tannen, “Mars: A system for publishing XML from mixed and redundant storage,” in _In Proc. of the 29th Intl Conf. on Very large data bases (VLDB)_ , 2003, pp. 201–212. * [24] C. Beeri and M. Y. Vardi, “The implication problem for data dependencies,” in _Proc. of the 8th Colloquim on Automata, Languages and Programming (ICALP)_ , 1981, pp. 73–85. * [25] A. Calı, G. Gottlob, and M. Kifer, “Taming the infinite chase: Query answering under expressive relational constraints,” in _Proc. of the 11th Intl Joint Conf. on Principles of Knowledge Representation and Reasoning (KR)_ , 2008, pp. 70–80. * [26] A. Deutsch, A. Nash, and J. Remmel, “The chase revisited,” in _Proc. of the 27th Symp. on Principles of Database Systems (PODS)_ , 2008, pp. 149–158. * [27] D. Maier, A. O. Mendelzon, and Y. Sagiv, “Testing implications of data dependencies,” _ACM Trans. on Database Systems_ , vol. 4, no. 4, pp. 455–469, 1979. * [28] D. S. Johnson and A. C. Klug, “Testing containment of conjunctive queries under functional and inclusion dependencies,” _Journal of Computer and System Sciences_ , vol. 28, no. 1, pp. 167–189, 1984. * [29] R. Fagin, P. Kolaitis, R. Miller, and L. Popa, “Data exchange: Semantics and query answering,” _Theoretical Computer Science_ , vol. 336, no. 1, pp. 89–124, 2005. * [30] B. Marnette, “Generalized schema-mappings: From termination to tractability,” in _In Proc. of the 28th Symp. on Principles of Database Systems (PODS)_ , 2009, pp. 13–22. * [31] A. Calì, G. Gottlob, and A. Pieris, “Tractable query answering over conceptual schemata,” in _Proc. of the 28th Intl Conf. on Conceptual Modeling (ER)_ , 2009, pp. 175–190. * [32] ——, “Query answering under expressive Entity-Relationship schemata,” in _Proc. of the 29th Intl Conf. on Conceptual Modeling (ER)_ , 2010, pp. 347–361. * [33] A. K. Chandra and M. Y. Vardi, “The implication problem for functional and inclusion dependencies is undecidable,” _SIAM Journal of Computing_ , vol. 14, no. 3, pp. 671–677, 1985. * [34] S. Abiteboul, R. Hull, and V. Vianu, _Foundations of Databases_. Addison-Wesley, 1995. * [35] A. Calì, D. Lembo, and R. Rosati, “On the decidability and complexity of query answering over inconsistent and incomplete databases,” in _Proc. of the 22nd Symp. on Principles of Database Systems (PODS)_ , 2003, pp. 260–271. * [36] R. de Virgilio, G. Orsi, L. Tanca, and R. Torlone, “Semantic data markets: a flexible environment for knowledge management,” in _Proc. of 20th Intl Conf. on Information and Knowledge Management (CIKM)_ , 2011, pp. 1559–1564.	arxiv-papers	2011-12-01T22:19:24	2024-09-04T02:49:24.885695	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Georg Gottlob and Giorgio Orsi and Andreas Pieris", "submitter": "Giorgio Orsi PhD", "url": "https://arxiv.org/abs/1112.0343" }
1112.0408	# Late-time symmetry near black hole horizons Kentaro Tanabe Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan Tetsuya Shiromizu Shunichiro Kinoshita Department of Physics, Kyoto University, Kyoto 606-8502, Japan ###### Abstract It is expected that black holes are formed dynamically under gravitational collapses and approach stationary states. In this paper, we show that the asymptotic Killing vector at late time should exist on the horizon and then that it can be extended outside black holes under the assumption of the analyticity of spacetimes. This fact implies that if there is another asymptotic Killing vector which becomes a stationary Killing at a far region and spacelike in the “ergoregion,” the rotating black holes may have the asymptotically axisymmetric Killing vector at late time. Thus, we may expect that the asymptotic rigidity of the black holes holds. ###### pacs: 04.20.-q, 04.20.Ha ††preprint: YITP-11-99 ## I introduction Black holes in our Universe are expected to be formed under gravitational collapses, and to finally approach stationary and vacuum states by radiating and absorbing energy, momentum, and angular momentum. Then the uniqueness theorem Israel guarantees that the black hole candidates in our Universe are the Kerr black holes. The key ingredient for the proof of the uniqueness theorem is the rigidity theorem Hawking:1973uf ; Hawking:1971vc ; Friedrich:1998wq ; Racz:1999ne . The rigidity theorem shows that the stationary rotating black holes have axisymmetric Killing vectors. The outline of the proof is as follows: The stationarity of spacetime implies that there are no gravitational waves around a black hole. Then the expansion and shear of the event horizon must vanish by virtue of the Raychaudhuri equation and stationarity. Using the Einstein equations, then, we can find that the null geodesic generator of the event horizon is a Killing vector. If the black hole is rotating, this new Killing vector may deviate from the stationary Killing vector which becomes spacelike in the ergoregion. This means that the stationary black holes might have two Killing vectors, that is, the stationary and axisymmetric Killing vectors. Hence the stationary black holes should rotate rigidly. However, the late-time phase of black holes produced by gravitational collapses would not be exactly stationary but nearly stationary. “Nearly stationary” means that the black holes are surrounded by the gravitational waves at late time. Then we cannot apply the rigidity theorem to such black holes because the expansion and shear of the null geodesic generator on the event horizon do not vanish due to the presence of the gravitational waves on the event horizon. Note that the late-time behaviors of the perturbations around the Schwarzschild and Kerr black holes were examined and it was shown that the perturbations decay at late time both on the horizon and null infinity at the same rate (for examples, see Refs. Gundlach:1993tp ; Barack:1999ma ; Barack:1999ya ). In the dynamical processes in gravitational collapses, however, it is quite nontrivial whether the formed black hole approaches the Kerr black hole. In this paper, we show that the asymptotic Killing vector, which will asymptotically approach a Killing vector at late time, should exist on the horizon without assuming any symmetries and then that it can be extended outside of the event horizon using the Einstein equations. This indicates that if there is another asymptotic Killing vector which will be an asymptotically stationary Killing vector at a far region, the rotating black hole may have the asymptotically axisymmetric Killing vector at late time. This paper is organized as follows. In the next section we provide the Einstein equations near the future event horizons. In Sec. III, we investigate the late-time behaviors of the metric on the event horizon and find the late- time symmetry. Then, under the assumption of the analyticity, we show that this late-time symmetry can be extended outside of the event horizon using the Einstein equation. In Sec. IV, we provide a summary and discussion. In Appendix A we perform the $(n+1)$ decompositions of the Einstein equation. Using them, we provide the explicit form of the Einstein equation in the current form of the metric in Appendix B. Almost a similar formulation is developed in the study on null infinity Tanabe2011 . In Appendix C, we present the details of the discussion associated with gauge freedom. ## II Bondi-like coordinate and Einstein equations In this section, introducing the Bondi-like coordinate near the event horizon, we investigate the initial value problem. For the details of these derivations, see Appendixes A and B. ### II.1 Bondi-like coordinate near event horizon We consider a dynamical black hole and investigate the late-time behavior of the near horizon geometry at late time. The black hole is defined as the region which is not contained in $J^{-}(\mathscr{I}^{+})$ where $\mathscr{I}^{+}$ is the future null infinity. $J^{-}(S)$ is the chronological past connecting a set $S$ by causal curves from $S$. The event horizon is the boundary of the black hole and it is a null hypersurface. See Ref. Hawking:1973uf for the precise definitions. Then, we introduce the Bondi-like (Gaussian null) coordinates $x^{A}=(u,r,x^{a})$ near the event horizon as $ds^{2}=g_{AB}dx^{A}dx^{B}=-Adu^{2}+2dudr+h_{ab}(dx^{a}+U^{a}du)(dx^{b}+U^{b}du),$ (1) where $u$ is a time coordinate. In this coordinate, the horizon position is taken to be $r=0$. We assume that a cross section of the event horizon is compact in the $u=\text{constant}$ hypersurface and its topology is $S^{2}$. $x^{a}$ are coordinates on $S^{2}$. Since the event horizon is a null hypersurface, $g_{uu}$ must vanish on the event horizon and $l=\partial/\partial u$ is the null geodesic generator on the event horizon. Furthermore we can choose the coordinate $u$ so that $l_{a}=0$ on the event horizon. Then, we have $A\,\hat{=}\,0,$ (2) and $U^{a}\,\hat{=}\,0,$ (3) where $\hat{=}$ means the evaluation on the event horizon $r=0$. To solve the vacuum Einstein equations, we formulate the initial value problem in the Bondi coordinates. For the convenience of our study in the following sections, we solve the Einstein equations as the evolution equations in the direction of $r$. Thus, the initial value of the metric should be set on the $r=\text{constant}$ surface. In this paper we take the event horizon $r=0$ as the initial surface. The evolution equations are given by $\displaystyle R_{rr}$ $\displaystyle=$ $\displaystyle-\frac{1}{2}(\log h)^{\prime\prime}-\frac{1}{4}h^{ac}h^{bd}(h_{ab})^{{}^{\prime}}(h_{cd})^{{}^{\prime}}\,=\,0,$ (4) $\displaystyle R_{rB}h^{aB}$ $\displaystyle=$ $\displaystyle\frac{1}{2}U^{a^{\prime\prime}}+\frac{1}{2}h^{ac}h_{bc}^{{}^{\prime}}U^{b^{\prime}}+\frac{1}{4}(\log h)^{\prime}U^{a^{\prime}}+\frac{1}{2}h^{ab}\bar{D}^{c}[h_{bc}^{{}^{\prime}}-h_{bc}(\log h)^{\prime}]\,=\,0$ (5) and $\displaystyle R_{AB}h^{A}_{~{}a}h^{B}_{~{}b}$ $\displaystyle=$ $\displaystyle-\frac{1}{2}Ah^{{}^{\prime\prime}}_{ab}-\frac{1}{2}A^{{}^{\prime}}h^{{}^{\prime}}_{ab}-(\dot{h}_{ab})^{\prime}+{}^{(h)}R_{ab}+\frac{A}{2}h^{cd}h^{{}^{\prime}}_{ac}h^{{}^{\prime}}_{bd}+\frac{1}{2}h^{cd}(h^{{}^{\prime}}_{ac}\dot{h}_{bd}+h^{{}^{\prime}}_{bd}\dot{h}_{ac})+\mathcal{L}_{U}h^{\prime}_{ab}$ (6) $\displaystyle~{}~{}-\frac{1}{2}h^{{}^{\prime}}_{ac}(\bar{D}_{b}U^{c}+\bar{D}^{c}U_{b})-\frac{1}{2}h^{{}^{\prime}}_{bc}(\bar{D}_{a}U^{c}+\bar{D}^{c}U_{a})-\frac{1}{2}h_{ac}h_{bd}U^{c}{}^{{}^{\prime}}U^{d}{}^{{}^{\prime}}-\frac{1}{4}[\dot{(\log{h})}-2\bar{D}_{a}U^{a}]h^{{}^{\prime}}_{ab}$ $\displaystyle~{}~{}-\frac{1}{4}(\log{h})^{{}^{\prime}}(Ah^{{}^{\prime}}_{ab}+\dot{h}_{ab}-\bar{D}_{a}U_{b}-\bar{D}_{b}U_{a})+\frac{1}{2}(h_{bc}\bar{D}_{a}U^{c}{}^{{}^{\prime}}+h_{ac}\bar{D}_{b}U^{c}{}^{{}^{\prime}})\,=\,0,$ where the prime and dot denote the $r$ and $u$ derivative, respectively, $\bar{D}_{a}$ is a covariant derivative with $h_{ab}$ and $h=\det{h_{ab}}$. Also, ${}^{(h)}R_{ab}$ is the Ricci tensor with respect to $h_{ab}$. The evolutions of the metric functions $A$, $U^{a}$ and $h_{ab}$ are determined by Eqs. (4), (5) and (6) completely. Note that the trace part of Eq. (6) gives us $\displaystyle A^{{}^{\prime}}(\log{h})^{{}^{\prime}}$ $\displaystyle=$ $\displaystyle 2{}^{(h)}R-\frac{A}{2}[(\log{h})^{{}^{\prime}}]^{2}-[\dot{(\log{h})}-2\bar{D}_{a}U^{a}](\log{h})^{{}^{\prime}}-h_{ab}U^{a^{\prime}}U^{b^{\prime}}$ (7) $\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-A(\log{h})^{{}^{\prime\prime}}-2\dot{(\log{h})}{}^{{}^{\prime}}+2(\bar{D}_{a}U^{a})^{{}^{\prime}}+U^{a}\bar{D}_{a}(\log h)^{\prime},$ which is used as the evolution equation for $A$. The other components of the Einstein equations, $R_{uu}=0$, $R_{ur}=0$ and $R_{ua}=0$, are related to the above evolution equations by the Bianchi identities. Therefore, once they are satisfied at the initial surface $r=0$, we do not need to solve them any more. In fact, if the evolution equations are satisfied, the Bianchi identities lead to $\left\\{\begin{aligned} (\sqrt{h}R_{ua})^{\prime}&=\,\sqrt{h}R_{ur},\\\ (\sqrt{h}R^{r}{}_{u})^{\prime}&=\,-\sqrt{h}\bar{D}_{a}R^{a}{}_{u}-\dot{(\sqrt{h})}R_{ur},\\\ (\log h)^{\prime}R_{ur}&=\,0,\end{aligned}\right.$ (8) where $R^{r}{}_{u}=R_{uu}+AR_{ur}-U^{a}R_{ua}$ and $R^{a}{}_{u}=-U^{a}R_{ur}+h^{ab}R_{ub}$. Thus, the evolution equations guarantee that $R_{uu}=0$, $R_{ur}=0$ and $R_{ua}=0$ will always be satisfied at $r\neq 0$ if $R_{uu}\,\hat{=}\,0$, $R_{ur}\,\hat{=}\,0$ and $R_{ua}\,\hat{=}\,0$ at the initial surface $r=0$. For convenience, at $r=0$ we will use the following constraint equations: $R_{uu}\hat{=}-\frac{1}{2}\ddot{(\log h)}+\frac{A^{\prime}}{4}\dot{(\log h)}-\frac{1}{4}h^{ac}h^{bd}\dot{h}_{ab}\dot{h}_{cd}=0,$ (9) and $R^{ra}\hat{=}-\frac{1}{2}(\dot{U}^{a})^{\prime}-\frac{1}{4}\dot{(\log h)}{U^{a}}^{\prime}-\frac{1}{2}h^{ab}{U^{c}}^{\prime}\dot{h}_{bc}-\frac{1}{2}\bar{D}^{a}A^{\prime}+\frac{1}{2}h^{ab}\bar{D}^{c}\dot{h}_{bc}-\frac{1}{2}\bar{D}^{a}\dot{(\log h)}=0$ (10) [see Eqs. (79) and (80)]. Moreover, since $A$ should vanish on the initial surface $r=0$ [see Eq. (2)], the evolution equations Eq. (6) become singular on $r=0$. Analyticity of $h_{ab}$ on $r=0$ gives us the regularity condition $\displaystyle-\frac{1}{2}A^{{}^{\prime}}h^{{}^{\prime}}_{ab}$ $\displaystyle-$ $\displaystyle\dot{h}^{{}^{\prime}}_{ab}+{}^{(h)}R_{ab}+\frac{1}{2}h^{cd}(h^{{}^{\prime}}_{ac}\dot{h}_{bd}+h^{{}^{\prime}}_{bd}\dot{h}_{ac})$ (11) $\displaystyle-$ $\displaystyle\frac{1}{2}h_{ac}h_{bd}U^{c}{}^{{}^{\prime}}U^{d}{}^{{}^{\prime}}-\frac{1}{4}\dot{(\log{h})}h^{{}^{\prime}}_{ab}$ $\displaystyle-$ $\displaystyle\frac{1}{4}(\log{h})^{{}^{\prime}}\dot{h}_{ab}+\frac{1}{2}(h_{bc}\bar{D}_{a}U^{c}{}^{{}^{\prime}}+h_{ac}\bar{D}_{b}U^{c}{}^{{}^{\prime}})\,\hat{=}\,0.$ If we give $h_{ab}\|_{r=0}$ on $r=0$, we can determine $h^{\prime}_{ab}\|_{r=0}$, ${U^{a}}^{\prime}\|_{r=0}$, and $A^{\prime}\|_{r=0}$ by solving Eqs. (9), (10) and (11). As a result, we will solve the evolution equations (4), (5) and (6) using the above initial values on $r=0$. ### II.2 Some explicit solutions near event horizon In this subsection, it is shown that we explicitly solve the constraint equations and evolution equations near the event horizon using power series expansion around $r=0$. To do this, we expand the metric functions with $r$ near the event horizon as $A\,=\,rA^{(1)}+\sum_{i\geq 2}r^{i}A^{(i)},$ (12) $U^{a}\,=\,rU^{(1)a}+\sum_{i\geq 2}r^{i}U^{(i)a},$ (13) and $h_{ab}\,=\,h^{(0)}_{ab}+rh^{(1)}_{ab}+\sum_{i\geq 2}r^{i}h^{(i)}_{ab},$ (14) where from the gauge conditions Eqs. (2) and (3), the expansions of $A$ and $U^{a}$ start from the first order of $r$. In the following, the tensor index of $h^{(i)}_{ab}$ and $U^{(i)a}$ is raised and lowered by $h^{(0)}_{ab}$. The trace and traceless parts are also taken by $h^{(0)}_{ab}$. First let us solve the constraint equations and regularity conditions in order to determine initial values. On the initial surface $r=0$, $h^{(0)}_{ab}$, $h^{(1)}_{ab}$, $A^{(1)}$ and $U^{(1)a}$ should be set on $r=0$ as initial values. The constraint equations for these initial values are $R_{uu}\,\hat{=}\,0$ and $R^{ra}\,\hat{=}\,0$. Now $R_{uu}\,\hat{=}\,0$ [Eq. (9)] is rewritten as $\ddot{h}^{(0)}-\frac{1}{2}A^{(1)}\dot{h}^{(0)}+\frac{1}{2}h^{(0)ac}h^{(0)bd}\dot{h}_{ab}^{(0)}\dot{h}_{cd}^{(0)}\,=\,0,$ (15) where $h^{(0)}=\det{h^{(0)}_{ab}}$. Thus, $A^{(1)}$ should be given to satisfy this equation for given $h^{(0)}_{ab}$. Also, $R^{ra}\,\hat{=}\,0$ [Eq. (10)] becomes $\dot{U}^{(1)a}\,=\,-D^{a}A^{(1)}+h^{(0)ac}D^{b}\dot{h}_{bc}^{(0)}-D^{a}\dot{(\log h^{(0)})}-h^{(0)ac}U^{(1)b}\dot{h}^{(0)}_{bc}-\frac{1}{2}U^{(1)a}\dot{(\log h^{(0)})},$ (16) where $D_{a}$ is the covariant derivative with respect to $h^{(0)}_{ab}$. Then the initial value $U^{(1)a}$ is given to satisfy the above. The regularity condition Eq. (11) becomes $\displaystyle-\dot{h}^{(1)}_{ab}$ $\displaystyle-$ $\displaystyle\frac{1}{2}A^{(1)}h^{(1)}_{ab}+{}^{(h^{(0)})}R_{ab}+\frac{1}{2}h^{(0)cd}(h^{(1)}_{ac}\dot{h}_{bd}^{(0)}+h^{(1)}_{bd}\dot{h}_{ac}^{(0)})-\frac{1}{2}h^{(0)}_{ac}h^{(0)}_{bd}U^{(1)c}U^{(1)d}$ (17) $\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-\frac{1}{4}\dot{(\log{h^{(0)}})}h^{(1)}_{ab}-\frac{1}{4}h^{(0)cd}h^{(1)}_{cd}\dot{h}^{(0)}_{ab}+\frac{1}{2}(D_{a}U^{(1)}_{b}+D_{b}U^{(1)}_{a})=0,$ where ${}^{(h^{(0)})}R_{ab}$ is the Ricci tensor of $h^{(0)}_{ab}$. We obtain $h^{(1)}_{ab}$ satisfying this equation. Hence, if we give $h^{(0)}_{ab}$ on the initial surface, the constraint equations and the regularity condition yield $h^{(1)}_{ab}$, $A^{(1)}$ and $U^{(1)a}$. Next we solve the evolution equations. Equation (4) becomes near the event horizon as $R_{rr}\,=\,-h^{(0)ab}h^{(2)}_{ab}+\frac{1}{4}(h^{(1)}_{ab})^{2}+O(r)\,=\,0.$ (18) This equation gives us the trace part of $h^{(2)}_{ab}$ as $h^{(0)ab}h^{(2)}_{ab}\,=\,\frac{1}{4}(h^{(1)}_{ab})^{2}.$ (19) The evolution equation of $U^{a}$ as $R_{rB}h^{aB}=0$ [Eq. (5)] can be expanded as $R_{rB}h^{aB}\,=\,U^{(2)a}+\frac{1}{2}h^{(0)ac}h^{(1)}_{bc}U^{(1)b}+\frac{1}{4}U^{(1)a}h^{(0)bc}h^{(1)}_{bc}+\frac{1}{2}h^{(1)ac}D^{b}\left(h^{(1)}_{bc}-h_{bc}^{(0)}h^{(0)de}h^{(1)}_{de}\right)+O(r).$ (20) Then $U^{(2)a}$ is given by $U^{(2)a}\,=\,-\frac{1}{2}h^{(0)ac}h^{(1)}_{bc}U^{(1)b}-\frac{1}{4}U^{(1)a}h^{(0)bc}h^{(1)}_{bc}-\frac{1}{2}h^{(1)ac}D^{b}\left(h^{(1)}_{bc}-h_{bc}^{(0)}h^{(0)de}h^{(1)}_{de}\right).$ (21) In the same way, expanding the evolution equations Eq. (6) near the event horizon, we can obtain the traceless part $h^{(2)}_{\langle ab\rangle}$ and $A^{(2)}$ in terms of $h^{(0)}_{ab},h^{(1)}_{ab},U^{(1)a}$ and $A^{(1)}$. Note that $A^{(2)}$ is given by the trace part of Eq. (6), namely Eq. (7). However, we do not provide its explicit form because its form is very cumbersome. Hence, we can determine $h^{(2)}_{ab}$, $U^{(2)a}$, and $A^{(2)}$ using the evolution equations. To determine the higher order quantities, $h^{(i)}_{ab}$, $U^{(i)a}$, and $A^{(i)}$ ($i>2$), we have to repeat the same procedure. ## III Late-time symmetry on and near event horizon In this section, we show that there is late-time symmetry on the event horizon. Then we will extend it outside of black hole regions. ### III.1 Late-time behaviors on event horizon To investigate late-time behaviors of the event horizon, we introduce the expansion and shear of the null geodesic generator $l=\partial/\partial u$ of the event horizon. The expansion $\theta$ and shear $\sigma_{ab}$ are defined as $\displaystyle\sigma_{ab}+\frac{1}{2}\theta h^{(0)}_{ab}$ $\displaystyle\hat{=}$ $\displaystyle h_{a}^{~{}A}h_{b}^{~{}B}\nabla_{A}l_{B}$ (22) $\displaystyle\hat{=}$ $\displaystyle\frac{1}{2}\dot{h}^{(0)}_{ab},$ where $\sigma_{ab}$ is the traceless part of $\dot{h}_{ab}^{(0)}$ with respect to $h^{(0)}_{ab}$. Then we can rewrite Eq. (9), one of the constraint equations, using $\theta$ and $\sigma_{ab}$ as $\dot{\theta}-\frac{1}{2}A^{(1)}\theta\,=\,-\frac{1}{2}\theta^{2}-\sigma_{ab}\sigma^{ab}.$ (23) We can regard $A^{(1)}/2$ as a surface gravity of the black hole with respect to the time coordinate $u$ because $l^{A}\nabla_{A}l^{B}\,\hat{=}\,\frac{A^{(1)}}{2}l^{B}$ (24) holds. Using the affine parameter $w$ defined by $\frac{dw}{du}\,=\,\exp\Bigl{(}\int^{u}\\!\frac{A^{(1)}}{2}du^{\prime}\Bigr{)},$ (25) we can obtain the Raychaudhuri equation $\partial_{w}\theta_{(w)}\,=\,-\frac{1}{2}\theta_{(w)}^{2}-\sigma_{(w)ab}\sigma_{(w)}^{ab},$ (26) where $\theta_{(w)}$ and $\sigma_{(w)ab}$ are expansion and shear with respect to $w$. We can see the relation between $\theta$, $\sigma_{ab}$ and $\theta_{(w)}$, $\sigma_{(w)ab}$ as $\theta\,=\,\theta_{(w)}\exp{\Bigl{(}\int^{u}\\!\frac{A^{(1)}}{2}du^{\prime}\Bigr{)}}\,,\quad\sigma_{ab}\,=\,\sigma_{(w)ab}\exp{\Bigl{(}\int^{u}\\!\frac{A^{(1)}}{2}du^{\prime}\Bigr{)}}.$ (27) Here we remember that the area law of the event horizon holds for spacetimes satisfying the null energy condition, that is, $\theta\geq 0$ and $\theta_{(w)}\geq 0$. Since $\sigma_{(w)ab}\sigma_{(w)}^{ab}\geq 0$, Eq. (26) implies the inequality $\partial_{w}\theta_{(w)}+\frac{1}{2}\theta_{(w)}^{2}\leq 0.$ (28) Then the integration over $w$ gives us $\theta_{(w)}\leq\frac{1}{1/\theta_{(0)}+(w-w_{0})/2}\to 0~{}~{}({\text{as}}~{}~{}w\to\infty),$ (29) where we used the fact of $\theta_{(0)}=\theta_{(w)}(w=w_{0})\geq 0$. In addition, Eq. (26) shows that the shear $\sigma_{(w)ab}$ should also vanish as $w\to\infty$. This is shown as a part of the proof of another theorem HSN . From now on, we assume that $w\to\infty$ corresponds to $u\to\infty$. Then, we see that $\theta$ and $\sigma_{ab}$ should also vanish as $u\to\infty$. It is natural to assume that the cross section of the event horizon is compact. Then the vanishing of the expansion implies that the horizon area approaches a constant and finite value. Altogether we see the behavior of the metric at late time as $\mathcal{L}_{l}g_{AB}\|_{\rm horizon}\to 0~{}~{}(u\to\infty).$ (30) Here we impose the following decaying condition on the event horizon for the metric: $\mathcal{L}_{l}g_{AB}\,\hat{=}\,O\left(\frac{1}{u^{n}}\right),$ (31) which explicitly means $\dot{h}^{(0)}_{ab}=O(u^{-n})$. This equation means that the null geodesic generator of the event horizon $l$ should be an asymptotic Killing vector at late time ($u\rightarrow\infty$). Thus there is a late-time symmetry on the event horizon. Since we consider the dynamics only near the event horizon, we cannot determine the decaying rate of the metric. However, the details of the decaying properties are not important for our argument, that is, our result does not depend on $n$. Note that it will be determined by the boundary conditions at a far region from the black holes as in Refs. Gundlach:1993tp ; Barack:1999ma ; Barack:1999ya . It should be remembered that the horizon which satisfies this condition is called a slowly evolving horizon in Refs. Booth:2003ji ; Booth:2006bn . If the right-hand side of Eq. (31) vanishes, the event horizon will be identical to the isolated horizon Ashtekar:2004cn . ### III.2 Extension of late-time symmetry The purpose of this subsection is to show that the decaying condition of Eq. (31) can be extended outside of the event horizon as $\mathcal{L}_{l}g_{AB}\,=\,O\left(\frac{1}{u^{n}}\right).$ (32) In the following we assume the analyticity of $g_{AB}$. Under the presence of the analyticity of spacetimes, the above is equivalent with $(\mathcal{L}_{n})^{m}\mathcal{L}_{l}g_{AB}\,\hat{=}\,O\left(\frac{1}{u^{n}}\right),$ (33) where $n=\partial/\partial r$ and $m=0,1,2,\cdots$. Note that “$\hat{=}$” means the evaluation on the event horizon $(r=0)$ again. First we will show the $m=1$ case: $\mathcal{L}_{n}\mathcal{L}_{l}g_{AB}\,\hat{=}\,O\left(\frac{1}{u^{n}}\right).$ (34) Substituting the explicit form of $g_{AB}$ to the above, we rewrite Eq. (34) as $-\mathcal{L}_{n}\mathcal{L}_{l}(A-U_{a}U^{a})(du)_{A}(du)_{B}+2\mathcal{L}_{n}\mathcal{L}_{l}U_{a}(du)_{(A}(dx^{a})_{B)}+\mathcal{L}_{n}\mathcal{L}_{l}h_{ab}(dx^{a})_{A}(dx^{b})_{B}\hat{=}\,O\left(\frac{1}{u^{n}}\right).$ (35) Using Eqs. (12), (13) and (14), the above will be equivalent with $\mathcal{L}_{l}A^{(1)}\,=\,O\left(\frac{1}{u^{n}}\right),$ (36) $\mathcal{L}_{l}U^{(1)a}\,=\,O\left(\frac{1}{u^{n}}\right),$ (37) and $\mathcal{L}_{l}h^{(1)}_{ab}\,=\,O\left(\frac{1}{u^{n}}\right).$ (38) Let us examine these conditions. First, we focus on Eq. (36). As a result, using the gauge freedom, we can show that the slightly strong condition, $\mathcal{L}_{l}A^{(1)}\,=\,O\ (1/u^{n+1})$, holds. To see this, we decompose $A^{(1)}$ into the $u$-independent term and others as $A^{(1)}\,=\,A^{(1)}_{0}(x^{a})+\tilde{A}^{(1)}(u,x^{a}).$ (39) As shown in Ref. Hollands:2006rj , we can choose the gauge so that $A^{(1)}_{0}$ is a constant. Furthermore, using the residual gauge, we can take $\tilde{A}^{(1)}=O(1/u^{n})$. Then, using the gauge freedom in our coordinate, we can make $A^{(1)}$ satisfy stronger decaying property as $\mathcal{L}_{l}A^{(1)}\,=\,O\left(\frac{1}{u^{n+1}}\right).$ (40) For the details, see Appendix C. Of course, Eq. (36) is satisfied. Now $U^{(1)a}$ should satisfy the constraint equation of Eq. (16) as $\mathcal{L}_{l}U^{(1)a}\,=\,-D^{a}\tilde{A}^{(1)}+h^{(0)ac}D^{b}\dot{h}_{bc}^{(0)}-D^{a}\dot{(\log h^{(0)})}-h^{(0)ac}U^{(1)b}\dot{h}^{(0)}_{bc}-\frac{1}{2}U^{(1)a}\dot{(\log h^{(0)})}.$ (41) Together with Eqs. (31) and (40), we can see that Eq. (37) holds from the above. Furthermore, $h^{(1)}_{ab}$ satisfy the following equation [see Eq. (17)] as a regularity condition $\displaystyle\dot{h}^{(1)}_{ab}+\frac{1}{2}A^{(1)}h^{(1)}_{ab}$ $\displaystyle=$ $\displaystyle{}^{(h^{(0)})}R_{ab}+\frac{1}{2}h^{(0)cd}(h^{(1)}_{ac}\dot{h}_{bd}^{(0)}+h^{(1)}_{bd}\dot{h}_{ac}^{(0)})-\frac{1}{2}h^{(0)}_{ac}h^{(0)}_{bd}U^{(1)c}U^{(1)d}$ (42) $\displaystyle~{}~{}-\frac{1}{4}\dot{(\log{h^{(0)}})}h^{(1)}_{ab}-\frac{1}{4}h^{(0)cd}h^{(1)}_{cd}\dot{h}^{(0)}_{ab}+\frac{1}{2}(D_{a}U^{(1)}_{b}+D_{b}U^{(1)}_{a}),$ Then multiplying $\mathcal{L}_{l}$ to the above and using Eq. (16), we can see $\frac{1}{2}A^{(1)}\mathcal{L}_{l}h^{(1)}_{ab}=\mathcal{L}_{l}{}^{(h^{(0)})}R_{ab}+O(u^{-n})$ (43) holds. In the above, the higher-order terms like $\mathcal{L}_{l}^{2}{h}^{(1)}_{ab},(\mathcal{L}_{l}h^{(0)}_{ab})^{2},(\mathcal{L}_{l}h^{(1)}_{ab})^{2}$, $\mathcal{L}_{l}h^{(0)}_{ab}\mathcal{L}_{l}h^{(1)}_{ab}$ and so on are contained in $O(u^{-n})$. Thus Eq. (31) tells us that $\mathcal{L}_{l}h^{(1)}_{ab}=O(u^{-n})$ holds. As a consequence, we can show the $m=1$ case of Eq. (33). For the $m>1$ cases of Eq. (33), we perform the same procedure inductively. Let $m>1$ be an integer and assume that the metric satisfies $(\mathcal{L}_{n})^{k}\mathcal{L}_{l}g_{AB}\,\hat{=}O\left(\frac{1}{u^{n}}\right)$ (44) for all $k(<m)$. Then $(\mathcal{L}_{n})^{m}\mathcal{L}_{l}g_{AB}\,\hat{=}O(u^{-n})$ is equivalent with $\displaystyle\mathcal{L}_{l}A^{(m)}\,=\,O\left(\frac{1}{u^{n}}\right),$ (45) $\displaystyle\mathcal{L}_{l}U^{(m)a}\,=\,O\left(\frac{1}{u^{n}}\right),$ (46) and $\mathcal{L}_{l}h^{(m)}_{ab}\,=\,O\left(\frac{1}{u^{n}}\right).$ (47) Since $U^{(m)a}$ is written by $U^{(1)a}$, $A^{(1)},h^{(0)}_{ab},\cdots,h^{(m-1)}_{ab}$ from Eq. (5) like Eq. (21), the induction assumption $\mathcal{L}_{l}h^{(i)}_{ab}=O\left(\frac{1}{u^{n}}\right)~{}~{}(i\leq m-1)$ (48) immediately shows us that Eq. (46) holds. From Eq. (7), $A^{(m)}$ is written by $U^{(1)},A^{(1)},h^{(i)}_{ab}(i<m)$ and the trace part of $h^{(m)}_{ab}$. Since the trace part of $h^{(m)}_{ab}$ is written by $U^{(1)},A^{(1)}$ and $h^{(i)}_{ab}$ ($i<m$) through Eq. (4), $A^{(m)}$ is written only by $U^{(1)},A^{(1)},h^{(i)}_{ab}~{}(i<m)$ in the end. Then, by the assumption of the induction, Eq. (45) holds. Next, the evolution equation for $h^{(m)}_{ab}$ is described by Eq. (6). Expanding Eq. (6) near the event horizon and multiplying, $(\mathcal{L}_{n})^{m-1}$, Eq. (6) becomes the following form on the event horizon $\dot{h}^{(m)}_{ab}+\frac{1}{2}A^{(1)}h^{(m)}_{ab}=F_{ab},$ (49) where $F_{ab}$ is a function of $h^{(i)}_{ab}\,(i\leq m)$, $\dot{h}^{(i)}_{ab}\,(i<m)$ and so on. Acting $\mathcal{L}_{l}$ to Eq. (49), then, we can see that $(1/2)A^{(1)}\dot{h}^{(m)}_{ab}=\dot{F}_{ab}\sim\dot{h}^{(j)}_{ab}=O(1/u^{n})$ for $j<m$. Thus Eq. (47) holds. Now we can confirm that the induction loop is closed. Then we can show that Eq. (33), equivalently Eq. (32) holds if the spacetime is real analytic. Therefore we can show that the null geodesic generator $l$ of the event horizon is the asymptotic Killing vector at late time in the sense of Eq. (32). ## IV Summary and discussion We have confirmed that the expansion and shear of the event horizon should decay at late time in the vacuum spacetimes. Then, assuming the compactness of the cross sections of the event horizon, the null geodesic generators on the horizon give us an asymptotic Killing vector $l$ at late time. This means that the horizon has late-time symmetry. By solving the Einstein equations, then, we have found that this late-time symmetry can be extended outside of the black holes. Therefore, at late time, there is the asymptotic symmetry outside of black holes. If the black hole rotates and there is another asymptotic Killing vector at late time, $k$, which will be a stationary Killing vector at a far distance and spacelike near the horizon, $k-l$ is also an asymptotic Killing vector expected to correspond to axisymmetry. In this sense, we would expect that the rigidity holds in gravitational collapse at late time. In these discussions, we assume the compactness of the event horizon. Thus this result cannot be applied to other null hypersurfaces which do not have a compact cross section. There is a remaining issue. In the proof of the rigidity theorem, the exact stationarity does show that the null geodesic generators of the horizon are Killing orbits. On the other hand, our argument could show us that the null geodesic generators of the horizon is a Killing orbit without assuming the presence of asymptotically stationary Killing vectors. It is likely that this difference suggests the existence of important and new points in black hole physics. ## Acknowledgment KT is supported by JSPS Grant-in-Aid for Scientific Research (No. 21-2105). TS is supported by Grant-in-Aid for Scientific Research from the Ministry of Education, Science, Sports and Culture of Japan (No. 21244033, No. 21111006, No. 20540258, and No. 19GS0219). This work is also supported by the Grant-in- Aid for the Global COE Program “The Next Generation of Physics, Spun from Universality and Emergence” from the Ministry of Education, Culture, Sports, Science and Technology (MEXT) of Japan. ## Appendix A Einstein equations near event horizon In Appendix A, using the suitable variables, we will show the derivations of the Einstein equations near the event horizon. ### A.1 $(n+1)$ decomposition First we describe the formula of the $(n+1)$ decomposition. The metric can be written as $g_{AB}\,=\,\epsilon n_{A}n_{B}+\gamma_{AB},$ (50) where $\gamma_{AB}$ is an $n$-dimensional induced metric. $n_{A}$ is the unit normal vector with $n_{A}n^{A}=\epsilon=+1$ ($n^{A}$: spacelike) or $-1$ ($n^{A}$: timelike). We define the extrinsic curvature as $K_{AB}\,=\,\frac{1}{2}\mathcal{L}_{n}\gamma_{AB}.$ (51) Now $n_{A}$ can be written as $n_{A}=\epsilon N(d\Omega)_{A}$, where $\Omega$ is a function which describes the hypersurface as $\Omega=\text{const.}$ and $N$ is the “lapse” function. Then the Riemann tensor is decomposed into $R_{EFGH}\gamma_{A}{}^{E}\gamma_{B}{}^{F}\gamma_{C}{}^{G}\gamma_{D}{}^{H}={}^{(\gamma)}R_{ABCD}-\epsilon K_{AC}K_{BD}+\epsilon K_{AD}K_{BC},$ (52) $R_{EFGD}\gamma_{A}{}^{E}\gamma_{B}{}^{F}\gamma_{C}{}^{G}n^{D}=\nabla_{A}K_{BC}-\nabla_{B}K_{AC},$ (53) $R_{ACBD}n^{C}n^{D}=-\mathcal{L}_{n}K_{AB}+K_{AC}K_{B}{}^{C}-\epsilon\frac{1}{N}\nabla_{A}\nabla_{B}N,$ (54) where $\nabla_{A}$ denotes the covariant derivative with respect to $\gamma_{AB}$. The Ricci tensor is decomposed into $R_{AB}n^{A}n^{B}=-\mathcal{L}_{n}K-K_{AB}K^{AB}-\epsilon\frac{1}{N}\nabla^{2}N,$ (55) $R_{AC}n^{A}\gamma_{B}{}^{C}=\nabla^{A}K_{AB}-\nabla_{B}K,$ (56) $R_{CD}\gamma_{A}{}^{C}\gamma_{B}{}^{D}={}^{(\gamma)}R_{AB}-\epsilon\mathcal{L}_{n}K_{AB}-\epsilon KK_{AB}+2\epsilon K_{AC}K_{B}{}^{C}-\frac{1}{N}\nabla_{A}\nabla_{B}N.$ (57) The Ricci scalar is written as $\displaystyle R=$ $\displaystyle{}^{(\gamma)}R-2\epsilon\mathcal{L}_{n}K-\epsilon K^{2}-\epsilon K_{AB}K^{AB}-\frac{2}{N}\nabla^{2}N$ (58) $\displaystyle=$ $\displaystyle{}^{(\gamma)}R+\epsilon K^{2}-\epsilon K_{AB}K^{AB}-\frac{2}{N}\nabla^{2}N-2\epsilon\nabla_{A}(Kn^{A}).$ The components of the Einstein tensor are $G_{AB}n^{A}n^{B}=\frac{1}{2}(-\epsilon{}^{(\gamma)}R+K^{2}-K_{AB}K^{AB}),$ (59) $G_{AC}n^{A}\gamma_{B}{}^{C}=\nabla^{A}K_{AB}-\nabla_{B}K,$ (60) $\displaystyle G_{CD}\gamma_{A}{}^{C}\gamma_{B}{}^{D}=$ $\displaystyle{}^{(\gamma)}G_{AB}-\epsilon KK_{AB}+2\epsilon K_{AC}K_{B}{}^{C}+\frac{\epsilon}{2}\gamma_{AB}(K_{CD}K^{CD}+K^{2})$ (61) $\displaystyle-\epsilon\mathcal{L}_{n}K_{AB}+\epsilon\gamma_{AB}\mathcal{L}_{n}K-\frac{1}{N}\nabla_{A}\nabla_{B}N+\frac{1}{N}\gamma_{AB}\nabla^{2}N.$ ### A.2 Einstein equations We apply the $(n+1)$ decomposition presented in the previous section to the $r$-constant surface in our current four dimensional metric form: $ds^{2}\,=\,-Adu^{2}+2dudr+h_{ab}(dx^{a}+U^{a}du)(dx^{b}+U^{b}du).$ (62) We express the above in the following form $ds^{2}\,=\,N^{2}dr^{2}+q_{\mu\nu}(dx^{\mu}+N^{\mu}dr)(dx^{\nu}+N^{\nu}dr),$ (63) where $N^{2}\,=\,\frac{1}{A},$ (64) $N^{u}\,=\,-\frac{1}{A},$ (65) $N^{a}\,=\,\frac{1}{A}U^{a}.$ (66) $q_{\mu\nu}$ is the induced metric on the $r$-const. hypersurface as $\displaystyle q_{\mu\nu}=\begin{pmatrix}-A+U^{a}U_{a}&U_{b}\\\ U_{a}&h_{ab}\end{pmatrix}.$ (67) Note that the Latin indices $a,b,...$ and the Greek indices $\mu,\nu,..$ are raised and lowered by $h_{ab}$ and $q_{\mu\nu}$ respectively. The unit normal vector to the $r$-const. hypersurface is given by $m_{A}=N(dr)_{A}$ and $m^{A}=N^{-1}(\partial_{r}-N^{\mu}\partial_{\mu})^{A}$. The extrinsic curvature on the $r$-const. hypersurface is defined as $K_{\mu\nu}\,=\,\frac{1}{2}\mathcal{L}_{m}q_{\mu\nu}\,=\,\frac{1}{2N}(\partial_{r}q_{\mu\nu}-\mathcal{D}_{\mu}N_{\nu}-\mathcal{D}_{\nu}N_{\mu}),$ (68) where $\mathcal{D}_{\mu}$ is the covariant derivative with respect to $q_{\mu\nu}$. We rewrite the induced metric as $q_{\mu\nu}dx^{\mu}dx^{\nu}=-\alpha^{2}du^{2}+h_{ab}(dx^{a}+\beta^{a}du)(dx^{b}+\beta^{b}du),$ (69) where $\alpha^{2}\,=\,A\,,\,\beta^{a}\,=\,U^{a}.$ (70) The timelike unit vector to the $u$-const. surface is given by $u=-\alpha du$ and $u^{\mu}=\alpha^{-1}(\partial_{u}-\beta^{a}\partial_{a})^{\mu}$. The extrinsic curvature on the $u$-const. surface is given by $k_{ab}\,=\,\frac{1}{2}\mathcal{L}_{u}h_{ab}\,=\,\frac{1}{2\alpha}(\partial_{u}h_{ab}-\bar{D}_{a}\beta_{b}-\bar{D}_{b}\beta_{a}),$ (71) where $\bar{D}_{a}$ is the covariant derivative with respect to $h_{ab}$. We introduce the following quantities for later convenience $\Theta\equiv K_{\,u\nu}u^{\mu}u^{\nu}\,=\,-\frac{1}{N}\partial_{r}(\log{\alpha})-\mathcal{L}_{u}\log{N},$ (72) $\rho^{a}\equiv K^{a}_{\mu}u^{\mu}\,=\,\frac{1}{2}\partial_{r}\beta^{a}+\bar{D}^{a}\log{\alpha},$ (73) $\kappa_{ab}\equiv K_{cd}h_{a}{}^{c}h_{b}{}^{d}\,=\,\frac{\alpha}{2}\partial_{r}h_{ab}+k_{ab},$ (74) and $\kappa\,=\,\kappa_{ab}h^{ab}\,=\,\frac{\alpha}{2}(\log{h})^{{}^{\prime}}+k,$ (75) where $\rho_{\mu}u^{\mu}=0=\kappa_{\mu\nu}u^{\mu}$, $h=\det{h}_{ab}$ and $k=k_{ab}h^{ab}$. The prime denotes the $r$ differentiation. Then $K_{\mu\nu}$ can be written as $K_{\mu\nu}\,=\,\Theta u_{\mu}u_{\nu}-2\rho_{(\mu}u_{\nu)}+\kappa_{\mu\nu}.$ (76) Using these quantities, we can decompose the four dimensional Ricci tensor $R_{AB}$ into the quantities on two dimensional space: $\displaystyle R_{AB}m^{A}m^{B}$ $\displaystyle=$ $\displaystyle\frac{1}{N}(\Theta-\kappa)^{{}^{\prime}}+\mathcal{L}_{u}(\Theta-\kappa)-\Theta^{2}+2\rho_{a}\rho^{a}-\kappa_{ab}\kappa^{ab}$ (77) $\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+\frac{1}{N}(\mathcal{L}_{u}\mathcal{L}_{u}N+k\mathcal{L}_{u}N-\bar{D}^{2}N-\bar{D}^{a}N\bar{D}_{a}\log{\alpha}),$ $R_{AB}m^{A}q^{B}{}_{C}u^{C}\,=\,-\mathcal{L}_{u}\kappa+\bar{D}^{a}\rho_{a}-k_{ab}\kappa^{ab}+2\rho^{a}\bar{D}_{a}\log{\alpha}-\Theta k,$ (78) $\displaystyle R_{AB}q^{A}{}_{C}q^{B}{}_{D}u^{C}u^{D}$ $\displaystyle=$ $\displaystyle-\frac{1}{N}\Theta^{{}^{\prime}}-\mathcal{L}_{u}\Theta+\Theta^{2}-\Theta\kappa-2\rho^{a}\rho_{a}-2\rho^{a}\bar{D}_{a}\log{\frac{N}{\alpha}}$ (79) $\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+\bar{D}^{a}\log{\alpha}\bar{D}_{a}\log{N}+\frac{1}{\alpha}\bar{D}^{2}\alpha-\mathcal{L}_{u}k-k_{ab}k^{ab}-\frac{1}{N}\mathcal{L}_{u}\mathcal{L}_{u}N,$ $R_{AB}m^{A}h^{Ba}\,=\,\Theta\bar{D}^{a}\log{\alpha}-2\rho_{b}k^{ab}-k\rho^{a}+\bar{D}_{b}\kappa^{ab}-\bar{D}^{a}\kappa+\bar{D}^{a}\Theta+\kappa^{ab}\bar{D}_{b}\log{\alpha}-\frac{1}{\alpha}(\partial_{u}\rho^{a}-\mathcal{L}_{\beta}\rho^{a}),$ (80) $\displaystyle R_{AB}q^{A}{}_{C}u^{C}h^{Bb}$ $\displaystyle=$ $\displaystyle\bar{D}_{a}k^{ab}-\bar{D}^{b}k-\rho^{b}\kappa-2\rho_{a}\kappa^{ab}-\frac{1}{N}\bar{D}^{b}\mathcal{L}_{u}N+k^{ab}\bar{D}_{a}\log{N}-\frac{1}{N}(\rho^{b})^{{}^{\prime}}$ (81) $\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-\Theta\bar{D}^{b}\log{\frac{N}{\alpha}}-\kappa^{ab}\bar{D}_{a}\log{\frac{N}{\alpha}}-\frac{1}{\alpha}(\partial_{u}\rho^{b}-\mathcal{L}_{\beta}\rho^{b}),$ and $\displaystyle R_{AB}h^{A}{}_{a}h^{B}{}_{b}\,=\,{}^{(h)}R_{ab}+\mathcal{L}_{u}k_{ab}+kk_{ab}$ $\displaystyle-$ $\displaystyle 2k_{ac}k_{b}{}^{c}-\frac{1}{\alpha}\bar{D}_{a}\bar{D}_{b}\alpha-\frac{1}{N}\kappa_{ab}^{{}^{\prime}}-\frac{1}{\alpha}\dot{\kappa}_{ab}+\frac{1}{\alpha}\mathcal{L}_{\beta}\kappa_{ab}-2\rho_{(a}\bar{D}_{b)}\log{\frac{N}{\alpha}}$ (82) $\displaystyle+$ $\displaystyle(\Theta-\kappa)\kappa_{ab}-2\rho_{a}\rho_{b}+2\kappa_{ac}\kappa_{b}{}^{c}-\frac{1}{N}\bar{D}_{a}\bar{D}_{b}N+k_{ab}\mathcal{L}_{u}\log{N},$ where the dot denotes $\partial_{u}$. The vacuum Einstein equation is given by $R_{AB}=0$. ## Appendix B Explicit form of Einstein equations In Appendix B, we describe the components of the Einstein equations in terms of the metric functions explicitly. Using $u=\alpha^{-1}(l-U^{a}\partial_{a})=\alpha^{-1}(\partial_{u}-U^{a}\partial_{a}),$ (83) $\Theta\,=\,-(A^{1/2})^{{}^{\prime}}+\frac{1}{2}\mathcal{L}_{u}\log{A},$ (84) $\rho^{a}\,=\,\frac{1}{2}U^{a}{}^{{}^{\prime}}+\frac{1}{2}\bar{D}^{a}\log{A},$ (85) and $\kappa_{ab}\,=\,\frac{A^{1/2}}{2}h^{{}^{\prime}}_{ab}+\frac{1}{2A^{1/2}}(\dot{h}_{ab}-\bar{D}_{a}U_{b}-\bar{D}_{b}U_{a}),$ (86) we can rewrite the Einstein equation $R_{AB}=0$ in terms of our metric form. We will not provide all components of the Einstein equation explicitly. For example, $R_{AB}h^{A}{}_{a}h^{B}{}_{b}=0$ becomes $\displaystyle\mathcal{L}_{l}h^{{}^{\prime}}_{ab}+\frac{1}{2}A^{{}^{\prime}}h^{{}^{\prime}}_{ab}$ $\displaystyle=$ $\displaystyle{}^{(h)}R_{ab}+\frac{A}{2}h^{cd}h^{{}^{\prime}}_{ac}h^{{}^{\prime}}_{bd}+\frac{1}{2}h^{cd}(h^{{}^{\prime}}_{ac}\dot{h}_{bd}+h^{{}^{\prime}}_{bd}\dot{h}_{ac})-\frac{1}{2}h^{{}^{\prime}}_{ac}(\bar{D}_{b}U^{c}+\bar{D}^{c}U_{b})+\mathcal{L}_{U}h^{\prime}_{ab}$ (87) $\displaystyle~{}~{}~{}~{}~{}-\frac{1}{2}h^{{}^{\prime}}_{bc}(\bar{D}_{a}U^{c}+\bar{D}^{c}U_{a})-\frac{1}{2}h_{ac}h_{bd}U^{c}{}^{{}^{\prime}}U^{d}{}^{{}^{\prime}}-\frac{1}{4}[\dot{(\log{h})}-2\bar{D}_{a}U^{a}]h^{{}^{\prime}}_{ab}$ $\displaystyle~{}~{}~{}~{}~{}-\frac{1}{4}(\log{h})^{{}^{\prime}}(Ah^{{}^{\prime}}_{ab}+\dot{h}_{ab}-\bar{D}_{a}U_{b}-\bar{D}_{b}U_{a})-\frac{1}{2}Ah^{{}^{\prime\prime}}_{ab}+\frac{1}{2}(h_{bc}\bar{D}_{a}U^{c}{}^{{}^{\prime}}+h_{ac}\bar{D}_{b}U^{c}{}^{{}^{\prime}}).$ This determines the evolutions of $h_{ab}$. For $R_{AB}m^{A}h^{Ba}=0$, we have $\displaystyle\mathcal{L}_{l}U^{a}{}^{{}^{\prime}}$ $\displaystyle=$ $\displaystyle-\bar{D}^{a}A^{{}^{\prime}}-h^{ac}U^{b}{}^{{}^{\prime}}(\dot{h}_{bc}-\bar{D}_{a}U_{b}-\bar{D}_{b}U_{a})+h^{ab}\bar{D}^{c}(Ah^{{}^{\prime}}_{bc}+\dot{h}_{bc}-\bar{D}_{b}U_{c}-\bar{D}_{c}U_{b})-A\bar{D}^{a}(\log{h})^{{}^{\prime}}$ (88) $\displaystyle~{}~{}~{}~{}-\frac{1}{2}U^{a}{}^{{}^{\prime}}[\dot{(\log{h})}-2\bar{D_{a}}U^{a}]-\frac{1}{2}(\log{h})^{{}^{\prime}}\bar{D}^{a}A-\bar{D}^{a}[\dot{(\log{h})}-2\bar{D}_{b}U^{b}]+\mathcal{L}_{U}U^{a}{}^{{}^{\prime}}.$ ## Appendix C The gauge issue for $A^{(1)}$ [Eq. (40)] In Appendix C we will show the presence of the gauge where Eq. (40) is satisfied. In our coordinate $x^{A}=(u,r,x^{a})$, the metric can be written as $ds^{2}\,=\,-Adu^{2}+2dudr+h_{ab}(dx^{a}+U^{a}du)(dx^{b}+U^{b}du).$ (89) The components of the metric are expanded near the event horizon $(r=0)$ as $A\,=\,rA^{(1)}+O(r^{2}),$ (90) $U^{a}\,=\,rU^{(1)a}+O(r^{2})$ (91) and $h_{ab}\,=\,h^{(0)}_{ab}+O(r).$ (92) Here $A^{(1)}$ can be decomposed as $A^{(1)}\,=\,A_{0}^{(1)}+\tilde{A}^{(1)}(u,x^{a}),$ (93) where $A_{0}^{(1)}$ is set to be a constant as shown in Ref. Hollands:2006rj . When we consider the gauge transformation $x^{A}\rightarrow x^{A}+\xi^{A}$, the metric is transformed as $g_{AB}\rightarrow g_{AB}+\mathcal{L}_{\xi}g_{AB}\equiv g_{AB}+\delta g_{AB}.$ (94) To keep our gauge, the following conditions will be imposed: $\displaystyle\delta g_{ur}\,=\,0\,,\,\delta g_{rr}\,=\,0\,,\,\delta g_{ra}\,=\,0\,,$ $\displaystyle\delta g_{uu}\,=\,O(r)\,,\,\delta g_{ua}\,=\,O(r)\,,\,\delta g_{ab}\,=\,O(1).$ (95) From $\delta g_{rr}=2\partial_{r}\xi^{u}=0$, we have $\xi^{u}=f(u,x^{a})$. Since $\delta g_{ra}$ and $\delta g_{ur}$ are given by $\delta g_{ra}\,=\,\partial_{a}\xi^{u}+U_{a}\partial_{r}\xi^{u}+h_{ab}\partial_{r}\xi^{b},$ (96) $\delta g_{ur}\,=\,(-A+U^{a}U_{a})\partial_{r}\xi^{u}+\partial_{r}\xi^{r}+\partial_{u}\xi^{u}+U_{a}\partial_{r}\xi^{a},$ (97) $\delta g_{ra}=0$ and $\delta g_{ur}=0$ lead to $\xi^{r}=-r\partial_{u}f+\partial_{a}f\int^{r}U^{a}dr^{\prime},\quad\xi^{a}=-\partial_{b}f\int^{r}h^{ab}dr^{\prime}.$ (98) Then $\delta g_{uu}$ becomes $\delta g_{uu}\,=\,r[-\partial_{u}(fA^{(1)})-\partial^{2}_{u}f]+O(r^{2}),$ (99) where we used Eqs. (90) and (91). This means that $A^{(1)}$ is transformed under the gauge transformation as $A^{(1)}\rightarrow A^{(1)}+\partial_{u}(fA^{(1)})+\partial_{u}^{2}f.$ (100) Thus if we choose $f$ as $f\,=\,-\frac{1}{A^{(1)}_{0}}\int^{u}du^{{}^{\prime}}\tilde{A}^{(1)},$ (101) $A^{(1)}$ is transformed as $A^{(1)}\rightarrow\bar{A}^{(1)}\,=\,A^{(1)}_{0}-\frac{1}{A^{(1)}_{0}}[\partial_{u}\tilde{A}^{(1)}+\partial_{u}(f\tilde{A}^{(1)})].$ (102) Let assume that $\tilde{A}^{(1)}$ decays as $u\to\infty$. For the moment, we write $\tilde{A}^{(1)}=O(1/u^{m})$, where $m$ is an integer. If $m\geq n$, Eq. (40) is already satisfied. Therefore, we suppose that $m$ is smaller than $n$. In the current gauge transformation, the transformed $\bar{A}^{(1)}$ satisfies $\bar{A}^{(1)}\,=\,A^{(1)}_{0}+O(1/u^{m+1}).$ (103) Repeating this procedure, we can always choose the gauge satisfying $\mathcal{L}_{l}A^{(1)}\,=\,O\left(\frac{1}{u^{n+1}}\right).$ (104) If one wishes, one can continue the same procedure and then achieve an arbitrary faster decaying rate. But, the above is enough for our current purpose. ## References * (1) W. Israel, Phys. Rev. 164, 1776 (1967); B. Carter, Phys. Rev. Lett. 26, 331 (1971); S. W. Hawking, Commun. Math. Phys. 25, 152 (1972); D. C. Robinson, Phys. Rev. Lett. 34, 905 (1975); P. O. Mazur, J. Phys. A15, 3173 (1982); For a review, see M. Heusler, Black Hole Uniqueness Theorems, (Cambridge University Press, London, 1996); P. O. Mazur, hep-th/0101012; G. L. Bunting, PhD thesis, Univ. of New England, Armidale (1983). * (2) S. W. Hawking and G. F. R. Ellis, The Large scale structure of space-time, (Cambridge Univ. Press, Cambridge, 1973). * (3) S. W. Hawking, Commun. Math. Phys. 25, 152-166 (1972). * (4) H. Friedrich, I. Racz, R. M. Wald, Commun. Math. Phys. 204, 691-707 (1999). * (5) I. Racz, Class. Quant. Grav. 17, 153-178 (2000). * (6) C. Gundlach, R. H. Price, J. Pullin, Phys. Rev. D49, 883-889 (1994). * (7) L. Barack, A. Ori, Phys. Rev. Lett. 82, 4388 (1999). * (8) L. Barack, A. Ori, Phys. Rev. D60, 124005 (1999). * (9) K. Tanabe, S. Kinoshita, T. Shiromizu, Phys. Rev. D84, 044055 (2011). * (10) S. A. Hayward, T. Shiromizu, K. -i. Nakao, Phys. Rev. D49, 5080-5085 (1994). * (11) I. Booth, S. Fairhurst, Phys. Rev. Lett. 92, 011102 (2004). * (12) I. Booth, S. Fairhurst, Phys. Rev. D75, 084019 (2007). * (13) A. Ashtekar, B. Krishnan, Living Rev. Rel. 7, 10 (2004). * (14) S. Hollands, A. Ishibashi, R. M. Wald, Commun. Math. Phys. 271, 699-722 (2007).	arxiv-papers	2011-12-02T09:30:17	2024-09-04T02:49:24.901658	{ "license": "Public Domain", "authors": "Kentaro Tanabe, Tetsuya Shiromizu, Shunichiro Kinoshita", "submitter": "Kentaro Tanabe", "url": "https://arxiv.org/abs/1112.0408" }
1112.0580	# Perfect Absorption in Ultrathin Epsilon-Near-Zero Metamaterials Induced by Fast-Wave Non-Radiative Modes Simin Feng simin.feng@navy.mil Klaus Halterman Michelson Lab, Physics Division, Naval Air Warfare Center, China Lake, California 93555 ###### Abstract Above-light-line surface plasmon polaritons can arise at the interface between a metal and $\epsilon$-near-zero metamaterial. This unique feature induces unusual fast-wave non-radiative modes in a $\epsilon$-near-zero material/metal bilayer. Excitation of this peculiar mode leads to wide-angle perfect absorption in low-loss ultrathin metamaterials. The ratio of the perfect absorption wavelength to the thickness of the $\epsilon$-near-zero metamaterial can be as high as $10^{4}$; the electromagnetic energy can be confined in a layer as thin as $\lambda/10000$. Unlike conventional fast-wave leaky modes, these fast-wave non-radiative modes have quasi-static capacitive features that naturally match with the space-wave field, and thus are easily accessible from free space. The perfect absorption wavelength can be tuned from mid- to far-infrared by tuning the $\epsilon\approx 0$ wavelength while keeping the thickness of the structure unchanged. ###### pacs: 42.25.Bs, 78.67.Pt, 42.82.Et A metamaterial is a composite structure with an electromagnetic (EM) response not readily observed in naturally occurring materials. Many remarkable phenomenon have been predicted Vesalago ; Feng ; Lai ; nguyen by tuning the permittivity $\epsilon$ and permeability $\mu$ in extraordinary ways. If the dielectric response is made vanishingly small, creating an epsilon-near-zero (ENZ) material, interesting radiative effects are expected to occur Enoch ; Silveirinha ; Halterman . On the other hand, by manipulating the EM response to achieve a small transmittance ($T$) and reflectance ($R$), enhanced absorption can ensue. Strong absorption in a thin layer typically requires high loss. One of the earliest absorbers, the Salisbury screen Salisbury , is based on the phenomenon of destructive wave interference, and is thus limited to a minimum thickness of one quarter wavelength. To overcome the thickness constraint, absorbing screens using metamaterials Engheta and high impedance ground planes Sievenpiper have recently been proposed. By exploring resonant enhancement, thin metamaterial and nanoplasmonic absorbers were demonstrated in structures having localized resonances Landy ; XLiu ; Brown ; Li ; Avitzour ; Ye ; Kazemzadeh ; Costa ; Kravets ; NLiu ; Kang ; Chern ; atwater . In those structures, the geometrical quality factor (GQF), i.e. the ratio of the perfect absorption wavelength to the thickness of the medium, is significantly improved compared to the standard Salisbury screen. The best GQF of those structures is about 40 Li . Based on a fundamentally different mechanism, in this paper we demonstrate wide-angle perfect absorption in a low-loss ultrathin ENZ-metamaterial/metal bilayer as shown in Fig. 1. In our structure, the GQF can be as high as $10^{4}$. This structure possesses fast-wave non-radiative (FWNR) modes due to unconventional above-light-line surface plasmon polaritons (ALL-SPPs) at the ENZ-metal interface, which is easily accessible from free space. Fast waves have phase velocities exceeding the speed of light in vacuum. Conventional fast waves in planar structures are radiative leaky modes that cannot be excited by plane wave incidence due to wave-vector mismatch at the boundary. For our ENZ-metal structure, the exotic FWNR mode is naturally matched to free space at the ENZ-air interface, and thus is easily accessible from free space. Unlike the Salisbury screen where the thickness of the absorbers increases with the absorption wavelength, in our structure the wavelength can be tuned from mid- to far-infrared (IR) by tuning the $\epsilon\approx 0$ wavelength while maintaining the thickness. Figure 1: (Color online) A plane wave is incident (at an angle $\theta$) on an anisotropic ($\epsilon_{1x}=1$ and $\epsilon_{1z}\approx 0$) ENZ/metal bilayer with thicknesses $d_{1}$ and $d_{2}$ for the ENZ medium and metal (Ag), respectively. Above-light-line surface plasmon polaritons (ALL-SPPs) are excited at the ENZ-metal interface. To investigate this absorption phenomenon in a general fashion, an anisotropic ENZ metamaterial ($\epsilon_{1x}=1$, $\epsilon_{1z}\approx 0$) is assumed (see Fig. 1). In the following, the subscripts 1 and 2 refer, respectively, to the ENZ medium and metal. The areas above the ENZ material and below the metal are free space and refer to regions 0 and 3, respectively. Since the $\epsilon_{1x}$ has no significant impact on the result, $\epsilon_{1x}=1$ is assumed throughout the paper. The permittivity of silver in the infrared region is obtained by curve fitting experimental data Palik with a Drude model, $\epsilon_{m}=1-\omega_{p}^{2}/\omega^{2}$, where the “plasma frequency” $\omega_{p}=5.38\,\mu$m-1. The propagation constant of SPPs at the interface of two semi-infinite anisotropic media is given by $\frac{k_{spp}}{k_{0}}=\sqrt{\dfrac{\epsilon_{1z}\epsilon_{2z}\bigl{(}\epsilon_{2x}\mu_{1y}-\epsilon_{1x}\mu_{2y}\bigr{)}}{\epsilon_{2x}\epsilon_{2z}-\epsilon_{1x}\epsilon_{1z}}}\approx\sqrt{\epsilon_{1z}\mu_{1y}\left(1-\delta\right)}\,,$ (1) where $k_{0}=\omega/c$ and $\delta=\epsilon_{1x}\mu_{2y}/(\epsilon_{2x}\mu_{1y})$. The approximation is taken when $\epsilon_{1z}\rightarrow 0$. In our case, $\mu_{1y}=\mu_{2y}=1$ and $\epsilon_{2x}=\epsilon_{2z}=\epsilon_{m}\sim-10^{5}$ (in the IR region). Thus, $k_{spp}/k_{0}\approx\sqrt{\epsilon_{1z}\mu_{1y}}<1$, which characterizes above light-line SPP dispersion, and is clearly different from conventional SPP dispersion. The absorption can be calculated from Maxwell’s equations. Assuming a harmonic time dependence $\exp(-i\omega t)$ for the EM field, we have $\displaystyle\begin{split}\nabla\times\bigl{(}\bar{\bar{\mu}}_{n}^{-1}\cdot\nabla\times{\bm{E}}\bigr{)}&=\,k_{0}^{2}\bigl{(}\bar{\bar{\epsilon}}_{n}\cdot{\bm{E}}\bigr{)}\,,\\\ \nabla\times\bigl{(}\bar{\bar{\epsilon}}_{n}^{-1}\cdot\nabla\times{\bm{H}}\bigr{)}&=\,k_{0}^{2}\bigl{(}\bar{\bar{\mu}}_{n}\cdot{\bm{H}}\bigr{)}\,,\end{split}$ (2) where $\bar{\bar{\epsilon}}_{n}$ and $\bar{\bar{\mu}}_{n}$ are, respectively, the permittivity and permeability tensors for a given (uniform) region ($n=0,1,2,\cdots$), which in the principal coordinates are described by, $\displaystyle\bar{\bar{\epsilon}}_{n}$ $\displaystyle=$ $\displaystyle\epsilon_{nx}\hat{\bm{x}}\hat{\bm{x}}+\epsilon_{ny}\hat{\bm{y}}\hat{\bm{y}}+\epsilon_{nz}\hat{\bm{z}}\hat{\bm{z}}\,,$ (3) $\displaystyle\bar{\bar{\mu}}_{n}$ $\displaystyle=$ $\displaystyle\mu_{nx}\hat{\bm{x}}\hat{\bm{x}}+\mu_{ny}\hat{\bm{y}}\hat{\bm{y}}+\mu_{nz}\hat{\bm{z}}\hat{\bm{z}}\,.$ (4) We consider TM modes, corresponding to non-zero field components $H_{y}$, $E_{x}$, and $E_{z}$. The magnetic field $H_{y}$ satisfies the following wave equation: $\frac{1}{\epsilon_{z}}\frac{\partial^{2}H_{y}}{\partial x^{2}}+\frac{1}{\epsilon_{x}}\frac{\partial^{2}H_{y}}{\partial z^{2}}+k_{0}^{2}\mu_{y}H_{y}=0\,,$ (5) which admits solutions of the form $\psi(z)\exp(i\beta x)$. The parallel wave- vector $\beta$ is determined by the incident wave, and is conserved across the interface, $\beta^{2}=k_{0}^{2}\epsilon_{nz}\mu_{ny}-\alpha_{n}^{2}\frac{\epsilon_{nz}}{\epsilon_{nx}}\,,\hskip 14.45377pt(n=0,1,2,\cdots)\,,$ (6) where $\alpha_{n}$ is the wave number in the $z$ direction. The functional form of $\psi(z)$ is either a simple exponential $\exp(i\alpha_{n}z)$ for the semi-infinite region or a superposition of $\cos(\alpha_{n}z)$ and $\sin(\alpha_{n}z)$ terms for the bounded region (along the $z$ direction). The other two components $E_{x}$ and $E_{z}$ are found using Maxwell’s equations. By matching boundary conditions at the interface, i.e., continuity of $H_{y}$ and $E_{x}$, the transmittance ($T$) and reflectance ($R$) can be calculated via the Poynting vector $\bm{S}$, given by $\bm{S}=c/8\pi\Re(\bm{E}\times\bm{H}^{})$. The fraction of energy that is absorbed by the system is determined by the absorptance ($A$), where $A=1-T-R$, consistent with energy conservation. Figure 2: (Color online) Absorptance vs. angle of incidence (see Fig. 1) at the wavelength $\lambda=10.94\,\mu$m (left panels) and $\lambda=201.66\,\mu$m (right panels) for two different thicknesses of the ENZ medium: $d_{1}=0.02\,\mu$m (top panels) and $d_{1}=0.2\,\mu$m (bottom panels). The Ag- layer thickness ($d_{2}$) is $20\,$nm. Figure 2 shows the absorptance of the ENZ-metal structure versus the angle of incidence (AOI) in the mid- to far-IR regime. Unless stated otherwise, the results that follow correspond to $\epsilon_{1z}=0.001+i0.01$. The AOI at which the perfect absorption occurs depends on the wavelength and the thickness of the ENZ layer. The full-width-half-maximum (FWHM) angular width can be as high as $\sim 60^{\circ}$ (top-left panel), while the GQF can reach $\sim 10^{4}$ (top-right panel). When the thickness increases, the angular bandwidth varies differently in the mid- and far-IR regions. Figure 3: (Color online) Absorptance (right panels) and direction of perfect absorption (left panels) when the ENZ-thickness $d_{1}=0.02\,\mu$m (top panels) and $d_{1}=0.2\,\mu$m (bottom panels). Color-bars represent the magnitude of the absorptance. The curves in the left panels are computed using three different methods (see text). Solid (blue): numerically extracted from the corresponding 2D plot in the right panels. Dashed (green): obtained from Eq. (7). Circles (red): computed from Eq. (8). The Ag thickness is $20\,$nm. In Fig. 3, we show how the absorptance varies with the AOI and incident wavelength (right panels). The left set of panels corresponds to the incident angle $\theta_{p}$ that results in perfect absorption, as a function of wavelength. Perfect absorption occurs when the $\theta$ dependent effective impedance (${\cal Z}_{e}$) of the ENZ-metal structure matches that of free space (${\cal Z}_{0}$), i.e. ${\cal Z}_{e}={\cal Z}_{0}$, in which case the reflection coefficient is zero. The effective impedance of the ENZ-metal structure can be expressed as ${\cal Z}_{e}={\cal Z}_{1}\frac{1-r_{12}\exp\bigl{(}i2\phi\bigr{)}}{1+r_{12}\exp\bigl{(}i2\phi\bigr{)}}={\cal Z}_{0}\,,$ (7) where $r_{12}=({\cal Z}_{1}-{\cal Z}_{2})/({\cal Z}_{1}+{\cal Z}_{2})$, ${\cal Z}_{j}=\alpha_{j}/\epsilon_{jx}\,(j=0,1,2)$, and $\phi=\alpha_{1}d_{1}$. Perfect absorption occurs at the particular AOI that satisfy Eq. (7). It is also informative to analyze the corresponding waveguide modes, which are solutions to the transcendental equation, $\tan\phi=-i\frac{{\cal Z}_{1}({\cal Z}_{0}+{\cal Z}_{2})}{{\cal Z}_{1}^{2}+{\cal Z}_{0}{\cal Z}_{2}},$ (8) which implicitly depends on the parallel complex propagation constant $\beta$ (Eq. (6)). The mode solutions of Eq. (8) have four branches due to the $\pm$ signs inherent to the square root of $\alpha_{0}$ and $\alpha_{2}$ in Eq. (6). The branch leading to perfect absorption corresponds to both $\alpha_{0}<0$ and $\alpha_{2}<0$. In the long wavelength regime and for ultrathin slabs, this branch provides fast-wave ($\beta/k_{0}<1$), non-radiative ($\beta$ real) modes that are guided along the ENZ layer. In the left panels of Fig. 3 we compute the direction of perfect absorption as a function of wavelength in three complimentary ways: The solid (blue) curves are numerically extracted from the absorptance corresponding to the right panels, the dashed (green) curves are obtained from Eq. (7), where the structure is impedance matched to free space, and the circles (red) are calculated using Eq. (8) and a root searching algorithm. The overlap of each set of data further demonstrates that the underlying mechanism of perfect absorption is the excitation of FWNR modes. We see also that perfect absorption occurs only at oblique incidence where the electric field has a nonzero component in the normal direction. The energy density, $U$, in lossy and dispersive anisotropic media is given by, $U=\frac{1}{16\pi}\Re\left[{\bm{E}}^{\dagger}\cdot\frac{\partial(\omega\bar{\bar{\epsilon}})}{\partial\omega}{\bm{E}}+{\bm{H}}^{\dagger}\cdot\frac{\partial(\omega\bar{\bar{\mu}})}{\partial\omega}{\bm{H}}\right].$ (9) Figure 4: (Color online) Spatial distribution of the energy density computed from Eq. (9) (normalized so that $H_{y}$ is unity at the ENZ-air interface) for $\lambda=10\,\mu$m and AOI = $16.1^{\circ}$ (left) and $\lambda=155\,\mu$m and AOI = $56.3^{\circ}$ (right). The ENZ-layer with $d_{1}=0.2\,\mu$m is located at $z\in[-0.1\ 0.1]$. The regions $z\in[0.1\ 0.15]$ and $z\in[-0.2\ -0.1]$ are, respectively, silver and air. Color-bars represent the magnitude ($\times 0.01$) of the energy density, indicating the energy is more confined to the ENZ medium at far-IR wavelengths. Figure 5: (Color online) Direction (top left) and FWHM bandwidth (bottom left) of perfect absorption vs. ENZ thickness. The curves are numerically extracted from the absorptance when $\lambda=10\,\mu$m (dots (blue)), $\lambda=100\,\mu$m (solid (green)), and $\lambda=300\,\mu$m (dashed (red)). The circles in the left-top panel are obtained from Eq. (8). The normalized propagation constant $\beta/k_{0}$ corresponding to these modes is shown in the right panels. Top-right: real part of $\beta/k_{0}$. Bottom-right: imaginary part of $\beta/k_{0}$. The Ag thickness ($d_{2}$) is $10\,$nm. We show in Fig. 4 the spatial distribution of the energy density in the ENZ- metal structure under conditions of perfect absorption at mid- and far-IR wavelengths. In the mid-IR (left panel), the field inside the ENZ-layer is concentrated at the two interfaces, and a very thin layer of ALL-SPPs can be seen at the ENZ-metal interface. The strong field at the ENZ-air interface is distinguishable from conventional surface waves, since in this case the exponential decay of the field occurs strictly inside the ENZ medium. Outside of the ENZ medium, the field neither decays like a surface wave nor does it increase like a leaky wave. The observed uniformity indicates that the field outside the structure is plane wave matching with the free-space wave, and thus can be easily excited by plane wave incidence – a striking difference from surface waves and leaky waves. In the far-IR (right panel), the field is even more uniform and confined to the ENZ-medium and also matched to the space-wave field at the ENZ-air boundary. Around the point where the real part of $\epsilon$ approaches zero, even with a small loss $\Im(\epsilon_{1z})=0.01$, the presence of the FWNR mode reinforces the increasing absorption due to the strong field inside the ENZ medium (by virtue of the electric displacement continuity) leading to perfect absorption in the ultrathin ENZ medium. The minimum thickness of ENZ materials is generally limited by fabrication methods except in some cases of naturally occurring resonances. Figure 5 shows the direction, $\theta_{p}$ (top left), and FWHM angular bandwidth (bottom left) of perfect absorption versus ENZ thickness for three wavelengths, as well as the corresponding normalized complex propagation constants (right panels). The circles in the top left panel represent solutions to Eq. (8), which are consistent with the absorption calculated using the Poynting vector. These curves also correlate with the solutions from the matched impedance expression (Eq. (7)) (not shown). When the thickness increases, the direction of perfect absorption is shifted towards the surface normal for both mid- and far-IR regions. There is an optimal thickness that yields the largest bandwidth. The real part of $\beta/k_{0}$ is above the vacuum light-line, indicating a fast-wave character for all three wavelengths shown. The imaginary component of $\beta$ is zero for the far-IR wavelengths. For $\lambda=10\,\mu$m, $\Im(\beta)=0$ when $d_{1}<0.2\,\mu$m. When $d_{1}>0.2\,\mu$m, $\beta$ starts to pick up a small imaginary part which increases with thickness: The fast-wave non-radiative mode transforms into a fast-wave leaky mode Halterman as the thickness increases. Figure 6: (Color online) Average energy density (left top), transport energy ($S_{x}/c$) per unit area (left bottom), voltage ($V$) across the ENZ-layer (middle top), the ratio of power to dissipation rate (middle bottom), the effective surface charge density $Q$ (right top) and the normalized capacitance $C/C_{0}$ (right bottom) vs. $d_{1}$ for $\lambda=10\,\mu$m (dots (blue)), $\lambda=100\,\mu$m (solid (green)), and $\lambda=300\,\mu$m (dashed (red)). Figure 7: (Color online) Color map of the absorptance versus AOI and loss ($\Im(\epsilon_{1z})$) for $\lambda=10\,\mu$m (top panels) and $\lambda=200\,\mu$m (bottom panels). The left and right set of panels correspond to $\Re(\epsilon_{1z})=0.001$ and $\Re(\epsilon_{1z})=0.01$ respectively. Here, $d_{1}=0.15\,\mu$m and $d_{2}=10\,$nm. The color bars represent the magnitude of the absorptance. To further understand the FWNR modes, we integrated the transport power $S_{x}$, dissipation rate $D$, and $\Re(E_{z})$ over the ENZ-thickness (normalized by the input $E_{z}$). The dissipation rate is given by, $D=\frac{\omega}{8\pi}\Im\Bigl{[}{\bm{E}}^{\dagger}\cdot\bar{\bar{\epsilon}}{\bm{E}}+{\bm{H}}^{\dagger}\cdot\bar{\bar{\mu}}{\bm{H}}\Bigr{]}.$ (10) The average energy density in the ENZ medium is also calculated (Eq. (9)), as well as the effective surface charge density $Q=\epsilon_{1z}E_{z}/4\pi$ (normalized by the input $E_{z}$) and capacitance $C=Q/V$ per unit area, which is much larger than the capacitance ($C_{0}=\epsilon_{1z}/d_{1}$) of a parallel-plate capacitor (of unit-area), due in part to the field enhancement inside the ENZ material. These results are presented in Fig. 6. The confinement is stronger for far-IR and decreases with the thickness (left- top). The linear responses of the voltage (middle-top) and constant behavior of the charge (right-top) as a function of thickness are hallmarks of capacitors. The deviation from the linear behavior for $\lambda=10\,\mu$m can be understood as a balance between radiative loss, which acquires more “charge” stored in the “capacitor” (right- and middle-top) and energy transport (left-bottom) (see right-bottom panel of Fig. 5 where $\Im(\beta)\neq 0$ when $d_{1}>0.2\,\mu$m for $\lambda=10\,\mu$m). The radiative loss reduces the enhancement of the capacitance at $\lambda=10\,\mu$m as shown in the right-bottom panel. In essence, the higher the confinement, the larger the capacitance. The effective capacitance is found to be equal to $cQ^{2}/(2S_{x})$ or $2S_{x}/(cV^{2})$, which implies the transport energy is driven by quasi-static “surface charges”. Thus, FWNR modes can be considered quasi-static, capacitively driven fast waves. The middle- bottom panel of Fig. 6 shows the ratio of transport power to dissipation power, suggesting longer wavelengths have better power handling capability if the structure is used as an ultrathin channel to transport light. The influence of loss on the absorption is illustrated in Fig. 7 for different $\epsilon_{1z}$ (real part) in the mid- and far-IR. In the far-IR, the smaller $\Im(\epsilon_{1z})$ has larger angular bandwidth, which is opposite to what occurs in the mid-IR. In summary, wide-angle perfect absorbers have been demonstrated in ultrathin ENZ-metal structures, and which have extraordinary high geometric quality factors. Such structures may have applications involving ultra light-weight infrared absorbers, cloaking, EM shielding, photovoltaic systems, and highly efficient infrared sensors and detectors. The authors gratefully acknowledge the sponsorship of ONR N-STAR and NAVAIR’s ILIR programs. ## References (1) V. G. Vesalago, Sov. Phys. Usp. 10, 509 (1968). * (2) S. Feng and K. Halterman, Phys. Rev. Lett. 100, 063901 (2008). * (3) Y. Lai, H. Chen, Z.-Q. Zhang, and C. T. Chan, Phys. Rev. Lett. 102, 093901 (2009). * (4) V. C. Nguyen et al., Phys. Rev. Lett. 105, 233908 (2010). * (5) S. Enoch, et al., Phys. Rev. Lett. 89, 213902 (2002). * (6) M. Silveirinha and N. Engheta, Phys. Rev. Lett. 97, 157403 (2006). * (7) K. Halterman, S. Feng, and V. C. Nguyen, Phys. Rev. B84, 075162 (2011). * (8) W. W. Salisbury, U.S. Patent No. 2,599,944 (1952). * (9) N. Engheta, IEEE Antennas and Propagation Society (AP-S) Int. Symp. v.2, p.392 (2002). * (10) D. Sievenpiper, et al., IEEE Trans. on Microwave Theory and Techniques, 47, 2059 (1999). * (11) N. I. Landy, et al., Phys. Rev. Lett. 100, 207402 (2008). * (12) X. Liu, et al., Phys. Rev. Lett. 104, 207403 (2010). * (13) J. R. Brown, et al., J. Appl. Phys. 104, 043105 (2008). * (14) H. Li, et al., J. Appl. Phys. 110, 014909 (2011). * (15) Y. Avitzour, Y. A. Urzhumov, and G. Shvets, Phys. Rev. B79, 045131 (2009). * (16) Y. Q. Ye, Y. Jin, and S. He J. Opt. Soc. Am. B27, 498 (2010). * (17) A. Kazemzadeh and A. Karlsson, IEEE Trans. Antennas Propag. 58, 3310 (2010). * (18) F. Costa, A. Monorchio, and G. Manara, IEEE Trans. Antennas Propag. 58, 1551 (2010). * (19) V. G. Kravets, F. Schedin, and A. N. Grigorenko, Phys. Rev. B78, 205405 (2008). * (20) N. Liu, et al., Nano Lett. 10, 2342 (2010). * (21) G. Kang, et al., Opt. Express 19, 770 (2011). * (22) R.-L. Chern and W.-T. Hong, Opt. Express 19, 8962 (2011). * (23) K. Aydin, et al., Nature Comm. 2, 517 (2011). * (24) E. D. Palik, Handbook of Optical Constants of Solids, Academic Press, 1998, San Diego, CA.	arxiv-papers	2011-12-01T17:29:16	2024-09-04T02:49:24.915826	{ "license": "Public Domain", "authors": "Simin Feng and Klaus Halterman", "submitter": "Simin Feng", "url": "https://arxiv.org/abs/1112.0580" }
1112.0715	# A dynamical characterization of $C$ sets John H. Johnson Department of Mathematics, James Madison University, Harrisonburg, VA 22807, USA john.j.jr@gmail.com ###### Abstract Furstenberg, using tools from topological dynamics, defined the notion of a central subset of positive integers, and proved a powerful combinatorial theorem about such sets. Using the algebraic structure of the Stone-Čech compactification, this combinatorial theorem has been generalized and extended to the Central Sets Theorem. The algebraic techniques also discovered many sets, which are not central, that satisfy the conclusion of the Central Sets Theorem. We call such sets $C$ sets. Since $C$ sets are defined combinatorially, it is natural to ask if this notion admits a dynamical characterization similar to Furstenberg’s original definition of a central set? In this paper we give a positive answer to this question by proving a dynamical characterization of $C$ sets. ## 1 Introduction Furstenberg, in his book connecting dynamical systems with combinatorial number theory, defined the concept of a central subset of positive integers [5, Definition 8.3] and proved several important properties of such sets, all using notions from topological dynamics. For instance, whenever a central set is finitely partitioned, at least one cell of the partition contains a central set [5, Theorem 8.8]. Many of the remaining important properties of central sets follows from a powerful combinatorial theorem [5, Proposition 8.21] also due to Furstenberg. Inspired by the fruitful interaction between Ramsey Theory and ultrafilters on semigroups, Bergelson and Hindman, with the assistance of B. Weiss, later proved an algebraic characterization of central sets in $\mathbb{N}$ [1, Section 6]. Using this algebraic characterization as a definition enabled them to easily extend the notion of a central set to any semigroup. The definition in [5] also extends naturally to arbitrary semigroups, and the algebraic and dynamical characterizations were proved to be equivalent in general by H. Shi and H. Yang in [11]. This algebraic definition turns out to have several advantages over the original dynamical definition. For instance, the fact that central sets are ‘preserved under finite partitions’ (this is a concise way of stating [5, Theorem 8.8]) easily follows from the algebraic definition. More importantly, the combinatorial result [5, Proposition 8.21]—and stronger combinatorial statements about central sets—follow from a relatively simple recursive construction. As an example, we state (currently) the strongest combinatorial theorem about central sets commonly used. We first state this theorem for commutative semigroups. (In Section 4 of this paper, we shall give the simplest statement of the Central Sets Theorem currently known for arbitrary semigroups. The statement of this ‘general’ version of the Central Sets Theorem is necessarily complicated because of noncommutativity.) In the statement of this theorem, and in the remainder of this paper, we let $\mathcal{P}_{\\!f}(X)$ denote the collection of all nonempty finite subsets of a set $X$; let ${}^{\hbox{$A$}}{\hskip-2.0ptB}$ denote the collection of all functions with domain $A$ and codomain $B$; and, for typographical convenience, we let $\mathcal{T}=\hbox{${}^{\hbox{$\mathbb{N}$}}{\hskip-2.0ptS}$}$ for a given set $S$. (Generally, the set $S$ in question will be clear from context.) ###### Theorem 1.1 (Central Sets Theorem). Let $(S,+)$ be a commutative semigroup and $A\subseteq S$ central. Then there exist functions $\alpha\colon\mathcal{P}_{\\!f}(\mathcal{T})\to S$ and $H\colon\mathcal{P}_{\\!f}(\mathcal{T})\to\mathcal{P}_{\\!f}(\mathbb{N})$ that satisfy the following two statements: * (1) If $F$, $G\in\mathcal{P}_{\\!f}(\mathcal{T})$ and $F\subsetneq G$, then $\max H(F)<\min H(G)$. * (2) Whenever $m\in\mathbb{N}$, $G_{1}$, $G_{2}$, …, $G_{m}$ is a finite sequence in $\mathcal{P}_{\\!f}(\mathcal{T})$ with $G_{1}\subsetneq G_{2}\subsetneq\cdots\subsetneq G_{m}$ and for each $i\in\\{1,2,\ldots,m\\}$, $f_{i}\in G_{i}$, then we have $\sum_{i=1}^{m}\Bigl{(}\alpha(G_{i})+\sum_{t\in H(G_{i})}f_{i}\Bigr{)}\in A.$ ###### Proof. This was proved by De, Hindman, and Strauss in [4, Theorem 2.2]. ∎ ###### Remark 1.2. It’s an accident of history that [5, Proposition 8.21] is also known in the literature as the Central Sets Theorem. Depending on how one counts, there are about four different versions of ‘the’ Central Sets Theorem. (Happily each newer version implies or easily reduces to the previous version.) From this point on in this paper we shall only refer to Theorem 1.1 and Theorem 4.3 as the Central Sets Theorem. We shall call sets that satisfy the conclusion of the Central Sets Theorem $C$ sets. Despite the combinatorial power (and its name), the Central Sets Theorem is not strong enough to combinatorially characterize central sets. In short there are $C$ sets which are not central sets. This somewhat surprising situation was first discovered in the context of an important type of $C$ sets called the quasi-central sets. The quasi-central sets were first defined algebraically [7, Definition 1.2] and given a combinatorially characterization [7, Theorem 3.7] in a paper of Hindman, Maleki, and Strauss. The fact that quasi-central sets also satisfy the conclusion of the Central Sets Theorem follows from the proof of [4, Theorem 2.2]. Since quasi-central sets are defined algebraically, it is natural to wonder if this notion admits a dynamical characterization similar to Furstenberg’s original definition of central sets. In their recent paper, Burns and Hindman prove such a dynamical characterization of quasi-central sets [2, Theorem 3.4]. However, their paper didn’t provide a dynamical characterization of $C$ sets. (The fact that the notions of $C$ sets and quasi-central sets are distinct follows from an example constructed in a recent paper [6] of Hindman.) In this paper we fill this lacuna and prove a dynamical characterization of $C$ sets in Theorem 4.8. This characterization will be a special case of a more general result in Theorem 3.3 that gives a dynamical characterization of members of idempotent ultrafilters in compact subsemigroups of the Stone-Čech compactification. ### Acknowledgements The characterization proved here is a generalization of part of the author’s dissertation research conducted under the guidance of Neil Hindman. I want to thank Dr. Hindman for his excellent advisement and helpful comments on this paper. ## 2 Preliminaries on Compact Subsemigroups In this section we state the basic definitions, conventions, and results we need to prove our dynamical characterization of members of certain idempotent ultrafilters. None of the results and definitions in this section are due to the author. We also omit any proofs, but we do give references to where proofs can be found. We start by giving a brief review of the algebraic structure of the Stone-Čech compactification of a discrete semigroup. Given a discrete nonempty space $S$ we take the points of $\beta S$ to be the collection of all ultrafilters on $S$. We identify points of $S$ with the principal ultrafilters in $\beta S$. (Thus we pretend that $S\subseteq\beta S$.) Given $A\subseteq S$, put $\overline{A}=\\{p\in\beta S:A\in p\\}$. Then the collection $\\{\overline{A}:A\subseteq S\\}$ is a basis for a compact Hausdorff topology on $\beta S$. This topology is the Stone-Čech compactification of the discrete space $S$. The proofs for all of these assertions can be found in [8, Sections 3.2 and 3.3]. Given a discrete semigroup $(S,\cdot)$, we can extend the semigroup operation to $\beta S$ [8, Theorem 4.1] such that for $p$, $q\in\beta S$ and $A\subseteq S$, we have $A\in p\cdot q$ if and only if $\\{x\in S:x^{-1}A\in q\\}\in p$ [8, Theorem 4.12] where $x^{-1}A=\\{y\in S:xy\in A\\}$. With this operation, $(\beta S,\cdot)$ becomes a compact Hausdorff right-topological semigroup. The word ‘right-topological’ means that for every $q\in\beta S$ the function $\rho_{q}\colon\beta S\to\beta S$, defined by $\rho_{q}(p)=p\cdot q$, is continuous. ###### Definition 2.1. Let $S$ be a nonempty discrete space and $\mathcal{K}$ a filter on $S$. * (a) $\overline{\mathcal{K}}=\\{p\in\beta S:\mathcal{K}\subseteq p\\}$. * (b) $\mathcal{L}(\mathcal{K})=\\{A\subseteq S:S\setminus A\not\in\mathcal{K}\\}$. As is well known, the function $\mathcal{K}\mapsto\overline{\mathcal{K}}$ is a bijection from the collection of all filters on $S$ onto the collection of all compact subspaces of $\beta S$ [8, Theorem 3.20]. We also have the following important theorem relating the above two concepts. ###### Theorem 2.2. Let $S$ be a nonempty discrete space and $\mathcal{K}$ a filter on $S$. * (a) $\overline{\mathcal{K}}=\\{p\in\beta S:\mbox{$A\in\mathcal{L}(\mathcal{K})$ for all $A\in p$}\\}$. * (b) Let $\mathcal{B}\subseteq\mathcal{L}(\mathcal{K})$ be closed under finite intersections. Then there exists a $p\in\beta S$ with $\mathcal{B}\subseteq p\subseteq\mathcal{L}(\mathcal{K})$. ###### Proof. Both of these assertions follow from [8, Theorem 3.11]. ∎ If $(S,\cdot)$ is discrete semigroup and $\mathcal{K}$ a filter on $S$, then there are precise conditions on $\mathcal{K}$ which guarantee that $\overline{\mathcal{K}}$ is a compact subsemigroup of $\beta S$ [3, Theorem 2.6]. Hence $\overline{\mathcal{K}}$ is a compact Hausdorff right-topological semigroup for a suitable filter $\mathcal{K}$. ###### Theorem 2.3. Let $T$ be a compact Hausdorff right-topological semigroup. * (a) $T$ contains at least one idempotent, that is, there exists $x\in T$ such that $x=x\cdot x$. * (b) $T$ contains an ideal, called the smallest ideal and denoted as $K(T)$, that is contained in every ideal of $T$. Additionally, $K(T)$ also contains at least one idempotent. ###### Proof. The proofs of statements (a) and (b) are given in [8, Theorem 2.5] and [8, Theorem 2.8], respectively. ∎ ###### Remark 2.4. It does follow that $c\ell_{T}K(T)$ is a compact subsemigroup of $T$, and hence by (a) this subsemigroup also contains an idempotent. While the smallest ideal $K(T)$ itself may not be closed, it is the union of all of the minimal left ideals of $T$, which are closed, so the fact that it contains an idempotent is also immediate. We are now in a position to give the algebraic definitions of a central set and quasi-central set in a semigroup. ###### Definition 2.5. Let $(S,\cdot)$ be a semigroup and $A\subseteq S$. * (a) We call $A$ a central set if and only if there exists an idempotent $p\in K(\beta S)$ such that $A\in p$. * (b) We call $A$ a quasi-central set if and only if there exists an idempotent $p\in c\ell_{\beta S}K(\beta S)$ such that $A\in p$. To finish this section, we give the definition of a dynamical system and relate this notion to the algebraic structure of the Stone-Čech compactification. ###### Definition 2.6. A pair $(X,\langle T_{s}\rangle_{s\in S})$ is a dynamical system if and only if it satisfies the following four conditions: * (1) $X$ is a compact Hausdorff space. * (2) $S$ is a semigroup. * (3) $T_{s}\colon X\to X$ is continuous for every $s\in S$. * (4) For every $s$, $t\in S$ we have $T_{st}=T_{s}\circ T_{t}$. ###### Theorem 2.7. Let $(X,\langle T_{s}\rangle_{s\in S})$ be a dynamical system. Then we can extend $(X,\langle T_{s}\rangle_{s\in S})$ to a semigroup action on $\beta S$. More precisely, for each $p\in\beta S\setminus S$ we can define $T_{p}\colon X\to X$ such that for every $q$, $r\in\beta S$, $T_{qr}=T_{q}\circ T_{r}$. Furthermore, given $p\in\beta S$, $x$ and $y$ in $X$, we have $T_{p}(x)=y$ if and only if for every neighborhood $U$ of $y$, $\\{s\in S:T_{s}(x)\in U\\}\in p$. ###### Proof. Both of these assertions follow from [8, Theorems 3.27 and Corollary 4.22]. ∎ Be warned that for $p\in\beta S\setminus S$, $T_{p}$ is usually not continuous. ## 3 Dynamical Characterization of Members of Idempotent Ultrafilters ###### Definition 3.1. Let $(X,\langle T_{s}\rangle_{s\in S})$ be a dynamical system, $x$ and $y$ points in $X$, and $\mathcal{K}$ a filter on $S$. The pair $(x,y)$ is called jointly $\mathcal{K}$-recurrent if and only if for every neighborhood $U$ of $y$ we have $\\{s\in S:\mbox{$T_{s}(x)\in U$ and $T_{s}(y)\in U$}\\}\in\mathcal{L}(\mathcal{K})$. ###### Lemma 3.2. Let $(X,\langle T_{s}\rangle_{s\in S})$ be a dynamical system, let $x$ and $y$ be points in $X$, and let $\mathcal{K}$ be a filter $S$ such that $\overline{\mathcal{K}}$ is a compact subsemigroup of $\beta S$. The following statements are equivalent. * (a) The pair $(x,y)$ is jointly $\mathcal{K}$-recurrent. * (b) There exists $p\in\overline{\mathcal{K}}$ such that $T_{p}(x)=y=T_{p}(y)$. * (c) There exists an idempotent $p\in\overline{\mathcal{K}}$ such that $T_{p}(x)=y=T_{p}(y)$. ###### Proof. (a) $\implies$ (b). For each neighborhood $U$ of $y$ put $B_{U}=\\{s\in S:\mbox{$T_{s}(x)\in U$ and $T_{s}(y)\in U$}\\}$. Observe that since $B_{U\cap V}=B_{U}\cap B_{V}$ for $U$ and $V$ neighborhoods of $y$, we have that the collection $\mathcal{B}=\\{B_{U}:\mbox{$U$ is a neighborhood of $y$}\\}$ is closed under finite intersections. Also, by assumption we have that $\mathcal{B}\subseteq\mathcal{L}(\mathcal{K})$. Hence by Theorem 2.2 we can pick $p\in\overline{\mathcal{K}}$ with $\mathcal{B}\subseteq p$. For every neighborhood $U$ of $y$ we have $B_{U}\subseteq\\{s\in S:T_{s}(x)\in U\\}$ and $B_{U}\subseteq\\{s\in S:T_{s}(y)\in U\\}$. Therefore $\\{s\in S:T_{s}(x)\in U\\}\in p$ and $\\{s\in S:T_{s}(y)\in U\\}\in p$. It now follows from Theorem 2.7 that $T_{p}(x)=y=T_{p}(y)$. (b) $\implies$ (c). Put $M=\\{p\in\overline{\mathcal{K}}:T_{p}(x)=y=T_{p}(y)\\}$. By Theorem 2.3 it suffices to show that $M$ is a compact subsemigroup of $\beta S$. $M$ is nonempty by assumption. To see that $M$ is compact, we simply show that $M$ is closed. Let $p\not\in M$, then either $T_{p}(x)\neq y$ or $T_{p}(y)\neq y$. First assume that $T_{p}(x)\neq y$. By Theorem 2.7 pick $U$ a neighborhood of $y$ such that $\\{s\in S:T_{s}(x)\in U\\}\not\in p$. Put $A=\\{s\in S:T_{s}(x)\in U\\}$ and note that $S\setminus A\in p$. We have that $(\overline{S\setminus A})\cap M=\emptyset$, that is, $\overline{S\setminus A}$ is a (basic) neighborhood of $p$ that misses $M$. (If $q\in(\overline{S\setminus A})\cap M$, then it follows that $A\in q$ and $S\setminus A\in q$, a contradiction.) The construction of a (basic) neighborhood of $p$ that misses $M$ when $T_{p}(y)\neq y$ is similar. Therefore $M$ is a closed subset of $\beta S$. To see that $M$ is a subsemigroup, let $q$, $r\in M$. Then by Theorem 2.7 and assumption we have $T_{qr}(x)=T_{q}\circ T_{r}(x)=T_{q}(y)=y=T_{q}\circ T_{r}(y)=T_{qr}(y)$. ∎ ###### Theorem 3.3 (Main Result). Let $(S,\cdot)$ be a semigroup, let $\mathcal{K}$ be a filter on $S$ such that $\overline{\mathcal{K}}$ is a compact subsemigroup of $\beta S$, and let $A\subseteq S$. Then $A$ is a member of an idempotent in $\overline{\mathcal{K}}$ if and only if there exists a dynamical system $(X,\langle T_{s}\rangle_{s\in S})$ with points $x$ and $y$ in $X$ and there exists a neighborhood $U$ of $y$ such that the pair $(x,y)$ is jointly $\mathcal{K}$-recurrent and $A=\\{s\in S:T_{s}(x)\in U\\}$. ###### Proof. ($\Rightarrow$) Let $R=S\cup\\{e\\}$ be the semigroup with an identity $e$ adjoined to $S$. (For expository convenience, we add this new identity even if $S$ already contains an identity.) Give $\\{0,1\\}$ the discrete topology and give $X=\hbox{${}^{\hbox{$R$}}{\hskip-2.0pt\\{0,1\\}}$}$ the product topology. Then $X$ is a compact Hausdorff space. For each $s\in S$, define $T_{s}\colon X\to X$ by $T_{s}(f)=f\circ\rho_{s}$. It’s a routine exercise, or see [8, Theorem 19.14], to show that $(X,\langle T_{s}\rangle_{s\in S})$ is a dynamical system. Now let $x=\mathbf{1}_{A}$ be the characteristic function of $A$, pick an idempotent $r\in\overline{\mathcal{K}}$ with $A\in r$, and put $y=T_{r}(x)$. Then we have that $T_{r}(y)=T_{r}\bigl{(}T_{r}(x)\bigr{)}=T_{rr}(x)=T_{r}(x)=y$. Therefore by Lemma 3.2 we have that the pair $(x,y)$ is jointly $\mathcal{K}$-recurrent. Put $U=\\{w\in X:w(e)=y(e)\\}$ and observe that $U$ is a (subbasic) open neighborhood of $y$. (The set $U$ is equal to the inverse image of $\\{y(e)\\}$ under the projection map.) To help us show that $U$ is the neighborhood of $y$ we are looking for, we first will show that $y(e)=1$. Since $y=T_{r}(x)$ we have that $\\{s\in S:T_{s}(x)\in U\\}\in r$ by Theorem 2.7. We can pick $s\in A$ such that $T_{s}(x)\in U$. Then by definition of $U$ we have that $y(e)=T_{s}(x)(e)=x\bigl{(}\rho_{s}(e)\bigr{)}=x(es)=x(s)$. Also by our choice of $s\in A$ we have $x(s)=\mathbf{1}_{A}(s)=1$. To finish up this direction observe that for $s\in S$ the following logical relation is true: $\displaystyle s\in A$ $\displaystyle\iff\mathbf{1}_{A}(s)=1,$ $\displaystyle\iff x(s)=1,$ $\displaystyle\iff x(es)=1,$ $\displaystyle\iff(x\circ\rho_{s})(e)=1,$ $\displaystyle\iff T_{s}(x)(e)=1=y(e),$ $\displaystyle\iff T_{s}(x)\in U.$ Hence $A=\\{s\in S:T_{s}(x)\in U\\}$. ($\Leftarrow$) Pick a dynamical system $(X,\langle T_{s}\rangle_{s\in S})$, pick points $x$ and $y$ in $X$, and pick $U$ a neighborhood of $y$ as guaranteed by assumption. By Lemma 3.2 pick an idempotent $r\in\overline{\mathcal{K}}$ such that $T_{r}(x)=y=T_{r}(y)$. Since $U$ is a neighborhood of $y$ and $T_{r}(x)=y$, we have $A=\\{s\in S:T_{s}(x)\in U\\}\in r$ by Theorem 2.7. ∎ ###### Remark 3.4. One remarkable thing about Lemma 3.2 and Theorem 3.3 is that, while the results are much more general, the proofs are essentially trivial modifications of the proofs of [2, Lemma 3.3 and Theorem 3.4]. ## 4 A Dynamical Characterization of $C$ sets In this section we give an application of our main result in Section 3 to prove a dynamical characterization of $C$ sets. We start by giving the combinatorial definition of a $C$ set. As mentioned in Section 1 this combinatorial definition is rather complicated; but we shall soon state an algebraic characterization showing that $C$ sets are members of idempotents in a certain compact subsemigroup. In the following definition, given an indexed family $\langle A_{i}:i\in I\rangle$ of sets, we let $\hbox{\bigmath\char 2\relax}_{i\in I}A_{i}$ represent its cartesian product. (We reserve the use of the symbol $\prod$ for our semigroup operation.) Recall from Section 1 that given a set $S$ we let $\mathcal{T}=\hbox{${}^{\hbox{$\mathbb{N}$}}{\hskip-2.0ptS}$}$ and $\mathcal{P}_{\\!f}(X)$ is the collection of all nonempty finite subsets of $X$. ###### Definition 4.1. Let $(S,\cdot)$ be a semigroup. * (a) For each positive integer $m$ put $\mathcal{J}_{m}=\\{(t_{1},t_{2},\ldots,t_{m})\in\mathbb{N}^{m}:t_{1}<t_{2}<\cdots<t_{m}\\}$. * (b) Given $m\in\mathbb{N}$, $a\in S^{m+1}$, $t\in\mathcal{J}_{m}$, and $f\in\mathcal{T}$, put $x(m,a,t,f)=\prod_{i=1}^{m}\bigr{(}a(i)f(t_{i})\bigl{)}a(m+1)$. * (c) We call a subset $A\subseteq S$ a $C$ set if and only if there exist functions $m\colon\mathcal{P}_{\\!f}(\mathcal{T})\to\mathbb{N}$, $\alpha\in\hbox{\bigmath\char 2\relax}_{F\in\mathcal{P}_{\\!f}(\mathcal{T})}S^{m(F)+1}$, and $\tau\in\hbox{\bigmath\char 2\relax}_{F\in\mathcal{P}_{\\!f}(\mathcal{T})}\mathcal{J}_{m(F)}$ such that the following two statements are satisfied: * (1) If $F$, $G\in\mathcal{P}_{\\!f}(\mathcal{T})$ and $F\subsetneq G$, then $\tau(F)\bigl{(}m(F)\bigr{)}<\tau(G)(1)$. * (2) Whenever $m\in\mathbb{N}$, $G_{1}$, $G_{2}$, …, $G_{m}$ is a finite sequence in $\mathcal{P}_{\\!f}(\mathcal{T})$ with $G_{1}\subsetneq G_{2}\subsetneq\cdots\subsetneq G_{m}$, and for each $i\in\\{1,2,\ldots,m\\}$, $f_{i}\in G_{i}$, then we have $\prod_{i=1}^{m}x(m(G_{i}),\alpha(G_{i}),\tau(G_{i}),f_{i})\in A.$ ###### Remark 4.2. This definition of a $C$ set is different from the original (and more complicated) definition given in [4, Definition 3.3(i)]. It is a result in the author’s dissertation, to be included in a forthcoming paper [10], that our simpler definition of a $C$ set is equivalent to the original definition. Before giving an algebraic characterization of $C$ sets, we pause to state the Central Sets Theorem. ###### Theorem 4.3 (Central Sets Theorem). Central sets in a semigroup are $C$ sets. ###### Proof. This is proved in [10] using the our definition of a $C$ set, and is proved in [4, Corollary 3.10] under the original definition. ∎ To give the algebraic definition of a $C$ set we shall need the following (curiously named) combinatorial notion closely related to $C$ sets. ###### Definition 4.4. Let $(S,\cdot)$ be a semigroup. * (a) We call a subset $A\subseteq S$ a $J$ set if and only if for every $F\in\mathcal{P}_{\\!f}(\mathcal{T})$, there exist $m\in\mathbb{N}$, $a\in S^{m+1}$, and $t\in\mathcal{J}_{m}$ such that for all $f\in F$, $x(m,a,t,f)\in A$. * (b) $J(S)=\\{p\in\beta S:\mbox{$A$ is a $J$ set for every $A\in p$}\\}$. ###### Remark 4.5. I must point out that $J$ sets are not named after the author! The term $J$ set is derived from the term ‘$J_{Y}$ set’ introduced as [7, Definition 2.4(b)] in a different and earlier paper. This definition of a $J$ set is also different from the original (and more complicated) definition given in [4, Definition 3.3(e)]. The fact that these two definitions are equivalent is proved by the author in his dissertation and in the forthcoming paper [10]. ###### Theorem 4.6. Let $(S,\cdot)$ be a semigroup and $\mathcal{K}=\\{A\subseteq S:\mbox{$S\setminus A$ is not a $J$ set}\\}$. Then $\mathcal{K}$ is a filter on $S$ with $J(S)=\overline{\mathcal{K}}$ and $J(S)$ is a compact subsemigroup of $\beta S$. ###### Proof. To show that $\mathcal{K}$ is nonempty, doesn’t contain the empty set, and is closed under supersets is a routine exercise. The fact that $\mathcal{K}$ is closed under finite intersections follows from [10] using the new definition or [9, Theorem 2.14] using the old equivalent definition of $J$ sets. Observe that, under the assumption that $\mathcal{K}$ is a filter, $\mathcal{L}(\mathcal{K})=\\{A\subseteq S:\mbox{$A$ is a $J$ set}\\}$. Hence it follows from [10] (new definition) or [8, Theorem 3.11] (old equivalent definition) that $J(S)=\overline{\mathcal{K}}$. Finally, the fact that $J(S)$ is a subsemigroup follows from [4, Theorem 3.5]. ∎ Since $J(S)$ is a compact subsemigroup, in fact an ideal, of $\beta S$, by Theorem 2.3 we have that $J(S)$ contains idempotents. The following theorem connects these idempotent elements with $C$ sets. ###### Theorem 4.7. Let $(S,\cdot)$ be a semigroup and $A\subseteq S$. Then $A$ is a $C$ set if and only if there exists an idempotent $p\in J(S)$ such that $A\in p$. ###### Proof. This is proved in [10] for the new definition or [4, Theorem 3.8] for the old definition. ∎ Using these facts and the general results in Section 3 we end with the following dynamical characterization of $C$ sets. ###### Theorem 4.8. Let $(S,\cdot)$ be a semigroup and $A\subseteq S$. Then $A$ is a $C$ set if and only if there exists a dynamical system $(X,\langle T_{s}\rangle_{s\in S})$ with points $x$ and $y$ in $X$ and there exists a neighborhood $U$ of $y$ such that $\\{s\in S:\mbox{$T_{s}(x)\in U$ and $T_{s}(y)\in U$}\\}$ is a $J$ set and $A=\\{s\in S:T_{s}(x)\in U\\}$. ###### Proof. Let $\mathcal{K}=\\{B\subseteq S:\mbox{$S\setminus B$ is not a $J$ set}\\}$ and note by Theorem 4.6 that $\overline{\mathcal{K}}=J(S)$ and $\mathcal{L}(\mathcal{K})=\\{A\subseteq S:\mbox{$A$ is a $J$ set}\\}$. Since Theorem 4.7 characterizes $C$ sets in terms of idempotents in $\overline{\mathcal{K}}$, we can apply Theorem 3.3 to prove our statement. ∎ ## References * [1] Vitaly Bergelson and Neil Hindman, Nonmetrizable topological dynamics and Ramsey theory, Transactions of the American Mathematical Society 320 (1990), no. 1, 293–320. * [2] Shea D. Burns and Neil Hindman, Quasi-central sets and their dynamical characterization, Topology Proceedings 31 (2007), no. 2, 445–455. * [3] Dennis Davenport, The minimal ideal of compact subsemigroups of $\beta S$, Semigroup Forum 41 (1990), 201–213. * [4] Dibyendu De, Neil Hindman, and Dona Strauss, A new and stronger central sets theorem, Fundamenta Mathematicae 199 (2008), 155–175. * [5] H. Furstenberg, Recurrence in Ergodic Theory and Combinatorial Number Theory, Princeton University Press, 1981. * [6] Neil Hindman, Small sets satisfying the Central Sets Theorem, Integers 9 Supplement (2009), article 5. * [7] Neil Hindman, Amir Maleki, and Dona Strauss, Central sets and their combinatorial characterization, Journal of Combinatorial Theory (Series A) 74 (1996), 188–208. * [8] Neil Hindman and Dona Strauss, Algebra in the Stone-Čech Compactification, De Gruyter expositions in mathematics, no. 27, Walter de Gruyter, 1998. * [9] , Cartesian products of sets satisfying the Central Sets Theorem, Topology Proceedings 35 (2010), 203–223. * [10] John H. Johnson, A new and simpler Central Sets Theorem, (in preparation), November 2011. * [11] Hong ting Shi and Hong wei Yang, Nonmetrizable topological dynamical characterization of central sets, Fundamenta Mathematicae (1996), no. 150, 1–9.	arxiv-papers	2011-12-04T04:10:01	2024-09-04T02:49:24.924950	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "John H. Johnson", "submitter": "John Johnson", "url": "https://arxiv.org/abs/1112.0715" }
1112.0722		arxiv-papers	2011-12-04T05:03:14	2024-09-04T02:49:24.931287	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Ion Olaru", "submitter": "Ion Olaru", "url": "https://arxiv.org/abs/1112.0722" }
1112.0724	# Trudinger-Moser inequalities on complete noncompact Riemannian manifolds Yunyan Yang yunyanyang@ruc.edu.cn Department of Mathematics, Renmin University of China, Beijing 100872, P. R. China ###### Abstract Let $(M,g)$ be a complete noncompact Riemannian $n$-manifold ($n\geq 2$). If there exist positive constants $\alpha$, $\tau$ and $\beta$ such that $\sup_{u\in W^{1,n}(M),\,\\|u\\|_{1,\tau}\leq 1}\int_{M}\left(e^{\alpha\|u\|^{\frac{n}{n-1}}}-\sum_{k=0}^{n-2}\frac{\alpha^{k}\|u\|^{\frac{nk}{n-1}}}{k!}\right)dv_{g}\leq\beta,$ where $\\|u\\|_{1,\tau}=\\|\nabla_{g}u\\|_{L^{n}(M)}+\tau\\|u\\|_{L^{n}(M)}$, then we say that Trudinger-Moser inequality holds. Suppose Trudinger-Moser inequality holds, we prove that there exists some positive constant $\epsilon$ such that ${\rm Vol}_{g}(B_{x}(1))\geq\epsilon$ for all $x\in M$. Also we give a sufficient condition under which Trudinger-Moser inequality holds, say the Ricci curvature of $(M,g)$ has lower bound and its injectivity radius is positive. Moreover, Adams inequality is discussed in this paper. For application of Trudinger-Moser inequalities, we obtain existence results for some quasilinear equations with nonlinearity of exponential growth. ###### keywords: Trudinger-Moser inequality, Adams inequality, exponential growth ###### MSC: 58E35, 35J60 ††journal: *** ## 1 Introduction Let $\Omega$ be a smooth bounded domain in $\mathbb{R}^{n}$ ($n\geq 2)$ and $C_{0}^{\infty}(\Omega)$ be a space of smooth functions with compact support in $\Omega$. Let $W_{0}^{m,p}(\Omega)$ be the completion of $C_{0}^{\infty}(\Omega)$ under the Sobolev norm $\\|u\\|_{W_{0}^{m,p}(\Omega)}:=\left(\sum_{l=0}^{m}\int_{\Omega}\|\nabla^{l}u\|^{p}dx\right)^{1/p}.$ (1.1) Assume that $m$ is an integer satisfying $1\leq m<n$. Then Sobolev embedding theorem asserts that $W_{0}^{m,p}(\Omega)\hookrightarrow L^{q}(\Omega)$, $1\leq q\leq{np}/{(n-mp)}$. Concerning the limiting case $mp=n$, one has $W_{0}^{m,n/m}(\Omega)\hookrightarrow L^{q}(\Omega)$ for all $q\geq 1$. But the embedding is not valid for $q=\infty$. To fill this gap, it is natural to find the maximal growth function $g:\mathbb{R}\rightarrow\mathbb{R}^{+}$ such that $\sup_{u\in W_{0}^{m,n/m}(\Omega),\,\\|u\\|_{W_{0}^{m,n/m}(\Omega)}\leq 1}\int_{\Omega}g(u)dx<\infty.$ In the case $m=1$, Trudinger [38] and Pohozaev [33] found independently that the maximal growth is of exponential type. More precisely, there exist two positive constants $\alpha_{0}$ and $C$ depending only on $n$ such that $\sup_{u\in W_{0}^{1,n}(\Omega),\,\\|u\\|_{W_{0}^{1,n}(\Omega)}\leq 1}\int_{\Omega}e^{\alpha_{0}\|u\|^{\frac{n}{n-1}}}dx\leq C\|\Omega\|,$ (1.2) where $\|\Omega\|$ denotes the Lebesgue measure of $\Omega$. Moser [30] obtained the best constant $\alpha_{n}=n\omega_{n-1}^{1/(n-1)}$ such that the above supremum is finite when $\alpha_{0}$ is replaced by $\alpha_{n}$, where $\omega_{n-1}$ is the area of the unit sphere in $\mathbb{R}^{n}$. Moser’s work relies on a rearrangement argument [17]. In literature the kind of inequalities like (1.2) are called Trudinger-Moser inequalities. Adams [2] generalized inequality (1.2) to the case of general $m:1\leq m<n$ as follows. For any $u\in W_{0}^{m,n/m}(\Omega)$, the $l$-th order gradient of $u$ reads $\nabla^{l}u=\left\\{\begin{array}[]{lll}\Delta^{\frac{l}{2}}u,&{\rm if}\,\,\,l\,\,\,{\rm is}\,\,\,{\rm even,}\\\\[6.45831pt] \nabla\Delta^{\frac{l-1}{2}}u,&{\rm if}\,\,\,l\,\,\,{\rm is}\,\,\,{\rm odd,}\end{array}\right.$ (1.3) there exits a positive constant $C_{m,n}$ such that $\sup_{u\in W_{0}^{m,n/m}(\Omega),\,\\|u\\|_{W_{0}^{m,n/m}(\Omega)}\leq 1}\int_{\Omega}e^{\beta_{0}\|u\|^{\frac{n}{n-m}}}dx\leq C_{m,n}\|\Omega\|,$ (1.4) where $\beta_{0}$ is the best constant depending only on $n$ and $m$, namely $\beta_{0}=\beta_{0}(m.n):=\left\\{\begin{array}[]{lll}\frac{n}{\omega_{n-1}}\left[\frac{\pi^{{n}/{2}}2^{m}\Gamma\left({(m+1)}/{2}\right)}{\Gamma\left({(n-m+1)}/{2}\right)}\right]&{\rm when}\,\,\,m\,\,\,{\rm is}\,\,\,odd\\\\[6.45831pt] \frac{n}{\omega_{n-1}}\left[\frac{\pi^{{n}/{2}}2^{m}\Gamma\left({m}/{2}\right)}{\Gamma\left({(n-m)}/{2}\right)}\right]&{\rm when}\,\,\,m\,\,\,{\rm is}\,\,\,even.\end{array}\right.$ (1.5) The inequality (1.4) is known as Adams inequality. Adams first represented a function $u$ in terms of its gradient function $\nabla^{m}u$ by using a convolution operator. Then using the O’Neil’s idea [31] of rearrangement of convolution of two functions and the idea which originally goes back to Garcia, he obtained (1.4). There are many types of extensions for Trudinger-Moser inequality and Adams inequality. One is to establish such inequalities on the whole euclidian space $\mathbb{R}^{n}$. Cao [8] employed the decreasing rearrangement argument to prove that for all $\alpha<4\pi$ and $A>0$, there exists a constant $C$ depending only on $\alpha$ and $A$ such that for all $u\in W^{1,2}(\mathbb{R}^{2})$ with $\int_{\mathbb{R}^{2}}\|\nabla u\|^{2}dx\leq 1,\,\int_{\mathbb{R}^{2}}u^{2}dx\leq A$, there holds $\int_{\mathbb{R}^{2}}\left(e^{\alpha u^{2}}-1\right)dx\leq C.$ (1.6) His argument was generalized to $n$-dimensional case by do Ó [12] and Panda [32] independently. Later, Adachi-Tanaka [1] gave another type of generalization. All these inequalities are subcritical ones since $\alpha<\alpha_{n}$. It was Ruf [35] who first proved the critical Trudinger- Moser inequality in the whole euclidian space $\mathbb{R}^{2}$ and gave out extremal functions via more delicate analysis. This result was generalized to $n$-dimensional case by Li-Ruf [25] through combining symmetrization and blow- up analysis. Subsequently, using the decreasing rearrangement argument and Young’s inequality, Adimurthi-Yang [4] derived an interpolation of Trudinger- Moser inequality and Hardy inequality in $\mathbb{R}^{n}$, which can be viewed as a singular Trudinger-Moser inequality. Another kind of singular Trudinger- Moser inequality was recently established by Wang-Ye [39] through the method of blow-up analysis. Substantial progresses on Adams inequality in $\mathbb{R}^{n}$ was also made recently. Following lines of Adams, Kozono et al. [19] obtained subcritical Adams inequality in the whole euclidian space $\mathbb{R}^{n}$. Based on rearrangement argument of Trombetti-Vazquez [37], Ruf-Sani [36] proved the critical Adams inequalities in $\mathbb{R}^{n}$ as follows. Let $m$ be an even integer less than $n$. Assume that $u\in W_{0}^{m,n/m}(\mathbb{R}^{n})$ and $\\|(-\Delta+I)^{m/2}u\\|_{L^{n/m}(\mathbb{R}^{n})}\leq 1$. There exists a constant $C>0$ depending only on $n$ and $m$ such that $\int_{\mathbb{R}^{n}}\left(e^{\beta_{0}\|u\|^{\frac{n}{n-m}}}-\sum_{k=0}^{j-2}\frac{\beta_{0}^{k}\|u\|^{\frac{nk}{n-m}}}{k!}\right)dx<C,$ where $j$ is the smallest integer great than or equal to $n/m$. Another extension is to establish Trudinger-Moser inequality and Adams inequality on compact Riemannian manifolds. Let $(M,g)$ be a compact Riemannian $n$-manifold. For $u\in W^{1,n}(M)$, it was shown by Aubin [5] that $\exp(\alpha\|u\|^{n/(n-1)}\\|u\\|_{W^{1,n}(M)}^{-n/(n-1)})$ is integrable for sufficiently small $\alpha>0$ which does not depend on $u$. In fact, this is an easy consequence of Trudinger-Moser inequality and finite partition of unity on $M$. Let $\tilde{\alpha}$ be the supremum of the above $\alpha$’s. It was first found by Cherrier [9] that $\tilde{\alpha}=\alpha_{n}$. Cherrier [10] obtained similar results for $u\in W_{m,n/m}(M)$. Following the lines of Adams, Fontana [15] obtained critical Adams inequality on $(M,g)$. In 1997, using the method of blow-up analysis, Ding et al. [11] established a nice Trudinger-Moser inequality on compact Riemannian surface and successfully applied it to deal with the prescribed Gaussian curvature problem. Adapting the argument of Ding et al., Li [21, 22] and Li-Liu [23] proved the existence of extremal functions for Trudinger-Moser inequalities. Their idea was also employed by the author [40, 41, 42] to find extremal functions for various Trudinger-Moser type inequalities. For vector bundles over a compact Riemannian 2-manifold, Li-Liu-Yang obtained Trudinger-Moser inequalities in [24]. Among other contributions, we mention the following results. Using the method of blow-up analysis, Adimurthi-Druet [3] proved that when $0\leq\alpha<\lambda_{1}(\Omega)$, there holds $\sup_{u\in W_{0}^{1,2}(\Omega),\,\\|\nabla u\\|_{2}\leq 1}\int_{\Omega}e^{4\pi u^{2}(1+\alpha\\|u\\|_{2}^{2})}dx<\infty,$ where $\lambda_{1}(\Omega)$ is the first eigenvalue of Laplacian on bounded smooth domain $\Omega\subset\mathbb{R}^{2}$. Moreover, the supremum is infinite when $\alpha\geq\lambda_{1}(\Omega)$. Later this result was generalized by the author [43] and Lu-Yang [27, 28, 29]. Although there are fruitful results on euclidian space and compact Riemannian manifolds, we know little about Trudinger-Moser inequalities on complete noncompact Riemannian manifolds. In this paper, we concern this problem. Let $(M,g)$ be any complete noncompact Riemannian $n$-manifold. Throughout this paper, all the manifolds are assumed to be without boundary, and of dimension $n\geq 2$. We say that Trudinger-Moser inequality holds on $(M,g)$ if there exist positive constants $\alpha$, $\tau$ and $\beta$ such that $\sup_{u\in W^{1,n}(M),\,\\|u\\|_{1,\tau}\leq 1}\int_{M}\left(e^{\alpha\|u\|^{\frac{n}{n-1}}}-\sum_{k=0}^{n-2}\frac{\alpha^{k}\|u\|^{\frac{nk}{n-1}}}{k!}\right)dv_{g}\leq\beta,$ (1.7) where $\\|u\\|_{1,\tau}=\left(\int_{M}\|\nabla_{g}u\|^{n}dv_{g}\right)^{1/n}+\tau\left(\int_{M}\|u\|^{n}dv_{g}\right)^{1/n}.$ (1.8) If the above supremum is infinite for all $\alpha>0$ and $\tau>0$, then we say that Trudinger-Moser inequality is not valid on $(M,g)$. Motivated by Sobolev embedding (Hebey [18], Chapter 3), in this paper, we propose and answer the following three questions. $(Q_{1})$ Which kind of complete noncompact Riemannian manifolds can possibly make Trudinger-Moser inequalities hold? $(Q_{2})$ What geometric assumptions should we consider in order to obtain Trudinger-Moser inequalities on complete noncompact Riemannian manifolds? $(Q_{3})$ Are those geometric assumptions in $(Q_{2})$ necessary? This paper is organized as follows: In Section 2, we state our main results. From section 3 to section 5, we answer the questions $(Q_{1})$-$(Q_{3})$, respectively. Adams inequalities are considered in section 6. Finally, Trudinger-Moser inequalities are applied to nonlinear analysis in section 7. ## 2 Main results In this section, we answer questions $(Q_{1})$-$(Q_{3})$, and give an application of Trudinger-Moser inequality. Throughout this paper, we denote for simplicity a function $\zeta:\mathbb{N}\times[0,\infty)\rightarrow\mathbb{R}$ by $\zeta(l,t)=e^{t}-\sum_{k=0}^{l-2}\frac{t^{k}}{k!},\quad\forall l\geq 2.$ (2.1) From ([44], lemma 2.1 and lemma 2.2), we know that $\left(\zeta(l,t)\right)^{q}\leq\zeta(l,qt)$ (2.2) and $\zeta(l,t)\leq\frac{1}{\mu}\zeta(l,\mu t)+\frac{1}{\nu}\zeta(l,\nu t).$ (2.3) for all $l\geq 2$, $q\geq 1$, $t\in[0,\infty)$, and $\mu>0$, $\nu>0$ satisfying $1/\mu+1/\nu=1$. The following proposition answers question $(Q_{1})$. Proposition 2.1. Let $(M,g)$ be a complete Riemannian $n$-manifold. Suppose that Trudinger-Moser inequality holds on $(M,g)$, i.e. there exist positive constants $\alpha$, $\tau$ and $\beta$ such that (1.7) holds. Then the Sobolev space $W^{1,n}(M)$ is embedded in $L^{q}(M)$ continuously for any $q\geq n$. Furthermore, for any $r>0$ there exists a positive constant $\epsilon$ depending only on $n$, $\alpha$, $\tau$, $\beta$ and $r$ such that ${\rm Vol}_{g}(B_{x}(r))\geq\epsilon$ for all $x\in M$, where $B_{x}(r)$ denotes the geodesic ball centered at $x$ with radius $r$. From proposition 2.1 we know that there are indeed complete noncompact Riemannian manifolds such that Trudinger-Moser inequalities are not valid, namely Corollary 2.2. For any integer $n\geq 2$, there exists a complete noncompact Riemannian $n$-manifold on which Trudinger-Moser inequality is not valid. To answer question $(Q_{2})$, we have the following: Theorem 2.3. Let $(M,g)$ be a complete noncompact Riemannian $n$-manifold. Suppose that its Ricci curvature has lower bound, namely ${\rm Rc}_{(M,g)}\geq Kg$ for some constant $K\in\mathbb{R}$, and its injectivity radius is strictly positive, namely ${\rm inj}_{(M,g)}\geq i_{0}$ for some constant $i_{0}>0$. Then we have $(i)$ for any $0\leq\alpha<\alpha_{n}=n\omega_{n-1}^{1/(n-1)}$, there exists positive constants $\tau$ and $\beta$ depending only on $n$, $\alpha$, $K$ and $i_{0}$ such that (1.7) holds. As a consequence, $W^{1,n}(M)$ is embedded in $L^{q}(M)$ continuously for any $q\geq n$; $(ii)$ for any $\alpha>\alpha_{n}$ and any $\tau>0$, the supremum in (1.7) is infinite; $(iii)$ for any $\alpha>0$ and any $u\in W^{1,n}(M)$, there holds $\zeta(n,\alpha\|u\|^{n/(n-1)})\in L^{1}(M)$. Now we turn to question $(Q_{3})$. The following proposition implies that one of the hypotheses of theorem 2.3, the injectivity radius is strictly positive, can not be removed. Proposition 2.4. For any integer $n\geq 2$, there exists a complete noncompact Riemannian $n$-manifold, whose Ricci curvature has lower bound, such that Trudinger-Moser inequality is not valid on it. We shall construct complete noncompact Riemannian manifolds on which Trudinger-Moser inequalities hold, but their Ricci curvatures are unbounded from below. This implies that the other hypothesis of theorem 2.3, Ricci curvature has lower bound, is not necessary. Namely Proposition 2.5. For any integer $n\geq 2$, there exists a complete noncompact Riemannian $n$-manifold on which Trudinger-Moser inequality holds, but its Ricci curvature is unbounded from below. Let us explain the idea of proving proposition 2.1 and theorem 2.3. The first part of conclusions of proposition 2.1, $W^{1,n}(M)\hookrightarrow L^{q}(M)$ for all $q\geq n$, is based on an observation $\int_{M}\zeta\left(n,\alpha\|u\|^{\frac{n}{n-1}}\right)dv_{g}=\sum_{k=n-1}^{\infty}\frac{\alpha^{k}}{k!}\int_{M}\|u\|^{\frac{nk}{n-1}}dv_{g}.$ To find some $\epsilon>0$ such that ${\rm Vol}_{g}(B_{x}(r))\geq\epsilon$ for all $x\in M$, we employ the method of Carron ([18], lemma 3.2) who obtained similar result for Sobolev embedding. For the proof of theorem 2.3, we first derive a uniform local Trudinger-Moser inequality (lemma 4.2 below). Then using harmonic coordinates and Gromov’s covering lemma, we get the desired global Trudinger-Moser inequality. The proofs of corollary 2.2, proposition 2.4 and proposition 2.5 are all based on construction of Riemannian manifolds. Concerning Adams inequalities on complete noncompact Riemannian manifolds, we have the following: Theorem 2.6. Let $(M,g)$ be a complete noncompact Riemannian $n$-manifold. Suppose that there exist positive constants $C(k)$ and $i_{0}$ such that $\|\nabla_{g}^{k}{\rm Rc}_{(M,g)}\|\leq C(k)$, $k=0,1,\cdots,m-1$, ${\rm inj}_{(M,g)}\geq i_{0}>0$. Let $j=n/m$ when $n/m$ is an integer, and $j=[n/m]+1$ when $n/m$ is not an integer, where $[n/m]$ denotes the integer part of $n/m$. Then we conclude the following: $(i)$ there exist positive constants $\alpha_{0}$ and $\beta$ depending only on $n$, $m$, $C(k)$, $k=1,\cdots,m-1$, and $i_{0}$ such that $\sup_{\\|u\\|_{W^{m,n/m}(M)}\leq 1}\int_{M}\zeta\left(j,\alpha_{0}\|u\|^{\frac{n}{n-m}}\right)dv_{g}\leq\beta.$ As a consequence, $W^{m,{n}/{m}}(M)$ is embedded in $L^{q}(M)$ continuously for any $q\geq{n}/{m}$; $(ii)$ for any $\alpha>0$ and any $u\in W^{m,n/m}(M)$, there holds $\zeta(j,\alpha\|u\|^{n/(n-m)})\in L^{1}(M)$. The proof of theorem 2.6 is similar to that of theorem 2.3. It should be remarked that the existing proofs of Trudinger-Moser inequalities or Adams inequalities for the euclidian space $\mathbb{R}^{n}$ are all based on rearrangement argument, which is difficult to be applied to complete noncompact Riemannian manifold case. Our method is from uniform local estimates to global estimates. It does not depend on the rearrangement theory directly. Trudinger-Moser inequality plays an important role in nonlinear analysis. Let $(M,g)$ be a complete noncompact Riemannian $n$-manifold. $\nabla_{g}$ denotes its covariant derivative, and ${\rm div}_{g}$ denotes its divergence operator. Assume the Ricci curvature of $(M,g)$ has lower bound and the injectivity radius is strictly positive. We consider the existence results for the following quasilinear equation. $-{\rm div}_{g}(\|\nabla_{g}u\|^{n-2}\nabla_{g}u)+v(x)\|u\|^{n-2}u=\phi(x)f(x,u),$ (2.4) where $v(x)$, $\phi(x)$ and $f(x,t)$ are all continuous functions, and $f(x,t)$ behaves like $e^{\alpha t^{n/(n-1)}}$ as $t\rightarrow+\infty$. In the case that $(M,g)$ is the standard euclidean space $\mathbb{R}^{n}$ and $\phi(x)=\|x\|^{-\beta}$ $(0\leq\beta<n)$, problem (2.4) has been studied by do Ó et. al. [13, 14], Adimurthi-Yang [4], Yang [44], Lam-Lu [20] and Zhao [45]. Let $O$ be a fixed point of $M$ and $d_{g}(\cdot,\cdot)$ be the geodesic distance between two points of $(M,g)$. Assume that $\phi(x)$ satisfies the following hypotheses. $(\phi_{1})$ $\phi(x)\in L^{p}_{\rm loc}(M)$ for some $p>1$, i. e., for any $R>0$ there holds $\phi(x)\in L^{p}(B_{O}(R))$; $(\phi_{2})$ $\phi(x)>0$ for all $x\in M$ and there exist positive constants $C_{0}$ and $R_{0}$ such that $\phi(x)\leq C_{0}$ for all $x\in M\setminus B_{O}(R_{0})$. The potential $v(x)$ is assumed to satisfy the following: $(v_{1})$ there exists some constant $v_{0}>0$ such that $v(x)\geq v_{0}$ for all $x\in M$; $(v_{2})$ either $v(x)\in L^{1/{(n-1)}}(M)$ or $v(x)\rightarrow+\infty$ as $d_{g}(O,x)\rightarrow+\infty$. The nonlinearity $f(x,t)$ satisfies the following hypotheses. $(f_{1})$ there exist constants $\alpha_{0}$, $b_{1}$, $b_{2}>0$ such that for all $(x,t)\in M\times\mathbb{R}^{+}$, $\|f(x,t)\|\leq b_{1}t^{n-1}+b_{2}\zeta\left(n,\alpha_{0}t^{n/(n-1)}\right);$ $(f_{2})$ there exists some constant $\mu>n$ such that for all $x\in M$ and $t>0$, $0<\mu F(x,t)\equiv\mu\int_{0}^{t}f(x,s)ds\leq tf(x,t);$ $(f_{3})$ there exist constants $R_{1}$, $A_{1}>0$ such that if $t\geq R_{1}$, then for all $x\in M$ there holds $F(x,t)\leq A_{1}f(x,t).$ Define a function space $E=\left\\{u\in W^{1,n}(M):\int_{M}v(x)\|u\|^{n}dv_{g}<\infty\right\\}.$ (2.5) We say that $u\in E$ is a weak solution of problem (2.4) if for all $\varphi\in E$ we have $\int_{M}\left(\|\nabla_{g}u\|^{n-2}\nabla_{g}u\nabla_{g}\varphi+v(x)\|u\|^{n-2}u\varphi\right)dv_{g}=\int_{M}\phi(x){f(x,u)}\varphi dv_{g}.$ Define a weighted eigenvalue for the $n$-Laplace operator by $\lambda_{\phi}=\inf_{u\in E,\,u\not\equiv 0}\frac{\int_{M}(\|\nabla_{g}u\|^{n}+v(x)\|u\|^{n})dv_{g}}{\int_{M}\phi(x)\|u\|^{n}dv_{g}}.$ (2.6) Then we state the following: Theorem 2.7. Let $(M,g)$ be a complete noncompact Riemannian $n$-manifold. Suppose that ${\rm Rc}_{(M,g)}\geq Kg$ for some constant $K\in\mathbb{R}$, and ${\rm inj}_{(M,g)}\geq i_{0}$ for some positive constant $i_{0}$. Assume that $v(x)$ is a continuous function satisfying $(v_{1})$ and $(v_{2})$, $\phi(x)$ is a continuous function satisfying $(\phi_{1})$ and $(\phi_{2})$, $f:M\times\mathbb{R}\rightarrow\mathbb{R}$ is a continuous function and the hypotheses $(f_{1})$, $(f_{2})$ and $(f_{3})$ are satisfied. Furthermore we assume $(f_{4})$ $\limsup_{t\rightarrow 0+}{nF(x,t)}/{t^{n}}<\lambda_{\phi}$ uniformly in $x\in M$; $(f_{5})$ there exist constants $q>n$ and $C_{q}$ such that for all $(x,t)\in M\times[0,\infty)$ $f(x,t)\geq C_{q}t^{q-1},$ where $C_{q}>\left(\frac{q-n}{q}\right)^{{(q-n)}/{n}}\left(\frac{p\alpha_{0}}{(p-1)\alpha_{n}}\right)^{(q-n)(n-1)/n}S_{q}^{q}$ and $S_{q}=\inf_{u\in E\setminus\\{0\\}}\frac{\left(\int_{M}(\|\nabla_{g}u\|^{n}+v(x)\|u\|^{n})dv_{g}\right)^{1/n}}{\left(\int_{M}\phi(x)\|u\|^{q}dv_{g}\right)^{1/q}}.$ (2.7) Then the problem (2.4) has a nontrivial nonnegative weak solution. Remark 2.8. We shall prove that $S_{q}$ can be attained (lemma 7.2 below). When $(M,g)$ is the standard euclidian space $\mathbb{R}^{n}$, $\phi(x)=\|x\|^{-\beta}$ for $0\leq\beta<n$, $(f_{1})$-$(f_{4})$ and $None$ uniformly in $x$, where $\mathcal{M}$ is some sufficiently large number, we obtained similar existence result in [44]. The following proposition implies that the set of functions satisfying $(f_{1})$-$(f_{5})$ is not empty and assumptions $(f_{1})$-$(f_{5})$ do not imply $(H_{5})$. Proposition 2.9. There exist continuous functions $f:M\times\mathbb{R}\rightarrow\mathbb{R}$ such that $(f_{1})$-$(f_{5})$ are satisfied, but $(H_{5})$ is not satisfied. We also consider multiplicity results for a perturbation of the problem (2.4), namely $-{\rm div}_{g}(\|\nabla_{g}u\|^{n-2}\nabla_{g}u)+v(x)\|u\|^{n-2}u=\phi(x)f(x,u)+\epsilon h(x),$ (2.8) where $h(x)\in E^{}$, the dual space of $E$. If $h\not\equiv 0$ and $\epsilon>0$ is sufficiently small, under some assumptions there exist at least two distinct weak solutions to (2.8). Precisely, we have the following theorem. Theorem 2.10. Let $(M,g)$ be a complete noncompact Riemannian $n$-manifold. Suppose that ${\rm Rc}_{(M,g)}\geq Kg$ for some constant $K\in\mathbb{R}$, and ${\rm inj}_{(M,g)}\geq i_{0}$ for some positive constant $i_{0}$. Assume $f(x,t)$ is continuous in $M\times\mathbb{R}$ and $(f_{1})$-$(f_{5})$ are satisfied. Both $v(x)$ and $\phi(x)$ are continuous in $M$ and $(v_{1})$, $(v_{2})$, $(\phi_{1})$, $(\phi_{2})$ are satisfied, $h$ belongs to $E^{}$, the dual space of $E$, with $h\geq 0$ and $h\not\equiv 0$. Then there exists $\epsilon_{0}>0$ such that if $0<\epsilon<\epsilon_{0}$, then the problem (2.8) has at least two distinct nonnegative weak solutions. The proofs of theorem 2.7 and theorem 2.10 are based on theorem 2.3, Mountain- pass theorem and Ekeland’s variational principle. Though similar idea was used in the case $(M,g)$ is the standard euclidian space $\mathbb{R}^{n}$ [4, 13, 14, 20, 44], technical difficulties caused by manifold structure must be smoothed. ## 3 Necessary conditions In this section, we consider the necessary conditions under which Trudinger- Moser inequality holds. Precisely we shall prove proposition 2.1 and corollary 2.2. Firstly we have the following: Lemma 3.1. Let $(M,g)$ be a complete Riemannian $n$-manifold. Suppose that there exist constants $q>n$, $A>0$ and $\tau>0$ such that for all $u\in W^{1,n}(M)$, there holds $\left(\int_{M}\|u\|^{q}dv_{g}\right)^{1/q}\leq A\\|u\\|_{1,\tau},$ (3.1) where $\\|u\\|_{1,\tau}$ is defined by (1.8). Then for any $r>0$ there exists some positive constant $\epsilon$ depending only on $A$, $n$, $q$, $\tau$, and $r$ such that for all $x\in M$, ${\rm Vol}_{g}(B_{x}(r))\geq\epsilon$. Proof. Let $r>0$, $x\in M$, and $\phi\in W^{1,n}(M)$ be such that $\phi=0$ on $M\setminus B_{x}(r)$. By Hölder’s inequality, $\left(\int_{M}\|\phi\|^{n}dv_{g}\right)^{1/n}\leq{\rm Vol}_{g}(B_{x}(r))^{\frac{1}{n}-\frac{1}{q}}\left(\int_{M}\|\phi\|^{q}dv_{g}\right)^{1/q}.$ This together with (3.1) gives $\left(1-\tau A{\rm Vol}_{g}(B_{x}(r))^{\frac{1}{n}-\frac{1}{q}}\right)\left(\int_{M}\|\phi\|^{q}dv_{g}\right)^{1/q}\leq A\left(\int_{M}(\|\nabla\phi\|^{n}dv_{g}\right)^{1/n}.$ (3.2) Fix $x\in M$ and $R>0$. Then either ${\rm Vol}_{g}(B_{x}(R))>\left(\frac{1}{2\tau A}\right)^{{nq}/{(q-n)}}$ (3.3) or ${\rm Vol}_{g}(B_{x}(R))\leq\left(\frac{1}{2\tau A}\right)^{{nq}/{(q-n)}}.$ (3.4) If (3.4) holds, then we have $1-\tau A{\rm Vol}_{g}(B_{x}(R))^{\frac{1}{n}-\frac{1}{q}}\geq{1}/{2},$ and whence for all $r\in(0,R]$ and all $\phi\in W^{1,n}(M)$ with $\phi=0$ on $M\setminus B_{x}(r)$, $\left(\int_{M}\|\phi\|^{q}dv_{g}\right)^{1/q}\leq 2A\left(\int_{M}(\|\nabla\phi\|^{n}dv_{g}\right)^{1/n}.$ (3.5) Now we set $\phi(y)=\left\\{\begin{array}[]{lll}r-d_{g}(x,y)&{\rm when}\quad d_{g}(x,y)\leq r\\\\[6.45831pt] 0&{\rm when}\quad d_{g}(x,y)>r.\end{array}\right.$ Clearly $\phi\in W^{1,n}(M)$, $\phi=0$ on $M\setminus B_{x}(r)$, $\phi\geq r/2$ on $B_{x}(r/2)$, and $\|\nabla\phi\|=1$ almost everywhere in $B_{x}(r)$. It then follows from (3.5) that $\frac{r}{2}{\rm Vol}_{g}(B_{x}(r/2))^{1/q}\leq 2A{\rm Vol}_{g}(B_{x}(r))^{1/n}.$ Hence we have for all $r\leq R$, ${\rm Vol}_{g}(B_{x}(r))\geq\left(\frac{r}{4A}\right)^{n}{\rm Vol}_{g}(B_{x}(r/2))^{n/q}.$ By induction we obtain for any positive integer $m$, ${\rm Vol}_{g}(B_{x}(R))\geq\left(\frac{R}{2A}\right)^{n\alpha(m)}\left(\frac{1}{2}\right)^{n\beta(m)}{\rm Vol}_{g}(B_{x}(R/2^{m}))^{(n/q)^{m}},$ (3.6) where $\alpha(m)=\sum_{j=1}^{m}({n}/{q})^{j-1},\quad\beta(m)=\sum_{j=1}^{m}j(n/q)^{j-1}.$ On one hand we know from ([7], Theorem 3.98) that ${\rm Vol}_{g}(B_{x}(r))=\frac{\omega_{n-1}}{n}r^{n}(1+o(r))$, where $\omega_{n-1}$ is the area of the euclidean unit sphere in $\mathbb{R}^{n}$, and $o(r)\rightarrow 0$ as $r\rightarrow 0$. One can see without any difficulty that $\lim_{m\rightarrow\infty}{\rm Vol}_{g}(B_{x}(R/2^{m}))^{(n/q)^{m}}=1.$ On the other hand we have $\sum_{j=1}^{\infty}({n}/{q})^{j-1}=\frac{q}{q-n},\quad\sum_{j=1}^{\infty}j(n/q)^{j-1}=\frac{q^{2}}{(q-n)^{2}}.$ Hence, passing to the limit $m\rightarrow\infty$ in (3.6), one concludes that ${\rm Vol}_{g}(B_{x}(R))\geq\left(\frac{R}{2^{(2q-n)/(q-n)}A}\right)^{nq/(q-n)}.$ This together with (3.3), (3.4) implies that ${\rm Vol}_{g}(B_{x}(R))\geq\min\left\\{\frac{1}{2\tau A},\frac{R}{2^{(2q-n)/(q-n)}A}\right\\}^{nq/(q-n)}$ and completes the proof of the lemma. $\hfill\Box$ It should be pointed out that the above argument is a modification of that of Carron ([18], lemma 3.2). Note that the condition (3.1) implies that $W^{1,n}(M)$ is continuously embedded in $L^{q}(M)$ for some $q>n$. This is different from the assumption of ([18], lemma 3.2). To prove proposition 2.1, we also need the following interpolation inequality. Lemma 3.2. Let $\tau$ be any positive real number. Suppose there exist positive constants $q_{1}$, $q_{2}$, $A_{1}$ and $A_{2}$ such that $q_{2}>q_{1}>0$ and $\left(\int_{M}\|u\|^{q_{i}}dv_{g}\right)^{1/{q_{i}}}\leq A_{i}\\|u\\|_{1,\tau}$ (3.7) for all $u\in W^{1,n}(M)$, $i=1,2$. Then for all $q:q_{1}<q<q_{2}$ there exists a positive constant $A=A(A_{1},A_{2},q_{1},q_{2})$ such that $\left(\int_{M}\|u\|^{q}dv_{g}\right)^{1/{q}}\leq A\\|u\\|_{1,\tau}$ (3.8) for all $u\in W^{1,n}(M)$. Proof. For any $u\in W^{1,n}(M)\setminus\\{0\\}$, we set $\widetilde{u}=u/\\|u\\|_{1,\tau}$. It follows from (3.7) that $\left(\int_{M}\|\widetilde{u}\|^{q_{i}}dv_{g}\right)^{1/{q_{i}}}\leq A_{i},\,\,\,i=1,2.$ Assume $q_{1}<q<q_{2}$. Since $\|\widetilde{u}\|^{q}\leq\|\widetilde{u}\|^{q_{1}}+\|\widetilde{u}\|^{q_{2}}$, there holds $\int_{M}\|\widetilde{u}\|^{q}dv_{g}\leq\int_{M}\|\widetilde{u}\|^{q_{1}}dv_{g}+\int_{M}\|\widetilde{u}\|^{q_{2}}dv_{g}\leq A_{1}^{q_{1}}+A_{2}^{q_{2}}.$ Hence $\left(\int_{M}\|{u}\|^{q}dv_{g}\right)^{1/q}\leq(A_{1}^{q_{1}}+A_{2}^{q_{2}})^{\frac{1}{q}}\\|u\\|_{1,\tau}.$ Take $A=\max\\{(A_{1}^{q_{1}}+A_{2}^{q_{2}})^{1/q_{1}},(A_{1}^{q_{1}}+A_{2}^{q_{2}})^{1/q_{2}}\\}$. Then (3.8) follows immediately. $\hfill\Box$ Proof of proposition 2.1. Assume there exist positive constants $\alpha$, $\tau$ and $\beta$ such that (1.7) holds. For any $u\in W^{1,n}(M)$ we set $\widetilde{u}=u/\\|u\\|_{1,\tau}$. It follows from (1.7) that $\int_{M}\sum_{k=n-1}^{\infty}\frac{\alpha^{k}\|\widetilde{u}\|^{\frac{nk}{n-1}}}{k!}dv_{g}\leq\beta.$ Particularly for any integer $k\geq n-1$ there holds $\int_{M}\frac{\alpha^{k}\|\widetilde{u}\|^{\frac{nk}{n-1}}}{k!}dv_{g}\leq\beta,$ and thus $\left(\int_{M}\|u\|^{\frac{nk}{n-1}}dv_{g}\right)^{\frac{n-1}{nk}}\leq\left(\frac{k!\beta}{\alpha^{k}}\right)^{\frac{n-1}{nk}}\\|u\\|_{1,\tau}.$ For any $q\geq n$, there exists some $k\geq n-1$ such that $\frac{nk}{n-1}\leq q<\frac{n(k+1)}{n-1}.$ In fact we can choose $k=[(n-1)p/n]$, the integer part of $(n-1)p/n$. By lemma 3.2, there exists a positive constant $A$ depending only on $n$, $q$, $\alpha$, and $\beta$ such that $\left(\int_{M}\|{u}\|^{q}dv_{g}\right)^{1/q}\leq A\\|u\\|_{1,\tau}.$ This implies that $W^{1,n}(M)\hookrightarrow L^{q}(M)$ continuously. Now we fix some $q>n$, say $q=n+1$. Then by lemma 3.1, there exists some constant $\epsilon>0$ depending only on $n$, $\alpha$, $\tau$, $\beta$ and $r$ such that for all $x\in M$, ${\rm Vol}_{g}(B_{x}(r))\geq\epsilon$. $\hfill\Box$ Proof of corollary 2.2. For any complete noncompact Riemannian $n$-manifold $(M,g)$, if Trudinger-Moser inequality holds, then by proposition 2.1, there exists some constant $\epsilon>0$ such that ${\rm Vol}_{g}(B_{x}(r))\geq\epsilon$ for all $x\in M$. Hence if there exists some complete noncompact Riemannian $n$-manifold $(M,g)$ such that $\inf_{x\in M}{\rm Vol}_{g}(B_{x}(r))=0,$ then we conclude that Trudinger-Moser inequality is not valid on it. Now we construct such complete Riemannian manifolds. Consider the warped product $M=\mathbb{R}\times N,\,\,\,g(t,\theta)=dt^{2}+f(t)ds_{N}^{2},$ where $(N,ds_{N}^{2})$ is a compact $(n-1)$-Riemannian manifold, $dt^{2}$ is the euclidian metric of $\mathbb{R}$, and $f$ is a smooth function satisfying $f(t)>0,\forall t\in\mathbb{R}$ and $\lim_{t\rightarrow+\infty}f(t)=0$. If $y=(t_{1},m_{1})$ and $z=(t_{2},m_{2})$ are two points of ${M}$, then $d_{g}(y,z)\geq\|t_{2}-t_{1}\|$. This together with the compactness of $N$ implies that $({M},g)$ is complete. In addition, for any $x=(t,m)\in M$, there holds $B_{x}(1)\subset(t-1,t+1)\times N.$ Therefore $\displaystyle{\rm Vol}_{g}(B_{x}(1))$ $\displaystyle\leq$ $\displaystyle{\rm Vol}_{g}\left((t-1,t+1)\times N\right){}$ (3.9) $\displaystyle\leq$ $\displaystyle{\rm Vol}_{ds_{N}^{2}}(N)\int_{t-1}^{t+1}f(t)dt{}$ $\displaystyle=$ $\displaystyle 2{\rm Vol}_{ds_{N}^{2}}(N)f(\xi)$ $\displaystyle\rightarrow$ $\displaystyle 0\,\,\,{\rm as}\,\,\,t\rightarrow+\infty,$ where we used the integral mean value theorem, $\xi$ is some point in $(t-1,t+1)$. This gives the desired result. $\hfill\Box$ ## 4 Sufficient conditions In this section, we investigate sufficient conditions under which Trudinger- Moser inequality holds. Precisely we shall prove theorem 2.3 and proposition 2.4. Firstly we have the following key observation: Lemma 4.1. Let $\mathbb{B}_{0}(\delta)\subset\mathbb{R}^{n}$ be a ball centered at $0$ with radius $\delta$. If $0\leq\alpha\leq\alpha_{n}=n\omega_{n-1}^{1/(n-1)}$, then there exists some constant $C$ depending only on $n$ such that for all $u\in W_{0}^{1,n}(\mathbb{B}_{0}(\delta))$ satisfying $\int_{\mathbb{B}_{0}(\delta)}\|\nabla u\|^{n}dx\leq 1$, there holds $\int_{\mathbb{B}_{0}(\delta)}\zeta\left(n,\alpha\|u\|^{\frac{n}{n-1}}\right)dx\leq C\delta^{n}\left(\frac{\alpha}{\alpha_{n}}\right)^{n-1}\int_{\mathbb{B}_{0}(\delta)}\|\nabla u\|^{n}dx.$ (4.1) Proof. Let $\widetilde{u}=u/\\|\nabla u\\|_{L^{n}(\mathbb{B}_{0}(\delta))}$. Since $\\|\nabla u\\|_{L^{n}(\mathbb{B}_{0}(\delta))}\leq 1$ and $0\leq\alpha\leq\alpha_{n}$, we have $\displaystyle\zeta\left(n,\alpha\|u\|^{\frac{n}{n-1}}\right)$ $\displaystyle=$ $\displaystyle\sum_{k=n-1}^{\infty}\frac{\alpha^{k}\|u\|^{\frac{nk}{n-1}}}{k!}{}$ (4.2) $\displaystyle=$ $\displaystyle\sum_{k=n-1}^{\infty}\left(\frac{\alpha}{\alpha_{n}}\right)^{k}\frac{\alpha_{n}^{k}\\|\nabla u\\|_{L^{n}(\mathbb{B}_{0}(\delta))}^{\frac{nk}{n-1}}\|\widetilde{u}\|^{\frac{nk}{n-1}}}{k!}{}$ $\displaystyle\leq$ $\displaystyle\\|\nabla u\\|_{L^{n}(\mathbb{B}_{0}(\delta))}^{n}\left(\frac{\alpha}{\alpha_{n}}\right)^{n-1}\zeta\left(n,\alpha_{n}\|\widetilde{u}\|^{\frac{n}{n-1}}\right).$ It follows from the classical Trudinger-Moser inequality ((1.2) with $\alpha_{0}$ replaced by $\alpha_{n}$) that $\int_{\mathbb{B}_{0}(\delta)}\zeta\left(n,\alpha_{n}\|\widetilde{u}\|^{\frac{n}{n-1}}\right)dx\leq C\delta^{n}$ (4.3) for some constant $C$ depending only on $n$. Integrating (4.2) on $\mathbb{B}_{0}(\delta)$, we immediately obtain (4.1) by using (4.3). This concludes the lemma. $\hfill\Box$ Let $(M,g)$ be a complete Riemannian $n$-manifold with ${\rm Ric}_{(M,g)}\geq Kg$ for some $K\in\mathbb{R}$ and ${\rm inj}_{(M,g)}\geq i_{0}$ for some $i_{0}>0$. Then we have the following local version of Trudinger-moser inequality which is the key estimate for the proof of theorem 2.3: Lemma 4.2. For any $\alpha:0<\alpha<\alpha_{n}$ there exists some constant $\delta$ depending only on $n$, $\alpha$, $K$ and $i_{0}$ such that for all $x\in M$ and all $u\in C_{0}^{\infty}(B_{x}(\delta))$ with $\\|\nabla_{g}u\\|_{L^{n}(B_{x}(\delta))}\leq 1$, there holds $\int_{M}\zeta\left(n,\alpha\|u\|^{\frac{n}{n-1}}\right)dv_{g}\leq C\int_{M}\|\nabla_{g}u\|^{n}dv_{g}$ for some constant $C$ depending only on $n$, $\alpha$, $K$ and $i_{0}$. Proof. By (Hebey [18], theorem 1.3), we know that for any $\epsilon>0$ there exists a positive constant $\delta$ depending only on $\epsilon$, $n$, $K$ and $i_{0}$ satisfying the following property: for any $x\in M$ there exists a harmonic coordinate chart $\phi:B_{x}(\delta)\rightarrow\mathbb{R}^{n}$ such that $\phi(x)=0$, and the components $(g_{jl})$ of $g$ in this chart satisfy $e^{-\epsilon}\delta_{jl}\leq g_{jl}\leq e^{\epsilon}\delta_{jl}$ as bilinear forms. One then has that $\phi(B_{x}(\delta))\subset\mathbb{B}_{0}(e^{\epsilon/2}\delta)$. Let $u$ be a function in $C_{0}^{\infty}(B_{x}(\delta))$ and $\\|\nabla_{g}u\\|_{L^{n}(B_{x}(\delta))}\leq 1$. It is not difficult to see that $\displaystyle\int_{B_{x}(\delta)}\|\nabla_{g}u\|^{n}dv_{g}$ $\displaystyle\geq$ $\displaystyle e^{-n\epsilon}\int_{\mathbb{B}_{0}(e^{\epsilon/2}\delta)}\|\nabla(u\circ\phi^{-1})(x)\|^{n}dx,$ (4.4) $\displaystyle\int_{M}\zeta\left(n,\alpha\|u\|^{\frac{n}{n-1}}\right)dv_{g}$ $\displaystyle\leq$ $\displaystyle e^{n\epsilon/2}\int_{\mathbb{B}_{0}(e^{\epsilon/2}\delta)}\zeta\left(n,\alpha\|(u\circ\phi^{-1})(x)\|^{\frac{n}{n-1}}\right)dx.$ (4.5) For any fixed $\alpha:0<\alpha<\alpha_{n}$, there exists some $\epsilon_{0}$ depending only on $n$ and $\alpha$ such that when $0<\epsilon\leq\epsilon_{0}$, it follows from (4.4) and $\\|\nabla_{g}u\\|_{L^{n}(B_{x}(\delta))}\leq 1$ that $\alpha\left(\int_{\mathbb{B}_{0}(e^{\epsilon/2}\delta)}\|\nabla(u\circ\phi^{-1})(x)\|^{n}dx\right)^{1/(n-1)}\leq\alpha e^{n\epsilon_{0}/(n-1)}<\alpha_{n}.$ Now let $\epsilon=\epsilon_{0}$ be fixed and $\delta$ depending only on $\epsilon_{0}$, $n$, $K$ and $i_{0}$ be chosen as above. By lemma 4.1, there exists a constant $C_{1}=C_{1}(n)$ depending only on $n$ such that $\int_{\mathbb{B}_{0}(e^{\epsilon_{0}/2}\delta)}\zeta\left(n,\alpha\|(u\circ\phi^{-1})(x)\|^{\frac{n}{n-1}}\right)dx\leq C_{1}e^{n\epsilon_{0}/2}\delta^{n}\int_{\mathbb{B}_{0}(e^{\epsilon_{0}/2}\delta)}\|\nabla(u\circ\phi^{-1})(x)\|^{n}dx.$ This together with (4.4) and (4.5) implies that $\int_{M}\zeta\left(n,\alpha\|u\|^{\frac{n}{n-1}}\right)dv_{g}\leq C_{1}e^{2n\epsilon_{0}}\delta^{n}\int_{M}\|\nabla u\|^{n}dv_{g}.$ Take $C=C_{1}e^{2n\epsilon_{0}}\delta^{n}$. We conclude that $C$ depends on $n$, $\alpha$, $K$ and $i_{0}$. $\hfill\Box$ Proof of theorem 2.3. $(i)$ For any $\alpha:0<\alpha<\alpha_{n}$, let $\delta=\delta(n,\alpha,K,i_{0})$ be chosen as in lemma 4.2. Independently, by Gromov’s covering lemma (Hebey [18], lemma 1.6), we can select a sequence $(x_{j})$ of points of $M$ such that $(a)$ $M=\cup_{j}B_{x_{j}}(\delta/2)$, and for any $j\not=l$ there holds $B_{x_{j}}(\delta/4)\cap B_{x_{l}}(\delta/4)=\varnothing$; $(b)$ there exists $N$ depending only on $n$, $K$ and $\delta$ such that each point of $M$ has a neighborhood which intersects at most $N$ of the $B_{x_{j}}(\delta)$’s. For any $j$, we take a cut-off function $\phi_{j}\in C_{0}^{\infty}(B_{x_{j}}(\delta))$ satisfying $0\leq\phi_{j}\leq 1$, $\phi_{j}\equiv 1$ on $B_{x_{j}}(\delta/2)$, and $\|\nabla_{g}\phi_{j}\|\leq 4/\delta$. It follows that for all $j$ $\|\nabla_{g}\phi_{j}^{2}\|=2\phi_{j}\|\nabla_{g}\phi_{j}\|\leq\frac{8}{\delta}\phi_{j}.$ (4.6) By the covering properties $(a)$ and $(b)$, we have $1\leq\sum_{j}\phi_{j}(x)\leq N\,\,\,{\rm for\,\,\,all}\,\,\,x\in M.$ (4.7) Set $\tau=8/\delta$. Assume $u\in C_{0}^{\infty}(M)$ satisfies $\\|u\\|_{1,\tau}=\left(\int_{M}\|\nabla u\|^{n}dv_{g}\right)^{1/n}+\tau\left(\int_{M}\|u\|^{n}dv_{g}\right)^{1/n}\leq 1.$ It follows from (4.6) and the Minkowvsky inequality that $\displaystyle\left(\int_{M}\|\nabla_{g}(\phi_{j}^{2}u)\|^{n}dv_{g}\right)^{1/n}\leq\left(\int_{M}\phi_{j}^{2n}\|\nabla_{g}u\|^{n}dv_{g}\right)^{1/n}+\left(\int_{M}\|\nabla_{g}\phi_{j}^{2}\|^{n}\|u\|^{n}dv_{g}\right)^{1/n}\leq\\|u\\|_{1,\tau}\leq 1.$ In view of lemma 4.2, this leads to $\displaystyle\int_{M}\zeta\left(n,\alpha\|u\|^{\frac{n}{n-1}}\right)dv_{g}$ $\displaystyle\leq$ $\displaystyle\sum_{j}\int_{B_{\delta/2}(x_{j})}\zeta\left(n,\alpha\|u\|^{\frac{n}{n-1}}\right)dv_{g}{}$ (4.8) $\displaystyle\leq$ $\displaystyle\sum_{j}\int_{B_{\delta}(x_{j})}\zeta\left(n,\alpha\|\phi_{j}^{2}u\|^{\frac{n}{n-1}}\right)dv_{g}$ $\displaystyle\leq$ $\displaystyle C\sum_{j}\int_{M}\|\nabla(\phi_{j}^{2}u)\|^{n}dv_{g}$ for some constant $C$ depending only on $n$, $\alpha$, $K$ and $i_{0}$. In addition we have by using (4.6) and $0\leq\phi_{j}\leq 1$ that $\displaystyle\int_{M}\|\nabla_{g}(\phi_{j}^{2}u)\|^{n}dv_{g}$ $\displaystyle\leq$ $\displaystyle 2^{n}\int_{M}\left(\phi_{j}^{2n}\|\nabla_{g}u\|^{n}+\|\nabla_{g}\phi_{j}^{2}\|^{n}\|u\|^{n}\right)dv_{g}{}$ $\displaystyle\leq$ $\displaystyle 2^{n}\int_{M}\phi_{j}\|\nabla_{g}u\|^{n}dv_{g}+\frac{16^{n}}{\delta^{n}}\int_{M}\phi_{j}\|u\|^{n}dv_{g}.$ In view of (4.7), it follows that $\displaystyle\sum_{j}\int_{M}\|\nabla_{g}(\phi_{j}^{2}u)\|^{n}dv_{g}$ $\displaystyle\leq$ $\displaystyle 2^{n}N\int_{M}\|\nabla_{g}u\|^{n}dv_{g}+\frac{16^{n}}{\delta^{n}}N\int_{M}\|u\|^{n}dv_{g}$ $\displaystyle\leq$ $\displaystyle 2^{n}N+\frac{16^{n}}{\tau\delta^{n}}N.$ This together with (4.8) implies $\int_{M}\zeta\left(n,\alpha\|u\|^{\frac{n}{n-1}}\right)dv_{g}\leq{C}$ for some constant ${C}$ depending only on $n$, $\alpha$, $K$ and $i_{0}$. By the density of $C_{0}^{\infty}(M)$ in $W^{1,n}(M)$, the inequality (1.7) holds for the above $\alpha$, $\tau$ and $C$. By proposition 2.1, we have that $W^{1,n}(M)$ is continuously embedded in $L^{q}(M)$ for any $q\geq n$. $(ii)$ Fix some point $z\in M$, let $r=r(x)=d_{g}(z,x)$ be the geodesic distance between $x$ and $z$. Without loss of generality, we may assume the injectivity radius of $(M,g)$ at $z$ is strictly larger than 1. Take a function sequence $\phi_{\epsilon}(x)=\left\\{\begin{array}[]{ll}1,&{\rm when}\quad r<\epsilon\\\\[6.45831pt] \left(\log\frac{1}{\epsilon}\right)^{-1}\log\frac{1}{r},&{\rm when}\quad\epsilon\leq r\leq 1\\\\[6.45831pt] 0,&{\rm when}\quad r>1.\end{array}\right.$ Then $\phi_{\epsilon}\in W^{1,n}(M)$ and for any constant $\tau>0$ there holds $\displaystyle\\|\phi_{\epsilon}\\|_{1,\tau}=\left(\log\frac{1}{\epsilon}\right)^{(1-n)/n}\omega_{n-1}^{1/n}\left(1+O\left(\frac{1}{\log{\epsilon}}\right)\right).$ Set $\widetilde{\phi}_{\epsilon}=\phi_{\epsilon}/\\|\phi_{\epsilon}\\|_{1,\tau}$. Then we have on the geodesic ball $B_{z}(\epsilon)\subset M$, $\zeta(n,\alpha\widetilde{\phi}_{\epsilon}^{\frac{n}{n-1}})=e^{\alpha\widetilde{\phi}_{\epsilon}^{\frac{n}{n-1}}}-\sum_{k=0}^{n-2}\frac{\alpha^{k}\widetilde{\phi}_{\epsilon}^{\frac{nk}{n-1}}}{k!}\geq\epsilon^{\alpha\omega_{n-1}^{-\frac{1}{n-1}}(1+O(1/\log\epsilon))}+O\left(\left(\log\frac{1}{\epsilon}\right)^{n-2}\right).$ Note that $\alpha\omega_{n-1}^{-\frac{1}{n-1}}>n$ for any $\alpha>\alpha_{n}$. Hence, when $\alpha>\alpha_{n}$, we have $\displaystyle\int_{M}\zeta(n,\alpha\|\widetilde{\phi}_{\epsilon}\|^{\frac{n}{n-1}})dv_{g}$ $\displaystyle\geq$ $\displaystyle\int_{B_{z}(\epsilon)}\zeta(n,\alpha\|\widetilde{\phi}_{\epsilon}\|^{\frac{n}{n-1}})dv_{g}$ $\displaystyle\geq$ $\displaystyle\frac{\omega_{n-1}}{n}(1+o_{\epsilon}(1))\epsilon^{n-\alpha\omega_{n-1}^{-1/(n-1)}(1+O(1/\log\epsilon))}+o_{\epsilon}(1).$ $\displaystyle\rightarrow$ $\displaystyle+\infty\quad{\rm as}\quad\epsilon\rightarrow 0.$ This ends the proof of $(ii)$. $(iii)$ Take $\alpha_{0}:0<\alpha_{0}<\alpha_{n}$. By $(i)$ there exists some $\tau_{0}=\tau_{0}(n,\alpha_{0},K,i_{0})>0$ such that $\Lambda_{\alpha_{0}}:=\sup_{\\|u\\|_{1,\tau_{0}}\leq 1}\int_{M}\zeta(n,\alpha_{0}\|u\|^{\frac{n}{n-1}})dv_{g}<\infty.$ Given any $\alpha>0$ and any $u\in W^{1,n}(M)$. Since $C_{0}^{\infty}(M)$ is dense in $W^{1,n}(M)$ under the norm $\\|\cdot\\|_{W^{1,n}(M)}$, which is equivalent to the norm $\\|\cdot\\|_{1,\tau_{0}}$, we can choose some $u_{0}\in C_{0}^{\infty}(M)$ such that $2^{\frac{n}{n-1}}\alpha\\|u-u_{0}\\|_{1,\tau_{0}}^{\frac{n}{n-1}}<\alpha_{0}.$ (4.9) Since $\zeta(n,t)$ is increasing in $t$ for $t\geq 0$, we obtain by using (2.3) $\displaystyle\int_{M}\zeta(n,\alpha\|u\|^{\frac{n}{n-1}})dv_{g}$ $\displaystyle\leq$ $\displaystyle\int_{M}\zeta(n,2^{\frac{n}{n-1}}\alpha\|u-u_{0}\|^{\frac{n}{n-1}}+2^{\frac{n}{n-1}}\alpha\|u_{0}\|^{\frac{n}{n-1}})dv_{g}{}$ (4.10) $\displaystyle\leq$ $\displaystyle\frac{1}{\mu}\int_{M}\zeta(n,2^{\frac{n}{n-1}}\alpha\mu\|u-u_{0}\|^{\frac{n}{n-1}})dv_{g}$ $\displaystyle\quad+\frac{1}{\nu}\int_{M}\zeta(n,2^{\frac{n}{n-1}}\alpha\nu\|u_{0}\|^{\frac{n}{n-1}})dv_{g},$ where $1/\mu+1/\nu=1$. In view of (4.9), we can take $\mu>1$ sufficiently close to $1$ such that $2^{\frac{n}{n-1}}\alpha\mu\\|u-u_{0}\\|_{1,\tau_{0}}^{\frac{n}{n-1}}<\alpha_{0}.$ Hence $\int_{M}\zeta(n,2^{\frac{n}{n-1}}\alpha\mu\|u-u_{0}\|^{\frac{n}{n-1}})dv_{g}\leq\Lambda_{\alpha_{0}}.$ (4.11) Since $u_{0}\in C_{0}^{\infty}(M)$, particularly $u_{0}$ has compact support, there holds $\int_{M}\zeta(n,2^{\frac{n}{n-1}}\alpha\nu\|u_{0}\|^{\frac{n}{n-1}})dv_{g}<\infty.$ (4.12) Combining (4.10), (4.11) and (4.12), we obtain $\int_{M}\zeta(n,\alpha\|u\|^{\frac{n}{n-1}})dv_{g}<\infty.$ This completes the proof of $(iii)$. $\hfill\Box$ Now we shall prove proposition 2.4. Let us recall some notations from Riemannian geometry. In any chart, the Christoffel symbols of the Levi-Civita connection are given by $\Gamma_{ij}^{k}=\frac{1}{2}g^{mk}\left(\partial_{i}g_{mj}+\partial_{j}g_{mi}-\partial_{m}g_{ij}\right),$ (4.13) where $g_{ij}$’s are the components of $g$, $(g^{ij})$ denotes the inverse matrix of $(g_{ij})$. Here and in the sequel the Einstein’s summation convention is adopted. Denote the Riemannian curvature of $(M,g)$, a $(4,0)$-type tensor field, by ${\rm Rm}_{(M,g)}$. The components of ${\rm Rm}_{(M,g)}$ are given by the relation $R_{ijkl}=g_{i\alpha}\left(\partial_{k}\Gamma_{jl}^{\alpha}-\partial_{l}\Gamma_{jk}^{\alpha}+\Gamma_{k\beta}^{\alpha}\Gamma_{jl}^{\beta}-\Gamma_{l\beta}^{\alpha}\Gamma_{jk}^{\beta}\right).$ (4.14) Similarly, the components of the Ricci curvature ${\rm Rc}_{(M,g)}$ of $(M,g)$ are given by the relation $R_{ij}=g^{\alpha\beta}R_{i\alpha j\beta}.$ (4.15) Proof of proposition 2.4. In view of proposition 2.1, it suffices to construct a complete noncompact Riemannian $n$-manifold $(M,g)$ such that its Ricci curvature has lower bound and there holds $\inf_{x\in M}{\rm Vol}_{g}(B_{x}(1))=0.$ Again we consider the warped product ${M}=\mathbb{R}\times N,\,\,\,g(x,\theta)=dx^{2}+f(x)ds_{N}^{2},$ where $(N,ds_{N}^{2})$ is a compact $(n-1)$-Riemannian manifold, $dx^{2}$ is the euclidean metric of $\mathbb{R}$, and $f$ is a smooth function satisfying $f(x)>0,\forall x\in\mathbb{R}$. In the following we calculate the Ricci curvature of $({M},g)$. In some product chart $(\mathbb{R}\times U,Id\times\phi)$ ($\\{x,y^{2},\cdots,y^{n}\\}$), $g_{11}=1$, $g_{1\alpha}=0$, $g_{\alpha\beta}=fh_{\alpha\beta}$, $g^{11}=1$, $g^{1,\alpha}=0$, and $g^{\alpha\beta}=f^{-1}h^{\alpha\beta}$. Equivalently $g=dx^{2}+f(x)h_{\alpha\beta}dy^{\alpha}dy^{\beta},$ where $(h_{\alpha\beta})$ denote components of the metric $ds_{N}^{2}$. Here and in the sequel, all indices $\alpha$, $\beta$, $\mu$, $\nu$ and $\lambda$ run from $2$ to $n$. In view of (4.13), the Christoffel symbols of the Levi- Civita connection was calculated as follows: $\displaystyle\Gamma_{11}^{1}=\Gamma_{11}^{\alpha}=\Gamma_{1\alpha}^{1}=0,\,\,\,\Gamma_{1\alpha}^{\beta}=\frac{1}{2}g^{\mu\beta}\partial_{1}g_{\mu\alpha}=\frac{f^{\prime}}{2f}\delta_{\alpha}^{\beta}$ $\displaystyle\Gamma_{\alpha\beta}^{1}=-\frac{1}{2}\partial_{1}g_{\alpha\beta}=-\frac{f^{\prime}}{2}h_{\alpha\beta},\quad\Gamma_{\alpha\beta}^{\gamma}=\widetilde{\Gamma}_{\alpha\beta}^{\gamma},$ where $\delta_{\alpha}^{\beta}$ is equal to $1$ when $\alpha=\beta$, and $0$ when $\alpha\not=\beta$, $\widetilde{\Gamma}_{\alpha\beta}^{\gamma}$’s are components of the Christoffel symbols of Levi-Civita connection on $(N,ds_{N}^{2})$. In view of (4.14), the components of the Riemannian curvature reads $\displaystyle R_{1\alpha 1\beta}$ $\displaystyle=$ $\displaystyle g_{11}\partial_{1}\Gamma_{\alpha\beta}^{1}$ $\displaystyle=$ $\displaystyle\frac{{f^{\prime}}^{2}-2ff^{\prime\prime}}{4f}h_{\alpha\beta}$ $\displaystyle R_{1\alpha\beta\gamma}$ $\displaystyle=$ $\displaystyle g_{11}\left(\partial_{\beta}\Gamma_{\alpha\gamma}^{1}-\partial_{\gamma}\Gamma_{\alpha\beta}^{1}+\Gamma_{\beta k}^{1}\Gamma_{\alpha\gamma}^{k}-\Gamma_{\gamma k}^{1}\Gamma_{\alpha\beta}^{k}\right)$ $\displaystyle=$ $\displaystyle\frac{f^{\prime}}{2}\left(-\partial_{\beta}h_{\alpha\gamma}+\partial_{\gamma}h_{\alpha\beta}-h_{\beta\mu}\widetilde{\Gamma}_{\alpha\gamma}^{\mu}+h_{\gamma\mu}\widetilde{\Gamma}_{\alpha\beta}^{\mu}\right)$ $\displaystyle R_{\alpha\beta\gamma\mu}$ $\displaystyle=$ $\displaystyle g_{\alpha\lambda}\left(\partial_{\gamma}\Gamma_{\beta\mu}^{\lambda}-\partial_{\mu}\Gamma_{\beta\gamma}^{\lambda}+\Gamma_{\gamma k}^{\lambda}\Gamma_{\beta\mu}^{k}-\Gamma_{\mu k}^{\lambda}\Gamma_{\beta\gamma}^{k}\right)$ $\displaystyle=$ $\displaystyle f\widetilde{R}_{\alpha\beta\gamma\mu}+g_{\alpha\lambda}\left(\Gamma_{\gamma 1}^{\lambda}\Gamma_{\beta\mu}^{1}-\Gamma_{\mu 1}^{\lambda}\Gamma_{\beta\gamma}^{1}\right)$ $\displaystyle=$ $\displaystyle f\widetilde{R}_{\alpha\beta\gamma\mu}+\frac{{f^{\prime}}^{2}}{4}\left(h_{\alpha\mu}h_{\beta\gamma}-h_{\alpha\gamma}h_{\beta\mu}\right),$ where $\widetilde{R}_{\alpha\beta\gamma\mu}$’s denote the components of Riemannian curvature of $(N,ds_{N}^{2})$. In view of (4.15), we get the components of the Ricci curvature as follows. $\displaystyle R_{11}$ $\displaystyle=$ $\displaystyle g^{\alpha\beta}R_{1\alpha 1\beta}$ $\displaystyle=$ $\displaystyle(n-1)\frac{{f^{\prime}}^{2}-2ff^{\prime\prime}}{4f^{2}}$ $\displaystyle R_{1\alpha}$ $\displaystyle=$ $\displaystyle g^{\beta\gamma}R_{1\beta\alpha\gamma}$ $\displaystyle=$ $\displaystyle\frac{f^{\prime}}{2f}h^{\beta\gamma}\left(-\partial_{\alpha}h_{\beta\gamma}+\partial_{\gamma}h_{\alpha\beta}-h_{\alpha\mu}\widetilde{\Gamma}_{\beta\gamma}^{\mu}+h_{\gamma\mu}\widetilde{\Gamma}_{\alpha\beta}^{\mu}\right)$ $\displaystyle R_{\alpha\beta}$ $\displaystyle=$ $\displaystyle g^{11}R_{\alpha 1\beta 1}+g^{\mu\nu}R_{\alpha\mu\beta\nu}$ $\displaystyle=$ $\displaystyle\frac{{f^{\prime}}^{2}-2ff^{\prime\prime}}{4f}h_{\alpha\beta}+\widetilde{R}_{\alpha\beta}+\frac{{f^{\prime}}^{2}}{4f}h^{\mu\nu}\left(h_{\alpha\nu}h_{\mu\beta}-h_{\alpha\beta}h_{\mu\nu}\right)$ $\displaystyle=$ $\displaystyle\frac{(2-n){f^{\prime}}^{2}-2ff^{\prime\prime}}{4f}h_{\alpha\beta}+\widetilde{R}_{\alpha\beta},$ where $\widetilde{R}_{\alpha\beta}$’s are components of the Ricci curvature of $(N,ds_{N}^{2})$. If we assume the functions $f$, $f^{\prime}/f$ and $f^{\prime\prime}/f$ are all bounded, then in the chart $(\mathbb{R}\times U,Id\times\phi)$, the eigenvalues of the matrix $(R_{jl})$ and the matrix $(g_{jl})$ are uniformly bounded. Thus there exists some constant $K_{1}\in\mathbb{R}$ such that $(R_{jl})\geq K_{1}(g_{jl})$. Note that $(N,ds_{N}^{2})$ is compact. There exists some constant $K\in\mathbb{R}$ such that $Ric_{(M,g)}\geq Kg$ as bilinear forms. If we further assume $\lim_{x\rightarrow+\infty}f(x)=0$, then by (LABEL:volume), we have ${\rm Vol}_{g}(B_{y}(1))\rightarrow 0$ as $x\rightarrow+\infty$, where $y=(x,m)\in\mathbb{R}\times N$. One can check that the following functions satisfy all the above assumptions on $f$. $\bullet$ $f$ is a smooth positive function defined on $\mathbb{R}$ and satisfies $f(x)=\left\\{\begin{array}[]{lll}(1+x^{2})e^{-x+\sin x}&{\rm when}&x>1\\\\[6.45831pt] 1,&{\rm when}&x<0\end{array}\right.$ $\bullet$ $f$ is a smooth positive function defined on $\mathbb{R}$ and satisfies $f(x)=\left\\{\begin{array}[]{lll}\frac{1}{\log x}&{\rm when}&x>2\\\\[6.45831pt] 1,&{\rm when}&x<0.\end{array}\right.$ This gives the desired result.$\hfill\Box$ ## 5 Proof of proposition 2.5 In this section, we shall construct complete noncompact Riemannain $n$-manifolds to show that the condition Ricci curvature has lower bound in theorem 2.3 is not necessarily needed. Proof of proposition 2.5. It suffices to construct a complete noncompact Riemannian $n$-manifold on which Trudinger-Moser embedding holds, but its Ricci curvature has no lower bound. For this purpose, we consider the Riemannian manifold $(\mathbb{R}^{n},g)$, where $\mathbb{R}^{n}$ is the euclidian space and $g=dx_{1}^{2}+f(x_{1})dx_{2}^{2}+\cdots+f(x_{1})dx_{n}^{2},$ and $f$ is a smooth function on $\mathbb{R}$ such that $a\leq f\leq b$ for two positive constants $a$ and $b$. Clearly $(\mathbb{R}^{n},g)$ is complete and noncompact. In view of Trudinger-Moser inequality on the standard euclidian space $\mathbb{R}^{n}$ [8, 12, 32], one can easily see that if $\alpha$ is chosen sufficiently small, then the supremum $\sup_{u\in W^{1,n}(M),\,\\|u\\|_{W^{1,n}}\leq 1}\int_{\mathbb{R}^{n}}\zeta\left(n,\alpha\|u\|^{\frac{n}{n-1}}\right)dv_{g}$ is finite, i.e. Trudinger-Moser inequality holds on the manifold $(\mathbb{R}^{n},g)$, where $\\|u\\|_{W^{1,n}}=\left(\int_{\mathbb{R}^{n}}(\|\nabla_{g}u\|^{n}+\|u\|^{n})dv_{g}\right).$ In the following, we shall further choose $f$ such that the Ricci curvature of $(\mathbb{R}^{n},g)$ is unbounded from below. By (4.15), $R_{11}=(n-1)\frac{{f^{\prime}}^{2}-2ff^{\prime\prime}}{4f^{2}}.$ (5.1) It suffices to find a sequence of points $(x^{(m)})$ of $\mathbb{R}^{n}$ such that $R_{11}(x^{(m)})\rightarrow-\infty$. One choice of $f$ is that $f(t)=2+\sin t^{2}$. In this case, we have $\displaystyle f^{\prime}(x_{1})=2+2x_{1}\cos x_{1}^{2},\quad f^{\prime\prime}(x_{1})=2\cos x_{1}^{2}-4x_{1}^{2}\sin x_{1}^{2}.$ Thus (5.1) implies $R_{11}(x)=(n-1)\frac{(2+2x_{1}\cos x_{1}^{2})^{2}-2(2+\sin x_{1}^{2})(2\cos x_{1}^{2}-4x_{1}^{2}\sin x_{1}^{2})}{4(2+\sin x_{1}^{2})^{2}}.$ Choosing $x^{(m)}=\left(\sqrt{2m\pi+3\pi/2},0,\cdots,0\right)$, we obtain $R_{11}({x^{(m)}})=-4m\pi-3\pi+n-1\rightarrow-\infty\quad{\rm as}\quad m\rightarrow\infty.$ Another choice of $f$ is that $f(t)=e^{\sin t^{2}}$. In this case, we have $\displaystyle f^{\prime}(x_{1})$ $\displaystyle=$ $\displaystyle 2x_{1}e^{\sin x_{1}^{2}}\cos x_{1}^{2},\quad f^{\prime\prime}(x_{1})=e^{\sin x_{1}^{2}}\left(-4x_{1}^{2}\sin x_{1}^{2}+4x_{1}^{2}\cos^{2}x_{1}^{2}+2\cos x_{1}^{2}\right).$ In view of (5.1), we obtain $R_{11}(x)=(n-1)(2x_{1}^{2}\sin x_{1}^{2}+x_{1}^{2}\cos^{2}x_{1}^{2}-2x_{1}^{2}\cos^{2}x_{1}^{2}-\cos x_{1}^{2}).$ Again, we select $x^{(m)}=\left(\sqrt{2m\pi+3\pi/2},0,\cdots,0\right)$ and conclude $R_{11}(x^{(m)})\rightarrow-\infty$ as $m\rightarrow\infty$. $\hfill\Box$ ## 6 Adams inequalities In this section, we concern Adams inequalities on complete noncompact Riemannian manifolds. Precisely we shall prove theorem 2.6. The method we adopted here is similar to that of theorem 2.3. Proof of theorem 2.6. $(i)$ Suppose that ${\rm inj}_{(M,g)}\geq i_{0}>0$ and there exist constants $C(k)$ such that $\|\nabla^{k}{\rm Rc}_{(M,g)}\|\leq C(k)$, $k=0,1,\cdots,m-1$. It follows from (Hebey [18], theorem 1.3) that for any $Q>1$ and $\alpha\in(0,1)$, the harmonic radius $r_{H}=r_{H}(Q,m,\alpha)$ is positive. Namely, for any $Q>1$, $\alpha\in(0,1)$, and $x\in M$, there exists a harmonic coordinate chart $\psi:B_{x}(r_{H})\rightarrow\mathbb{R}^{n}$ such that $\left\\{\begin{array}[]{lll}Q^{-1}\delta_{lq}\leq g_{lq}\leq Q\delta_{lq}\quad{\rm as\,\,a\,\,bilinear\,\,form};\\\\[6.45831pt] \sum_{1\leq\|\beta\|\leq m}\\|\partial^{\beta}g_{lq}\\|_{C^{0}(B_{x}(r_{H}))}+\sum_{\|\beta\|=m}\\|\partial^{\beta}g_{lq}\\|_{C^{\alpha}(B_{x}(r_{H}))}\leq Q-1.\end{array}\right.$ (6.1) Now we fix $Q>1$ and $\alpha\in(0,1)$. Without loss of generality, we may assume $\psi(x)=0$. Particularly we have that for any $r:0<r\leq r_{H}$ $\mathbb{B}_{0}(r/\sqrt{Q})\subset\psi(B_{x}(r))\subset\mathbb{B}_{0}({\sqrt{Q}}r).$ Let $\eta\in C_{0}^{\infty}(\mathbb{R}^{n})$ be such that $0\leq\eta\leq 1$, and $\eta=\left\\{\begin{array}[]{lll}1&\,\,\,{\rm on}&\mathbb{B}_{0}(r_{H}/(4\sqrt{Q})),\\\\[6.45831pt] 0&\,\,\,{\rm on}&\mathbb{R}^{n}\setminus\mathbb{B}_{0}(r_{H}/(2\sqrt{Q})).\end{array}\right.$ Then $\eta\circ\psi\in C_{0}^{\infty}(M)$ satisfies $0\leq\eta\circ\psi\leq 1$, $\eta\circ\psi\equiv 1$ on $B_{x}(r_{H}/(4Q))$, and $\eta\circ\psi\equiv 0$ on $M\setminus B_{x}(r_{H}/2)$. By Gromov’s covering lemma (Hebey [18], lemma 1.6), there exists a sequence of points $(x_{k})$ of $M$ such that $M=\cup_{k}B_{x_{k}}(r_{H}/(4Q))$ (6.2) and there exists some integer $N$ such that for any $x\in M$, $x$ belongs to at most $N$ balls in the covering. Let $\psi_{k}:B_{x_{k}}(r_{H})\rightarrow\mathbb{R}^{n}$ be as the above $\psi$ and set $\eta_{k}=\eta\circ\psi_{k}$. By (6.1), the components of the metric tensor are $C^{m}$-controlled in the charts $(B_{x_{k}}(r_{H}),\psi_{k})$. It then follows that there exists some constant $C_{1}>0$ depending only on $r_{H}$ and $Q$ such that $\|\nabla_{g}^{l}\eta_{k}\|\leq C_{1}$ for all all $l:0\leq l\leq m$ and all $k\in\mathbb{N}$, where $\nabla_{g}^{l}$ is defined by (1.3). Assume $u\in C^{\infty}(M)$ satisfies $\\|u\\|_{W^{m,n/m}(M)}\leq 1$. Then we get $\eta_{k}^{m+1}u\in C_{0}^{\infty}(B_{x_{k}}(r_{H}/2))$ and $\\|\nabla_{g}^{m}(\eta_{k}^{m+1}u)\\|_{L^{\frac{n}{m}}(B_{x_{k}}(r_{H}/2))}\leq C_{2}$ (6.3) for some constant $C_{2}$ depending only on $n$, $m$, and $C_{1}$. By the standard elliptic estimates (Gilbarg-Trudinger [16], Chapter 9), one can see that $\\|\nabla_{\mathbb{R}^{n}}^{m}((\eta_{k}^{m+1}u)\circ\psi_{k}^{-1})\\|_{L^{\frac{n}{m}}(\mathbb{B}_{0}(\sqrt{Q}r_{H}))}\leq C_{3}$ (6.4) for some constant $C_{3}$ depending only on $n$, $m$, $Q$, $r_{H}$ and $C_{1}$. Let $j$ be the smallest integer great than or equal to $n/m$. Similarly as we derived (4.8), we calculate by using (6.2), (6.3) and the relation ${(j-1)n}/{(n-m)}\geq{n}/{m}$ $\displaystyle\int_{M}\zeta\left(j,\alpha\|u\|^{\frac{n}{n-m}}\right)dv_{g}$ $\displaystyle\leq$ $\displaystyle\sum_{k}\int_{B_{x_{k}}(r_{H}/(4Q))}\zeta\left(j,\alpha\|u\|^{\frac{n}{n-m}}\right)dv_{g}{}$ (6.5) $\displaystyle\leq$ $\displaystyle\sum_{k}\int_{B_{x_{k}}(r_{H}/2)}\zeta\left(j,\alpha\|\eta_{k}^{m+1}u\|^{\frac{n}{n-m}}\right)dv_{g}{}$ $\displaystyle\leq$ $\displaystyle\sum_{k}\left(\frac{\\|\nabla_{g}^{m}(\eta_{k}^{m+1}u)\\|_{L^{\frac{n}{m}}(B_{x_{k}}(r_{H}/2))}}{C_{2}}\right)^{\frac{(j-1)n}{n-m}}\int_{B_{x_{k}}(r_{H}/2)}\zeta\left(j,\alpha C_{2}^{\frac{n}{n-m}}\|\eta_{k}^{m+1}u\|^{\frac{n}{n-m}}\right)dv_{g}{}$ $\displaystyle\leq$ $\displaystyle\sum_{k}\frac{\\|\nabla_{g}^{m}(\eta_{k}^{m+1}u)\\|_{L^{\frac{n}{m}}(B_{x_{k}}(r_{H}/2))}^{\frac{n}{m}}}{C_{2}^{\frac{n}{m}}}\int_{B_{x_{k}}(r_{H}/2)}\zeta\left(j,\alpha C_{2}^{\frac{n}{n-m}}\|\eta_{k}^{m+1}u\|^{\frac{n}{n-m}}\right)dv_{g}.$ Noting that $Q^{-1}\delta_{lq}\leq g_{lq}\leq Q\delta_{lq}$ as a bilinear form, we have $\int_{B_{x_{k}}(r_{H}/2)}\zeta\left(j,\alpha C_{2}^{\frac{n}{n-m}}\|\eta_{k}^{m+1}u\|^{\frac{n}{n-m}}\right)dv_{g}\leq Q^{\frac{n}{2}}\int_{\mathbb{B}_{0}(\sqrt{Q}r_{H})}\zeta\left(j,\alpha C_{2}^{\frac{n}{n-m}}\|(\eta_{k}^{m+1}u)\circ\psi_{k}^{-1}\|^{\frac{n}{n-m}}\right)dx.$ (6.6) In view of (6.4), we take $\alpha_{0}=\beta_{0}/(C_{2}C_{3})^{\frac{n}{n-m}}.$ (6.7) Then for any $\alpha:0<\alpha\leq\alpha_{0}$, it follows from Adams inequality (1.4) that $\int_{\mathbb{B}_{0}(\sqrt{Q}r_{H})}\zeta\left(j,\alpha C_{2}^{\frac{n}{n-m}}\|(\eta_{k}^{m+1}u)\circ\psi_{k}^{-1}\|^{\frac{n}{n-m}}\right)dx\leq C_{m,n}\|\mathbb{B}_{0}(\sqrt{Q}r_{H})\|.$ (6.8) Clearly there exists some constant $C_{4}>0$ depending only on $n$, $m$, $Q$ and $r_{H}$ such that $\|\nabla_{g}^{l}\eta_{k}^{m+1}\|^{\frac{n}{m}}\leq C_{4}\eta_{k},\quad\forall l=0,1,\cdots,m.$ (6.9) Since $1\leq\sum_{k}\eta_{k}(x)\leq N$ for all $x\in M$, we obtain by combining (6.5)-(6.9) that $\displaystyle\int_{M}\zeta\left(j,\alpha\|u\|^{\frac{n}{n-m}}\right)dv_{g}$ $\displaystyle\leq$ $\displaystyle C_{5}\sum_{k}\int_{M}\|\nabla_{g}^{m}(\eta_{k}^{m+1}u)\|^{\frac{n}{m}}dv_{g}$ $\displaystyle\leq$ $\displaystyle C_{5}\sum_{k}\sum_{l=0}^{m}(C_{m}^{l})^{\frac{n}{m}}\int_{M}\|\nabla_{g}^{m-k}\eta_{k}^{m+1}\nabla_{g}^{l}u\|^{\frac{n}{m}}dv_{g}$ $\displaystyle\leq$ $\displaystyle C_{4}C_{5}\sum_{l=0}^{m}(C_{m}^{l})^{\frac{n}{m}}\int_{M}(\sum_{k}\eta_{k})\|\nabla_{g}^{l}u\|^{\frac{n}{m}}dv_{g}$ $\displaystyle\leq$ $\displaystyle C_{4}C_{5}N\sum_{l=0}^{m}(C_{m}^{l})^{\frac{n}{m}}\int_{M}\|\nabla_{g}^{l}u\|^{\frac{n}{m}}dv_{g}$ $\displaystyle\leq$ $\displaystyle C_{6}$ for constants $C_{5}$ and $C_{6}$ depending only on $n$, $m$, $Q$ and $r_{H}$, where $C_{m}^{l}=\frac{m!}{l!\,(m-l)!}$. According to (Hebey [18], theorem 2.8), $C_{0}^{\infty}(M)$ is dense in $W^{m,\frac{n}{m}}(M)$. Hence for any $u\in W^{m,\frac{n}{m}}(M)$, there exists a sequence $(u_{k})$ in $C_{0}^{\infty}(M)$ such that $\\|u_{k}-u\\|_{W^{m,\frac{n}{m}}(M)}\rightarrow 0$ as $k\rightarrow\infty$. Assume $\\|u\\|_{W^{m,\frac{n}{m}}(M)}\leq 1$. Then for any $\alpha:0<\alpha<\alpha_{0}$ there holds $\displaystyle\int_{M}\zeta\left(j,\alpha\|u\|^{\frac{n}{n-m}}\right)dv_{g}$ $\displaystyle\leq$ $\displaystyle\lim_{k\rightarrow\infty}\int_{M}\zeta\left(j,\alpha\|u_{k}\|^{\frac{n}{n-m}}\right)dv_{g}\leq C_{6}.$ Using the same method of deriving $W^{1,n}(M)\hookrightarrow L^{q}(M)$ continuously for all $q\geq n$ in theorem 2.3, we obtain the continuous embedding $W^{m,{n}/{m}}(M)\hookrightarrow L^{q}(M)$ for any $q\geq{n}/{m}$. $(ii)$ Let $\alpha>0$ be any real number and $u$ be any function belonging to the space $W^{m,\frac{n}{m}}(M)$. Since $C_{0}^{\infty}(M)$ is dense in $W^{m,\frac{n}{m}}(M)$, there exists some $u_{0}\in C_{0}^{\infty}(M)$ such that $\alpha\\|u-u_{0}\\|_{W^{m,\frac{n}{m}}(M)}^{\frac{n}{n-m}}<\alpha_{0}/2,$ (6.10) where $\alpha_{0}$ is defined by (6.7). Using (2.3) and an elementary inequality $\|a\|^{p}\leq(1+\epsilon)\|a-b\|^{p}+c(\epsilon,p)\|b\|^{p},$ where $\epsilon>0$, $p>1$ and $c(\epsilon,p)$ is a constant depending only on $\epsilon$ and $p$, we have $\displaystyle\int_{M}\zeta\left(j,\alpha\|u\|^{\frac{n}{n-m}}\right)dv_{g}$ $\displaystyle\leq$ $\displaystyle\int_{M}\zeta\left(j,(1+\epsilon)\alpha\|u-u_{0}\|^{\frac{n}{n-m}}+c(\epsilon,n/(n-m))\alpha\|u_{0}\|^{\frac{n}{n-m}}\right)dv_{g}$ (6.11) $\displaystyle\leq$ $\displaystyle\frac{1}{\mu}\int_{M}\zeta\left(j,\mu(1+\epsilon)\alpha\|u-u_{0}\|^{\frac{n}{n-m}}\right)dv_{g}$ $\displaystyle+\frac{1}{\nu}\int_{M}\zeta\left(j,\nu c(\epsilon,n/(n-m))\alpha\|u_{0}\|^{\frac{n}{n-m}}\right)dv_{g},$ where $\mu>1$, $\nu>1$ and $1/\mu+1/\nu=1$. Choosing $\epsilon$ sufficiently small and $\mu$ sufficiently close to $1$ such that $\mu(1+\epsilon)\alpha_{0}/2\leq\alpha_{0}$, in view of (6.10), we have by part $(i)$ $\int_{M}\zeta\left(j,\mu(1+\epsilon)\alpha\|u-u_{0}\|^{\frac{n}{n-m}}\right)dv_{g}\leq C_{6}.$ (6.12) Note that $u_{0}\in C_{0}^{\infty}(M)$, particularly $u_{0}$ has compact support. It follows that $\int_{M}\zeta\left(j,\nu c(\epsilon,n/(n-m))\alpha\|u_{0}\|^{\frac{n}{n-m}}\right)dv_{g}<\infty.$ (6.13) Inserting (6.12) and (6.13) into (6.11), we complete the proof of part $(ii)$. $\hfill\Box$ ## 7 Applications of Trudinger-Moser inequalities In this section, we consider applications of theorem 2.3, namely the existence and multiplicity results for the problem (2.4) and its perturbation (2.8). Specifically we shall prove theorem 2.7 and theorem 2.10. Throughout this section, we use the notations introduced in section 2. Let $(M,g)$ be a complete noncompact Riemannian $n$-manifold with ${\rm Rc}_{(M,g)}\geq Kg$ for some $K\in\mathbb{R}$ and ${\rm inj}_{(M,g)}\geq i_{0}>0$. Assume $\phi(x)$ satisfies the hypotheses $(\phi_{1})$ and $(\phi_{2})$, $v(x)$ satisfies the hypotheses $(v_{1})$ and $(v_{2})$. Let $E$ be a function space defined by (2.5). If $u\in E$, then the $E$-norm of $u$ is defined by $\\|u\\|_{E}=\left(\int_{M}(\|\nabla_{g}u\|^{n}+v\|u\|^{n})dv_{g}\right)^{1/n}.$ The following compact embedding result is very important in our analysis. Proposition 7.1. For any $q\geq n$, the function space $E$ is compactly embedded in $L^{q}(M)$. Proof. Let $(u_{k})$ be a sequence of functions with $\\|u_{k}\\|_{E}\leq C$ for some constant $C$. It suffices to prove that up to a subsequence, $(u_{k})$ converges in $L^{q}(M)$ for any $q\geq n$. Clearly $(u_{k})$ is bounded in $W^{1,n}(M)$, and thus we can assume that for any $q>1$, up to a subsequence $\displaystyle u_{k}\rightharpoonup u_{0}\quad{\rm weakly\,\,in}\quad E{}$ $\displaystyle u_{k}\rightarrow u_{0}\quad{\rm strongly\,\,in}\quad L^{q}_{\rm loc}(M)$ (7.1) $\displaystyle u_{k}\rightarrow u_{0}\quad{\rm a.\,e.}\,\,\,{\rm in}\quad M.$ If $v(x)\in L^{1/(n-1)}(M)$, using the same argument of ([44], Lemma 2.4), we conclude that $E\hookrightarrow L^{q}(M)$ compactly for any $q>1$. So, in view of $(v_{2})$, we may assume $v(x)\rightarrow\infty$ as $d_{g}(O,x)\rightarrow\infty$, where $O$ is a fixed point of $M$. Given any $\epsilon>0$, there exists some $R>0$ such that $v(x)>(2C)^{n}/\epsilon$ when $d_{g}(O,x)\geq R$. Hence $\frac{(2C)^{n}}{\epsilon}\int_{M\setminus B_{O}(R)}\|u_{k}-u_{0}\|^{n}dv_{g}<\int_{M}v\|u_{k}-u_{0}\|^{n}dv_{g}\leq(2C)^{n}.$ This gives $\int_{M\setminus B_{O}(R)}\|u_{k}-u_{0}\|^{n}dv_{g}<\epsilon.$ By (LABEL:Lploc), we have $\lim_{k\rightarrow\infty}\int_{B_{O}(R)}\|u_{k}-u_{0}\|^{n}dv_{g}=0.$ Hence for the above $\epsilon$, there exists some $l\in\mathbb{N}$ such that when $k>l$, $\int_{M}\|u_{k}-u_{0}\|^{n}dv_{g}<2\epsilon.$ This implies $u_{k}\rightarrow u_{0}$ strongly in $L^{n}(M)$ as $k\rightarrow\infty$. It follows from $(i)$ of theorem 2.3 that $(u_{k})$ is bounded in $L^{q}(M)$ for any $q\geq n$. Now fixing $q>n$, we get by Hölder’s inequality $\int_{M}\|u_{k}-u_{0}\|^{q}dv_{g}\leq\left(\int_{M}\|u_{k}-u_{0}\|^{n}dv_{g}\right)^{1/n}\left(\int_{M}\|u_{k}-u_{0}\|^{\frac{n(q-1)}{n-1}}dv_{g}\right)^{1-1/n}.$ This together with the fact that $u_{k}\rightarrow u_{0}$ in $L^{n}(M)$ implies $u_{k}\rightarrow u_{0}$ in $L^{q}(M)$. $\hfill\Box$ Let $S_{q}$ be defined by (2.7). Then we have the following: Proposition 7.2. For any $q>n$, $S_{q}$ is attained by some nonnegative function $u\in E\setminus\\{0\\}$. Proof. Assume $q>n$. It is easy to see that $S_{q}^{n}=\inf_{\int_{M}\phi\|u\|^{q}dv_{g}=1}\int_{M}\left(\|\nabla u\|^{n}+v\|u\|^{n}\right)dv_{g}.$ Choosing a sequence of functions $(u_{k})\subset E$ such that $\int_{M}\phi\|u_{k}\|^{q}dv_{g}=1$ and $\lim_{k\rightarrow\infty}\int_{M}\left(\|\nabla u_{k}\|^{n}+v\|u_{k}\|^{n}\right)dv_{g}=S_{q}^{n}.$ By proposition 7.1, there exists some $u\in E$ such that up to a subsequence, $u_{k}\rightharpoonup u$ weakly in $E$, $u_{k}\rightarrow u$ strongly in $L^{q}(M)$ for any $q\geq n$, and $u_{k}\rightarrow u$ almost everywhere in $M$. Since $u_{k}\rightarrow u$ strongly in $L^{s}(B_{O}(R_{0}))$ for all $s>1$ and $\phi\in L^{p}(B_{O}(R_{0}))$, we have by using Hölder’s inequality that $\lim_{k\rightarrow\infty}\int_{B_{O}(R_{0})}\phi\|u_{k}\|^{q}dv_{g}=\int_{B_{O}(R_{0})}\phi\|u\|^{q}dv_{g}.$ (7.2) In view of $(v_{2})$, we have $\displaystyle\int_{M\setminus B_{O}(R_{0})}\phi\|\|u_{k}\|^{q}-\|u\|^{q}\|dv_{g}$ $\displaystyle\leq$ $\displaystyle qC_{0}\int_{M}(\|u_{k}\|^{q-1}+\|u\|^{q-1})\|u_{k}-u\|dv_{g}$ $\displaystyle\leq$ $\displaystyle qC_{0}\left\\{\left(\int_{M}\|u_{k}\|^{q}dv_{g}\right)^{1-1/q}+\left(\int_{M}\|u\|^{q}dv_{g}\right)^{1-1/q}\right\\}$ $\displaystyle\quad\times\left(\int_{M}\|u_{k}-u\|^{q}dv_{g}\right)^{1/q}$ $\displaystyle\rightarrow$ $\displaystyle 0\,\,\,{\rm as}\,\,\,k\rightarrow\infty.$ This together with (7.2) implies $\int_{M}\phi\|u\|^{q}dv_{g}=\lim_{k\rightarrow\infty}\int_{M}\phi\|u_{k}\|^{q}dv_{g}=1.$ (7.3) Since $u_{k}\rightharpoonup u$ weakly in $E$, we have $\displaystyle\int_{M}\|\nabla u\|^{n}dv_{g}=\lim_{k\rightarrow\infty}\int_{M}\|\nabla u\|^{n-2}\nabla u\nabla u_{k}dv_{g}\leq\limsup_{k\rightarrow\infty}\left(\int_{M}\|\nabla u_{k}\|^{n}dv_{g}\right)^{\frac{1}{n}}\left(\int_{M}\|\nabla u\|^{n}dv_{g}\right)^{1-\frac{1}{n}},$ from which we obtain $\int_{M}\|\nabla u\|^{n}dv_{g}\leq\limsup_{k\rightarrow\infty}\int_{M}\|\nabla u_{k}\|^{n}dv_{g}.$ (7.4) In addition, we have by Fatou’s lemma $\int_{M}v\|u\|^{n}dv_{g}\leq\limsup_{k\rightarrow\infty}\int_{M}v\|u_{k}\|^{n}dv_{g}.$ (7.5) Combining (7.3), (7.4) and (7.5), we conclude that $S_{q}$ is attained by $u\in E\setminus\\{0\\}$. Since $\|u\|\in E$, one can easily see that $S_{q}$ is also attained by $\|u\|$. $\hfill\Box$ Now we get back to the problem (2.4). Since we are interested in nonnegative weak solutions, without loss of generality we may assume $f(x,t)\equiv 0$ for all $(x,t)\in M\times(-\infty,0]$. By $(f_{1})$, we have for all $(x,t)\in M\times\mathbb{R}$, $\|F(x,t)\|\leq\frac{b_{1}}{n}\|t\|^{n}+b_{2}t\zeta\left(n,\|t\|^{\frac{n}{n-1}}\right).$ This together with $(\phi_{1})$, $(\phi_{2})$ and (2.2) implies that for any $u\in E$ there holds $\displaystyle\int_{M}\phi F(x,u)dv_{g}$ $\displaystyle\leq$ $\displaystyle\\|\phi\\|_{L^{p}(B_{O}(R_{0}))}\\|F(x,u)\\|_{L^{q}(M)}+C_{0}\int_{M}F(x,u)dv_{g}$ $\displaystyle\leq$ $\displaystyle\\|\phi\\|_{L^{p}(B_{O}(R_{0}))}\left(\frac{b_{1}}{n}\\|u\\|_{L^{qn}(M)}^{n}+b_{2}\\|u\zeta(n,\|u\|^{\frac{n}{n-1}})\\|_{L^{q}(M)}\right)$ $\displaystyle+C_{0}\frac{b_{1}}{n}\\|u\\|_{L^{n}(M)}^{n}+C_{0}b_{2}\\|u\zeta(n,\|u\|^{\frac{n}{n-1}})\\|_{L^{1}(M)}$ $\displaystyle\leq$ $\displaystyle C\left(\\|u\\|_{L^{qn}(M)}^{n}+\\|u\\|_{L^{qn}(M)}\\|\zeta(n,\frac{qn}{n-1}\|u\|^{\frac{n}{n-1}})\\|_{L^{1}(M)}^{1-\frac{1}{n}}\right.$ $\displaystyle\left.\\|u\\|_{L^{n}(M)}^{n}+\\|u\\|_{L^{n}(M)}\\|\zeta(n,\frac{n}{n-1}\|u\|^{\frac{n}{n-1}})\\|_{L^{1}(M)}\right),$ where $C$ is a constant depending only on $n$, $b_{1}$, $b_{2}$, $C_{0}$ and $\\|\phi\\|_{L^{p}(B_{O}(R_{0}))}$, and $1/p+1/q=1$. By theorem 2.3, $u\in L^{s}(M)$ for all $s\geq n$, and for any $\alpha>0$ there holds $\zeta(n,\alpha\|u\|^{\frac{n}{n-1}})\in L^{1}(M)$. Hence $\int_{M}\phi F(x,u)dv_{g}<+\infty,\quad\forall u\in E.$ Based on this, we can define a functional on $E$ by $J(u)=\frac{1}{n}\\|u\\|_{E}^{n}-\int_{M}\phi F(x,u)dv_{g}.$ (7.6) By ([13], proposition 1) and the standard argument [34], we have $J\in\mathcal{C}^{1}(E,\mathbb{R})$. Clearly the critical point of $J$ is a weak solution to (2.4). Concerning the geometry of $J$, the following two lemmas imply that $J$ has a mountain pass structure. Lemma 7.3. Assume that $(f_{1})$, $(f_{2})$, and $(f_{3})$ are satisfied. Then for any nonnegative, compactly supported function $u\in E\setminus\\{0\\}$, there holds $J(tu)\rightarrow-\infty$ as $t\rightarrow+\infty$. Proof. By $(f_{2})$ and $(f_{3})$, there exist $c_{1}$, $c_{2}>0$ and $\mu>n$ such that $F(x,s)\geq c_{1}s^{\mu}-c_{2}$ for all $(x,s)\in M\times[0,+\infty)$. Assume ${\rm supp}\,u\subset B_{O}(R_{1})$ for some $R_{1}>0$. We have $\displaystyle J(tu)$ $\displaystyle=$ $\displaystyle\frac{t^{n}}{n}\\|u\\|_{E}^{n}-\int_{B_{O}(R_{1})}\phi F(x,tu)dv_{g}$ $\displaystyle\leq$ $\displaystyle\frac{t^{n}}{n}\\|u\\|_{E}^{n}-c_{1}t^{\mu}\int_{B_{O}(R_{1})}\phi u^{\mu}dv_{g}-c_{2}\int_{B_{O}(R_{1})}\phi dv_{g}.$ This gives the desired result since $\phi(x)>0$ for all $x\in M$ and $\mu>n$.$\hfill\Box$ Lemma 7.4. Assume that $(f_{1})$ and $(f_{4})$ are satisfied. Then there exist sufficiently small constants $r>0$ and $\delta>0$ such that $J(u)\geq\delta$ for all $u$ with $\\|u\\|_{E}=r$. Proof. By $(f_{1})$ and $(f_{4})$, there exists some constants $\theta\in(0,1)$ and $C>0$ such that $F(x,s)\leq\frac{(1-\theta)\lambda_{\phi}}{n}\|s\|^{n}+C\|s\|^{n+1}\zeta\left(n,\alpha_{0}\|s\|^{\frac{n}{n-1}}\right)$ for all $(x,s)\in M\times\mathbb{R}$. By definition of $\lambda_{\phi}$, $\frac{(1-\theta)\lambda_{\phi}}{n}\int_{M}\phi\|u\|^{n}dv_{g}\leq\frac{1-\theta}{n}\\|u\\|_{E}^{n}.$ (7.7) Note that $\phi$ satisfies $(\phi_{1})$ and $(\phi_{2})$. We have by Hölder’s inequality and (2.2) that $\displaystyle\int_{M}\phi\|u\|^{n+1}\zeta\left(n,\alpha_{0}\|u\|^{\frac{n}{n-1}}\right)dv_{g}$ $\displaystyle\leq$ $\displaystyle\\|\phi\\|_{L^{p}(B_{O}(R_{0}))}\left(\int_{M}\|u\|^{(n+1)q}dv_{g}\right)^{1/q}\left(\int_{M}\zeta\left(n,q^{\prime}\alpha_{0}\|u\|^{\frac{n}{n-1}}\right)dv_{g}\right)^{1/q^{\prime}}$ (7.8) $\displaystyle+C_{0}\left(\int_{M}\|u\|^{(n+1)\beta}dv_{g}\right)^{1/\beta}\left(\int_{M}\zeta\left(n,\gamma\alpha_{0}\|u\|^{\frac{n}{n-1}}\right)dv_{g}\right)^{1/\gamma},$ where $1/p+1/q+1/q^{\prime}=1$ and $1/\beta+1/\gamma=1$. Fix $\alpha=\beta_{0}/2$, where $\beta_{0}$ is defined by (1.5). It follows from $(i)$ of theorem 2.3 that there exists some constant $\tau$ depending only on $\alpha$, $n$, $K$ and $i_{0}$ such that $\Lambda_{\alpha}:=\sup_{\\|u\\|_{1,\tau}\leq 1}\int_{M}\zeta\left(n,\alpha\|u\|^{\frac{n}{n-1}}\right)dv_{g}<+\infty.$ (7.9) Let $r$ be a positive constant to be determined later. Now suppose $\\|u\\|_{E}=r$. It is easy to see that $\\|u\\|_{1,\tau}\leq r+\tau r/v_{0}^{1/n}$. Clearly one can select $r$ sufficiently small such that $q^{\prime}\alpha_{0}\\|u\\|_{1,\tau}^{n/(n-1)}<\alpha$ and $\gamma\alpha_{0}\\|u\\|_{1,\tau}^{n/(n-1)}<\alpha$. It follows from (7.9) that $\sup_{\\|u\\|_{E}=r}\int_{M}\zeta\left(n,q^{\prime}\alpha_{0}\|u\|^{\frac{n}{n-1}}\right)dv_{g}\leq\Lambda_{\alpha}$ and $\sup_{\\|u\\|_{E}=r}\int_{M}\zeta\left(n,\gamma\alpha_{0}\|u\|^{\frac{n}{n-1}}\right)dv_{g}\leq\Lambda_{\alpha},$ provided that $r$ is chosen sufficiently small. Inserting these two inequalities into (7.8), then using the embedding $E\hookrightarrow L^{s}(M)$ for all $s\geq n$ (proposition 7.1) and (7.7), we obtain $J(u)\geq\frac{\theta}{n}\\|u\\|_{E}^{n}-\tilde{C}\\|u\\|_{E}^{n+1}$ for some constant $\tilde{C}$ depending only on $\alpha$, $n$, $K$ and $i_{0}$, provided that $\\|u\\|_{E}$ is sufficiently small. This gives the desired result. $\hfill\Box$ To estimate the min-max level of $J$, we state the following: Lemma 7.5. Assume $(f_{5})$. There exists some nonnegative function $u^{}\in E$ such that $\sup_{t\geq 0}J(tu^{})<\frac{1}{n}\left(\frac{(p-1)\alpha_{n}}{p\alpha_{0}}\right)^{n-1}.$ Proof. Let $u^{}$ be given by proposition 7.2, namely $u^{}\geq 0$, $\\|u^{}\\|_{E}=S_{q}$, and $\int_{M}\phi\|u^{}\|^{q}dv_{g}=1$. Then for any $t\geq 0$ there holds $\displaystyle J(tu^{})$ $\displaystyle=$ $\displaystyle\frac{1}{n}\\|tu^{}\\|_{E}^{n}-\int_{M}\phi(x)F(x,tu^{})dv_{g}$ $\displaystyle\leq$ $\displaystyle\frac{S_{q}^{n}}{n}t^{n}-\frac{C_{q}}{q}t^{q}$ $\displaystyle\leq$ $\displaystyle\frac{q-n}{nq}\frac{S_{q}^{{nq}/{(q-n)}}}{C_{q}^{{n}/{(q-n)}}}$ $\displaystyle<$ $\displaystyle\frac{1}{n}\left(\frac{(p-1)\alpha_{n}}{p\alpha_{0}}\right)^{n-1}.$ Here we have used the hypothesis $(f_{5})$. $\hfill\Box$ Adapting the proof of ([44], lemma 3.4), we obtain the following compactness result. Lemma 7.6. Assume $(f_{1})$, $(f_{2})$ and $(f_{3})$. Let $(u_{j})\subset E$ be an arbitrary Palais-Smale sequence of $J$, i.e., $J(u_{j})\rightarrow c,\,\,J^{\prime}(u_{j})\rightarrow 0\,\,{\rm in}\,\,E^{}\,\,{\rm as}\,\,j\rightarrow\infty,$ (7.10) where $E^{}$ denotes the dual space of $E$. Then there exist a subsequence of $(u_{j})$ (still denoted by $(u_{j})$) and $u\in E$ such that $u_{j}\rightharpoonup u$ weakly in $E$, $u_{j}\rightarrow u$ strongly in $L^{q}(M)$ for all $q\geq n$, and $\displaystyle\left\\{\begin{array}[]{lll}\nabla u_{j}(x)\rightarrow\nabla u(x)\quad{\rm a.\,\,e.\,\,\,in}\quad M\\\\[6.45831pt] {\phi(x)F(x,\,u_{j})}\rightarrow{\phi(x)F(x,\,u)}\,\,{\rm strongly\,\,in}\,\,L^{1}(M).\end{array}\right.$ Furthermore $u$ is a weak solution of (2.4). Proof. Assume $(u_{j})$ is a Palais-Smale sequence of $J$. By $(\ref{PS})$, we have $\displaystyle\frac{1}{n}\\|u_{j}\\|_{E}^{n}-\int_{M}\phi(x)F(x,u_{j})dv_{g}\rightarrow c\,\,{\rm as}\,\,j\rightarrow\infty,$ (7.12) $\displaystyle\left\|\int_{M}\left(\|\nabla_{g}u_{j}\|^{n-2}\nabla_{g}u_{j}\nabla_{g}\psi+v\|u_{j}\|^{n-2}u_{j}\psi\right)dv_{g}-\int_{M}\phi(x)f(x,u_{j})\psi dv_{g}\right\|\leq\sigma_{j}\\|\psi\\|_{E}\qquad\quad$ (7.13) for all $\psi\in E$, where $\sigma_{j}\rightarrow 0$ as $j\rightarrow\infty$. Note that $f(x,s)\equiv 0$ for all $(x,s)\in M\times(-\infty,0]$. By $(f_{2})$, we have $0\leq\mu F(x,u_{j})\leq u_{j}f(x,u_{j})$ for some $\mu>n$. Taking $\psi=u_{j}$ in (7.13) and multiplying (7.12) by $\mu$, we have $\displaystyle\left(\frac{\mu}{n}-1\right)\\|u_{j}\\|_{E}^{n}$ $\displaystyle\leq$ $\displaystyle\mu\|c\|+\int_{M}\phi(x)\left(\mu F(x,u_{j})-f(x,u_{j})u_{j}\right)dv_{g}+\sigma_{j}\\|u_{j}\\|_{E}$ $\displaystyle\leq$ $\displaystyle\mu\|c\|+\sigma_{j}\\|u_{j}\\|_{E}.$ Therefore $\\|u_{j}\\|_{E}$ is bounded. It then follows from (7.12) and (7.13) that $\int_{M}\phi(x)f(x,u_{j})u_{j}dv_{g}\leq C,\quad\int_{M}\phi(x)F(x,u_{j})dv_{g}\leq C$ (7.14) for some constant $C$ depending only on $\mu$, $n$ and $c$. By proposition 7.1, there exists some $u\in E$ such that $u_{j}\rightharpoonup u$ weakly in $E$, $u_{j}\rightarrow u$ strongly in $L^{q}(M)$ for any $q\geq n$, and $u_{j}\rightarrow u$ almost everywhere in $M$. By $(f_{3})$, there exist positive constants $A_{1}$ and $R_{1}$ such that $F(x,s)\leq A_{1}f(x,s)$ for all $s\geq R_{1}$. Particularly for any $A>R_{1}$ there holds $F(x,s)\leq A_{1}f(x,s),\quad\forall s\geq A.$ (7.15) Now we prove that $\phi(x)F(x,u_{j})\rightarrow\phi F(x,u)$ strongly in $L^{1}(M)$. To this end, for any $\epsilon>0$, we take $A>\max\\{A_{1}C/\epsilon,R_{1}\\}$, where $C$ is given by (7.14). Then we have by (7.15) $\int_{\|u_{j}\|>A}\phi(x)F(x,u_{j})dv_{g}\leq\frac{A_{1}}{A}\int_{M}\phi(x)f(x,u_{j})u_{j}dv_{g}<\epsilon.$ (7.16) In the same way $\int_{\|u\|>A}\phi(x)F(x,u)dv_{g}<\epsilon.$ (7.17) By $(f_{1})$, we have for $(x,s)\in M\times[0,\infty)$ $\displaystyle f(x,s)$ $\displaystyle\leq$ $\displaystyle b_{1}s^{n-1}+b_{2}\zeta\left(n,\alpha_{0}s^{\frac{n}{n-1}}\right)$ $\displaystyle=$ $\displaystyle b_{1}s^{n-1}+b_{2}s^{n}\sum_{k=n-1}^{\infty}\frac{\alpha_{0}^{k}s^{\frac{n}{n-1}(k-n+1)}}{k!}$ $\displaystyle\leq$ $\displaystyle b_{1}s^{n-1}+b_{2}s^{n}\alpha_{0}^{n-1}e^{\alpha_{0}s^{\frac{n}{n-1}}}.$ Hence for all $(x,s)\in M\times[0,A]$ there holds $f(x,s)\leq\left(b_{1}+b_{2}\alpha_{0}^{n-1}Ae^{\alpha_{0}A^{\frac{n}{n-1}}}\right)s^{n-1}.$ It follows that $F(x,s)\leq\frac{b_{1}+b_{2}\alpha_{0}^{n-1}Ae^{\alpha_{0}A^{\frac{n}{n-1}}}}{n}s^{n},\quad\forall s\in[0,A].$ for all $(x,s)\in M\times[0,A]$, which implies $\|\phi(x)\chi_{\\{\|u_{j}\|\leq A\\}}(x)F(x,u_{j})\|\leq C_{1}\phi(x)\|u_{j}\|^{n},$ (7.18) where $C_{1}={(b_{1}+b_{2}\alpha_{0}^{n-1}Ae^{\alpha_{0}A^{{n}/{(n-1)}}})}/{n}$ and $\chi_{\\{\|u_{j}\|\leq A\\}}(x)$ denotes the characteristic function of the set $\\{x\in M:\|u_{j}(x)\|\leq A\\}$. By an inequality $\|\|a\|^{n}-\|b\|^{n}\|\leq n\|a-b\|(\|a\|^{n-1}+\|b\|^{n-1})\,(\forall a,b\in\mathbb{R})$ and Hölder’s inequality, we get $\displaystyle\int_{M}\phi\|\|u_{j}\|^{n}-\|u\|^{n}\|dv_{g}$ $\displaystyle\leq$ $\displaystyle n\int_{M}\phi\|u_{j}-u\|(\|u_{j}\|^{n-1}+\|u\|^{n-1})dv_{g}$ $\displaystyle\leq$ $\displaystyle n\left(\int_{M}\phi\|u_{j}-u\|^{n}dv_{g}\right)^{\frac{1}{n}}\left\\{\left(\int_{M}\phi\|u_{j}\|^{n}dv_{g}\right)^{1-\frac{1}{n}}+\left(\int_{M}\phi\|u\|^{n}dv_{g}\right)^{1-\frac{1}{n}}\right\\}.$ Hence $\phi\|u_{j}\|^{n}\rightarrow\phi\|u\|^{n}$ in $L^{1}(M)$ since $u_{j}\rightarrow u$ strongly in $L^{n}(M)$. In view of (7.18), we conclude from the generalized Lebesgue’s dominated convergence theorem $\lim_{j\rightarrow\infty}\int_{M}\phi(x)\chi_{\\{\|u_{j}\|\leq A\\}}(x)F(x,u_{j})dv_{g}=\int_{M}\phi(x)\chi_{\\{\|u\|\leq A\\}}(x)F(x,u)dv_{g}.$ This together with (7.16) and (7.17) implies that there exists some $m\in\mathbb{N}$ such that when $j>m$ there holds $\left\|\int_{M}\phi F(x,u_{j})dv_{g}-\int_{M}\phi F(x,u)dv_{g}\right\|<3\epsilon.$ Therefore $\lim_{j\rightarrow\infty}\int_{M}\phi F(x,u_{j})dv_{g}=\int_{M}\phi F(x,u)dv_{g}.$ Using the same method as that of proving ([4], (4.26)), we have $\nabla_{g}u_{j}(x)\rightarrow\nabla_{g}u(x)$ for almost every $x\in M$ and $\|\nabla_{g}u_{j}\|^{n-2}\nabla_{g}u_{j}\rightharpoonup\|\nabla_{g}u\|^{n-2}\nabla_{g}u\quad{\rm weakly\,\,in}\quad\left(L^{\frac{n}{n-1}}(M)\right)^{n}.$ Passing to the limit $j\rightarrow\infty$ in $(\ref{2})$, we obtain $\int_{M}\left(\|\nabla_{g}u\|^{n-2}\nabla_{g}u\nabla\psi+v\|u\|^{n-2}u\psi\right)dv_{g}-\int_{M}\phi(x)f(x,u)\psi dv_{g}=0$ for all $\psi\in C_{0}^{\infty}(M)$. Since $C_{0}^{\infty}(M)$ is dense in $E$ under the norm $\\|\cdot\\|_{E}$, $u$ is a weak solution of $(\ref{equa})$. $\hfill\Box$ We say more words on lemma 7.6. Suppose $(M,g)$ is the standard euclidian space $\mathbb{R}^{n}$ and $\phi(x)=\|x\|^{-\beta}$, $0\leq\beta<n$. The author [44] proved that $\phi F(x,u_{j})\rightarrow\phi F(x,u)$ in $L^{1}(\mathbb{R}^{n})$ under the assumption $E\hookrightarrow L^{q}(\mathbb{R}^{n})$ compactly for all $q\geq 1$. While Lam-Lu [20] observed that the convergence still holds under the assumption $E\hookrightarrow L^{q}(\mathbb{R}^{n})$ for all $q\geq n$. Here we generalized these two situations. The following lemma is a nontrivial consequence of theorem 2.3. It is sufficient for our use when we consider the existence and multiplicity results for problems (2.4) and (2.8). Lemma 7.7. Let $(u_{j})\subset E$ be any sequence of functions satisfying $\\|u_{j}\\|_{E}\leq 1$, $u_{j}\rightharpoonup u_{0}$ weakly in $E$, $\nabla_{g}u_{j}\rightarrow\nabla_{g}u_{0}$ almost everywhere in M, and $u_{j}\rightarrow u_{0}$ strongly in $L^{n}(M)$ as $j\rightarrow\infty$. Then $(i)$ for any $\alpha:0<\alpha<\alpha_{n}$, there holds $\sup_{j}\int_{M}\zeta\left(n,\alpha\|u_{j}\|^{\frac{n}{n-1}}\right)dv_{g}<\infty;$ (7.19) $(ii)$ for any $\alpha:0<\alpha<\alpha_{n}$ and $q:0<q<(1-\\|u_{0}\\|_{E}^{n})^{-1/(n-1)}$, there holds $\sup_{j}\int_{M}\zeta\left(n,q\alpha\|u_{j}\|^{\frac{n}{n-1}}\right)dv_{g}<\infty.$ (7.20) Proof. $(i)$ For any fixed $\alpha:0<\alpha<\alpha_{n}$, it follows from part $(i)$ of theorem 2.3 that there exists a positive constant $\tau_{\alpha}$ depending only on $\alpha$, $n$, $K$ and $i_{0}$ such that $\mathcal{B}_{\alpha}=\sup_{u\in W^{1,n}(M),\,\\|u\\|_{1,\tau_{\alpha}}\leq 1}\int_{M}\zeta\left(n,\alpha\|u\|^{\frac{n}{n-1}}\right)dv_{g}<\infty.$ (7.21) Note that $v\geq v_{0}$ in $M$. Since $\\|u_{j}\\|_{E}\leq 1$, we get $\displaystyle\\|u_{j}\\|_{1,\tau_{\alpha}}=\left(\int_{M}\|\nabla_{g}u_{j}\|^{n}dv_{g}\right)^{\frac{1}{n}}+\tau_{\alpha}\left(\int_{M}\|u_{j}\|^{n}dv_{g}\right)^{\frac{1}{n}}\leq 1+\frac{\tau_{\alpha}}{v_{0}^{1/n}}.$ There exists some small positive number $\alpha_{0}$ such that $\alpha_{0}\\|u_{j}\\|_{1,\tau_{\alpha}}^{\frac{n}{n-1}}\leq\alpha$. Hence by (7.21), there holds $\sup_{j}\int_{M}\zeta\left(n,\alpha_{0}\|u_{j}\|^{\frac{n}{n-1}}\right)dv_{g}\leq\sup_{j}\int_{M}\zeta\left(n,\alpha\left\|\frac{u_{j}}{\\|u_{j}\\|_{1,\tau_{\alpha}}}\right\|^{\frac{n}{n-1}}\right)dv_{g}\leq\mathcal{B}_{\alpha}.$ This allows us to define $\alpha^{}=\sup\left\\{\alpha:\sup_{j}\int_{M}\zeta\left(n,\alpha\|u_{j}\|^{\frac{n}{n-1}}\right)dv_{g}<\infty\right\\}.$ To prove (7.19), it suffices to prove that $\alpha^{}\geq\alpha_{n}$. Suppose not, we have $\alpha^{}<\alpha_{n}$. Take two constants $\alpha^{\prime}$ and $\alpha^{\prime\prime}$ such that $\alpha^{}<\alpha^{\prime}<\alpha^{\prime\prime}<\alpha_{n}$. By part $(i)$ of theorem 2.3 again, there exists some constant $\tau_{\alpha^{\prime\prime}}$ depending only on $\alpha^{\prime\prime}$, $n$, $K$ and $i_{0}$ such that $\mathcal{B}_{\alpha^{\prime\prime}}=\sup_{u\in W^{1,n}(M),\,\\|u\\|_{1,\tau_{\alpha^{\prime\prime}}}\leq 1}\int_{M}\zeta\left(n,\alpha^{\prime\prime}\|u\|^{\frac{n}{n-1}}\right)dv_{g}<\infty.$ (7.22) Since $u_{j}\rightarrow u_{0}$ strongly in $L^{n}(M)$ and $\nabla_{g}u_{j}\rightarrow\nabla_{g}u_{0}$ a. e. in M, we obtain by using Brezis-Lieb’s lemma [6] $\\|u_{j}-u_{0}\\|_{1,\tau_{\alpha\,^{\prime\prime}}}=\left(\int_{M}\|\nabla_{g}u_{j}\|^{n}dv_{g}-\int_{M}\|\nabla_{g}u_{0}\|^{n}dv_{g}\right)^{{1}/{n}}+o_{j}(1),$ where $o_{j}(1)\rightarrow 0$ as $j\rightarrow\infty$. Since $u_{j}\rightharpoonup u_{0}$ weakly in $E$, there holds $\lim_{j\rightarrow+\infty}\int_{M}\|\nabla_{g}u_{0}\|^{n-2}\nabla_{g}u_{0}\nabla_{g}u_{j}\,dv_{g}=\int_{M}\|\nabla_{g}u_{0}\|^{n}dv_{g}.$ This immediately implies that $\int_{M}\|\nabla_{g}u_{0}\|^{n}dv_{g}\leq\limsup_{j\rightarrow+\infty}\int_{M}\|\nabla_{g}u_{j}\|^{n}dv_{g}\leq 1.$ Hence $\\|u_{j}-u_{0}\\|_{1,\tau_{\alpha\,^{\prime\prime}}}\leq 1+o_{j}(1).$ It follows from (2.3) that for any $\epsilon>0$ there exists some constant $\tilde{c}$ depending only on $\epsilon$ and $n$ such that $\displaystyle\zeta\left(n,\alpha^{\prime}\|u_{j}\|^{\frac{n}{n-1}}\right)\leq\frac{1}{\mu}\zeta\left(n,\alpha^{\prime}(1+\epsilon)\mu\|u_{j}-u_{0}\|^{\frac{n}{n-1}}\right)+\frac{1}{\nu}\zeta\left(n,\alpha^{\prime}\tilde{c}\nu\|u_{0}\|^{\frac{n}{n-1}}\right),$ (7.23) where $1/\mu+1/\nu=1$. Choosing $\epsilon$ sufficiently small and $\mu$ sufficiently close to $1$ such that $\alpha^{\prime}(1+\epsilon)\mu\\|u_{j}-u_{0}\\|_{1,\tau_{\alpha^{\prime\prime}}}^{\frac{n}{n-1}}<\alpha^{\prime\prime},$ provided that $j$ is sufficiently large. This together with (7.22) implies that $\sup_{j}\int_{M}\zeta\left(n,\alpha^{\prime}(1+\epsilon)\mu\|u_{j}-u_{0}\|^{\frac{n}{n-1}}\right)dv_{g}\leq\mathcal{B}_{\alpha^{\prime\prime}}.$ (7.24) In addition, we have by part $(iii)$ of theorem 2.3 that $\int_{M}\zeta\left(n,\alpha^{\prime}\tilde{c}\nu\|u_{0}\|^{\frac{n}{n-1}}\right)dv_{g}<+\infty.$ (7.25) Inserting (7.24) and (7.25) into (7.23), we get $\sup_{j}\int_{M}\zeta\left(n,\alpha^{\prime}\|u_{j}\|^{\frac{n}{n-1}}\right)dv_{g}<+\infty,$ which contradicts the definition of $\alpha^{}$ and thus ends the proof of part $(i)$. $(ii)$ Given any $\alpha:0<\alpha<\alpha_{n}$ and any $q:0<q<(1-\\|u_{0}\\|_{E}^{n})^{-1/(n-1)}$. By (2.3), $\forall\epsilon>0$, there exist constants $\tilde{c}>0$, $\mu>1$ and $\nu>1$ $(1/\mu+1/\nu=1)$ such that $\displaystyle\int_{M}\zeta\left(n,q\alpha\|u_{j}\|^{\frac{n}{n-1}}\right)dv_{g}\leq\frac{1}{\mu}\int_{M}\zeta\left(n,q\alpha(1+\epsilon)\mu\|u_{j}-u_{0}\|^{\frac{n}{n-1}}\right)dv_{g}+\frac{1}{\nu}\int_{M}\zeta\left(n,q\alpha\tilde{c}\nu\|u_{0}\|^{\frac{n}{n-1}}\right)dv_{g}.$ By Brezis-Lieb’s lemma [6], $\\|u_{j}-u_{0}\\|_{E}^{\frac{n}{n-1}}\leq(1-\\|u_{0}\\|_{E}^{n})^{\frac{1}{n-1}}+o_{j}(1).$ If we choose $\epsilon$ sufficiently small and $\mu$ sufficiently close to $1$ such that $q\alpha(1+\epsilon)\mu\\|u_{j}-u_{0}\\|_{E}^{\frac{n}{n-1}}\leq(\alpha+\alpha_{n})/2,$ provided that $j$ is sufficiently large. It then follows from part $(i)$ that $\sup_{j}\int_{M}\zeta\left(n,q\alpha(1+\epsilon)\mu\|u_{j}-u_{0}\|^{\frac{n}{n-1}}\right)dv_{g}<+\infty.$ By part $(iii)$ of theorem 2.3, we have $\int_{M}\zeta\left(n,q\alpha\tilde{c}\nu\|u_{0}\|^{\frac{n}{n-1}}\right)dv_{g}<+\infty.$ Therefore (7.20) holds. $\hfill\Box$ Remark 7.8. In lemma 7.7, if $u_{0}\equiv 0$, then the conclusion of $(ii)$ is weaker than that of $(i)$. If $u_{0}\not\equiv 0$, then the conclusion of $(i)$ is a special case of that of $(ii)$. If $(M,g)$ has dimension two, the assumption $\nabla_{g}u_{j}\rightarrow\nabla_{g}u_{0}$ almost everywhere in $M$ can be removed. Proof of theorem 2.7. It follows from lemma 7.3 and lemma 7.4 that $J$ satisfies all the hypothesis of the mountain-pass theorem except for the Palais-Smale condition: $J\in\mathcal{C}^{1}(E,\mathbb{R})$; $J(0)=0$; $J(u)\geq\delta>0$ when $\\|u\\|_{E}=r$; $J(e)<0$ for some $e\in E$ with $\\|e\\|_{E}>r$. Then using the mountain-pass theorem without the Palais-Smale condition [34], we can find a sequence $(u_{j})$ in $E$ such that $J(u_{j})\rightarrow c>0,\quad J^{\prime}(u_{j})\rightarrow 0\,\,{\rm in}\,\,E^{},$ where $c=\min_{\gamma\in\Gamma}\max_{u\in\gamma}J(u)\geq\delta$ is the min-max value of $J$, where $\Gamma=\\{\gamma\in\mathcal{C}([0,1],E):\gamma(0)=0,\gamma(1)=e\\}$. This is equivalent to (7.12) and $(\ref{2})$. By lemma 7.6, up to a subsequence, there holds $\left\\{\begin{array}[]{lll}u_{j}\rightharpoonup u\,\,{\rm weakly\,\,in}\,\,E\\\\[6.45831pt] u_{j}\rightarrow u\,\,{\rm strongly\,\,in}\,\,L^{q}(M),\,\,\forall q\geq n\\\\[6.45831pt] \lim\limits_{j\rightarrow\infty}\int_{M}\phi(x){F(x,u_{j})}dv_{g}=\int_{M}\phi(x){F(x,u)}dv_{g}\\\\[6.45831pt] u\,\,{\rm is\,\,a\,\,weak\,\,solution\,\,of}\,\,(\ref{equa}).\end{array}\right.$ (7.26) Now suppose by contradiction $u\equiv 0$. Since $F(x,0)=0$ for all $x\in M$, it follows from (7.12) and (7.26) that $\lim_{j\rightarrow\infty}\\|u_{j}\\|_{E}^{n}=nc>0.$ (7.27) By lemma 7.5, $0<c<\frac{1}{n}\left(\frac{(p-1)\alpha_{n}}{p\alpha_{0}}\right)^{n-1}$. Thus there exists some $\eta_{0}>0$ and $m>0$ such that $\\|u_{j}\\|_{E}^{n}\leq\left(\frac{p-1}{p}\frac{\alpha_{n}}{\alpha_{0}}-\eta_{0}\right)^{n-1}$ for all $j>m$. Choose $q>1$ sufficiently close to $1$ such that $q\alpha_{0}\\|u_{j}\\|_{E}^{\frac{n}{n-1}}\leq(1-1/p)\alpha_{n}-\alpha_{0}\eta_{0}/2$ for all $j>m$. By $(f_{1})$, $\|f(x,u_{j})u_{j}\|\leq b_{1}\|u_{j}\|^{n}+b_{2}\|u_{j}\|\zeta\left(n,\alpha_{0}\|u_{j}\|^{\frac{n}{n-1}}\right).$ It follows from (2.2), Hölder’s inequality, and part $(i)$ of lemma 7.7 that $\displaystyle\int_{M}\phi{\|f(x,u_{j})u_{j}\|}dv_{g}$ $\displaystyle\leq$ $\displaystyle b_{1}\int_{M}\phi\|u_{j}\|^{n}dv_{g}+b_{2}\int_{M}\phi\|u_{j}\|\zeta\left(n,\alpha_{0}\|u_{j}\|^{\frac{n}{n-1}}\right)dv_{g}$ $\displaystyle\leq$ $\displaystyle b_{1}\int_{M}\phi{\|u_{j}\|^{n}}dv_{g}+b_{2}\left(\int_{M}\phi\|u_{j}\|^{q^{\prime}}dv_{g}\right)^{1/{q^{\prime}}}\left(\int_{M}\phi\zeta\left(n,q\alpha_{0}\|u_{j}\|^{\frac{n}{n-1}}\right)dv_{g}\right)^{1/{q}}$ $\displaystyle\leq$ $\displaystyle b_{1}\int_{M}\phi{\|u_{j}\|^{n}}dv_{g}+C\left(\int_{M}\phi{\|u_{j}\|^{q^{\prime}}}dv_{g}\right)^{1/{q^{\prime}}}\rightarrow 0\quad{\rm as}\quad j\rightarrow\infty,$ where $1/q+1/q^{\prime}=1$ and $C$ is some constant which is independent of $j$. Here we have used (7.26) again (precisely $u_{j}\rightarrow u$ in $L^{r}(\mathbb{R}^{N})$ for all $r\geq n$) in the above estimates. Inserting this into (7.13) with $\psi=u_{j}$, we have $\\|u_{j}\\|_{E}\rightarrow 0\quad{\rm as}\quad j\rightarrow\infty,$ which contradicts (7.27). Therefore $u\not\equiv 0$ and we obtain a nontrivial weak solution of (2.4). Finally $u$ is nonnegative since $f(x,s)\equiv 0$ for all $(x,s)\in M\times(-\infty,0]$. $\hfill\Box$ Proof of theorem 2.10. Since the proof is very similar to that of ([44], theorem 1.2), we only give its sketch and emphasize the difference between these two situations. Instead of $J:E\rightarrow\mathbb{R}$ defined by (7.6), we consider functionals for all $u\in E$ and $\epsilon>0$ $J_{\epsilon}(u)=\frac{1}{n}\\|u\\|_{E}^{n}-\int_{M}\phi(x){F(x,u)}dv_{g}-\epsilon\int_{M}hudv_{g}.$ Firstly, lemma 7.6 still holds if we replace $J$ by $J_{\epsilon}$. Namely for any Palais-Smale sequence $(u_{j})\subset E$ of $J_{\epsilon}$, there exist a subsequence of $(u_{j})$ (still denoted by $(u_{j})$) and $u\in E$ such that $u_{j}\rightharpoonup u$ weakly in $E$, $u_{j}\rightarrow u$ strongly in $L^{q}(M)$ for all $q\geq n$, and $\left\\{\begin{array}[]{lll}\nabla_{g}u_{j}(x)\rightarrow\nabla_{g}u(x)\quad{\rm a.\,\,e.\,\,\,in}\quad M\\\\[6.45831pt] {\phi(x)F(x,\,u_{j})}\rightarrow{\phi(x)F(x,\,u)}\,\,{\rm strongly\,\,in}\,\,L^{1}(M)\\\\[6.45831pt] u\,\,{\rm is\,\,a\,\,weak\,\,solution\,\,of}\,\,(\ref{equa1}).\end{array}\right.$ (7.28) Secondly, using the same method in the first two steps of the proof of ([44], theorem 1.2), we have the following: $(a)$ there exist constants $\epsilon_{1}>0$, $\delta>0$, and a sequence of functions $(v_{j})\subset E$ such that $J_{\epsilon}(v_{j})\rightarrow c_{M}$ and $J^{\prime}_{\epsilon}(v_{j})\rightarrow 0$ as $j\rightarrow\infty$, provided that $0<\epsilon<\epsilon_{1}$. In addition, $v_{j}$ is bounded in $E$, $v_{j}\rightharpoonup u_{M}$ weakly in $E$ and $u_{M}$ is a weak solution of (2.8). Here $c_{M}$ is the min-max value of $J_{\epsilon}$ and satisfies $0<c_{M}<\frac{1}{n}\left(1-\frac{1}{p}\right)^{n-1}\left(\frac{\alpha_{n}}{\alpha_{0}}\right)^{n-1}-\delta;$ (7.29) $(b)$ there exists a constant $\epsilon_{2}:0<\epsilon_{2}<\epsilon_{1}$ such that for any $\epsilon:0<\epsilon<\epsilon_{2}$, there exist positive constant $r_{\epsilon}$ with $r_{\epsilon}\rightarrow 0$ as $\epsilon\rightarrow 0$ and sequence $(u_{j})\subset E$ such that $J_{\epsilon}(u_{j})\rightarrow c_{\epsilon}=\inf_{\\|u\\|_{E}\leq r_{\epsilon}}J_{\epsilon}(u)<0,\quad J_{\epsilon}^{\prime}(u_{j})\rightarrow 0\quad{\rm in}\quad E^{}\quad{\rm as}\quad j\rightarrow\infty.$ In addition, $u_{j}\rightarrow u_{0}$ strongly in $E$, where $u_{0}$ is a weak solution of (2.8) with $J_{\epsilon}(u_{0})=c_{\epsilon}$. Thirdly, there exists $\epsilon_{0}:0<\epsilon_{0}<\epsilon_{2}$ such that if $0<\epsilon<\epsilon_{0}$, then $u_{M}\not\equiv u_{0}$. Suppose by contradiction that $u_{M}\equiv u_{0}$. Then $v_{j}\rightharpoonup u_{0}$ weakly in $E$. By $(a)$, $J_{\epsilon}(v_{j})\rightarrow c_{M}>0,\quad\|\langle J_{\epsilon}^{\prime}(v_{j}),\varphi\rangle\|\leq\gamma_{j}\\|\varphi\\|_{E}$ (7.30) with $\gamma_{j}\rightarrow 0$ as $j\rightarrow\infty$. On one hand we have by (7.28), $\int_{M}\phi(x)F(x,v_{j})dv_{g}\rightarrow\int_{M}\phi(x)F(x,u_{0})dv_{g}\quad{\rm as}\quad j\rightarrow\infty.$ (7.31) On the other hand, since $v_{j}\rightharpoonup u_{0}$ weakly in $E$ and $h\in E^{}$, it follows that $\int_{M}hv_{j}dv_{g}\rightarrow\int_{M}hu_{0}dv_{g}\quad{\rm as}\quad j\rightarrow\infty.$ (7.32) Inserting (7.31) and (7.32) into the first equality of (7.30), we obtain $\frac{1}{n}\\|v_{j}\\|_{E}^{n}=c_{M}+\int_{M}\phi(x)F(x,u_{0})dv_{g}+\epsilon\int_{M}hu_{0}dv_{g}+o_{j}(1).$ (7.33) In the same way, one can derive $\frac{1}{n}\\|u_{j}\\|_{E}^{n}=c_{\epsilon}+\int_{M}\phi(x)F(x,u_{0})dv_{g}+\epsilon\int_{M}hu_{0}dv_{g}+o_{j}(1).$ (7.34) Combining (7.33) and (7.34), we have $\\|v_{j}\\|_{E}^{n}-\\|u_{0}\\|_{E}^{n}=n\left(c_{M}-c_{\epsilon}+o_{j}(1)\right).$ (7.35) From $(b)$, we know that $c_{\epsilon}\rightarrow 0$ as $\epsilon\rightarrow 0$. This together with (7.29) leads to the existence of $\epsilon_{0}:0<\epsilon_{0}<\epsilon_{2}$ such that if $0<\epsilon<\epsilon_{0}$, then $0<c_{M}-c_{\epsilon}<\frac{1}{n}\left(\frac{p-1}{p}\frac{\alpha_{n}}{\alpha_{0}}\right)^{n-1}.$ (7.36) Write $w_{j}=\frac{v_{j}}{\\|v_{j}\\|_{E}},\quad w_{0}=\frac{u_{0}}{\left(\\|u_{0}\\|_{E}^{n}+n(c_{M}-c_{\epsilon})\right)^{1/n}}.$ It follows from (7.35) and $v_{j}\rightharpoonup u_{0}$ weakly in $E$ that $w_{j}\rightharpoonup w_{0}$ weakly in $E$. Note that $\int_{M}\phi(x)\zeta\left(n,\alpha_{0}\|v_{j}\|^{n/(n-1)}\right)dv_{g}=\int_{M}\phi(x)\zeta\left(n,\alpha_{0}\\|v_{j}\\|_{E}^{{n}/{(n-1)}}\|w_{j}\|^{n/(n-1)}\right)dv_{g}.$ By (7.35) and (7.36), a straightforward calculation shows $\lim_{j\rightarrow\infty}\alpha_{0}\\|v_{j}\\|_{E}^{\frac{n}{n-1}}\left(1-\\|w_{0}\\|_{E}^{n}\right)^{\frac{1}{n-1}}<\left(1-\frac{1}{p}\right)\alpha_{n}.$ Hence lemma 7.7 together with (2.3) implies that $\phi(x)\zeta\left(n,\alpha_{0}\|v_{j}\|^{n/(n-1)}\right)$ is bounded in $L^{q}(M)$ for some $q:1<q<n/(n-1)$. By $(f_{1})$, $\|f(x,v_{j})\|\leq b_{1}\|v_{j}\|^{n-1}+b_{2}\zeta(n,\alpha_{0}\|v_{j}\|^{\frac{n}{n-1}}).$ By the definition of $\zeta$ there exists a constant $c>0$ such that $\|f(x,v_{j})\chi_{\\{\|v_{j}\|\leq 1\\}}(x)\|\leq c\|v_{j}\|^{n-1},\quad\|f(x,v_{j})\chi_{\\{\|v_{j}\|>1\\}}(x)\|\leq c\zeta(n,\alpha_{0}\|v_{j}\|^{\frac{n}{n-1}}),$ where $\chi_{B}$ denotes the characteristic function of $B\subset M$. Hence $\displaystyle\left\|\int_{M}\phi(x)f(x,v_{j})(v_{j}-u_{0})dv_{g}\right\|$ $\displaystyle\leq$ $\displaystyle c\int_{M}\phi(x)\left(\|v_{j}\|^{n-1}+\zeta(n,\alpha_{0}\|v_{j}\|^{\frac{n}{n-1}})\right)\|v_{j}-u_{0}\|dv_{g}$ $\displaystyle\leq$ $\displaystyle c\left\\|\phi\|v_{j}\|^{n-1}\right\\|_{L^{\frac{n}{n-1}}(M)}\\|v_{j}-u_{0}\\|_{L^{n}(M)}$ $\displaystyle+c\left\\|\phi\zeta(n,\alpha_{0}\|v_{j}\|^{\frac{n}{n-1}})\right\\|_{L^{q}(M)}\\|v_{j}-u_{0}\\|_{L^{q^{\prime}}(M)}.$ Since $1<q<n/(n-1)$, we have $q^{\prime}>n$. Then it follows from the compact embedding $E\hookrightarrow L^{r}(M)$ for all $r\geq n$ that $\lim_{j\rightarrow\infty}\int_{M}\phi(x)f(x,v_{j})(v_{j}-u_{0})dv_{g}=0.$ (7.37) Taking $\varphi=v_{j}-u_{0}$ in (7.30), we have by using (7.32) and (7.37) that $\int_{M}\left(\|\nabla_{g}v_{j}\|^{n-2}\nabla_{g}v_{j}\nabla_{g}(v_{j}-u_{0})+v(x)\|v_{j}\|^{n-2}v_{j}(v_{j}-u_{0})\right)dv_{g}\rightarrow 0.$ (7.38) However the fact $v_{n}\rightharpoonup u_{0}$ weakly in $E$ leads to $\int_{M}\left(\|\nabla_{g}u_{0}\|^{n-2}\nabla_{g}u_{0}\nabla_{g}(v_{j}-u_{0})+v(x)\|u_{0}\|^{n-2}u_{0}(v_{j}-u_{0})\right)dv_{g}\rightarrow 0.$ (7.39) Subtracting (7.39) from (7.38), using the well known inequality (see [26], chapter 10) $2^{n-1}\|b-a\|^{n}\leq\langle\|b\|^{n-2}b-\|a\|^{n-2}a,b-a\rangle,\quad\forall a,b\in\mathbb{R}^{n},$ we have $\\|v_{j}-u_{0}\\|_{E}^{n}\rightarrow 0$ as $j\rightarrow\infty$. This together with (7.35) implies that $c_{M}=c_{\epsilon}$, which is absurd since $c_{M}>0$ and $c_{\epsilon}<0$. Therefore $u_{M}\not\equiv u_{0}$. Since $f(x,s)\equiv 0$ for all $(x,s)\in M\times(-\infty,0]$, one can easily see that $u_{M}\geq 0$ and $u_{0}\geq 0$. This completes the proof of the theorem. $\hfill\Box$ Finally we shall construct examples of $f$’s to show that $(f_{1})$-$(f_{5})$ do not imply $(H_{5})$. Proof of proposition 2.9. Let $\phi$ satisfies the hypotheses $(\phi_{1})$ and $(\phi_{2})$, $p>1$ be given in $(\phi_{1})$, $l$ be an integer satisfying $l\geq n$, $q=nl/(n-1)+1$ and $S_{q}$ be defined by (2.7). In view of lemma 7.2, $S_{q}$ is attained by some nonnegative function $u\in E$. Let $C_{q}$ be a positive number such that $C_{q}>\left(\frac{q-n}{q}\right)^{{(q-n)}/{n}}\left(\frac{p\alpha_{0}}{(p-1)\alpha_{n}}\right)^{(q-n)(n-1)/n}S_{q}^{q}.$ Let $\chi:[0,\infty)\rightarrow\mathbb{R}$ be a smooth function such that $0\leq\chi\leq 1$, $\chi\equiv 0$ on $[0,A]$, $\chi\equiv 1$ on $[2A,\infty)$, and $\|\chi\,^{\prime}\|\leq 2/A$, where $A$ is a positive constant to be determined later. We set $\displaystyle f(t)=\left\\{\begin{array}[]{lll}2^{l}l!C_{q}\sum_{k=l}^{\infty}\frac{(t^{\frac{n}{n-1}}-\chi(t)t^{\frac{1}{n-1}})^{k}}{k!},&t\geq 0\\\\[6.45831pt] 0,&t<0.\end{array}\right.$ Now we check $(f_{1})$-$(f_{5})$ for appropriate choice of $A$ as follows. $(f_{1})$: If $A>1$, then $0\leq t^{n/(n-1)}-\chi(t)t^{1/(n-1)}\leq t^{n/(n-1)}$ for all $t\geq 0$. Thus $\displaystyle f(t)$ $\displaystyle=$ $\displaystyle 2^{l}l!C_{q}\left(e^{t^{n/(n-1)}-\chi(t)t^{1/(n-1)}}-\sum_{k=0}^{l-1}\frac{(t^{\frac{n}{n-1}}-\chi(t)t^{\frac{1}{n-1}})^{k}}{k!}\right)$ $\displaystyle\leq$ $\displaystyle 2^{l}l!C_{q}\left(e^{t^{n/(n-1)}}-\sum_{k=0}^{l-1}\frac{t^{\frac{nk}{n-1}}}{k!}\right)$ $\displaystyle\leq$ $\displaystyle 2^{l}l!C_{q}\zeta(n,t^{n/(n-1)})$ for all $t\geq 0$. So $(f_{1})$ is satisfied when $A>1$. $(f_{2})$: When $t\in[0,A]$, we have $\chi(t)=0$ and $\int_{0}^{t}f(t)dt=2^{l}l!C_{q}\sum_{k=l}^{\infty}\int_{0}^{t}\frac{t^{\frac{nk}{n-1}}}{k!}dt\leq{2^{l}l!C_{q}t}\sum_{k=l}^{\infty}\frac{t^{\frac{nk}{n-1}}}{k!}=tf(t).$ (7.41) When $t\geq A$, we claim that if $A$ is chosen sufficiently large, say $A\geq 4^{n-1}$, then $\int_{A}^{t}\frac{(t^{\frac{n}{n-1}}-\chi(t)t^{\frac{1}{n-1}})^{k}}{k!}dt\leq\frac{(t^{\frac{n}{n-1}}-\chi(t)t^{\frac{1}{n-1}})^{k+1}}{(k+1)!}-\frac{A^{\frac{n(k+1)}{n-1}}}{(k+1)!}.$ (7.42) In fact, if we set $\gamma(t)=\int_{A}^{t}\frac{(t^{\frac{n}{n-1}}-\chi(t)t^{\frac{1}{n-1}})^{k}}{k!}dt-\frac{(t^{\frac{n}{n-1}}-\chi(t)t^{\frac{1}{n-1}})^{k+1}}{(k+1)!}+\frac{A^{\frac{n(k+1)}{n-1}}}{(k+1)!},$ then $\gamma(A)=0$ and $\gamma\,^{\prime}(t)=\frac{(t^{\frac{n}{n-1}}-\chi(t)t^{\frac{1}{n-1}})^{k}}{k!}-\frac{(t^{\frac{n}{n-1}}-\chi(t)t^{\frac{1}{n-1}})^{k}}{k!}\left(\frac{n}{n-1}t^{\frac{1}{n-1}}-\chi\,^{\prime}(t)t^{\frac{1}{n-1}}-\frac{1}{n-1}\chi(t)t^{\frac{1}{n-1}-1}\right).$ Let $A\geq 4^{n-1}$. Then for $t\in[A,\infty)$ there holds $\displaystyle\frac{n}{n-1}t^{\frac{1}{n-1}}-\chi\,^{\prime}(t)t^{\frac{1}{n-1}}-\frac{1}{n-1}\chi(t)t^{\frac{1}{n-1}-1}$ $\displaystyle\geq$ $\displaystyle\left(\frac{n}{n-1}-\frac{2}{A}\right)A^{\frac{1}{n-1}}-\frac{1}{n-1}A^{\frac{1}{n-1}-1}$ $\displaystyle\geq$ $\displaystyle 4\left(\frac{n}{n-1}-\frac{2}{4(n-1)}-\frac{1}{4(n-1)^{2}}\right)$ $\displaystyle>$ $\displaystyle 1.$ Hence $\gamma\,^{\prime}(t)\leq 0$ and thus our claim (7.42) holds. Note that $\int_{0}^{A}\frac{t^{\frac{nk}{n-1}}}{k!}dt=\frac{A^{\frac{n(k+1)}{n-1}}}{(k+1)!}\frac{k+1}{\frac{nk}{n-1}+1}A^{-\frac{1}{n-1}}\leq\frac{A^{\frac{n(k+1)}{n-1}}}{(k+1)!}.$ (7.43) It follows from (7.42) and (7.43) that when $t\geq A$, $\int_{0}^{t}\frac{(t^{\frac{n}{n-1}}-\chi(t)t^{\frac{1}{n-1}})^{k}}{k!}dt\leq\frac{(t^{\frac{n}{n-1}}-\chi(t)t^{\frac{1}{n-1}})^{k+1}}{(k+1)!},$ and whence $\int_{0}^{t}f(t)dt\leq f(t)\leq\frac{1}{\mu}tf(t)$ (7.44) for some $\mu>n$. This together with (7.41) implies that $(f_{2})$ holds. $(f_{3})$: Let $A\geq 4^{n-1}$. In view of (7.44), when $t\geq A$, $F(t)=\int_{0}^{t}f(t)dt\leq f(t).$ Hence $(f_{3})$ is satisfied. $(f_{4})$: Since $l>n$, we get $F(t)/t^{n}\rightarrow 0$ as $t\rightarrow 0+$. Hence $(f_{4})$ holds. $(f_{5})$: Note that $t^{n/(n-1)}-t^{1/(n-1)}\geq t^{n/(n-1)}/2$ for all $t\geq 2$. Let $A\geq 2$. Then for all $t\geq A$ there holds $f(t)\geq 2^{l}l!C_{q}\frac{(t^{\frac{n}{n-1}}-\chi(t)t^{\frac{1}{n-1}})^{l}}{l!}\geq 2^{l}C_{q}(t^{\frac{n}{n-1}}/2)^{l}=C_{q}t^{q-1}.$ When $t\in[0,A]$, we get $f(t)\geq 2^{l}l!C_{q}\frac{t^{\frac{nl}{n-1}}}{l!}=2^{l}C_{q}t^{q-1}.$ Hence $(f_{5})$ is satisfied. In short, $f(t)$ satisfies $(f_{1})$-$(f_{5})$ if $A\geq 4^{n-1}$. Finally we check that $(H_{5})$ does not hold. When $t\geq 2A$, we have $f(t)=2^{l}l!C_{q}\left(e^{t^{\frac{n}{n-1}}-t^{\frac{1}{n-1}}}-\sum_{k=0}^{l-1}\frac{(t^{\frac{n}{n-1}}-t^{\frac{1}{n-1}})^{k}}{k!}\right).$ It follows that $\lim_{t\rightarrow+\infty}tf(t)e^{-t^{\frac{n}{n-1}}}=0.$ Thus $f(t)$ does not satisfy $(H_{5})$. $\hfill\Box$ Acknowledgements. This work was partly supported by the NSFC 11171347 and the NCET program 2008-2011. ## References [1] S. Adachi, K. Tanaka, Trudinger type inequalities in $\mathbb{R}^{N}$ and their best exponents, Proc. Amer. Math. Soc. 128 (2000) 2051-2057. * [2] D. Adams, A sharp inequality of J. Moser for higher order derivatives, Ann. Math. 128 (1988) 385-398. * [3] Adimurthi, O. Druet, Blow-up analysis in dimension 2 and a sharp form of Trudinger-Moser inequality, Commun. Partial Differential Equation 29 (2004) 295-322. * [4] Adimurthi, Y. Yang, An interpolation of Hardy inequality and Trudinger-Moser inequality in $\mathbb{R}^{N}$ and its applications, Internat. Mathematics Research Notices 13 (2010) 2394-2426. * [5] T. Aubin, Sur la function exponentielle, C. R. Acad. Sci. Paris, Series A 270 (1970) 1514-1514. * [6] H. Brézis, E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc. 88 (1983) 486-490. * [7] S. Gallot, D. Hulin, J. Lafontaine, Riemannian geometry, Springer, 2004. * [8] D. Cao, Nontrivial solution of semilinear elliptic equations with critical exponent in $\mathbb{R}^{2}$, Commun. Partial Differential Equations 17 (1992) 407-435. * [9] P. Cherrier, Une inégalité de Sobolev sur les variétés Riemanniennes, Bull. Sc. Math. 103 (1979) 353-374. * [10] P. Cherrier, Cas d’exception du théorème d’inclusion de Sobolev sur les variétés Riemanniennes et applications, Bull. Sc. Math. 105 (1981) 235-288. * [11] W. Ding, J. Jost, J. Li, G. Wang, The differential equation $\Delta u=8\pi-8\pi he^{u}$ on a compact Riemann surface, Asian J. Math. 1 (1997) 230-248. * [12] J. M. do Ó, $N$-Laplacian equations in $\mathbb{R}^{N}$ with critical growth, Abstr. Appl. Anal. 2 (1997) 301-315. * [13] J. M. do Ó, E. Medeiros, U. Severo, On a quasilinear nonhomogeneous elliptic equation with critical growth in $\mathbb{R}^{N}$, J. Diff. Equations 246 (2009) 1363-1386. * [14] J. M. do Ó, M. de Souza, On a class of singular Trudinger-Moser type inequalities and its applications, Math. Nachr. 284 (2011) 1754-1776. * [15] L. Fontana, Sharp borderline Sobolev inequalities on compact Riemannian manifolds, Comm. Math. Helv. 68 (1993) 415- 454\. * [16] D. Gilbarg, N. Trudinger, Elliptic partial differential equations of second order, Springer, 2001. * [17] G. Hardy, J. Littlewood, G. Polya, Inequalities. Cambridge University Press, 1952. * [18] E. Hebey, Sobolev spaces on Riemannian maifolds, Lecture notes in mathematics 1635, Springer, 1996. * [19] H. Kozono, T. Sato, H. Wadade, Upper Bound of the best constant of the Trudinger-Moser inequality and its application to the Gagliardo-Nirenberg inequality, Indiana Univ. Math. J. 55 (2006) 1951-1974. * [20] N. Lam, G. Lu, Existence and multiplicity of solutions to equations of $N$-Laplacian type with critical exponential growth in $\mathbb{R}^{N}$, J. Funct. Anal. 2012 (In press), arXiv: 1107.0375v1. * [21] Y. Li, Moser-Trudinger inequality on compact Riemannian manifolds of dimension two, J. Partial Differential Equations 14 (2001) 163-192. * [22] Y. Li, The extremal functions for Moser-Trudinger inequality on compact Riemannian manifolds, Sci. China, Ser. A 48 (2005) 618-648. * [23] Y. Li, P. Liu, Moser-Trudinger inequality on the boundary of compact Riemannian surface, Math. Z. 250 (2005) 363-386 * [24] Y. Li, P. Liu, Y. Yang, Moser-Trudinger inequalities of vector bundle over a compact Riemannian manifold of dimension 2, Calc. Var. 28 (2007) 59-83. * [25] Y. Li, B. Ruf, A sharp Trudinger-Moser type inequality for unbounded domains in $\mathbb{R}^{N}$, Ind. Univ. Math. J. 57 (2008) 451-480. * [26] P. Lindqvist, Notes on the $p$-Laplace equation, Preprint. * [27] G. Lu, Y. Yang, Adams’ inequalities for bi-Laplacian and extremal functions in dimension four, Advances in Mathematics 220 (2009) 1135-1170. * [28] G. Lu, Y. Yang, The sharp constant and extremal functions for Moser-Trudinger inequalities involving $L^{p}$ norms, Discrete and continuous dynamical systems 25 (2009) 963-979. * [29] G. Lu, Y. Yang, A sharpened Moser-Pohozaev-Trudinger inequality with mean value zero in $\mathbb{R}^{2}$, Nonlinear Anal. 70 (2009) 2992-3001. * [30] J. Moser, A sharp form of an inequality by N. Trudinger, Ind. Univ. Math. J. 20 (1971) 1077-1091. * [31] R. O Neil, Convolution operators and $L(p,q)$ spaces, Duke Math. J. 30 (1963) 129-142. * [32] R. Panda, Nontrivial solution of a quasilinear elliptic equation with critical growth in $\mathbb{R}^{n}$, Proc. Indian Acad. Sci. (Math. Sci.) 105 (1995) 425-444. * [33] S. Pohozaev, The Sobolev embedding in the special case $pl=n$, Proceedings of the technical scientific conference on advances of scientific reseach 1964-1965, Mathematics sections, 158-170, Moscov. Energet. Inst., Moscow, 1965. * [34] P. H. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys. 43 (1992) 270-291. * [35] B. Ruf, A sharp Trudinger-Moser type inequality for unbounded domains in $\mathbb{R}^{2}$, J. Funct. Anal. 219 (2005) 340-367. * [36] B. Ruf, F. Sani, Sharp Adams-type inequalities in $\mathbb{R}^{n}$. Trans. Amer. Math. Soc. (In press). * [37] G. Trombetti, J. L. Vázquez, A Symmetrization Result for Elliptic Equations with Lower Order Terms, Ann. Fac. Sci. Toulouse Math. 7 (1985) 137-150. * [38] N. S. Trudinger, On embeddings into Orlicz spaces and some applications, J. Math. Mech. 17 (1967) 473-484. * [39] G. Wang, D. Ye, A Hardy-Moser-Trudinger inequality, arXiv: 1012.5591v1. * [40] Y. Yang, Extremal functions for a sharp Moser-Trudinger inequality. Internatinal J. Math. 17 (2006) 331-338. * [41] Y. Yang, A sharp form of Moser-Trudinger inequality on compact Riemannian surface, Trans. Amer. Math. Soc. 359 (2007) 5761-5776. * [42] Y. Yang, A sharp form of trace Moser-Trudinger inequality on compact Riemannian surface with boundary, Math. Z. 255 (2007) 373-392. * [43] Y. Yang, A sharp form of Moser-Trudinger inequality in high dimension, J. Funct. Anal. 239 (2006) 100-126. * [44] Y. Yang, Existence of positive solutions to quasi-linear elliptic equations with exponential growth in the whole Euclidean space, J. Funct. Anal. 2012 (In press), arXiv: 1106.4622v1. * [45] L. Zhao, A multiplicity result for a singular and nonhomogeneous elliptic problem in $\mathbb{R}^{n}$, J. Partial Differential equations (In press).	arxiv-papers	2011-12-04T06:17:05	2024-09-04T02:49:24.937014	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Yunyan Yang", "submitter": "Yunyan Yang", "url": "https://arxiv.org/abs/1112.0724" }
1112.0739	# On the UMD constants for a class of iterated $L_{p}(L_{q})$ spaces Yanqi QIU Inst. Math. Jussieu, Équipe d’Analyse Fonctionnelle Université Paris VI, 75252 Paris Cedex 05, France yanqi-qiu@math.jussieu.fr ###### Abstract. Let $1<p\neq q<\infty$ and $(D,\mu)=(\\{\pm 1\\},\frac{1}{2}\delta_{-1}+\frac{1}{2}\delta_{1})$. Define by recursion: $X_{0}=\mathbb{C}$ and $X_{n+1}=L_{p}(\mu;L_{q}(\mu;X_{n}))$. In this paper, we show that there exist $c_{1}=c_{1}(p,q)>1$ depending only on $p,q$ and $c_{2}=c_{2}(p,q,s)$ depending on $p,q,s$, such that the $\text{UMD}_{s}$ constants of $X_{n}$’s satisfy $c_{1}^{n}\leq C_{s}(X_{n})\leq c_{2}^{n}$ for all $1<s<\infty$. Similar results will be showed for the analytic UMD constants. We mention that the first super-reflexive non-UMD Banach lattices were constructed by Bourgain. Our results yield another elementary construction of super-reflexive non-UMD Banach lattices, i.e. the inductive limit of $X_{n}$, which can be viewed as iterating infinitely many times $L_{p}(L_{q})$. ###### Key words and phrases: UMD property, analytic UMD property, iterated $L_{p}(L_{q})$ spaces, super- reflexive non-UMD Banach lattices The author was partially supported by ANR grant 2011 BS01 008 01 ## 1\. Introduction A Banach space $X$ is UMD if for all (or equivalently, for some) $1<s<\infty$ there is a constant $C>0$ depending only on $s$ and $X$ such that (1) $\displaystyle\sup_{\varepsilon_{k}\in\\{-1,1\\}}\\|\sum_{k=0}^{n}\varepsilon_{k}df_{k}\\|_{L_{s}(X)}\leq C\\|\sum_{k=0}^{n}df_{k}\\|_{L_{s}(X)}$ for all $n\geq 0$ and all $X$-valued martingale difference sequences $(df_{k})_{k=0}^{n}$. The best such $C$ is called the $\text{UMD}_{s}$ constant of $X$ and will be denoted by $C_{s}(X)$ in the sequel. It is well- known that in the above definition, we can restrict to the dyadic martingale differences and the best constant remains the same. The UMD property for Banach spaces was introduced by Maurey and Pisier. The reader is refered to Burkholder’s papers [5, 7] for the details of the UMD property. Let $\mathbb{T}=\\{z\in\mathbb{C}:\|z\|=1\\}$ be the one dimensional torus equipped with the normalised Haar measure $m$. Consider the canonical filtration on the probability space $(\mathbb{T}^{\mathbb{N}},m^{\otimes\mathbb{N}})$ defined by $\sigma(z_{0})\subset\sigma(z_{0},z_{1})\subset\cdots\subset\sigma(z_{0},z_{1},\cdots,z_{n})\subset\cdots.$ By definition, a Hardy martingale in $L_{s}(\mathbb{T}^{\mathbb{N}};X)$ is a martingale $f=(f_{n})_{n\geq 0}$ with respect to the canonical filtration such that $\sup_{n}\\|f_{n}\\|_{L_{s}}<\infty$, and such that the martingale difference $df_{n}=f_{n}-f_{n-1}$ (by convention, $df_{0}:=f_{0}$) is analytic in the last variable $z_{n}$, i.e., $df_{n}$ has the form: $df_{n}(z_{0},\cdots,z_{n-1},z_{n})=\sum_{k\geq 1}\phi_{n,k}(z_{0},\cdots,z_{n-1})z_{n}^{k}.$ In the above definition of UMD spaces, if the Banach space is over the complex field $\mathbb{C}$, and if we restrict to the Hardy martingales, then a different class of Banach spaces is defined, i.e. the analytic UMD class (AUMD by abreviation). The best constant is called the $\text{AUMD}_{s}$ constant of $X$ and will be denoted by $C_{s}^{a}(X)$. Note that UMD implies AUMD but not conversely, for instance, $L_{1}(\mathbb{T},m)$ is an AUMD space which is not UMD (cf. [9]). It is well-known that UMD implies super-reflexivity but not conversely. The first super-reflexive non-UMD Banach space was constructed by Pisier in [11]. Super-reflexive non-UMD Banach lattices were later constructed by Bourgain in [2, 3]. We refer to Rubio de Francia’s paper [13] for some open problems related to the super-reflexive non-UMD Banach lattices. The main topic of this paper is the investigation of the UMD constants of a family of iterated $L_{p}(L_{q})$-spaces. As a consequence of our results, we give an elementary construction of super-reflexive non-UMD Banach lattices. ## 2\. Some elementary inequalities We will use the following lemma. ###### Lemma 2.1. Let $(\Omega,\nu)$ be a measure space such that $\nu$ is finite. Suppose that $\alpha\neq 1$ and $0<\alpha<\infty$. If $F,f\in L_{\alpha}(\Omega,\nu)\bigcap L_{1}(\Omega,\nu)$ satisfy $\int(\|F\|+\|g\|)^{\alpha}d\nu\leq\int(\|f\|+\|g\|)^{\alpha}d\nu$ for all $g\in L_{\infty}(\Omega,\nu)$. Then $\|F\|\leq\|f\|$ a.e.. ###### Proof. Consider first those $g\in L_{\infty}(\Omega,\nu)$ such that there exists $\delta>0$ and $\|g\|\geq\delta$ a.e.. If $F,f$ satisfy the condition in the statement, then for all $\varepsilon>0$, we have (2) $\displaystyle\int(\varepsilon\|F\|+\|g\|)^{\alpha}d\nu\leq\int(\varepsilon\|f\|+\|g\|)^{\alpha}d\nu.$ By the mean value theorem, there exists $\theta=\theta_{\varepsilon}\in(0,1)$, such that $\frac{(\varepsilon\|f\|+\|g\|)^{\alpha}-\|g\|^{\alpha}}{\varepsilon}=\alpha(\theta\varepsilon\|f\|+\|g\|)^{\alpha-1}\|f\|.$ If $\alpha<1$, then $(\theta\varepsilon\|f\|+\|g\|)^{\alpha-1}\|f\|\leq\|g\|^{\alpha-1}\|f\|\in L_{1}(\Omega,\nu)$ and if $\alpha>1$, then for $0<\varepsilon<1$, we have $0<\theta\varepsilon<1$ and hence $(\theta\varepsilon\|f\|+\|g\|)^{\alpha-1}\|f\|\leq 2^{\alpha-1}(\|f\|^{\alpha}+\|g\|^{\alpha-1}\|f\|)\in L_{1}(\Omega,\nu)$. By the dominated convergence theorem, we have $\lim_{\varepsilon\to 0^{+}}\frac{\int(\varepsilon\|f\|+\|g\|)^{\alpha}d\nu-\int\|g\|^{\alpha}d\nu}{\varepsilon}=\alpha\int\|f\|\|g\|^{\alpha-1}d\nu.$ The same equality holds for $F$. Combining this with (2), we get $\int\|F\|\|g\|^{\alpha-1}d\nu\leq\int\|f\|\|g\|^{\alpha-1}d\nu.$ Replacing $g$ by $\|g\|^{\frac{1}{\alpha-1}}$ yields $\int\|F\|\|g\|d\nu\leq\int\|f\|\|g\|d\nu.$ By approximation, the above inequality holds for all $g\in L_{\infty}(\Omega,\nu)$. Hence $\|F\|\leq\|f\|$ a.e., as announced. ∎ ###### Proposition 2.2. Let $(\Omega,\nu)$ be a measure space such that $\nu$ is finite. Suppose that $1\leq p\neq q<\infty$. If $F,f\in L_{p}(\Omega,\nu)\bigcap L_{q}(\Omega,\nu)$ satisfy $\int(\|F\|^{q}+\|g\|^{q})^{p/q}d\nu\leq\int(\|f\|^{q}+\|g\|^{q})^{p/q}d\nu$ for all $g\in L_{\infty}(\Omega,\nu)$. Then $\|F\|\leq\|f\|$ a.e.. ###### Proof. This is just a reformulation of Lemma 2.1. ∎ Let $D=\\{-1,1\\}$ be the Bernoulli probability space equipped with the measure $\mu=\frac{1}{2}\delta_{-1}+\frac{1}{2}\delta_{1}$. For any $1\leq q\leq\infty$, the 2-dimensional $\ell_{q}$-space will be denoted by $\ell_{q}^{(2)}$. ###### Proposition 2.3. Suppose that $1\leq p\neq q\leq\infty$. Let $P$ be the projection on $L_{p}(\mu;\ell_{q}^{(2)})$ defined by $\begin{array}[]{cccc}P:&L_{p}(\mu;\ell_{q}^{(2)})&\rightarrow&L_{p}(\mu;\ell_{q}^{(2)})\\\ &(f,g)&\mapsto&(\mathbb{E}f,g)\end{array},$ where $\mathbb{E}$ is the expectation. Then $P$ is not contractive. ###### Proof. Assume first that both $p,q$ are finite. If $P$ is contractive, then for any two functions $f$ and $g$, we have $\int(\|\mathbb{E}f\|^{q}+\|g\|^{q})^{p/q}d\mu\leq\int(\|f\|^{q}+\|g\|^{q})^{p/q}d\mu.$ By Proposition 2.2, it follows that $\|\mathbb{E}(f)\|\leq\|f\|$, which is a contradiction, hence $P$ is not contractive. If $p=\infty$ and $1<q<\infty$, then $p^{\prime}=1$ and $1<q^{\prime}<\infty$. Since the adjoint map $P^{}$ on $L_{1}(\mu;\ell_{q^{\prime}}^{(2)})$ has the same form as $P$, the preceding argument shows that $P^{}$ and hence $P$ is not contractive. If $p=\infty$ and $q=1$. Assume $P$ is contractive, then we have (3) $\displaystyle\big{\\|}\|\mathbb{E}f\|+\|g\|\big{\\|}_{\infty}\leq\big{\\|}\|f\|+\|g\|\big{\\|}_{\infty}.$ Consider $f=1+\varepsilon,g=1-\varepsilon$, where $\varepsilon:D\rightarrow D$ is the identity function. Then the left hand side of (3) equals to 3 while the right hand side equals to 2. This contradiction shows that $P$ is not contractive. If $1\leq p<\infty$ and $q=\infty$, then $1<p^{\prime}\leq\infty$ and $q^{\prime}=1$, hence $P^{}$ is not contractive. It follows that $P$ is not contractive. ∎ The norm of $P$ on $L_{p}(\mu;\ell_{q}^{(2)})$ will be denoted by $c(p,q)$ in the sequel. If $p=q$, then $c(p,p)=1$. If $1\leq p\neq q\leq\infty$, then (4) $\displaystyle c(p,q)>1.$ ###### Remark 2.4. It is not difficult to check that $c(\infty,1)=c(1,\infty)=\frac{3}{2}$. But we do not know the exact value of $c(p,q)$ for general $p\neq q$. As usual, we set $H_{p}(\mathbb{T})=\\{f\in L_{p}(\mathbb{T},m):\hat{f}(k)=0,\forall k\in\mathbb{Z}_{<0}\\}.$ We will say that a measurable function $f:\mathbb{T}\rightarrow\mathbb{C}$ is bounded from below, if there exists $\delta>0$, such that $\|f\|\geq\delta$ a.e. on $\mathbb{T}$. If $f\in L_{p}(\mathbb{T})$ is bounded from below, then the geometric mean $M(\|f\|)$ of $\|f\|$ is defined by $\log M(\|f\|)=\int_{\mathbb{T}}\log\|f(z)\|dm(z).$ In particular, if $f:\mathbb{D}\rightarrow\mathbb{C}$ is an outer function, then (5) $\displaystyle M(\|f\|)=\|f(0)\|=\|\mathbb{E}f\|.$ The following elementary proposition will be used in §4 when we treat the analytic UMD property. ###### Proposition 2.5. Suppose that $1\leq p\neq q<\infty$. Define $\kappa(p,q)$ to be the best constant $C$ satisfying the property: For any measurable partition $\mathbb{T}=A\dot{\cup}B$ with $m(A)=m(B)=\frac{1}{2}$, for any function $f=f_{1}\chi_{A}+f_{2}\chi_{B}$ with $f_{1}>0,f_{2}>0$ and any function $g=g_{1}\chi_{A}+g_{2}\chi_{B}$, we have $\int_{\mathbb{T}}(M(\|f\|)^{q}+\|g\|^{q})^{p/q}dm\leq C^{p}\int_{\mathbb{T}}(\|f\|^{q}+\|g\|^{q})^{p/q}dm.$ Then $\kappa(p,q)>1$. ###### Proof. Assume $k(p,q)\leq 1$. Fix any measurable partition $\mathbb{T}=A\dot{\cup}B$ such that $m(A)=m(B)=\frac{1}{2}$. Consider the 2-valued functions $f=f_{1}\chi_{A}+f_{2}\chi_{B}$ and $g=g_{1}\chi_{A}+g_{2}\chi_{B}$ with $f_{1},f_{2}$ positive scalars. By Proposition 2.2, $M(f)\leq f$. However, one can easily check that $M(f)=f_{1}^{1/2}f_{2}^{1/2}$. If $f_{1}>f_{2}$, then $M(f)>f_{2}^{1/2}f_{2}^{1/2}=f_{2}$, which contradicts to $M(f)\leq f$. Whence the announced statement. ∎ ## 3\. UMD constants of iterated $L_{p}(L_{q})$ spaces The following definition is essential in the sequel. ###### Definition 3.1. Consider a Banach space $X$ with a fixed family of vectors $\\{x_{i}\\}_{i\in I}$. We define $S(X;\\{x_{i}\\})$ to be the best constant $C$ such that (6) $\displaystyle\Big{\\|}\sum_{k=0}^{N}\mathbb{E}^{\mathcal{A}_{k}}(\theta_{k})x_{i_{k}}\Big{\\|}_{L_{1}(\Omega,\mathbb{P};X)}\leq C\Big{\\|}\sum_{k=0}^{N}\theta_{k}x_{i_{k}}\Big{\\|}_{L_{\infty}(\Omega,\mathbb{P};X)}$ holds for any $N\in\mathbb{N}$, any probability space $(\Omega,\mathcal{F},\mathbb{P})$ equipped with a filtration $\mathcal{A}_{0}\subset\mathcal{A}_{1}\subset\cdots\subset\mathcal{A}_{n}\subset\cdots\subset\mathcal{F}$, any $N+1$ distinct indices $\\{i_{0},i_{1},\cdots,i_{N}\\}\subset I$ and any $N+1$ functions $\theta_{0},\theta_{1},\cdots,\theta_{N}$ in $L_{\infty}(\Omega,\mathcal{F},\mathbb{P})$. If there does not exist such constant, we set $S(X;\\{x_{i}\\})=\infty$. In what follows, we are mostly interested in the special case when $\\{x_{i}\\}$ is a 1-unconditional basic sequence, since in this case we can relate $S(X;\\{x_{i}\\})$ to the UMD constants of $X$. If $\\{x_{i}\\}$ is clear from the context and there is no confusion, we will use the simplified notation $S(X)$ for $S(X;\\{x_{i}\\})$. In particular, if $X$ has a natural basis, then $S(X)$ will always mean to be calculated with this basis. We will need the following well-known Stein inequality in UMD spaces, which was originally proved by Bourgain [4]. For the sake of completeness, we include the proof. ###### Theorem 3.2. Let $X$ be a UMD space. Then for any $1<s<\infty$, any finite sequences of functions $(F_{k})_{k\geq 0}$ in $L_{s}(\Omega,\mathbb{P};X)$ and any filtration $\mathcal{A}_{0}\subset\mathcal{A}_{1}\subset\cdots\subset\mathcal{A}_{n}\subset\cdots$ on $(\Omega,\mathbb{P})$, we have (7) $\displaystyle\Big{\\|}\sum_{k}\varepsilon_{k}\mathbb{E}_{k}(F_{k})\Big{\\|}_{L_{s}(\mu_{\infty}\times\mathbb{P};X)}\leq C_{s}(X)\Big{\\|}\sum_{k}\varepsilon_{k}F_{k}\Big{\\|}_{L_{s}(\mu_{\infty}\times\mathbb{P};X)},$ where $\mathbb{E}_{k}=\mathbb{E}^{\mathcal{A}_{k}}$ and $(\varepsilon_{k})_{k\geq 0}$ is the usual Rademacher sequence on $(D^{\mathbb{N}},\mu_{\infty})$, $\mu_{\infty}=\mu^{\otimes\mathbb{N}}$. ###### Proof. Let $f=\sum_{k}\varepsilon_{k}F_{k}$ and $f^{\prime}=\sum_{k}\varepsilon_{k}\mathbb{E}_{k}(F_{k})$. Then if $\mathcal{C}_{2j}=\mathcal{A}_{j}\otimes\sigma(\varepsilon_{0},\cdots,\varepsilon_{j})$ and $\mathcal{C}_{2j-1}=\mathcal{A}_{j}\otimes\sigma(\varepsilon_{0},\cdots,\varepsilon_{j-1})$, we have $f^{\prime}=\sum_{j}(\mathbb{E}^{\mathcal{C}_{2j}}-\mathbb{E}^{\mathcal{C}_{2j-1}})(f).$ Indeed, $\mathbb{E}^{\mathcal{C}_{2j}}(f)=\sum_{0}^{j}\varepsilon_{k}\mathbb{E}_{j}(F_{k})$ and $\mathbb{E}^{\mathcal{C}_{2j-1}}(f)=\sum_{0}^{j-1}\varepsilon_{k}\mathbb{E}_{j}(F_{k})$. Hence $(\mathbb{E}^{\mathcal{C}_{2j}}-\mathbb{E}^{\mathcal{C}_{2j-1}})(f)=\varepsilon_{j}\mathbb{E}_{j}(F_{j})$. It follows (see the next remark) that $\\|f^{\prime}\\|_{L_{s}(\mu_{\infty}\times\mathbb{P};X)}\leq C_{s}(X)\\|f\\|_{L_{s}(\mu_{\infty}\times\mathbb{P};X)},$ whence (7). ∎ ###### Remark 3.3. By an extreme point argument, we have $\sup_{-1\leq\alpha_{k}\leq 1}\\|\sum_{k=0}^{n}\alpha_{k}df_{k}\\|_{L_{s}(X)}=\sup_{\varepsilon_{k}\in\\{-1,1\\}}\\|\sum_{k=0}^{n}\varepsilon_{k}df_{k}\\|_{L_{s}(X)}.$ Hence we have $\sup_{-1\leq\alpha_{k}\leq 1}\\|\sum_{k=0}^{n}\alpha_{k}df_{k}\\|_{L_{s}(X)}\leq C_{s}(X)\\|\sum_{k=0}^{n}df_{k}\\|_{L_{s}(X)}.$ ###### Proposition 3.4. Let $X$ be a UMD space. Assume that $\\{x_{i}\\}_{i\in I}$ is a 1-unconditional basic sequence in $X$. Then for any $1<s<\infty$, any finite sequence of functions $(\theta_{k})_{k\geq 0}$ in $L_{s}(\Omega,\mathbb{P})$ and any filtration $\mathcal{A}_{0}\subset\mathcal{A}_{1}\subset\cdots\subset\mathcal{A}_{n}\subset\cdots$ on $(\Omega,\mathbb{P})$, we have (8) $\displaystyle\Big{\\|}\sum_{k}\mathbb{E}_{k}(\theta_{k})x_{i_{k}}\Big{\\|}_{L_{s}(\Omega,\mathbb{P};X)}\leq C_{s}(X)\Big{\\|}\sum_{k}\theta_{k}x_{i_{k}}\Big{\\|}_{L_{s}(\Omega,\mathbb{P};X)}.$ ###### Proof. For any $i_{k}$’s, consider the sequence $(F_{k})_{k\geq 0}$ in $L_{s}(\Omega,\mathbb{P};X)$ defined by $F_{k}(w)=\theta_{k}(w)x_{i_{k}}$. Then $\mathbb{E}_{k}(F_{k})=\mathbb{E}_{k}(\theta_{k})x_{i_{k}}$. By the 1-unconditionality of $\\{x_{i}\\}_{i\in I}$, for any fixed choice of signs $\varepsilon_{k}\in\\{-1,1\\}$ and $w\in\Omega$, we have $\Big{\\|}\sum_{k}\varepsilon_{k}F_{k}(w)\Big{\\|}_{X}=\Big{\\|}\sum_{k}\varepsilon_{k}\theta_{k}(w)x_{i_{k}}\Big{\\|}_{X}=\Big{\\|}\sum_{k}\theta_{k}(w)x_{i_{k}}\Big{\\|}_{X}.$ It follows that $\Big{\\|}\sum_{k}\varepsilon_{k}F_{k}\Big{\\|}_{L_{s}(\mu_{\infty}\times\mathbb{P};X)}=\Big{\\|}\sum_{k}\theta_{k}x_{i_{k}}\Big{\\|}_{L_{s}(\Omega,\mathbb{P};X)}.$ Similarly, we have $\Big{\\|}\sum_{k}\varepsilon_{k}\mathbb{E}_{k}(F_{k})\Big{\\|}_{L_{s}(\mu_{\infty}\times\mathbb{P};X)}=\Big{\\|}\sum_{k}\mathbb{E}_{k}(\theta_{k})x_{i_{k}}\Big{\\|}_{L_{s}(\Omega,\mathbb{P};X)}.$ By these equalities, (8) follows from (7). ∎ Let $X$ be as in Proposition 3.4, $\\{x_{i}\\}_{i\in I}$ is a 1-unconditional basic sequence in $X$. Assume that $\theta_{k}\in L_{\infty}(\Omega,\mathbb{P})$. By an application of the contractive inclusions $L_{\infty}(\Omega,\mathbb{P};X)\subset L_{s}(\Omega,\mathbb{P};X)\subset L_{1}(\Omega,\mathbb{P};X)$, we have (9) $\displaystyle\Big{\\|}\sum_{k}\mathbb{E}_{k}(\theta_{k})x_{i_{k}}\Big{\\|}_{L_{1}(\Omega,\mathbb{P};X)}\leq C_{s}(X)\Big{\\|}\sum_{k}\theta_{k}x_{i_{k}}\Big{\\|}_{L_{\infty}(\Omega,\mathbb{P};X)}.$ Hence (10) $\displaystyle S(X;\\{x_{i}\\})\leq C_{s}(X)$ for all $1<s<\infty$. ###### Theorem 3.5. Let $E$ be a Banach space with a 1-unconditional basis $\\{e_{i}:i\in I\\}$, let $F$ be another Banach space. By definition, $E(F)$ is the completion of the algebraic tensor product $E\otimes F$ under the norm defined as follows: if $x=\sum_{i}e_{i}\otimes x_{i}\in E\otimes F$, where $(x_{i})$ is a finite supported sequence in $F$, then $\\|x\\|_{E(F)}:=\Big{\\|}\sum_{i}e_{i}\big{\\|}x_{i}\big{\\|}_{F}\Big{\\|}_{E}.$ For any fixed family of vectors $\\{f_{j}:j\in J\\}$ in $F$, consider the family of vectors $\\{e_{i}\otimes f_{j}:i\in I,j\in J\\}$. Then we have $S(E(F))\geq S(E)S(F),$ where $S(E(F))$, $S(E)$ and $S(F)$ are defined with respect to the mentioned families of vectors respectively. ###### Proof. From the definition, for any $\varepsilon>0$, there exist finite number of distinct indices $\\{i_{k}:1\leq k\leq N_{1}\\}\subset I$ and $\\{j_{n}:1\leq n\leq N_{2}\\}\subset J$, and there exist functions $\theta_{k}\in L_{\infty}(\Omega^{\prime},\mathbb{P}^{\prime}),1\leq k\leq N_{1}$ and functions $\xi_{n}\in L_{\infty}(\Omega_{0},\mathbb{P}_{0}),1\leq n\leq N_{2}$ satisfying $\\|\sum_{k}\theta_{k}e_{i_{k}}\\|_{L_{\infty}(\Omega^{\prime},\mathbb{P}^{\prime};E)}\leq 1$ and $\\|\sum_{n}\xi_{n}f_{j_{n}}\\|_{L_{\infty}(\Omega_{0},\mathbb{P}_{0};F)}\leq 1$ such that $\Big{\\|}\sum_{k}\mathbb{E}^{\mathcal{A}_{k}}(\theta_{k})e_{i_{k}}\Big{\\|}_{L_{1}(\Omega^{\prime},\mathbb{P}^{\prime};E)}\geq S(E)-\varepsilon$ and $\Big{\\|}\sum_{n}\mathbb{E}^{\mathcal{B}_{n}}(\xi_{n})f_{j_{n}}\Big{\\|}_{L_{1}(\Omega_{0},\mathbb{P}_{0};F)}\geq S(F)-\varepsilon.$ Let $(\Omega,\mathbb{P})=(\Omega^{\prime}\times\Omega_{0}^{\mathbb{N}},\mathbb{P}^{\prime}\otimes\mathbb{P}_{0}^{\otimes\mathbb{N}})$, the general element in $\Omega$ will be denoted by $w=(w^{\prime},(w_{l})_{l\geq 0})$. Consider the $\sigma$-algebras $\mathcal{F}_{k,n}$ defined on $(\Omega,\mathbb{P})$ by $\mathcal{F}_{k,n}:=\mathcal{A}_{k}\otimes\underbrace{\mathcal{B}_{\infty}\otimes\cdots\otimes\mathcal{B}_{\infty}}_{k-1\text{ times }}\otimes\mathcal{B}_{n}\otimes\mathcal{C}_{\geq k+1},$ where $\mathcal{B}_{\infty}=\sigma(\mathcal{B}_{n}:n\geq 0)$ is a $\sigma$-algebra on $(\Omega_{0},\mathbb{P}_{0})$, $\mathcal{B}_{0}$ is assumed to be trivial and $\mathcal{C}_{\geq k+1}$ is the trivial $\sigma$-algebra on $(\Omega_{0}^{\mathbb{N}_{\geq{k+1}}},\mathbb{P}_{0}^{\mathbb{N}_{\geq k+1}})$. It is easy to check that $\mathcal{F}_{k,n}$ is a filtration with respect to the lexigraphic order, i.e. if $(k,n)<(k^{\prime},n^{\prime})$ (that is $k<k^{\prime}$ or $k=k^{\prime}$ but $n<n^{\prime}$), then $\mathcal{F}_{k,n}\subset\mathcal{F}_{k^{\prime},n^{\prime}}$. Now let us define $h:\Omega\rightarrow E(F)$ by $h(w)=h(w^{\prime},(w_{l}))=\sum_{k,n}\theta_{k}(w^{\prime})\xi_{n}(w_{k})e_{i_{k}}\otimes f_{j_{n}}.$ Let $h_{k,n}(w)=\theta_{k}(w^{\prime})\xi_{n}(w_{k})$, then $h=\sum_{k,n}h_{k,n}e_{i_{k}}\otimes f_{j_{n}}$. Clearly, we have (11) $\displaystyle\mathbb{E}^{\mathcal{F}_{k,n}}(h_{k,n})(w)=\Big{[}\mathbb{E}^{\mathcal{A}_{k}}(\theta_{k})\Big{]}(w^{\prime})\Big{[}\mathbb{E}^{\mathcal{B}_{n}}(\xi_{n})\Big{]}(w_{k})\quad a.e..$ By the 1-unconditionality of $\\{e_{i}:i\in I\\}$, for a.e. $w\in\Omega$, we have $\displaystyle\\|h(w)\\|_{E(F)}$ $\displaystyle=$ $\displaystyle\Big{\\|}\sum_{k,n}\theta_{k}(w^{\prime})\xi_{n}(w_{k})e_{i_{k}}\otimes f_{j_{n}}\Big{\\|}_{E(F)}$ $\displaystyle=$ $\displaystyle\Big{\\|}\sum_{k}e_{i_{k}}\big{\\|}\sum_{n}\theta_{k}(w^{\prime})\xi_{n}(w_{k})f_{j_{n}}\big{\\|}_{F}\Big{\\|}_{E}$ $\displaystyle=$ $\displaystyle\Big{\\|}\sum_{k}e_{i_{k}}\|\theta_{k}(w^{\prime})\|\big{\\|}\sum_{n}\xi_{n}(w_{k})f_{j_{n}}\big{\\|}_{F}\Big{\\|}_{E}$ $\displaystyle\leq$ $\displaystyle\Big{\\|}\sum_{k}e_{i_{k}}\|\theta_{k}(w^{\prime})\|\Big{\\|}_{E}=\Big{\\|}\sum_{k}e_{i_{k}}\theta_{k}(w^{\prime})\Big{\\|}_{E}\leq 1.$ Hence $\\|h\\|_{L_{\infty}(\Omega,\mathbb{P};E(F))}\leq 1$. If we denote $\widetilde{h}=\sum_{k,n}\mathbb{E}^{\mathcal{F}_{k,n}}(h_{k,n})e_{i_{k}}\otimes f_{j_{n}},$ then by (11), $\displaystyle\\|\widetilde{h}(w)\\|_{E(F)}$ $\displaystyle=$ $\displaystyle\Big{\\|}\sum_{k}e_{i_{k}}\|\mathbb{E}^{\mathcal{A}_{k}}(\theta_{k})(w^{\prime})\|\big{\\|}\sum_{n}\mathbb{E}^{\mathcal{B}_{n}}(\xi_{n})(w_{k})f_{j_{n}}\big{\\|}_{F}\Big{\\|}_{E}.$ By Jensen’s inequality, we have $\displaystyle\int\Big{\\|}\sum_{k}e_{i_{k}}\|\mathbb{E}^{\mathcal{A}_{k}}(\theta_{k})(w^{\prime})\|\big{\\|}\sum_{n}\mathbb{E}^{\mathcal{B}_{n}}(\xi_{n})(w_{k})f_{j_{n}}\big{\\|}_{F}\Big{\\|}_{E}d\mathbb{P}_{0}^{\otimes\mathbb{N}}((w_{l}))$ $\displaystyle\geq$ $\displaystyle\Big{\\|}\int\sum_{k}e_{i_{k}}\|\mathbb{E}^{\mathcal{A}_{k}}(\theta_{k})(w^{\prime})\|\big{\\|}\sum_{n}\mathbb{E}^{\mathcal{B}_{n}}(\xi_{n})(w_{k})f_{j_{n}}\big{\\|}_{F}d\mathbb{P}_{0}^{\otimes\mathbb{N}}((w_{l}))\Big{\\|}_{E}$ $\displaystyle=$ $\displaystyle\Big{\\|}\sum_{k}e_{i_{k}}\|\mathbb{E}^{\mathcal{A}_{k}}(\theta_{k})(w^{\prime})\|\Big{\\|}_{E}\cdot\Big{\\|}\sum_{n}\mathbb{E}^{\mathcal{B}_{n}}(\xi_{n})f_{j_{n}}\Big{\\|}_{L_{1}(\Omega_{0},\mathbb{P}_{0};F)}$ $\displaystyle=$ $\displaystyle\Big{\\|}\sum_{k}e_{i_{k}}\mathbb{E}^{\mathcal{A}_{k}}(\theta_{k})(w^{\prime})\Big{\\|}_{E}\cdot\Big{\\|}\sum_{n}\mathbb{E}^{\mathcal{B}_{n}}(\xi_{n})f_{j_{n}}\Big{\\|}_{L_{1}(\Omega_{0},\mathbb{P}_{0};F)}.$ Note that in the last equality, we used the 1-unconditionality assumption on $\\{e_{i}:i\in I\\}$. By integrating both sides with respect to $\int d\mathbb{P}^{\prime}(w^{\prime})$, we get $\displaystyle\Big{\\|}\sum_{k,n}\mathbb{E}^{\mathcal{F}_{k,n}}(h_{k,n})e_{i_{k}}\otimes f_{j_{n}}\Big{\\|}_{L_{1}(\Omega,\mathbb{P};E(F))}$ $\displaystyle\geq$ $\displaystyle\Big{\\|}\sum_{k}\mathbb{E}^{\mathcal{A}_{k}}(\theta_{k})e_{i_{k}}\Big{\\|}_{L_{1}(\Omega^{\prime},\mathbb{P}^{\prime};E)}\cdot\Big{\\|}\sum_{n}\mathbb{E}^{\mathcal{B}_{n}}(\xi_{n})f_{j_{n}}\Big{\\|}_{L_{1}(\Omega_{0},\mathbb{P}_{0};F)}$ $\displaystyle\geq$ $\displaystyle(S(E)-\varepsilon)(S(F)-\varepsilon).$ Therefore $S(E(F))\geq(S(E)-\varepsilon)(S(F)-\varepsilon)$. Since $\varepsilon>0$ is arbitrary, it follows that $S(E(F))\geq S(E)S(F)$ as desired. ∎ ###### Remark 3.6. If $E$ is a Banach lattice which is $p$-convex and $q$-concave $($see [10] for the details$)$ with $1\leq p\leq q\leq\infty$ and $F$ is a Banach space. Then the preceding proof is valid with $S_{q,p}(E)$ and $S_{q,p}(F)$ defined using (6) with $L_{p}$-norm on the left hand side and $L_{q}$-norm on the right hand side. ###### Remark 3.7. Let $1\leq p<q\leq\infty$. If we define $C_{q,p}(X)$ as the best constant $C$ in (1) with $L_{p}$-norm on the left hand side and $L_{q}$-norm on the right hand side, it is well-known that $X$ is in the UMD class if and only if $C_{q,p}(X)<\infty$. The preceding argument shows that under the same assumption of Theorem 3.5, we have $C_{\infty,1}(E(F))\geq S(E)C_{\infty,1}(F)$. Moreover, if $E$ is $p$-convex and $q$-concave we have $C_{q,p}(E(F))\geq S_{q,p}(E)C_{q,p}(F)$. ###### Lemma 3.8. Suppose that $1\leq p\neq q\leq\infty$. If $E_{1}=\ell_{p}^{(2)}(\ell_{q}^{(2)})$, then $S(E_{1})\geq c(p,q)>1.$ ###### Proof. Denote by $\\{e_{1}^{p},e_{2}^{p}\\}$, $\\{e_{1}^{q},e_{2}^{q}\\}$ the canonical basis of $\ell_{p}^{(2)}$ and $\ell_{q}^{(2)}$ respectively .Then $\\{e_{1}^{p}\otimes e_{1}^{q},e_{1}^{p}\otimes e_{2}^{q},e_{2}^{p}\otimes e_{1}^{q},e_{2}^{p}\otimes e_{2}^{q}\\}$ is the canonical 1-unconditional basis of $\ell_{p}^{(2)}(\ell_{q}^{(2)})$. Consider the probability space $(D,\mu)$ equipped with the filtration $\\{\phi,D\\}\subset\sigma(\varepsilon)$, where $\varepsilon$ is the identity function on $D$. Define a linear map $T:L_{\infty}(D;E_{1})\rightarrow L_{1}(D;E_{1})$ by setting $T\Big{[}a_{ij}(\varepsilon)e_{i}^{p}\otimes e_{j}^{q}\Big{]}=\left\\{\begin{array}[]{lc}\mathbb{E}(a_{ij})e_{i}^{p}\otimes e_{j}^{q},\text{ if }j=1\\\ a_{ij}(\varepsilon)e_{i}^{p}\otimes e_{j}^{q},\text{ if }j=2\end{array}.\right.$ By definition of $S(E_{1})$ we have $S(E_{1})\geq\\|T\\|_{L_{\infty}(D;E_{1})\rightarrow L_{1}(D;E_{1})}$. Now for any $a,b$ two scalar functions on $D$ , consider $f(\varepsilon)=e_{1}^{p}\otimes\Big{[}a(\varepsilon)e_{1}^{q}+b(\varepsilon)e_{2}^{q}\Big{]}+e_{2}^{p}\otimes\Big{[}a(-\varepsilon)e_{1}^{q}+b(-\varepsilon)e_{2}^{q}\Big{]}.$ Then $(Tf)(\varepsilon)=e_{1}^{p}\otimes\Big{[}\mathbb{E}(a)e_{1}^{q}+b(\varepsilon)e_{2}^{q}\Big{]}+e_{2}^{p}\otimes\Big{[}\mathbb{E}(a)e_{1}^{q}+b(-\varepsilon)e_{2}^{q}\Big{]}.$ If $p,q$ are both finite, then for any fixed $\varepsilon\in D$, we have $\displaystyle\\|f(\varepsilon)\\|_{E_{1}}\\!\\!\\!$ $\displaystyle=$ $\displaystyle\\!\\!\Big{\\{}(\|a(\varepsilon)\|^{q}+\|b(\varepsilon)\|^{q})^{p/q}+(\|a(-\varepsilon)\|^{q}+\|b(-\varepsilon)\|^{q})^{p/q}\Big{\\}}^{1/p}$ $\displaystyle=$ $\displaystyle\\!\\!\Big{\\{}(\|a(1)\|^{q}+\|b(1)\|^{q})^{p/q}+(\|a(-1)\|^{q}+\|b(-1)\|^{q})^{p/q}\Big{\\}}^{1/p}$ $\displaystyle=$ $\displaystyle\\!\\!2^{1/p}\Big{\\{}\frac{1}{2}(\|a(1)\|^{q}+\|b(1)\|^{q})^{p/q}+\frac{1}{2}(\|a(-1)\|^{q}+\|b(-1)\|^{q})^{p/q}\Big{\\}}^{1/p}$ $\displaystyle=$ $\displaystyle\\!\\!2^{1/p}\Big{\\{}\int(\|a(\varepsilon)\|^{q}+\|b(\varepsilon)\|^{q})^{p/q}d\mu(\varepsilon)\Big{\\}}^{1/p}$ $\displaystyle=$ $\displaystyle\\!\\!2^{1/p}\big{\\|}(a,b)\big{\\|}_{L_{p}(\mu;\ell_{q}^{(2)})}.$ Similarly, $\\|(Tf)(\varepsilon)\\|_{E_{1}}=2^{1/p}\big{\\|}(\mathbb{E}a,b)\big{\\|}_{L_{p}(\mu;\ell_{q}^{(2)})}.$ It follows that $\\|f\\|_{L_{\infty}(D;E_{1})}=2^{1/p}\big{\\|}(a,b)\big{\\|}_{L_{p}(\mu;\ell_{q}^{(2)})}$ and $\\|Tf\\|_{L_{1}(D;E_{1})}=2^{1/p}\big{\\|}(\mathbb{E}a,b)\big{\\|}_{L_{p}(\mu;\ell_{q}^{(2)})}.$ Hence (12) $\displaystyle\\|T\\|_{L_{\infty}(D;E_{1})\rightarrow L_{1}(D;E_{1})}\geq\frac{\\|Tf\\|_{L_{1}(D;E_{1})}}{\\|f\\|_{L_{\infty}(D;E_{1})}}=\frac{\big{\\|}(\mathbb{E}a,b)\big{\\|}_{L_{p}(\mu;\ell_{q}^{(2)})}}{\big{\\|}(a,b)\big{\\|}_{L_{p}(\mu;\ell_{q}^{(2)})}}.$ Similarly, if $q=\infty$ and $p$ is finite, then $\\|f\\|_{L_{\infty}(D;E_{1})}=2^{1/p}\\|(a,b)\\|_{L_{p}(\mu;\ell_{\infty}^{(2)})}$ and $\\|Tf\\|_{L_{1}(D;E_{1})}=2^{1/p}\big{\\|}(\mathbb{E}a,b)\big{\\|}_{L_{p}(\mu;\ell_{\infty}^{(2)})}.$ If $p=\infty$ and $q$ is finite, then $\\|f\\|_{L_{\infty}(D;E_{1})}=\\|(a,b)\\|_{L_{\infty}(\mu;\ell_{q}^{(2)})}$ and $\\|Tf\\|_{L_{1}(D;E_{1})}=\big{\\|}(\mathbb{E}a,b)\big{\\|}_{L_{\infty}(\mu;\ell_{q}^{(2)})}$. Therefore, (12) holds in full generality. By Proposition 2.3, we have $\\|T\\|_{L_{\infty}(D;E_{1})\rightarrow L_{1}(D;E_{1})}\geq\\|P\\|=c(p,q).$ Hence $S(E_{1})\geq c(p,q)>1$, as announced. ∎ ###### Remark 3.9. Let $(e_{k})_{k\geq 0}$ be the canonical basis of $\ell_{p}=\ell_{p}(\mathbb{N})$, then $S(\ell_{p})=1$. Indeed, if $(\theta_{k})_{k\geq 0}$ is a finite sequence of functions, then $\displaystyle\Big{\\|}\sum_{k}\mathbb{E}_{k}(\theta_{k})e_{k}\Big{\\|}_{L_{1}(\ell_{p})}$ $\displaystyle\leq$ $\displaystyle\Big{\\|}\sum_{k}\mathbb{E}_{k}(\theta_{k})e_{k}\Big{\\|}_{L_{p}(\ell_{p})}=\Big{\\|}(\sum_{k}\|\mathbb{E}_{k}(\theta_{k})\|^{p})^{1/p}\Big{\\|}_{L_{p}}$ $\displaystyle=$ $\displaystyle\Big{\\|}\sum_{k}\|\mathbb{E}_{k}(\theta_{k})\|^{p}\Big{\\|}_{L_{1}}^{1/p}=(\sum_{k}\big{\\|}\mathbb{E}_{k}(\theta_{k})\big{\\|}_{p}^{p})^{1/p}$ $\displaystyle\leq$ $\displaystyle(\sum_{k}\big{\\|}\theta_{k}\big{\\|}_{p}^{p})^{1/p}=\Big{\\|}\sum_{k}\theta_{k}e_{k}\Big{\\|}_{L_{p}(\ell_{p})}$ $\displaystyle\leq$ $\displaystyle\Big{\\|}\sum_{k}\theta_{k}e_{k}\Big{\\|}_{L_{\infty}(\ell_{p})}.$ ###### Theorem 3.10. Suppose that $1\leq p,q\leq\infty$. Let $E_{1}=\ell_{p}^{(2)}(\ell_{q}^{(2)})$ and define by recursion: $E_{n+1}=\ell_{p}^{(2)}(\ell_{q}^{(2)}(E_{n}))$. Then for any $1<s<\infty$, we have $C_{s}(E_{n})\geq S(E_{n})\geq c(p,q)^{n},$ where $S(E_{n})$ is computed with respect to the canonical basis of $E_{n}$. In particular, if $p\neq q$, then $C_{s}(E_{n})$ has at least an exponential growth with respect to $n$. ###### Proof. By Theorem 3.5, $S(E_{n+1})\geq S(\ell_{p}^{(2)}(\ell_{q}^{(2)}))S(E_{n}).$ By Lemma 3.8, we have $S(E_{n+1})\geq c(p,q)S(E_{n})$. It follows that $S(E_{n})\geq c(p,q)^{n}$. Since the canonical basis of $E_{n}$ is 1-unconditional, by (10), for any $1<s<\infty$, we have $C_{s}(E_{n})\geq S(E_{n})$. ∎ The following simple observation shows that the exponential growth of $C_{s}(E_{n})$ is optimal. ###### Proposition 3.11. Suppose $1<p\neq q<\infty$. Let $X$ be a Banach space. Define by recursion: $Y_{0}=X$ and $Y_{n+1}=L_{p}(\mathbb{T};L_{q}(\mathbb{T};Y_{n}))$. Then for all $1<s<\infty$, there exists $\chi=\chi(p,q,s)$, such that $C_{s}(Y_{n})\leq\chi^{n}C_{s}(X).$ ###### Proof. We will use the following well-known fact (see e.g. [5, 6]) about UMD constants: for any $1<r,s<\infty$, there exist $\alpha(r,s)$ and $\beta(r,s)$ such that for all Banach space $X$, (13) $\displaystyle\alpha(r,s)C_{s}(X)\leq C_{r}(X)\leq\beta(r,s)C_{s}(X).$ We will also use the elementary identity $C_{s}(L_{s}(X))=C_{s}(X)$. Combining these, we have $\displaystyle C_{s}(Y_{n+1})$ $\displaystyle=$ $\displaystyle C_{s}(L_{p}(L_{q}(Y_{n})))\leq\beta(s,p)C_{p}(L_{p}(L_{q}(Y_{n})))$ $\displaystyle=$ $\displaystyle\beta(s,p)C_{p}(L_{q}(Y_{n}))\leq\beta(s,p)\beta(p,q)C_{q}(L_{q}(Y_{n}))$ $\displaystyle=$ $\displaystyle\beta(s,p)\beta(p,q)C_{q}(Y_{n})\leq\beta(s,p)\beta(p,q)\beta(q,s)C_{s}(Y_{n}).$ Let $\chi=\beta(s,p)\beta(p,q)\beta(q,s)$, then $C_{s}(E_{n})\leq\chi^{n}C_{s}(X).$ ∎ ###### Remark 3.12. Even if one of $p,q$ is infinite or equals to 1, then since $\dim(E_{n})=4^{n}$, we have $C_{s}(E_{n})\lesssim\sqrt{\dim E_{n}}=2^{n}$. Indeed, the Banach-Mazur distance between $E_{n}$ and $\ell_{2}^{\dim E_{n}}$ is $\leq\sqrt{\dim E_{n}}$ $($cf. e.g. [14]$)$. ## 4\. Analytic UMD constants The main idea in §3 can be easily adapted for treating the analytic UMD property. In this section, all spaces are over $\mathbb{C}$. Denote the general element in $\mathbb{T}^{\mathbb{N}}$ be $z=(z_{n})_{n\geq 0}$ and let $m_{\infty}=m^{\otimes\mathbb{N}}$ be the Haar measure on $\mathbb{T}^{\mathbb{N}}$. Recall the canonical filtration on $(\mathbb{T}^{\mathbb{N}},m_{\infty})$ defined by $\sigma(z_{0})\subset\sigma(z_{0},z_{1})\subset\cdots\subset\sigma(z_{0},z_{1},\cdots,z_{n})\subset\cdots.$ From now on, we will denote $\mathcal{G}_{n}=\sigma(z_{0},z_{1},\cdots,z_{n})$. Recall that $H_{s}(\mathbb{T}^{\mathbb{N}})$ is the subspace of $L_{s}(\mathbb{T}^{\mathbb{N}},m_{\infty})$ consisting of limit values of Hardy martingales, i.e. $f\in H_{s}(\mathbb{T}^{\mathbb{N}})$ if and only if $f\in L_{s}(\mathbb{T}^{\mathbb{N}},m_{\infty})$ and the associated martingale $(\mathbb{E}^{\mathcal{G}_{n}}f)_{n\geq 0}$ is a Hardy martingale. For convenience, we always assume $z_{0}\equiv 1$ such that $\mathcal{G}_{0}$ is a trivial $\sigma$-algebra. ###### Definition 4.1. Let $X$ be a Banach space and let $\\{x_{i}\\}_{i\in I}$ be a family of vectors in $X$. The number $S^{a}(X;\\{x_{i}\\})$ is defined to be the best constant $C$ such that for any $N\in\mathbb{N}$ and any finite sequence of functions $(\theta_{k})_{k=0}^{N}$ in $H_{\infty}(\mathbb{T}^{\mathbb{N}})$, we have $\Big{\\|}\sum_{k}\mathbb{E}^{\mathcal{G}_{k}}(\theta_{k})x_{i_{k}}\Big{\\|}_{L_{1}(m_{\infty};X)}\leq C\Big{\\|}\sum_{k}\theta_{k}x_{i_{k}}\Big{\\|}_{L_{\infty}(m_{\infty};X)}$ If there does not exist such constant, we set $S^{a}(X;\\{x_{i}\\})=\infty$. If $\\{x_{i}\\}$ is clear from the context, then $S^{a}(X;\\{x_{i}\\})$ will be simplified as $S^{a}(X)$. The Stein type inequality still holds in this setting, more precisely, we have ###### Proposition 4.2. Let $X$ be an AUMD space. For any $1\leq s<\infty$, let $(F_{k})_{k\geq 0}$ be an arbitrary finite sequence in $H_{s}(\mathbb{T}^{\mathbb{N}};X)$. Then we have (14) $\displaystyle\Big{\\|}\sum_{k}\zeta_{k}\mathbb{E}^{\mathcal{G}_{k}}(F_{k})(z)\Big{\\|}_{L_{s}(X)}\leq C_{s}^{a}(X)\Big{\\|}\sum_{k}\zeta_{k}F_{k}(z)\Big{\\|}_{L_{s}(X)},$ where $\zeta=(\zeta_{k})_{k\geq 0}$ is an independent copy of $z=(z_{k})_{k\geq 0}$ and $L_{s}(X)=L_{s}(\mathbb{T}_{z}^{\mathbb{N}}\times\mathbb{T}_{\zeta}^{\mathbb{N}},m_{\infty}\times m_{\infty};X)$. ###### Proof. Consider the filtration on $\mathbb{T}_{z}^{\mathbb{N}}\times\mathbb{T}_{\zeta}^{\mathbb{N}}$ defined by $\mathcal{B}_{2j}=\sigma(z_{0},\cdots,z_{j})\otimes\sigma(\zeta_{0},\cdots,\zeta_{j})$ and $\mathcal{B}_{2j-1}=\sigma(z_{0},\cdots,z_{j})\otimes\sigma(\zeta_{0},\cdots,\zeta_{j-1})$. Then $f=\sum_{k}\zeta_{k}F_{k}(z)$ is a Hardy martingale with respect to the above filtration. Let $f^{\prime}=\sum_{k}\zeta_{k}\mathbb{E}^{\mathcal{G}_{k}}(F_{k})$. Then we have $f^{\prime}=\sum_{j}(\mathbb{E}^{\mathcal{B}_{2j}}-\mathbb{E}^{\mathcal{B}_{2j-1}})(f)$. It follows (see Remark 3.3) that $\\|f^{\prime}\\|_{L_{s}(X)}\leq C_{s}^{a}(X)\\|f\\|_{L_{s}(X)}$, whence (14). ∎ ###### Proposition 4.3. Let $X$ be an AUMD space. Assume that $\\{x_{i}\\}_{i\in I}$ is a 1-unconditional basic sequence in $X$. Then for any $1\leq s<\infty$ and any finite sequence of functions $(\theta_{k})_{k\geq 0}$ in $H_{s}(\mathbb{T}^{\mathbb{N}})$, $\displaystyle\Big{\\|}\sum_{k}\mathbb{E}^{\mathcal{G}_{k}}(\theta_{k})x_{i_{k}}\Big{\\|}_{L_{s}(m_{\infty};X)}\leq C_{s}^{a}(X)\Big{\\|}\sum_{k}\theta_{k}x_{i_{k}}\Big{\\|}_{L_{s}(m_{\infty};X)}.$ ###### Proof. It follows verbatim the proof of Proposition 3.4. ∎ Let $X$ be as in Proposition 4.3, $\\{x_{i}\\}$ is a 1-unconditional basic sequence in $X$. Then for all $1\leq s<\infty$, we have $\displaystyle\Big{\\|}\sum_{k}\mathbb{E}^{\mathcal{G}_{k}}(\theta_{k})x_{i_{k}}\Big{\\|}_{L_{1}(m_{\infty};X)}\leq C_{s}^{a}(X)\Big{\\|}\sum_{k}\theta_{k}x_{i_{k}}\Big{\\|}_{L_{\infty}(m_{\infty};X)}.$ Hence $S^{a}(X;\\{x_{i}\\})\leq C_{s}^{a}(X)$ for all $1\leq s<\infty$. ###### Theorem 4.4. Let $E$ be a Banach space with a 1-unconditional basis $\\{e_{i}:i\in I\\}$, let $F$ be another Banach space. Let $E(F)$ be defined as in Theorem 3.5. For any fixed family of vectors $\\{f_{j}:j\in J\\}$ in $F$, consider the family of vectors $\\{e_{i}\otimes f_{j}:i\in I,j\in J\\}$ in $E(F)$, then we have $S^{a}(E(F))\geq S^{a}(E)S^{a}(F),$ where $S^{a}(E(F))$, $S^{a}(E)$ and $S^{a}(F)$ are defined with respect to the mentioned families of vectors respectively. ###### Proof. The proof is similar to the proof of Theorem 3.5. We mention the slight difference concerning the filtration. Consider the infinite tensor product $L_{\infty}(\mathbb{T}^{\mathbb{N}})\otimes L_{\infty}(\mathbb{T}^{\mathbb{N}})\otimes\cdots$, define $z_{k,n}=\underbrace{1\otimes\cdots\otimes 1}_{k\quad\text{times}}\otimes z_{n}\otimes 1\otimes\cdots,\text{if }n\geq 1$ and $z_{k,0}=z_{k}\otimes 1\otimes 1\otimes\cdots.$ Then the filtration defined by $\mathcal{F}^{a}_{k,n}:=\sigma\left(z_{j}:j\leq(k,n)\right)$ is an analytic filtration, where the order on $\mathbb{N}\times\mathbb{N}$ is the lexigraphic order as defined in the proof of Theorem 3.5. This filtration plays the role similar to that of $(\mathcal{F}_{k,n})_{k,n}$ in the proof of Theorem 3.5. Note that we may restrict to the functions $\theta_{k},\xi_{n}$ depending only on finitely many variables. Thus only a finite subset of $\mathbb{N}\times\mathbb{N}$ is used. ∎ The following lemma requires slightly more efforts than Lemma 3.8. ###### Lemma 4.5. Suppose that $1\leq p\neq q<\infty$. If $E_{1}=\ell_{p}^{(2)}(\ell_{q}^{(2)})$, then $S^{a}(E_{1})\geq\kappa(p,q)>1.$ ###### Proof. We will use the notations in the proof of Lemma 3.8. Define a linear map $U:H_{\infty}(\mathbb{T},m;E_{1})\rightarrow H_{1}(\mathbb{T},m;E_{1})$ by $U\Big{[}a_{ij}(z)e_{i}^{p}\otimes e_{j}^{q}\Big{]}=\left\\{\begin{array}[]{lc}\mathbb{E}(a_{ij})e_{i}^{p}\otimes e_{j}^{q},\text{ if }j=1\\\ a_{ij}(z)e_{i}^{p}\otimes e_{j}^{q},\text{ if }j=2\end{array}.\right.$ If $C=\\|U\\|_{H_{\infty}(E_{1})\rightarrow H_{1}(E_{1})}$, then $S^{a}(E_{1})\geq C$. By definition, for any $a,b,c,d$ functions in $H_{\infty}(\mathbb{T})$, we have $\displaystyle\int_{\mathbb{T}}\Big{\\{}(\|\mathbb{E}a\|^{q}+\|b(z)\|^{q})^{p/q}+(\|\mathbb{E}c\|^{q}+\|d(z)\|^{q})^{p/q}\Big{\\}}^{1/p}dm(z)$ $\displaystyle\leq$ $\displaystyle C\sup_{z\in\mathbb{T}}\Big{\\{}(\|a(z)\|^{q}+\|b(z)\|^{q})^{p/q}+(\|c(z)\|^{q}+\|d(z)\|^{q})^{p/q}\Big{\\}}^{1/p}.$ Note that if $a,c$ are outer functions, then by (5), we have $\|\mathbb{E}a\|=M(\|a\|)$ and $\|\mathbb{E}c\|=M(\|c\|)$. So for any functions $a,b,c,d\in H_{\infty}(\mathbb{T})$ such that $a,c$ are outer, we have $\displaystyle\int_{\mathbb{T}}\Big{\\{}(M(\|a\|)^{q}+\|b(z)\|^{q})^{p/q}+(M(\|c\|)^{q}+\|d(z)\|^{q})^{p/q}\Big{\\}}^{1/p}dm(z)$ $\displaystyle\leq$ $\displaystyle C\sup_{z\in\mathbb{T}}\Big{\\{}(\|a(z)\|^{q}+\|b(z)\|^{q})^{p/q}+(\|c(z)\|^{q}+\|d(z)\|^{q})^{p/q}\Big{\\}}^{1/p}.$ By the classical Szegö’s condition, if $a^{\prime},b^{\prime},c^{\prime},d^{\prime}$ are functions in $L_{\infty}(\mathbb{T})$ which are bounded from below, then there are outer functions $a,b,c,d\in H_{\infty}(\mathbb{T})$, such that $\|a^{\prime}\|=\|a\|,\|b^{\prime}\|=\|b\|,\|c^{\prime}\|=\|c\|,\|d^{\prime}\|=\|d\|$ a.e.. Hence (4) still holds for any 2-valued non-vanishing functions $a,b,c,d\in L_{\infty}(\mathbb{T})$ (note that for a function taking only two values, non- vanishing is the same as bounded from below). By approximation, we can further relax the non-vanishing condition on $b,d$. Now consider any measurable partition $\mathbb{T}=A\dot{\cup}B$, such that $m(A)=m(B)=\frac{1}{2}$. If $a=u\chi_{A}+v\chi_{B}$, $c=v\chi_{A}+u\chi_{B}$, $b=w\chi_{A}+t\chi_{B}$ and $d=t\chi_{A}+w\chi_{B}$, then it is easy to check that for all $z\in\mathbb{T}$, we have $\displaystyle\Big{\\{}(\|a(z)\|^{q}+\|b(z)\|^{q})^{p/q}+(\|c(z)\|^{q}+\|d(z)\|^{q})^{p/q}\Big{\\}}^{1/p}$ $\displaystyle=$ $\displaystyle\Big{\\{}(\|u\|^{q}+\|w\|^{q})^{p/q}+(\|v\|^{q}+\|t\|^{q})^{p/q}\Big{\\}}^{1/p}$ $\displaystyle=$ $\displaystyle 2^{1/p}\Big{\\{}\int_{\mathbb{T}}(\|a\|^{q}+\|b\|^{q})^{p/q}dm\Big{\\}}^{1/p}.$ Similarly for all $z\in\mathbb{T}$, we have $\displaystyle\Big{\\{}(M(\|a\|)^{q}+\|b(z)\|^{q})^{p/q}+(M(\|c\|)^{q}+\|d(z)\|^{q})^{p/q}\Big{\\}}^{1/p}$ $\displaystyle=$ $\displaystyle 2^{1/p}\Big{\\{}\int_{\mathbb{T}}(M(\|a\|)^{q}+\|b\|^{q})^{p/q}dm\Big{\\}}^{1/p}.$ Substituting these equalities to (4), we get $\Big{\\{}\int_{\mathbb{T}}(M(\|a\|)^{q}+\|b\|^{q})^{p/q}dm\Big{\\}}^{1/p}\leq C\Big{\\{}\int_{\mathbb{T}}(\|a\|^{q}+\|b\|^{q})^{p/q}dm\Big{\\}}^{1/p}.$ By Proposition 2.5, we have $C\geq\kappa(p,q)$. This completes the proof. ∎ ###### Theorem 4.6. Suppose that $1\leq p\neq q<\infty$. If $E_{n}$’s are defined as in Theorem 3.10, then for any $1\leq s<\infty$, we have $C_{s}^{a}(E_{n})\geq S^{a}(E_{n})\geq\kappa(p,q)^{n}.$ Moreover, if $1<p,q<\infty$, then there exists $\kappa_{2}=\kappa_{2}(p,q,s)$, such that $C_{s}^{a}(E_{n})\leq\kappa_{2}^{n}.$ ###### Proof. The first part of proof is identical to the proof of Theorem 3.10. The second part follows from the fact that $C_{s}^{a}(E_{n})\leq C_{s}(E_{n})$ and Proposition 3.11. ∎ ## 5\. Construction and further discussions For the sake of clearness, we introduce the family $X_{n}(p,q)$, which is defined as follows: Let $X_{0}(p,q)=\mathbb{R}$, and define by recursion that $X_{n+1}(p,q)=L_{p}(D,\mu;L_{q}(D,\mu;X_{n}(p,q))).$ In the complex case, $X^{\mathbb{C}}_{n}(p,q)$ is defined similarly. Obviously, $X_{n}(p,q)$ is isometric to $E_{n}$ defined in the previous sections using $p,q$. Our main purpose for introducing $X_{n}$’s is the existence of canonical isometric inclusion $X_{n}(p,q)\subset X_{n+1}(p,q)$. By these inclusions, the union $\cup_{n}X_{n}(p,q)$ is a normed space and its completion will be denoted by $X(p,q)$. We have $X(p,q):=\overline{\cup_{n}X_{n}(p,q)}\simeq\lim_{\longrightarrow}X_{n}(p,q),$ where the last term is the inductive limit of $X_{n}(p,q)$’s associated to the canonical inclusions. In the complex case, $X^{\mathbb{C}}(p,q)$ is defined similarly. ###### Remark 5.1. If $1\leq p=q<\infty$, then $X(p,p)$ is the real space $L^{p}_{\mathbb{R}}(D^{\mathbb{N}},\mu^{\otimes\mathbb{N}})$ and $X^{\mathbb{C}}(p,p)$ is the complex space $L_{\mathbb{C}}^{p}(D^{\mathbb{N}},\mu^{\otimes\mathbb{N}})$. We have the following complex interpolation result. ###### Proposition 5.2. Let $1<p_{0},p_{1},q_{0},q_{1}<\infty$ and $0<\theta<1$. Then we have the following isometric isomorphism: $X^{\mathbb{C}}(p_{\theta},q_{\theta})=[X^{\mathbb{C}}(p_{0},q_{0}),X^{\mathbb{C}}(p_{1},q_{1})]_{\theta},$ with $\frac{1}{p}=\frac{\theta}{p_{1}}+\frac{1-\theta}{p_{0}}$ and $\frac{1}{q}=\frac{\theta}{q_{1}}+\frac{1-\theta}{q_{0}}$. ###### Proof. Note that $X(p,q)$ is a Banach lattice of functions on $(D^{\mathbb{N}},\mu^{\otimes\mathbb{N}})$. Clearly, $X(p,q)$ is $\min(p,q)$-convex and $\max(p,q)$-concave in the sense of §1.d in [10], and hence by Theorem 1.f.1 (p. 80) and Proposition 1.e.3 (p. 61) in [10] it is reflexive. Then the above result is a particular case of a classical formula going back to Calderón ([8], p. 125). ∎ Recall that a Banach space $X$ over the complex field is $\theta$-Hilbertian ($0\leq\theta\leq 1$) if there exists an interpolation pair $(X_{0},X_{1})$ of Banach spaces such that $X$ is isometric with $[X_{0},X_{1}]_{\theta}$ and $X_{1}$ is a Hilbert space. ###### Corollary 5.3. Let $1<p\neq q<\infty$. Then $X(p,q)$ is non-UMD and $X^{\mathbb{C}}(p,q)$ is non-AUMD. Moreover, there exists $0<\theta<1$ such that $X^{\mathbb{C}}(p,q)$ is $\theta$-Hilbertian. In particular, $X^{\mathbb{C}}(p,q)$ and a fortiori $X(p,q)$ is super-reflexive. ###### Proof. It follows easily from Theorem 3.10 and Theorem 4.6 that $X(p,q)$ is non-UMD and $X^{\mathbb{C}}(p,q)$ is non-AUMD. For $0<\theta<1$ small enough, such that $\max(\frac{1/p-\theta/2}{1-\theta},\frac{1/q-\theta/2}{1-\theta})<1$, we can find $1<\tilde{p},\tilde{q}<\infty$ satisfying the equalities: $\frac{1}{p}=\frac{\theta}{2}+\frac{1-\theta}{\tilde{p}},\quad\frac{1}{q}=\frac{\theta}{2}+\frac{1-\theta}{\tilde{q}}.$ By Proposition 5.2, we have $X^{\mathbb{C}}(p,q)=[X^{\mathbb{C}}(\tilde{p},\tilde{q}),X^{\mathbb{C}}(2,2)]_{\theta}.$ Since $X^{\mathbb{C}}(2,2)=L^{2}_{\mathbb{C}}(D^{\mathbb{N}},\mu^{\otimes\mathbb{N}})$ is Hilbertian, $X^{\mathbb{C}}(p,q)$ is $\theta$-Hilbertian. The super- reflexivity of $X^{\mathbb{C}}(p,q)$ follows from the well-known fact that any $\theta$-Hilbertian space is super-reflexive for $\theta>0$ (cf.[12]). ∎ ###### Remark 5.4. Let $1<p\neq q<\infty$. For any $0<\eta<1$, let $\frac{1}{p_{\eta}}=\frac{1-\eta}{p}+\frac{\eta}{q}$ and $\frac{1}{q_{\eta}}=\frac{1-\eta}{q}+\frac{\eta}{p}$. By Proposition 5.2, we have $X^{\mathbb{C}}(p_{\eta},q_{\eta})=[X^{\mathbb{C}}(p,q),X^{\mathbb{C}}(q,p)]_{\eta}.$ Note that in this interpolation scale, there is only one UMD space corresponding to $\eta=\frac{1}{2}$. For futher discussions, let us now turn to the non-atomic case and modify slightly the definitions. For any $1<p,q<\infty$, consider the family of spaces $Z_{n}=Z_{n}(p,q)$ defined by recursion: $Z_{0}=\mathbb{C}$ and $Z_{n+1}=Z_{n}(L_{p}(\mathbb{T},m;L_{q}(\mathbb{T},m))$. From the definition, we have $Z_{n}(p,q)\subset Z_{n+1}(p,q).$ Thus we can define $Z(p,q)=\lim_{\longrightarrow}Z_{n}(p,q).$ To avoid ambiguity, let us emphasize the inclusions $Z_{n}(p,q)\subset Z_{n+1}(p,q)$ used to define the inductive limit. For simplicity of notations, we will write $L_{p_{1}}L_{p_{2}}=L_{p_{1}}(L_{p_{2}})$, $L_{p_{1}}L_{p_{2}}L_{p_{3}}=L_{p_{1}}(L_{p_{2}}(L_{p_{3}}))$, etc. With these notations, one can easily see the difference between $X_{n}$ and $Z_{n}$ as follows: $X_{n+1}=L_{p}(L_{q}(X_{n}))=L_{p}L_{q}\underbrace{L_{p}L_{q}\cdots L_{p}L_{q}}_{X_{n}},$ where $L_{p}=L_{p}(D,\mu)$ and $L_{q}=L_{q}(D,\mu)$ are two dimensional. And $Z_{n+1}=Z_{n}(L_{p}(L_{q}))=\underbrace{L_{p}L_{q}\cdots L_{p}L_{q}}_{Z_{n}}L_{p}L_{q},$ where $L_{p}=L_{p}(\mathbb{T},m)$ and $L_{q}=L_{q}(\mathbb{T},m)$. ###### Remark 5.5. The main purpose of introducing the spaces $Z_{n}(p,q)$ is that we have lattice isometric isomorphisms $L_{p}(Z_{n}(p,q))\simeq Z_{n}(p,q)$ for all $n$ and moreover, these isomorphisms are compatible with the inclusion of $Z_{n}(p,q)\subset Z_{n+1}(p,q)$ (the word “compatible” will be explained by a commutative diagram in the sequel) and this will be used to show some additional properties for $Z(p,q)$. The family of $X_{n}(p,q)$’s shares the property of having lattice isometric isomorphisms $L_{p}(X_{n}(p,q))\simeq X_{n}(p,q)$ for all $n$, but the isomorphisms are not compatible with the inclusions $X_{n}(p,q)\subset X_{n+1}(p,q)$. The $Z(p,q)$’s are Banach lattices of functions on the infinite torus $\mathbb{T}^{\mathbb{N}}$, they have the following properties. ###### Proposition 5.6. Let $1<p,q<\infty$. We have isomorphisms $Z(p,q)\simeq Z(q,p)$ and $L_{p}(Z(p,q))\simeq L_{q}(Z(p,q)).$ If $p\neq q$, then $Z(p,q)$ does not have unconditional basis. ###### Proof. Since $L_{p}(\mathbb{T})$ and $L_{p}(\mathbb{T}\times\mathbb{T})$ are isometric as Banach lattices, we have isometric isomorphisms which are compatible with the inclusions $Z_{n}\subset Z_{n+1}$, that is we have the commutative diagram $\begin{CD}Z_{n}(p,q)@>{\text{ inclusion }}>{}>Z_{n+1}(p,q)\\\ @V{\text{ isometric }}V{\simeq}V@V{\simeq}V{\text{ isometric}}V\\\ L_{p}(Z_{n}(p,q))@ >\text{ inclusion }>>L_{p}(Z_{n+1}(p,q)).\end{CD}$ By taking Banach space inductive limit, we have $Z(p,q)\xrightarrow[\text{isometric}]{\simeq}L_{p}(Z(p,q)).$ If $p\neq q$, then $Z(p,q)$ and hence $L_{p}(Z(p,q))$ is non-UMD. By a result of D.J. Aldous (see [1], Proposition 4), $Z(p,q)$ has no unconditional basis. It is easy to see that $Z(p,q)$ and $Z(q,p)$ complementably embed into each other. Since $\ell_{p}^{(2)}(L_{p})=L_{p}$ as Banach lattices, we have $\ell_{p}^{(2)}(L_{p}(Z(p,q)))=L_{p}(Z(p,q)).$ Moreover, since $L_{p}(Z(p,q))=Z(p,q)$, the above isometry implies that as Banach space $Z(p,q)=Z(p,q)\oplus Z(p,q)$. Similarly, $Z(q,p)=Z(q,p)\oplus Z(q,p)$. By the classical Pełcyński decomposition method, we have $Z(p,q)\simeq Z(q,p)$. Hence $\qquad\quad L_{p}(Z(p,q))=Z(p,q)\simeq Z(q,p)=L_{q}(Z(q,p))\simeq L_{q}(Z(p,q)).$ ∎ Let $(p_{i})_{i\geq 1}$ be a sequence of real numbers such that $1<p_{i}<\infty$. Define $X[(p_{i})]=\lim_{\longrightarrow}L_{p_{n}}\cdots L_{p_{2}}L_{p_{1}}$ and $Z[(p_{i})]=\lim_{\longrightarrow}L_{p_{1}}L_{p_{2}}\cdots L_{p_{n}}.$ ###### Problem. Under which condition is $X[(p_{i})]$ or $Z[(p_{i})]$ in the UMD class ? We have the following observations on the necessary condition: (i) A trivial necessary condition is that there exist $1<p_{0},p_{\infty}<\infty$, such that $p_{0}\leq p_{i}\leq p_{\infty}$ for all $i\geq 1$. * (ii) If the above condition is satisfied, then the sequence $(p_{i})$ has at least one cluster point $1<p<\infty$. Then a necessary condition is that the sequence has only one cluster point, i.e. $\lim_{i\to\infty}p_{i}=p$. Indeed, assume that the sequence $(p_{i})$ has two cluster points $1<p\neq q<\infty$, so that there exist two subsequences of $(p_{i})$ which tend to $p,q$ respectively. Then one can easily show that by Theorem 3.10, both $X[(p_{i})]$ and $Z[(p_{i})]$ are non-UMD (they are in fact non-AUMD). * (iii) Now the speed of convergence of $(p_{i})$ will play a role. Since $\ell_{p_{1}}^{(2)}(\ell_{p_{2}}^{(2)}(\cdots(\ell_{p_{n}}^{(2)})\cdots))$ embeds isometrically into $L_{p_{1}}L_{p_{2}}\cdots L_{p_{n}}$. A necessary condition for $Z[(p_{i})]$ to be UMD is $\prod_{i}c(p_{2i},p_{2i+1})<\infty$. Similarly, it is necessary that $\prod_{i}c(p_{2i+1},p_{2i+2})<\infty$. Combining these, a necessary condition for $Z[(p_{i})]$ to be in the UMD class is $\prod_{i}c(p_{i},p_{i+1})<\infty.$ The same statement remains true for $X[(p_{i})]$. Note that by (4), $c(p_{i},p_{i+1})>1$ if $p_{i}\neq p_{i+1}$. Intuitively, if $p_{i}$ tends to $p$ sufficiently fast, then both $X[(p_{i})]$ and $Z[(p_{i})]$ are in the UMD class. The author obtained some partial results in this direction, which will be treated elsewhere. ###### Remark 5.7. Let $1<p<q<\infty$. We have the following Banach lattices isometries $L_{p}L_{q}=L_{p}L_{p}L_{q},\quad L_{p}L_{q}=L_{p}L_{q}L_{q}.$ Since $L_{p}L_{r}L_{q}$ is an interpolation space between $L_{p}L_{p}L_{q}$ and $L_{p}L_{q}L_{q}$ for any $p\leq r\leq q$, the $\text{UMD}_{s}$ constant of $L_{p}L_{r}L_{q}$ is actually the same as that of $L_{p}(L_{q})$. The same argument shows that $L_{p}L_{u}L_{r}L_{v}L_{q}$ has the same $\text{UMD}_{s}$ constant with $L_{p}L_{q}$, provided $p\leq u\leq r\leq v\leq q$. More generally, if $(p_{i})_{i=1}^{n}$ is a finite sequence, assume that $(p_{i})_{i=k}^{l}$ is consecutive monotone (non-increasing or non-decreasing) subsequence, then $L_{p_{1}}\cdots L_{p_{k}}\cdots L_{p_{l}}\cdots L_{p_{n}}$ and $L_{p_{1}}\cdots L_{p_{k}}L_{p_{l}}\cdots L_{p_{n}}$ have the same $\text{UMD}_{s}$ constant for all $1<s<\infty$. Our results have some applications in the non-commutative setting, i.e. on the operator space UMD property, which will appear in a future publication. ## Acknowledgements This work was carried out while the author was visiting at Texas A&M University. The author would like to acknowledge the hospitality provided by Department of Mathematics of Texas A&M. He would like to thank his advisor G. Pisier for suggesting this problem and for the constant and stimulating discussions. The author also appreciates the careful review of the paper by the referee who suggested many changes to enhance the readability of the paper. ## References * [1] D. J. Aldous. Unconditional bases and martingales in $L_{p}(F)$. Math. Proc. Cambridge Philos. Soc., 85(1):117–123, 1979. * [2] J. Bourgain. Some remarks on Banach spaces in which martingale difference sequences are unconditional. Ark. Mat., 21(2):163–168, 1983. * [3] J. Bourgain. On martingales transforms in finite-dimensional lattices with an appendix on the $K$-convexity constant. Math. Nachr., 119:41–53, 1984. * [4] J. Bourgain. Vector-valued singular integrals and the $H^{1}$-BMO duality. In Probability theory and harmonic analysis (Cleveland, Ohio, 1983), volume 98 of Monogr. Textbooks Pure Appl. Math., pages 1–19. Dekker, New York, 1986. * [5] D. L. Burkholder. A geometrical characterization of Banach spaces in which martingale difference sequences are unconditional. Ann. Probab., 9(6):997–1011, 1981. * [6] D. L. Burkholder. A geometric condition that implies the existence of certain singular integrals of Banach-space-valued functions. In Conference on harmonic analysis in honor of Antoni Zygmund, Vol. I, II (Chicago, Ill., 1981), Wadsworth Math. Ser., pages 270–286. Wadsworth, Belmont, CA, 1983. * [7] Donald L. Burkholder. Martingales and singular integrals in Banach spaces. In Handbook of the geometry of Banach spaces, Vol. I, pages 233–269. North-Holland, Amsterdam, 2001. * [8] A.-P. Calderón. Intermediate spaces and interpolation, the complex method. Studia Math., 24:113–190, 1964. * [9] D. J. H. Garling. On martingales with values in a complex Banach space. Math. Proc. Cambridge Philos. Soc., 104(2):399–406, 1988. * [10] Joram Lindenstrauss and Lior Tzafriri. Classical Banach spaces. II, volume 97 of Ergebnisse der Mathematik und ihrer Grenzgebiete. Springer-Verlag, Berlin, 1979. Function spaces. * [11] G. Pisier. Un exemple concernant la super-réflexivité. In Séminaire Maurey-Schwartz 1974–1975: Espaces $L^{p}$ applications radonifiantes et géométrie des espaces de Banach, Annexe No. 2, page 12. Centre Math. École Polytech., Paris, 1975\. * [12] G. Pisier. Some applications of the complex interpolation method to Banach lattices. J. Analyse Math., 35:264–281, 1979. * [13] José L. Rubio de Francia. Martingale and integral transforms of Banach space valued functions. In Probability and Banach spaces (Zaragoza, 1985), volume 1221 of Lecture Notes in Math., pages 195–222. Springer, Berlin, 1986. * [14] Nicole Tomczak-Jaegermann. Banach-Mazur distances and finite-dimensional operator ideals, volume 38 of Pitman Monographs and Surveys in Pure and Applied Mathematics. Longman Scientific & Technical, Harlow, 1989.	arxiv-papers	2011-12-04T09:46:28	2024-09-04T02:49:24.951273	{ "license": "Public Domain", "authors": "Yanqi Qiu", "submitter": "Yanqi Qiu", "url": "https://arxiv.org/abs/1112.0739" }
1112.0854	# New approach for normalization and photon-number distributions of photon- added (-subtracted) squeezed thermal states Li-Yun Hu1,2† and Zhi-Ming Zhang2‡ 1College of Physics & Communication Electronics, Jiangxi Normal University, Nanchang 330022, China 2Key Laboratory of Photonic Information Technology of Guangdong Higher Education Institutes, SIPSE & LQIT, South China Normal University, Guangzhou 510006, China ${\dagger}$E-mail: hlyun2008@126.com; ‡ E-mail: zmzhang@scnu.edu.cn. ###### Abstract Using the thermal field dynamics theory to convert the thermal state to a “pure” state in doubled Fock space, it is found that the average value of $e^{fa^{{\dagger}}a}$ under squeezed thermal state (STS) is just the generating function of Legendre polynomials, a remarkable result. Based on this point, the normalization and photon-number distributions of m-photon added (or subtracted) STS are conviently obtained as the Legendre polynomials. This new concise method can be expanded to the entangled case. ## I Introduction Nonclassicality of optical fields is helpful in understanding fundamentals of quantum optics and have many applications in quantum information processing 1 . To generate and manipulate various nonclassical optical fields, subtracting or adding photons from/to traditional quantum states or Gaussian states is proposed 2 ; 2a ; 3 ; 4 ; 5 ; 6 ; 7 ; 8 . For example, the photon addition and subtraction have been successfully demonstrated experimentally for probing quantum commutation rules by Parigi et al. 6 . Recently, photon-added (-subtracted) Gaussian states have received more attention from both experimentalists and theoreticians 9 ; 10 ; 11 ; 11a ; 12 ; 13 ; 14 ; 15 ; 16 ; 17 , since these states exhibit an abundant of nonclassical properties and may give access to a complete engineering of quantum states and to fundamental quantum phenomena. Theoretically, the normalization factors of such quantum states are essential for studying their nonclassical properties. Very recent, Fan and Jiang 18 present a new concise approach for normalizing m-photon-added (-subtracted) squeezed vacuum state (pure state) by constructing generating function. However, most systems are not isolated, but are immersed in a thermal reservoir, thus it is often the case that we have no enough information to specify completely the state of a system. In such a situation, the system only can be described by mixed states, such as thermal states. In addition, the squeezed thermal states (STSs) can be considered as the generalized Gaussian states. In this paper, we shall extend this case to the mixed state, i.e., by using the thermal field dynamics (TFD) theory to convert the thermal state to a “pure” state in doubled Fock space, we present a new concise method for normalizing photon-added (-subtracted) squeezed thermal states (PASTSs, PSSTSs) and for deriving their photon-number distributions (PNDs) which have been a major topic of studies on quantum optics and quantum statistics. It is found that the normalization factors and PNDs are related to the Legendre polynomials in a compact form. Our paper is arranged as follows. In section 2, based on Takahashi-Umezawa TFD, we convert the thermal state $\rho_{th}$ to a pure state in doubled Fock space by the partial trace, $\rho_{th}=\mathtt{t\tilde{r}}\left[\left\|0(\beta)\right\rangle\left\langle 0(\beta)\right\|\right]$ (see Eq.(4) below). In section 3, we introduce the STS $S_{1}\left\|0(\beta)\right\rangle$ with $S_{1}$ being the single-mode squeezing operator for the real mode. It is shown that the average value of $e^{fa^{{\dagger}}a}$ under STS is just the generating function of Legendre polynomials, a remarkable result. Based on this point, in sections 4 and 5, the normalization factors and PNDs of m-photon added (or subtracted) STS are obtained as the Legendre polynomials, respectively. The last section is devoted to drawing a conclusion. ## II Representation of thermal state in doubled Fock space We begin with briefly reviewing the properties of thermal state. For a single field mode with frequency $\omega$ in a thermal equilibrium state corresponding to absolute temperature $T$, the density operator is $\rho_{th}=\sum_{n=0}^{\infty}\frac{n_{c}^{n}}{\left(n_{c}+1\right)^{n+1}}\left\|n\right\rangle\left\langle n\right\|,$ (1) where $n_{c}=[\exp(\hbar\omega/(kT))-1]^{-1}$ being the average photon number of the thermal state $\rho_{th}$ and $k$ being Bltzmann’s constant. Note $\left\|n\right\rangle=a^{\dagger n}/\sqrt{n!}\left\|0\right\rangle$ and the normally ordering form of vacuum projector $\left\|0\right\rangle\left\langle 0\right\|=\colon\exp(-a^{\dagger}a)\colon$(the symbol $\colon\colon$ denotes the normal ordering), one can put Eq.(1) into the following form $\rho_{th}=\colon\frac{1}{n_{c}+1}e^{-\frac{1}{n_{c}+1}a^{{\dagger}}a}\colon=\frac{1}{n_{c}+1}e^{a^{{\dagger}}a\ln\frac{n_{c}}{n_{c}+1}},$ (2) where in the last step, the operator identity $\exp\left(\lambda a^{{\dagger}}a\right)=\colon\exp\left[\left(e^{\lambda}-1\right)a^{{\dagger}}a\right]\colon$is used. Recalling the thermal field dynamics (TFD) introduced by Takahashi and Umezawa 19 ; 20 ; 21 , its elemental spirit is to convert the calculations of ensemble averages for a mixed state $\rho$, $\left\langle A\right\rangle=\mathtt{tr}\left(A\rho\right)/\mathtt{tr}\left(\rho\right),$ to equivalent expectation values with a pure state $\left\|0(\beta)\right\rangle$, i.e., $\left\langle A\right\rangle=\left\langle 0(\beta)\right\|A\left\|0(\beta)\right\rangle,$ (3) where $\beta=1/kT$, $k$ is the Boltzmann constant. Thus, for the density operator $\rho_{th}$, by using the partial trace method 22 , i.e., $\rho_{th}=\widetilde{\mathtt{tr}}\left[\left\|0(\beta)\right\rangle\left\langle 0(\beta)\right\|\right]$ where $\widetilde{\mathtt{tr}}$ denotes the trace operation over the environment freedom (denoted as operator $\tilde{a}^{{\dagger}}$), one can obtain the explicit expression of $\left\|0(\beta)\right\rangle$ in doubled Fock space, $\left\|0(\beta)\right\rangle=\text{sech}\theta\exp\left[a^{\dagger}\tilde{a}^{\dagger}\tanh\theta\right]\left\|0\tilde{0}\right\rangle=S\left(\theta\right)\left\|0\tilde{0}\right\rangle,$ (4) where $\left\|0\tilde{0}\right\rangle$ is annihilated by $\tilde{a}$ and $a$, $[\tilde{a},\tilde{a}^{{\dagger}}]=1$, and $S\left(\theta\right)$ is the thermal operator, $S\left(\theta\right)\equiv\exp\left[\theta\left(a^{\dagger}\tilde{a}^{\dagger}-a\tilde{a}\right)\right]\ $with a similar form to the a two-mode squeezing operator except for the tilde mode, and $\theta$ is a parameter related to the temperature by $\tanh\theta=\exp\left(-\frac{\hbar\omega}{2kT}\right)$. $\left\|0(\beta)\right\rangle$ is named thermal vacuum state. Let $\mathtt{Tr}$ denote the trace operation over both the system freedom (expressed by $\mathtt{tr}$) and the environment freedom by $\widetilde{\mathtt{tr}}$, i.e., $\mathtt{Tr}=\mathtt{tr}\widetilde{\mathtt{tr}}$, then we have $\displaystyle\mathtt{tr}\left(A\rho_{th}\right)$ $\displaystyle=\mathtt{Tr}\left[A\left\|0(\beta)\right\rangle\left\langle 0(\beta)\right\|\right]$ $\displaystyle=\mathtt{tr}\left[A\widetilde{\mathtt{tr}}\left\|0(\beta)\right\rangle\left\langle 0(\beta)\right\|\right],$ (5) and the average photon number of the thermal state $\rho_{th}$ is $n_{c}=\mathtt{Tr}\left[a^{\dagger}a\left\|0(\beta)\right\rangle\left\langle 0(\beta)\right\|\right]=\sinh^{2}\theta.$ (6) Here we should emphasize that $\widetilde{\mathtt{tr}}\left\|0(\beta)\right\rangle\left\langle 0(\beta)\right\|\neq\left\langle 0(\beta)\right\|\left.0(\beta)\right\rangle,$ since $\left\|0(\beta)\right\rangle$ involves both real mode $a$ and fictitious mode $\tilde{a}$. From Eqs.(3) and (4) one can see that the worthwhile convenience in Eq.(4) is at the expense of introducing a fictitious field (or called a tilde-conjugate field) in the extended Hilbert space, i.e., the original optical field state $\left\|n\right\rangle$ in the Hilbert space $\mathcal{H}$ is accompanied by a tilde state $\left\|\tilde{n}\right\rangle$ in $\mathcal{\tilde{H}}$. A similar rule holds for operators: every Bose annihilation operator $a$ acting on $\mathcal{H}$ has an image $\tilde{a}$ acting on $\mathcal{\tilde{H}}$. These operators in $\mathcal{H}$ are commutative with those in $\mathcal{\tilde{H}}$. ## III Suqeezed thermal vacuum state To realize our purpose, we first introduce the squeezed thermal vacuum state, defined as $S_{1}\left(r\right)\left\|0(\beta)\right\rangle$, where $S_{1}\left(r\right)=\exp[r(a^{2}-a^{\dagger 2})/2]$ is the single-mode squeezing operator for the real mode with $r$ being squeezing parameter. Note that Eq.(4) and the Baker-Hausdorf lemma $S_{1}\left(r\right)a^{{\dagger}}S_{1}^{{\dagger}}\left(r\right)=a^{{\dagger}}\cosh r+a\sinh r,$ (7) then we get $\displaystyle S_{1}\left(r\right)\left\|0(\beta)\right\rangle$ $\displaystyle=S_{1}\left(r\right)\text{sech}\theta\exp\left[a^{\dagger}\tilde{a}^{\dagger}\tanh\theta\right]\left\|0\tilde{0}\right\rangle$ $\displaystyle=\text{sech}\theta\text{sech}^{1/2}r\exp\left[\left(a^{{\dagger}}\cosh r+a\sinh r\right)\tilde{a}^{\dagger}\tanh\theta\right]$ $\displaystyle\times\exp\left[-\frac{a^{\dagger 2}}{2}\tanh r\right]\left\|0\tilde{0}\right\rangle,$ (8) where we have used $S_{1}\left(\lambda\right)\left\|0\right\rangle=$sech${}^{1/2}\lambda\exp[-a^{\dagger 2}/2\tanh\lambda]\left\|0\right\rangle.$ Further, note $e^{\tau a\tilde{a}^{\dagger}}a^{\dagger}e^{-\tau a\tilde{a}^{\dagger}}=a^{\dagger}+\tau\tilde{a}^{\dagger},$ and for operators $A,B$ satisfying the conditions $\left[A,[A,B]\right]=\left[B,[A,B]\right]=0,$ we have $e^{A+B}=e^{A}e^{B}e^{-[A,B]/2},$ thus Eq.(8) can be put into the following form $\displaystyle S_{1}\left(r\right)\left\|0(\beta)\right\rangle$ $\displaystyle=\text{sech}\theta\text{sech}^{1/2}r\exp\left\\{\frac{\tanh\theta}{\cosh r}a^{{\dagger}}\tilde{a}^{\dagger}\right.$ $\displaystyle+\left.\frac{\tanh r}{2}\left(\tilde{a}^{\dagger 2}\tanh^{2}\theta-a^{\dagger 2}\right)\right\\}\left\|0\tilde{0}\right\rangle.$ (9) Next, we shall use Eq.(9) to derive the average of operator $e^{fa^{\dagger}a}$ under the suqeezed thermal vacuum state $S_{1}\left(r\right)\left\|0(\beta)\right\rangle$, which is a bridge for our whole calculations. Notice $e^{f/2a^{\dagger}a}a^{\dagger}e^{-f/2a^{\dagger}a}=a^{\dagger}e^{f/2}$ and $e^{-f/2a^{\dagger}a}ae^{f/2a^{\dagger}a}=ae^{f/2}$, so we have $\displaystyle e^{f/2a^{\dagger}a}S_{1}\left(r\right)\left\|0(\beta)\right\rangle$ $\displaystyle=\text{sech}\theta\text{sech}^{1/2}r\exp\left\\{\frac{\tanh\theta}{\cosh r}a^{{\dagger}}\tilde{a}^{\dagger}e^{f/2}\right.$ $\displaystyle+\left.\frac{\tanh r}{2}\left(\tilde{a}^{\dagger 2}\tanh^{2}\theta-a^{\dagger 2}e^{f}\right)\right\\}\left\|0\tilde{0}\right\rangle,$ (10) which leads to $\displaystyle\left\langle e^{fa^{\dagger}a}\right\rangle$ $\displaystyle\equiv\left\langle 0(\beta)\right\|S_{1}^{{\dagger}}\left(r\right)e^{fa^{\dagger}a}S_{1}\left(r\right)\left\|0(\beta)\right\rangle$ $\displaystyle=\text{sech}^{2}\theta\text{sech}r\left\langle 0\tilde{0}\right\|\exp\left\\{\frac{e^{f/2}\tanh\theta}{\cosh r}a\tilde{a}\right.$ $\displaystyle+\left.\frac{\tanh r}{2}\left(\tilde{a}^{2}\tanh^{2}\theta-a^{2}e^{f}\right)\right\\}$ $\displaystyle\times\exp\left\\{\frac{\tanh\theta}{\cosh r}a^{{\dagger}}\tilde{a}^{\dagger}e^{f/2}\right.$ $\displaystyle+\left.\frac{\tanh r}{2}\left(\tilde{a}^{\dagger 2}\tanh^{2}\theta-a^{\dagger 2}e^{f}\right)\right\\}\left\|0\tilde{0}\right\rangle$ $\displaystyle=\left[Ce^{2f}-2Be^{f}+A\right]^{-1/2},$ (11) where we have set $\displaystyle A$ $\displaystyle=\cosh^{4}\theta+\cosh 2\theta\sinh^{2}r$ $\displaystyle=n_{c}^{2}+\left(2n_{c}+1\right)\cosh^{2}r,$ $\displaystyle B$ $\displaystyle=\sinh^{2}\theta\cosh^{2}\allowbreak\theta=n_{c}\left(n_{c}+1\right),$ $\displaystyle C$ $\displaystyle=\cosh^{4}\theta-\cosh 2\theta\cosh^{2}r$ $\displaystyle=n_{c}^{2}-\left(2n_{c}+1\right)\sinh^{2}r,$ (12) and we have used the completness relation of coherent state $\int d^{2}zd^{2}\tilde{z}\left\|z\tilde{z}\right\rangle\left\langle z\tilde{z}\right\|/\pi^{2}=1,$ here $\left\|z\right\rangle$ and $\left\|\tilde{z}\right\rangle$ is the coherent state in real and fictitious modes, respectively, and the formula 23 $\displaystyle\int\frac{d^{2}z}{\pi}\exp\left(\zeta\left\|z\right\|^{2}+\xi z+\eta z^{\ast}+fz^{2}+gz^{\ast 2}\right)$ $\displaystyle=\frac{1}{\sqrt{\zeta^{2}-4fg}}\exp\left[\frac{-\zeta\xi\eta+\xi^{2}g+\eta^{2}f}{\zeta^{2}-4fg}\right],$ (13) whose convergent condition is Re$\left(\zeta\pm f\pm g\right)<0,\ $Re($\zeta^{2}-4fg)/(\zeta\pm f\pm g)<0.$ Eq.(11) is very important for later calculation of photon-number distribution (PND) and normalization of photon- added (-subtracted) squeezed thermal states (PASTS, PSSTS). It is interesting to notice that the standard generating function of Legendre polynomials 25 is given by $\frac{1}{\sqrt{1-2xt+t^{2}}}=\sum_{m=0}^{\infty}P_{m}\left(x\right)t^{m},$ (14) thus comparing Eq.(11) with Eq.(14) we find $\displaystyle\left\langle e^{fa^{\dagger}a}\right\rangle$ $\displaystyle=A^{-1/2}\left[\frac{C}{A}e^{2f}-2\frac{B}{A}e^{f}+1\right]^{-1/2}$ $\displaystyle=A^{-1/2}\sum_{m=0}^{\infty}P_{m}\left(B/\sqrt{AC}\right)\left(\sqrt{C/A}e^{f}\right)^{m},$ (15) which indicates that the average value of $e^{fa^{{\dagger}}a}$ under squeezed thermal state (STS) is just the generating function of Legendre polynomials, a remarkable result. Next, we shall examine the normalizations and PNDs of PASTS and PSSTS by using Eqs.(11) and (15). ## IV Normalization and PND of PASTS The $m$-photon-added scheme, denoted by the mapping $\rho\rightarrow a^{{\dagger}m}\rho a^{m},$ was first proposed by Agarwal and Tara 2 . Here, we introduce the PASTS. Theoretically, the PASTS can be obtained by repeatedly operating the photon creation operator $a^{\dagger}$ on a STS, so its density operator is given by $\rho_{ad}=C_{a,m}^{-1}a^{{\dagger}m}S_{1}\rho_{th}S_{1}^{\dagger}a^{m},$ (16) where $m$ is the added photon number (a non-negative integer), $C_{a,m}^{-1}$ is the normalization constant to be determined. ### IV.1 Normalization of PASTS To fully describe a quantum state, its normalization is usually necessary. Next, we shall employ the fact (5) and (11), (15) to realize our aim. According to the normalization condition tr$\rho_{ad}=1$ and the TFD, we have $\displaystyle C_{a,m}$ $\displaystyle=\mathtt{tr}\left[a^{{\dagger}m}S_{1}\rho_{th}S_{1}^{\dagger}a^{m}\right]$ $\displaystyle=\mathtt{Tr}\left[a^{{\dagger}m}S_{1}\left\|0(\beta)\right\rangle\left\langle 0(\beta)\right\|S_{1}^{\dagger}a^{m}\right]$ $\displaystyle=\left\langle 0(\beta)\right\|S_{1}^{\dagger}a^{m}a^{{\dagger}m}S_{1}\left\|0(\beta)\right\rangle,$ (17) which implies that the calculation of normaliation factor $C_{a,m}$ is converted to a matrix element after introducing the thermal vacuum state $\left\|0(\beta)\right\rangle$. Note the operator identity 24 $e^{\tau a^{{\dagger}}a}=e^{-\tau}\vdots\exp[(1-e^{-\tau})a^{{\dagger}}a]\vdots$, we see $\sum_{m=0}^{\infty}\frac{\tau^{m}}{m!}a^{m}a^{{\dagger}m}=\vdots e^{\tau a^{{\dagger}}a}\vdots=\left(\frac{1}{1-\tau}\right)^{a^{{\dagger}}a+1},$ (18) where the symbol $\vdots$ $\vdots$ denotes antinormally ordering. Thus using Eqs.(11), (17) and (18), ($e^{f}\rightarrow\frac{1}{1-\tau}$) we have $\displaystyle\sum_{m=0}^{\infty}\frac{\tau^{m}}{m!}C_{a,m}$ $\displaystyle=\frac{1}{1-\tau}\left\langle 0(\beta)\right\|S_{1}^{\dagger}e^{a^{{\dagger}}a\ln\frac{1}{1-\tau}}S_{1}\left\|0(\beta)\right\rangle$ $\displaystyle=\left[A\tau^{2}-2D\tau+1\right]^{-1/2},$ (19) where we have set $\displaystyle D$ $\displaystyle=\cosh^{2}\theta\cosh 2r-\sinh^{2}r$ $\displaystyle=n_{c}\cosh 2r+\cosh^{2}r.$ (20) Comparing Eq.(19) with Eq.(14), and taking $\tau^{\prime}\rightarrow\sqrt{A}\tau$, we obtain $\displaystyle\sum_{m=0}^{\infty}\tau^{\prime m}\frac{C_{a,m}}{m!A^{m/2}}$ $\displaystyle=\left[\tau^{\prime 2}-2D/\sqrt{A}\tau^{\prime}+1\right]^{-1/2}$ $\displaystyle=\sum_{m=0}^{\infty}P_{m}\left(D/\sqrt{A}\right)\tau^{\prime m},$ (21) thus the normalization constant of PASTS is given by $C_{a,m}=m!A^{m/2}P_{m}\left(D/\sqrt{A}\right),$ (22) which is identical with the result in Ref.26 . It is noted that, for the case of no-photon-addition with $m=0$, $C_{a,0}=1$ as expected. Under the case of $m$-photon-addition thermal state (no squeezing) with $D=\allowbreak n_{c}+1$, $A=\allowbreak\left(n_{c}+1\right)^{2},$ and $P_{m}\left(1\right)=1$, then $C_{a,m}=m!\left(n_{c}+1\right)^{m}.$ The same result as Eq.(32) found in Ref.27 . In addition, when $r=0$ corresponding to photon-added thermal state, Eq.(22) just reduces to $C_{a,m}=m!\cosh^{2m}\theta$ 27 . ### IV.2 PND of PASTS The photon-number distribution (PND) is a key characteristic of every optical field. The PND, i.e., the probability of finding $n$ photons in a quantum state described by the density operator $\rho$, is $\mathcal{P}(n)=\mathtt{tr}\left[\left\|n\right\rangle\left\langle n\right\|\rho\right]$. In a similar spirit of deriving Eq.(22), noting $a^{m}\left\|n\right\rangle=\sqrt{n!/(n-m)!}\left\|n-m\right\rangle$ and $\left\|n\right\rangle=a^{\dagger n}/\sqrt{n!}\left\|0\right\rangle,$ the PND of the PASTS can be calculated as $\displaystyle\mathcal{P}_{a}(n)$ $\displaystyle=C_{a,m}^{-1}\mathtt{tr}\left[\left\|n\right\rangle\left\langle n\right\|a^{{\dagger}m}S_{1}\rho_{th}S_{1}^{\dagger}a^{m}\right]$ $\displaystyle=C_{a,m}^{-1}\mathtt{Tr}\left[\left\|n\right\rangle\left\langle n\right\|a^{{\dagger}m}S_{1}\left\|0(\beta)\right\rangle\left\langle 0(\beta)\right\|S_{1}^{\dagger}a^{m}\right]$ $\displaystyle=\frac{n!C_{a,m}^{-1}}{l!}\left\langle 0(\beta)\right\|S_{1}^{\dagger}\left\|l\right\rangle\left\langle l\right\|S_{1}\left\|0(\beta)\right\rangle$ $\displaystyle=\frac{n!C_{a,m}^{-1}}{\left(l!\right)^{2}}\left\langle 0(\beta)\right\|S_{1}^{\dagger}a^{\dagger l}\left\|0\right\rangle\left\langle 0\right\|a^{l}S_{1}\left\|0(\beta)\right\rangle$ (23) which leads to $\displaystyle\sum_{l=0}^{\infty}\tau^{l}\frac{l!}{n!}C_{a,m}\mathcal{P}_{a}(n)$ $\displaystyle=\sum_{l=0}^{\infty}\frac{\tau^{l}}{l!}\left\langle 0(\beta)\right\|S_{1}^{\dagger}\colon\left(a^{\dagger}a\right)^{l}e^{-a^{\dagger}a}\colon S_{1}\left\|0(\beta)\right\rangle$ $\displaystyle=\left\langle 0(\beta)\right\|S_{1}^{\dagger}e^{a^{\dagger}a\ln\tau}S_{1}\left\|0(\beta)\right\rangle,$ (24) where $l=n-m$ and the vacuum projector operator $\left\|0\right\rangle\left\langle 0\right\|=\colon e^{-a^{\dagger}a}\colon$ and the operator identity $e^{\lambda a^{\dagger}a}=\colon\exp[(e^{\lambda}-1)a^{\dagger}a]\colon$ are used. Using Eq.(11) again ($e^{f}\rightarrow\tau$) and comparing Eq.(24) with Eq. (14) we see $\displaystyle\sum_{l=0}^{\infty}\tau^{l}\frac{l!}{n!}C_{a,m}\mathcal{P}_{a}(n)$ $\displaystyle=A^{-1/2}\left[\frac{C}{A}\tau^{2}-2\frac{B}{A}\tau+1\right]^{-1/2}$ $\displaystyle=A^{-1/2}\sum_{l=0}^{\infty}P_{l}\left(B/\sqrt{AC}\right)\left(\sqrt{C/A}\tau\right)^{l},$ (25) which leads to the PND of PASTS $\mathcal{P}_{a}(n)=\frac{n!C_{a,m}^{-1}\left(C/A\right)^{\left(n-m\right)/2}}{\left(n-m\right)!\sqrt{A}}P_{n-m}\left(B/\sqrt{AC}\right),$ (26) a Legendre polynomial with a condition $n\geqslant m$ which implies that the photon-number ($n$) involved in PASTS is always no-less than the photon-number ($m$) operated on the STS, and there is no photon distribution when $n<m$). It is obvious that when $m=0$ corresponding to the STS, then the PND of STS is also a Legendre distribution 28 . ## V Normalization and PND of PSSTS Next, we turn our attention to discussing the photon-subtracted squeezed thermal state (PSSTS), defined as $\rho_{sb}=C_{s,m}^{-1}a^{m}S_{1}\rho_{th}S_{1}^{\dagger}a^{{\dagger}m},$ (27) where $m$ is the subtracted photon number (a non-negative integer), and $C_{s,m}$ is a normalized constant. In a similar way to deriving Eq.(22), we have $C_{s,m}=\left\langle 0(\beta)\right\|S_{1}^{\dagger}a^{{\dagger}m}a^{m}S_{1}\left\|0(\beta)\right\rangle,$ (28) so employing $e^{\lambda a^{\dagger}a}=\colon\exp[(e^{\lambda}-1)a^{\dagger}a]\colon$ and Eq.(11) ($e^{f}\rightarrow 1+\tau$) we see $\displaystyle\sum_{m=0}^{\infty}\frac{\tau^{m}}{m!}C_{s,m}$ $\displaystyle=\left\langle 0(\beta)\right\|S_{1}^{\dagger}e^{a^{{\dagger}}a\ln(1+\tau)}S_{1}\left\|0(\beta)\right\rangle$ $\displaystyle=\left[C\tau^{2}-2E\tau+1\right]^{-1/2},$ (29) where $\displaystyle E$ $\displaystyle=\cosh 2r\cosh^{2}\theta-\cosh^{2}r$ $\displaystyle=\frac{1}{2}\left[(2n_{c}+1)\cosh 2r-1\right].$ (30) Comparing Eq.(29) with Eq.(14) yields $C_{s,m}=m!C^{m/2}P_{m}\left(E/\sqrt{C}\right),$ (31) which is the normalization factor of PSSTS. When $r=0$ corresponding to photon-subtracted thermal state, Eq.(31) just reduces to $C_{s,m}=m!\sinh^{2m}\theta$ 27 . Using the same procession as obtaining Eq.(26), the PND of PSSTS is given by $\displaystyle\mathcal{P}_{s}(n)$ $\displaystyle=C_{s,m}^{-1}\left\langle 0(\beta)\right\|S_{1}^{\dagger}a^{{\dagger}m}\left\|n\right\rangle\left\langle n\right\|a^{m}S_{1}\left\|0(\beta)\right\rangle$ $\displaystyle=\frac{1}{n!}C_{s,m}^{-1}\left\langle 0(\beta)\right\|S_{1}^{\dagger}\colon a^{{\dagger}m+n}a^{m+n}e^{-a^{{\dagger}}a}\colon S_{1}\left\|0(\beta)\right\rangle,$ (32) so $\left(k=m+n\right)$ $\displaystyle\sum_{k=0}^{\infty}\frac{\tau^{k}}{k!}n!C_{s,m}\mathcal{P}_{s}(n)$ $\displaystyle=\left\langle 0(\beta)\right\|S_{1}^{\dagger}e^{a^{{\dagger}}a\ln\tau}S_{1}\left\|0(\beta)\right\rangle$ $\displaystyle=\text{R.H.S of (\ref{1.21}),}$ (33) whcih leads to the PND of PSSTS $\mathcal{P}_{s}(n)=\frac{\left(m+n\right)!}{n!C_{s,m}\sqrt{A}}\left(C/A\right)^{m+n/2}P_{m+n}\left(B/\sqrt{AC}\right),$ (34) a Legendre polynomial, which is same as the result of Ref.28 . ## VI Conclusion In this paper, we present a new concise approach for normalizing m-photon- added (-subtracted) STS and for deriving the PNDs, which improve the method used in Refs. 26 ; 28 . That is to say, using the thermal field dynamics theory, we convert the thermal state to a pure state in doubled Fock space in which the calculations of ensemble averages under a mixed state $\rho$, $\left\langle A\right\rangle=\mathtt{tr}\left(A\rho\right)/\mathtt{tr}\left(\rho\right)\ $is replaced by an equivalent expectation values with a pure state $\left\|0(\beta)\right\rangle$, i.e., $\left\langle A\right\rangle=\left\langle 0(\beta)\right\|A\left\|0(\beta)\right\rangle$. It is shown that the average value of $e^{fa^{{\dagger}}a}$ under STS is just the generating function of Legendre polynomials, a remarkable result. Based on this point, the normalization and PNDs of m-photon added (or subtracted) STS are easily obtained as the Legendre polynomials. The generating function of the Legendre polynomials and the average value of $e^{fa^{{\dagger}}a}$ under STS are used in the whole calculation. ACKNOWLEDGEMENTS: Work supported by the National Natural Science Foundation of China (Grant Nos. 11047133, 60978009), the Major Research Plan of the National Natural Science Foundation of China (Grant No. 91121023), and the “973” Project (Grant No. 2011CBA00200), as well as the Natural Science Foundation of Jiangxi Province of China (No. 2010GQW0027). ## References * (1) D. Bouwmeester, A. Ekert and A. Zeilinger, The Physics of Quantum Information (Springer-Verlag, 2000). * (2) G. S. Agarwal and K. Tara, Phys. Rev. A 43, 492 (1991); Phys. Rev. A 46, 485 (1992). * (3) Z. X. Zhang and H. Y. Fan, Phys. Lett. A 165, 14 (1992). * (4) A. Zavatta, S. Viciani, and M. Bellini, Science, 306, 660 (2004). * (5) A. Zavatta, V. Parigi, and M. Bellini, Phys. Rev. A 75, 052106 (2007). * (6) A. Zavatta, S. Viciani, and M. Bellini, Phys. Rev. A 72, 023820 (2005). * (7) V. Parigi, A. Zavatta, M. S. Kim, and M. Bellini, Science, 317, 1890 (2007). * (8) A. Ourjoumtsev, R. Tualle-Brouri, J. Laurat, Ph. Grangier, Science 312, 83 (2006). * (9) A. Ourjoumtsev, A. Dantan, R. Tualle-Brouri, and Ph.Grangier, Phys. Rev. Lett. 98, 030502 (2007). * (10) P. Marek, H. Jeong and M. S. Kim, Phys. Rev. A 78, 063811 (2008). * (11) S. L. Zhang and P. van Loock, Phys. Rev. A 82, 06216 (2010). * (12) Li-yun Hu and Hong-yi Fan, J. Opt. Soc. Am. B, 25, 1955 (2008). * (13) Li-yun Hu and Hong-yi Fan, Chin. Phys. B 18, 4657 (2009). * (14) T. Opatrný, G. Kurizki and D-G. Welsch, Phys. Rev. A 61, 032302 (2000). * (15) S. Olivares and M. G. A. Paris, J. Opt. B: Quantum Semiclassical Opt. 7, S392 (2005). * (16) A. Kitagawa,M. Takeoka,M. Sasaki, and A. Chefles, Phys. Rev. A 73, 042310 (2006). * (17) Y. Yang and F. L. Li, J. Opt. Soc. Am. B, 26, 830 (2009). * (18) A. Biswas and G. S. Agarwal, Phys. Rev. A 75, 032104 (2007). * (19) S. Y. Lee and H. Nha, Phys. Rev. A 82, 053812 (2010). * (20) H. Y. Fan and N. Q. Jiang, Chin. Phys. Lett. 27, 044206 (2010). * (21) Y. Takahashi and H. Umezawa, Collective Phenomena 2, 55 (1975). * (22) Memorial Issue for H. Umezawa, Int. J. Mod. Phys. B 10, 1695 (1996) memorial issue and references therein. * (23) H. Umezawa, Advanced Field Theory—–Micro; Macro; and Thermal Physics (AIP, 1993). * (24) Li-yun Hu and Hong-yi Fan, Chin. Phys. Lett. 26, 090307 (2009). * (25) R. R. Puri, Mathematical Methods of Quantum Optics (Springer-Verlag, Berlin/Heidelberg/New York, 2001), p. 269 (A.29). * (26) George B. Arfken, Hans J. Weber, Mathematical Methods for Physicists, Elsevier Academic Press, p. 743, (2005). * (27) Hongyi Fan, Representation and Transformation Theory in Quantum Mechanics—–Progress of Dirac’s symbolic method (Shanghai Scientific and technical press, Shanghai, 1997). * (28) Li-yun Hu and Z. M. Zhang, [quant-ph] arXiv: 1110.6587. * (29) Li-yun Hu and Hong-yi Fan, Mod. Phys. Lett. A 24, 2263 (2009). * (30) Li-yun Hu, Xue-xiang Xu, Zi-sheng Wang, and Xue-fen Xu, Phys. Rev. A 82, 043842 (2010).	arxiv-papers	2011-12-05T07:55:27	2024-09-04T02:49:24.962879	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Li-Yun Hu and Zhi-Ming Zhang", "submitter": "Liyun Hu", "url": "https://arxiv.org/abs/1112.0854" }
1112.0938	The LHCb collaboration # Evidence for $C\\!P$ violation in time-integrated $\bm{D^{0}\rightarrow h^{-}h^{+}}$ decay rates R. Aaij23, C. Abellan Beteta35,n, B. Adeva36, M. Adinolfi42, C. Adrover6, A. Affolder48, Z. Ajaltouni5, J. Albrecht37, F. Alessio37, M. Alexander47, G. Alkhazov29, P. Alvarez Cartelle36, A.A. Alves Jr22, S. Amato2, Y. Amhis38, J. Anderson39, R.B. Appleby50, O. Aquines Gutierrez10, F. Archilli18,37, L. Arrabito53, A. Artamonov 34, M. Artuso52,37, E. Aslanides6, G. Auriemma22,m, S. Bachmann11, J.J. Back44, D.S. Bailey50, V. Balagura30,37, W. Baldini16, R.J. Barlow50, C. Barschel37, S. Barsuk7, W. Barter43, A. Bates47, C. Bauer10, Th. Bauer23, A. Bay38, I. Bediaga1, S. Belogurov30, K. Belous34, I. Belyaev30,37, E. Ben-Haim8, M. Benayoun8, G. Bencivenni18, S. Benson46, J. Benton42, R. Bernet39, M.-O. Bettler17, M. van Beuzekom23, A. Bien11, S. Bifani12, T. Bird50, A. Bizzeti17,h, P.M. Bjørnstad50, T. Blake37, F. Blanc38, C. Blanks49, J. Blouw11, S. Blusk52, A. Bobrov33, V. Bocci22, A. Bondar33, N. Bondar29, W. Bonivento15, S. Borghi47,50, A. Borgia52, T.J.V. Bowcock48, C. Bozzi16, T. Brambach9, J. van den Brand24, J. Bressieux38, D. Brett50, M. Britsch10, T. Britton52, N.H. Brook42, H. Brown48, A. Büchler-Germann39, I. Burducea28, A. Bursche39, J. Buytaert37, S. Cadeddu15, O. Callot7, M. Calvi20,j, M. Calvo Gomez35,n, A. Camboni35, P. Campana18,37, A. Carbone14, G. Carboni21,k, R. Cardinale19,i,37, A. Cardini15, L. Carson49, K. Carvalho Akiba2, G. Casse48, M. Cattaneo37, Ch. Cauet9, M. Charles51, Ph. Charpentier37, N. Chiapolini39, K. Ciba37, X. Cid Vidal36, G. Ciezarek49, P.E.L. Clarke46,37, M. Clemencic37, H.V. Cliff43, J. Closier37, C. Coca28, V. Coco23, J. Cogan6, P. Collins37, A. Comerma-Montells35, F. Constantin28, A. Contu51, A. Cook42, M. Coombes42, G. Corti37, G.A. Cowan38, R. Currie46, C. D’Ambrosio37, P. David8, P.N.Y. David23, I. De Bonis4, S. De Capua21,k, M. De Cian39, F. De Lorenzi12, J.M. De Miranda1, L. De Paula2, P. De Simone18, D. Decamp4, M. Deckenhoff9, H. Degaudenzi38,37, L. Del Buono8, C. Deplano15, D. Derkach14,37, O. Deschamps5, F. Dettori24, J. Dickens43, H. Dijkstra37, P. Diniz Batista1, F. Domingo Bonal35,n, S. Donleavy48, F. Dordei11, A. Dosil Suárez36, D. Dossett44, A. Dovbnya40, F. Dupertuis38, R. Dzhelyadin34, A. Dziurda25, S. Easo45, U. Egede49, V. Egorychev30, S. Eidelman33, D. van Eijk23, F. Eisele11, S. Eisenhardt46, R. Ekelhof9, L. Eklund47, Ch. Elsasser39, D. Elsby55, D. Esperante Pereira36, L. Estève43, A. Falabella16,14,e, E. Fanchini20,j, C. Färber11, G. Fardell46, C. Farinelli23, S. Farry12, V. Fave38, V. Fernandez Albor36, M. Ferro-Luzzi37, S. Filippov32, C. Fitzpatrick46, M. Fontana10, F. Fontanelli19,i, R. Forty37, M. Frank37, C. Frei37, M. Frosini17,f,37, S. Furcas20, A. Gallas Torreira36, D. Galli14,c, M. Gandelman2, P. Gandini51, Y. Gao3, J-C. Garnier37, J. Garofoli52, J. Garra Tico43, L. Garrido35, D. Gascon35, C. Gaspar37, N. Gauvin38, M. Gersabeck37, T. Gershon44,37, Ph. Ghez4, V. Gibson43, V.V. Gligorov37, C. Göbel54, D. Golubkov30, A. Golutvin49,30,37, A. Gomes2, H. Gordon51, M. Grabalosa Gándara35, R. Graciani Diaz35, L.A. Granado Cardoso37, E. Graugés35, G. Graziani17, A. Grecu28, E. Greening51, S. Gregson43, B. Gui52, E. Gushchin32, Yu. Guz34, T. Gys37, G. Haefeli38, C. Haen37, S.C. Haines43, T. Hampson42, S. Hansmann-Menzemer11, R. Harji49, N. Harnew51, J. Harrison50, P.F. Harrison44, T. Hartmann56, J. He7, V. Heijne23, K. Hennessy48, P. Henrard5, J.A. Hernando Morata36, E. van Herwijnen37, E. Hicks48, K. Holubyev11, P. Hopchev4, W. Hulsbergen23, P. Hunt51, T. Huse48, R.S. Huston12, D. Hutchcroft48, D. Hynds47, V. Iakovenko41, P. Ilten12, J. Imong42, R. Jacobsson37, A. Jaeger11, M. Jahjah Hussein5, E. Jans23, F. Jansen23, P. Jaton38, B. Jean-Marie7, F. Jing3, M. John51, D. Johnson51, C.R. Jones43, B. Jost37, M. Kaballo9, S. Kandybei40, M. Karacson37, T.M. Karbach9, J. Keaveney12, I.R. Kenyon55, U. Kerzel37, T. Ketel24, A. Keune38, B. Khanji6, Y.M. Kim46, M. Knecht38, R. Koopman24, P. Koppenburg23, A. Kozlinskiy23, L. Kravchuk32, K. Kreplin11, M. Kreps44, G. Krocker11, P. Krokovny11, F. Kruse9, K. Kruzelecki37, M. Kucharczyk20,25,37,j, T. Kvaratskheliya30,37, V.N. La Thi38, D. Lacarrere37, G. Lafferty50, A. Lai15, D. Lambert46, R.W. Lambert24, E. Lanciotti37, G. Lanfranchi18, C. Langenbruch11, T. Latham44, C. Lazzeroni55, R. Le Gac6, J. van Leerdam23, J.-P. Lees4, R. Lefèvre5, A. Leflat31,37, J. Lefrançois7, O. Leroy6, T. Lesiak25, L. Li3, L. Li Gioi5, M. Lieng9, M. Liles48, R. Lindner37, C. Linn11, B. Liu3, G. Liu37, J. von Loeben20, J.H. Lopes2, E. Lopez Asamar35, N. Lopez-March38, H. Lu38,3, J. Luisier38, A. Mac Raighne47, F. Machefert7, I.V. Machikhiliyan4,30, F. Maciuc10, O. Maev29,37, J. Magnin1, S. Malde51, R.M.D. Mamunur37, G. Manca15,d, G. Mancinelli6, N. Mangiafave43, U. Marconi14, R. Märki38, J. Marks11, G. Martellotti22, A. Martens8, L. Martin51, A. Martín Sánchez7, D. Martinez Santos37, A. Massafferri1, Z. Mathe12, C. Matteuzzi20, M. Matveev29, E. Maurice6, B. Maynard52, A. Mazurov16,32,37, G. McGregor50, R. McNulty12, M. Meissner11, M. Merk23, J. Merkel9, R. Messi21,k, S. Miglioranzi37, D.A. Milanes13,37, M.-N. Minard4, J. Molina Rodriguez54, S. Monteil5, D. Moran12, P. Morawski25, R. Mountain52, I. Mous23, F. Muheim46, K. Müller39, R. Muresan28,38, B. Muryn26, B. Muster38, M. Musy35, J. Mylroie- Smith48, P. Naik42, T. Nakada38, R. Nandakumar45, I. Nasteva1, M. Nedos9, M. Needham46, N. Neufeld37, C. Nguyen-Mau38,o, M. Nicol7, V. Niess5, N. Nikitin31, A. Nomerotski51, A. Novoselov34, A. Oblakowska-Mucha26, V. Obraztsov34, S. Oggero23, S. Ogilvy47, O. Okhrimenko41, R. Oldeman15,d, M. Orlandea28, J.M. Otalora Goicochea2, P. Owen49, K. Pal52, J. Palacios39, A. Palano13,b, M. Palutan18, J. Panman37, A. Papanestis45, M. Pappagallo47, C. Parkes50,37, C.J. Parkinson49, G. Passaleva17, G.D. Patel48, M. Patel49, S.K. Paterson49, G.N. Patrick45, C. Patrignani19,i, C. Pavel-Nicorescu28, A. Pazos Alvarez36, A. Pellegrino23, G. Penso22,l, M. Pepe Altarelli37, S. Perazzini14,c, D.L. Perego20,j, E. Perez Trigo36, A. Pérez-Calero Yzquierdo35, P. Perret5, M. Perrin-Terrin6, G. Pessina20, A. Petrella16,37, A. Petrolini19,i, A. Phan52, E. Picatoste Olloqui35, B. Pie Valls35, B. Pietrzyk4, T. Pilař44, D. Pinci22, R. Plackett47, S. Playfer46, M. Plo Casasus36, G. Polok25, A. Poluektov44,33, E. Polycarpo2, D. Popov10, B. Popovici28, C. Potterat35, A. Powell51, J. Prisciandaro38, V. Pugatch41, A. Puig Navarro35, W. Qian52, J.H. Rademacker42, B. Rakotomiaramanana38, M.S. Rangel2, I. Raniuk40, G. Raven24, S. Redford51, M.M. Reid44, A.C. dos Reis1, S. Ricciardi45, K. Rinnert48, D.A. Roa Romero5, P. Robbe7, E. Rodrigues47,50, F. Rodrigues2, P. Rodriguez Perez36, G.J. Rogers43, S. Roiser37, V. Romanovsky34, M. Rosello35,n, J. Rouvinet38, T. Ruf37, H. Ruiz35, G. Sabatino21,k, J.J. Saborido Silva36, N. Sagidova29, P. Sail47, B. Saitta15,d, C. Salzmann39, M. Sannino19,i, R. Santacesaria22, C. Santamarina Rios36, R. Santinelli37, E. Santovetti21,k, M. Sapunov6, A. Sarti18,l, C. Satriano22,m, A. Satta21, M. Savrie16,e, D. Savrina30, P. Schaack49, M. Schiller24, S. Schleich9, M. Schlupp9, M. Schmelling10, B. Schmidt37, O. Schneider38, A. Schopper37, M.-H. Schune7, R. Schwemmer37, B. Sciascia18, A. Sciubba18,l, M. Seco36, A. Semennikov30, K. Senderowska26, I. Sepp49, N. Serra39, J. Serrano6, P. Seyfert11, M. Shapkin34, I. Shapoval40,37, P. Shatalov30, Y. Shcheglov29, T. Shears48, L. Shekhtman33, O. Shevchenko40, V. Shevchenko30, A. Shires49, R. Silva Coutinho44, T. Skwarnicki52, A.C. Smith37, N.A. Smith48, E. Smith51,45, K. Sobczak5, F.J.P. Soler47, A. Solomin42, F. Soomro18, B. Souza De Paula2, B. Spaan9, A. Sparkes46, P. Spradlin47, F. Stagni37, S. Stahl11, O. Steinkamp39, S. Stoica28, S. Stone52,37, B. Storaci23, M. Straticiuc28, U. Straumann39, V.K. Subbiah37, S. Swientek9, M. Szczekowski27, P. Szczypka38, T. Szumlak26, S. T’Jampens4, E. Teodorescu28, F. Teubert37, C. Thomas51, E. Thomas37, J. van Tilburg11, V. Tisserand4, M. Tobin39, S. Topp-Joergensen51, N. Torr51, E. Tournefier4,49, M.T. Tran38, A. Tsaregorodtsev6, N. Tuning23, M. Ubeda Garcia37, A. Ukleja27, P. Urquijo52, U. Uwer11, V. Vagnoni14, G. Valenti14, R. Vazquez Gomez35, P. Vazquez Regueiro36, S. Vecchi16, J.J. Velthuis42, M. Veltri17,g, B. Viaud7, I. Videau7, X. Vilasis-Cardona35,n, J. Visniakov36, A. Vollhardt39, D. Volyanskyy10, D. Voong42, A. Vorobyev29, H. Voss10, S. Wandernoth11, J. Wang52, D.R. Ward43, N.K. Watson55, A.D. Webber50, D. Websdale49, M. Whitehead44, D. Wiedner11, L. Wiggers23, G. Wilkinson51, M.P. Williams44,45, M. Williams49, F.F. Wilson45, J. Wishahi9, M. Witek25, W. Witzeling37, S.A. Wotton43, K. Wyllie37, Y. Xie46, F. Xing51, Z. Xing52, Z. Yang3, R. Young46, O. Yushchenko34, M. Zavertyaev10,a, F. Zhang3, L. Zhang52, W.C. Zhang12, Y. Zhang3, A. Zhelezov11, L. Zhong3, E. Zverev31, A. Zvyagin37. 1Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, Brazil 2Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil 3Center for High Energy Physics, Tsinghua University, Beijing, China 4LAPP, Université de Savoie, CNRS/IN2P3, Annecy-Le-Vieux, France 5Clermont Université, Université Blaise Pascal, CNRS/IN2P3, LPC, Clermont- Ferrand, France 6CPPM, Aix-Marseille Université, CNRS/IN2P3, Marseille, France 7LAL, Université Paris-Sud, CNRS/IN2P3, Orsay, France 8LPNHE, Université Pierre et Marie Curie, Université Paris Diderot, CNRS/IN2P3, Paris, France 9Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany 10Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany 11Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg, Germany 12School of Physics, University College Dublin, Dublin, Ireland 13Sezione INFN di Bari, Bari, Italy 14Sezione INFN di Bologna, Bologna, Italy 15Sezione INFN di Cagliari, Cagliari, Italy 16Sezione INFN di Ferrara, Ferrara, Italy 17Sezione INFN di Firenze, Firenze, Italy 18Laboratori Nazionali dell’INFN di Frascati, Frascati, Italy 19Sezione INFN di Genova, Genova, Italy 20Sezione INFN di Milano Bicocca, Milano, Italy 21Sezione INFN di Roma Tor Vergata, Roma, Italy 22Sezione INFN di Roma La Sapienza, Roma, Italy 23Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands 24Nikhef National Institute for Subatomic Physics and Vrije Universiteit, Amsterdam, The Netherlands 25Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of Sciences, Kraców, Poland 26AGH University of Science and Technology, Kraców, Poland 27Soltan Institute for Nuclear Studies, Warsaw, Poland 28Horia Hulubei National Institute of Physics and Nuclear Engineering, Bucharest-Magurele, Romania 29Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia 30Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia 31Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow, Russia 32Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN), Moscow, Russia 33Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State University, Novosibirsk, Russia 34Institute for High Energy Physics (IHEP), Protvino, Russia 35Universitat de Barcelona, Barcelona, Spain 36Universidad de Santiago de Compostela, Santiago de Compostela, Spain 37European Organization for Nuclear Research (CERN), Geneva, Switzerland 38Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland 39Physik-Institut, Universität Zürich, Zürich, Switzerland 40NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine 41Institute for Nuclear Research of the National Academy of Sciences (KINR), Kyiv, Ukraine 42H.H. Wills Physics Laboratory, University of Bristol, Bristol, United Kingdom 43Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom 44Department of Physics, University of Warwick, Coventry, United Kingdom 45STFC Rutherford Appleton Laboratory, Didcot, United Kingdom 46School of Physics and Astronomy, University of Edinburgh, Edinburgh, United Kingdom 47School of Physics and Astronomy, University of Glasgow, Glasgow, United Kingdom 48Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom 49Imperial College London, London, United Kingdom 50School of Physics and Astronomy, University of Manchester, Manchester, United Kingdom 51Department of Physics, University of Oxford, Oxford, United Kingdom 52Syracuse University, Syracuse, NY, United States 53CC-IN2P3, CNRS/IN2P3, Lyon-Villeurbanne, France, associated member 54Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de Janeiro, Brazil, associated to 2 55University of Birmingham, Birmingham, United Kingdom 56Physikalisches Institut, Universität Rostock, Rostock, Germany, associated to 11 aP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS), Moscow, Russia bUniversità di Bari, Bari, Italy cUniversità di Bologna, Bologna, Italy dUniversità di Cagliari, Cagliari, Italy eUniversità di Ferrara, Ferrara, Italy fUniversità di Firenze, Firenze, Italy gUniversità di Urbino, Urbino, Italy hUniversità di Modena e Reggio Emilia, Modena, Italy iUniversità di Genova, Genova, Italy jUniversità di Milano Bicocca, Milano, Italy kUniversità di Roma Tor Vergata, Roma, Italy lUniversità di Roma La Sapienza, Roma, Italy mUniversità della Basilicata, Potenza, Italy nLIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain oHanoi University of Science, Hanoi, Viet Nam ###### Abstract A search for time-integrated $C\\!P$ violation in $D^{0}\rightarrow h^{-}h^{+}$ ($h=K$, $\pi$) decays is presented using 0.62 $\mbox{\,fb}^{-1}$ of data collected by LHCb in 2011. The flavor of the charm meson is determined by the charge of the slow pion in the $D^{+}\rightarrow D^{0}\pi^{+}$ and $D^{-}\rightarrow\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}\pi^{-}$ decay chains. The difference in $C\\!P$ asymmetry between $D^{0}\rightarrow K^{-}K^{+}$ and $D^{0}\rightarrow\pi^{-}\pi^{+}$, $\Delta A_{C\\!P}\equiv A_{C\\!P}(K^{-}K^{+})\,-\,A_{C\\!P}(\pi^{-}\pi^{+})$, is measured to be $\left[-0.82\pm 0.21(\mathrm{stat.})\pm 0.11(\mathrm{syst.})\right]\%$. This differs from the hypothesis of $C\\!P$ conservation by $3.5$ standard deviations. ###### pacs: 11.30.Er, 13.25.Ft EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN) \| \| ---\|---\|--- \| \| LHCb-PAPER-2011-023 \| \| CERN-PH-EP-2011-208 The charm sector is a promising place to probe for the effects of physics beyond the Standard Model (SM). There has been a resurgence of interest in the past few years since evidence for $D^{0}$ mixing was first seen bib:babar_mixing_moriond ; bib:belle_mixing_moriond . Mixing is now well- established bib:hfag at a level which is consistent with, but at the upper end of, SM expectations falk_grossman_ligeti_nir_petrov . By contrast, no evidence for $C\\!P$ violation in charm decays has yet been found. The time-dependent $C\\!P$ asymmetry $A_{C\\!P}(f;\,t)$ for $D^{0}$ decays to a CP eigenstate $f$ (with $f=\bar{f}$) is defined as $A_{C\\!P}(f;\,t)\equiv\frac{\Gamma(D^{0}(t)\rightarrow f)-\Gamma(\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}(t)\rightarrow f)}{\Gamma(D^{0}(t)\rightarrow f)+\Gamma(\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}(t)\rightarrow f)},$ (1) where $\Gamma$ is the decay rate for the process indicated. In general $A_{C\\!P}(f;\,t)$ depends on $f$. For $f=K^{-}K^{+}$ and $f=\pi^{-}\pi^{+}$, $A_{C\\!P}(f;\,t)$ can be expressed in terms of two contributions: a direct component associated with $C\\!P$ violation in the decay amplitudes, and an indirect component associated with $C\\!P$ violation in the mixing or in the interference between mixing and decay. In the limit of U-spin symmetry, the direct component is equal in magnitude and opposite in sign for $K^{-}K^{+}$ and $\pi^{-}\pi^{+}$, though the size of U-spin breaking effects remains to be quantified precisely bib:grossman_kagan_nir . The magnitudes of $C\\!P$ asymmetries in decays to these final states are expected to be small in the SM bib:cicerone ; bib:lenz ; bib:grossman_kagan_nir ; bib:petrov , with predictions of up to $\mathcal{O}(10^{-3})$. However, beyond the SM the rate of $C\\!P$ violation could be enhanced bib:grossman_kagan_nir ; bib:littlest_higgs . The asymmetry $A_{C\\!P}(f;\,t)$ may be written to first order as bib:cdf_paper ; bib:bigi_d2hh $A_{C\\!P}(f;\,t)=a^{\mathrm{dir}}_{C\\!P}(f)\,+\,\frac{t}{\tau}a^{\mathrm{ind}}_{C\\!P},$ (2) where $a^{\mathrm{dir}}_{C\\!P}(f)$ is the direct $C\\!P$ asymmetry, $\tau$ is the $D^{0}$ lifetime, and $a^{\mathrm{ind}}_{C\\!P}$ is the indirect $C\\!P$ asymmetry. To a good approximation this latter quantity is universal bib:grossman_kagan_nir ; bib:kagan_sokoloff . The time-integrated asymmetry measured by an experiment, $A_{C\\!P}(f)$, depends upon the time-acceptance of that experiment. It can be written as $A_{C\\!P}(f)=a^{\mathrm{dir}}_{C\\!P}(f)\,+\,\frac{\langle t\rangle}{\tau}a^{\mathrm{ind}}_{C\\!P},$ (3) where $\langle t\rangle$ is the average decay time in the reconstructed sample. Denoting by $\Delta$ the differences between quantities for $D^{0}\rightarrow K^{-}K^{+}$ and $D^{0}\rightarrow\pi^{-}\pi^{+}$ it is then possible to write $\displaystyle\Delta A_{C\\!P}$ $\displaystyle\equiv$ $\displaystyle A_{C\\!P}(K^{-}K^{+})\,-\,A_{C\\!P}(\pi^{-}\pi^{+})$ $\displaystyle=$ $\displaystyle\left[a^{\mathrm{dir}}_{C\\!P}(K^{-}K^{+})\,-\,a^{\mathrm{dir}}_{C\\!P}(\pi^{-}\pi^{+})\right]\,+\,\frac{\Delta\langle t\rangle}{\tau}a^{\mathrm{ind}}_{C\\!P}.$ In the limit that $\Delta\langle t\rangle$ vanishes, $\Delta A_{C\\!P}$ is equal to the difference in the direct $C\\!P$ asymmetry between the two decays. However, if the time-acceptance is different for the $K^{-}K^{+}$ and $\pi^{-}\pi^{+}$ final states, a contribution from indirect $C\\!P$ violation remains. The most precise measurements to date of the time-integrated $C\\!P$ asymmetries in $D^{0}\rightarrow K^{-}K^{+}$ and $D^{0}\rightarrow\pi^{-}\pi^{+}$ were made by the CDF, BaBar, and Belle collaborations bib:cdf_paper ; bib:babar_paper2008 ; bib:belle_paper2008 . The Heavy Flavor Averaging Group (HFAG) has combined time-integrated and time- dependent measurements of $C\\!P$ asymmetries, taking account of the different decay time acceptances, to obtain world average values for the indirect $C\\!P$ asymmetry of $a_{C\\!P}^{\mathrm{ind}}=(-0.03\pm 0.23)\%$ and the difference in direct $C\\!P$ asymmetry between the final states of $\Delta a_{C\\!P}^{\mathrm{dir}}=(-0.42\pm 0.27)\%$ bib:hfag . In this Letter, we present a measurement of the difference in time-integrated $C\\!P$ asymmetries between $D^{0}\rightarrow K^{-}K^{+}$ and $D^{0}\rightarrow\pi^{-}\pi^{+}$, performed with 0.62 $\mbox{\,fb}^{-1}$ of data collected at LHCb between March and June 2011. The flavor of the initial state ($D^{0}$ or $\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}$) is tagged by requiring a $D^{+}\rightarrow D^{0}\pi_{\mathrm{s}}^{+}$ decay, with the flavor determined by the charge of the slow pion ($\pi_{\mathrm{s}}^{+}$). The inclusion of charge-conjugate modes is implied throughout, except in the definition of asymmetries. The raw asymmetry for tagged $D^{0}$ decays to a final state $f$ is given by $A_{\mathrm{raw}}(f)$, defined as $A_{\mathrm{raw}}(f)\equiv\frac{N(D^{+}\rightarrow D^{0}(f)\pi_{s}^{+})\,-\,N(D^{-}\rightarrow\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}(f)\pi_{s}^{-})}{N(D^{+}\rightarrow D^{0}(f)\pi_{s}^{+})\,+\,N(D^{-}\rightarrow\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}(f)\pi_{s}^{-})},$ (5) where $N(X)$ refers to the number of reconstructed events of decay $X$ after background subtraction. To first order the raw asymmetries may be written as a sum of four components, due to physics and detector effects: $A_{\mathrm{raw}}(f)=A_{C\\!P}(f)\,+\,A_{\mathrm{D}}(f)\,+\,A_{\mathrm{D}}(\pi_{\mathrm{s}}^{+})\,+\,A_{\mathrm{P}}(D^{+}).$ (6) Here, $A_{\mathrm{D}}(f)$ is the asymmetry in selecting the $D^{0}$ decay into the final state $f$, $A_{\mathrm{D}}(\pi_{\mathrm{s}}^{+})$ is the asymmetry in selecting the slow pion from the $D^{+}$ decay chain, and $A_{\mathrm{P}}(D^{+})$ is the production asymmetry for $D^{+}$ mesons. The asymmetries $A_{\mathrm{D}}$ and $A_{\mathrm{P}}$ are defined in the same fashion as $A_{\mathrm{raw}}$. The first-order expansion is valid since the individual asymmetries are small. For a two-body decay of a spin-0 particle to a self-conjugate final state there can be no $D^{0}$ detection asymmetry, i.e. $A_{\mathrm{D}}(K^{-}K^{+})=A_{\mathrm{D}}(\pi^{-}\pi^{+})=0.$ Moreover, $A_{\mathrm{D}}(\pi_{\mathrm{s}}^{+})$ and $A_{\mathrm{P}}(D^{+})$ are independent of $f$ and thus in the first-order expansion of equation 5 those terms cancel in the difference $A_{\mathrm{raw}}(K^{-}K^{+})\,-\,A_{\mathrm{raw}}(\pi^{-}\pi^{+})$, resulting in $\Delta A_{C\\!P}=A_{\mathrm{raw}}(K^{-}K^{+})\,-\,A_{\mathrm{raw}}(\pi^{-}\pi^{+}).$ (7) To minimize second-order effects that are related to the slightly different kinematic properties of the two decay modes and that do not cancel in $\Delta A_{C\\!P}$, the analysis is performed in bins of the relevant kinematic variables, as discussed later. The LHCb detector is a forward spectrometer covering the pseudorapidity range $2<\eta<5$, and is described in detail in Ref. LHCb . The Ring Imaging Cherenkov (RICH) detectors are of particular importance to this analysis, providing kaon-pion discrimination for the full range of track momenta used. The nominal downstream beam direction is aligned with the $+z$ axis, and the field direction in the LHCb dipole is such that charged particles are deflected in the horizontal ($xz$) plane. The field polarity was changed several times during data taking: about 60% of the data were taken with the down polarity and 40% with the other. Selections are applied to provide samples of $D^{+}\rightarrow D^{0}\pi_{\mathrm{s}}^{+}$ candidates, with $D^{0}\rightarrow K^{-}K^{+}$ or $\pi^{-}\pi^{+}$. Events are required to pass both hardware and software trigger levels. A loose $D^{0}$ selection is applied in the final state of the software trigger, and in the offline analysis only candidates that are accepted by this trigger algorithm are considered. Both the trigger and offline selections impose a variety of requirements on kinematics and decay time to isolate the decays of interest, including requirements on the track fit quality, on the $D^{0}$ and $D^{+}$ vertex fit quality, on the transverse momentum ($\mbox{$p_{\mathrm{T}}$}>2$ GeV$/c$) and decay time ($ct>100\,\,\upmu\rm m$) of the $D^{0}$ candidate, on the angle between the $D^{0}$ momentum in the lab frame and its daughter momenta in the $D^{0}$ rest frame ($\|\cos\theta\|<0.9$), that the $D^{0}$ trajectory points back to a primary vertex, and that the $D^{0}$ daughter tracks do not. In addition, the offline analysis exploits the capabilities of the RICH system to distinguish between pions and kaons when reconstructing the $D^{0}$ meson, with no tracks appearing as both pion and kaon candidates. A fiducial region is implemented by imposing the requirement that the slow pion lies within the central part of the detector acceptance. This is necessary because the magnetic field bends pions of one charge to the left and those of the other charge to the right. For soft tracks at large angles in the $xz$ plane this implies that one charge is much more likely to remain within the 300 mrad horizontal detector acceptance, thus making $A_{\mathrm{D}}(\pi_{\mathrm{s}}^{+})$ large. Although this asymmetry is formally independent of the $D^{0}$ decay mode, it breaks the assumption that the raw asymmetries are small and it carries a risk of second-order systematic effects if the ratio of efficiencies of $D^{0}\rightarrow K^{-}K^{+}$ and $D^{0}\rightarrow\pi^{-}\pi^{+}$ varies in the affected region. The fiducial requirements therefore exclude edge regions in the slow pion $(p_{x},p)$ plane. Similarly, a small region of phase space in which one charge of slow pion is more likely to be swept into the beampipe region in the downstream tracking stations, and hence has reduced efficiency, is also excluded. After the implementation of the fiducial requirements about 70% of the events are retained. The invariant mass spectra of selected $K^{-}K^{+}$ and $\pi^{-}\pi^{+}$ pairs are shown in Fig. 1. The half-width at half-maximum of the signal lineshape is 8.6 MeV$/c^{2}$ for $K^{-}K^{+}$ and 11.2 MeV$/c^{2}$ for $\pi^{-}\pi^{+}$, where the difference is due to the kinematics of the decays and has no relevance for the subsequent analysis. The mass difference ($\delta m$) spectra of selected candidates, where $\delta m\equiv m(h^{-}h^{+}\pi_{\mathrm{s}}^{+})-m(h^{-}h^{+})-m(\pi^{+})$ for $h=K,\pi$, are shown in Fig. 2. Figure 1: Fits to the (a) $m(K^{-}K^{+})$ and (b) $m(\pi^{-}\pi^{+})$ spectra of $D^{+}$ candidates passing the selection and satisfying $0<\delta m<15$ MeV$/c^{2}$. The dashed line corresponds to the background component in the fit, and the vertical lines indicate the signal window of 1844–1884 MeV$/c^{2}$. Candidates are required to lie inside a wide $\delta m$ window of 0–15 MeV$/c^{2}$, and in Fig. 2 and for all subsequent results candidates are in addition required to lie in a mass signal window of 1844–1884 MeV$/c^{2}$. The $D^{+}$ signal yields are approximately $1.44\times 10^{6}$ in the $K^{-}K^{+}$ sample, and $0.38\times 10^{6}$ in the $\pi^{-}\pi^{+}$ sample. Charm from $b$-hadron decays is strongly suppressed by the requirement that the $D^{0}$ originate from a primary vertex, and accounts for only 3% of the total yield. Of the events that contain at least one $D^{+}$ candidate, 12% contain more than one candidate; this is expected due to background soft pions from the primary vertex and all candidates are accepted. The background- subtracted average decay time of $D^{0}$ candidates passing the selection is measured for each final state, and the fractional difference $\Delta\langle t\rangle/\tau$ is obtained. Systematic uncertainties on this quantity are assigned for the uncertainty on the world average $D^{0}$ lifetime $\tau$ $(0.04\%)$, charm from $b$-hadron decays $(0.18\%$), and the background- subtraction procedure $(0.04\%)$. Combining the systematic uncertainties in quadrature, we obtain $\Delta\langle t\rangle/\tau=\left[9.83\pm 0.22(\mathrm{stat.})\pm 0.19(\mathrm{syst.})\right]\%$. The $\pi^{-}\pi^{+}$ and $K^{-}K^{+}$ average decay time is $\langle t\rangle=\left(0.8539\pm 0.0005\right)$ ps, where the error is statistical only. Figure 2: Fits to the $\delta m$ spectra, where the $D^{0}$ is reconstructed in the final states (a) $K^{-}K^{+}$ and (b) $\pi^{-}\pi^{+}$, with mass lying in the window of 1844–1884 MeV$/c^{2}$. The dashed line corresponds to the background component in the fit. Fits are performed on the samples in order to determine $A_{\mathrm{raw}}(K^{-}K^{+})$ and $A_{\mathrm{raw}}(\pi^{-}\pi^{+})$. The production and detection asymmetries can vary with $p_{\mathrm{T}}$ and pseudorapidity $\eta$, and so can the detection efficiency of the two different $D^{0}$ decays, in particular through the effects of the particle identification requirements. The analysis is performed in 54 kinematic bins defined by the $p_{\mathrm{T}}$ and $\eta$ of the $D^{+}$ candidates, the momentum of the slow pion, and the sign of $p_{x}$ of the slow pion at the $D^{+}$ vertex. The events are further partitioned in two ways. First, the data are divided between the two dipole magnet polarities. Second, the first 60% of data are processed separately from the remainder, with the division aligned with a break in data taking due to an LHC technical stop. In total, 216 statistically independent measurements are considered for each decay mode. In each bin, one-dimensional unbinned maximum likelihood fits to the $\delta m$ spectra are performed. The signal is described as the sum of two Gaussian functions with a common mean $\mu$ but different widths $\sigma_{i}$, convolved with a function $B(\delta m;s)=\Theta(\delta m)\,\delta m^{s}$ taking account of the asymmetric shape of the measured $\delta m$ distribution. Here, $s\simeq-0.975$ is a shape parameter fixed to the value determined from the global fits shown in Fig. 2, $\Theta$ is the Heaviside step function, and the convolution runs over $\delta m$. The background is described by an empirical function of the form $1-e^{-(\delta m-\delta m_{0})/\alpha}$, where $\delta m_{0}$ and $\alpha$ are free parameters describing the threshold and shape of the function, respectively. The $D^{+}$ and $D^{-}$ samples in a given bin are fitted simultaneously and share all shape parameters, except for a charge-dependent offset in the central value $\mu$ and an overall scale factor in the mass resolution. The raw asymmetry in the signal yields is extracted directly from this simultaneous fit. No fit parameters are shared between the 216 subsamples of data, nor between the $K^{-}K^{+}$ and $\pi^{-}\pi^{+}$ final states. The fits do not distinguish between the signal and backgrounds that peak in $\delta m$. Such backgrounds can arise from $D^{+}$ decays in which the correct slow pion is found but the $D^{0}$ is partially mis-reconstructed. These backgrounds are suppressed by the use of tight particle identification requirements and a narrow $D^{0}$ mass window. From studies of the $D^{0}$ mass sidebands (1820–1840 and 1890–1910 MeV$/c^{2}$), this contamination is found to be approximately 1% of the signal yield and to have small raw asymmetry (consistent with zero asymmetry difference between the $K^{-}K^{+}$ and $\pi^{-}\pi^{+}$ final states). Its effect on the measurement is estimated in an ensemble of simulated experiments and found to be negligible; a systematic uncertainty is assigned below based on the statistical precision of the estimate. A value of $\Delta A_{C\\!P}$ is determined in each measurement bin as the difference between $A_{\mathrm{raw}}(K^{-}K^{+})$ and $A_{\mathrm{raw}}(\pi^{-}\pi^{+})$. Testing these 216 measurements for mutual consistency, we obtain $\chi^{2}/\mathrm{ndf}=211/215$ ($\chi^{2}$ probability of 56%). A weighted average is performed to yield the result $\Delta A_{C\\!P}=(-0.82\pm 0.21)\%$, where the uncertainty is statistical only. Numerous robustness checks are made. The value of $\Delta A_{C\\!P}$ is studied as a function of the time at which the data were taken (Fig. 3) and found to be consistent with a constant value ($\chi^{2}$ probability of 57%). The measurement is repeated with progressively more restrictive RICH particle identification requirements, finding values of $(-0.88\pm 0.26)\%$ and $(-1.03\pm 0.31)\%$; both of these values are consistent with the baseline result when correlations are taken into account. Table 1 lists $\Delta A_{C\\!P}$ for eight disjoint subsamples of data split according to magnet polarity, the sign of $p_{x}$ of the slow pion, and whether the data were taken before or after the technical stop. The $\chi^{2}$ probability for consistency among the subsamples is 45%. The significances of the differences between data taken before and after the technical stop, between the magnet polarities, and between $p_{x}>0$ and $p_{x}<0$ are $0.4$, $0.6$, and $0.7$ standard deviations, respectively. Other checks include applying electron and muon vetoes to the slow pion and to the $D^{0}$ daughters, use of different kinematic binnings, validation of the size of the statistical uncertainties with Monte Carlo pseudo-experiments, tightening of kinematic requirements, testing for variation of the result with the multiplicity of tracks and of primary vertices in the event, use of other signal and background parameterizations in the fit, and imposing a full set of common shape parameters between $D^{+}$ and $D^{-}$ candidates. Potential biases due to the inclusive hardware trigger selection are investigated with the subsample of data in which one of the signal final-state tracks is directly responsible for the hardware trigger decision. In all cases good stability is observed. For several of these checks, a reduced number of kinematic bins are used for simplicity. No systematic dependence of $\Delta A_{C\\!P}$ is observed with respect to the kinematic variables. Figure 3: Time-dependence of the measurement. The data are divided into 19 disjoint, contiguous, time-ordered blocks and the value of $\Delta A_{C\\!P}$ measured in each block. The horizontal red dashed line shows the result for the combined sample. The vertical dashed line indicates the technical stop referred to in Table 1. Table 1: Values of $\Delta A_{C\\!P}$ measured in subsamples of the data, and the $\chi^{2}/\mathrm{ndf}$ and corresponding $\chi^{2}$ probabilities for internal consistency among the 27 bins in each subsample. The data are divided before and after a technical stop (TS), by magnet polarity (up, down), and by the sign of $p_{x}$ for the slow pion (left, right). The consistency among the eight subsamples is $\chi^{2}/\mathrm{ndf}=6.8/7$ (45%). Subsample \| $\Delta A_{C\\!P}~{}[\%]$ \| $\chi^{2}/\mathrm{ndf}$ ---\|---\|--- Pre-TS, up, left \| $-1.22\pm 0.59$ \| $13/26~{}(98\%)$ Pre-TS, up, right \| $-1.43\pm 0.59$ \| $27/26~{}(39\%)$ Pre-TS, down, left \| $-0.59\pm 0.52$ \| $19/26~{}(84\%)$ Pre-TS, down, right \| $-0.51\pm 0.52$ \| $29/26~{}(30\%)$ Post-TS, up, left \| $-0.79\pm 0.90$ \| $26/26~{}(44\%)$ Post-TS, up, right \| $+0.42\pm 0.93$ \| $21/26~{}(77\%)$ Post-TS, down, left \| $-0.24\pm 0.56$ \| $34/26~{}(15\%)$ Post-TS, down, right \| $-1.59\pm 0.57$ \| $35/26~{}(12\%)$ All data \| $-0.82\pm 0.21$ \| $211/215~{}(56\%)$ Systematic uncertainties are assigned by: loosening the fiducial requirement on the slow pion; assessing the effect of potential peaking backgrounds in Monte Carlo pseudo-experiments; repeating the analysis with the asymmetry extracted through sideband subtraction in $\delta m$ instead of a fit; removing all candidates but one (chosen at random) in events with multiple candidates; and comparing with the result obtained without kinematic binning. In each case the full value of the change in result is taken as the systematic uncertainty. These uncertainties are listed in Table 2. The sum in quadrature is $0.11\%$. Combining statistical and systematic uncertainties in quadrature, this result is consistent at the $1\sigma$ level with the current HFAG world average bib:hfag . Table 2: Summary of absolute systematic uncertainties for $\Delta A_{C\\!P}$. Source \| Uncertainty ---\|--- Fiducial requirement \| 0.01% Peaking background asymmetry \| 0.04% Fit procedure \| 0.08% Multiple candidates \| 0.06% Kinematic binning \| 0.02% Total \| 0.11% In conclusion, the time-integrated difference in $C\\!P$ asymmetry between $D^{0}\rightarrow K^{-}K^{+}$ and $D^{0}\rightarrow\pi^{-}\pi^{+}$ decays has been measured to be $\Delta A_{C\\!P}=\left[-0.82\pm 0.21(\mathrm{stat.})\pm 0.11(\mathrm{syst.})\right]\%$ with 0.62 $\mbox{\,fb}^{-1}$ of 2011 data. Given the dependence of $\Delta A_{C\\!P}$ on the direct and indirect $C\\!P$ asymmetries, shown in Eq. (Evidence for $C\\!P$ violation in time-integrated $\bm{D^{0}\rightarrow h^{-}h^{+}}$ decay rates), and the measured value $\Delta\langle t\rangle/\tau=\left[9.83\pm 0.22(\mathrm{stat.})\pm 0.19(\mathrm{syst.})\right]\%$, the contribution from indirect $C\\!P$ violation is suppressed and $\Delta A_{C\\!P}$ is primarily sensitive to direct $C\\!P$ violation. Dividing the central value by the sum in quadrature of the statistical and systematic uncertainties, the significance of the measured deviation from zero is $3.5\sigma$. This is the first evidence for $C\\!P$ violation in the charm sector. To establish whether this result is consistent with the SM will require the analysis of more data, as well as improved theoretical understanding. ## Acknowledgements We express our gratitude to our colleagues in the CERN accelerator departments for the excellent performance of the LHC. We thank the technical and administrative staff at CERN and at the LHCb institutes, and acknowledge support from the National Agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil); CERN; NSFC (China); CNRS/IN2P3 (France); BMBF, DFG, HGF and MPG (Germany); SFI (Ireland); INFN (Italy); FOM and NWO (The Netherlands); SCSR (Poland); ANCS (Romania); MinES of Russia and Rosatom (Russia); MICINN, XuntaGal and GENCAT (Spain); SNSF and SER (Switzerland); NAS Ukraine (Ukraine); STFC (United Kingdom); NSF (USA). We also acknowledge the support received from the ERC under FP7 and the Region Auvergne. ## References * (1) BaBar collaboration, B. Aubert et al., Evidence for $D^{0}$–$\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ Mixing, Phys. Rev. Lett. 98 (2007) 211802, [arXiv:hep-ex/0703020] * (2) Belle collaboration, M. Staric et al., Evidence for $D^{0}$–$\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ Mixing, Phys. Rev. Lett. 98 (2007) 211803, [arXiv:hep-ex/0703036] * (3) Heavy Flavor Averaging Group, D. Asner et al., Averages of b-hadron, c-hadron, and tau-lepton Properties, arXiv:1010.1589 * (4) A. F. Falk, Y. Grossman, Z. Ligeti, Y. Nir, and A. A. Petrov, $D^{0}$–$\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ mass difference from a dispersion relation, Phys. Rev. D69 (2004) 114021, [arXiv:hep-ph/0402204] * (5) Y. Grossman, A. L. Kagan, and Y. Nir, New physics and CP violation in singly Cabibbo suppressed $D$ decays, Phys. Rev. D75 (2007) 036008, [arXiv:hep-ph/0609178] * (6) S. Bianco, F. L. Fabbri, D. Benson, and I. Bigi, A Cicerone for the physics of charm, Riv. Nuovo Cim. 26N7 (2003) 1, [arXiv:hep-ex/0309021] * (7) M. Bobrowski, A. Lenz, J. Riedl, and J. Rohrwild, How large can the SM contribution to CP violation in $D^{0}-\bar{D}^{0}$ mixing be?, JHEP 1003 (2010) 009, [arXiv:1002.4794] * (8) A. A. Petrov, Searching for New Physics with Charm, PoS BEAUTY2009 (2009) 024, [arXiv:1003.0906] * (9) I. I. Bigi, M. Blanke, A. J. Buras, and S. Recksiegel, CP Violation in $D^{0}$–$\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ Oscillations: General Considerations and Applications to the Littlest Higgs Model with T-Parity, JHEP 0907 (2009) 097, [arXiv:0904.1545] * (10) CDF collaboration, T. Aaltonen et al., Measurement of CP–violating asymmetries in $D^{0}\rightarrow\pi^{+}\pi^{-}$ and $D^{0}\rightarrow K^{+}K^{-}$ decays at CDF, arXiv:1111.5023. (submitted to Phys. Rev. D) * (11) I. I. Bigi, A. Paul, and S. Recksiegel, Conclusions from CDF Results on CP Violation in $D^{0}\rightarrow\pi^{+}\pi^{-}$, $K^{+}K^{-}$ and Future Tasks, JHEP 1106 (2011) 089, [arXiv:1103.5785] * (12) A. L. Kagan and M. D. Sokoloff, On Indirect CP Violation and Implications for $D^{0}$–$\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ and $B^{0}_{s}$–$\overline{B}^{0}_{s}$ mixing, Phys. Rev. D80 (2009) 076008, [arXiv:0907.3917] * (13) BaBar collaboration, B. Aubert et al., Search for CP violation in the decays $D^{0}\rightarrow K^{-}K^{+}$ and $D^{0}\rightarrow\pi^{-}\pi^{+}$, Phys. Rev. Lett. 100 (2008) 061803, [arXiv:0709.2715] * (14) Belle collaboration, M. Staric et al., Measurement of CP asymmetry in Cabibbo suppressed $D^{0}$ decays, Phys. Lett. B670 (2008) 190, [arXiv:0807.0148] * (15) LHCb collaboration, A. A. Alves Jr et al., The LHCb Detector at the LHC, JINST 3 (2008) S08005	arxiv-papers	2011-12-05T14:30:17	2024-09-04T02:49:24.971618	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "LHCb Collaboration: R. Aaij, C. Abellan Beteta, B. Adeva, M. Adinolfi,\n C. Adrover, A. Affolder, Z. Ajaltouni, J. Albrecht, F. Alessio, M. Alexander,\n G. Alkhazov, P. Alvarez Cartelle, A. A. Alves Jr, S. Amato, Y. Amhis, J.\n Anderson, R. B. Appleby, O. Aquines Gutierrez, F. Archilli, L. Arrabito, A.\n Artamonov, M. Artuso, E. Aslanides, G. Auriemma, S. Bachmann, J. J. Back, D.\n S. Bailey, V. Balagura, W. Baldini, R. J. Barlow, C. Barschel, S. Barsuk, W.\n Barter, A. Bates, C. Bauer, Th. Bauer, A. Bay, I. Bediaga, S. Belogurov, K.\n Belous, I. Belyaev, E. Ben-Haim, M. Benayoun, G. Bencivenni, S. Benson, J.\n Benton, R. Bernet, M.-O. Bettler, M. van Beuzekom, A. Bien, S. Bifani, T.\n Bird, A. Bizzeti, P. M. Bj{\\o}rnstad, T. Blake, F. Blanc, C. Blanks, J.\n Blouw, S. Blusk, A. Bobrov, V. Bocci, A. Bondar, N. Bondar, W. Bonivento, S.\n Borghi, A. Borgia, T. J. V. Bowcock, C. Bozzi, T. Brambach, J. van den Brand,\n J. Bressieux, D. Brett, M. Britsch, T. Britton, N. H. Brook, H. Brown, A.\n B\\\"uchler-Germann, I. Burducea, A. Bursche, J. Buytaert, S. Cadeddu, O.\n Callot, M. Calvi, M. Calvo Gomez, A. Camboni, P. Campana, A. Carbone, G.\n Carboni, R. Cardinale, A. Cardini, L. Carson, K. Carvalho Akiba, G. Casse, M.\n Cattaneo, Ch. Cauet, M. Charles, Ph. Charpentier, N. Chiapolini, K. Ciba, X.\n Cid Vidal, G. Ciezarek, P. E. L. Clarke, M. Clemencic, H. V. Cliff, J.\n Closier, C. Coca, V. Coco, J. Cogan, P. Collins, A. Comerma-Montells, F.\n Constantin, A. Contu, A. Cook, M. Coombes, G. Corti, G. A. Cowan, R. Currie,\n C. D'Ambrosio, P. David, P. N. Y. David, I. De Bonis, S. De Capua, M. De\n Cian, F. De Lorenzi, J. M. De Miranda, L. De Paula, P. De Simone, D. Decamp,\n M. Deckenhoff, H. Degaudenzi, L. Del Buono, C. Deplano, D. Derkach, O.\n Deschamps, F. Dettori, J. Dickens, H. Dijkstra, P. Diniz Batista, F. Domingo\n Bonal, S. Donleavy, F. Dordei, A. Dosil Su\\'arez, D. Dossett, A. Dovbnya, F.\n Dupertuis, R. Dzhelyadin, A. Dziurda, S. Easo, U. Egede, V. Egorychev, S.\n Eidelman, D. van Eijk, F. Eisele, S. Eisenhardt, R. Ekelhof, L. Eklund, Ch.\n Elsasser, D. Elsby, D. Esperante Pereira, L. Est\\`eve, A. Falabella, E.\n Fanchini, C. F\\\"arber, G. Fardell, C. Farinelli, S. Farry, V. Fave, V.\n Fernandez Albor, M. Ferro-Luzzi, S. Filippov, C. Fitzpatrick, M. Fontana, F.\n Fontanelli, R. Forty, M. Frank, C. Frei, M. Frosini, S. Furcas, A. Gallas\n Torreira, D. Galli, M. Gandelman, P. Gandini, Y. Gao, J-C. Garnier, J.\n Garofoli, J. Garra Tico, L. Garrido, D. Gascon, C. Gaspar, N. Gauvin, M.\n Gersabeck, T. Gershon, Ph. Ghez, V. Gibson, V.V. Gligorov, C. G\\\"obel, D.\n Golubkov, A. Golutvin, A. Gomes, H. Gordon, M. Grabalosa G\\'andara, R.\n Graciani Diaz, L. A. Granado Cardoso, E. Graug\\'es, G. Graziani, A. Grecu, E.\n Greening, S. Gregson, B. Gui, E. Gushchin, Yu. Guz, T. Gys, G. Haefeli, C.\n Haen, S. C. Haines, T. Hampson, S. Hansmann-Menzemer, R. Harji, N. Harnew, J.\n Harrison, P. F. Harrison, T. Hartmann, J. He, V. Heijne, K. Hennessy, P.\n Henrard, J. A. Hernando Morata, E. van Herwijnen, E. Hicks, K. Holubyev, P.\n Hopchev, W. Hulsbergen, P. Hunt, T. Huse, R. S. Huston, D. Hutchcroft, D.\n Hynds, V. Iakovenko, P. Ilten, J. Imong, R. Jacobsson, A. Jaeger, M. Jahjah\n Hussein, E. Jans, F. Jansen, P. Jaton, B. Jean-Marie, F. Jing, M. John, D.\n Johnson, C. R. Jones, B. Jost, M. Kaballo, S. Kandybei, M. Karacson, T. M.\n Karbach, J. Keaveney, I. R. Kenyon, U. Kerzel, T. Ketel, A. Keune, B. Khanji,\n Y. M. Kim, M. Knecht, R. Koopman, P. Koppenburg, A. Kozlinskiy, L. Kravchuk,\n K. Kreplin, M. Kreps, G. Krocker, P. Krokovny, F. Kruse, K. Kruzelecki, M.\n Kucharczyk, T. Kvaratskheliya, V. N. La Thi, D. Lacarrere, G. Lafferty, A.\n Lai, D. Lambert, R. W. Lambert, E. Lanciotti, G. Lanfranchi, C. Langenbruch,\n T. Latham, C. Lazzeroni, R. Le Gac, J. van Leerdam, J.-P. Lees, R. Lef\\`evre,\n A. Leflat, J. Lefran\\c{c}ois, O. Leroy, T. Lesiak, L. Li, L. Li Gioi, M.\n Lieng, M. Liles, R. Lindner, C. Linn, B. Liu, G. Liu, J. von Loeben, J. H.\n Lopes, E. Lopez Asamar, N. Lopez-March, H. Lu, J. Luisier, A. Mac Raighne, F.\n Machefert, I. V. Machikhiliyan, F. Maciuc, O. Maev, J. Magnin, S. Malde, R.\n M. D. Mamunur, G. Manca, G. Mancinelli, N. Mangiafave, U. Marconi, R.\n M\\\"arki, J. Marks, G. Martellotti, A. Martens, L. Martin, A. Mart\\'in\n S\\'anchez, D. Martinez Santos, A. Massafferri, Z. Mathe, C. Matteuzzi, M.\n Matveev, E. Maurice, B. Maynard, A. Mazurov, G. McGregor, R. McNulty, M.\n Meissner, M. Merk, J. Merkel, R. Messi, S. Miglioranzi, D. A. Milanes, M.-N.\n Minard, J. Molina Rodriguez, S. Monteil, D. Moran, P. Morawski, R. Mountain,\n I. Mous, F. Muheim, K. M\\\"uller, R. Muresan, B. Muryn, B. Muster, M. Musy, J.\n Mylroie-Smith, P. Naik, T. Nakada, R. Nandakumar, I. Nasteva, M. Nedos, M.\n Needham, N. Neufeld, C. Nguyen-Mau, M. Nicol, V. Niess, N. Nikitin, A.\n Nomerotski, A. Novoselov, A. Oblakowska-Mucha, V. Obraztsov, S. Oggero, S.\n Ogilvy, O. Okhrimenko, R. Oldeman, M. Orlandea, J. M. Otalora Goicochea, P.\n Owen, K. Pal, J. Palacios, A. Palano, M. Palutan, J. Panman, A. Papanestis,\n M. Pappagallo, C. Parkes, C. J. Parkinson, G. Passaleva, G. D. Patel, M.\n Patel, S. K. Paterson, G. N. Patrick, C. Patrignani, C. Pavel-Nicorescu, A.\n Pazos Alvarez, A. Pellegrino, G. Penso, M. Pepe Altarelli, S. Perazzini, D.\n L. Perego, E. Perez Trigo, A. P\\'erez-Calero Yzquierdo, P. Perret, M.\n Perrin-Terrin, G. Pessina, A. Petrella, A. Petrolini, A. Phan, E. Picatoste\n Olloqui, B. Pie Valls, B. Pietrzyk, T. Pila\\v{r}, D. Pinci, R. Plackett, S.\n Playfer, M. Plo Casasus, G. Polok, A. Poluektov, E. Polycarpo, D. Popov, B.\n Popovici, C. Potterat, A. Powell, J. Prisciandaro, V. Pugatch, A. Puig\n Navarro, W. Qian, J. H. Rademacker, B. Rakotomiaramanana, M. S. Rangel, I.\n Raniuk, G. Raven, S. Redford, M. M. Reid, A. C. dos Reis, S. Ricciardi, K.\n Rinnert, D. A. Roa Romero, P. Robbe, E. Rodrigues, F. Rodrigues, P. Rodriguez\n Perez, G. J. Rogers, S. Roiser, V. Romanovsky, M. Rosello, J. Rouvinet, T.\n Ruf, H. Ruiz, G. Sabatino, J. J. Saborido Silva, N. Sagidova, P. Sail, B.\n Saitta, C. Salzmann, M. Sannino, R. Santacesaria, C. Santamarina Rios, R.\n Santinelli, E. Santovetti, M. Sapunov, A. Sarti, C. Satriano, A. Satta, M.\n Savrie, D. Savrina, P. Schaack, M. Schiller, S. Schleich, M. Schlupp, M.\n Schmelling, B. Schmidt, O. Schneider, A. Schopper, M.-H. Schune, R.\n Schwemmer, B. Sciascia, A. Sciubba, M. Seco, A. Semennikov, K. Senderowska,\n I. Sepp, N. Serra, J. Serrano, P. Seyfert, M. Shapkin, I. Shapoval, P.\n Shatalov, Y. Shcheglov, T. Shears, L. Shekhtman, O. Shevchenko, V.\n Shevchenko, A. Shires, R. Silva Coutinho, T. Skwarnicki, A. C. Smith, N. A.\n Smith, E. Smith, K. Sobczak, F. J. P. Soler, A. Solomin, F. Soomro, B. Souza\n De Paula, B. Spaan, A. Sparkes, P. Spradlin, F. Stagni, S. Stahl, O.\n Steinkamp, S. Stoica, S. Stone, B. Storaci, M. Straticiuc, U. Straumann, V.\n K. Subbiah, S. Swientek, M. Szczekowski, P. Szczypka, T. Szumlak, S.\n T'Jampens, E. Teodorescu, F. Teubert, C. Thomas, E. Thomas, J. van Tilburg,\n V. Tisserand, M. Tobin, S. Topp-Joergensen, N. Torr, E. Tournefier, M. T.\n Tran, A. Tsaregorodtsev, N. Tuning, M. Ubeda Garcia, A. Ukleja, P. Urquijo,\n U. Uwer, V. Vagnoni, G. Valenti, R. Vazquez Gomez, P. Vazquez Regueiro, S.\n Vecchi, J. J. Velthuis, M. Veltri, B. Viaud, I. Videau, X. Vilasis-Cardona,\n J. Visniakov, A. Vollhardt, D. Volyanskyy, D. Voong, A. Vorobyev, H. Voss, S.\n Wandernoth, J. Wang, D. R. Ward, N. K. Watson, A. D. Webber, D. Websdale, M.\n Whitehead, D. Wiedner, L. Wiggers, G. Wilkinson, M. P. Williams, M. Williams,\n F. F. Wilson, J. Wishahi, M. Witek, W. Witzeling, S. A. Wotton, K. Wyllie, Y.\n Xie, F. Xing, Z. Xing, Z. Yang, R. Young, O. Yushchenko, M. Zavertyaev, F.\n Zhang, L. Zhang, W. C. Zhang, Y. Zhang, A. Zhelezov, L. Zhong, E. Zverev, A.\n Zvyagin", "submitter": "Vincenzo Maria Vagnoni", "url": "https://arxiv.org/abs/1112.0938" }
1112.0960	# Antisymmetric tensor unparticle and the radiative lepton flavor violating decays E. O. Iltan, Physics Department, Middle East Technical University Ankara, Turkey E-mail address: eiltan@metu.edu.tr ###### Abstract We study the contribution of the tensor unparticle mediation to the branching ratios of the radiative lepton flavor violating decays and predict a restriction region for free parameters of the scenario by using experimental upper limits. We observe that the branching ratios of the radiative lepton flavor violating decays are sensitive to the fundamental mass scales of the scenario and to the scale dimension of antisymmetric tensor unparticle. We obtain a more restricted set for the free parameters in the case of the $\mu\rightarrow e\gamma$ decay. The radiative lepton flavor violating (LFV) decays $l_{i}\rightarrow l_{j}\gamma$ reach great interest since their branching ratios (BRs) in the framework of standard model (SM) are much below the experimental upper limits, and, therefore, they are candidates to search and to test more fundamental models beyond. The current experimental upper limits of the BRs read BR $(\mu\rightarrow e\gamma)=2.4\,(1.2)\times 10^{-12}\,(10^{-11})\,(\,\,90\%CL)$ [1] ([2]), BR $(\tau\rightarrow e\gamma)=3.3\times 10^{-8}\,(\,\,90\%CL)$ [3] and BR $(\tau\rightarrow\mu\gamma)=4.4\times 10^{-8}\,(\,\,90\%CL)$ [3]. There is an extensive theoretical work in the literature in order to enhance BRs of these decays. They were studied in the SM with the extended Higgs sector, so called two Higgs doublet model (2HDM) [4]-[10], in supersymmetric models [11]-[18], in a model independent way [19], in the framework of the 2HDM and the supersymmetric model [20], in the SM including effective operators coming from the possible unparticle effects [21]-[22], in little Higgs models [23]\- [26], in seesaw models [27] -[30], in models with A(4) and S(4) flavor symmetries [31], using the effective field theory with Higgs mediation [32], in the Higgs triplet model [33], in the framework of Higgs-induced lepton flavor violation [34]. In the present work, we consider the contribution of the antisymmetric tensor unparticle mediation to the BRs of the radiative LFV decays (see [35] for the contribution of the antisymmetric tensor unparticle mediation to the muon anomalous magnetic dipole moment, to the electroweak precision observable $S$, its effects in $Z$ invisible decays and see [36] for the contribution of the antisymmetric tensor unparticle mediation to the lepton electric dipole moment). Unparticles [37, 38], being massless due to the scale invariance and having non integral scaling dimension $d_{U}$ around the scale $\Lambda_{U}\sim 1.0\,TeV$, come out with the interaction of SM-ultraviolet sector at some scale $M_{U}$: ${\cal{L}}_{eff}=\frac{C_{n}}{M_{U}^{d_{UV}+n-4}}\,O_{SM}\,O_{UV}\,,$ (1) where $d_{UV}$ is the scaling dimension of the UV operator [39]. Around the scale $\Lambda_{U}$ the effective interaction becomes [40] ${\cal{L}}_{eff}=\frac{C^{i}_{n}}{\Lambda_{n}^{d_{U}+n-4}}\,O_{SM,i}\,O_{U}\,.$ (2) Here $O_{SM,i}$ is type $i$ SM operator, $n$ is its scaling dimension and $\Lambda_{n}$ is the mass scale (see [35, 40] for details) which reads $\Lambda_{n}=\Bigg{(}\frac{M_{U}^{d_{UV}+n-4}}{\Lambda_{U}^{d_{UV}-d_{U}}}\Bigg{)}^{\frac{1}{d_{U}+n-4}}\,.$ (3) The antisymmetric tensor unparticle mediation induces these LFV decays at tree level and we consider the case that the scale invariance is broken at some scale $\mu$ after the electroweak symmetry breaking (see for example [41, 42] for a possible interaction which causes that scale invariance is broken). The effective lagrangian [35] which can drive the radiative LFV decays is $\displaystyle{\cal{L}}_{eff}$ $\displaystyle=$ $\displaystyle\frac{g^{\prime}\,\lambda_{B}}{\Lambda_{2}^{d_{U}-2}}\,B_{\mu\nu}\,O^{\mu\nu}_{U}+\frac{g\,\lambda_{W}}{\Lambda_{4}^{d_{U}}}\,(H^{\dagger}\,\tau_{a}\,H)\,W^{a}_{\mu\nu}\,O^{\mu\nu}_{U}$ (4) $\displaystyle+$ $\displaystyle\frac{\lambda_{ij}}{\Lambda_{4}^{d_{U}}}\,y_{ij}\,\bar{l}_{i}\,H\,\sigma_{\mu\nu}\,l_{j}\,O^{\mu\nu}_{U}\,,$ where $l_{i(j)}$ is the lepton field, $H$ is the Higgs doublet, $g$ and $g^{\prime}$ are weak couplings, $\lambda_{B}$, $\lambda_{W}$ and $\lambda_{ij}$ are the unparticle-field tensor and unparticle-lepton-lepton couplings, $B_{\mu\nu}$ is the field strength tensor of the $U(1)_{Y}$ gauge boson with $B_{\mu}=c_{W}\,A_{\mu}+s_{W}\,Z_{\mu}$ and $W^{a}_{\mu\nu}$, $a=1,2,3$, are field strength tensors of the $SU(2)_{L}$ gauge bosons with $W^{3}_{\mu}=s_{W}\,A_{\mu}-c_{W}\,Z_{\mu}$ and $A_{\mu}$ ($Z_{\mu}$) is photon (Z boson) field. The couplings $y_{ij}$ are responsible for the LF violation and after the electroweak symmetry breaking we introduce modified couplings $\xi_{ij}=\frac{v}{\sqrt{2}}\,y_{ij}$ which respect the mass hierarchy of charged leptons. The process $l_{i}\rightarrow l_{j}\gamma$ appears in the tree level with the communication of two vertices111The first vertex arises from the last term of the effective lagrangian and leads to the $l_{i}\rightarrow l_{j}$ transition. The second one arises from the first and second terms of the effective lagrangian and results in the $O^{\mu\nu}_{U}\rightarrow A_{\nu}$ transition $\lambda_{ij}\,\frac{\xi_{ij}}{\Lambda_{4}^{d_{U}}}\,\bar{l}_{i}\,\sigma_{\mu\nu}\,l_{j}\,O^{\mu\nu}_{U}$ and $\Big{(}\,2\,i\,\frac{g^{\prime}\,c_{W}\,\lambda_{B}}{\Lambda_{2}^{d_{U}-2}}-i\,\frac{g\,v^{2}\,s_{W}\,\lambda_{W}}{2\,\Lambda_{4}^{d_{U}}}\,\Big{)}\,k_{\mu}\,\epsilon_{\nu}\,O^{\mu\nu}_{U}$, by the antisymmetric tensor unparticle propagator (see Appendix and eq.(12)) and the matrix element of this process reads $\displaystyle M=a_{ij}\,\bar{l}_{i}\,\sigma_{\mu\nu}\,l_{j}\,k_{\mu}\,\epsilon_{\nu}\,,$ (5) where $\displaystyle a_{ij}=\frac{i\,e\,\mu^{2\,(d_{U}-2)}\,\,A_{d_{U}}\,\lambda_{ij}\,\xi_{ij}}{sin\,(d_{U}\pi)\,\Lambda_{4}^{d_{U}}}\,\Bigg{(}\frac{\lambda_{B}}{\Lambda_{2}^{d_{U}-2}}-\frac{v^{2}\,\lambda_{W}}{4\,\Lambda_{4}^{d_{U}}}\Bigg{)}\,.$ (6) Finally the decay width $\Gamma(l_{i}\rightarrow l_{j}\gamma)$ becomes $\displaystyle\Gamma(l_{i}\rightarrow l_{j}\gamma)=\frac{1}{8\,\pi}\,m_{i}^{3}\,\|a_{ij}\|^{2}\,,$ (7) where $m_{i}$ is the mass of incoming lepton. Notice that, in this expression, we ignore the mass of outgoing one. Discussion In this section we study the intermediate antisymmetric tensor unparticle contribution to the radiative LFV decays $l_{i}\rightarrow l_{j}\gamma$ which exist at tree level (see Fig.1). There are various free parameters in this scenario and we restrict them by using the current experimental upper limits of BRs of LFV decays. Now, we would like to present the free parameters and discuss the restrictions predicted. The SM sector interacts with the UV one and it appears as unparticle sector at a lower scale. The corresponding UV (unparticle) operator $O_{UV}$ ($O_{U}$) has the scaling dimension $d_{UV}$ ($d_{U}$) which is among the free parameters. We choose the scale dimension $d_{U}$ in the range $1<d_{U}<2$. Notice that the scale dimension must satisfy $d_{U}>2$ for antisymmetric tensor unparticle in order not to violate the unitarity [43]. Our assumption is based on the fact that the scale invariance is broken at some scale $\mu$ and one reaches to the particle sector. This results in a relaxation on the values of $d_{U}$ and we choose $d_{U}$ in the range $1<d_{U}<2$ so that the propagator for particle sector is obtained when $d_{U}$ tends to one. Furthermore we choose the numerical value of $d_{UV}$ as $d_{UV}=3$ which satisfies $d_{UV}>d_{U}$ (see [40]). The SM-ultraviolet sector interaction scale $M_{U}$, the SM-unparticle sector interaction scale $\Lambda_{U}$ and the scale $\mu$ which is the one that scale invariance is broken belong to the free parameter set of the present scenario. Here we choose $\mu\sim 1.0\,GeV$ and predict the restrictions for the others by using the experimental upper limits of LFV decays. Finally, for the couplings $\lambda_{B}$, $\lambda_{W}$ and $\lambda_{ij}$ we consider $\lambda_{B}=\lambda_{W}=\lambda_{ij}=1$ and for $\xi_{ij}$ we respect the mass hierarchy of charged leptons, namely we choose $\xi_{\tau\mu}>\xi_{\tau e}>\xi_{\mu e}$ and we take $\xi_{\tau\mu}=0.1\,GeV$, $\xi_{\tau e}=0.01\,GeV$ and $\xi_{\mu e}=0.001\,GeV$ in our numerical calculations. In Fig.2, we present the BR$(\mu\rightarrow e\,\gamma$) with respect to the mass scale $M_{U}$ for $r_{U}=\frac{\Lambda_{U}}{M_{U}}=0.1$. Here, the solid (long dashed-short dashed) line represents the BR for $d_{U}=1.7\,(1.8-1.9)$. We observe that the BR is sensitive to the mass scale $M_{U}$ especially for the large values of the scale dimension $d_{U}$ and it decreases almost three orders in the range of $2000\,GeV<M_{U}<10000\,GeV$. The experimental upper limit is reached for $d_{U}\sim 1.8\,(1.9)$ and $M_{U}\sim 8000\,(4500)\,GeV$. Fig.3 is devoted to the BR$(\mu\rightarrow e\,\gamma$) with respect to the scale parameter $d_{U}$ for $r_{U}=0.1$. Here the solid (long dashed-short dashed-dotted) line represents the BR for $M_{U}=3000\,(5000-8000-10000)\,GeV$. The BR strongly depends on $d_{U}$ and decreases with the increasing values of $d_{U}$. The experimental upper limit is observed in the range of $d_{U}\sim 1.78-1.88$ for $M_{U}\sim 5000-10000\,GeV$. In Fig.4 we show the parameter $r_{U}$ with respect to $d_{U}$ for BR$(\mu\rightarrow e\,\gamma)=2.4\times 10^{-12}$. Here the solid (long dashed-short dashed) line represents $r_{U}$ for $M_{U}=5000\,(8000-10000)\,GeV$. We see that the scale dimension $d_{U}$ and $r_{U}$ can take values in the range $1.73-1.90$ and $0.05-0.12$, respectively for $M_{U}=5000\,GeV$. For $M_{U}=10000\,GeV$ we have the range $1.65-1.9$ for $d_{U}$ and $0.05-0.20$ for $r_{U}$. Fig.5 represents the BR$(\tau\rightarrow e\,\gamma$) with respect to the mass scale $M_{U}$. Here, the solid (long dashed-short dashed-dotted) line represents the BR for $r_{U}=0.1$, $d_{U}=1.3$ ($r_{U}=0.1$, $d_{U}=1.4$-$r_{U}=0.5$, $d_{U}=1.6$-$r_{U}=0.5$, $d_{U}=1.7$). It is observed that the sensitivity of the BR to the mass scale $M_{U}$ increases with the increasing values of the ratio $r_{U}$. The experimental upper limit is reached for $r_{U}=0.1$ and $d_{U}\sim 1.3$ when the mass scale $M_{U}$ reads $M_{U}\sim 4000\,GeV$. For $r_{U}=0.5$ one reaches the experimental limit in the case of $d_{U}\sim 1.6$ and $M_{U}\sim 4000\,GeV$. Fig.5 shows the BR$(\tau\rightarrow e\,\gamma$) with respect to the scale parameter $d_{U}$ for $r_{U}=0.1$. Here the solid (long dashed-short dashed) line represents the BR for $M_{U}=2000\,(5000-10000)\,GeV$. The BR strongly depends on $d_{U}$ and decreases with the increasing values of $d_{U}$ similar to the $\mu\rightarrow e\,\gamma$ decay. One reaches the experimental upper limit in the range of $d_{U}\sim 1.26-1.32$ for $M_{U}\sim 2000-10000\,GeV$. Fig.7 is devoted to the parameter $r_{U}$ with respect to $d_{U}$ for BR$(\tau\rightarrow e\,\gamma)=3.3\times 10^{-8}$. Here the solid (long dashed-short dashed) line represents $r_{U}$ for $M_{U}=5000\,(8000-10000)\,GeV$. This figure shows that the experimental upper limit is reached for $M_{U}=5000\,GeV$ if the scale dimension $d_{U}$ and $r_{U}$ can take values in the range $1.10-1.68$ and $0.05-1.00$, respectively. For $M_{U}=10000\,GeV$ we have the range $1.10-1.64$ for $d_{U}$ and $0.05-1.00$ for $r_{U}$. In Fig.8, we present the BR$(\tau\rightarrow\mu\,\gamma$) with respect to the mass scale $M_{U}$ for $r_{U}=0.1$. Here, the solid (long dashed-short dashed- dotted) line represents the BR for $d_{U}=1.4\,(1.5-1.6-1.7)$. We observe that the BR decreases more than one order in the range of $2000\,GeV<M_{U}<10000\,GeV$. The experimental upper limit is reached for $d_{U}\sim 1.4$ and $M_{U}\sim 9000\,GeV$. Fig.9 represents the BR$(\tau\rightarrow\mu\,\gamma)$ with respect to the scale parameter $d_{U}$ for $r_{U}=0.1$. Here the solid (long dashed-short dashed) line represents the BR for $M_{U}=2000\,(5000-10000)\,GeV$. The experimental upper limit of the BR is observed in the range of $d_{U}\sim 1.38-1.47$ for $M_{U}\sim 2000-10000\,GeV$. In Fig. 10 we show the parameter $r_{U}$ with respect to $d_{U}$ for BR$(\tau\rightarrow\mu\,\gamma)=4.4\times 10^{-8}$. Here the solid (long dashed-short dashed) line represents $r_{U}$ for $M_{U}=2000\,(5000-10000)\,GeV$. We see that the scale dimension $d_{U}$ and $r_{U}$ can take values in the range $1.48-1.80\,(1.3-1.8)$ and $0.10-0.55\,(0.05-1.00)$, respectively for $M_{U}=2000\,(5000)\,GeV$. For $M_{U}=10000\,GeV$ we have the range $1.30-1.75$ for $d_{U}$ and $0.05-1.00$ for $r_{U}$. As a summary, the BRs of radiative LFV decays are sensitive to the mass scale $M_{U}$ especially for the large values of the scale dimension $d_{U}$ and this sensitivity increases with the increasing values of the ratio $r_{U}$. The experimental upper limit of the BR$(\mu\rightarrow e\,\gamma$) can be reached for $r_{U}\sim 0.1$, $d_{U}>1.7$ and for larger values of $M_{U}$, namely $M_{U}\sim 7000-9000\,GeV$. For BR$(\tau\rightarrow e\,\gamma$) one reaches the experimental upper limit for $r_{U}\sim 0.1$, $d_{U}\sim 1.3$ and $M_{U}>2000\,GeV$ . If we consider the $\tau\rightarrow\mu\,\gamma$ decay the experimental upper limit of BR is obtained for $r_{U}\sim 0.1$, when $d_{U}$ is in the range $d_{U}\sim 1.4-1.5$ and $M_{U}>2000\,GeV$. We see that the free parameters of this scenario are more restricted if the $\mu\rightarrow e\,\gamma$ decay is considered. However the future more accurate measurements of the upper limits of the LFV decays make it possible to obtain a more restricted range for the free parameters of this scenario and they stimulate to search the role and the nature of unparticle physics which is a candidate to drive the lepton flavor violation. Appendix The scalar unparticle propagator reads [38, 44] $\displaystyle\int\,d^{4}x\,e^{ip.x}<0\|T\Big{(}O_{U}(x)\,O_{U}(0)\Big{)}0>=i\frac{A_{d_{U}}}{2\,\pi}\,\int_{0}^{\infty}\\!\\!\\!ds\,\frac{s^{d_{U}-2}}{p^{2}-s+i\epsilon}\\!=i\,\frac{A_{d_{U}}}{2\,sin\,(d_{U}\pi)}\,(-p^{2}-i\epsilon)^{d_{U}-2}$ (8) where the factor $A_{d_{U}}$ is $\displaystyle A_{d_{U}}=\frac{16\,\pi^{5/2}}{(2\,\pi)^{2\,d_{U}}}\,\frac{\Gamma(d_{U}+\frac{1}{2})}{\Gamma(d_{U}-1)\,\Gamma(2\,d_{U})}\,.$ (9) Now the tensor unparticle propagator is obtained by using the projection operator $\Pi^{\mu\nu\alpha\beta}$ $\displaystyle\Pi_{\mu\nu\alpha\beta}=\frac{1}{2}(g_{\mu\alpha}\,g_{\nu\beta}-g_{\nu\alpha}\,g_{\mu\beta})\,,$ (10) with the transverse and the longitudinal parts $\displaystyle\Pi^{T}_{\mu\nu\alpha\beta}=\frac{1}{2}(P^{T}_{\mu\alpha}\,P^{T}_{\nu\beta}-P^{T}_{\nu\alpha}\,P^{T}_{\mu\beta})\,,\,\,\,\,\,\,\Pi^{L}_{\mu\nu\alpha\beta}=\Pi_{\mu\nu\alpha\beta}-\Pi^{T}_{\mu\nu\alpha\beta}\,,$ (11) where $P^{T}_{\mu\nu}=g_{\mu\nu}-p_{\mu}\,p_{\nu}/{p^{2}}$ (see for example [35] and references therein) and the propagator of antisymmetric tensor unparticle becomes $\displaystyle\int\,d^{4}x\,e^{ipx}\,<0\|T\Big{(}O^{\mu\nu}_{U}(x)\,O^{\alpha\beta}_{U}(0)\Big{)}0>=i\,\frac{A_{d_{U}}}{2\,sin\,(d_{U}\pi)}\,\Pi^{\mu\nu\alpha\beta}(-p^{2}-i\epsilon)^{d_{U}-2}\,.$ On the other hand the propagator is modified if the scale invariance broken at a certain scale and this modification is model dependent. Following the the simple model [41, 45, 46] we take $\displaystyle\int\,d^{4}x\,e^{ipx}\,<0\|T\Big{(}O^{\mu\nu}_{U}(x)\,O^{\alpha\beta}_{U}(0)\Big{)}0>=i\,\frac{A_{d_{U}}}{2\,sin\,(d_{U}\pi)}\,\Pi^{\mu\nu\alpha\beta}(-(p^{2}-\mu^{2})-i\epsilon)^{d_{U}-2}\,.$ (12) where $\mu$ is the scale that the scale invariance broken and the particle sector comes out. ## References * [1] J. Adam et.al., MEG Collaboration, hep-ex/1107.5547, (2011). * [2] M. L. Brooks et. al., MEGA Collaboration, Phys. Rev. Lett. 83, 1521 (1999). * [3] B. Aubert et. al., BABAR Collaboration, BABAR-PUB-09/026, SLAC-PUB-13753, Phys. Rev. Lett. 104 021802, (2010). * [4] E. O. Iltan, Phys. Rev. D64 115005, (2001). * [5] E. O. Iltan,Phys. Rev. D64 013013, (2001) * [6] R. Diaz, R. Martinez and J-Alexis Rodriguez, Phys.Rev. D63 095007, (2001). * [7] E. O. Iltan, JHEP 0402 20, (2004). * [8] E. O. Iltan, JHEP 0408 20, (2004) * [9] E. O. Iltan, Mod. Phys. Lett. A22 819, (2007). * [10] E. O. Iltan, Int. J. Mod. Phys. A23 1055, (2008). * [11] R. Barbieri and L. J. Hall, Phys. Lett. B338 212, (1994). * [12] R. Barbieri, L. J. Hall and A. Strumia, Nucl. Phys. B445 219, (1995). * [13] R. Barbieri, L. J. Hall and A. Strumia, Nucl. Phys. B449 437, (1995). * [14] P. Ciafaloni, A. Romanino and A. Strumia, IFUP-YH-42-95. * [15] T. V. Duong, B. Dutta and E. Keith, Phys. Lett. B378 128, (1996). * [16] G. Couture, et. al., Eur. Phys. J. C7 135, (1999). * [17] Y. Okada, K. Okumara and Y. Shimizu, Phys. Rev. D61 094001, (2000). * [18] S. Khalil, Phys. Rev. D81 035002, (2010). * [19] D. Chang, W. S. Hou and W. Y. Keung, Phys. Rev. D48 217, (1993). * [20] P. Paradisi, JHEP 0602 050, (2006). * [21] G. J. Ding, M. L. Yan, Phys. Rev. D77 014005, (2008). * [22] A. Hektor, Y. Kajiyama, K. Kannike, Phys. Rev. D78 053008, (2008). * [23] F. del Aguila, J. I. Illana, M. D. Jenkins, JHEP 0901 080, (2009). * [24] J. I. Illana, M. D. Jenkins, Acta Phys. Polon. B40 3143, (2009). * [25] T. Goto, Y. Okada, Y. Yamamoto, Phys. Rev. D83 053011, (2011). * [26] F. del Aguila, J. I. Illana, M. D. Jenkins, JHEP 1103 080, (2011). * [27] X. G. He, S. Oh, JHEP 0909 027, (2009). * [28] D. Ibanez, S. Morisi, J.W.F. Valle, Phys. Rev. D80 053015, (2009). * [29] C. Hagedorn, E. Molinaro, S. T. Petcov, JHEP 1002 047, (2010). * [30] F. F. Deppisch, F. Plentinger, G. Seidl, JHEP 1101 004, (2011). * [31] G. J. Ding, J. F. Liu , JHEP 1105 029, (2010). * [32] J.I. Aranda, A. F. Tlalpa, F. R. Zavaleta , F. J. Tlachino, J. J. Toscano, E. S. Tututi, Phys. Rev. D79 093009, (2009). * [33] A. G. Akeroyd, M. Aoki, H. Sugiyama, Phys. Rev. D79 113010, (2009). * [34] A. Goudelis, O. Lebedev, J. H. Park, hep-ph/1111.1715, (2011). * [35] T. Hur, P. Ko, X. H. Wu, Phys. Rev. D76, 096008 (2007) * [36] E. Iltan, J. Phys. G38 , 105001 (2011). * [37] H. Georgi, Phys. Rev. Lett. 98, 221601 (2007). * [38] H. Georgi, Phys. Lett. B650, 275 (2007). * [39] T. Banks, A. Zaks, Nucl. Phys. B196, 189 (1982). * [40] M. Bander, J. L. Feng, A. Rajaraman, Y. Shirman, Phys. Rev. D76, 115002 (2007). * [41] P. J. Fox, A. Rajaraman, Y. Shirman, Phys. Rev. D76, 075004 (2007). * [42] T. Kikuchi, N. Okada, Phys. Lett. B661, 360 (2008). * [43] B. Grinstein, K. A. Intriligator, I. Z. Rothstein Phys. Lett. B662, 367 (2008). * [44] K. Cheung, W. Y. Keung and T. C. Yuan, Phys. Rev. D76, 055003 (2007).. * [45] A. Rajaraman, Phys. Lett. B671, 411 (2009). * [46] A. Delgado, J. R. Espinosa, J. M. No and M. Quiros, JHEP 0804, 028 (2008) Figure 1: Tree level diagram contributing to the $l_{i}\rightarrow l_{j}\,\gamma$ decay due to antisymmetric tensor unparticle. Wavy (solid) line represents the electromagnetic field (lepton field) and double dashed line the antisymmetric tensor unparticle field. Figure 2: $M_{U}$ dependence of the BR $(\mu\rightarrow e\,\gamma)$ for $r_{U}=0.1$. Here, the solid (long dashed-short dashed) line represents the BR for $d_{U}=1.7\,(1.8-1.9)$. Figure 3: $d_{U}$ dependence of the BR $(\mu\rightarrow e\,\gamma)$ for $r_{U}=0.1$. Here the solid (long dashed- short dashed-dotted) line represents the BR for $M_{U}=3000\,(5000-8000-10000)\,GeV$. Figure 4: $r_{U}$ with respect to $d_{U}$ for BR$(\mu\rightarrow e\,\gamma)=2.4\times 10^{-12}$. Here the solid (long dashed-short dashed) line represents $r_{U}$ for $M_{U}=5000\,(8000-10000)\,GeV$ Figure 5: $M_{U}$ dependence of the BR $(\tau\rightarrow e\,\gamma)$. Here, the solid (long dashed-short dashed- dotted) line represents the BR for $r_{U}=0.1$, $d_{U}=1.3$ ($r_{U}=0.1$, $d_{U}=1.4$-$r_{U}=0.5$, $d_{U}=1.6$-$r_{U}=0.5$, $d_{U}=1.7$). Figure 6: $d_{U}$ dependence of the BR $(\tau\rightarrow e\,\gamma)$ for $r_{U}=0.1$. Here the solid (long dashed-short dashed) line represents the BR for $M_{U}=2000\,(5000-10000)\,GeV$. Figure 7: $r_{U}$ with respect to $d_{U}$ for BR$(\tau\rightarrow e\,\gamma)=3.3\times 10^{-8}$. Here the solid (long dashed-short dashed) line represents $r_{U}$ for $M_{U}=5000\,(8000-10000)\,GeV$. Figure 8: $M_{U}$ dependence of the BR$(\tau\rightarrow\mu\,\gamma)$. Here, the solid (long dashed-short dashed- dotted) line represents the BR for $d_{U}=1.4\,(1.5-1.6-1.7)$. Figure 9: $d_{U}$ dependence of the BR$(\tau\rightarrow\mu\,\gamma)$ for $r_{U}=0.1$. Here the solid (long dashed-short dashed) line represents the BR for $M_{U}=2000\,(5000-10000)\,GeV$. Figure 10: $r_{U}$ with respect to $d_{U}$ for BR$(\tau\rightarrow\mu\,\gamma)=4.4\times 10^{-8}$. Here the solid (long dashed-short dashed) line represents $r_{U}$ for $M_{U}=2000\,(5000-10000)\,GeV$.	arxiv-papers	2011-12-05T15:10:44	2024-09-04T02:49:24.980859	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "E. Iltan", "submitter": "Erhan Iltan", "url": "https://arxiv.org/abs/1112.0960" }
1112.1070	# Mechanism for puddle formation in graphene S. Adam Center for Nanoscale Science and Technology, National Institute of Standards and Technology, Gaithersburg, MD 20899, USA Suyong Jung Center for Nanoscale Science and Technology, National Institute of Standards and Technology, Gaithersburg, MD 20899, USA Maryland NanoCenter, University of Maryland, College Park, MD 20472, USA Nikolai N. Klimov Center for Nanoscale Science and Technology, National Institute of Standards and Technology, Gaithersburg, MD 20899, USA Maryland NanoCenter, University of Maryland, College Park, MD 20472, USA Physical Measurement Laboratory, National Institute of Standards and Technology, Gaithersburg, MD 20899, USA Nikolai B. Zhitenev Center for Nanoscale Science and Technology, National Institute of Standards and Technology, Gaithersburg, MD 20899, USA Joseph A. Stroscio Center for Nanoscale Science and Technology, National Institute of Standards and Technology, Gaithersburg, MD 20899, USA M. D. Stiles Center for Nanoscale Science and Technology, National Institute of Standards and Technology, Gaithersburg, MD 20899, USA ###### Abstract When graphene is close to charge neutrality, its energy landscape is highly inhomogeneous, forming a sea of electron-like and hole-like puddles, which determine the properties of graphene at low carrier density. However, the details of the puddle formation have remained elusive. We demonstrate numerically that in sharp contrast to monolayer graphene, the normalized autocorrelation function for the puddle landscape in bilayer graphene depends only on the distance between the graphene and the source of the long-ranged impurity potential. By comparing with available experimental data, we find quantitative evidence for the implied differences in scanning tunneling microscopy measurements of electron and hole puddles for monolayer and bilayer graphene in nominally the same disorder potential. ###### pacs: 73.22.Pr,68.37.Ef,81.05.ue ## I Introduction In monolayer graphene, the hexagonal arrangement of carbon atoms dictates that in the absence of atomic-scale disorder, graphene is a gapless semiconductorDas Sarma et al. (2011); Castro Neto et al. (2009) that is always metallic at low temperature.Fuhrer and Adam (2009) This metallic behavior holds even in the presence of quantum interference and strong disorder,Bardarson et al. (2007) in stark contrast to most other materials, which undergo a metal-to-insulator transition at low carrier density.Anderson (1958); Tanatar and Ceperley (1989) The physical origin for this robust metallic state is that the ground-state of graphene at vanishing mean carrier density becomes spatially inhomogeneous, breaking up into electron-rich and hole-rich metallic regions connected by highly conducting p-n junctions.Katsnelson et al. (2006) Bilayer graphene comprising two sheets of graphene that become strongly coupled due to the $AB$ stacking arrangementMcCann and Fal’ko (2006) shares some properties with regular semiconductors (such as the parabolic band dispersion) and in other ways behaves like monolayer graphene, including having chiral wavefunctions and forming electron and hole puddles at low density. These electron and hole puddles have now been observed in several experiments of exfoliated graphene on an insulating SiO2 substrate including Refs. Martin et al., 2008; Zhang et al., 2009; Deshpande et al., 2009, 2011; Jung et al., 2011; Rutter et al., 2011. While these authors suggest that long-range charged impurities in the substrate could be responsible for the spatial inhomogeneity, detailed comparisons to microscopic models have not been made. In this paper, we demonstrate that differences between the spatial properties of puddles in monolayer and bilayer graphene can be quantitatively explained by the differences in the screening properties of the two systems (that ultimately arises from the differences in their band-structure). Numerical results show that the correlation length for bilayer graphene is relatively independent of density and significantly smaller than that of monolayer graphene for a typical range of impurity densities. Finally, we find good quantitative agreement when comparing our results with available experimental data. The rest of the paper is organized as follows. In Sec. II, we outline the the theoretical model, providing a heuristic understanding of our results using the Thomas-Fermi (TF) screening theory. However, the TF significantly underestimates the effect of electronic screening in both monolayer and bilayer graphene. It is therefore necessary to use the Random Phase Approximation (RPA) screening theory, which we discuss in Sec. III. Our main finding is that the puddle correlation length in bilayer graphene $(\xi\approx 3.5~{}{\rm nm})$ is relatively insensitive to the impurity concentration and carrier doping. This is in contrast to monolayer graphene, where the puddle correlation length varies from $3~{}{\rm nm}$ in dirty samples to more than $35~{}{\rm nm}$ in clean samples. The comparison with experiment is done in Sec. IV, where we examine three different experimental results: (1) We consider first the experimentally determined normalized correlation function $A(r)$ (see definition below) obtained from the scanning tunneling microscopy (STM) data reported for exfoliated bilayer graphene in Ref. Rutter et al., 2011. The full functional form of $A(r)$ agrees with the theory where the only adjusted parameter in the theory is the distance $d$ of the impurities from the graphene sheet. In particular the experimentally determined correlation length $\xi=(3.68\pm 0.03)~{}{\rm nm}$, defined here as the half-width at half-maximum (HWHM) decay length of $A(r)$ agrees well with the $d=1~{}{\rm nm}$ RPA theory value of $\xi=3.4~{}{\rm nm}$. This value of $d$ is both reasonable and consistent with those determined from other transport measurements on bilayer graphene.Adam and Das Sarma (2008) (2) From the monolayer graphene STM experimental data reported in Ref. Jung et al., 2011, we extract a correlation length $6~{}{\rm nm}<\xi<11~{}{\rm nm}$. Since the measurements Jung et al. (2011); Rutter et al. (2011) were made on the same exfoliated graphene sample containing both single layer and bilayer graphene regions, we expect that the extrinsic disorder potential is statistically identical for the two samples. Therefore, using the value of $d=1~{}{\rm nm}$ (discussed above) and the disorder induced Dirac point shift reported in Ref. Rutter et al., 2011, we calculate theoretically (without any adjustable parameters) that a monolayer graphene sample in the same disorder environment would have a puddle correlation length $\xi=8~{}{\rm nm}$, in reasonable agreement with the experiment. (3) We then compare $A(r)$ obtained using scanning Coulomb blockade spectroscopy reported in Ref. Deshpande et al., 2011 with our theoretical results for monolayer graphene at the Dirac point. The parameters used in the theory were obtained from separate transport measurements on the same experimental sample.Deshpande et al. (2011) The agreement between theory and experiment is remarkable since it involves no adjustable parameters. Finally in Sec. V, we conclude by making predictions for future experiments involving monolayer and bilayer graphene on BN substrates. ## II Formalism The doping level of graphene can be measured in a variety of different ways. In transport measurements, the gate voltage potential that yields the resistivity maximum identifies the extrinsic doping level due to extraneous sources, such as charged impurities in the substrate impurities with density, $n_{imp}$. While the width of the resistivity maximum is a measure of the homogeneity of the sample, Adam et al. (2007) or the electron-hole puddle distribution. In local probe measurements, such as STM, the Dirac point energy relative to the Fermi-level can be observed as a minimum in the tunneling differential conductance, $dI/dV$ as a function of tunneling bias. Knowing the electronic dispersion relation (see details below), for a particular gate voltage $V_{g}$, this spatial map $V(r)$ of the Dirac point variation can then be used to extract the spatial distribution of the local carrier density (characterized by a width $n_{\rm rms}$). We can also characterize the puddles through the radially averaged autocorrelation function $C(r)=\frac{1}{2\pi}\int\limits_{0}^{2\pi}d\phi\langle\langle V({\bf r})V(0)\rangle\rangle,$ (1) where the angular brackets denote an average over the image area and the $\phi$-integration averages over orientations. Notice that while $C(0)=V_{\rm rms}^{2}$ (which is related to $n_{\rm rms}$) characterizes the fluctuations in the puddle depth, $C(r)$ describes the spatial profile of the electron and hole puddles. We find it useful to consider the normalized correlation function $A(r)=C(r)/C(0)$. We will argue below that $A(r)$ and $C(0)$ are quite different physical quantities that depends quite differently on the parameters of the extrinsic impurity potential. (In addition, for a typical STM experiment, where the shifts in the Dirac point are determined Zhang et al. (2009) from the shifts in $dI/dV$ at fixed $V_{g}$, the determination of $C(0)$ is complicated by the experimental uncertainty in converting spatial maps of $dI/dV$ to Dirac point energy shifts $V({\bf r})$. By contrast, the much smaller uncertainty in $A(r)$ is mostly determined by the spatial resolution and image area.) To theoretically compute the correlation functions for puddles in graphene, we make two assumptions. First, the impurity potential comes from a random two dimensional distribution of charged impurities displaced by a distance $d$ from the plane with density $n_{\rm imp}$. This model has been highly successful in describing the effect of disorder in semiconductor heterojunctionsAndo et al. (1982) and in graphene.Das Sarma et al. (2011) This spatially varying potential gives rise to a varying charge density and local variations in the screening of the potential. Second, we assume that it is possible to find a global screening function $\epsilon(q,n_{\rm eff})$ that adequately describes the effects of these local screening variations. Here, the screening depends on the disorder potential only through an effective carrier density $n_{\rm eff}$. This self-consistent screening model has been used previously to understand the minimum conductivity problem in both monolayerAdam et al. (2007) and bilayer graphene.Adam and Das Sarma (2008) If these two assumptions are satisfied, the correlation function, aside from a prefactor of $n_{\rm imp}$, then depends only on the screened impurity potential $\displaystyle C(r)$ $\displaystyle=$ $\displaystyle 2\pi n_{\rm imp}\left(\frac{e^{2}}{\kappa}\right)^{2}\int_{0}^{\infty}dq\frac{q\exp(-2qd)}{[q\epsilon(q,n_{\rm eff})]^{2}}J_{0}(qr),$ (2) where $\kappa$ is the bulk (3D) dielectric constant, $\epsilon(q)$ is the surface (2D) screening function in the plane, $-e$ the electron charge, and $J_{0}(x)$ is a Bessel function. As an illustration, consider the Thomas-Fermi (TF) screening for which the surface dielectric function is given by $\epsilon(q,n_{\rm eff})=1+q_{\rm TF}(n_{\rm eff})/q$, where $q_{\rm TF}(n_{\rm eff})$ (discussed below) is the Thomas-Fermi screening wavevector. Shown in Fig. 1 is a calculation of $C(r)$ for different values of $q_{\rm TF}$ and $d$. Notice that for fixed $q_{\rm TF}$, the spatial dependence of $C(r)$ depends on $d$ and $q_{\rm TF}$, but not on $n_{\rm imp}$. On the other-hand, the function $C(0)$ (which, within the TF can be calculated analytically) depends on $n_{\rm imp}$, $\kappa$, $d$ and $q_{\rm TF}$. This is why we find it useful to use the normalized correlation function $A(r)=C(r)/C(0)$ that describes the spatial profile of the screened impurity potential. We also emphasize that $A(r)$ contains different information than the typical puddle size – for example, the puddle correlation length $\xi$ (recall that $\xi$ is defined as the HWHM of $A(r)$) describes the width of the screened impurity potential, and not the mean impurity separation. For example, in the low impurity density limit, spatial maps of the puddles would show isolated impurities, but $A(r)$ would not change (for fixed $q_{\rm TF}$). Moreover, since $A(r)$ scales differently with $q_{\rm TF}$ and $d$, in principle, both of these length scales can be extracted from a measurement of $A(r)$. Figure 1: Theoretical calculations for the correlation function $C(r)$ using the Thomas-Fermi approximation. This autocorrelation function depends separately on the typical distance $d$ of long-ranged impurities from the graphene sheet, and $q_{\rm TF}$ the inverse effective screening length, allowing them to be determined independently. Symbols show the correlation length $\xi$, defined as the HWHM length. Within the TF screening theory, any differences between monolayer and bilayer graphene can only arise from differences in $q_{\rm TF}(n_{\rm eff})$. The linear dispersion in monolayer graphene and the hyperbolic dispersion in bilayer graphene gives rise to these differences. The low energy linearly dispersing bands of monolayer graphene can be modeled by a single parameter, the Fermi velocity $v_{\rm F}$, or equivalently, the effective fine-structure constant $r_{s}=e^{2}/(\kappa\hbar v_{\rm F})\approx 0.8$. $r_{s}$ characterizes the strength of the electron-electron interaction for graphene on a SiO2 substrateDas Sarma et al. (2011) and is useful because we are interested in the screening properties of graphene. The Thomas-Fermi screening wavevector is related to the density of states, and for monolayer graphene is given by $q_{\rm TF}(n_{\rm eff})=4r_{s}\sqrt{\pi n_{\rm eff}}$, where $n_{\rm eff}$ is the effective carrier density. Bilayer graphene can be modeled with a hyperbolic dispersion with two parameters $v_{\rm F}$ (throughout this manuscript, $v_{\rm F}$ is the Fermi velocity of a single decoupled graphene sheet), and the low-energy effective mass $m_{\rm eff}$. For simplicity we use for the two parameters $r_{s}$ (defined above) and $n_{0}=m_{\rm eff}^{2}v_{\rm F}^{2}/(\hbar^{2}\pi)\approx 2.3\times 10^{12}~{}{\rm cm}^{-2}$ which is the characteristic density scale for the crossover from a (low density) parabolic to a (high density) linear dispersion. The Thomas-Fermi screening wavevector for bilayer graphene is given by $q_{\rm TF}(n_{\rm eff})=4r_{s}\sqrt{\pi n_{0}}\sqrt{1+n_{\rm eff}/n_{0}}$. As the system approaches the Dirac point, the fluctuations in carrier density become larger than the average density. In this case, screening varies spatially with the density fluctuations. We assume that it is possible to describe the effect of this screening by using the screening for an ideal system and an effective carrier density $n_{\rm eff}$ obtained self- consistently.Adam et al. (2007) This is done by equating the squared Fermi level shift with respect to the Dirac point with the square of the potential fluctuations, $E^{2}[n=n_{\rm eff}]=C(0)$ where $C(0)$ is defined in Eq. 2 and $E[n]=\hbar v_{\rm F}\sqrt{\pi n}$ for monolayer graphene, and $E[n]=v_{\rm F}^{2}m_{\rm eff}\left[\sqrt{1+n/n_{0}}-1\right]$ for bilayer graphene. The result of this procedure are shown in Fig. 2. Figure 2: Effective carrier density as a function of impurity density assuming $d=1~{}{\rm nm}$, $r_{s}=0.8$, and $n_{0}=2.3\times 10^{12}~{}{\rm cm}^{-2}$. For bilayer graphene, the blue circles show the Thomas-Fermi approximation and the red squares are RPA results. The empirical relation $n_{\rm eff}=\sqrt{n_{\rm imp}~{}n_{1}}$ adequately captures the RPA results, with $n_{1}=6.8\times 10^{11}~{}{\rm cm}^{-2}$. Within the TF theory, we can now qualitatively discuss the main differences between monolayer and bilayer graphene. For bilayer graphene, the inverse screening length changes only slightly from the low density value of $q_{\rm BLG}\approx 4r_{s}\sqrt{\pi n_{0}}$ that is set entirely by the band parameters. As a consequence, the puddle correlation length does not change with the impurity concentration or carrier density, and depends only on the distance $d$ of the bilayer graphene sheet from the source of the long-ranged impurity potential. In contrast, for monolayer graphene, $q_{\rm MLG}=4r_{s}\sqrt{\pi n_{\rm eff}}$ depends essentially on $C(0)$ (and therefore on $n_{\rm imp}$). This heuristic description (which we make more quantitative below) implies that depending on the sample quality, choice of substrate, or doping, the puddle correlation length in monolayer graphene (but not bilayer graphene) could vary by more than an order of magnitude. ## III Random Phase Approximation While the TF screening theory discussed in the previous section is useful to obtain a qualitative picture, we find that it significantly underestimates the effect of electronic screening. In both monolayer and bilayer graphene it gives larger values for $C(0)$ and smaller values for $\xi$. In what follows we use the the random phase approximation (RPA) where the screening function is obtained using $\epsilon(q)=1+q_{\rm TF}{\tilde{\Pi}}(q)/q$. The normalized polarizability ${\tilde{\Pi}}(q)$ for monolayerHwang and Das Sarma (2007) and bilayerGamayun (2011) graphene are both available in the literature. We note that for monolayer graphene ${\tilde{\Pi}}(q)$ depends only on the dimensionless variable $x=q/(2\sqrt{\pi n_{\rm eff}})$ $\displaystyle{\tilde{\Pi}}(x)$ $\displaystyle=$ $\displaystyle 1+\theta(x-1)\left[\frac{\pi x}{4}-\frac{x}{2}{\rm arccsc}(x)-\frac{\sqrt{1-x^{-2}}}{2}\right],$ (3) $\displaystyle\approx$ $\displaystyle\theta(1-x)+\theta(x-1)\frac{\pi x}{4},$ where $\theta(x)$ is a step-function and $D_{0}$ is the density of states with $q_{\rm TF}=2\pi(e^{2}/\kappa)D_{0}=4\sqrt{\pi n_{\rm eff}}r_{s}$. In contrast, for bilayer graphene, the polarizability depends both on the scaled momentum transfer $x$, and on $\eta=n_{\rm eff}/n_{0}<8$, which parameterizes the bilayer hyperbolic dispersion relation. For $\eta\ll 1$, the bilayer graphene dispersion is quadratic, while for $1\ll\eta\leq 8$, the dispersion is linear. For $\eta>8$, one must consider the effects of a second higher-energy band that provides additional screeningMin et al. (2011) and is not considered here. By restricting the density to $n\leq 8n_{0}$, we can simplify the expression for the bilayer polarizability reported in Ref. Gamayun, 2011. Using the bilayer density of states $D=D_{0}\sqrt{1+\eta}$, with $D_{0}=2m/(\pi\hbar^{2})$, the normalized polarizability function ${\tilde{\Pi}}(x,\eta)=\Pi(q)/D_{0}$ is given by $\displaystyle{\tilde{\Pi}}(x,\eta)$ $\displaystyle=$ $\displaystyle f(x,\eta)+\theta(x-1)g(x,\eta),$ $\displaystyle f(x,\eta)$ $\displaystyle=$ $\displaystyle\left[1-x^{2}\eta+x^{4}(2+\eta+2\sqrt{1+\eta})\right]^{1/2}-\ln\left[\frac{2\sqrt{1+\eta x^{2}}}{-1+\sqrt{1+\eta}}\right]-\frac{1}{2}+\frac{3x^{2}\eta-1}{2x\sqrt{\eta}}\arctan(x\sqrt{\eta})$ $\displaystyle\mbox{}+\sqrt{1-x^{2}\eta}\left(2{\rm arctanh}(\sqrt{1-\eta x^{2}})-{\rm arcsinh}\left[\frac{\sqrt{1-x^{2}\eta}(-1+\sqrt{1+\eta})}{x^{2}\eta}\right]\right)+\sqrt{1+\eta}$ $\displaystyle g(x,\eta)$ $\displaystyle=$ $\displaystyle\frac{-\sqrt{x^{2}-1}(1+\eta+2x^{2}\eta-\sqrt{1+\eta})}{2x(\sqrt{1+\eta}-1)}+\frac{(3x^{2}\eta-1)}{2x\sqrt{\eta}}\arccos\left[\sqrt{\frac{1+\eta}{1+x^{2}\eta}}\right]$ (4) $\displaystyle\mbox{}+{\rm arctanh}\left[\frac{x\sqrt{x^{2}-1}\eta}{1+x^{2}\eta-\sqrt{1+\eta}}\right].$ The polarizability functions for monolayer and bilayer graphene are shown in Fig. 3. What is left is to calculate the effective residual density $n_{\rm eff}$ as a function of impurity concentration. As discussed earlier, this is obtained by first calculating the autocorrelation function $C(0)$ from Eq. 2, using the RPA results shown in Fig. 3. Figure 4 shows the autocorrelation function $C(0)$ obtained using the RPA results (solid lines) as well the Thomas-Fermi results (dashed lines). We note that except at very high density (where both monolayer and bilayer graphene approach the “complete screening” limit, with $C(0)=(4k_{\rm F}r_{s}d)^{-2}$), the Thomas-Fermi approximation grossly underestimates the effect of screening. Moreover, for typical densities in bilayer graphene $C(0)$ is approximately constant and independent of carrier density consistent with the heuristic picture discussed at the end of Sec. II. Figure 3: The Random Phase Approximation polarizability function $\Pi(q)$ normalized by the density of states for monolayer graphene and for bilayer graphene with a hyperbolic dispersion. Also shown is the parabolic approximation for the bilayer, which can be obtained from the hyperbolic dispersion when $\eta\ll 1$. The Thomas-Fermi approximation discussed in the text corresponds to the assumption that the normalized $\Pi(q)=1$ for all $q$. Figure 4: Potential autocorrelation function $C[0]$ for monolayer and bilayer graphene. At the Dirac point, $k_{\rm F}$ is the Fermi wavevector arising from the effective carrier density i.e. $k_{\rm F}=\sqrt{\pi n_{\rm eff}}$. For large density $4k_{\rm F}r_{s}d\gg 1$, both the monolayer and bilayer results approach the “complete screening” limit, defined here as $C(0)=(4k_{\rm F}r_{s}d)^{-2}$. Notice that the Thomas-Fermi approximation shown as dashed lines captures the correct qualitative behavior, but can give significantly larger values for $C[0]$, and is therefore unsuitable for quantitative comparisons. The effective density calculated within the RPA is shown in Fig. 2. For bilayer graphene, we find that the following empirical relationship adequately describes the numerical results $n_{\rm eff}=\sqrt{n_{\rm imp}~{}n_{1}},$ (5) where $n_{1}=11.5\times 10^{11}~{}{\rm cm}^{-2}$ for the Thomas-Fermi approximation, and $n_{1}=6.8\times 10^{11}~{}{\rm cm}^{-2}$ for the RPA results. The scaling of the bilayer effective density $n_{\rm eff}\sim\sqrt{n_{\rm imp}}$ can be anticipated for the TF approximation in the limit $n_{\rm eff}\ll n_{0}$. However, it is surprising that the simple empirical relation continues to hold both for the RPA screening theory, and for larger values of $n_{\rm eff}$. This dependence of $n_{\rm eff}\sim\sqrt{n_{\rm imp}}$ for bilayer graphene should be contrasted with similar results obtained previously for monolayer graphene,Adam et al. (2007) where $n_{\rm eff}=2r_{s}^{2}C[0]n_{\rm imp}$ cannot be captured by a similar empirical fit. For comparison, these earlier results are also shown in Fig. 2, where we emphasize that for a given impurity concentration ($n_{\rm imp}$), bilayer graphene exhibits larger density fluctuations ($n_{\rm eff}$) than monolayer graphene. Finally, using Eq. 2, we can also calculate the puddle correlation function, and the corresponding HWHM correlation length, $\xi$, that is shown in Fig. 5 Figure 5: Theoretical results for the puddle correlation length at the Dirac point as a function of impurity concentration. While the puddle size in bilayer graphene ($\xi\approx 3.5~{}{\rm nm}$) is relatively insensitive to the disorder concentration, the size of the puddles in monolayer graphene varies from $3~{}{\rm nm}$ in dirty samples to more than $35~{}{\rm nm}$ in clean samples. ## IV Comparison With Experiments We now compare the calculated correlation functions with experiment. Figure 6(a) shows that for bilayer graphene, $A(r)$ extracted from the data reported in Ref. Rutter et al., 2011 agrees with the calculation for $d=1~{}{\rm nm}$.kn: (a) The circles show the experimental data and the RPA theory for bilayer graphene is shown for $d=1~{}{\rm nm}$ (solid curve) and $d=0.5~{}{\rm nm}$ (dashed curve). The theoretical results are insensitive to the impurity concentration $n_{\rm imp}$ and to how far the doping is away from the Dirac point. Consequently, the only free parameter in the theory is the distance $d$ of the impurities from the graphene sheet. Figure 6: Comparison of theoretical results with experimental data. Top panel shows the normalized correlation function $A(r)=C(r)/C(0)$ for bilayer graphene. The circles are from the experimental data and solid curve is the theory for bilayer graphene with $d=1~{}{\rm nm}$. The theory curve is insensitive to impurity concentration and doping away from the Dirac point. The error bars indicate single standard deviation uncertainties.kn: (a) The small oscillation in the data over the monotonic decrease is a result of the finite size of the experimental image. Bottom panel is the normalized puddle correlation function in monolayer graphene at the Dirac point. Note the change in $x$-axis scale from bilayer graphene in top panel. The solid curve is obtained from the self-consistent screening theory. The black squares are the results of a numerical mesoscopic density functional theory calculation for the ground-state properties of monolayer graphene,Rossi and Das Sarma (2008) while the circles are experimental data taken from Deshpande et al.(Ref. Deshpande et al., 2011). Transport measurements on that same device set $n_{\rm imp}=10^{11}{\rm cm}^{-2}$ which is the value used for the theory curves. The theory also uses $d=1~{}{\rm nm}$, which is the typical distance of the impurities from the graphene sheet extracted from transport measurements of graphene on SiO2.Tan et al. (2007) In Ref. Rutter et al., 2011, we reported a maximum peak-to-peak carrier density fluctuation of $3.6\times 10^{11}~{}{\rm cm}^{-2}$. To extract an impurity density (using the results in Fig. 2), we need to estimate $n_{\rm rms}$ from this peak-to-peak value. By assuming that the carrier density has a Gaussian distribution, and estimating that the peak-to-peak corresponds to a measurement of $4\sigma$, we roughly estimate that $n_{\rm eff}=n_{\rm rms}\approx 10^{11}~{}{\rm cm}^{-2}$ and that $n_{\rm imp}=n_{\rm rms}^{2}/n_{1}\approx 1.2\times 10^{10}~{}{\rm cm}^{-2}$ for the substrate induced impurities in that experiment. We can use this value to predict theoretically the corresponding density fluctuations in the adjacent monolayer sample reported in Ref. Jung et al., 2011. However, one complication is that the theory discussed in Sec. III was developed for monolayer graphene at the Dirac point, while the experimental data was taken at a backgate induced density $n_{g}=1.4~{}\times 10^{12}~{}{\rm cm}^{-2}$. Very far from the Dirac point, i.e. when $n_{\rm imp}/n_{g}\rightarrow 0$, the potential fluctuations $V_{\rm rms}$ can be obtained from Eq. 2 by setting $n_{\rm eff}=n_{g}$ on the right-hand side. In this case, the density fluctuations are $n_{\rm rms}=\frac{2VV_{\rm rms}}{\pi\hbar^{2}v_{\rm F}^{2}}.$ (6) We note that when $z\sim d\sqrt{n_{g}}\gg 1$, we can use the result $C_{0}(z)=z^{-2}$ (see Fig. 4) to obtain $n_{\rm rms}\approx\sqrt{n_{\rm imp}/(8\pi d^{2})}$. However, these constraints are not fully satisfied in the experimental data. Calculating $n_{\rm rms}$ in the crossover between the limits $n_{g}=0$ and $n_{g}\gg n_{\rm imp}$ is more complicated. For our purposes, it is sufficient to extrapolate between the low-density and high- density limits by adding the two contributions in quadrature, and solving for $n_{\rm rms}$ self-consistently. This procedure gives $\displaystyle n_{\rm rms}=2r_{s}\sqrt{n_{\rm imp}C^{\rm RPA}(0)}\left[2n_{g}+3r_{s}^{2}n_{\rm imp}C^{\rm RPA}(0)\right]^{1/2},$ (7) where the superscript indicates that the RPA screening approximation has been used. In the limit that $n_{g}\gg n_{\rm imp}$, Eq. 7 reduces to Eq. 6, while in the opposite limit $n_{g}\rightarrow 0$, Eq. 7 reduces to results shown in Fig. 2. Using the values for $n_{\rm imp}$ and $d$ determined from the bilayer data discussed above, and using Eq. 7 for $n_{\rm rms}$ and Eq. 2 to calculate $\xi$, we find theoretically (without any adjustable parameter) that $n_{\rm rms}\approx 6~{}\times 10^{10}~{}{\rm cm}^{-2}$ and $\xi=8~{}{\rm nm}$, which should be compared to the experimental values extracted from the data reported in Ref. Jung et al., 2011. The area surveyed in Ref. Jung et al., 2011 was not large enough to obtain $\xi$ accurately. However, by looking at different real-space cuts of the autocorrelation function, we conclude that the experimental data is consistent with a correlation length $6~{}{\rm nm}<\xi<11~{}{\rm nm}$. This is in qualitative agreement with our theoretical calculations. This result should be contrasted with bilayer graphene shown in Fig. 6(a) where the experimentally determined $\xi=(3.68\pm 0.03)~{}{\rm nm}$, and the $d=1~{}{\rm nm}$ RPA theory gives $\xi=3.4~{}{\rm nm}$. To further confirm our results, we compare our calculations to measurements made using scanning Coulomb blockade spectroscopy on a sample of monolayer graphene.Deshpande et al. (2011) The circles in Fig. 6(b) are experimental data for the normalized correlation and the solid line is the self-consistent theory discussed above using $n_{\rm imp}=10^{11}{\rm cm}^{-2}$ and $d=1~{}{\rm nm}$ for monolayer graphene at the Dirac point. The impurity concentration and impurity distance were determined from transport measurementsDeshpande et al. (2011); Adam et al. (2007) and as such, no adjustable parameters were used in the calculation. In Fig. 6 we also show (black squares) the results extracted from a numerical mesoscopic density functional theoryRossi and Das Sarma (2008) using the same parameters. The agreement between the two calculations provides a posteriori justification for our assumption of a global screening function characterized by the density $n_{\rm eff}$. ## V Conclusions We conclude with the observation that our results require only that the source of the disorder potential be uncorrelated charged impurities, and as such should apply to graphene on other substrates. For example, recently graphene devices with hexagonal BN gate insulators have been fabricated showing transport properties similar to suspended grapheneDean et al. (2010) and larger puddles than on SiO2 substrates.Xue et al. (2011); Decker et al. (2011) These observations are consistent with both a much smaller charged impurity density $n_{\rm imp}$ on the BN substrate and with a larger distance $d$ of the impurities from the graphene layer. Both these scenarios are possible because the BN substrate is typically placed on top of the usual SiO2 wafer which would have similar charged disorder to the samples we study here. We argue that an analysis similar to what we have performed here would be able to uniquely determine both $d$ and $n_{\rm imp}$. Moreover, if a similar experiment is done with bilayer graphene on BN substrates, we predict that the puddle characteristics will not change much from what we find here with bilayer graphene on SiO2. ## Acknowledgements This work is supported in part by the NIST-CNST/UMD-NanoCenter Cooperative Agreement. It is a pleasure to thank W. G. Cullen, M. S. Fuhrer, and M. Polini for discussions, and P. W. Brouwer, E. Cockayne, G. Gallatin, J. McClelland and R. McMichael for comments on the manuscript. ## References * Das Sarma et al. (2011) S. Das Sarma, S. Adam, E. H. Hwang, and E. Rossi, Rev. Mod. Phys. 83, 407 (2011). * Castro Neto et al. (2009) A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, Rev. Mod. Phys. 81, 109 (2009). * Fuhrer and Adam (2009) M. S. Fuhrer and S. Adam, Nature 458, 38 (2009). * Bardarson et al. (2007) J. H. Bardarson, J. Tworzydlo, P. W. Brouwer, and C. W. J. Beenakker, Phys. Rev. Lett. 99, 106801 (2007). * Anderson (1958) P. W. Anderson, Phys. Rev. 109, 1492 (1958). * Tanatar and Ceperley (1989) B. Tanatar and D. M. Ceperley, Phys. Rev. B 39, 5005 (1989). * Katsnelson et al. (2006) M. I. Katsnelson, K. S. Novoselov, and A. K. Geim, Nature Phys. 2, 620 (2006). * McCann and Fal’ko (2006) E. McCann and V. Fal’ko, Phys. Rev. Lett. 96, 086805 (2006). * Martin et al. (2008) J. Martin, N. Akerman, G. Ulbricht, T. Lohmann, J. H. Smet, K. von Klitzing, and A. Yacobi, Nature Phys. 4, 144 (2008). * Zhang et al. (2009) Y. Zhang, V. Brar, C. Girit, A. Zettl, and M. Crommie, Nature Phys. 5, 722 (2009). * Deshpande et al. (2009) A. Deshpande, W. Bao, Z. Zhao, C. N. Lau, and B. J. LeRoy, Appl. Phys. Lett. 95, 243502 (2009). * Deshpande et al. (2011) A. Deshpande, W. Bao, Z. Zhao, C. N. Lau, and B. J. LeRoy, Phys. Rev. B 83, 155409 (2011). * Jung et al. (2011) S. Jung, G. M. Rutter, N. N. Klimov, D. B. Newell, I. Calizo, A. R. Hight-Walker, N. B. Zhitenev, and J. A. Stroscio, Nature Phys. 7, 245 (2011). * Rutter et al. (2011) G. R. Rutter, S. Jung, N. N. Klimov, D. B. Newell, N. B. Zhitenev, and J. A. Stroscio, Nature Phys. 7, 649 (2011). * Adam and Das Sarma (2008) S. Adam and S. Das Sarma, Phys. Rev. B 77, 115436 (2008). * Adam et al. (2007) S. Adam, E. H. Hwang, V. M. Galitski, and S. Das Sarma, Proc. Natl. Acad. Sci. USA 104, 18392 (2007). * Ando et al. (1982) T. Ando, A. B. Fowler, and F. Stern, Rev. Mod. Phys. 54, 437 (1982). * Hwang and Das Sarma (2007) E. H. Hwang and S. Das Sarma, Phys. Rev. B 75, 205418 (2007). * Gamayun (2011) O. V. Gamayun, Phys. Rev. B 84, 085112 (2011). * Min et al. (2011) H. Min, P. Jain, S. Adam, and M. D. Stiles, Phys. Rev. B p. 195117 (2011). * kn: (a) The dominant uncertainty in $A(r)$ and $\xi$ comes from the finite size of the experimental puddle images. The error in $A(r)$ is estimated by dividing the image area into $N=4$ patches and analyzing each patch separately. Treating the $N$ patches as independent samples, we compute the variance of the set and estimate the variance of the whole sample as the variance of the $N$ independent samples, divided by $N-1$. The uncertainty in $A(r)$ is square root of the variance, and the uncertainty in $\xi$ then follows. We say that the experimentally determined $A(r)$ agrees with the calculated $A(r)$ for $d=1~{}{\rm nm}$ in Fig. 6a because theory curve lies within the experimental uncertainty of $A(r)$. * Rossi and Das Sarma (2008) E. Rossi and S. Das Sarma, Phys. Rev. Lett. 101, 166803 (2008). * Tan et al. (2007) Y.-W. Tan, Y. Zhang, K. Bolotin, Y. Zhao, S. Adam, E. H. Hwang, S. Das Sarma, H. L. Stormer, and P. Kim, Phys. Rev. Lett. 99, 246803 (2007). * Dean et al. (2010) C. R. Dean, A. F. Young, I. Meric, C. Lee, L. Wang, S. Sorgenfrei, K. Watanabe, T. Taniguchi, P. Kim, K. L. Shepard, et al., Nature Nano. 5, 722 (2010). * Xue et al. (2011) J. Xue, J. Sanchez-Yamagishi, D. Bulmash, P. Jacquod, A. Deshpande, K. Watanabe, T. Taniguchi, P. Jarillo-Herrero, and B. J. Leroy, Nature Mat. 10, 282 (2011). * Decker et al. (2011) R. Decker, Y. Wang, V. W. Brar, W. Regan, H.-Z. Tsai, Q. Wu, W. Gannett, A. Zettl, and M. F. Crommie, Nano Lett. 11, 2291 (2011).	arxiv-papers	2011-12-05T21:00:02	2024-09-04T02:49:24.989095	{ "license": "Public Domain", "authors": "Shaffique Adam, Suyong Jung, Nikolai N. Klimov, Nikolai B. Zhitenev,\n Joseph A. Stroscio, M. D. Stiles", "submitter": "Shaffique Adam", "url": "https://arxiv.org/abs/1112.1070" }
1112.1111	# Investigating the origin of the long range pseudo rapidity correlation in 2d di-hadron measurements from STAR††thanks: Presented at Strange Quark Matter conference, September, 2011, Krakow, Poland. L. C. De Silva The Department of Physics, University of Houston, 617 SR Building 1, Houston, Texas 77204-5005 ###### Abstract Angular di-hadron correlations reveal novel structures in central Au+Au collisions at $\sqrt{S_{NN}}$ = 200 GeV. One of them, known as the ridge, is elongated in pseudo rapidity and peaks on the near side ($\Delta\phi$ $\approx$ 0). Investigating the origin of the ridge structure helps to understand the hot dense matter that is created in ultra relativistic heavy ion collisions. Results showing the $\langle$$p_{T}$$\rangle$ dependence of the ridge structure are presented. Evidence for possible jet and non-jet contributions to the ridge structure will be discussed. ## 1 Introduction Recent triggered di-hadron correlation studies by STAR report a ridge structure in two dimensions ($\Delta\eta$,$\Delta\phi$)[1]. Based on their studies, the near side was assumed to consist of two independent structures, a jet-like structure and the ridge. A complementary approach, presented here, is carried out by using all possible charged particle pairs. The approach does not require a trigger particle but with appropriate kinematic cuts reproduces qualitatively similar correlation structure to that of triggered analysis. In this approach, two empirical 2d fit functional models are used to extract ridge properties. Possible model dependent quantitative evidence for jet and non-jet phenomena contribution to the near side structure can be extracted from the fits. ## 2 Data and analysis The data used in this analysis were collected during run 4, year 2004, from the STAR detector at Relativistic Heavy Ion Collider (RHIC) Brookhaven National Lab (BNL), Long Island, New York. 32M Au+Au collisions at $\sqrt{S_{NN}}$ = 200 GeV were analyzed. Charged tracks reconstructed using the Time Projection Chamber (TPC) with 2$\pi$ azimuthal coverage and $\|\eta\|$ $\leq$ 1 in pseudo rapidity were selected. An earlier centrality evolution analysis[2] was based on particles in the full transverse momentum range above 0.15 GeV/c. For the study presented here the lower threshold of the transverse momentum was raised for both particles. The selected event vertexes are within $\pm$ 25cm of the detector center and primary tracks are selected to come from within a distance of closet approach of 3cm to the event vertex. The correlation function for this analysis is a construct of Pearson s correlation coefficient. Mathematically it can be derived as[2]: $\displaystyle\frac{\Delta\rho}{\sqrt{\rho_{ref}}}$ $\displaystyle=\frac{\rho_{sib}\\--\rho_{mix}}{\sqrt{\rho_{ref}}}$ (1) $\rho_{sib}$ is the normalized charged particle pair density within a single event, which carries both correlated and uncorrelated pairs. The uncorrelated background is subtracted by mixed event pairs. The resulting uncorrelated density is denoted as $\rho_{mix}$. The difference is the number of correlated pairs which is normalized by the square root of mixed pair density to yield the _correlated particle pairs per final state charged particle_ as the equivalent correlation measure to the Pearson s correlation coefficient. In order to avoid possible artifacts due to pseudo rapidity acceptance, the correlation measure is carried out in sub bins of the z-vertex. The event mixing artifacts were eliminated by mixing events of a particular centrality within a multiplicity window of 50 charged tracks. To correct for pair loss a pair cut is implemented on both mixed and same event pairs. The effects due to pileup events were removed by one of the standard pileup eliminating procedures in STAR[7]. Finally the conversion electron positron pair background is reduced by using a 1.5$\sigma$ dE/dX cut on electrons in momentum ranges, 0.2 <$p_{T}$ <0.45 GeV/c and 0.7 <$p_{T}$ <0.8 GeV/c. Figure 1: Evolution of the raw normalized correlation structure for selected $\langle$$p_{T}$$\rangle$ bins at 0-1% centrality bin. The lower momentum threshold of both particles have been increased in generating the spectrum of correlation plots. ## 3 Fit functions The initial empirical fit function carries three mathematical model components that lead to seven free fit parameters. F = $c_{0}$ \+ $c_{1}$cos($\Delta\phi$) + $c_{2}$exp(-0.5(($\Delta\phi$/$c_{3}$)2 \+ ($\Delta\eta$/$c_{4}$)2) + $c_{5}$exp(-0.5(($\Delta\phi$/$c_{6}$)2 \+ ($\Delta\eta$/$c_{6}$)2)) The first component of the fit is an offset parameter followed by a cos($\Delta\phi$) term to extract the away side structure as seen in Fig. 1. The assumption is that this structure corresponds to correlations due to a recoiling jet[3]. The two remaining terms are introduced based on $\langle$$p_{T}$$\rangle$ evolution of data (see Fig. 1). The asymmetric 2d Gaussian attempts to address the long range correlation properties excluding the peak structure around (0,0) that appears as $\langle$$p_{T}$$\rangle$ increases. This structure is described via the symmetric 2d Gaussian component of the fit. Modifications to the model were introduced after indications of higher order harmonics were first predicted [4, 5] and then observed (see Fig. 2). F = $c_{0}$ \+ $c_{1}$cos($\Delta\phi$) + $c_{2}$cos(2$\Delta\phi$) + $c_{3}$cos(3$\Delta\phi$) + $c_{4}$cos(4$\Delta\phi$) + $c_{5}$cos(5$\Delta\phi$) + $c_{6}$exp(-0.5(($\Delta\phi$/$c_{7}$)2 \+ ($\Delta\eta$/$c_{8}$)2)) The resulting fit carries higher order harmonics up to 5th order and an asymmetric 2d Gaussian. It is important to note that in such a Fourier expansion, the first order term serves as a momentum conserving term (correlation due to recoil jet). In essence, the new model help in further constraining the knowledge gathered from the initial model study. Figure 2: Correlations at high $p_{T}$ very central data show evidence for the existence of higher order harmonics. ## 4 Results The focus of comparing the initial model study to the study which includes fit components for higher harmonics based on initial energy density fluctuations is to determine whether any information can be extracted from the asymmetric 2d Gaussian fit to the away side structure. ### 4.1 Initial model study Resulting asymmetric 2d Gaussian parameters are reported in Fig. 3. The long correlation strength approaches zero at higher $\langle$$p_{T}$$\rangle$ as indicated by the asymmetric 2d Gaussian amplitude and volume parameters. The $\Delta\eta$ and $\Delta\phi$ width evolution clearly indicate the asymmetry shown in data. Also the $\Delta\eta$ width suggests that within the STAR acceptance the ridge is flat whereas the $\phi$ width shows a very smooth narrowing. An important observation is that the $\langle$$p_{T}$$\rangle$ of the ridge ($\langle$$p_{T}$$\rangle$ = 0.72 GeV/c) is greater than that of the inclusive spectrum ($\langle$$p_{T}$$\rangle$ $\approx$ 0.42 GeV/c [8]). The hardness of the ridge spectrum is an indication of possible Jet contributions to the ridge. (a) Amplitude (b) $\Delta\eta$ width (c) $\Delta\phi$ width (d) Symmetric width Figure 3: Asymmetric and Symmetric 2d Gaussian parameter evolution as a function of $\langle$$p_{T}$$\rangle$. On the other hand, the symmetric 2d Gaussian component reveals important information about an unmodified jet emerging at higher momenta (Fig. 3). The width is comparable to the jet width in PP collisions. In order to further constrain the above conclusion, amplitude and yield comparisons need to be made. (a) $\frac{v_{3}}{v_{2}}$ ratio (b) $v_{n}\hskip 2.84544ptscaling\hskip 2.84544ptat\hskip 2.84544pt0\\--10\%$ Figure 4: Comparison to predicted higher harmonic ratios[5] and scaling relations[4] ### 4.2 Higher harmonics study Recent advancements in heavy ion collision models, predict significant contributions from higher order harmonics to the observed correlation spectra due to initial energy density fluctuations[4, 5]. STAR data also support the predicted observation in high $p_{T}$, very central AuAu 200 GeV data(Fig. 2). The $\frac{v_{3}}{v_{2}}$ ratio predictions were tested and reasonable agreement to theory is evident (see Fig. 4a). However it is to be noted at low $p_{T}$, deviations from the trends are observed for $N_{part}$ <150\. One should note, thought, that the data is extracted fitting 2d di-hadron correlations, whereas, theory uses single particle spectra and event plane calculations. A predicted hydro scaling relation[6] using extracted higher order Fourier coefficients is studied, and reasonable agreement to theory has been observed (Fig. 4b).In summary, given the different approaches used by theory predictions tested in this proceedings and experiment, the data trends seem support the hydro based theory predictions to a reasonable extent. The remaining structure on the near side, after subtracting the Fourier components still shows a $\Delta\eta-\Delta\phi$ asymmetry (see Fig. 3, red data markers). This $\langle$$p_{T}$$\rangle$ evolution indicates that this $\Delta\eta$ elongation is visible up to 2.7 GeV/c and is strongest at 0.9 GeV/c. The structure thus suggests possible jet modification at low $\langle$$p_{T}$$\rangle$ which evolves to an unmodified jet at high $p_{T}$. The amplitude and volume of the asymmetric structure scales lineary with Glauber scaling as a function of centrality, which hints at a jet origin. The rise of amplitude at high $\langle$$p_{T}$$\rangle$ is expected due to an increase in per charge particle pair yield in jets. Further studies will be carried out using the comparison to p+p collisions. ## 5 Summary and discussion Un-triggered 2d di-hadron correlation studies reproduce qualitatively similar results to that of a triggered analysis. The observed near side correlation is modeled via an empirical fit function which extracts short and long range structure properties. The $\langle$$p_{T}$$\rangle$ evolution of the extracted parameters suggests a possible correlation between jets and the observed long range correlation. Further constraint to data has been imposed modifying the empirical model to incorporate higher order Fourier model components cos(n$\Delta\phi$), n = 1– 5, based on theoretical predictions and evidence in the data. The extracted higher order component strengths show reasonable agreement to predicted hydrodynamical trends. The remainder on the near side still reveals an asymmetric 2d Gaussian suggestive of possible modified jet phenomena. A Glauber linear scaling was successfully applied to the un- triggered centrality evolution (see Fig. 5) which indicates a jet origin for the observed near side Gaussian. Figure 5: Comparison of near side asymmetric 2D Gaussian amplitude to Glauber linear scaling as a function of centrality. ## References [1] B. I. Abelev et al. (STAR Collaboration), Phys. Rev. C 80, 064912 (2009) * [2] M Daugherity et al. (STAR Collaboration), J. Phys. G 35, 104090 (2008) * [3] J. Porter and T. Trainor, J. Phys.: Conf. Ser. 27, 98 (2005) * [4] Xin Niang Wang et al., Phys. Rev. Lett. 106, 162301(2011) * [5] B. Alver et al., Phys. Rev. C 81, 054905 (2010) * [6] C. Gombeaud et al., Phys. Rev. C 81, 014901 (2010) * [7] Nuclear Phys. Lab. Annual Report, University of Washington (2009) p. 58 * [8] J. Adams et al. STAR Collaboration	arxiv-papers	2011-12-05T22:13:47	2024-09-04T02:49:24.997087	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "L. C. De Silva (for the STAR collaboration)", "submitter": "Chanaka De Silva Mr.", "url": "https://arxiv.org/abs/1112.1111" }
1112.1139	# Quantum Verification of Minimum Spanning Tree Mark Heiligman Intelligence Advanced Research Projects Activity, Office of the Director of National Intelligence, Washington, D.C. mark.i.heiligmanugov.gov (January 29, 2011 ; 1926 ; 2006 ; 1997 ; 1997 ; 1985 ) Previous studies has shown that for a weighted undirected graph having $n$ vertices and $m$ edges, a minimal weight spanning tree can be found with $O^{}\bigl{(}\sqrt{mn}\bigr{)}$ calls to the weight oracle. The present note shows that a given spanning tree can be verified to be a minimal weight spanning tree with only $O\bigl{(}n\bigr{)}$ calls to the weight oracle and $O\bigl{(}n+\sqrt{m}\log n\bigr{)}$ total work. quantum algorithms, graph theory, spanning tree ††support: Disclaimer. All statements of fact, opinion, or analysis expressed in this paper are solely those of the author and do not necessarily reflect the official positions or views of the Office of the Director of National Intelligence (ODNI), the Intelligence Advanced Research Projects Activity (IARPA), or any other government agency. Nothing in the content should be construed as asserting or implying U.S. Government authentication of information or ODNI endorsement of the author’s views. ## Introduction ### Problem Statement The determination of a minimal weight spanning tree of a weighted undirected graph is a central problem in computational graph theory and a number of well known classical algorithms address the problem quite efficiently. This problem has also shown up in the realm of quantum algorithms and the paper [DHHM] provides nearly matching upper and lower bounds for the problem. (The term “nearly matching” as used here means that the upper and lower bounds agree to withing a power of the logarithm of the problem size.) The algorithm in [DHHM] uses some of the constructs that occur in the classical minimal spanning tree algorithms, along with a somewhat sophisticated version of the quantum minimum algorithm (which itself is based on Grover’s algorithm). A closely related problem deals with the verification of minimal spanning tree. In this formulation of the problem, both a weighted graph and a spanning tree of that graph are given as inputs, and the problem is to decide whether the given spanning tree is of minimal weight (and if not to give a lower weight spanning tree). Based on work of [Ko], a simple classical verification algorithm was given in [Ki]. ###### Problem Given a graph $G=(V,E)$ consisting of $n={\left\|{G}\right\|}$ vertices and $m={\left\|{E}\right\|}$ edges along with a weight function on the edges $w:\,E\rightarrow{\mathbb{R}}^{+}$, and a spanning tree $T=(V,F)$ with $F\subset E$ and ${\left\|{F}\right\|}=n-1$, verify that $T$ is a minimal weight spanning tree. The goal of this paper is to develop a quantum algorithm for the verification problem. We build heavily on the graph theory methods given in [Ki] and [KPRS]. Our quantum tool in this case is a fairly simple version of Grover’s algorithm. Nevertheless we are able to show that verification is simpler than finding the solution ab initio. ### Computational Models There is a basic question of how the graphs $G$ and $T$ are presented, and this can critically affect the efficiency of the algorithm. A graph can be presented by an adjacency matrix or by a simple listing of its edges (and this may be either a sorted or an unsorted list). In the classical world, the problem statement is fairly simple. In the quantum world, the graph is presented to the algorithm as an oracle, and the complexity of the algorithm is measured in the number of oracle calls necessary to solve the problem. Oracles can be applied in the classical world, as well, but they are less indicative of the computational complexity of the problem than in the quantum world. In the classical world, the entire graph needs to be made available to the algorithm, so in the adjacency matrix model there would be $O\bigl{(}n^{2}\bigr{)}$ calls to the oracle specifying the graph, while in the edge list model there would be $m$ calls to the oracle simply to get the entire graph into the computer. There also has to be an oracle that gives the weight of an edge, and in the adjacency matrix model or the edge list model, there would be $m$ calls to the weight oracle. It is useful to combine the graph oracle with the weight oracle. In the adjacency matrix model, the graph is extended to a complete graph and weight $+\infty$ is assigned to all non-graph edges, with the oracle being given as a function $w:\,V\times V\rightarrow{\mathbb{R}}^{+}\cup\\{\infty\\}$. In the edge list model the oracle is a function, $e:\,[1,m]\rightarrow V\times V\times{\mathbb{R}}^{+}$ where the first two components give the endpoints of the $i^{\hbox{\sevenrm th}}$ edge of the graph $G$ and the last component gives the weight of that edge. In this note, we consider both of these models, but from the quantum perspective, the oracle has to be viewed as a reversible function that then operates on quantum states. The two models to be considered here are: (1) There is the weight oracle in the adjacency matrix model. For this model, a call to the quantum weight oracle is ${\mid{a,b,x}\rangle}\rightarrow{\mid{a,b,x\oplus w(a,b)}\rangle}$ where $a,b\in V$ are a pair of vertices. (Note that $x$ here is just some arbitrary initial bit string.) For finding minimum weight spanning trees, this is bad if the graph is moderately sparse. For checking the minimality of a spanning tree, the input would consist of a simple listing of the edges and would be of length $n-1$. (2) There is the combined edge list and edge weight in the edge list model. For this model, a call to the quantum oracle is ${\mid{i,x,y,z}\rangle}\rightarrow{\mid{i,x\oplus a,y\oplus b,z\oplus w}\rangle}$ where $a,b\in V$ are a pair of vertices such that $(a,b)$ is the $i^{\hbox{\sevenrm th}}$ edge of the graph $G$ and $w=w(a,b)$ is its weight. (Note that $x$, $y$, and $z$ here is just some arbitrary initial bit strings.) Note that the result for model (1) above will give an upper bound for model (2), but both models will be considered in this note. Thus in the adjacency matrix model, there will be a weight oracle given $w:\,V\times V\rightarrow{\mathbb{R}}^{+}\cup\\{+\infty\\}$ and the spanning tree to be checked for minimality will be (classically) input as a list of edges $T=\bigl{\\{}(a_{1},b_{1}),(a_{2},b_{2}),\ldots,(a_{n-1},b_{n-1})\bigr{\\}}$ with $(a_{i},b_{i})\in V\times V$ for $i=1,\ldots,n-1$. We will also consider the $e$ oracle in the edge list model. However, even there, the spanning tree to be checked for minimality will still be classically input as a list of edges, only now the spanning tree to be checked for minimality will be (classically) input as a list of edge indices $T=\bigl{\\{}e_{i},e_{2}\,\ldots e_{n-1}\bigr{\\}}$ with $e_{i}\in[1,m]$. In both of the above formulations, the subtree to be tested for minimality by the quantum algorithm is input classically. This gives a lower bound for the complexity of the quantum algorithm of $O(n)$, since the algorithm has to at least read in all the (classical) input. However, there are other possible statements of the problem. (3) Given an oracle for the weights of $G$ (which is by default, also an oracle for querying whether a given pair of points of $V$ is an edge of $G$), the input could be by an oracle for the putative minimal spanning tree. Thus, in the adjacency matrix model, there is a function $mst:V\times V\rightarrow\\{0,1\\}$ where $mst(a,b)=1$ if $(a,b)$ is an edge in $T$ and $mst(a,b)=0$ if $(a,b)$ is not an edge in $T$, while in the edge list model,there is a function $mst:V\times[1,m]\rightarrow\\{0,1\\}$ where $mst(i)=1$ if $i$ is an edge index in $T$ and $mst(i)=0$ if $i$ is not an edge in $T$. In either case, the problem then becomes to determine whether $mst$ is a correct oracle. The complication is that there are now two oracles to count calls to, and in principle there could be an operation curve of tradeoffs. In fact, this is almost certainly the case, because on one extreme the minimal spanning tree can be found simply by computing it with a quantum algorithm and then checking the $mst$ oracle for mismatches with the minimal spanning tree found. This reduces to the problem: Given a set $S$ and a subset $U\subset S$ and a (quantum) oracle $p:\,S\rightarrow\\{0,1\\}$, is it the case that $p(s)=1$ if and only if $s\in U$? Counting oracle calls here would seem to be a simple application of Grover’s algorithm. ## Minimal Weight Spanning Trees ### Checking a Spanning Tree The key observation from [Ki] is the following. For a graph $G=(V,E)$ and any spanning tree $T$ of $G$, there is a unique path between any two edges $u,v\in V$. $T$ is a minimal weight spanning tree if and only if the weight of each edge $(u,v)\in E-T$ is greater than or equal to the the heaviest edge in the path in $T$ between $u$ and $v$. What is needed is an easy way to find the weight of the heaviest edge in the path in $T$ between $u$ and $v$. The idea for checking a putative spanning tree $T$ for minimality is to show that for any other edge of $G$ not in $T$, in the cycle formed by including this edge, the highest weight edge in the cycle is exactly this edge. There is no way that this edge can be part of a minimum weight spanning tree. This is to be checked for all edges of $G$, so by invoking Grover’s algorithm in the quantum setting, the total work is $O\bigl{(}\sqrt{m}\bigr{)}$ times the work of checking an edge. The problem is that for checking an edge $(u,v)\in E-T$, the length of the path in $T$ between $u$ and $v$ could be very large, perhaps even as big as $n-1$, so even using Grover’s algorithm to find the maximal weight edge on this path is not adequately efficient. ### Boruvka Trees What is needed is a a new data structure that allows the maximal weight edge on any path in $T$ to be found efficiently. The basic idea for this comes from one of the earliest papers in computational graph theory [B], that was the forerunner of several modern spanning tree algorithms. The key properties of the Boruvka tree built from a putative minimal spanning tree come from [Ki] and will be summarized here without proof. The idea of a Boruvka tree built from a spanning tree $T$ is that $B_{T}$ is a tree whose leaves are the vertices $V$ and whose internal nodes are to be viewed as subsets of $V$. In general for any graph $G=(V,E)$ and any spanning tree $T$ of $G$, the Boruvka graph is a rooted tree of depth at most $\lceil\log_{2}({\left\|{V}\right\|})\rceil$. This Boruvka graph consists of successively larger aggregations of elements of $V$. All nodes in a Boruvka tree are subsets of $V$. The leaves are all the singleton sets $\\{v_{i}\\}$ as $v_{i}$ runs over all the elements of $V$, and eventually the root is formed, which will be $V$ itself. Any intermediate node in a Boruvka tree is the union of its children. A Boruvka tree is built from the bottom up. At each stage or level, every node computes its nearest neighbor (i.e. the node that it is closest to), and an edge is formed for all such nodes. The nodes of the next level up are then the connected components of the graph of the previous level. The weight of each branch is just the weight of the edge of $G$ that was just added. The result is a rooted tree with at most $2\,n$ nodes and $n$ leaves. The key property of $B_{T}$ is that if $u,v\in V$ are a pair of vertices and if $B_{T}(u,v)$ is the smallest subtree of $B_{T}$ that has both $u$ and $v$ as leaves, then the weight of the heaviest edge that connects $u$ and $v$ in the original spanning tree $T$ is equal to the weight of the heaviest edge in $B_{T}(u,v)$. The Boruvka tree $B_{T}$ is a full branching tree, which means that it has a specified root, all its leaves are at the same level, and each internal node has at least two children. The height of $B_{T}$ is at most $\lceil\log_{2}n\rceil$. Therefore once $B_{T}$ has been constructed, finding the heaviest edge in $B_{T}(u,v)$ costs at most $O\bigl{(}\log n\bigr{)}$ operations. In fact, if $B_{T}$ has already been built, then finding the heaviest edge in $B_{T}(u,v)$ requires no queries of the edge weight oracle. Therefore, to check any edge in the original graph requires only one oracle query, and total work at most $O\bigl{(}\log n\bigr{)}$. ## The Quantum Algorithm The Boruvka tree $B_{T}$ can be made with work $O\bigl{(}n\bigr{)}$, (not just $O\bigl{(}n\log n\bigr{)}$ work), and can be done classically (see [Ki] and [Ko]), the total number of oracle queries of the weight function being $O\bigl{(}n\bigr{)}$, as well. This is what makes this algorithm so effective. Once the Boruvka tree of the input spanning tree has been formed, it is possible to check whether the input spanning tree is minimal. To check any edge $(a,b)\in E$ from the original graph $G$, the maximal weight of the edge in the path in $T$ that connects $a$ and $b$ is easily found. Simply start with the leaves $a$ and $b$ and go up $B_{T}$ one level at a time until they meet at a common internal node (which might be the root). Recording the maximal weight found in the set of edges traversed in $B_{T}$ up the their common internal node gives the maximal weight of the edge in the path that connects $a$ and $b$. Since the height of $B_{T}$ is bounded by $\lceil\log_{2}n\rceil$, the total work for this is $O\bigl{(}\log n\bigr{)}$, and no oracle calls are required since $B_{T}$ is already built classically. To check if $(a,b)$ is of lower weight than the maximal weight of the edge in the path in $T$ that connects $a$ and $b$ only require one invocation of the weight oracle. Of course, if this weight is less, then a lower weight spanning tree than $T$ has been found by swapping out the maximum weight edge in the path that connects $a$ and $b$ in $T$ with the edge $(a,b)$. Using Grover’s algorithm over all vertex pairs $V\times V$ for the weight oracle, therefore requires $O\bigl{(}n\bigr{)}$ oracle queries, and $O\bigl{(}n\log n\bigr{)}$ total work. If an edge weight oracle is given, then it is possible to run Grover’s algorithm over the original edge set $E$, which only requires $O\bigl{(}\sqrt{m}\bigr{)}$ oracle queries, and $O\bigl{(}\sqrt{m}\log n\bigr{)}$ total work. Since $m<n^{2}$, it follows that the number of oracle queries for the quantum part of the algorithm is less than the number of classical oracle queries needed to construct $B_{T}$. Therefore the total number of oracle queries is $O\bigl{(}n\bigr{)}$ and the total work is $O\bigl{(}n+\sqrt{m}\log n\bigr{)}$. In conclusion, it is interesting that the verification of a putatively correct answer can be accomplished with considerably less work than that of finding the answer. ## References B Otakar Borűvka, O jistém problému minimálním (About a certain minimal problem), Práca Moravské Pr̆írodoc̆edecké Spolec̆nosti 3, 37-58 . (in Czech, with German summary) * DHHM Christoph Dürr, Mark Heiligman, Peter Høyer, and Mehdi Mhalla, Quantum query complexity of some graph problems, SIAM Journal on Computing 35(6), 1310–1328 , see also http://xxx.lanl.gov/abs/quant-ph/0401091. * Ki Valerie King, A simpler minimum spanning tree verification algorithm., Algorithmica 18, 263-270 . * KPRS V. King, C.K. Poon, V. Ramachandran, and S. Sinha, An optimal EREW PRAM algorithm for minimum spanning tree verification., Information Processing Letters 62, 153-159 . * Ko Komlos, Linear verifcation for spanning trees, Combinatorica 5, 57-65 .	arxiv-papers	2011-12-05T15:26:51	2024-09-04T02:49:25.006298	{ "license": "Public Domain", "authors": "Mark Heiligman", "submitter": "Mark Heiligman", "url": "https://arxiv.org/abs/1112.1139" }
1112.1159	# Evolution of the single-mode squeezed vacuum state in amplitude dissipative channel Hong-Yi Fan1, Shuai Wang1 and Li-Yun Hu2∗ E-mail: hlyun2008@126.com. 1Department of Physics, Shanghai Jiao Tong University, Shanghai 200240,China 2College of Physics & Communication Electronics, Jiangxi Normal University, Nanchang 330022, China $\ast$Corresponding author.E-mail: hlyun2008@126.com. ###### Abstract Using the way of deriving infinitive sum representation of density operator as a solution to the master equation describing the amplitude dissipative channel by virtue of the entangled state representation, we show manifestly how the initial density operator of a single-mode squeezed vacuum state evolves into a definite mixed state which turns out to be a squeezed chaotic state with decreasing-squeezing. We investigate average photon number, photon statistics distributions for this state. ## I Introduction Squeezed states are such for which the noise in one of the chosen pair of observables is reduced below the vacuum or ground-state noise level, at the expense of increased noise in the other observable. The squeezing effect indeed improves interferometric and spectroscopic measurements, so in the context of interferometric detection and of gravitational waves the squeezed state is very useful 01 ; 02 . In a very recently published paper, Agarwal r1 revealed that a vortex state of a two-mode system can be generated from a squeezed vacuum by subtracting a photon, such a subtracting mechanism may happen in a quantum channel with amplitude damping. Usually, in nature every system is not isolated, dissipation or dephasing usually happens when a system is immersed in a thermal environment, or a signal (a quantum state) passes through a quantum channel which is described by a master equation 03 . For example, when a pure state propagates in a medium, it inevitably interacts with it and evolves into a mixed state 04 . Dissipation or dephasing will deteriorate the degree of nonclassicality of photon fields, so physicists pay much attention to it 05 ; 06 ; 07 . In this present work we investigate how an initial single-mode squeezed vacuum state evolves in an amplitude dissipative channel (ADC). When a system is described by its interaction with a channel with a large number of degrees of freedom, master equations are set up for a better understanding how quantum decoherence is processed to affect unitary character in the dissipation or gain of the system. In most cases people are interested in the evolution of the variables associated with the system only. This requires us to obtain the equations of motion for the system of interest only after tracing over the reservoir variables. A quantitative measure of nonclassicality of quantum fields is necessary for further investigating the system’s dynamical behavior. For this channel, the associated loss mechanism in physical processes is governed by the following master equation 03 $\frac{d\rho\left(t\right)}{dt}=\kappa\left(2a\rho a^{\dagger}-a^{\dagger}a\rho-\rho a^{\dagger}a\right),$ (1) where $\rho$ is the density operator of the system, and $\kappa$ is the rate of decay. We have solved this problem with use of the thermo entangled state representation 08 . Our questions are: What kind of mixed state does the initial squeezed state turns into? How does the photon statistics distributions varies in the ADC? Thus solving master equations is one of the fundamental tasks in quantum optics. Usually people use various quasi-probability representations, such as P-representation, Q-representation, complex P-representation, and Wigner functions, etc. for converting the master equations of density operators into their corresponding c-number equations. Recently, a new approach 08 ; 09 , using the thermal entangled state representation 10 ; 11 to convert operator master equations to their c-number equations is presented which can directly lead to the corresponding Kraus operators (the infinitive representation of evolved density operators) in many cases. The work is arranged as follows. In Sec. 2 by virtue of the entangled state representation we briefly review our way of deriving the infinitive sum representation of density operator as a solution of the master equation. In Sec. 3 we show that a pure squeezed vacuum state (with squeezing parameter $\lambda)$ will evolves into a mixed state (output state), whose exact form is derived, which turns out to be a squeezed chaotic state. We investigate average photon number, photon statistics distributions for this state. The probability of finding $n$ photons in this mixed state is obtained which turns out to be a Legendre polynomial function relating to the squeezing parameter $\lambda$ and the decaying rate $\kappa$. In Sec. 4 we discuss the photon statistics distributions of the output state. In Sec. 5 and 6 we respectively discuss the Wigner function and tomogram of the output state. ## II Brief review of deducing the infinitive sum representation of $\rho\left(t\right)$ For solving the above master equation, in a recent review paper 12 we have introduced a convenient approach in which the two-mode entangled state 10 ; 11 $\|\eta\rangle=\exp(-\frac{1}{2}\|\eta\|^{2}+\eta a^{{\dagger}}-\eta^{\ast}\tilde{a}^{{\dagger}}+a^{{\dagger}}\tilde{a}^{{\dagger}})\|0\tilde{0}\rangle,$ (2) is employed, where $\tilde{a}^{{\dagger}}$ is a fictitious mode independent of the real mode $a^{\dagger},$ $[\tilde{a},a^{\dagger}]=0$. $\|\eta=0\rangle$ possesses the properties $\displaystyle a\|\eta$ $\displaystyle=0\rangle=\tilde{a}^{{\dagger}}\|\eta=0\rangle,$ $\displaystyle a^{{\dagger}}\|\eta$ $\displaystyle=0\rangle=\tilde{a}\|\eta=0\rangle,$ (3) $\displaystyle(a^{{\dagger}}a)^{n}\|\eta$ $\displaystyle=0\rangle=(\tilde{a}^{{\dagger}}\tilde{a})^{n}\|\eta=0\rangle.$ Acting the both sides of Eq.(1) on the state $\|\eta=0\rangle\equiv\left\|I\right\rangle$, and denoting $\left\|\rho\right\rangle=\rho\left\|I\right\rangle$, we have $\displaystyle\frac{d}{dt}\left\|\rho\right\rangle$ $\displaystyle=\kappa\left(2a\rho a^{\dagger}-a^{\dagger}a\rho-\rho a^{\dagger}a\right)\left\|I\right\rangle$ $\displaystyle=\kappa\left(2a\tilde{a}-a^{\dagger}a-\tilde{a}^{\dagger}\tilde{a}\right)\left\|\rho\right\rangle,$ (4) so its formal solution is $\left\|\rho\right\rangle=\exp\left[\kappa t\left(2a\tilde{a}-a^{\dagger}a-\tilde{a}^{\dagger}\tilde{a}\right)\right]\left\|\rho_{0}\right\rangle,$ (5) where $\left\|\rho_{0}\right\rangle\equiv\rho_{0}\left\|I\right\rangle,$ $\rho_{0}$ is the initial density operator. Noticing that the operators in Eq.(5) obey the following commutative relation, $\left[a\tilde{a},a^{\dagger}a\right]=\left[a\tilde{a},\tilde{a}^{\dagger}\tilde{a}\right]=\tilde{a}a$ (6) and $\left[\frac{a^{\dagger}a+\tilde{a}^{\dagger}\tilde{a}}{2},a\tilde{a}\right]=-\tilde{a}a,$ (7) as well as using the operator identity 13 $e^{\lambda\left(A+\sigma B\right)}=e^{\lambda A}e^{\sigma\left(1-e^{-\lambda\tau}\right)B/\tau},$ (8) (which is valid for $\left[A,B\right]=\tau B$), we have $e^{-2\kappa t\left(\frac{a^{\dagger}a+\tilde{a}^{\dagger}\tilde{a}}{2}-a\tilde{a}\right)}=e^{-\kappa t\left(a^{\dagger}a+\tilde{a}^{\dagger}\tilde{a}\right)}e^{T^{\prime}a\tilde{a}},$ (9) where $T^{\prime}=1-e^{-2\kappa t}.$ Then substituting Eq.(9) into Eq.(5) yields 12 $\displaystyle\left\|\rho\right\rangle$ $\displaystyle=e^{-\kappa t\left(a^{\dagger}a+\tilde{a}^{\dagger}\tilde{a}\right)}\sum_{n=0}^{\infty}\frac{T^{\prime n}}{n!}a^{n}\tilde{a}^{n}\left\|\rho_{0}\right\rangle$ $\displaystyle=e^{-\kappa ta^{\dagger}a}\sum_{n=0}^{\infty}\frac{T^{\prime n}}{n!}a^{n}\rho_{0}a^{{\dagger}n}e^{-\kappa t\tilde{a}^{\dagger}\tilde{a}}\left\|I\right\rangle$ $\displaystyle=\sum_{n=0}^{\infty}\frac{T^{\prime n}}{n!}e^{-\kappa ta^{\dagger}a}a^{n}\rho_{0}a^{{\dagger}n}e^{-\kappa ta^{\dagger}a}\left\|I\right\rangle,$ (10) which leads to the infinitive operator-sum representation of$\ \rho$, $\rho=\sum_{n=0}^{\infty}M_{n}\rho_{0}M_{n}^{\dagger},$ (11) where $M_{n}\equiv\sqrt{\frac{T^{\prime n}}{n!}}e^{-\kappa ta^{\dagger}a}a^{n}.$ (12) We can prove $\displaystyle\sum_{n}M_{n}^{\dagger}M_{n}$ $\displaystyle=\sum_{n}\frac{T^{\prime n}}{n!}a^{{\dagger}n}e^{-2\kappa ta^{\dagger}a}a^{n}$ $\displaystyle=\sum_{n}\frac{T^{\prime n}}{n!}e^{2n\kappa t}\colon a^{{\dagger}n}a^{n}\colon e^{-2\kappa ta^{\dagger}a}$ $\displaystyle=\left.:e^{T^{\prime}e^{2\kappa t}a^{\dagger}a}:\right.e^{-2\kappa ta^{\dagger}a}$ $\displaystyle=\left.:e^{\left(e^{2\kappa t}-1\right)a^{\dagger}a}:\right.e^{-2\kappa ta^{\dagger}a}=1,$ (13) where $\colon\colon$ stands for the normal ordering. Thus $M_{n}$ is a kind of Kraus operator, and $\rho$ in Eq.(11) is qualified to be a density operator, i.e., $Tr\left[\rho\left(t\right)\right]=Tr\left[\sum_{n=0}^{\infty}M_{n}\rho_{0}M_{n}^{\dagger}\right]=Tr\rho_{0}.$ (14) Therefore, for any given initial state $\rho_{0}$, the density operator $\rho\left(t\right)$ can be directly calculated from Eq.(11). The entangled state representation provides us with an elegant way of deriving the infinitive sum representation of density operator as a solution of the master equation. ## III Evolving of an initial single-mode squeezed vacuum state in ADC It is seen from Eq.(11) that for any given initial state $\rho_{0}$, the density operator $\rho\left(t\right)$ can be directly calculated. When $\rho_{0}$ is a single-mode squeezed vacuum state, $\rho_{0}=\text{sech}\lambda\exp\left(\frac{\tanh\lambda}{2}a^{{\dagger}2}\right)\left\|0\right\rangle\left\langle 0\right\|\exp\left(\frac{\tanh\lambda}{2}a^{2}\right),$ (15) we see $\displaystyle\rho\left(t\right)$ $\displaystyle=$ $\displaystyle\text{sech}\lambda\sum_{n=0}^{\infty}\frac{T^{\prime n}}{n!}e^{-\kappa ta^{\dagger}a}a^{n}\exp\left(\frac{\tanh\lambda}{2}a^{{\dagger}2}\right)\left\|0\right\rangle$ (16) $\displaystyle\times\left\langle 0\right\|\exp\left(\frac{\tanh\lambda}{2}a^{2}\right)a^{{\dagger}n}e^{-\kappa ta^{\dagger}a}.$ Using the Baker-Hausdorff lemma 14 , $e^{\lambda\hat{A}}\hat{B}e^{-\lambda\hat{A}}=\hat{B}+\lambda\left[\hat{A},\hat{B}\right]+\frac{\lambda^{2}}{2!}\left[\hat{A},\left[\hat{A},\hat{B}\right]\right]+\cdots.$ (17) we have $\displaystyle a^{n}\exp\left(\frac{\tanh\lambda}{2}a^{{\dagger}2}\right)\left\|0\right\rangle$ $\displaystyle=$ $\displaystyle e^{\frac{\tanh\lambda}{2}a^{{\dagger}2}}e^{-\frac{\tanh\lambda}{2}a^{{\dagger}2}}a^{n}e^{\frac{\tanh\lambda}{2}a^{{\dagger}2}}\left\|0\right\rangle$ (18) $\displaystyle=$ $\displaystyle e^{\frac{\tanh\lambda}{2}a^{{\dagger}2}}\left(a+a^{\dagger}\tanh\lambda\right)^{n}\left\|0\right\rangle.$ Further employing the operator identity 15 $\left(\mu a+\nu a^{\dagger}\right)^{m}=\left(-i\sqrt{\frac{\mu\nu}{2}}\right)^{m}\colon H_{m}\left(i\sqrt{\frac{\mu}{2\nu}}a+i\sqrt{\frac{\nu}{2\mu}}a^{\dagger}\right)\colon,$ (19) where $H_{m}(x)$ is the Hermite polynomial, we know $\displaystyle\left(a+a^{\dagger}\tanh\lambda\right)^{n}$ (20) $\displaystyle=$ $\displaystyle\left(-i\sqrt{\frac{\tanh\lambda}{2}}\right)^{n}\colon H_{n}\left(i\sqrt{\frac{1}{2\tanh\lambda}}a+i\sqrt{\frac{\tanh\lambda}{2}}a^{\dagger}\right)\colon.$ From Eq.(18), it follows that $\displaystyle a^{n}e^{\frac{\tanh\lambda}{2}a^{{\dagger}2}}\left\|0\right\rangle$ $\displaystyle=$ $\displaystyle\left(-i\sqrt{\frac{\tanh\lambda}{2}}\right)^{n}e^{\frac{\tanh\lambda}{2}a^{{\dagger}2}}$ (21) $\displaystyle\times H_{n}\left(i\sqrt{\frac{\tanh\lambda}{2}}a^{\dagger}\right)\left\|0\right\rangle.$ On the other hand, noting $e^{-\kappa ta^{\dagger}a}a^{\dagger}e^{\kappa ta^{\dagger}a}=a^{\dagger}e^{-\kappa t},e^{\kappa ta^{\dagger}a}ae^{-\kappa ta^{\dagger}a}=ae^{-\kappa t}$ and the normally ordered form of the vacuum projector $\left\|0\right\rangle\left\langle 0\right\|=\colon e^{-a^{\dagger}a}\colon,$ we have $\displaystyle\rho\left(t\right)$ $\displaystyle=\text{sech}\lambda\sum_{n=0}^{\infty}\frac{T^{\prime n}}{n!}e^{-\kappa ta^{\dagger}a}a^{n}e^{\frac{\tanh\lambda}{2}a^{{\dagger}2}}\left\|0\right\rangle$ $\displaystyle\times\left\langle 0\right\|e^{\frac{\tanh\lambda}{2}a^{2}}a^{{\dagger}n}e^{-\kappa ta^{\dagger}a}$ $\displaystyle=\text{sech}\lambda\sum_{n=0}^{\infty}\frac{\left(T^{\prime}\tanh\lambda\right)^{n}}{2^{n}n!}e^{\frac{e^{-2\kappa t}a^{{\dagger}2}\tanh\lambda}{2}}$ $\displaystyle\times H_{n}\left(i\sqrt{\frac{\tanh\lambda}{2}}a^{\dagger}e^{-\kappa t}\right)\left\|0\right\rangle\left\langle 0\right\|$ $\displaystyle\times H_{n}\left(-i\sqrt{\frac{\tanh\lambda}{2}}ae^{-\kappa t}\right)e^{\frac{e^{-2\kappa t}a^{2}\tanh\lambda}{2}}$ $\displaystyle=\text{sech}\lambda\sum_{n=0}^{\infty}\frac{\left(T^{\prime}\tanh\lambda\right)^{n}}{2^{n}n!}\colon e^{\frac{e^{-2\kappa t}\left(a^{2}+a^{{\dagger}2}\right)\tanh\lambda}{2}-a^{\dagger}a}$ $\displaystyle\times H_{n}\left(i\sqrt{\frac{\tanh\lambda}{2}}a^{\dagger}e^{-\kappa t}\right)H_{n}\left(-i\sqrt{\frac{\tanh\lambda}{2}}ae^{-\kappa t}\right)\colon$ (22) then using the following identity 16 $\displaystyle\sum_{n=0}^{\infty}\frac{t^{n}}{2^{n}n!}H_{n}\left(x\right)H_{n}\left(y\right)$ (23) $\displaystyle=$ $\displaystyle\left(1-t^{2}\right)^{-1/2}\exp\left[\frac{t^{2}\left(x^{2}+y^{2}\right)-2txy}{t^{2}-1}\right],$ and $e^{\lambda a^{{\dagger}}a}=\colon e^{\left(e^{\lambda}-1\right)a^{{\dagger}}a}\colon,$ we finally obtain the expression of the output state $\rho\left(t\right)=We^{\frac{\text{\ss}}{2}a^{{\dagger}2}}e^{a^{\dagger}a\ln\left(\text{\ss}T^{\prime}\tanh\lambda\right)}e^{\frac{\text{\ss}}{2}a^{2}},$ (24) with $T^{\prime}=1-e^{-2\kappa t}$ and $W\equiv\frac{\text{sech}\lambda}{\sqrt{1-T^{\prime 2}\tanh^{2}\lambda}},\text{\ss}\equiv\frac{e^{-2\kappa t}\tanh\lambda}{1-T^{\prime 2}\tanh^{2}\lambda}.$ (25) By comparing Eq.(15) with (23) one can see that after going through the channel the initial squeezing parameter $\tanh\lambda$ in Eq.( 15) becomes to ß$\equiv\frac{e^{-2\kappa t}\tanh\lambda}{1-T^{\prime 2}\tanh^{2}\lambda},$ and $\left\|0\right\rangle\left\langle 0\right\|\rightarrow\frac{1}{\sqrt{1-T^{\prime 2}\tanh^{2}\lambda}}e^{a^{\dagger}a\ln\left(\text{\ss}T^{\prime}\tanh\lambda\right)},$ a chaotic state (mixed state), due to $T^{\prime}>0,$ we can prove $\frac{e^{-2\kappa t}}{1-T^{\prime 2}\tanh^{2}\lambda}<1,$ which means a squeezing-decreasing process. When $\kappa t=0$, then $T^{\prime}=0$ and ß $=\tanh\lambda$, Eq.(22) becomes the initial squeezed vacuum state as expected. It is important to check: if Tr$\rho(t)=1$. Using Eq.(22) and the completeness of coherent state $\int\frac{d^{2}z}{\pi}\left\|z\right\rangle\left\langle z\right\|=1$ as well as the following formula 17 $\int\frac{d^{2}z}{\pi}e^{\zeta\left\|z\right\|^{2}+\xi z+\eta z^{\ast}+fz^{2}+gz^{\ast 2}}=\frac{1}{\sqrt{\zeta^{2}-4fg}}e^{\frac{-\zeta\xi\eta+f\eta^{2}+g\xi^{2}}{\zeta^{2}-4fg}},$ (26) whose convergent condition is Re$\left(\zeta\pm f\pm g\right)<0$ and$\ \mathtt{Re}\left(\frac{\zeta^{2}-4fg}{\zeta\pm f\pm g}\right)<0$, we really see $\displaystyle\text{Tr}\rho\left(t\right)$ $\displaystyle=$ $\displaystyle W\int\frac{d^{2}z}{\pi}\left\langle z\right\|e^{\frac{\text{\ss}}{2}a^{{\dagger}2}}e^{a^{\dagger}a\ln\left(\text{\ss}T^{\prime}\tanh\lambda\right)}e^{\frac{\text{\ss}}{2}a^{2}}\left\|z\right\rangle$ (27) $\displaystyle=$ $\displaystyle\frac{W}{\sqrt{\left(\text{\ss}T^{\prime}\tanh\lambda-1\right)^{2}-\text{\ss}^{2}}}=1.$ so $\rho\left(t\right)$ is qualified to be a mixed state, thus we see an initial pure squeezed vacuum state evolves into a squeezed chaotic state with decreasing-squeezing after passing through an amplitude dissipative channel. ## IV Average photon number Using the completeness relation of coherent state and the normally ordering form of $\rho\left(t\right)$ in Eq. (22), and using $e^{\frac{\text{\ss}}{2}a^{2}}a^{\dagger}e^{-\frac{\text{\ss}}{2}a^{2}}=a^{\dagger}+$ß$a$, as well as $e^{a^{\dagger}a\ln\left(\text{\ss}T^{\prime}\tanh\lambda\right)}a^{\dagger}e^{-a^{\dagger}a\ln\left(\text{\ss}T^{\prime}\tanh\lambda\right)}$=$a^{\dagger}$ß$T^{\prime}\tanh\lambda,$ we have $\displaystyle\mathtt{Tr}\left(\rho\left(t\right)a^{\dagger}a\right)$ $\displaystyle=W\int\frac{d^{2}z}{\pi}\left\langle z\right\|e^{\frac{\text{\ss}}{2}a^{{\dagger}2}}e^{a^{\dagger}a\ln\left(\text{\ss}T^{\prime}\tanh\lambda\right)}e^{\frac{\text{\ss}}{2}a^{2}}a^{\dagger}a\left\|z\right\rangle$ $\displaystyle=W\int\frac{d^{2}z}{\pi}\left\langle z\right\|e^{\frac{\text{\ss}}{2}a^{{\dagger}2}}e^{a^{\dagger}a\ln\left(\text{\ss}T^{\prime}\tanh\lambda\right)}e^{\frac{\text{\ss}}{2}a^{2}}za^{\dagger}\left\|z\right\rangle$ $\displaystyle=W\int\frac{d^{2}z}{\pi}z\left\langle z\right\|e^{\frac{\text{\ss}}{2}a^{{\dagger}2}}e^{a^{\dagger}a\ln\left(\text{\ss}T^{\prime}\tanh\lambda\right)}\left(a^{\dagger}+\text{\ss}a\right)e^{\frac{\text{\ss}}{2}a^{2}}\left\|z\right\rangle$ $\displaystyle=W\text{\ss}\int\frac{d^{2}z}{\pi}ze^{\frac{\text{\ss}}{2}\left(z^{\ast 2}+z^{2}\right)}\left\langle z\right\|\left(a^{\dagger}T^{\prime}\tanh\lambda+z\right)e^{a^{\dagger}a\ln\left(\text{\ss}T^{\prime}\tanh\lambda\right)}\left\|z\right\rangle$ $\displaystyle=W\text{\ss}\int\frac{d^{2}z}{\pi}\left(\|z\|^{2}T^{\prime}\tanh\lambda+z^{2}\right)$ $\displaystyle\times\exp\left[\left(\text{\ss}T^{\prime}\tanh\lambda-1\right)\|z\|^{2}+\frac{\text{\ss}}{2}\left(z^{\ast 2}+z^{2}\right)\right].$ (28) In order to perform the integration, we reform Eq.(28) as $\displaystyle\mathtt{Tr}\left(\rho\left(t\right)a^{\dagger}a\right)$ $\displaystyle=$ $\displaystyle W\text{\ss}\left\\{T^{\prime}\tanh\lambda\frac{\partial}{\partial f}+\frac{2}{\text{ \ss}}\frac{\partial}{\partial s}\right\\}$ (29) $\displaystyle\times\int\frac{d^{2}z}{\pi}\exp\left[\left(\text{\ss}T^{\prime}\tanh\lambda-1+f\right)\|z\|^{2}\right.$ $\displaystyle+\left.\frac{\text{\ss}}{2}\left(z^{\ast 2}+\left(1+s\right)z^{2}\right)\right]_{f=s=0}$ $\displaystyle=$ $\displaystyle\frac{1-\text{\ss}T^{\prime}\tanh\lambda}{\left(\text{\ss}T^{\prime}\tanh\lambda-1\right)^{2}-\text{\ss}^{2}}-1$ in the last step, we have used Eq.(27). Using Eq.(29), we present the time evolution of the average photon number in Fig. 1, from which we find that the average photon number of the single-mode squeezed vacuum state in the amplitude damping channel reduces gradually to zero when decay time goes. Figure 1: (Color online) The average $\bar{n}\left(\kappa t\right)$ as the function of $\kappa t$ for different values of squeezing parameter $\lambda$ (from bottom to top $\lambda=0,0.1,0.3,0.5,1$.) ## V Photon statistics distribution Next, we shall derive the photon statistics distributions of $\rho\left(t\right)$. The photon number is given by $p\left(n,t\right)=\left\langle n\right\|\rho\left(t\right)\left\|n\right\rangle$. Noticing $a^{{\dagger}m}\left\|n\right\rangle=\sqrt{(m+n)!/n!}\left\|m+n\right\rangle$ and using the un-normalized coherent state $\left\|\alpha\right\rangle=\exp[\alpha a^{{\dagger}}]\left\|0\right\rangle$, 18 ; 19 leading to $\left\|n\right\rangle=\frac{1}{\sqrt{n!}}\frac{\mathtt{d}^{n}}{\mathtt{d}\alpha^{n}}\left\|\alpha\right\rangle\left\|{}_{\alpha=0}\right.,$ $\left(\left\langle\beta\right.\left\|\alpha\right\rangle=e^{\alpha\beta^{\ast}}\right)$, as well as the normal ordering form of $\rho\left(t\right)$ in Eq. (22), the probability of finding $n$ photons in the field is given by $\displaystyle p\left(n,t\right)$ (30) $\displaystyle=$ $\displaystyle\left\langle n\right\|\rho\left(t\right)\left\|n\right\rangle$ $\displaystyle=$ $\displaystyle\frac{W}{n!}\frac{\mathtt{d}^{n}}{\mathtt{d}\beta^{\ast n}}\frac{\mathtt{\ d}^{n}}{\mathtt{d}\alpha^{n}}\left.\left\langle\beta\right\|e^{\frac{\text{\ss}}{2}\beta^{\ast 2}}e^{a^{\dagger}a\ln\left(\text{\ss}T^{\prime}\tanh\lambda\right)}e^{\frac{\text{\ss}}{2}\alpha^{2}}\left\|\alpha\right\rangle\right\|_{\alpha,\beta^{\ast}=0}$ $\displaystyle=$ $\displaystyle\frac{W}{n!}\frac{\mathtt{d}^{n}}{\mathtt{d}\beta^{\ast n}}\frac{\mathtt{\ d}^{n}}{\mathtt{d}\alpha^{n}}\left.\exp\left[\beta^{\ast}\alpha\text{ \ss}T^{\prime}\tanh\lambda+\frac{\text{\ss}}{2}\beta^{\ast 2}+\frac{\text{\ss}}{2}\alpha^{2}\right]\right\|_{\alpha,\beta^{\ast}=0}.$ Note that $\left[e^{\frac{\text{\ss}}{2}a^{\dagger 2}}e^{a^{\dagger}a\ln\left(\text{\ss}T^{\prime}\tanh\lambda\right)}e^{\frac{\text{\ss}}{2}\alpha^{2}}\right]^{\dagger}=e^{\frac{\text{\ss}}{2}a^{\dagger 2}}e^{a^{\dagger}a\ln\left(\text{\ss}T^{\prime}\tanh\lambda\right)}e^{\frac{\text{\ss}}{2}\alpha^{2}}$ so $\left\langle n\right\|\rho\left(t\right)\left\|n\right\rangle^{\ast}=\left\langle n\right\|\rho\left(t\right)^{\dagger}\left\|n\right\rangle=\left\langle n\right\|\rho\left(t\right)\left\|n\right\rangle$ $\displaystyle\frac{\partial^{n+n}}{\partial t^{n}\partial t^{\prime n}}\exp\left[2xtt^{\prime}-t^{2}-t^{\prime 2}\right]_{t=t^{\prime}=0}$ (31) $\displaystyle=$ $\displaystyle 2^{n}n!\sum_{m=0}^{[n/2]}\frac{n!}{2^{2m}\left(m!\right)^{2}(n-2m)!}x^{n-2m},$ we derive the compact form for $\mathfrak{p}\left(n,t\right)$, i.e., $\displaystyle p\left(n,t\right)$ (32) $\displaystyle=$ $\displaystyle\frac{W}{n!}\left(-\frac{\text{\ss}}{2}\right)^{n}\frac{\mathtt{d}^{n}}{\mathtt{d}\beta^{\ast n}}\frac{\mathtt{d}^{n}}{\mathtt{d}\alpha^{n}}\left.e^{-2T^{\prime}\tanh\lambda\beta^{\ast}\alpha-\beta^{\ast 2}-\alpha^{2}}\right\|_{\alpha,\beta^{\ast}=0}$ $\displaystyle=$ $\displaystyle W\left(\text{\ss}T^{\prime}\tanh\lambda\right)^{n}\sum_{m=0}^{[n/2]}\frac{n!\left(T^{\prime}\tanh\lambda\right)^{-2m}}{2^{2m}\left(m!\right)^{2}(n-2m)!}.$ Using the newly expression of Legendre polynomials found in Ref. 20 $x^{n}\sum_{m=0}^{[n/2]}\frac{n!}{2^{2m}\left(m!\right)^{2}(n-2m)!}\left(1-\frac{1}{x^{2}}\right)^{m}=P_{n}\left(x\right),$ (33) we can formally recast Eq.(32) into the following compact form, i.e., $p\left(n,t\right)=W\left(e^{-\kappa t}\sqrt{-\text{\ss}\tanh\lambda}\right)^{n}P_{n}\left(e^{\kappa t}T^{\prime}\sqrt{-\text{\ss}\tanh\lambda}\right)$ note that since $\sqrt{-\text{\ss}\tanh\lambda}$ is pure imaginary, while $p\left(n,t\right)$ is real, so we must still use the power-series expansion on the right-hand side of Eq.(32) to depict figures of the variation of $p\left(n,t\right)$. In particular, when $t=0$, Eq.(32 ) reduces to $\displaystyle p\left(n,0\right)$ $\displaystyle=$ $\displaystyle\text{sech}\lambda\left(\tanh\lambda\right)^{n}\lim_{T^{\prime}\rightarrow 0}\sum_{m=0}^{[n/2]}\frac{n!\left(T^{\prime}\tanh\lambda\right)^{n-2m}}{2^{2m}\left(m!\right)^{2}(n-2m)!}$ (36) $\displaystyle=$ $\displaystyle\left\\{\begin{array}[]{cc}\frac{\left(2k\right)!}{2^{2k}k!k!}\text{sech}\lambda\tanh^{2k}\lambda,&n=2k\\\ 0&n=2k+1\end{array}\right.,$ which just correspond to the number distributions of the squeezed vacuum state 21 ; 22 . From Eq.(36) it is not difficult to see that the photocount distribution decreases as the squeezing parameter $\lambda$ increases. While for $\kappa t\rightarrow\infty,$ we see that $p\left(n,\infty\right)=0.$ This indicates that there is no photon when a system interacting with a amplitude dissipative channel for enough long time, as expected. In Fig. 2, the photon number distribution is shown for different $\kappa t$. Figure 2: (Color online) Photon number distribution of the squeezed vacuum state in amplitude damping channel for $\lambda=1$, and different $\kappa t$: ($a)$ $\kappa t=0$, ($b$) $\kappa t=0.5,$ ($c$) $\kappa t=1$ and ($d$) $\kappa t=2$. ## VI Wigner functions In this section, we shall use the normally ordering for of density operators to calculate the analytical expression of Wigner function. For a single-mode system, the WF is given by 23 $W\left(\alpha,\alpha^{\ast},t\right)=e^{2\left\|\alpha\right\|^{2}}\int\frac{d^{2}\beta}{\pi^{2}}\left\langle-\beta\right\|\rho\left(t\right)\left\|\beta\right\rangle e^{-2\left(\beta\alpha^{\ast}-\beta^{\ast}\alpha\right)},$ (37) where $\left\|\beta\right\rangle$ is the coherent state 18 ; 19 . From Eq.(22) it is easy to see that once the normal ordered form of $\rho\left(t\right)$ is known, we can conveniently obtain the Wigner function of $\rho\left(t\right)$. On substituting Eq.(24) into Eq.(37) we obtain the WF of the single-mode squeezed state in the ADC, $\displaystyle W\left(\alpha,\alpha^{\ast},t\right)$ (38) $\displaystyle=$ $\displaystyle We^{2\left\|\alpha\right\|^{2}}\int\frac{d^{2}\beta}{\pi^{2}}\exp\left[-\left(1+\text{\ss}T^{\prime}\tanh\lambda\right)\left\|\beta\right\|^{2}\right.$ $\displaystyle\left.-2\left(\beta\alpha^{\ast}-\beta^{\ast}\alpha\right)+\frac{\text{\ss}}{2}\beta^{\ast 2}+\frac{\text{\ss}}{2}\beta^{2}\right]$ $\displaystyle=$ $\displaystyle\frac{W}{\pi\sqrt{\left(1+\text{\ss}T^{\prime}\tanh\lambda\right)^{2}-\text{\ss}^{2}}}\exp\left[2\left\|\alpha\right\|^{2}\right]$ $\displaystyle\times\exp\left[2\frac{-2\left(1+\text{\ss}T^{\prime}\tanh\lambda\right)\left\|\alpha\right\|^{2}+\text{\ss}\left(\alpha^{\ast 2}+\alpha^{2}\right)}{\left(1+\text{\ss}T^{\prime}\tanh\lambda\right)^{2}-\text{\ss}^{2}}\right]$ In particular, when $t=0$ and $t\rightarrow\infty$, Eq.(38) reduces to $W\left(\alpha,\alpha^{\ast},0\right)=\frac{1}{\pi}\exp[-2\left\|\alpha\right\|^{2}\cosh 2\lambda+\left(\alpha^{\ast 2}+\alpha^{2}\right)\sinh 2\lambda]$, and $W\left(\alpha,\alpha^{\ast},\infty\right)=\frac{1}{\pi}\exp\left[-2\left\|\alpha\right\|^{2}\right]$, which are just the WF of the single-mode squeezed vacuum state and the vacuum state, respectively. In Fig. 3, the WF of the single-mode squeezed vacuum state in the amplitude damping channel is shown for different decay time $\kappa t$. Figure 3: (Color online) Wigner function of the squeezed vacuum state in amplitude damping channel for $\lambda=1.0$, different $\kappa t$: ($a)$ $\kappa t=0.0$, ($b$) $\kappa t=0.5$, ($c$) $\kappa t=1$, and ($d$) $\kappa t=2$. ## VII Tomogram As we know, once the probability distributions $P_{\theta}\left(\hat{x}_{\theta}\right)$ of the quadrature amplitude are obtained, one can use the inverse Radon transformation familiar in tomographic imaging to obtain the WF and density matrix 24 . Thus the Radon transform of the WF is corresponding to the probability distributions $P_{\theta}\left(\hat{x}_{\theta}\right)$. In this section we derive the tomogram of $\rho\left(t\right)$. For a single-mode system, the Radon transform of WF, denoted as $\mathcal{R}$ is defined by 25 $\displaystyle\mathcal{R}\left(q\right)_{f,g}$ $\displaystyle=$ $\displaystyle\int\delta\left(q-fq^{\prime}-gp^{\prime}\right)Tr\left[\Delta\left(\beta\right)\rho\left(t\right)\right]dq^{\prime}dp^{\prime}$ (39) $\displaystyle=$ $\displaystyle Tr\left[\left\|q\right\rangle_{f,g\text{ }f,g}\left\langle q\right\|\rho\left(t\right)\right]=_{f,g}\left\langle q\right\|\rho\left(t\right)\left\|q\right\rangle_{f,g}$ where the operator $\left\|q\right\rangle_{f,g\text{ }f,g}\left\langle q\right\|$ is just the Radon transform of single-mode Wigner operator $\Delta\left(\beta\right)$, and $\left\|q\right\rangle_{f,g}=A\exp\left[\frac{\sqrt{2}qa^{{\dagger}}}{B}-\frac{B^{\ast}}{2B}a^{{\dagger}2}\right]\left\|0\right\rangle,$ (40) as well as $B=f-ig,$ $A=\left[\pi\left(f^{2}+g^{2}\right)\right]^{-1/4}\exp[-q^{2}/2\left(f^{2}+g^{2}\right)]$. Thus the tomogram of a quantum state $\rho\left(t\right)$ is just the quantum average of $\rho\left(t\right)$ in $\left\|q\right\rangle_{f,g}$ representation (a kind of intermediate coordinate-momentum representation) 26 . Substituting Eqs.(24) and (40) into Eq.(39), and using the completeness relation of coherent state, we see that the Radom transform of WF of $\rho\left(t\right)$ is given by $\displaystyle\mathcal{R}\left(q\right)_{f,g}$ (41) $\displaystyle=$ $\displaystyle W_{f,g}\left\langle q\right\|e^{\frac{\text{\ss}}{2}a^{{\dagger}2}}e^{a^{\dagger}a\ln\left(\text{\ss}T^{\prime}\tanh\lambda\right)}e^{\frac{\text{\ss}}{2}a^{2}}\left\|q\right\rangle_{f,g}$ $\displaystyle=$ $\displaystyle\frac{WA^{2}}{\sqrt{E}}\exp\left\\{\frac{q^{2}\text{\ss}}{E\left\|B\right\|^{4}}\left(B^{2}+B^{\ast}{}^{2}\right)\right.$ $\displaystyle+\left.\frac{2q^{2}\text{\ss}}{E\left\|B\right\|^{2}}\left(T\tanh\lambda+\text{\ss}-\text{\ss}T^{2}\tanh^{2}\lambda\right)\right\\},$ where we have used the formula (26) and $\left\langle\alpha\right\|\left.\gamma\right\rangle=\exp[-\left\|\alpha\right\|^{2}/2-\left\|\gamma\right\|^{2}/2+\alpha^{\ast}\gamma]$, as well as $\displaystyle E$ $\displaystyle=$ $\displaystyle\left(1+\text{\ss}\frac{B}{B^{\ast}}\right)\left(1+\frac{B^{\ast}}{B}\text{\ss}-B^{\ast}\frac{\left(\text{\ss}T^{\prime}\tanh\lambda\right)^{2}}{B^{\ast}+\text{\ss}B}\right)$ (42) $\displaystyle=$ $\displaystyle\left\|1+\frac{\text{\ss}B}{B^{\ast}}\right\|^{2}-\left(\text{ \ss}T^{\prime}\tanh\lambda\right)^{2}.$ In particular, when $t=0,$ ($T=0$), then Eq.(41) reduces to ($\frac{B}{B^{\ast}}=e^{2i\phi}$) $\displaystyle\mathcal{R}\left(q\right)_{f,g}$ $\displaystyle=$ $\displaystyle\frac{A^{2}\text{sech}\lambda}{\left\|1+e^{2i\phi}\tanh\lambda\right\|}$ (43) $\displaystyle\times\exp\left\\{\frac{q^{2}\left(B^{2}+B^{\ast}{}^{2}+2\left\|B\right\|^{2}\tanh\lambda\right)\tanh\lambda}{\left\|1+e^{2i\phi}\left\|B\right\|^{4}\tanh\lambda\right\|^{2}}\right\\},$ which is a tomogram of single-mode squeezed vacuum state; while for $\kappa t\rightarrow\infty,$($T=1$), then $\mathcal{R}\left(q\right)_{f,g}=A^{2},$ which is a Gaussian distribution corresponding to the vacuum state. In summary, using the way of deriving infinitive sum representation of density operator by virtue of the entangled state representation describing, we conclude that in the amplitude dissipative channel the initial density operator of a single-mode squeezed vacuum state evolves into a squeezed chaotic state with decreasing-squeezing. We investigate average photon number, photon statistics distributions, Wigner functions and tomogram for the output state. ## Acknowledgments This work is supported by the National Natural Science Foundation of China (Grant No.11175113 and 11047133), Shandong Provincial Natural Science Foundation in China (Gant No.ZR2010AQ024), and a grant from the Key Programs Foundation of Ministry of Education of China (Grant No. 210115),as well as Jiangxi Provincial Natural Science Foundation in China (No. 2010GQW0027). ## References * (1) Caves C M 1981 Phys. Rev. D 23 1693 * (2) Loudon R 1981 Phys. Rev. Lett. 47 815 * (3) Agarwal G S 2011 New J. Phys. 13 073008 * (4) Gardner C W and Zoller P 2000 Quantum Noise (Berlin: Spinger) * (5) Louisell W H 1973 Quantum Statistical Properties of Radiation (New York: Wiley) * (6) Biswas A and Agarwal G S 2007 Phys. Rev. A 75 032104 * (7) Hu L Y and Fan H Y 2008 J. Opt. Soc. Am. B 25 1955 * (8) Hu L Y, Xu X X, Wang Z S and Xu X F 2010 Phys. Rev. A 82 043842 * (9) Fan H Y and Hu L Y 2008 Opt. Commun. 281 5571 * (10) Hu L Y and Fan H Y, Opt. Commun. 282, 4379 (2009) * (11) Fan H Y and Fan Y 1998 Phys. Lett. A 246 242 * (12) Fan H Y and Fan Y 2001 Phys. Lett. A 282 269 * (13) Fan H Y and Hu L Y 2008 Mod. Phys. Lett. B 22 2435 * (14) Fan H Y 1997 Representation and Transformation Theory in Quantum Mechanics (Shanghai: Shanghai Scientific and Technical) (in Chinese) * (15) Klauder J R and Skargerstam B S 1985 Coherent States (Singapore: World Scientific) * (16) Xu X X, Yuan H C, Hu L Y and Fan H Y 2011 J. Phys. A: Math. Theor. 44 445306 * (17) Rainville E D 1960 Special Functions (New York: MacMillan Company) * (18) Puri R R 2001 Mathematical Methods of Quantum Optics (Berlin/Heidelberg/New York: Springer-Verlag) * (19) Glauber R J 1963 Phys. Rev. 130 2529 * (20) Glauber R J 1963 Phys. Rev. 131 2766 * (21) Fan H Y, Hu L Y and Xu X X 2009 Mod. Phys. Lett. A 24 1597 * (22) Kim M S, de Oliveira F A M and Knight P L 1989 Phys. Rev. A 40 2494 * (23) Paulina Marian 1992 Phys. Rev. A 45 2044 * (24) Fan H Y andZaidi H R 1987 Phys. Lett. A 124 303 * (25) Vogel K and Risken H 1989 Phys. Rev. A 40 2847 * (26) Fan H Y and Niu J B 2010 Opt. Commun. 283 3296 * (27) Fan H Y and Hu L Y 2009 Opt. Commun. 282 3734	arxiv-papers	2011-12-06T05:04:48	2024-09-04T02:49:25.012889	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Hong-Yi Fan, Shuai Wang and Li-Yun Hu", "submitter": "Liyun Hu", "url": "https://arxiv.org/abs/1112.1159" }
1112.1248	$\rm{D^{0}}$ cross section in pp collisions at $\sqrt{s}=7$ TeV, measured with the ALICE experiment Xianbao Yuan, for the ALICE Collaboration Institute of Particle Physics, Central China Normal University University and INFN, Padova, Italy Contact:yuanxb@iopp.ccnu.edu.cn or xianbao.yuan@pd.infn.it The measurement of the cross-section for charm production in pp collisions at the LHC is not only a fundamental reference to investigate medium properties in heavy-ion collisions, but also a key test of pQCD predictions in a new energy domain. The ALICE [1] experiment has measured the D meson production in pp collisions at $\sqrt{s}=7$ TeV. We present the analysis procedure for $\rm D^{0}\to\rm K^{-}\rm\pi^{+}$ and for the calculation of efficiency and acceptance corrections. Finally, we show the preliminary results on $\rm D^{0}$ cross section in pp collisions at $\sqrt{s}=7$ TeV, measured in the region $2<p_{\rm t}<12$ GeV/$c$ at central rapidity $\|y\|<0.5$. These results are compared to perturbative QCD predictions. The analysis is based on an invariant mass analysis of opposite-charge pairs of reconstructed tracks that can represent a $\rm D^{0}$ with a displaced vertex (the mean proper decay length of the $\rm D^{0}$ is $c\tau\approx 123~{}\mu$m). The selection is based on topological cuts and particle identification via specific energy deposit and time-of-flight measurement. The cross section is calculated from the raw signal yield extracted with the invariant mass analysis, $N^{\rm D^{0}~{}raw}(p_{\rm t})$, using the following formula: $\left.\frac{{\rm d}\sigma^{\rm D^{0}}}{{\rm d}p_{\rm t}}\right\|_{\|y\|<0.5}=\frac{1}{2}\frac{1}{2\,y_{\rm acc}\Delta p_{\rm t}}\frac{\left.f_{prompt}\cdot N^{\rm D^{0}~{}raw}(p_{\rm t})\right\|_{\|y\|<y_{\rm acc}}}{\epsilon_{prompt}\cdot{\rm BR}\cdot\mathit{L}_{int}}\cdot$ (1) Here, $\epsilon_{prompt}$ means the efficiency of prompt mesons, which accounts for selection cuts, for track and primary vertex reconstruction efficiency, and for detector acceptance. The $f_{prompt}$ is the prompt fraction of raw yield. Figure 1 (Left) shows the invariant mass distribution for $p_{\rm t}>2$ GeV/$c$ after applying the cuts, which corresponds to $1.1\times 10^{8}$ minimum bias events collected by ALICE in 2010 at $\sqrt{s}=7$ TeV. Figure 1 (Right) shows the efficiencies for $\rm D^{0}\to\rm K^{-}\rm\pi^{+}$ with all the decay particles in the acceptance $\left\|\eta\right\|<0.9$. The efficiencies increase and flatten at about 0.1 at $p_{\rm t}>2$ GeV/$c$. The efficiency without particle identification selection, shown for comparison, is the same as with particle identification for $p_{\rm t}>2$ GeV/$c$, indicating that this selection is essentially fully efficient for the signal. The efficiencies for $\rm D^{0}$ meson from B meson decay, also shown for comparison, are larger by a factor about 2, because this feed-down component is more displaced from the primary vertex, due to the longer B life time. The $10-15\%$ feed-down from B decays is subtracted based on pQCD prediction [2]. Several sources of systematic uncertainties were considered, namely those affecting the signal extraction from the invariant mass spectra and all the correction factors applied to obtain the $p_{\rm t}$-differential cross sections. A summary of the estimated relative systematic errors is shown in Fig 2 (Left). The $p_{\rm t}$-differential cross section for prompt $\rm D^{0}$, obtained from the yields extracted by fitting the invariant mass spectra and corrected for efficiency and B feed-down, is shown in Fig 2 (Right). The error bars represent the statistical errors, while the systematic errors are plotted as rectangle areas around the data points. The measured $\rm D^{0}$ meson production cross section is compared to two theoretical predictions, namely FONLL [2] and GM-VFNS [3]. Our measurement of $\rm D^{0}$ at $\sqrt{s}=7$ TeV is reproduced by both models within their theoretical uncertainties. Figure 1: Left: $p_{\rm t}>$2 GeV/$c$ invariant mass distribution. Right: efficiencies for $\rm D^{0}$ as a function of $p_{\rm t}$. ## References * [1] B. Abelev, A. Abrahantes Quintana, $\mathit{et~{}al}$., [ALICE Coll.], __Arxiv: hep-ex/1111.1553v1. * [2] M. Cacciari $\mathit{et~{}al}$., __JHEP 0407 (2004) 033; Private communication. * [3] B.A. Kniehl $\mathit{et~{}al}$., __Phys. Rev. D77 (2008) 014011; Private communication. Figure 2: Left: systematic errors summary plot. Right: $p_{\rm t}$-differential cross section for prompt $\rm D^{0}$ in pp collisions at $\sqrt{s}=7$ TeV compared with FONLL [2] and GM-VFNS [3] theoretical predictions.	arxiv-papers	2011-12-06T12:17:04	2024-09-04T02:49:25.021371	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Xianbao Yuan", "submitter": "Xianbao Yuan", "url": "https://arxiv.org/abs/1112.1248" }
1112.1387	# Systematic enumeration of configuration classes for entropic sampling of Ising models BRUNO JEFERSON LOURENÇO 111brunojl@fisica.ufmg.br and RONALD DICKMAN 222dickman@fisica.ufmg.br _Departamento de Física, Instituto de Ciências Exatas, and National Institute of Science and Technology for Complex Systems, Universidade Federal de Minas Gerais C.P. 702, 30123-970, Belo Horizonte, MG, Brazil _ ## Abstract We describe a systematic method for complete enumeration of configuration classes (CCs) of the spin-1/2 Ising model in the energy-magnetization plane. This technique is applied to the antiferromagnetic Ising model in an external magnetic field on the square lattice, which is simulated using the tomographic entropic sampling algorithm. We estimate the number of configurations, $\Omega(E,m,L)$, and related microcanonical averages, for all allowed energies $E$ and magnetizations $m$ for $L=10$ to 30, with $\Delta L=2$. With prior knowledge of the CCs, we can be sure that all allowed classes are sampled during the simulation. Complete enumeration of CCs also enables us to use the final estimate of $\Omega(E,m,L)$ to obtain good initial estimates, $\Omega_{0}(E,m,L^{\prime})$, for successive system sizes ($L^{\prime}>L$) through a two-dimensional interpolation. Using these results we calculate canonical averages of the thermodynamic quantities of interest as continuous functions of temperature $T$ and external field $h$. In addition, we determine the critical line in the $h$-$T$ plane using finite-size scaling analysis, and compare these results with several approximate theoretical expressions. Keywords: Ising model; antiferromagnet; configuration classes; Monte Carlo simulation; phase diagram. ## 1 Introduction The most important task of entropic sampling algorithms [1]-[6] is to visit the full configuration space (CS) to obtain good estimates of the number of configurations, $\Omega$, as functions of the energy E and other quantities of interest. In studies of the Ising model in an external field, for example, we require $\Omega(E,m)$ with $m$ the magnetization; each allowed $(E,m)$ pair defines a class of configurations (CC). Only if we know beforehand the possible values of $(E,m)$ for a given system size, can we be sure that all CCs are sampled during the simulation. Figure 1 shows the CCs in the $n-m$ plane for the spin-1/2 Ising model with nearest-neighbor (NN) interactions, on a square lattice of $L\times L$ sites with periodic boundaries. [We use $n$ to denote the number of NN pairs of spins with the same orientation; the interaction energy of the antiferromagnetic (AF) Ising model is $E=-2(L^{2}-n)$.] Although the CCs tend to fill in a triangular region, some gaps are evident near the lower apex and along the upper edge. Knowing just which $(n,m)$ values are allowed for a given lattice size is important if we are to implement entropic sampling with confidence. In this paper we present a method for systematically enumerating all CCs of Ising models in the $(n,m)$ plane. Figure 1: Allowed configuration classes for system size $L=10$; $\eta\equiv n/L^{2}$ and $\nu\equiv m/L^{2}$. We study the spin-1/2 antiferromagnetic Ising model in an external magnetic field, whose energy is given by $\mathcal{H}=-J\sum_{<i,j>}\sigma_{i}\sigma_{j}-h\sum_{i=1}^{N}\sigma_{i}=-E-hm,$ (1) where $\sigma_{i}=\pm 1$, $<i,j>$ indicates a sum over NN pairs of spins, $h$ is the external field, and $N$ is the number of spins; the model is defined on a square lattice of $L\times L$ sites, with periodic boundary conditions. Unlike the ferromagnetic Ising model ($J>0$), which exhibits a unique critical point in the $h-T$ plane and has an exact solution [7], the AF model ($J<0$) possesses a critical line, which is not completely understood. Various approximate methods have been applied to determine the critical line of the AF Ising model on the square lattice [8]-[18]; these results, however, do not agree altogether. Binder and Landau [19] estimated the critical line via Monte Carlo simulation, obtaining very good agreement with the approximate closed-form expression of Müller-Hartmann and Zittartz [8], raising the possibility that the latter expression was in fact exact. (It was later shown that this is not so [12].) Hwang _et al._ [20] studied the AF Ising model on the square and triangular lattices using a microcanonical transfer matrix method and the Wang-Landau algorithm [3]. They performed an exact enumeration of the number of configurations, $\Omega(E,m)$, and found CSs in the $E-m$ plane similar to that shown in Fig. 1. Using the tomographic entropic sampling (TES) algorithm [6] we estimate $\Omega(n,m,L)$, and associated microcanonical averages, for lattice sizes $L=10$ to 30, with an increment $\Delta L=2$. We then calculate the canonical averages of the thermodynamic quantities of interest. Using these data we map out the critical line in the $h-T$ plane, and compare our results with several theoretical expressions. Prior determination of the set of allowed CCs is an important tool to verify the quality of the sampling: we want to be sure that all CCs are visited during the simulation. Since this algorithm uses an initial guess, $\Omega_{0}(n,m,L)$, to begin the study, it is convenient to use the final estimate $\Omega_{N}(n,m,L)$ (after the $N$-th iteration) to obtain the initial guess $\Omega_{0}(n,m,L_{0})$ of the next system size to be studied $(L_{0}>L)$. As we will show, good initial estimates, $\Omega_{0}(n,m,L)$, can be obtained using a two-dimensional interpolation because we know a priori the set of allowed CCs. This paper is organized as follows. In Sec. 2 we define the basic CCs and the respective allowed values of $(n,m)$ for the spin-1/2 Ising model on the square lattice; the main goal of this section is to find all gaps in the $(n,m)$ plane. In Sec. 3, this information is used in simulations of the AF Ising model via the TES algorithm. There we describe the method used to determine $\Omega_{0}(n,m,L_{2})$ via two-dimensional interpolation of the final estimate, $\Omega_{N}(n,m,L_{1})$, of the previous system size studied. Simulation results are reported in Sec. 4, for the order parameter and the staggered susceptibility as functions of $h$ and $T$. Points along the critical line in the $h-T$ plane are obtained using finite size scaling analysis, and the results compared with several theoretical expressions. We summarize our findings in Sec. 5. ## 2 Configuration Classes As pointed out above, one of the main problems in entropic sampling methods is the prior determination of the complete set of configuration classes for a given system size. Let us denote by $N_{+}$ and $N_{-}$ the number of up and down spins, respectively, on a square lattice with $N=N_{+}+N_{-}=L^{2}$ spins. The number of pairs of opposite spins, $u$, and $N_{+}$ are related to $n$ and $m$, respectively, via $n=2L^{2}-u$ (2) and $m=2N_{+}-L^{2}.$ (3) Thus, once the possible values of $(N_{+},u)$ are determined, so are those of $(n,m)$. Note that $u$, $n$ and $m$ can only take even values. The gaps in CS fall near the maximum and minimum values of $n$ ($n_{max}$ and $n_{min}$, respectively) for a given $m$. Therefore, we will identify the possible values of $u$ near its maximum, $u_{max}$, and minimum, $u_{min}$, for a given $N_{+}\in[0,\frac{L^{2}}{2}]$; note that the number of configurations is symmetric under interchange of $N_{+}$ and $N_{-}$. ### 2.1 Determining $u_{min}$ and nearby classes #### 2.1.1 Compact configurations Compact configurations consist of a square or rectangular cluster with $N_{+}$ up spins. To begin, consider the case of a square cluster of size $l\times l$ $(2<l<L)$ with $N_{+}=l^{2}$, as is illustrated in Fig. 2. It is evident that this configuration corresponds to the minimum value of $u$, $u^{(0)}=4l$. A configuration with the same number of up spins, and $u=N_{+}+2$ is obtained by transferring an up spin from one of the corners of the square to an edge, as shown in Fig. 3. From this configuration, further rearrangements leading to $u=u_{min}+4$, etc., are possible. When $N_{+}$ is not a square number, the most compact configuration (i.e., with the smallest perimeter) is a rectangle, or a square or rectangle with an incomplete layer of sites along one edge. For $N_{+}=l(l-1)$ we have $u_{min}=4l-2$, while for $l(l-1)<N_{+}<l^{2}$, $u_{min}=4l$. In all cases, moving a corner site to an edge, one constructs a configuration with $u=u_{min}+2$, and further arrangements yield additional increases in $u$. Figure 2: Basic compact configuration. Up and down spins are represented by “$\times$” and “$\bullet$”, respectively; wavy lines represent pairs of opposite spins. Figure 3: Modified compact configuration. The new up and down spins are represented by “$\otimes$” (previously “$\bullet$”) and “$\odot$” (previously “$\times$”), respectively; double straight and double wavy lines represent new pairs of identical and opposite NNs, respectively. #### 2.1.2 Extended configurations Thus there are no gaps in the large-$n$ region due to compact configurations. Such compact configurations, however, are not necessarily the ones that minimize $u$ for a given value of $N_{+}$. Consider for example the case $N_{+}=kL$ with $k$ an integer. The cluster of up spins can be arranged to wrap around the lattice, yielding $u_{min}=2L$, independent of $k$. We call such a configuration extended. For $k$ greater than a certain value, on the order of $L/4$, the configurations that minimize $u$ are extended rather than compact. Suppose now that for a given $N_{+}$, $u_{min}$ is obtained with an extended configuration, and that all compact configurations have $u$ strictly greater than $u_{min}$. If $N_{+}$ lies between $kL$ and $(k+1)L$, then the configuration that minimizes $u$ has at least two corners, and by the previous construction, configurations with $u=u_{min}+2$ exist. But if $N_{+}=kL$, the minimizing configuration has no corners and any modification yields a configuration with $u\geq u_{min}+4$. This is how the gaps near the maximum values of $n$ arise. Summarizing, if $L$ and $N_{+}$ are such that $u_{min}$ is obtained with a compact configuration, then there are configurations with $u_{min}+2$, $u_{min}+4$, …, etc., and no gap exists. If, on the other hand, $u_{min}$ is realized only for an extended configuration, and $N_{+}$ is an integer multiple of $L$, then there are no configurations with $u=u_{min}+2$. ### 2.2 Determining $u_{max}$ and nearby classes The largest possible value of $u$, $u_{max}$, occurs in a configuration such that $N_{+}=L^{2}/2$ with spins arranged in a chess board (CB) configuration, such that all up spins have down spins as NNs and vice versa. One readily verifies that for $0\leq N+\leq L^{2}/2$, the maximum number of unlike NN pairs is $u_{max}=4N_{+}$. Starting from the CB configuration, we can reduce $u$ by exchanging an up and a down spin. If the exchanged spins are NNs, the resulting configuration has $u=u_{max}-6$; otherwise one has $u=u_{max}-8$. Thus, for $N_{+}=L^{2}/2$, there are no configurations with $u=u_{max}-2$ or $u_{max}-4$. One readily verifies that configurations with $u=u_{max}-10$, $u_{max}-12$, etc., can be obtained via further exchanges of spins. For $N_{+}=L^{2}/2-1$, we have $u_{max}=2L^{2}-4$. Such configurations can be constructed by flipping one up spin in the CB configuration. Starting from this configuration, exchanging a pair of spins, one can reduce $u$ by 4, 6 or 8, but there is no rearrangement which reduces $u$ by just two. Thus for $N_{+}=L^{2}/2-1$, there is no configuration having $u=u_{max}-2$; configurations with $u=u_{max}-4$, $u_{max}-6$, etc. do exist. Finally, we note that for $1<N+<L^{2}/2-1$, there are no gaps in the neighborhood of $u_{max}$. To verify this, consider a configuration obtained by flipping $k$ up spins in the CB configuration, so that $N_{+}=L^{2}/2-k$, where $1<k<L^{2}/2$. By hypothesis there are now at least four more sites with down spins than with up spins. Starting from a configuration with $u=u_{max}$, in which each up spin is surrounded by down spins, we can create a single NN pair of up spins, with all six neighbors down, and with all remaining up spins completely surrounded by down spins. In this manner, $u$ is reduced by two. Configurations with $u=u_{min}-4$, $u_{min}-6$, etc. can be obtained by further exchanges of up and down spins. Using the facts summarized above, it is straightforward to construct an algorithm that determines which values of $u$ are possible, for a given $L$ and $N_{+}$, and thereby which values $(n,m)$ are accessible for a given system size. In the simulations reported below, we have verified that our entropic sampling scheme converges to visit all allowed classes. ## 3 Implementation Using tomographic sampling, we study the antiferromagnetic Ising model in an external field on the square lattice; we consider periodic boundary conditions and NN interactions. The CCs of the systems are defined in the energy- magnetization space $(n,m)$. The TES method is applied in order to generate estimates of $\Omega(n,m,L)$. For the smallest system size $(L=10)$ we begin with a guess, $\Omega_{0}(n,m)$, obtained using a mean-field approximation. For subsequent system sizes, however, we use a two-dimensional interpolation of $\Omega_{N}(n,m,L)$ (the final estimate of $(n,m)$ for the smaller system size, $L$) to construct $\Omega_{0}(n,m,L^{\prime})$, where $L^{\prime}>L$. For most studies we use five iterations, each one with $N_{sim}=10$ initial configurations, which are simulated for $N_{U}=10^{7}$ lattice updates or Monte Carlo steps. Let us denote by $\Gamma(n,m)$ and $\Gamma^{\prime}(n_{0}=n+\Delta n,m_{0}=m+\Delta m)$ the CCs that contain configurations $\mathcal{C}$ and $\mathcal{C^{\prime}}$, respectively. The simulation uses a single spin-flip dynamics, so that the possible variations of $n$ and $m$ are $\Delta n=0,\pm 2,\pm 4$ and $\Delta m=\pm 2$. At iteration $j$, the acceptance probability for the transition $(\mathcal{C}\to\mathcal{C^{\prime}})$ is $p_{j}(\Gamma\to\Gamma^{\prime})=\textrm{min}\bigg{[}\frac{\Omega_{j-1}(\Gamma)}{\Omega_{j-1}(\Gamma^{\prime})},1\bigg{]}.$ (4) These probabilities are stored in a table. For each configuration generated, be it a new one (if it is accepted) or the same (if it is rejected), we update the sums used to calculate the microcanonical and canonical averages of $\phi$, $\|\phi\|$, $\phi^{2}$, and $\phi^{4}$, where $\phi\equiv m_{A}-m_{B}$ (5) is the order parameter (staggered magnetization); $m_{A,B}$ are the magnetizations of the two sublattices. At the end of each iteration $j$ the estimate of $\Omega_{j}(n,m)$ is refined according to $\Omega_{j}(\Gamma)=\frac{H_{j}(\Gamma)}{\overline{H}_{j}(\Gamma)}\,\Omega_{j-1}(\Gamma),$ (6) where $H_{j}(\Gamma)$ is the histogram containing the number of times the CC $\Gamma$ is visited during the sampling, and $\overline{H}_{j}(\Gamma)$ is the average of $H_{j}(\Gamma)$ over all accessible CCs; the acceptance probability [Eq. (4)] is updated using the new estimate of $\Omega_{j}(n,m)$ and the histogram, $H_{j}(n,m)$, is set to zero. A single iteration of our method consists of ten independent simulations, each involving $10^{7}$ lattice updates, and each beginning from a different initial configuration. (By a lattice update we mean one attempted flip per spin; the initial configurations include, high, low, and intermediate interaction energies and both signs of the magnetization.) ### 3.1 Determining $\Omega_{0}(n,m,L)$: mean field approach As noted above, for the smallest system size we use an estimate of $\Omega_{0}(n,m,L)$ obtained via a mean-field approximation, specifically $\frac{1}{L^{2}}\ln\Omega_{0}(n,m,L)=\frac{1}{L^{2}}\ln\Omega(m)-\frac{(n-\langle n\rangle)^{2}}{2L^{2}\sigma^{2}}-\frac{\ln\sigma}{L^{2}}+const.\,,$ (7) where $\sigma\equiv\sigma(m)=\sqrt{\textrm{var}(n)}$, and $\frac{\ln\Omega(m)}{L^{2}}\simeq\ln 2-\frac{1}{2}[(1+m/L^{2})\ln(1+m/L^{2})+(1-m/L^{2})\ln(1-m/L^{2})].$ (8) To obtain this expression, we first note that $\Omega(m,L)=\sum_{n}\Omega(n,m,L)=\binom{L}{N_{+}}$, and use Stirling’s approximation. The dependence on $n$ is then obtained by estimating $\langle n\rangle$ and $\textrm{var}(n)$ for a given $m$ and $L$, using a random-mixing approximation, and supposing that $n$ follows a Gaussian distribution. In Fig. 4 we plot the estimate of $\ln\Omega_{0}(n,m,L=10)$ given by Eq. (7). We present in Fig. 5 the final estimated value of $\ln\Omega_{5}(n,m,L=10)$ after the 5$th$ iteration of the simulation; this result is quite similar to that presented by Hwang _et al._ [20] for the square lattice, which was obtained using Wang-Landau sampling [3]. We note that the differences between the initial estimate obtained via mean field approximation and the final simulation result of $\ln\Omega(n,m,L=10)$ are more evident near the edges of the CS, specially near to the maximum values of $n$. To have a better idea of how close this initial estimate is to the final simulation result, we plot in Fig. 6 the difference between that estimate and the final result of the simulation for $L=10$. Analyzing this figure, it is again clear that the differences are larger near the edges of the CS. Figure 4: Estimate of $\ln\Omega_{0}(n,m,L=10)$ via mean-field approximation. Figure 5: Final estimate of $\ln\Omega_{5}(n,m,L=10)$ after the 5$th$ iteration of the simulation. Figure 6: Difference between $\ln\Omega_{0}(n,m,L=10)$, estimated via mean-field approximation, and the final simulation result after the 5th iteration, $\ln\Omega_{5}(n,m,L=10)$. ### 3.2 Determining $\Omega_{0}(n,m,L)$: interpolation We expect that the closer the estimate $\Omega_{0}(n,m)$ is to the (unknown) exact value, $\Omega_{e}(n,m)$, the faster the simulation will converge, and the more accurate our final estimate will be. Since the mean-field estimate worsens as the system size grows, we only use it for the smallest system size studied; initial estimates of subsequent system sizes are obtained by interpolating the final estimate, $\Omega_{N}(n,m,L)$, of the previous system size studied. Our procedure is based on the existence of the limiting microcanonical entropy density as a function of the intensive parameters $\eta$ and $\nu$ (the bond and magnetization densities, respectively), $s(\eta,\nu)=\lim_{L\to\infty}\frac{1}{L^{2}}\ln\Omega(n_{L},m_{L},L),$ (9) where $n_{L}\simeq\eta L^{2}$ and $m_{L}\simeq\nu L^{2}$. (Since $n$ and $m$ are restricted to even integers we have $n_{L}=\eta L^{2}+{\cal O}(1/L^{2})$ and similarly for $m_{L}$.) The idea is then to write $\frac{1}{L^{\prime 2}}\ln\Omega_{0}(n^{\prime},m^{\prime},L^{\prime})=\frac{1}{L^{2}}\ln\Omega_{N}(\eta L^{2},\nu L^{2},L),$ (10) where $\eta=n^{\prime}/L^{\prime 2}$, $\nu=m^{\prime}/L^{\prime 2}$, and the r.h.s. is evaluated by extending $\ln\Omega_{N}$ to noninteger $n$ and $m$ via extrapolation and interpolation. Using this approach, we obtain better estimates, as is shown in Fig. 7. (Note that the largest differences continue to fall along the edges of the CS.) Figure 7: Difference between $\ln\Omega_{0}(n,m,L=18)$, estimated by interpolating the final result of $L=16$, and the final simulation result after the 5th iteration, $\ln\Omega_{5}(n,m,L=18)$. ### 3.3 Extrapolation and interpolation Suppose we wish to construct the initial estimate $\Omega_{0}(n,m,L_{2})$ on the basis of the simulation results for a smaller system, $\Omega_{N}(n,m,L_{1})$. We could do this via interpolation if every CC of the larger system were surrounded by four points of the smaller one in the $\eta$-$\mu$ plane. Fig. 8 shows, however, that along the edges of the CS, the points corresponding to classes of the larger system are not surrounded by four points of the smaller one. For those points, one could in principle use extrapolation rather than interpolation. We found, however, that direct extrapolation yields poor estimates for $\Omega$. Figure 8: Region of the $(\eta,\nu)$ plane with every possible configuration classes for system sizes $L_{1}=26$ and $L_{2}=28$. We obtain better estimates by first extrapolating the points along the edges of the CS for the smaller system, such that every point of the larger system is surrounded by four points of the smaller. Figure 9 shows the CS with accessible and extrapolated CCs for $L=10$. Following this extrapolation we perform a linear two-dimensional interpolation as per Eq. (10). Figure 9: Accessible and extrapolated configuration classes of a system of size $L=10$. ## 4 Results In this section we present results of the AF Ising model on the square lattice in an external field; periodic boundary conditions are employed. We use TES to simulate systems of sizes $L=10$ to 30, with $\Delta L=2$. To calculate the uncertainties we perform five independent studies for each system size. We plot in Fig. 10 the staggered magnetization (order parameter) per site, $\phi$, as a function of $h$ at $T=0.2$. We can see that $\phi$ decreases considerably between $h=3.85$ and $h=3.90$; this behavior suggests a critical point, $h_{c}$, marking a phase transition from the AF to the paramagnetic state. Figure 11 shows the staggered susceptibility per site, $\chi_{\phi}\equiv\frac{1}{L^{2}}(\langle\phi^{2}\rangle-\langle\phi\rangle^{2}),$ (11) as a function of $h$ at $T=0.02$. As $L$ grows the peaks tend to the critical point, $h_{c}$. The specific heat per site, $c$, (not shown) has a similar behavior. Figure 10: Staggered magnetization per site as a function of $h$ at $T=0.2$, for $L=10$ to 30. The absolute uncertainties are plotted in the inset. Figure 11: Staggered susceptibility per site as a function of $h$ at $T=0.02$, for $L=10$ to 30. The highest peaks correspond to the largest system sizes. The absolute uncertainties are plotted in the inset. ### 4.1 Phase diagram Using finite size scaling analysis [21] we estimate the critical line, $h_{c}(T)$, or, equivalently, $T_{c}(h)$, via the relations: $h_{c}(Y_{max},T,L)=h_{c}(Y_{max},T)+a_{1}/L+a_{2}/L^{2},$ (12) and $T_{c}(Y_{max},h,L)=T_{c}(Y_{max},h)+b_{1}/L+b_{2}/L^{2},$ (13) where $h_{c}(Y_{max},T,L)$ is the field at which $Y$ (the specific heat or the staggered susceptibility) takes its maximum for a given temperature and system size; $T_{c}(Y_{max},h,L)$ is defined in an analogous manner. The estimates obtained using the maximum of $c$ and $\phi$ are averaged to yield $h_{c}(T)$ and $T_{c}(h)$; Figure 12 illustrates the procedure for estimating $h_{c}$, for $T=0.02$. Our estimated points for the phase boundary are shown in Fig. 13 along with the approximate expression derived by Müller-Hartmann and Zittartz [8]: $\cosh\bigg{(}\frac{h}{T_{c}}\bigg{)}=\sinh^{2}\bigg{(}\frac{2J}{T_{c}}\bigg{)}.$ (14) Our simulation results are in good agreement with their expression. For the critical field, the greatest relative difference between theory and simulation is about 0.9%, which occurs at $T_{c}=1.8$. Figure 12: Critical field determination using finite size scaling analysis: $\overline{h}_{c}(T=0.02)=3.98666(3)$. Figure 13: Phase diagram of the Ising AF on the square lattice. Comparison between simulation and the theoretical expression of Müller-Hartmann and Zittartz. Asterisks denote points obtained varying $h$ with $T$ fixed; circles denote points obtained varying $T$, with $h$ fixed. The error bars of our results are smaller than the symbols. In Table 1 we compare our simulation estimates for the critical magnetic field $h_{c}(T)$ with some theoretical approximations. For temperature $T\leq 1$ we find good agreement with the estimates of Monroe [17] (whose analysis involves a free parameter $\omega$), Wu and Wu [12], and Blöte and Wu [13], whereas there are significant discrepancies in relation to the other approximations. At higher temperatures, differences appear between simulation and the predictions of Monroe, Wu and Wu, and Blöte and Wu. These may reflect a small systematic error or an underestimate of simulation uncertainties. We intend to examine this issue in greater detail in future work. Table 1: Comparison between our simulation estimates of $h_{c}(T)$ with some theoretical approximations. $T$ \| TES \| Monroe \| Monroe \| MHZ \| WW \| BW \| WK ---\|---\|---\|---\|---\|---\|---\|--- \| \| $(\omega=0.92484)$ \| $(\omega=0.93895)$ \| \| \| \| 0.1 \| 3.93304(16) \| 3.93307 \| 3.93318 \| 3.93069 \| 3.93329 \| 3.93330 \| 3.93372 0.5 \| 3.6648(8) \| 3.66506 \| 3.66561 \| 3.65309 \| 3.66611 \| 3.66614 \| 3.67589 1.0 \| 3.2906(14) \| 3.29303 \| 3.29391 \| 3.26843 \| 3.29200 \| 3.29261 \| 3.31764 1.5 \| 2.7258(14) \| 2.73243 \| 2.73396 \| 2.70401 \| 2.73094 \| 2.73176 \| 2.75099 2.0 \| 1.696(2) \| 1.71629 \| 1.71872 \| 1.69490 \| 1.71492 \| 1.71499 \| 1.71512 TES: Tomographic entropic sampling [6] MHZ: Müller-Hartmann and Zittartz [8] WW: Wu and Wu [12] BW: Blöte and Wu [13] WK: Wang and Kim [16] ## 5 Conclusions The complete enumeration of CCs is of fundamental importance to study the antiferromagnetic Ising model using tomographic entropic sampling. The determination of CCs for entropic sampling of Ising models also enables us to obtain good initial estimates for the configuration numbers $\Omega_{0}(n,m,L)$, using two-dimensional linear interpolation; the initial estimate is reasonably close to the final estimate $\Omega_{N}$. Despite the relatively small system sizes used in this study, we obtain a good estimate for the critical line in the temperature-magnetic field plane. Further details on critical behavior will be published elsewhere [22]. ## Acknowledgments We are grateful to CNPq and CAPES, Brazil, for financial support. ## References * [1] B. A. Berg and T. Neuhaus, Phys. Rev. Lett. 68, 9 (1992). * [2] J. Lee, Phys. Rev. Lett. 71, 211 (1993). * [3] F. Wang and D.P. Landau, _Phys. Rev. Lett._ , 84, 10 (2001). * [4] F. Wang and D.P. Landau, _Phys. Rev. E_ , 63, 056101 (2001). * [5] M.E.J Newman and G.T. Barkema. _Monte Carlo Methods in Statistical Physics_ (Oxford University Press, New York, 2001), p. 161 - 169. * [6] R. Dickman and A.G. Cunha-Netto, _Phis. Rev. E_ , 84, 026701 (2011). * [7] L. Onsager, _Phys. Rev._ , 65, 117 (1944). * [8] E. Müller-Hartmann and J. Zittartz, _Z. Phys._ , 73, 261 (1977). * [9] L. Sneddon, _J. Phys. C_ , 12, 3051 (1979). * [10] R.R. Santos, _J. Phys. C_ , 18, L1067 (1985). * [11] M. Kaufman, _Phys. Rev. B_ , 36, 3697 (1987). * [12] X.N. Wu and F.Y. Wu, _Phys. Lett. A_ , 144, 123 (1990). * [13] H.W.J. Blöte and X-N. Wu, _J. Phys. A_ , 23, L627 (1990). * [14] M. Badehdah, A. Benyoussef and M. Touzani, _J. Magn. Magn. Mater._ , 172, 254 (1997). * [15] X-Z. Wang and J.S. Kim, _Phys. Rev. Lett._ , 78, 413 (1997). * [16] X-Z. Wang and J.S. Kim, _Phys. Rev. E_ , 56, 2793 (1997). * [17] J.L. Monroe, _Phys. Rev. E_ , 64, 016126 (2003). * [18] S.J. Penney, V.K. Cumyn and D.D. Betts, _Physica A_ , 330, 507 (2003). * [19] K. Binder and D.P. Landau, _Phys. Rev. B_ , 21, 5 (1980). * [20] C-O. Hwang, S-Y. Kim, D. Kang and J.M Kim, _J. Stat. Mech: Theory Exp._ , 2007, L05001 (2007). * [21] V. Privman. _Finite Size Scaling Analysis and Numerical Simulations of Statistical Systems_ (World Scientific, London, 1990). * [22] B.J. Lourenço and R. Dickman, _in preparation_.	arxiv-papers	2011-12-06T19:51:20	2024-09-04T02:49:25.028746	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Bruno Jeferson Louren\\c{c}o and Ronald Dickman", "submitter": "Bruno Louren\\c{c}o", "url": "https://arxiv.org/abs/1112.1387" }
1112.1409	# Separating the BL Lac and Cluster X-ray Emissions in Abell 689 with _Chandra_ P. A. Giles, B. J. Maughan, M. Birkinshaw, D. M. Worrall, K. Lancaster HH Wills Physics Laboratory, Tyndall Avenue, Bristol, BS8 1TL, UK E-mail: P.Giles@bristol.ac.uk (Accepted 2011 August 26. Received 2011 August 4; in original form 2011 March 10) ###### Abstract We present the results of a _Chandra_ observation of the galaxy cluster Abell 689 (z=0.279). Abell 689 is one of the most luminous clusters detected in the ROSAT All Sky Survey (RASS), but was flagged as possibly including significant point source contamination. The small PSF of the _Chandra_ telescope allows us to confirm this and separate the point source from the extended cluster X-ray emission. For the cluster we determine a bolometric luminosity of Lbol = (3.3$\pm$0.3)$\times$1044 erg s-1 and a temperature of kT=5.1${}^{+2.2}_{-1.3}$ keV when including a physically motivated background model. We compare our measured luminosity for A689 to that quoted in the Rosat All Sky Survey (RASS) and find L0.1-2.4,keV = 2.8$\times$1044 erg s-1, a value $\sim$10 times lower than the ROSAT measurement. Our analysis of the point source shows evidence for significant pileup, with a pile-up fraction of $\simeq$60%. SDSS spectra and HST images lead us to the conclusion that the point source within Abell 689 is a BL Lac object. Using radio and optical observations from the VLA and HST archives, we determine $\alpha_{\rm ro}$=0.50, $\alpha_{\rm ox}$=0.77 and $\alpha_{\rm rx}$=0.58 for the BL Lac, which would classify it as being of ‘High-energy peak BL Lac’ (HBL) type. Spectra extracted of A689 show a hard X-ray excess at energies above 6 keV that we interpret as inverse Compton emission from aged electrons that may have been transported into the cluster from the BL Lac. ###### keywords: galaxies: clusters: general – galaxies: clusters: individual: Abell 689 – BL Lacertae objects: general ††pubyear: 2011 ## 1 Introduction Studies of clusters of galaxies, including measurements of their number density and growth from the highest density perturbations in the early Universe, offer insight into the underlying cosmology (Vikhlinin et al., 2003; Allen et al., 2004). However, in order to use clusters as a cosmological probe three essential tools are required (Del Popolo et al., 2010): (a) an efficient method to find clusters over a wide redshift range, (b) an observational method of determining the cluster mass, and (c) a method to compute the selection function or the survey volume in which clusters are found. These requirements are met by large surveys with well understood selection criteria. Arguably the most effective method of building large, well defined cluster samples has been via X-ray selection. The high X-ray luminosities of clusters make it relatively easy to detect and study clusters out to high redshifts. Many cluster samples have been constructed based upon large X-ray surveys such as the Einstein Medium Sensitivity Survey (EMSS; Gioia et al., 1990) and the ROSAT All Sky Survey (RASS; Voges, 1992). However due to the relatively poor angular resolution of these X-ray observatories, observations of clusters were susceptible to point source contamination. Indeed, within the ROSAT Brightest Cluster Sample (BCS; Ebeling et al., 1998) and its low-flux extension (eBCS; Ebeling et al., 2000), 9 out of 201 clusters and 8 out of 99 clusters respectively were flagged as probably having a significant fraction of the quoted flux from embedded point sources. Being able to resolve these point sources is of crucial importance for the reliable estimation of cluster properties, and indeed the nature of the point source contamination is of independent interest. The study of galaxy clusters has been transformed with the launch of powerful X-ray telescopes such as _Chandra_ and XMM Newton, which have allowed the study of the X-ray emitting intracluster medium (ICM) with unprecedented detail and accuracy. With the launch of this new generation of X-ray telescopes, we are able to uncover interesting features in the morphologies of individual clusters. In particular, _Chandra’s_ high angular resolution provides the means to examine individual cluster features with great detail. Abell 689 (hereafter A689; Abell, 1958), at z=0.279 (Collins et al., 1995), was detected in the RASS in an accumulated exposure time of 317s. It is included in the BCS, with a measured X-ray luminosity of 3.0$\times$1045 erg s-1 in the 0.1–2.4 keV band. This luminosity is the third highest in the BCS, and thus A689 meets the selection criteria for various highly X-ray luminous cluster samples (e.g. Dahle et al., 2002). However this cluster was noted as having possible point source contamination, and for this reason has often been rejected from many flux limited samples. In this study we present results of _Chandra_ observations designed to separate any point sources and determine uncontaminated cluster properties. The outline of this paper is as follows. In $\S$ 2 we discuss the observation and data analysis. Results of the X-ray cluster analysis is presented in $\S$ 3. In $\S$ 4 we present our analysis of the X-ray point source through X-ray, optical and radio observations. We interpret our results in $\S$ 5 and the conclusions are presented in $\S$ 6. Throughout this paper we adopt a cosmology with $\Omega_{M}$= 0.3, $\Omega_{\Lambda}$ = 0.7 and H0 = 70 km s-1 Mpc-1, so that 1′′ corresponds to 4.22 kpc at the redshift of A689. We define spectral index, $\alpha$, in the sense S $\propto$ $\nu^{-\alpha}$. ## 2 Observations and data reduction The _Chandra_ observation of A689 (ObsID 10415) was carried out January 01, 2009. A summary of the cluster’s properties is presented in Table 1. The observation was taken in VFAINT mode, and the source was observed in an ACIS-I configuration at the aim point of the I3 chip, with the ACIS S2 chip also turned on. Name \| RA \| Dec \| z \| NH,Gal \| RASS LX,0.1-2.4keV ---\|---\|---\|---\|---\|--- Abell 689 \| 08h 37m 24s.70 \| +14o 58′ 20′′.78 \| 0.279 \| 3.66$\times$1020 cm-2 \| 30.4$\times$1044 erg s-1 Table 1: Basic properties. Columns: (1) = Source name; (2) = Right Ascension at J2000 from Chandra; (3) = Declination at J2000 from Chandra; (4) = Redshift; (5) = Galactic column density; (6) = Intrinsic X-ray luminosity in the 0.1–2.4 keV band based upon ROSAT observations (Ebeling et al., 1998). For the imaging analysis of the cluster we used the CIAO111See http://cxc.harvard.edu/ciao/ 4.2 software package with CALDB222See http://cxc.harvard.edu/caldb/ version 4.3.0 and followed standard reduction methods. Since our observation was telemetered in VFAINT mode additional background screening was applied by removing events with significantly positive pixels at the border of the 5$\times$5 event island333See http://cxc.harvard.edu/ciao/why/aciscleanvf.html. We inspected background light curves of the observation following the recommendations given in Markevitch et al. (2003), to search for possible background fluctuations. The light curve was cleaned by 3$\sigma$ clipping and periods with count rates $>$20$\%$ above the mean rate were rejected. curve with rejected periods showed in red. The final level-2 event file had a total cleaned exposure time of 13.862 ks. As discussed in Sect. 4, there is a bright point source at the center of the extended cluster emission which is affected by pileup. For the analysis of this source we followed the same reduction method, with the exception that VFAINT cleaning was not applied. Applying VFAINT cleaning leads to incorrect rejection of piled-up events, introducing artifacts in the data. Figure 1: Background light curve for the observation of A689 in the 0.3-12.0 keV band. The CCD on which the cluster falls (ACIS-I3) and all point sources are excluded. The red bands show periods excluded by the Good Time Interval (GTI) file. ## 3 X-RAY CLUSTER ANALYSIS In this section we determine global cluster properties of Abell 689. Figure 2 shows a Gaussian smoothed image of the cleaned level 2 events file in the 0.7-2.0 keV band (the readout streak is removed using the CIAO tool ACISREADCORR), with an inset image of the point source which lies at the center of the cluster. The extent of the diffuse cluster emission was determined by plotting an exposure-corrected radial surface brightness profile (Fig 3), in the 0.7-2.0 keV band, of both the observation and blank-sky background to determine where the cluster emission is lost against the background. Figure 3 demonstrates that the diffuse cluster emission is detectable to a radius r $\approx$ 570′′ ($\approx$ 2.41 Mpc). At large radii (r$\geq$700′′) the curves rise due to vignetting corrections (larger at larger radii) applied to all the counts, whereas in reality each curve contains a component from particles that have not been focused by the telescope. Figure 2: 0.7-2.0 keV image of A689, smoothed by a Gaussian ($\sigma$=1.5 pixels, where 1 pixel = 3.94′′), cleaned in VFAINT mode and with the readout streak removed. Inset is a zoomed-in unbinned image of the central point source within A689, cleaned in FAINT mode (see $\S$ 2). The inner black circle (r = 208′′) is the region in which we extract the spectra for our analysis of the cluster emission (see $\S$ 3.2), the outer black circle (r = 570′′) represents the detected cluster radius (see $\S$ 3), and the region between this and the black box was used for the local background (see $\S$ 3.2). The white box displays the size of the inset, and the inner white circle (r = 26′′) shows the region excluded due to the central point source. Many point sources are seen in the observation and are excluded from our analysis. An extended source on the NE chip can be seen, that is unrelated to A689 and is also excluded from our analysis. Figure 3: Exposure-corrected radial surface brightness profiles (0.7-2.0 keV band) of the cluster (red) and the blank-sky background (blue), with the green line representing the radius beyond which no significant cluster emission is detected. ### 3.1 Background Subtraction In order to take the background of the observation into account, appropriate period E _Chandra_ blank-sky backgrounds were obtained, processed identically to the cluster, and reprojected onto the sky to match the cluster observation. We then followed a method similar to that outlined in Vikhlinin et al. (2006), in order to improve the accuracy of the background by applying small adjustments to the baseline model. Firstly we corrected for the rate of charged particle events, which has a secular and short-term variation by as much as 30%. We renormalise the background in the 9.5–12 keV band, where the _Chandra_ effective area is nearly zero and the observed flux is due entirely to the particle background events. The renormalisation factor was derived by taking the ratio of the observed count rate in the source and background observations respectively. The normalised spectrum of the blank-sky background is shown in Figure 4, over-plotted on the local background for comparison. The spectra agree well in the 9.5–12.0 keV band, and across the whole spectrum with only slight differences. In addition to the particle background, the blank-sky and source observations contain differing contributions from the soft X-ray background, containing a mixture of the Galactic and geocoronal backgrounds, significant at energies $\leq$1 keV. To take into account any difference in this background component between the blank-sky and source observations, the spectra were subtracted and residuals were modeled in the 0.4-1keV band using an APEC thermal plasma model (Smith et al., 2001). This model was included in the spectral fitting for the cluster analysis. As can be seen in Figure 4 this component is very weak in the case of A689. Figure 4: Comparison of the local (black) and blank-sky (red) background spectra, normalised to match in the 9.5-12.0 keV band. ### 3.2 X-ray Cluster Properties The analysis of the diffuse X-ray emission allows us to determine the X-ray environment surrounding the cluster central point source. Throughout this process we excluded the central point source (Fig 2; r$\leq$26′′) and the associated readout streaks to avoid contaminating the cluster emission. To determine cluster properties we extract spectra out to a radius chosen so that the cluster has the maximum possible signal-to-noise (SNR). The net number of counts, corrected for background, is then 414 (with SNR = 15) and the extraction annulus is 26′′$<$r$<$208′′ (see Fig 2), centered on the cluster (at $\alpha$, $\delta$ = 08h 37m 24s.70, 14o 58′ 20′′.78). We fitted the extracted spectrum in XSPEC with an absorbed thermal plasma model (WABS$\times$APEC) and subtracted the background described in $\S$ 3.1. We obtain a temperature of 13.6${}^{+13.2}_{-5.1}$ keV and a bolometric luminosity of Lbol=(10.2$\pm$2.9)$\times$$10^{44}$ ergs s-1. The measured temperature is far above what we would expect from the luminosity. Figure 5 shows the luminosity-temperature relation for a sample of 115 galaxy clusters (Maughan et al., 2008), along with the luminosity and temperature derived for A689 from our values above (pink square). As the Maughan et al. (2008) sample of clusters covers a wide redshift range, the luminosities of the clusters were corrected for the expected self-similar evolution, given by LX$\times$E(z)-1, where: $E(z)=\Omega_{\rm M0}(1+z)^{3}+(1-\Omega_{\rm M0}-\Omega_{\Lambda})(1+z)^{2}+\Omega_{\Lambda}.$ (1) The same correction was also applied to the A689 data for the plot. Our luminosity was derived within the same annular region as the cluster temperature and extrapolated both inward and outward in radius between (0-1)r500 (where r500 represents the radius at which the density of the cluster is 500 times the critical density of the Universe at that redshift) using parameters from a $\beta$-profile fit to the surface brightness profile. This takes into account our exclusion of the region near the central point source within the cluster, and the extrapolation to r500 is in order to compare with the Maughan et al. (2008) sample. Our r500 value was determined from the temperature and using the relation between r500 and T given in Vikhlinin et al. (2006). The high temperature for A689 could be an indication that our observation suffers from background flaring which would lead to an overestimate of the cluster temperature. However, no evidence is found that the spectrum of the blank-sky background differs from that of the local background (Figure 4), or that the background of the observation suffers from periods of flaring (Figure 1). To investigate the sensitivity of the temperature to our choice of background, we repeated the analysis using a local background region far from the cluster emission (see Fig 2). We obtained a temperature of 10.0${}^{+13.8}_{-3.3}$ keV. This temperature is again anomalous given the luminosity-temperature relation (Fig 5, green triangle. The luminosity is derived using the same method as above, only this time using this new value for the temperature to derive r500. The spectrum with a local background subtracted is shown in Figure 6. We see an excess of photons in the $\sim$6-9 keV band, which might have an effect on the spectral fit. We perform the same fit using a local background, however this time fitting in the 0.6-6.0 keV band to ignore these excess high energy photons. We obtain a temperature of 9.4${}^{+8.7}_{-2.9}$ keV. Figure 5 shows that the temperature is high for the luminosity, or the cluster is X-ray under-luminous. Before considering a physical interpretation for the apparently high cluster temperature, we investigated possible systematic effects from the background subtraction. This was done by independently modeling the background. Figure 5: Luminosity-Temperature relation of a sample of 115 clusters of Maughan et al. (2008)(blue open circles). The luminosities are measured within [0 $<$ r $<$ 1]r500 and the temperatures within [0.15 $<$ r $<$ 1.0]r500, in order to minimize the effect of cool cores on the derived cluster temperature. Our derived temperatures for A689 are overplotted for comparison (pink square, green triangle, red diamond) (see $\S$ 3.2). Figure 6: Spectrum of the cluster with the local background subtracted fitted with an absorbed thermal plasma model, with a reduced statistic of $\chi^{2}_{\rm\nu}$=1.45 ($\nu$=79). We model the background based upon a physical representation of its components. Our model consists of a thermal plasma APEC model, two power-law components and five Gaussian components. The APEC model and one of the power- law components are convolved with files describing the telescope and instrument response (the Auxiliary Response File (ARF) and Redistribution Matrix File (RMF)), and are intended to model soft X-ray thermal emission from the Galaxy and unresolved X-ray background. The second power-law component is not convolved with the ARF as it is used to describe the high energy particle background, which does not vary with effective area. The five Gaussian components are similarly not convolved, as these are used to model line features in the spectrum caused by the fluorescence of material in the telescope and focal plane caused by high energy particles. As described below and summarized in Table 2, the parameters of this background model were fit to the blank-sky background, local background or high energy cluster regions or taken from the literature, in order to build the most reliable model possible. We start by modeling blank-sky background in the region we used to extract our cluster spectrum, with the model outlined above. This allows us to place reasonable constraints on the line energy and widths of each Gaussian component, and the slope of the unconvolved power-law. We find line features at energies of 1.48, 1.75, 2.16, 7.48 and 8.29 keV. We also model the convolved power-law component, fixing the slope at a value of 1.48 taken from Hickox & Markevitch (2006). The normalisation of this power- law component will then be used throughout our modeling process, scaled by area where necessary. A spectrum of the blank-sky background and a fit using our model are shown in Figure 7. Next we further constrained model parameters by fitting our high-energy Gaussian and unconvolved power-law components in the 5.0-9.0 keV band within the cluster region. At these energies, the emission is dominated by particle background. We fix the line energies and widths to those found in the blank- sky background and fit for the normalisations. This fit finds a slope of the power-law consistent with that found in the blank-sky background. We therefore fix the slope of the power-law at $\Gamma$=0.0061, as found for the blank-sky background, and fit for the normalisation. We finally fit the low energy Gaussians and APEC normalisation model parameters in a local background region far from the cluster emission. The high energy Gaussians and unconvolved power-law components are frozen at the values found in the blank-sky and cluster regions, with the normalisations scaled by area. The convolved power- law component is frozen at the values found in the blank-sky background and the normalisation is scaled by area. The Gaussian features at 1.48, 1.75 and 2.16 keV are frozen at the energies and widths found in the blank-sky background. The temperature of the APEC model is frozen at 0.177 keV (taken from Hickox & Markevitch, 2006). We note that our APEC temperature is not well constrained by our data, but this is a weak component. A spectrum of the local background and the corresponding fit with the model are shown in Figure 8. We now model the cluster with an absorbed thermal plasma (WABS$\times$APEC) model, including the background model outlined above. The normalisations of the background APEC component and the Gaussians at energies 1.48, 1.75 and 2.16 keV were scaled by the ratio of the areas from the local background region to the cluster region. The normalisations of the fluorescent lines also vary with detector position. To account for this effect in the low energy ($<$3 keV) lines (where we must fit the normalisations in the off-axis local background region), we measure the relative change in the normalisations of each line between the local background and source regions in the blank-sky background data. The normalisation of each low energy Gaussian component in the fit to the cluster data is scaled for the different detector of the cluster region by the relative change in normailsiation determined above (in addition to the geometrical scaling for size of the extraction region). The unconvolved power-law and Gaussians at 7.48 and 8.29 keV are all frozen at the values found in the 5.0-9.0 keV cluster region fit. The normalisation of the convolved power-law is frozen at the value found in the blank-sky background. All parameters of the model used to describe the background are frozen in the corresponding cluster fit, we also freeze the redshift at 0.279 and the abundance at 0.3. Our fit yields a temperature of 5.1${}^{+2.2}_{-1.3}$ keV ($\chi^{2}_{\rm\nu}$=1.15 ($\nu$=79)). We measure a bolometric luminosity of Lbol = 1.7$\times$1044 erg s-1. The result is shown in Figure 5 (red diamond). The spectrum with the corresponding fit to the cluster including the background model is shown in Figure 9. Component \| Represents \| Parameter \| Value \| Where measured ---\|---\|---\|---\|--- Convolved power-law \| Unresolved X-ray bg \| slope \| 1.48 \| Hickox & Markevitch (2006) normalisation \| 4.11$\times$10-6 \| blank-sky bg Unconvolved power-law \| Particle bg \| slope \| 0.061 \| blank-sky bg normalisation \| 0.015 \| 5.0-9.0 keV cluster region Gaussian 1 \| Al K$\alpha$ fluorescence \| energy \| 1.48 keV \| blank-sky bg width \| 0.022 keV \| blank-sky bg normalisation \| 1.82$\times$10-4 \| local bg Gaussian 2 \| Si K$\alpha$ fluorescence \| energy \| 1.75 keV \| blank-sky bg width \| 0.95 keV \| blank-sky bg normalisation \| 1.45$\times$10-2 \| local bg Gaussian 3 \| Au M$\alpha\beta$ fluorescence \| energy \| 2.16 keV \| blank-sky bg width \| 0.045 keV \| blank-sky bg normalisation \| 2.24$\times$10-3 \| local bg Gaussian 4 \| Ni K$\alpha$ fluorescence \| energy \| 7.48 keV \| blank-sky bg width \| 0.022 keV \| blank-sky bg normalisation \| 5.73$\times$10-3 \| 5.0-9.0 keV cluster region Gaussian 5 \| Cu + Ni fluorescence \| energy \| 8.29 keV \| blank-sky bg width \| 0.168 keV \| blank-sky bg normalisation \| 4.56$\times$10-3 \| 5.0-9.0 keV cluster region APEC \| Galactic foreground emission \| kT \| 0.177 keV \| Hickox & Markevitch (2006) abundance \| 1.0 \| solar abundance redshift \| 0 \| Galactic normalisation \| 2.9$\times$10-5 \| local bg Table 2: Table of the individual model components used to represent the background, with a brief interpretation of each component, individual component parameters, parameter values and where each value is calculated. All normalisations are measured in photons/keV/cm2/s at 1 keV, and scaled to the cluster region. Blank-sky background parameters were derived in the same region and the local background were measured in an area 2.45 times that of the cluster and therefore the normalisations reduced by this factor. The low energy Gaussians were also corrected due to the normalisation dependence with position on the detector (see text). Figure 7: Spectrum of the blank-sky background in the source region and corresponding the fit (see Sect. 3.2), $\chi^{2}_{\rm\nu}$=1.17 ($\nu$=585). Figure 8: Spectrum of the local background and corresponding fit (see Sect 3.2). $\chi^{2}_{\rm\nu}$=0.79 ($\nu$=120). Figure 9: Spectrum of the cluster plus background fit with an absorbed thermal plasma model including a background model (see Sect 3.2). $\chi^{2}_{\rm\nu}$=1.15 ($\nu$=79). ## 4 The Central Point Source The point source is displayed in Figure 10. The presence of strong readout streaks indicate that the point source is likely to be affected by pile-up. The readout streaks occur as X-rays from the source are received during the ACIS parallel frame transfer, which provides 40$\mu$s exposure per frame in each pixel along the streak. We detail our analysis of the point source and estimate of the pileup fraction in the following section. Figure 10: _Chandra_ image of A689 showing the regions used for extracting spectra of the readout streak (inner rectangles) and the corresponding background regions for the readout streak (outer rectangles). ### 4.1 X-ray Analysis of the Point Source As a first test of the predicted pile-up, we compared the image of the point source to the _Chandra_ Point Spread Function (PSF). We made use of CIAO tool MKPSF to create an image of the on-axis PSF following the method outlined in Donato et al. (2003). This consists of merging 7 different monochromatic PSFs chosen and weighted on the basis of the source energy spectrum between 0.3 and 8 keV. This method can be summarized as follows: 1. 1. We first extract the energy distributions of the photons from a circular region centered on the peak brightness of the source with a radii of 2.5′′. 2. 2. We choose seven discrete energy values at which to creating each PSF, with the number of counts at each energy corresponding to that PSF’s ‘weight’. 3. 3. Using MKPSF we create seven monochromatic PSFs at the position of the point source on the detector and co-added them. Each PSF is weighted by its relative normalisation (found in the previous step). Figure 11: Surface brightness profiles of the point source (red squares) and the composite PSF (blue circles) in the 0.3-8.0 keV band, normalised to agree in the 2.46-4.92 arcsec radii region. Once we obtained the composite PSF, we normalised it to the counts within an annulus (inner and outer radius 2.46 and 4.92 arcsec respectively) in order to avoid any effects of pile-up in the central region. We then compare surface brightness profiles of the point source and PSF to look for evidence of pile- up in the core of the point source image. Figure 11 shows the radial surface brightness profiles of the point source (red) and the composite PSF (blue). We find that the point source and PSF agree well in the wings of the PSF ($>$ 2.46′′) and that there is an excess in the PSF surface brightness above that of the point source at the peak of the source. This is consistent with the flattening of the source profile relative to that of the PSF due to pileup in the core. The PSF then gives an estimate of the non piled up count rate. Given this count rate we use PIMMS444http://cxc.harvard.edu/toolkit/pimms.jsp to estimate a pile-up fraction of 65%. We also compute a second estimate of the core count rate using the ACIS readout streak. By fitting a model to the spectrum extracted from the readout streak we can compare this to a spectrum extracted in the core and fitted using a pileup model. We follow the method outlined in Marshall et al. (2005) in order to correct the exposure time of the readout streak spectrum. For an observation of live time tlive, a section of the readout streak that is $\theta_{s}$ arcsec long accumulates an exposure time of ts = 4 $\times$10-5tlive$\theta_{\rm s}$/(tf$\theta_{\rm x}$) s, where $\theta_{\rm x}$ = 0.492′′ is the angular size of an ACIS pixel. For our observation, tlive = 13.862 ks and the frame time parameter tf = 3.1 s, giving ts = 165s in a streak segment that is 454′′ long. Figure 10 shows the regions used for extracting spectra of the readout and an adjacent background region. This choice of background region ensures the cluster emission is subtracted from the readout streak spectrum. Using the SHERPA package (Refsdal et al., 2009) we fitted an absorbed 1-D power-law model (WABS$\times$POWER-LAW) to the extracted spectrum of the readout streak. We obtain fit parameters for the photon index = 2.33${}^{+0.34}_{-0.30}$ and a normalisation of 0.0033$\pm 0.0005$ photons keV-1 cm-2 s-1 ($\chi^{2}_{\rm\nu}$=0.34 ($\nu$=63)). We then extract a spectrum of the point source in a region of radius 5′′, and subtracted the same background as for the readout streak. We once again fitted an absorbed power-law model, including this time a pileup model (jdpileup). We obtain fit parameters for the photon index = 2.22${}^{+0.05}_{-0.04}$, consistent to that found from the readout streak, and a normalisation of 0.0015$\pm$0.0001 photons keV-1 cm-2 s-1 ($\chi^{2}_{\rm\nu}$=1.6 ($\nu$=119)). The extracted spectrum and corresponding fit is shown in Figure 12. The pileup fraction is estimated to be 60%, which is consistent with that found using the PSF count rate. The normalisation found in the fit can be converted to an X-ray flux density for this source, f${}_{\rm 1~{}keV}$=0.99$\pm$0.07 $\mu$Jy. Figure 12: Spectra of the point source fitted with an absorbed power-law, including a pileup model. $\chi^{2}_{\rm\nu}$=1.6 ($\nu$=119). ### 4.2 Optical Observations Abell 689 was observed with the Hubble Space Telescope (HST) with the F606W filter (Ṽ-band) on January 20, 2008. Marking the position of the peak X-ray emission of the point source on the HST image, we find that this corresponds to an object resembling an active nucleus in a relatively bright galaxy (Fig 13). We searched the SDSS DR7 archive for information on the spectral properties of this object. At the coordinates of the X-ray point source (SDSS coordinates of $\alpha$,$\delta$ = 08h 37m 24.7, 14o 58′ 19′′.8) we find a blue object with a corresponding relatively featureless spectrum (Fig 14). From SDSS we quote an r-band magnitude of 17.18. The spectrum resembles that of a BL Lac object, a type of AGN orientated such that the relativistic jet is closely aligned to the line of sight. From the H and K lines in the spectrum (dotted green lines in Fig 14), thought to be from the host galaxy, the redshift is determined to be z=0.279, consistent with the redshift assigned to the cluster (Collins et al., 1995). Using the HST observation we measure an optical flux for the BL Lac of f5997Å=112mJy. Figure 13: HST image of the point source with a cross marking the position of the peak of the X-ray emission. Figure 14: Spectra of the BL Lac from SDSS (SDSS J083724.71+145819.8). The vertical green lines represent the H and K lines thought to be from the host galaxy and the green line at the bottom represents the error spectrum. ### 4.3 Radio Observations Archival radio observations of Abell 689 are available, allowing us to determine the radio spectral index, $\alpha$. We obtained 8.46 GHz data taken in March 1998 from the VLA archive which we mapped using AIPS. The source is unresolved at 8.46 GHz, and we measure a flux density of 18.6$\pm$0.27 mJy. We also obtained a 1.4 GHz radio image from the FIRST survey, from which we measure a flux of 62.7$\pm$0.25 mJy. From these data we obtain a spectral index of $\alpha_{\rm r}$$\sim$0.67$\pm$0.01. We note that a slight angular extension in the FIRST survey suggests that the 8.46 GHz image may be missing some flux density. ## 5 Discussion ### 5.1 The ICM properties of Abell 689 \| r500 \| TX \| LX,bol \| \| ---\|---\|---\|---\|---\|--- Background Subtraction \| (arcsec) \| (keV) \| ($\times$1044 erg s-1) \| reduced $\chi^{2}$ \| degrees of freedom Blank-sky \| 1130 \| 13.6${}^{+13.2}_{-5.1}$ \| 10.1$\pm$2.9 \| 1.19 \| 74 Local \| 696 \| 10.0${}^{+13.8}_{-3.3}$ \| 6.2$\pm$1.4 \| 1.45 \| 79 Physically motivated model† \| 266 \| 5.1${}^{+2.2}_{-1.3}$ \| 3.3$\pm$0.3 \| 1.15 \| 79 Table 3: Table listing the derived spectral properties of A689 for each of the background treatments we employ in our analysis. † indicates our favored method for determining the cluster properties of A689. We have derived the ICM properties of A689 using three methods of background subtraction. Table 3 shows the spectral properties of the ICM for each of the background treatments we employ with our favored method coming from a physically motivated model of the background components. Through detailed modeling of the local and blank-sky backgrounds for A689, and including this background model in a spectral fit to the cluster, we have determined a temperature and luminosity for A689 of T = 5.1${}^{+2.2}_{-1.3}$ keV and Lbol = 3.3$\times$1044 erg s-1. Plotting these values on the luminosity-temperature plot (Fig 5) and comparing to the large X-ray sample of Maughan et al. (2008), we find that A689 is observed to be at the edge of the observed scatter in the luminosity-temperature relation. This suggests that either the temperature of the ICM has been enhanced or suppression of the luminosity has occurred. It has been shown that systems that host a radio source are likely to have higher temperatures at a given luminosity. Croston et al. (2005) showed that this is the case for radio-loud galaxy groups, and Magliocchetti & Brüggen (2007) showed that for clusters that host a radio source there is a departure from the typical luminosity-temperature relation, particularly in the case of low mass systems. Croston et al. (2005) also showed, through analysis of Chandra and XMM-Newton observations, evidence for radio-source interaction with the surrounding gas for many of the radio-loud groups. A similar process could be occurring within A689, which has a confirmed radio source at the center of the cluster. We note that more detailed X-ray and radio observations would to be needed in order to test any interaction between the BL Lac and ICM. The other possible explanation for the offset of radio-loud systems from the luminosity-temperature relation is the suppression of the luminosity, which could be caused by displacement of large amounts of gas due to the interaction of the radio source with the ICM. The interaction of the radio source with the ICM will cause an increase in the entropy of the local ICM. This higher entropy gas will be displaced so that it is in entropy equilibrium with the surrounding gas. However, Magliocchetti & Brüggen (2007) showed that there is a correlation between the radio luminosity and the heat input required to produce the observed temperature increment in clusters hosting radio sources. They note that this correlation favors an enhanced temperature scenario caused by the radio galaxy induced heating. ### 5.2 Comparison with the BCS A689 was noted in the BCS as having a significant fraction of its flux coming from embedded point sources. We have confirmed that A689 contains a point source at the center of the cluster and is that of a BL Lac object. We stated that the measured X-ray luminosity for A689 as quoted in the BCS (L0.1-2.4keV = 3$\times$1045 erg s-1, see $\S$ 1), is the third brightest in the BCS. From our follow up observation with Chandra we calculate the luminosity and compare with the BCS value. The luminosities in the BCS are calculated within a standard radius of 1.43 Mpc, which at the redshift of A689 corresponds to a radius of 338′′. We therefore employ the same method as in Sect 3.2 and integrate under a beta model fitted to the derived surface brightness profile and extrapolate inward and outward from 26-206′′ to 0-338′′. The unabsorbed flux in the 0.1 - 2.4 keV band (observed frame) was f0.1-2.4,keV = 5.8$\times$10-13 erg s-1 cm-2. After k-correction the X-ray luminosity in the 0.1 - 2.4 keV band (rest frame) was L0.1-2.4,keV = 2.8$\times$1044 erg s-1. Note that we assume an H0 of 50 for consistency with the BCS catalog. This value is $\sim$10 times lower than that quoted in the BCS, and A689 is now ranked 110th out of 201 in luminosity. ### 5.3 Classifying the BL Lac BL Lac objects may be split into ‘High-energy peak BL Lacs’ (HBL) and ‘Low- energy peak BL Lacs’ (LBL), for objects which emit most of their synchrotron power at high (UV–soft-X-ray) or low (far-IR, near-IR) frequencies respectively (Padovani & Giommi, 1995). HBL and LBL objects have radio-to-X- ray spectral indies of $\alpha_{rx}$$\leq$0.75 and $\alpha_{rx}$$\geq$0.75, respectively. We calculate a radio-to-X-ray spectral index for the BL Lac in A689 of $\alpha_{rx}$=0.58$\pm$0.04. From this value we classify our BL Lac as an HBL type. We also compared our BL Lac with those of Fossati et al. (1998), who investigated the properties of large samples of BL lacs at radio to $\gamma$-ray wavelengths. Our value of $\alpha_{rx}$=0.58 falls into the region 0.35$\leq$$\alpha_{rx}$$\leq$0.7, dominated by X-ray selected BL Lacs (XBL). This is consistent with the X-ray selection of this cluster. Using the measured HST flux, we also calculate $\alpha_{ro}$=0.50 and $\alpha_{ox}$=0.77. These values are not atypical for BL Lac objects (Figure 7 in Worrall et al., 1999). Donato et al. (2003) tried to determine whether HBLs and LBLs were characterized by different environments. They found that of 5 sources exhibiting diffuse X-ray emission that 4 were HBLs and 1 was an LBL. The BL Lac in A689 continues this trend, as it appears to be an HBL embedded within a cluster environment. ### 5.4 Evidence for Inverse-Compton Emission Our initial analysis of the extended emission in A689 yielded temperature estimates that were significantly higher than expected based upon its X-ray luminosity. We also found evidence for an hard excess of X-ray photons in the 6.0-9.0 keV band of the cluster spectrum. Before assigning a physical cause we must be careful to eliminate systematic effects in the background subtraction, as underestimating the particle background will give a hard excess. This is unlikely to be the case here as a high temperature was obtained when either a blank-sky and local background was used. When using our background model instead of subtracting a background spectrum, the cluster temperature was more consistent with its luminosity (Fig 5). This appears to be due to the unconvolved power law fitting a hard excess in the cluster region. In order to assess what impact the high-energy particle component has on the cluster temperature, we varied the normalisation of the unconvolved power-law component of our background model (sect. 3.2) within its 1$\sigma$ errors. The temperature ranged from 3.86 to 8.51 keV. Thus small changes in the particle background have a significant influence on the cluster temperature. Our power-law component was derived within the cluster region in the 5.0-9.0 keV band, so our background model may be removing a physical hard X-ray component associated with the cluster. We assess the possible properties of such a component by re-deriving the normalisation of the unconvolved power- law component in the local background region, scaling this value by the ratio of the areas, and using this value for the unconvolved power-law normalisation in the background model for our cluster fit. We find a power-law normalisation of 0.0129 photons keV-1 cm-2 s-1 (as opposed to 0.0149 photons keV-1 cm-2 s-1 in the cluster region). Using this value and re-fitting (Fig 15), we find a cluster temperature of 14.3${}^{+15.6}_{-5.4}$ keV. This suggests that the hard component is spatially associated with the cluster. Figure 15: Spectrum of the cluster including our physically motivated model, with the normalisation of the unconvolved power-law component derived in the local background region. $\chi^{2}_{\rm\nu}$=1.17 ($\nu$=79) Bonamente et al. (2007) find a similar excess of X-ray emission in the cluster A3112, which is known to have a central AGN. It is argued that this excess may be due to emission of a non-thermal component. Relativistic electrons in the intergalactic medium will cause CMB photons to scatter into the X-ray band (inverse Compton scattering). The same process could occur in A689, with the relativistic particles responsible for the inverse Compton scattering provided by the jets of the BL Lac. In order to test the assumption of inverse-Compton emission, we add a convolved power-law component with $\alpha$=1.5 (appropriate for modeling IC emission from aged electrons). We fit for the normalisation of this added power-law component and freeze all other parameters of the background and cluster model. We note that we use the normalisation of the unconvolved power- law found in the local background region as found above (a value of 0.0129). When fit, the $\chi$2 increases slightly but the fit is still acceptable at the 95% confidence level. From the additional power-law component we measure a 1 keV flux density of $\sim$7 nJy. If the X-ray emission is from scattering of the CMB by an aged population of electrons of power-law number index 4.0, we can determine what the implied synchrotron emission would be at 1.4 GHz, given plausible magnetic fields. Clusters typically have magnetic fields of a few $\mu$G (Carilli & Taylor, 2002). The NVSS survey would have detected a flux density of $\sim$10 mJy over the entire cluster. Adopting a 1.4 GHz flux density limit of 10 mJy, we require a magnetic field of B $\leq$ 2 $\mu$G over the cluster to avoid over-predicting radio emission. We also calculate a value for the minimum-energy magnetic field, BE_min, that would give a 1.4 GHz flux density of S${}_{\rm 1.4~{}GHz}$ = 10 mJy, which in this case is BE_min = 7.5 $\mu$G. For a flux density of S${}_{\rm 1.4~{}GHz}$ = 5 mJy, we would require a magnetic field of 1.6 $\mu$G and BE_min = 6.5 $\mu$G. Note that B $\propto$ S${}_{\rm 1.4~{}GHz}$1/1+α and BE_min $\propto$ S${}_{\rm 1.4~{}GHz}$1/3+α. We conclude that the excess X-ray emission can be attributed to inverse-Compton scattering without over-predicting the radio emission if the magnetic field strength is in a range typical of clusters and within a factor of a few of the minimum energy value. ## 6 Conclusions We have used a 14 ks _Chandra_ observation of the galaxy cluster A689 in order to determine the nature of the cluster’s point source contamination and to analyze the cluster properties excluding the central point source. Our main conclusions are as follows. * 1\. Using background subtraction of both local and blank-sky backgrounds, we obtain temperatures which are high relative to the luminosity-temperature relation. * 2\. We construct a physically motivated model for the background and include this model in a fit to the cluster spectrum. If the particle background in the cluster is allowed to exceed that in the local and blank-sky backgrounds we obtain a temperature of 5.1${}^{+2.2}_{-1.3}$ keV. However, there is no reason for there to be a higher particle rate in the specific region of the CCD in which the cluster lies. A hard excess needed to bring the temperature to a reasonable value must have a different origin. * 3\. We confirm the presence of a point source within A689 as suspected in the BCS. When excluding the point source and using our derive background model we find a luminosity of L0.1-2.4,keV = 2.8$\times$1044 erg s-1, a value $\sim$10 times lower than quoted in the BCS. * 4\. From the X-ray analysis of the point source we find a “flat-topped” point source with a pileup fraction of $\simeq$ 60%. * 5\. Optical observations of the cluster from SDSS and HST lead us to conclude that the point source is a BL Lac type AGN. * 6\. We classify the BL Lac as an ‘High-energy peak BL Lac’ with $\alpha$rx=0.58$\pm$0.04. * 7\. We interpret the hard X-ray excess needed to bring the cluster temperature to a reasonable value as inverse-Compton emission from aged electrons that may have been transported into the cluster from the BL Lac. We have shown here not only the importance of resolving and excluding point sources in cluster observations, but also the effect these point sources can have when determining the ICM properties of galaxy clusters. The detailed analysis we have performed here may not however be suitable for all clusters as it is unclear whether this analysis can be performed at higher redshifts. Separating the point source and cluster emissions becomes increasingly difficult at high redshifts, however Chandra has proved capable at resolving point source emission in clusters to z$\sim$2 (e.g. Andreon et al., 2009). Resolving point source and cluster emission out to high redshift becomes more important with redshift due to the evolution of the number density of point sources within clusters (Galametz et al., 2009). The area used to model the background components associated with the high energy cluster regions will decrease spatially with redshift, and separating the between thermal and inverse Compton emissions at higher redshift will become increasingly difficult (Fabian et al., 2003), due to the increase in the energy density of the CMB with redshift. ## Acknowledgments We thank J.Price for useful discussions regarding the SDSS and Hubble data. We thank Ewan O’Sullivan and Dominique Sluse for useful discussions on the nature of the point source. We thank the anonymous referee for valuable comments and suggestions. PG also acknowledges support from the UK Science and Technology Facilities Council. ## References * Abell (1958) Abell, G. O. 1958, ApJS, 3, 211 * Allen et al. (2004) Allen S. W., Schmidt R. W., Ebeling H., Fabian A. C., van Speybroeck L., 2004, MNRAS, 353, 457 * Andreon et al. (2009) Andreon, S., Maughan, B., Trinchiere, G., & Kurk, J. 2009, A&A, 507, 147 * Bonamente et al. (2007) Bonamente, M., Nevalainen, J., & Lieu, R. 2007, ApJ, 668, 796 * Carilli & Taylor (2002) Carilli, C. L., & Taylor, G. B. 2002, ARA&A, 40, 319 * Collins et al. (1995) Collins C. A., Guzzo L., Nichol R. C., Lumsden S. L., 1995, MNRAS, 274, 1071 * Croston et al. (2005) Croston, J. H., Hardcastle, M. J., & Birkinshaw, M. 2005, MNRAS, 357, 279 * Dahle et al. (2002) Dahle H., Kaiser N., Irgens R. J., Lilje P. B., Maddox S. J., 2002, ApJS, 139, 313 * Del Popolo et al. (2010) Del Popolo A., Costa V., Lanzafame G., 2010, A&A, 514, A80+ * Donato et al. (2003) Donato D., Gliozzi M., Sambruna R. M., Pesce J. E., 2003, A&A, 407, 503 * Ebeling et al. (2000) Ebeling H., Edge A. C., Allen S. W., Crawford C. S., Fabian A. C., Huchra J. P., 2000, MNRAS, 318, 333 * Ebeling et al. (1998) Ebeling H., Edge A. C., Bohringer H., Allen S. W., Crawford C. S., Fabian A. C., Voges W., Huchra J. P., 1998, MNRAS, 301, 881 * Fabian et al. (2003) Fabian, A. C., Sanders, J. S., Crawford, C. S., & Ettori, S. 2003, MNRAS, 341, 729 * Fossati et al. (1998) Fossati, G., Maraschi, L., Celotti, A., Comastri, A., & Ghisellini, G. 1998, MNRAS, 299, 433 * Galametz et al. (2009) Galametz, A., et al. 2009, ApJ, 694, 1309 * Gioia et al. (1990) Gioia I. M., Maccacaro T., Schild R. E., Wolter A., Stocke J. T., Morris S. L., Henry J. P., 1990, ApJS, 72, 567 * Hickox & Markevitch (2006) Hickox, R. C., & Markevitch, M. 2006, ApJ, 645, 95 * Magliocchetti & Brüggen (2007) Magliocchetti, M., & Brüggen, M. 2007, MNRAS, 379, 260 * Markevitch et al. (2003) Markevitch M., Bautz M. W., Biller B., Butt Y., Edgar R., Gaetz T., Garmire G., Grant C. E., Green P., Juda M., Plucinsky P. P., Schwartz D., Smith R., Vikhlinin A., Virani S., Wargelin B. J., Wolk S., 2003, ApJ, 583, 70 * Marshall et al. (2005) Marshall, H. L., et al. 2005, ApJS, 156, 13 * Maughan et al. (2008) Maughan, B. J., Jones, C., Forman, W., & Van Speybroeck, L. 2008, ApJS, 174, 117 * Padovani & Giommi (1995) Padovani, P., & Giommi, P. 1995, MNRAS, 277, 1477 * Refsdal et al. (2009) Refsdal B. L., Doe S. M., Nguyen D. T., Siemiginowska A. L., Bonaventura N. R., Burke D., Evans I. N., Evans J. D., Fruscione A., Galle E. C., Houck J. C., Karovska M., Lee N. P., Nowak M. A., 2009, in Varoquaux G., van der Walt S., Millman J., eds, Proceedings of the 8th Python in Science Conference Sherpa: 1d/2d modeling and fitting in python. Pasadena, CA USA, pp 51 – 57 * Smith et al. (2001) Smith R. K., Brickhouse N. S., Liedahl D. A., Raymond J. C., 2001, ApJ, 556, L91 * Vikhlinin et al. (2006) Vikhlinin A., Kravtsov A., Forman W., Jones C., Markevitch M., Murray S. S., Van Speybroeck L., 2006, ApJ, 640, 691 * Vikhlinin et al. (2003) Vikhlinin A., Voevodkin A., Mullis C. R., van Speybroeck L., Quintana H., McNamara B. R., Gioia I., Hornstrup A., Henry J. P., Forman W. R., Jones C., 2003, ApJ, 590, 15 * Voges (1992) Voges W., 1992, Technical report, The ROSAT all-sky X ray survey * Worrall et al. (1999) Worrall, D. M., Birkinshaw, M., Remillard, R. A., Prestwich, A., Tucker, W. H., & Tananbaum, H. 1999, ApJ, 516, 163	arxiv-papers	2011-12-06T21:00:04	2024-09-04T02:49:25.036485	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "P. A. Giles, B. J. Maughan, M. Birkinshaw, D. M. Worrall, K. Lancaster", "submitter": "Paul Giles", "url": "https://arxiv.org/abs/1112.1409" }
1112.1472	# Decaying Holographic Dark Energy and Emergence of Friedmann Universe Titus K Mathew Department of Physics, Cochin University of Science and Technology, Kochi-22, India. E-mail: tituskmathew@gmail.com, titus@cusat.ac.in ###### Abstract A universe started in almost de Sitter phase, with time varying holographic dark energy corresponding to a time varying cosmological term is considered. The time varying cosmological dark energy and the created matter are consistent with the Einstein’s equation. The general conservation law for the decaying dark energy and the created matter is stated. By assuming that the initial matter were created is in relativistic form, we have analyzed the possibility of evolving the universe from de Sitter phase to Friedmann universe. Keywords: dark energy, Friedmann Universe, cosmology ## 1 Introduction Recent Astrophysical data have shown that the present universe is undergoing an accelerated expansion [1]. This shows that the present universe is dominated by some kind of very smooth form of energy with negative pressure has been called dark energy, which accounts for about 75$\%$ of the total energy density of the present universe. Various models have been proposed to explain this phenomenon, for example there are models based on the dynamics of scalar or multi-scalar filed, called quintessence models [2]. Another dark energy candidate is the cosmological constant, which was initially introduced by Einstein. In the cosmological constant model, the dark energy density, $\rho_{\Lambda}$, remains constant throughout the entire history of the universe, while the matter density decreases during the expansion. The equation of the state for cosmological constant as dark energy is $w=p/\rho_{\Lambda}=-1$. While in Phantom models [7], it is possible to have an equation of state with $w<-1.$ An alternative approach to the dark energy problem arises from the holographic principle. According to the principle of holography the number of degrees of freedom in a bounded system should be finite and has relations with area of its boundary. By applying the principle to cosmology, one can obtain the upper bound of the entropy contained in the universe. For a system with size $L$ and UV cut-off $\Lambda$, without decaying into a black hole, it is required that the total energy in a region of size $L$ should not exceed the mass of a black hole of the same size, thus $L^{3}\rho_{\Lambda}\leq LM_{P}^{2}$ . The largest L allowed is the one saturating this inequality, thus $\rho_{\Lambda}=3c^{2}M_{P}^{2}L^{-2}$ (1) where $c$ is numerical constant having value close to one, we will take it as one in our analysis. and $M_{P}$ is the reduced Planck Mass $M_{P}^{-2}=8\pi G.$ When we take the whole universe into account,the vacuum energy related to this holographic principle can be viewed as dark energy. $L$ can be taken as the large scale of the universe, for example Hubble horizon, future event horizon or particle horizon which were discussed by many [5, 6, 7, 8, 9]. In this paper we assume a decaying cosmological term. We also assume that the universe is started in de-Sitter phase. While in the de-Sitter phase the universe is completely dominated with the cosmological term. As the universe expands the dynamical cosmological term decaying in to matter and the universe will subsequently enter the Friedman phase. As it expands further, the universe enter a matter dominated phase with decelerated expansion. In section two we have shown that the decaying cosmological dark energy and created matter are consistent with the Einstein’s equation. In section 3, we have obtained the Friedmann equations for the decaying dark energy, and analyzed the possibility of the evolution of the universe in to the Friedmann phase. We have also obtained the time evolution of the decaying dark energy and its equation of state. In section 4, we have presented a comprehensive discussion of our analysis. ## 2 Dynamical dark energy and horizon In the presence of cosmological constant the Einstein’s field equation is $G^{\mu\nu}=R^{\mu\nu}-\frac{1}{2}Rg^{\mu\nu}={8\pi G\over c^{4}}T^{\mu\nu}_{total}$ (2) where $G^{\mu\nu}$ is the Einstein tensor, $R^{\mu\nu}$ is Ricci tensor, $R$ is the Ricci scalar (except in this equation, we will refer $R$ as the scale factor of the expanding universe)and $T^{\mu\nu}_{total}$ is the total energy momentum tensor comprising matter and cosmological term, and is $T^{\mu\nu}_{total}=T^{\mu\nu}+\rho_{\Lambda}g^{\mu\nu}$ (3) in which $T^{\mu\nu}$ is the energy momentum tensor due to matter in perfect fluid form and $\rho_{\Lambda}$ is the density due to cosmological term, given as, $\rho_{\Lambda}={c^{4}\Lambda\over 8\pi G}$ (4) with $\Lambda$ as the so called “cosmological constant”. Einstein’s equation satisfies the covarient conservation condition, $\nabla_{\mu}G^{\mu\nu}=0$ (5) In the conventional case this implies that, $\nabla_{\mu}T^{\mu\nu}=0.$ As such this condition doesn’t give any time conserved charge. If the matter is being created form an independent source, say form the cosmological term, the conservation law will then take the general form $\nabla_{\mu}\left(T^{\mu\nu}+\rho_{\Lambda}g^{\mu\nu}\right)=0$ (6) This conservation law implies that the energy and momentum of matter alone is not conserved, bur energy and momentum of matter and cosmological term or dark energy are together be conserved. This general conservation law allows the exchange of energy and momentum between matter and dark energy and acting as a controlling condition for this exchange. The existing theories predicts a very large value for the cosmological term [3, 4] in the early stage of the universe, but the present observations points towards a very low value for the cosmological term for the late universe. In this light it is inevitable to consider that, there must be a transference of energy form the dark energy or cosmological term sector to the matter sector. Let us assume that, the term $\Lambda$ correspondingly $\rho_{\Lambda}$ is a function of time, since a space dependent $\Lambda$ will lead to an anisotropic universe. The covarient conservation law will then give the equations, $\nabla_{\mu}T^{\mu i}=0$ (7) and $\nabla_{\mu}T^{\mu 0}=-{c^{3}\over 8\pi G}{d\Lambda\over dt}$ (8) where $i=1,2,3$ for the spatial part and $i=0$ for the time part. In reference [10] authors have considered the energy transference between decaying cosmological term and matter. It is important to realise that the covarient conservation law given above is drastically different from that appearing in in some quintessence model [11, 12, 13], where energy-momentum tensor of the scalar field that replaces the cosmological term is itself covariently conserved, but no matter creation. In the present paper we have considered that the cosmological term decaying into matter which is consistent with the Friedmann model of the universe. The energy density $\rho_{\Lambda}$ corresponds to the time varying cosmological term is taken as the holographic dark energy as defined in equation (1). A simple holographic dark energy model is by taking $L=H^{-1}$, where $H$ is the Hubble’s constant is considered by Hsu et al [5] and they have shown that the Friedmann model with $\rho_{\Lambda}=3c^{2}M^{2}_{p}H^{2}$ makes the dark energy behave like ordinary matter rather than a negative pressure fluid, and prohibits accelerating expansion of the universe. We have adopt an equation for holographic dark energy energy, where the future event horizon $(R_{h})$ is used instead of the Hubble horizon as the IR cut-off L, which was shown to lead an accelerating universe by Li [14]. Thus the time varying cosmological energy density is $\rho_{\Lambda}=3c^{2}M^{2}_{p}R_{h}^{-2}$ (9) where $c$ is a constant have values $O(1)$ and the event horizon $R_{h}(t)$, a function of cosmological time, is given by $R_{h}(t)=R(t)\int^{\infty}_{t}{dR(t^{{}^{\prime}})\over H(t^{{}^{\prime}})R(t^{{}^{\prime}})^{2}}$ (10) where $R(t)$ is the expansion factor and $H(t)$ is the Hubble constant. ## 3 Cosmic evolution of dark energy and Friedmann Universe Let us consider an empty universe in de Sitter phase, with very large [15] decaying cosmological term, corresponds to dark energy density as given by equation (LABEL:eq:rho12). If the matter and energy are created from the decaying dark energy term are homogeneous and isotropic, then the geometry of the universe can be that of Friedmann-Robertson-Walker form, $ds^{2}=c^{2}dt^{2}-R^{2}\left[{dr^{2}\over 1-kr^{2}}+r^{2}\left(d\theta^{2}+\sin^{2}\theta\,d\phi^{2}\right)\right]$ (11) where $k$ is the curvature parameter $R$ is the scale factor of expansion, $t$ is the cosmological time and $(r,\theta,\phi)$ are the co-moving coordinates. By taking the energy momentum tensor of matter as $T^{\mu}_{\nu}=(\rho+p)u^{\mu}u_{\nu}-p\delta^{\mu}_{\nu}$ (12) where $\rho$ is the sum of energy densities of the created components of due to the decay of cosmological term and $p$ is the pressure of the matter components. Under these conditions, the covarient conservation law (8) leads to (here we consider only one component of matter) ${d\rho_{m}\over dt}+3H\left(\rho_{m}+p_{m}\right)=-{d\rho_{\Lambda}\over dt}$ (13) where $H={dR\over dt}/R$ the Hubble parameter, $\rho_{m}$ is the density of the created matter and $p_{m}$ is its pressure. This equation obtained form the general conservation law is found to followed from the combinations the standard Friedmann equations, $\left({dR\over dt}\right)^{2}={8\pi G\over 3c^{2}}\left(\rho_{m}+\rho_{\Lambda}\right)R^{2}-kR^{2}$ (14) and ${d^{2}R\over dt^{2}}={8\pi G\over 3c^{2}}\left(\rho_{\Lambda}-\frac{1}{2}\left(\rho_{m}+3p_{m}\right)\right)R$ (15) provided the cosmological term $\rho_{\Lambda}$ is time dependent. If one assumes ordinary pressureless matter as $\rho_{m}=\rho_{m0}R^{-3}$ (16) where $\rho_{m0}$ is the present density of matter. then equation (13) will lead to the result that, the cosmological term will be independent of time. On the other hand this shows that the time dependent cosmological term does not decay in to pressureless matter. Let us assume that the cosmological term can possibly decay into some form of matter with equation of state $p_{m}=\omega_{m}\rho_{m},$ where the parameter $\omega_{m}$ is assumed to be in the range $0\leq\omega_{m}\leq 1,$ the exact value is depends on particular matter component. In this paper we are considering only one component of matter. The covarient conservation law (8) can now be written for the possibility of cosmological term decaying into matter as ${d\rho_{m}\over dt}+3H\left(1+\omega_{m}\right)\rho_{m}=-{d\rho_{\Lambda}\over dt}$ (17) Since density behaviour of ordinary pressureless matter does not work for a varying cosmological dark energy, we will assume the form for $\rho_{m}$ which is slightly different from its canonical form, as [16, 17] $\rho_{m}=\rho_{m0}R^{-3+\delta}$ (18) where $\rho_{m0}$ is the present value of $\rho_{m}$ and $\delta$ is a parameter which is effectively depends upon the state of the universe. From equation (18) and (LABEL:eq:rho12), equation (17) become, $\rho_{\Lambda}\left({HR_{h}-1\over HR_{h}}\right)=\left({3\omega_{m}+\delta\over 2}\right)\rho_{m}$ (19) where we have assumed a vanishing integration constant and also with the condition that lim t$\rightarrow\infty$, $R(t)\rightarrow\infty$ and equation for time rate of horizon can be cast into the diﬀerential form ${dR_{h}\over dt}=HR_{h}-1.$ (20) The equation (19) suggest that depending on the parameters $\delta$ and $\omega_{m}$ the energy densities $\rho_{m}$ and $\rho_{\Lambda}$ may eventually be of the same order, as suggested by the present observations [18]. This relation rather suggest the relation between dark energy and matter, when $\delta=3$, the case corresponds to a constant matter density at which there exist a equilibrium between matter creation and universe expansion In general event horizon is not existing for Friedmann universe. But for de Sitter universe there exists event horizon, which satisfies the relation $R_{h}\sim H^{-1}.$ Consequently for de Sitter universe, $HR_{h}\sim 1$ which implies that in de Sitter phase the energy density is almost completely dominated by the cosmological term or dark energy. For Friedmann universe we will hence take $HR_{h}$ as very large In general we will take the value of $HR_{h}$ is equal to one or large. ### 3.1 Friedmann Universe Friedmann universe is a homogeneous and istropic universe, satisfying the conditions (14) and (15). With the relation between decaying cosmological term and created matter (19), the second Firedmann equation become, ${d^{2}R\over dt^{2}}={(\left(1+3\omega_{m}\right)\beta^{2}\over 2}\left[{3\omega_{m}+\delta\over 1+3\omega_{m}}\left({HR_{h}\over HR_{h}-1}\right)-1\right]R^{\delta-2}$ (21) where $\beta^{2}={8\pi G\rho_{mo}/3c^{2}}$ . For de Sitter phase the acceleration is very large, for which $HR_{h}\sim 1.$ As it enters the Friedmann phase by the decay of cosmological or the dark energy, the acceleration can be negative or positive, depends on the value of the term in the parenthesis of the right hand side of the above equation. The condition for acceleration is, ${3\omega_{m}+\delta\over 3\omega_{m}+1}>1-{1\over HR_{h}}$ (22) The factor $\displaystyle{1-{1\over HR_{h}}}$ is in the range $\displaystyle 0\leq{1-{1\over HR_{h}}}\leq 1.$ The extreme limits are corresponds to de Sitter phase and matter dominated universes respectively. For the transition period from de Sitter phase to Friedmann phase, $HR_{h}$ is near to one, and assuming that the created matter have the equation state $\omega_{m}=\frac{1}{3}$, where the created matter is in relativistic form then $\delta>2\alpha-1$ (23) where $\displaystyle\alpha=1-{1\over HR_{h}}$ having value less than one. This implies that during the period of decay of the cosmological dark energy term the density of created matter is diluted slowly as the universe expand, than the decreasing of the density of non-relativistic matter in the Friedmann universe. This shows that even in the friedmann universe it is possible to have an initial accelerating phase, where the cosmological dark energy is start its decay into matter and is still dominating over matter. As universe proceeds, the created matter will subsequently dominate and hence the universe will come to a matter dominated phase, at which the universe is expanding with deceleration. For decelerating expansion, where matter is dominating over the cosmological term, the condition in the limit where $HR_{h}$ is very large is, $\delta<1$ (24) This condition is true irrespective of whether the created matter is relativistic or non-relativistic. However as the universe enter the decelerating phase, the matter will become non-relativistic, satisfying the extreme condition that $\delta\rightarrow 0$ so that $\rho_{m}\sim R^{-3}$ ### 3.2 Flat Universe For flat universe, where the curvature parameter $k=0,$ the Friedmann equation (LABEL:eq:frw1) become $\left({dR\over dt}\right)^{2}=\beta^{2}\left({3\omega_{m}+\delta\over 2}{HR_{h}\over HR_{h}-1}+1\right)R^{\delta-1}$ (25) On integration and avoiding the integration constant, the solution would be, $R\sim t^{2\over 3-\delta}$ (26) By considering the relation for matter creating out of decaying dark energy, i.e. $\rho_{m}=\rho_{m0}R^{-3+\delta},$ then the relation between dark energy density and matter will have beheaviour $\rho_{m}\sim\rho_{\Lambda}\sim t^{-2}$ (27) This is the time dependence of $\rho_{\Lambda}$ for any value of $\omega_{m}$ and $\delta.$ This time dependence shows that $\rho_{\Lambda}$ diverge at the initial time, which implies the existence of initial singularity. ### 3.3 An equation of state for the decaying dark energy An equation of state for the time decaying cosmological term can be written as [19] $\omega_{\Lambda}^{eff}=-1-{1\over 3}\left({dln\rho_{\Lambda}\over d\ln R}\right)$ (28) With the equation for the relation between decaying dark energy and creating matter [fang], the equation of state become $\omega_{\Lambda}^{eff}=-{\delta\over 3}-{1\over 3\left(HR_{h}-1\right)^{2}}$ (29) This shows that for large values of $HR_{h}$ the equation of sate become $\omega_{\Lambda}=-{\delta\over 3}$. From the above analysis, it is seen that for an accelerating universe $\omega_{\Lambda}$ is less than $-{1\over 3},$ but for a decelerating Friedmann universe it is greater than $-{1\over 3},$ which is similar to the latest analysis by many. ## 4 Discussion In the presence of a time varying cosmological term, assumed to be of holographic dark energy form, it is possible that the universe may starts with the de Sitter phase, exhibiting horizon and where the energy density is completely dominated by the dark energy. The decaying holographic dark energy cause the primordial inflation. If the horizon $R_{h}$ is assumed to be equal to the plank length at very early stage, then $\Lambda$ would have a value of the order $\Lambda\sim 10^{66}\,cm^{-2}$ (30) the corresponding dark energy density would be $\rho_{\Lambda}\sim 10^{112}\,ergcm^{-3}.$ This enormous dark energy decay into matter as the universe is evolved to the Friedmann phase, and the dark energy reached the present value $\rho_{\Lambda}^{0}=10^{-8}\,erg\,cm^{-3}$ (31) The evolution of the de Sitter phase in to the Friedmann universe is in such a way that the total energy density comprising the dark energy, created matter and the gravitational field together be conserved. In this paper we have considered the decay of the dark energy into matter. During the initial phase of decay, the universe might be in the accelerating phase, where the parameter $\delta$ characterizing the equation $\rho_{m}\sim R^{-3+\delta},$ is greater than one. This implies that the dilution in the density of created matter in slower compared to the non-relativistic matter. As the universe proceeds with expansion, the matter created will come to dominate, and the universe eventually go over to matter dominated phase, with the condition $\delta<1.$ In the extreme limit this condition may go the limit $\delta\rightarrow 0$, which emphasises that, the created matted will eventually become non- relativistic, with behaviour, $\rho_{m}\sim R^{-3}.$ In section 3.3, the equation of the state of the decaying holographic dark energy, shows that, in the early phase of universe too, $\omega_{\Lambda}<-{1\over 3}$ as it’s equation of state in late accelerating universe [20]. To explain why $\rho_{\Lambda}$ and $\rho_{m}$ are of the same order today, it is essential to have a specific time evolution for dark energy. We argued that a dynamical dark energy, endowed with an appropriate time evolution can contain the possibility of the development of a Friedmann universe from a de Sitter universe. As the de Sitter phase evolved in to the Friedmann universe, the value of the dark energy is decreased gradually to a low value, which eventually lead to matter dominating phase with decelerating expansion. But on the other hand the recent observations indicating that the present universe is in a accelerating expansion. This fact indicating the possibility that at present the dark energy is increasing at the expense of decaying matter. This time increasing dark energy, in other words, implies that the universe may evolving in to stage where the whole energy density is coming to dominate completely by the dark energy. The ultimate clarity regarding these, of course be given by the proper quantum effects, which is still an open question. ## References * [1] Perlmutter S et al., Measurements of Omega and Lambda form 42 High-redshift supernovae, Astrophys. J. 517 (1999) 565 Knop R et al , New Constraints on $\Omega_{M}\,\Omega_{\Lambda},$ and $\omega$ from an independent Set of Eleven Hogh-Redshift Supernovae observed with Astrophys. J. 598 (2003) 102 [SPIRES] Riess A G et al., Type Ia Supernovae Discoveries at $Z>1$ From the Hubble Space telescope Evidence of Past Deceleration and Constraints on Dark Energy Evolution Astrophys. J. 607 (2004)665 [SPIRES] Jassal H, Bagla J and Padmanabhan T, Observational Constraints on Low Redshift Evolution of Dark Energy: How consistent are Diffrent Observations Phys. Rev. D 72 (2005) 103503 [SPIRES] * [2] Ratra B and Peebles P J E, Cosmological Consequences of a Rolling Homogeneous Scalar field Phys. Rev. D 37 (1998) 3406 [SPIRES] Zlatev I, Wang L and Steinhardt P J, Quintessence, Cosmic Coincidence and Cosmological Constant Phys. Rev. Lett. 82 (1999) 896 [SPIRES] Steinhardt P J, Wang L and Zlatev I, Cosmological Tracking Solutions 1999 Phys. Rev. D 59 (1999) 123504 [SPIRES] * [3] Weinberg S, Rev. Mod. Phys. The Cosmological Constant Problem, 61 (1989) 1 * [4] A. Linde, Particle Physics and Inflationar Cosmology, Harwood, New York, 1990. * [5] S.D.H. Hsu, Entropy bounds and dark energy, Phys. Lett. B 594 (2004) 13 [hep-th/0403052] [SPIRES]. * [6] M. Li, A model of holographic dark energy, Phys. Lett. B 603 (2004) 1 [hep-th/0403127] [SPIRES]; * [7] Q.-G. Huang and M. Li, Anthropic principle favors the holographic dark energy, JCAP 03 (2005) 001 [hep-th/0410095] [SPIRES]; Q. -G. Huang and M. Li, The holographic dark energy in a non-ﬂat universe, JCAP 08 (2004) 013 [astro-ph/0404229] [SPIRES]; B. Chen, M. Li and Y. Wang, Inﬂation with Holographic Dark Energy, Nucl. Phys. B 774 (2007) 256 [astro-ph/0611623] [SPIRES]; J.-f. Zhang, X. Zhang and H.-y. Liu, Holographic dark energy in a cyclic universe, Eur. Phys. J. C 52 (2007) 693 [arXiv:0708.3121] [SPIRES] S. Nojiri and S.D. Odintsov, Unifying phantom inﬂation with late-time acceleration: Scalar phantom-non-phantom transition model and generalized holographic dark energy, Gen. Rel. Grav. 38 (2006) 1285 [hep-th/0506212] [SPIRES]. * [8] P. Hoˇava and D. Minic, Probable values of the cosmological constant in a holographic theory, Phys. Rev. Lett. 85 (2000) 1610 [hep-th/0001145] [SPIRES]. * [9] R. Horvat, Holography and variable cosmological constant, Phys. Rev. D 70 (2004) 087301 [astro-ph/0404204] [SPIRES]. * [10] R.Aldrovandi, J.P.Beltran Almeida and J.G.Pereira, Time-Varying Cosmological Term: Emergence and Fate of a FRW Universe, Grav. Cosmol. 11 (2005) 277, arXiV:gr-qc/0312017v3 12 Apr 2005. * [11] B.Ratra and P.J.E.Peebles, Cosmological Consequences of a Rolling Homogeneous Scalar field Phys. Rev. D 37 (1998) 3406 [SPIRES] * [12] J.Friman, C.T.Hill and R.Watkins, Late Time Cosmological Phase Transitions,1. Particle Physics Models and Cosmic Evolution Phys. Rev D 46 (1992) 1226 * [13] V.Sahni and A.A.Starobisky, The Case for a Positive Cosmological Lambda-term Int. J. Mod. Phys. D9 (2000) 373. * [14] Huang Q-G and Li M, Holographic Dark Energy in Non-Flat Universe 2004 J. Cosmol. Astropart. Phys. JCAP 0408 (2004) 013 [SPIRES] * [15] B.Mashhoon and P.S.Wesson, Gauge Dependent Cosmological Constant Class. Quant. Grav. 21, (2004) 3611 * [16] B.Guberina, R.Horvat and Hrvoje Nikolic, Non-Structured Holographic Dark Energy, JCAP 0701 (2007) 012 * [17] B.Guberina, R.Horvat and Hrvoje Nikolic, Dynamical Dark Energy with a constant vacuum energy density, Phys. Lett.B 636: (2006) 80-85 * [18] S M Carrol, why is the universe Accelerating, in - Measuring and modelling the universe. ed. by W.L.Freeman, Cambridge University Press, Cambridge, 2003 [astro-ph/0310342] * [19] C J Fang, Holographic Cosmological Constant and Dark Energy, Phys. Lett. B663 (2008) 367. * [20] E J Copeland, M Sami, S Tsujikawa, Dynamics of Dark Energy, Int. J. Mod. Phys. D15 (2006) 1753 * [21] S.Weinberg, Gravitation and Cosmology, Wiley, New York, 1972	arxiv-papers	2011-12-07T05:18:22	2024-09-04T02:49:25.047143	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Titus K. Mathew", "submitter": "Titus K Mathew", "url": "https://arxiv.org/abs/1112.1472" }
1112.1482	# Omnidirectionally Bending to the Normal in $\epsilon$-near-Zero Metamaterials Simin Feng simin.feng@navy.mil Michelson Lab, Physics Division, Naval Air Warfare Center, China Lake, California 93555 ###### Abstract Contrary to conventional wisdom that light bends away from the normal at the interface when it passes from high to low refractive index media, here we demonstrate an exotic phenomenon that the direction of electromagnetic power bends towards the normal when light is incident from arbitrary high refractive index medium to $\epsilon$-near-zero metamaterial. Moreover, the direction of the transmitted beam is close to the normal for all angles of incidence. In other words, the electromagnetic power coming from different directions in air or arbitrary high refractive index medium can be redirected to the direction almost parallel to the normal upon entering the $\epsilon$-near-zero metamaterial. This phenomenon is counterintuitive to the behavior described by conventional Snell’s law and resulted from the interplay between $\epsilon$-near-zero and material loss. This property has potential applications in communications to increase acceptance angle and energy delivery without using optical lenses and mechanical gimbals. ###### pacs: 42.25.Bs, 42.79.Wc, 78.67.Pt Bending of light towards the normal when it passes from low to high refractive index media is one of the fundamental phenomena in optics. As a manifestation of this phenomenon, directive emission into air by a source inside the material with vanishingly small permittivity, known as $\epsilon$-near-zero (ENZ) metamaterials, has been demonstrated Enoch . With other intriguing properties, such as ultrathin waveguides Silveir ; Liu ; Edwards ; Adams ; Alu1 , diffraction-suppressed propagation and self-collimation Feng ; Mocella1 ; Mocella2 ; Polles , the ENZ materials have gained prominence as useful components to tailor antenna radiations Alu2 ; Halterman ; Saenz ; Mart . Previous studies on ENZ-directive emission have been focused on the radiation from low ($\epsilon\approx 0$) to high (air) refractive index media Enoch ; Ziolkow ; Yuan ; Jin ; Lovat , where the directive transmission can be intuitively understood from Snell’s law that dictates the light bending towards the normal as it passes from low to high refractive index media. From the reciprocal theorem, for the radiation from high to low refractive index materials, the transmitted beam should spread out into grazing angles as the result of bending away from the normal. Contrary to this conventional behavior, in this paper we will demonstrate an exotic phenomenon that electromagnetic (EM) power can bend towards the normal when light passes from arbitrary high ($\epsilon_{1}\gg 1$) to low ($\epsilon_{2}\approx 0$) refractive index media as shown in Fig. 1a. Furthermore, the direction of the transmitted beam is close to the normal for all angles of incidence. More interestingly, this counterintuitive to conventional Snell’s law behavior is induced by material loss. The interplay between ENZ and loss leads to unusual wave interaction. This phenomenon can be used to project EM power coming from different directions to one direction to the receivers as shown in Fig. 1b, where the incoming waves all bend to the normal pointing to the receptors upon entering the ENZ medium. A plasmonic thin film is superimposed on the ENZ material to enhance the transmission through structural resonances. For all the incoming directions including grazing angles, the transmitted powers impinge normally to the receptors or photocells embedded in the ENZ medium, effectively increasing the acceptance angle and energy transfer. Figure 1: (Color online) (a) A plane wave is incident from arbitrary high permittivity ($\epsilon_{1}\gg 1$) medium to ENZ ($\epsilon_{2}\approx 0$) metamaterial. (b) Incoming waves in air from different directions all bend to the normal upon entering the ENZ medium. A nanoplasmonic thin film is superimposed on the ENZ material to enhance the transmission. Receptors or photocells are embedded in the ENZ metamaterial. Our derivation is based on anisotropic media. The results can be applied to isotropic materials. Assuming a harmonic time dependence $\exp(-i\omega t)$ for the EM field, from Maxwell’s equations, we have $\displaystyle\begin{split}\nabla\times\bigl{(}\bar{\bar{\mu}}_{n}^{-1}\cdot\nabla\times{\bm{E}}\bigr{)}&=\,k_{0}^{2}\bigl{(}\bar{\bar{\epsilon}}_{n}\cdot{\bm{E}}\bigr{)}\,,\\\ \nabla\times\bigl{(}\bar{\bar{\epsilon}}_{n}^{-1}\cdot\nabla\times{\bm{H}}\bigr{)}&=\,k_{0}^{2}\bigl{(}\bar{\bar{\mu}}_{n}\cdot{\bm{H}}\bigr{)}\,,\end{split}$ (1) where $k_{0}=\omega/c$; and the $\bar{\bar{\epsilon}}_{n}$ and $\bar{\bar{\mu}}_{n}$ are, respectively, the permittivity and permeability tensors for each uniform region ($n=1,2,\cdots$), which in the principal coordinates can be described by $\displaystyle\bar{\bar{\epsilon}}_{n}$ $\displaystyle=$ $\displaystyle\epsilon_{nx}\hat{\bm{x}}\hat{\bm{x}}+\epsilon_{ny}\hat{\bm{y}}\hat{\bm{y}}+\epsilon_{nz}\hat{\bm{z}}\hat{\bm{z}}\,,$ (2) $\displaystyle\bar{\bar{\mu}}_{n}$ $\displaystyle=$ $\displaystyle\mu_{nx}\hat{\bm{x}}\hat{\bm{x}}+\mu_{ny}\hat{\bm{y}}\hat{\bm{y}}+\mu_{nz}\hat{\bm{z}}\hat{\bm{z}}\,.$ (3) Consider transverse magnetic (TM) modes, corresponding to non-zero field components $H_{y}$, $E_{x}$, and $E_{z}$. The magnetic field $H_{y}$ satisfies the following wave equation: $\frac{1}{\epsilon_{z}}\frac{\partial^{2}H_{y}}{\partial x^{2}}+\frac{1}{\epsilon_{x}}\frac{\partial^{2}H_{y}}{\partial z^{2}}+k_{0}^{2}\mu_{y}H_{y}=0\,,$ (4) which permits solutions of the form $\psi(z)\exp(i\beta x)$. Here the transverse wave number $\beta$ is determined by the incident wave, and is conserved across the interface of different regions, $\beta^{2}=k_{0}^{2}\epsilon_{nz}\mu_{ny}-\alpha_{n}^{2}\frac{\epsilon_{nz}}{\epsilon_{nx}}\,,\hskip 14.45377pt(n=1,2,\cdots)\,,$ (5) where $\alpha_{n}$ is the wave number in the $z$ direction. The functional form of $\psi(z)$ is either a simple exponential $\exp(i\alpha_{n}z)$ for the semi-infinite regions or a superposition of $\cos(\alpha_{n}z)$ and $\sin(\alpha_{n}z)$ terms for the bounded regions along the $z$ direction. The other two components $E_{x}$ and $E_{z}$ can be solved from $H_{y}$ using Maxwell’s equations. By matching boundary conditions at the interfaces, i.e., the continuity of $H_{y}$ and $E_{x}$, the electromagnetic field can be derived in each region; and then the Poynting vector $\bm{S}$ can be computed from $\bm{S}=\Re(\bm{E}\times\bm{H}^{})$. In anisotropic materials, the direction of the Poynting vector is different from that of the phase front of the field. Here, only the direction of the Poynting vector is considered since it is associated with the energy transport. The angle ($\theta_{S}$) of the Poynting vector is measured from the Poynting vector to the surface normal, and is given by $\theta_{S}=\tan^{-1}(S_{x}/S_{z})$. In Fig. 1a, the input medium is isotropic material with permittivity $\epsilon_{1}$; the output medium is ENZ material ($\epsilon_{2}\approx 0$). In the following, both anisotropic ($\epsilon_{2x}\neq\epsilon_{2z}$) and isotropic ($\epsilon_{2x}=\epsilon_{2z}$) ENZ materials will be considered. Figure 2 illustrates the effect of loss of the ENZ-materials on the transmission angle (TA), which is plotted against angle of incidence (AOI) with and without loss for the different permittivity ($\epsilon_{1}$) of the input medium. In the top panels, when the loss is zero $\bigl{(}\Im(\epsilon_{2z})=0\bigr{)}$, the transmission angle is the grazing angle $90^{\circ}$ except for the normal incidence. This behavior is complied with conventional Snell’s law. In the bottom panels, with a moderate loss $\Im(\epsilon_{2z})=0.6$, the transmission angle switches to near zero (normal direction) for all angles of incidence, leading to collimated transmission in the normal direction. This switching phenomenon persists even for the much higher permittivity ($\epsilon_{1}=100$) of the input medium (middle and right panels). Figure 2: (Color online) Transmission angle of the Poynting vector versus AOI. Left and middle panels: anisotropic ENZ material with $\epsilon_{2x}=1$ and $\epsilon_{2z}=0.001+i\epsilon_{2z}^{i}$. Right panels: isotropic ENZ material with $\epsilon_{2x}=\epsilon_{2z}=0.001+i\epsilon_{2z}^{i}$. Top panels: $\epsilon_{2z}^{i}=0$. Bottom panels: $\epsilon_{2z}^{i}=0.6$. Left panels: $\epsilon_{1}=1$. Middle and right panels: $\epsilon_{1}=100$. A good agreement between the numerical (blue-solid) and analytical (green-circles) results. The material loss switches the TA from the grazing angle $90^{\circ}$ (top panels) to the near-zero angle (bottom panels) for all the AOI. To understand this loss-induced switching behavior, let’s analyze the transmission angle ($\theta_{S}$), which is given by $\tan(\theta_{S})=\frac{S_{x}}{S_{z}}=\frac{\Re\left(\dfrac{\bar{\beta}}{\epsilon_{2z}}\right)}{\Re\left[\sqrt{\dfrac{\mu_{2y}}{\epsilon_{2x}}-\dfrac{\bigl{(}\bar{\beta}\bigr{)}^{2}}{\epsilon_{2x}\epsilon_{2z}}}\right]}\,,$ (6) where $\bar{\beta}\equiv\beta/k_{0}$, and $\bar{\beta}$ (real) is determined by the incidence angle. The transmission angle of the Poynting vector depends only on the input and output media. In the case of $\epsilon_{2x}\rightarrow 0$ and $\epsilon_{2z}$ finite, Eq. (6) indicates $\theta_{S}\rightarrow 0^{\circ}$ (normal direction). For the case of $\epsilon_{2z}\rightarrow 0$ and $\epsilon_{2x}$ finite and the case of isotropic ENZ material with $\epsilon_{2x}=\epsilon_{2z}\rightarrow 0$, the analysis is more involved. The numerator of Eq. (6) can be written as $\Re\left(\frac{\bar{\beta}}{\epsilon_{2z}}\right)=\frac{\bar{\beta}\,\epsilon_{2z}^{r}}{\|\epsilon_{2z}\|^{2}}\,,$ (7) where $\epsilon_{2z}^{r}\equiv\Re(\epsilon_{2z})$. Assuming $\mu_{2y}$ is real, the denominator of Eq. (6) becomes $\Re\left[\sqrt{\dfrac{\mu_{2y}}{\epsilon_{2x}}-\dfrac{\bigl{(}\bar{\beta}\bigr{)}^{2}}{\epsilon_{2x}\epsilon_{2z}}}\right]=\frac{a\,\bar{\beta}}{\|\epsilon_{2x}\epsilon_{2z}\|}\,,$ (8) where $a^{2}=\frac{1}{2}\left(A\epsilon_{2x}^{r}+B\|\epsilon_{2x}\|-\epsilon_{2x}^{r}\epsilon_{2z}^{r}+\epsilon_{2x}^{i}\epsilon_{2z}^{i}\right)\,,$ (9) where $\epsilon_{2z}^{i}\equiv\Im(\epsilon_{2z})$, $\epsilon_{2x}^{r}\equiv\Re(\epsilon_{2x})$, $\epsilon_{2x}^{i}\equiv\Im(\epsilon_{2x})$, and $A\equiv\frac{\|\epsilon_{2z}\|^{2}\mu_{2y}}{\bigl{(}\bar{\beta}\bigr{)}^{2}}\,,\hskip 8.67204ptB=\sqrt{\|\epsilon_{2z}\|^{2}-2A\epsilon_{2z}^{r}+A^{2}}\,.$ (10) Thus, the transmission angle ($\theta_{S}$) becomes $\tan(\theta_{S})=\frac{\|\epsilon_{2x}\|\epsilon_{2z}^{r}}{a\,\|\epsilon_{2z}\|}\,.$ (11) The loss-induced angle switching observed in Fig. 2 can be explained from Eq. (11). For the anisotropic material with $\epsilon_{2x}\neq\epsilon_{2z}$ and finite $\epsilon_{2x}$, if $\epsilon_{2z}^{i}=0$, when $\epsilon_{2z}^{r}\rightarrow 0$, $\epsilon_{2z}^{r}/\|\epsilon_{2z}\|\rightarrow 1$ and $a\rightarrow 0$, thus $\theta_{S}\rightarrow 90^{\circ}$. If $\epsilon_{2z}^{i}\neq 0$, when $\epsilon_{2z}^{r}\rightarrow 0$, $\epsilon_{2z}^{r}/\|\epsilon_{2z}\|\rightarrow 0$ and $a$ is finite, thus $\theta_{S}\rightarrow 0^{\circ}$. On the other hand, if $\epsilon_{2z}$ is finite, when $\epsilon_{2x}\rightarrow 0$, $a\rightarrow\sqrt{\epsilon_{2x}}$, thus $\theta_{S}\rightarrow 0^{\circ}$. For the isotropic case, let $\epsilon_{2x}=\epsilon_{2z}\equiv\epsilon_{2}^{r}+i\epsilon_{2}^{i}$. If $\epsilon_{2}^{i}=0$, when $\epsilon_{2}^{r}\rightarrow 0$, $\epsilon_{2z}^{r}/\|\epsilon_{2z}\|\rightarrow 1$ and $a\rightarrow(\epsilon_{2}^{r})^{3/2}$, thus $\theta_{S}\rightarrow 90^{\circ}$. If $\epsilon_{2}^{i}\neq 0$, when $\epsilon_{2}^{r}\rightarrow 0$, $\epsilon_{2z}^{r}/\|\epsilon_{2z}\|\rightarrow 0$ and $a$ is finite, therefore $\theta_{S}\rightarrow 0^{\circ}$. To validate Eq. (11), in Fig. 2 the TAs calculated from Eq. (11) (green-circles) are compared to those computed numerically (blue-solid), showing a perfect agreement. To validate the loss-induced switching behavior is a robust feature, in Fig. 3 the transmission angle versus AOI is plotted for the different real parts of $\epsilon_{2z}$ and $\epsilon_{2x}$ and the material loss. In essence, the transmission angle decreases with increasing the loss $\Im(\epsilon_{2z})$ and decreasing the $\Re(\epsilon_{2z})$. When $\Re(\epsilon_{2z})\rightarrow 0$, the angular width of the transmission can be estimated from $\Delta\theta_{S}\approx\left\\{\begin{matrix}\dfrac{\sqrt{2}\,\,\|\epsilon_{x}\|\,\epsilon_{z}^{r}}{\|\epsilon_{z}\|^{3/2}\sqrt{\|\epsilon_{x}\|+\epsilon_{x}^{i}+\eta\,\epsilon_{x}^{r}}}\,,&\mbox{if }\eta\leq 1\\\ \\\ \dfrac{\sqrt{2}\,\,\|\epsilon_{x}\|\,\epsilon_{z}^{r}}{\|\epsilon_{z}\|^{3/2}\sqrt{\epsilon_{x}^{i}+\eta\bigl{(}\|\epsilon_{x}\|+\epsilon_{x}^{r}\bigr{)}}}\,,&\mbox{if }\eta\geq 1\end{matrix}\right.\,,$ (12) where $\eta\equiv\dfrac{\|\epsilon_{z}\|\mu_{y}}{\epsilon_{1}\mu_{1}}$, and the subscript $2$ in the $\epsilon_{x}$, $\epsilon_{z}$, and $\mu_{y}$ was omitted in above equation. Figure 3: (Color online) Transmission angle versus AOI when the $\Re(\epsilon_{2z})=0.001$ (blue-solid) and $\Re(\epsilon_{2z})=0.01$ (green- dashed). Top panels: $\Im(\epsilon_{2z})=1.5$. Bottom panels: $\Im(\epsilon_{2z})=3.2$. Left and middle panels: anisotropic ENZ material with $\epsilon_{2x}=1.0$ (left panels) and $\epsilon_{2x}=2.0$ (middle panels). Right panels: isotropic ENZ material. The permittivity of the input medium $\epsilon_{1}=36$. Figure 4: (Color online) Transmission angle versus material loss $\Im(\epsilon_{2z})$ computed for AOI $=0.1^{\circ}$ (blue- solid) and AOI $=89^{\circ}$ (green-dashed). Top panels: $\Re(\epsilon_{2z})=0.001$. Bottom panels: $\Re(\epsilon_{2z})=0.01$. Left and middle panels: anisotropic ENZ material with $\epsilon_{2x}=1.0$ (left panels) and $\epsilon_{2x}=2.0$ (middle panels). Right panels: isotropic ENZ material. The permittivity of the input medium $\epsilon_{1}=36$. TA quickly converges to zero in all the scenarios. Figure 4 demonstrates how rapidly the transmission angle converges to zero as the loss $\Im(\epsilon_{2z})$ increases for the different values of $\Re(\epsilon_{2z})$ and $\epsilon_{2x}$. The blue-solid curves represent the transmission angles calculated for the near-zero angle of incidence, while the green-dashed curves for the grazing angle of incidence. The difference between the green-dashed and blue-solid curves corresponds to the angular width of the transmission. The angular width in the isotropic ENZ medium (right panels) is usually smaller than that in the anisotropic medium (left and middle panels). This is implicated in Eq. (12) as well. It is well-known that loss is inextricable to metamaterial. Many fascinating effects diminish as the result of high loss Dimmock ; Nieto . However, for the effect demonstrated here, the material loss plays a positive role, resulting in the omnidirectional bending of light towards the normal upon entering the ENZ medium. This phenomenon may have many applications, such as directive antennas. Instead of radiation applications, we will explore this phenomenon from a receiving perspective, i.e., redirect the incoming EM power from different directions to the direction of the receivers to enhance the acquisition power, as illustrated in Fig. 1b. To increase the coupling, a matching coating can be deposited on the surface of the ENZ medium such that the effective impedance of the overall structure is matched to the free-space impedance. For simplicity, here a dielectric-metal-dielectric thin film is superimposed on the ENZ material. This sandwich structure possesses coupled surface plasmon modes due to closely spaced two dielectric-metal interfaces. By exciting the plasmonic resonances of the structure, the transmission can be enhanced. The resonant frequency of the transmission can be tuned by changing the thickness of the layers. In our simulation, the materials of the dielectric and metallic layers are, respectively, amorphous polycarbonate (APC) and silver (Ag). The refractive index of the APC is given by Roberts $n_{p}=1.5567+\frac{8.0797\times 10^{-3}}{\lambda^{2}}+\frac{3.5971\times 10^{-4}}{\lambda^{4}}\,,$ (13) where $\lambda$ is the wavelength in $\mu m$. The loss of the APC is very small in the wavelength range of the simulation, and thus is neglected Roberts . The absorption of Ag is included through the complex permittivity given from Palik Palik . Figure 5: (Color online) Top panels: Transmittance (blue-solid) and reflectance (green-dashed) of the APC-Ag-APC thin film versus AOI when the medium after the film is the anisotropic ENZ material with $\epsilon_{2x}=1$ (left panels) or the isotropic ENZ material (right panels). $\epsilon_{2z}=0.001+0.6i$ for both cases. Bottom panels: transmission angle (corresponding to the top panels) versus AOI. The thickness of the APC $d=100\,$nm (left panels) and $d=80\,$nm (right panels). The thickness of Ag is 10 nm for both cases. Figure 6: (Color online) Transmittance of the APC-Ag- APC thin film versus AOI and wavelength when the medium after the film is the anisotropic (left panel) or the isotropic (right panel) ENZ material. Color- bars represent the magnitude of the transmittance. Simulation parameters are the same as those in Fig. 5. Shown in Fig. 5 are the transmission and reflection (top panels) of a plane wave incident from air to the APC-Ag-APC structure, along with the corresponding transmission angle (bottom panels). At the resonance, the thickness of the APC $d=100\,$nm with the resonant wavelength $\lambda=0.95\,\mu$m for the anisotropic ENZ medium (left panels); and the $d=80\,$nm with the $\lambda=0.64\,\mu$m for the isotropic ENZ medium (right panels). About $90\%$ transmittance are achieved for a wide range of the incidence angle up to $70^{\circ}$ (see top panels) with nearly-collimated transmission in the normal direction (see bottom panels). Transmittance of the APC-Ag-APC thin film as a function of AOI and wavelength is presented in Fig. 6 when the medium at the back of the film is the anisotropic (left panel) or the isotropic (right panel) ENZ material. In both cases, wide-angle $90\%$ transmittance are observed. It is worth mentioning that the loss of the ENZ medium does not affect the transmittance of the APC-Ag-APC structure since the transmitted power is computed right after the thin film, i.e., before traveling through the ENZ medium. If the receptors are embedded very close to the back of the film, the propagation loss in the ENZ medium can be minimized. However, the loss of the ENZ material plays an important role on controlling the direction of the transmission, no matter where the receptors are located. In conclusions, we have demonstrated omnidirectionally transmitting the electromagnetic power to one direction in the ENZ materials. This phenomenon is realized based on two mechanisms. One is the loss-assistant bending of the EM power to the normal for all angles of incidence. The other is the enhanced transmission through structural resonances. This phenomenon may have applications in communications, directive antennas, as well as detectors and sensors to increase acceptance angle and redirect electromagnetic power without using optical lenses and mechanical gimbals. The concept of employing metamaterial loss to control the direction of the transmission brings a positive perspective for material loss and may open up a new avenue for metamaterial designs and applications. The author gratefully acknowledge the sponsorship of ONR’s N-STAR and NAVAIR’s ILIR programs. ## References (1) S. Enoch, G. Tayeb, P. Sabouroux, N. Gueŕin, and P. Vincent, Phys. Rev. Lett. 89, 213902 (2002). * (2) M. G. Silveirinha and N. Engheta, Phys. Rev. Lett. 97, 157403 (2006). * (3) R. Liu, Q. Cheng, T. Hand, J. J. Mock, S. A. Cummer, and D. R. Smith, Phys. Rev. Lett. 100, 023903 (2008). * (4) B. Edwards, A. Alù, M. E. Young, M. Silveirinha, and N. Engheta, Phys. Rev. Lett. 100, 033903 (2008). * (5) D. C. Adams, et al., Phys. Rev. Lett. 107, 133901 (2011). * (6) A. Alù and N. Engheta, IEEE Trans. Antennas Propag. 58, 328 (2010). * (7) S. Feng and J. M. Elson, Opt. Express 14, 216 (2006). * (8) V. Mocella, et al., Phys. Rev. Lett. 102, 133902 (2009). * (9) V. Mocella, P. Dardano, I. Rendina, and S. Cabrini, Opt. Express 18, 25068 (2010). * (10) R. Pollès, E. Centeno, J. Arlandis, and A. Moreau, Opt. Express 19, 6149 (2011). * (11) A. Alù, M. G. Silveirinha, A. Salandrino, and N. Engheta, Phys. Rev. B 75, 155410 (2007). * (12) K. Halterman, S. Feng, and V. C. Nguyen, Phys. Rev. B 84, 075162 (2011). * (13) E. Saenz, K. Guven, E. Ozbay, I. Ederra, and R. Gonzalo, Appl. Phys. Lett. 92, 204103 (2008). * (14) A. Martínez, M. A. Piqueras, and J. Martí, Appl. Phys. Lett. 89, 131111 (2006). * (15) R. W. Ziolkowski, Phys. Rev. E 70, 046608 (2004). * (16) Y. Yuan, L. Shen, L. Ran, T. Jiang, J. Huangfu, J. A. Kong, Phys. Rev. A 77, 053821 (2008). * (17) Y. Jin and S. He, Opt. Express 18, 16587 (2010). * (18) G. Lovat, P. Burghignoli, F. Capolino, D. R. Jackson, and D. R. Wilton, IEEE Trans. Antennas Propag. 54, 1017 (2006). * (19) J. O. Dimmock, Opt. Express 11, 2397 (2003). * (20) M. Nieto-Vesperinas, J. Opt. Soc. Am. A 21, 491 (2004). * (21) M. J. Roberts, S. Feng, M. Moran, and L. Johnson, J. Nanophoton. 4, 043511 (2010). * (22) E. D. Palik, Handbook of Optical Constants of Solids (Academic, San Diego, 1998).	arxiv-papers	2011-12-07T06:18:04	2024-09-04T02:49:25.054327	{ "license": "Public Domain", "authors": "Simin Feng", "submitter": "Simin Feng", "url": "https://arxiv.org/abs/1112.1482" }
1112.1600	EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN) CERN-PH-EP-2011-186 LHCb-PAPER-2011-025 Search for the rare decays $B^{0}_{s}\rightarrow\mu^{+}\mu^{-}$ and $B^{0}\rightarrow\mu^{+}\mu^{-}$ The LHCb Collaboration111Authors are listed on the following pages. Abstract A search for the decays $B^{0}_{s}\rightarrow\mu^{+}\mu^{-}$ and $B^{0}\rightarrow\mu^{+}\mu^{-}$ is performed with 0.37 fb-1 of $pp$ collisions at $\sqrt{s}$ = 7 TeV collected by the LHCb experiment in 2011. The upper limits on the branching fractions are ${\cal B}(B^{0}_{s}\rightarrow\mu^{+}\mu^{-})$ $<1.6\times 10^{-8}$ and ${\cal B}(B^{0}\rightarrow\mu^{+}\mu^{-})$ $<3.6\times 10^{-9}$ at 95 % confidence level. A combination of these results with the LHCb limits obtained with the 2010 dataset leads to ${\cal B}(B^{0}_{s}\rightarrow\mu^{+}\mu^{-})$ $<1.4\times 10^{-8}$ and ${\cal B}(B^{0}\rightarrow\mu^{+}\mu^{-})$ $<3.2\times 10^{-9}$ at 95 % confidence level. Keywords: LHC, $b$-hadron, FCNC, rare decays, leptonic decays. R. Aaij23, C. Abellan Beteta35,n, B. Adeva36, M. Adinolfi42, C. Adrover6, A. Affolder48, Z. Ajaltouni5, J. Albrecht37, F. Alessio37, M. Alexander47, G. Alkhazov29, P. Alvarez Cartelle36, A.A. Alves Jr22, S. Amato2, Y. Amhis38, J. Anderson39, R.B. Appleby50, O. Aquines Gutierrez10, F. Archilli18,37, L. Arrabito53, A. Artamonov 34, M. Artuso52,37, E. Aslanides6, G. Auriemma22,m, S. Bachmann11, J.J. Back44, D.S. Bailey50, V. Balagura30,37, W. Baldini16, R.J. Barlow50, C. Barschel37, S. Barsuk7, W. Barter43, A. Bates47, C. Bauer10, Th. Bauer23, A. Bay38, I. Bediaga1, S. Belogurov30, K. Belous34, I. Belyaev30,37, E. Ben-Haim8, M. Benayoun8, G. Bencivenni18, S. Benson46, J. Benton42, R. Bernet39, M.-O. Bettler17, M. van Beuzekom23, A. Bien11, S. Bifani12, T. Bird50, A. Bizzeti17,h, P.M. Bjørnstad50, T. Blake37, F. Blanc38, C. Blanks49, J. Blouw11, S. Blusk52, A. Bobrov33, V. Bocci22, A. Bondar33, N. Bondar29, W. Bonivento15, S. Borghi47,50, A. Borgia52, T.J.V. Bowcock48, C. Bozzi16, T. Brambach9, J. van den Brand24, J. Bressieux38, D. Brett50, M. Britsch10, T. Britton52, N.H. Brook42, H. Brown48, A. Büchler-Germann39, I. Burducea28, A. Bursche39, J. Buytaert37, S. Cadeddu15, O. Callot7, M. Calvi20,j, M. Calvo Gomez35,n, A. Camboni35, P. Campana18,37, A. Carbone14, G. Carboni21,k, R. Cardinale19,i,37, A. Cardini15, L. Carson49, K. Carvalho Akiba2, G. Casse48, M. Cattaneo37, Ch. Cauet9, M. Charles51, Ph. Charpentier37, N. Chiapolini39, K. Ciba37, X. Cid Vidal36, G. Ciezarek49, P.E.L. Clarke46,37, M. Clemencic37, H.V. Cliff43, J. Closier37, C. Coca28, V. Coco23, J. Cogan6, P. Collins37, A. Comerma-Montells35, F. Constantin28, G. Conti38, A. Contu51, A. Cook42, M. Coombes42, G. Corti37, G.A. Cowan38, R. Currie46, B. D’Almagne7, C. D’Ambrosio37, P. David8, P.N.Y. David23, I. De Bonis4, S. De Capua21,k, M. De Cian39, F. De Lorenzi12, J.M. De Miranda1, L. De Paula2, P. De Simone18, D. Decamp4, M. Deckenhoff9, H. Degaudenzi38,37, M. Deissenroth11, L. Del Buono8, C. Deplano15, D. Derkach14,37, O. Deschamps5, F. Dettori24, J. Dickens43, H. Dijkstra37, P. Diniz Batista1, F. Domingo Bonal35,n, S. Donleavy48, F. Dordei11, P. Dornan49, A. Dosil Suárez36, D. Dossett44, A. Dovbnya40, F. Dupertuis38, R. Dzhelyadin34, A. Dziurda25, S. Easo45, U. Egede49, V. Egorychev30, S. Eidelman33, D. van Eijk23, F. Eisele11, S. Eisenhardt46, R. Ekelhof9, L. Eklund47, Ch. Elsasser39, D. Elsby55, D. Esperante Pereira36, L. Estève43, A. Falabella16,14,e, E. Fanchini20,j, C. Färber11, G. Fardell46, C. Farinelli23, S. Farry12, V. Fave38, V. Fernandez Albor36, M. Ferro-Luzzi37, S. Filippov32, C. Fitzpatrick46, M. Fontana10, F. Fontanelli19,i, R. Forty37, M. Frank37, C. Frei37, M. Frosini17,f,37, S. Furcas20, A. Gallas Torreira36, D. Galli14,c, M. Gandelman2, P. Gandini51, Y. Gao3, J-C. Garnier37, J. Garofoli52, J. Garra Tico43, L. Garrido35, D. Gascon35, C. Gaspar37, N. Gauvin38, M. Gersabeck37, T. Gershon44,37, Ph. Ghez4, V. Gibson43, V.V. Gligorov37, C. Göbel54, D. Golubkov30, A. Golutvin49,30,37, A. Gomes2, H. Gordon51, M. Grabalosa Gándara35, R. Graciani Diaz35, L.A. Granado Cardoso37, E. Graugés35, G. Graziani17, A. Grecu28, E. Greening51, S. Gregson43, B. Gui52, E. Gushchin32, Yu. Guz34, T. Gys37, G. Haefeli38, C. Haen37, S.C. Haines43, T. Hampson42, S. Hansmann-Menzemer11, R. Harji49, N. Harnew51, J. Harrison50, P.F. Harrison44, J. He7, V. Heijne23, K. Hennessy48, P. Henrard5, J.A. Hernando Morata36, E. van Herwijnen37, E. Hicks48, K. Holubyev11, P. Hopchev4, W. Hulsbergen23, P. Hunt51, T. Huse48, R.S. Huston12, D. Hutchcroft48, D. Hynds47, V. Iakovenko41, P. Ilten12, J. Imong42, R. Jacobsson37, A. Jaeger11, M. Jahjah Hussein5, E. Jans23, F. Jansen23, P. Jaton38, B. Jean-Marie7, F. Jing3, M. John51, D. Johnson51, C.R. Jones43, B. Jost37, M. Kaballo9, S. Kandybei40, M. Karacson37, T.M. Karbach9, J. Keaveney12, I.R. Kenyon55, U. Kerzel37, T. Ketel24, A. Keune38, B. Khanji6, Y.M. Kim46, M. Knecht38, P. Koppenburg23, A. Kozlinskiy23, L. Kravchuk32, K. Kreplin11, M. Kreps44, G. Krocker11, P. Krokovny11, F. Kruse9, K. Kruzelecki37, M. Kucharczyk20,25,37,j, T. Kvaratskheliya30,37, V.N. La Thi38, D. Lacarrere37, G. Lafferty50, A. Lai15, D. Lambert46, R.W. Lambert24, E. Lanciotti37, G. Lanfranchi18, C. Langenbruch11, T. Latham44, C. Lazzeroni55, R. Le Gac6, J. van Leerdam23, J.-P. Lees4, R. Lefèvre5, A. Leflat31,37, J. Lefrançois7, O. Leroy6, T. Lesiak25, L. Li3, L. Li Gioi5, M. Lieng9, M. Liles48, R. Lindner37, C. Linn11, B. Liu3, G. Liu37, J.H. Lopes2, E. Lopez Asamar35, N. Lopez-March38, H. Lu38,3, J. Luisier38, A. Mac Raighne47, F. Machefert7, I.V. Machikhiliyan4,30, F. Maciuc10, O. Maev29,37, J. Magnin1, S. Malde51, R.M.D. Mamunur37, G. Manca15,d, G. Mancinelli6, N. Mangiafave43, U. Marconi14, R. Märki38, J. Marks11, G. Martellotti22, A. Martens8, L. Martin51, A. Martín Sánchez7, D. Martinez Santos37, A. Massafferri1, Z. Mathe12, C. Matteuzzi20, M. Matveev29, E. Maurice6, B. Maynard52, A. Mazurov16,32,37, G. McGregor50, R. McNulty12, C. Mclean14, M. Meissner11, M. Merk23, J. Merkel9, R. Messi21,k, S. Miglioranzi37, D.A. Milanes13,37, M.-N. Minard4, J. Molina Rodriguez54, S. Monteil5, D. Moran12, P. Morawski25, R. Mountain52, I. Mous23, F. Muheim46, K. Müller39, R. Muresan28,38, B. Muryn26, B. Muster38, M. Musy35, J. Mylroie-Smith48, P. Naik42, T. Nakada38, R. Nandakumar45, I. Nasteva1, M. Nedos9, M. Needham46, N. Neufeld37, C. Nguyen-Mau38,o, M. Nicol7, V. Niess5, N. Nikitin31, A. Nomerotski51, A. Novoselov34, A. Oblakowska-Mucha26, V. Obraztsov34, S. Oggero23, S. Ogilvy47, O. Okhrimenko41, R. Oldeman15,d, M. Orlandea28, J.M. Otalora Goicochea2, P. Owen49, K. Pal52, J. Palacios39, A. Palano13,b, M. Palutan18, J. Panman37, A. Papanestis45, M. Pappagallo47, C. Parkes47,37, C.J. Parkinson49, G. Passaleva17, G.D. Patel48, M. Patel49, S.K. Paterson49, G.N. Patrick45, C. Patrignani19,i, C. Pavel-Nicorescu28, A. Pazos Alvarez36, A. Pellegrino23, G. Penso22,l, M. Pepe Altarelli37, S. Perazzini14,c, D.L. Perego20,j, E. Perez Trigo36, A. Pérez-Calero Yzquierdo35, P. Perret5, M. Perrin-Terrin6, G. Pessina20, A. Petrella16,37, A. Petrolini19,i, A. Phan52, E. Picatoste Olloqui35, B. Pie Valls35, B. Pietrzyk4, T. Pilař44, D. Pinci22, R. Plackett47, S. Playfer46, M. Plo Casasus36, G. Polok25, A. Poluektov44,33, E. Polycarpo2, D. Popov10, B. Popovici28, C. Potterat35, A. Powell51, T. du Pree23, J. Prisciandaro38, V. Pugatch41, A. Puig Navarro35, W. Qian52, J.H. Rademacker42, B. Rakotomiaramanana38, M.S. Rangel2, I. Raniuk40, G. Raven24, S. Redford51, M.M. Reid44, A.C. dos Reis1, S. Ricciardi45, K. Rinnert48, D.A. Roa Romero5, P. Robbe7, E. Rodrigues47,50, F. Rodrigues2, P. Rodriguez Perez36, G.J. Rogers43, S. Roiser37, V. Romanovsky34, M. Rosello35,n, J. Rouvinet38, T. Ruf37, H. Ruiz35, G. Sabatino21,k, J.J. Saborido Silva36, N. Sagidova29, P. Sail47, B. Saitta15,d, C. Salzmann39, M. Sannino19,i, R. Santacesaria22, C. Santamarina Rios36, R. Santinelli37, E. Santovetti21,k, M. Sapunov6, A. Sarti18,l, C. Satriano22,m, A. Satta21, M. Savrie16,e, D. Savrina30, P. Schaack49, M. Schiller24, S. Schleich9, M. Schlupp9, M. Schmelling10, B. Schmidt37, O. Schneider38, A. Schopper37, M.-H. Schune7, R. Schwemmer37, B. Sciascia18, A. Sciubba18,l, M. Seco36, A. Semennikov30, K. Senderowska26, I. Sepp49, N. Serra39, J. Serrano6, P. Seyfert11, B. Shao3, M. Shapkin34, I. Shapoval40,37, P. Shatalov30, Y. Shcheglov29, T. Shears48, L. Shekhtman33, O. Shevchenko40, V. Shevchenko30, A. Shires49, R. Silva Coutinho44, T. Skwarnicki52, A.C. Smith37, N.A. Smith48, E. Smith51,45, K. Sobczak5, F.J.P. Soler47, A. Solomin42, F. Soomro18, B. Souza De Paula2, B. Spaan9, A. Sparkes46, P. Spradlin47, F. Stagni37, S. Stahl11, O. Steinkamp39, S. Stoica28, S. Stone52,37, B. Storaci23, M. Straticiuc28, U. Straumann39, V.K. Subbiah37, S. Swientek9, M. Szczekowski27, P. Szczypka38, T. Szumlak26, S. T’Jampens4, E. Teodorescu28, F. Teubert37, C. Thomas51, E. Thomas37, J. van Tilburg11, V. Tisserand4, M. Tobin39, S. Topp-Joergensen51, N. Torr51, E. Tournefier4,49, M.T. Tran38, A. Tsaregorodtsev6, N. Tuning23, M. Ubeda Garcia37, A. Ukleja27, P. Urquijo52, U. Uwer11, V. Vagnoni14, G. Valenti14, R. Vazquez Gomez35, P. Vazquez Regueiro36, S. Vecchi16, J.J. Velthuis42, M. Veltri17,g, B. Viaud7, I. Videau7, X. Vilasis-Cardona35,n, J. Visniakov36, A. Vollhardt39, D. Volyanskyy10, D. Voong42, A. Vorobyev29, H. Voss10, S. Wandernoth11, J. Wang52, D.R. Ward43, N.K. Watson55, A.D. Webber50, D. Websdale49, M. Whitehead44, D. Wiedner11, L. Wiggers23, G. Wilkinson51, M.P. Williams44,45, M. Williams49, F.F. Wilson45, J. Wishahi9, M. Witek25, W. Witzeling37, S.A. Wotton43, K. Wyllie37, Y. Xie46, F. Xing51, Z. Xing52, Z. Yang3, R. Young46, O. Yushchenko34, M. Zavertyaev10,a, F. Zhang3, L. Zhang52, W.C. Zhang12, Y. Zhang3, A. Zhelezov11, L. Zhong3, E. Zverev31, A. Zvyagin37. 1Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, Brazil 2Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil 3Center for High Energy Physics, Tsinghua University, Beijing, China 4LAPP, Université de Savoie, CNRS/IN2P3, Annecy-Le-Vieux, France 5Clermont Université, Université Blaise Pascal, CNRS/IN2P3, LPC, Clermont- Ferrand, France 6CPPM, Aix-Marseille Université, CNRS/IN2P3, Marseille, France 7LAL, Université Paris-Sud, CNRS/IN2P3, Orsay, France 8LPNHE, Université Pierre et Marie Curie, Université Paris Diderot, CNRS/IN2P3, Paris, France 9Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany 10Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany 11Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg, Germany 12School of Physics, University College Dublin, Dublin, Ireland 13Sezione INFN di Bari, Bari, Italy 14Sezione INFN di Bologna, Bologna, Italy 15Sezione INFN di Cagliari, Cagliari, Italy 16Sezione INFN di Ferrara, Ferrara, Italy 17Sezione INFN di Firenze, Firenze, Italy 18Laboratori Nazionali dell’INFN di Frascati, Frascati, Italy 19Sezione INFN di Genova, Genova, Italy 20Sezione INFN di Milano Bicocca, Milano, Italy 21Sezione INFN di Roma Tor Vergata, Roma, Italy 22Sezione INFN di Roma La Sapienza, Roma, Italy 23Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands 24Nikhef National Institute for Subatomic Physics and Vrije Universiteit, Amsterdam, The Netherlands 25Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of Sciences, Kraców, Poland 26AGH University of Science and Technology, Kraców, Poland 27Soltan Institute for Nuclear Studies, Warsaw, Poland 28Horia Hulubei National Institute of Physics and Nuclear Engineering, Bucharest-Magurele, Romania 29Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia 30Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia 31Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow, Russia 32Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN), Moscow, Russia 33Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State University, Novosibirsk, Russia 34Institute for High Energy Physics (IHEP), Protvino, Russia 35Universitat de Barcelona, Barcelona, Spain 36Universidad de Santiago de Compostela, Santiago de Compostela, Spain 37European Organization for Nuclear Research (CERN), Geneva, Switzerland 38Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland 39Physik-Institut, Universität Zürich, Zürich, Switzerland 40NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine 41Institute for Nuclear Research of the National Academy of Sciences (KINR), Kyiv, Ukraine 42H.H. Wills Physics Laboratory, University of Bristol, Bristol, United Kingdom 43Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom 44Department of Physics, University of Warwick, Coventry, United Kingdom 45STFC Rutherford Appleton Laboratory, Didcot, United Kingdom 46School of Physics and Astronomy, University of Edinburgh, Edinburgh, United Kingdom 47School of Physics and Astronomy, University of Glasgow, Glasgow, United Kingdom 48Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom 49Imperial College London, London, United Kingdom 50School of Physics and Astronomy, University of Manchester, Manchester, United Kingdom 51Department of Physics, University of Oxford, Oxford, United Kingdom 52Syracuse University, Syracuse, NY, United States 53CC-IN2P3, CNRS/IN2P3, Lyon-Villeurbanne, France, associated member 54Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de Janeiro, Brazil, associated to 2 55University of Birmingham, Birmingham, United Kingdom aP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS), Moscow, Russia bUniversità di Bari, Bari, Italy cUniversità di Bologna, Bologna, Italy dUniversità di Cagliari, Cagliari, Italy eUniversità di Ferrara, Ferrara, Italy fUniversità di Firenze, Firenze, Italy gUniversità di Urbino, Urbino, Italy hUniversità di Modena e Reggio Emilia, Modena, Italy iUniversità di Genova, Genova, Italy jUniversità di Milano Bicocca, Milano, Italy kUniversità di Roma Tor Vergata, Roma, Italy lUniversità di Roma La Sapienza, Roma, Italy mUniversità della Basilicata, Potenza, Italy nLIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain oHanoi University of Science, Hanoi, Viet Nam ## 1 Introduction Measurements of low-energy processes can provide indirect constraints on particles that are too heavy to be produced directly. This is particularly true for Flavour Changing Neutral Current (FCNC) processes which are highly suppressed in the Standard Model (SM) and can only occur through higher-order diagrams. The SM predictions for the branching fractions of the FCNC decays222Inclusion of charged conjugated processes is implied throughout. $B^{0}_{s}\rightarrow\mu^{+}\mu^{-}$ and $B^{0}\rightarrow\mu^{+}\mu^{-}$ are ${\cal B}(B^{0}_{s}\rightarrow\mu^{+}\mu^{-})$ = $(3.2\pm 0.2)\times 10^{-9}$ and ${\cal B}(B^{0}\rightarrow\mu^{+}\mu^{-})$ = $(0.10\pm 0.01)\times 10^{-9}$ [1, Buras:2010wr]. However, contributions from new processes or new heavy particles can significantly enhance these values. For example, within Minimal Supersymmetric extensions of the SM (MSSM), in the large $\tan\beta$ regime, ${\cal B}(B^{0}_{s}\rightarrow\mu^{+}\mu^{-})$ is found to be approximately proportional to $\tan^{6}\beta$ [3, Hamzaoui:1998nu, Babu:1999hn, Hall:1993gn, Huang:1998vb], where $\tan\beta$ is the ratio of the vacuum expectation values of the two neutral $C\\!P$-even Higgs fields. The branching fractions could therefore be enhanced by orders of magnitude for large values of $\tan\beta$. The best published limits from the Tevatron are ${\cal B}(B^{0}_{s}\rightarrow\mu^{+}\mu^{-})$ $<~{}5.1~{}\times~{}10^{-8}$ at 95% confidence level (CL) by the D0 collaboration using 6.1 fb-1 of data [8], and ${\cal B}(B^{0}\rightarrow\mu^{+}\mu^{-})$ $<6.0\times 10^{-9}$ at 95% CL by the CDF collaboration using 6.9 fb-1 of data [9]. In the same dataset the CDF collaboration observes an excess of $B^{0}_{s}\rightarrow\mu^{+}\mu^{-}$ candidates compatible with ${\cal B}(B^{0}_{s}\rightarrow\mu^{+}\mu^{-})$ = $(1.8^{+1.1}_{-0.9})\times 10^{-8}$ and with an upper limit of ${\cal B}(B^{0}_{s}\rightarrow\mu^{+}\mu^{-})$ $<4.0\times 10^{-8}$ at 95% CL. The CMS collaboration has recently published ${\cal B}(B^{0}_{s}\rightarrow\mu^{+}\mu^{-})$ $<1.9\times 10^{-8}$ at 95% CL and ${\cal B}(B^{0}\rightarrow\mu^{+}\mu^{-})$ $<4.6\times 10^{-9}$ at 95% CL using 1.14 fb-1 of data [10]. The LHCb collaboration has published the limits [11] ${\cal B}(B^{0}_{s}\rightarrow\mu^{+}\mu^{-})$ $<5.4\times 10^{-8}$ and ${\cal B}(B^{0}\rightarrow\mu^{+}\mu^{-})$ $<1.5\times 10^{-8}$ at 95$\%$ CL based on about 37 pb-1 of integrated luminosity collected in the 2010 run. This Letter presents an analysis of the data recorded by LHCb in the first half of 2011 which correspond to an integrated luminosity of $\sim$ 0.37 fb-1. The results of this analysis are then combined with those published from the 2010 dataset. ## 2 The LHCb detector The LHCb detector [12] is a single-arm forward spectrometer designed to study production and decays of hadrons containing $b$ or $c$ quarks. The detector consists of a vertex locator (VELO) providing precise locations of primary $pp$ interaction vertices and detached vertices of long lived hadrons. The momenta of charged particles are determined using information from the VELO together with the rest of the tracking system, composed of a large area silicon tracker located before a warm dipole magnet with a bending power of $\sim$ 4 Tm, and a combination of silicon strip detectors and straw drift chambers located after the magnet. Two Ring Imaging Cherenkov (RICH) detectors are used for charged hadron identification in the momentum range 2–100 ${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. Photon, electron and hadron candidates are identified by electromagnetic and hadronic calorimeters. A muon system of alternating layers of iron and drift chambers provides muon identification. The two calorimeters and the muon system provide the energy and momentum information to implement a first level (L0) hardware trigger. An additional trigger level (HLT) is software based, and its algorithms are tuned to the experimental operating condition. Events with a muon final states are triggered using two L0 trigger decisions: the single-muon decision, which requires one muon candidate with a transverse momentum $p_{\rm T}$ larger than 1.5 ${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, and the di-muon decision, which requires two muon candidates with transverse momenta $p_{{\rm T},1}$ and $p_{{\rm T},2}$ satisfying the relation $\sqrt{p_{{\rm T},1}\cdot p_{{\rm T},2}}>1.3$ ${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. The single muon trigger decision in the second trigger level (HLT) includes a cut on the impact parameter (${\rm IP}$) with respect to the primary vertex, which allows for a lower $p_{\rm T}$ requirement ($p_{\rm T}>1.0$ GeV/$c$, $\rm IP>0.1$ mm). The di-muon trigger decision requires muon pairs of opposite charge with $p_{\rm T}>500$ MeV/c, forming a common vertex and with an invariant mass $m_{\mu\mu}>4.7$ GeV/$c^{2}$. A second trigger decision, primarily to select ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ events, requires $2.97<m_{\mu\mu}<3.21$ GeV/$c^{2}$. The remaining region of the di- muon invariant mass range is also covered by trigger decisions that in addition require the di-muon secondary vertex to be well separated from the primary vertex. Events with purely hadronic final states are triggered by the L0 trigger if there is a calorimeter cluster with transverse energy $E_{\rm T}>3.6$ GeV. Other HLT trigger decisions select generic displaced vertices, providing high efficiency for purely hadronic decays. ## 3 Analysis strategy Assuming the branching fractions predicted by the SM, and using the $b\bar{b}$ cross-section measured by LHCb in the pseudorapidity interval $2<\eta<6$ and integrated over all transverse momenta of $\sigma_{b\overline{b}}=75\pm 14\,\mu$b [13], approximately $3.9$ $B^{0}_{s}\rightarrow\mu^{+}\mu^{-}$ and 0.4 $B^{0}\rightarrow\mu^{+}\mu^{-}$ events are expected to be triggered, reconstructed and selected in the analysed sample embedded in a large background. The general structure of the analysis is based upon the one described in Ref. [11]. First a very efficient selection removes the biggest amount of background while keeping most of the signal within the LHCb acceptance. The number of observed events is compared to the number of expected signal and background events in bins of two independent variables, the invariant mass and the output of a multi-variate discriminant. The discriminant is a Boosted Decision Tree (BDT) constructed using the TMVA package [14]. It supersedes the Geometrical Likelihood (GL) used in the previous analysis [11] as it has been found more performant in discriminating between signal and background events in simulated samples. No data were used in the choice of the multivariate discriminant in order not to bias the result. The combination of variables entering the BDT discriminant is optimized using simulated events. The probability for a signal or background event to have a given value of the BDT output is obtained from data using $B^{0}_{(s)}\rightarrow h^{+}h^{{}^{\prime}-}$ candidates (where $h^{(^{\prime})}$ can be a pion or a kaon) as signal and sideband $B^{0}_{(s)}\rightarrow\mu^{+}\mu^{-}$ candidates as background. The invariant mass line shape of the signals is described by a Crystal Ball function [15] with parameters extracted from data control samples. The central values of the masses are obtained from $B^{0}\rightarrow K^{+}\pi^{-}$ and $B^{0}_{s}\rightarrow K^{+}K^{-}$ samples. The $B^{0}_{s}$ and $B^{0}$ mass resolutions are estimated by interpolating those obtained with di-muon resonances ($J/\psi,\psi(2S)$ and $\Upsilon(1S,2S,3S)$) and cross-checked with a fit to the invariant mass distributions of both inclusive $B^{0}_{(s)}\rightarrow h^{+}h^{{}^{\prime}-}$ decays and exclusive $B^{0}\rightarrow K^{+}\pi^{-}$ decays. The central values of the masses and the mass resolution are used to define the signal regions. The number of expected signal events, for a given branching fraction hypothesis, is obtained by normalizing to channels of known branching fractions: $B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$, $B^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\phi$ and $B^{0}\rightarrow K^{+}\pi^{-}$. These channels are selected in a way as similar as possible to the signals in order to minimize the systematic uncertainty related to the different phase space accessible to each final state. The BDT output and invariant mass distributions for combinatorial background events in the signal regions are obtained using fits of the mass distribution of events in the mass sidebands in bins of the BDT output. The two-dimensional space formed by the invariant mass and the BDT output is binned. For each bin we count the number of candidates observed in the data, and compute the expected number of signal events and the expected number of background events. The binning is unchanged with respect to the 2010 analysis [11]. The compatibility of the observed distribution of events in all bins with the distribution expected for a given branching fraction hypothesis is computed using the $\textrm{CL}_{\textrm{s}}$ method [16], which allows a given hypothesis to be excluded at a given confidence level. ## 4 Selection The $B^{0}_{(s)}\rightarrow\mu^{+}\mu^{-}$ selections require two muon candidates of opposite charge. Tracks are required to be of good quality and to be displaced with respect to any primary vertex. The secondary vertex is required to be well fitted ($\chi^{2}/{\rm nDoF}<9$) and must be separated from the primary vertex in the forward direction by a distance of flight significance ($L/\sigma(L)$) greater than 15. When more than one primary vertex is reconstructed, the one that gives the minimum impact parameter significance for the candidate is chosen. The reconstructed candidate has to point to this primary vertex (${\rm IP}/\sigma({\rm IP})<5$). Improvements have been made to the selection developed for 2010 data [11]. The RICH is used to identify kaons in the $B^{0}_{s}\rightarrow J/\psi\phi$ normalization channel and the Kullback-Leibler (KL) distance [17, KLdistance2] is used to suppress duplicated tracks created by the reconstruction. This procedure compares the parameters and correlation matrices of the reconstructed tracks and where two are found to be similar, in this case with a symmetrized KL divergence less than 5000, only the one with the higher track fit quality is considered. The inclusive $B^{0}_{(s)}\rightarrow h^{+}h^{{}^{\prime}-}$ sample is the main control sample for the determination from data of the probability distribution function (PDF) of the BDT output. This sample is selected in exactly the same way as the $B^{0}_{(s)}\rightarrow\mu^{+}\mu^{-}$ signals apart from the muon identification requirement. The same selection is also applied to the $B^{0}\rightarrow K^{+}\pi^{-}$ normalization channel. The muon identification efficiency is uniform within $\sim 1\%$ in the considered phase space therefore no correction is added to the BDT PDF extracted from the $B^{0}_{(s)}\rightarrow h^{+}h^{{}^{\prime}-}$ sample. The remaining phase space dependence of the muon identification efficiency is instead taken into account in the computation of the normalization factor when the $B^{0}\rightarrow K^{+}\pi^{-}$ channel is considered. The $J/\psi\rightarrow\mu\mu$ decay in the $B^{+}\rightarrow J/\psi K^{+}$ and $B^{0}_{s}\rightarrow J/\psi\phi$ normalization channels is selected in a very similar way to the $B^{0}_{(s)}\rightarrow\mu^{+}\mu^{-}$ channels, apart from the pointing requirement. $K^{\pm}$ candidates are required to be identified by the RICH detector and to pass track quality and impact parameter cuts. To avoid pathological events, all tracks from selected candidates are required to have a momentum less than 1 TeV/$c$. Only $B$ candidates with decay times less than $5\,\tau_{B_{(s)}^{0}}$, where $\tau_{B^{0}_{(s)}}$ is the $B$ lifetime [19], are accepted for further analysis. Di-muon candidates coming from elastic di-photon production are removed by requiring a minimum transverse momentum of the $B$ candidate of 500${\mathrm{\,Me\kern-1.00006ptV\\!/}c}$. ## 5 Determination of the mass and BDT distributions The variables entering the BDT discriminant are the six variables used as input to the ${\rm GL}$ in the 2010 analysis plus three new variables. The six variables used in the 2010 analysis are the $B$ lifetime, impact parameter, transverse momentum, the minimum impact parameter significance (${\rm IP}/\sigma({\rm IP})$) of the muons, the distance of closest approach between the two muons and the isolation of the two muons with respect to any other track in the event. The three new variables are: 1. 1. the minimum $p_{\rm T}$ of the two muons; 2. 2. the cosine of the angle between the muon momentum in the $B$ rest frame and the vector perpendicular to the $B$ momentum and the beam axis: $\cos P=\frac{p_{y,\mu 1}\,p_{x,B}-p_{x,\mu 1}\,p_{y,B}}{p_{{\rm T},B}\,(m_{\mu\mu}/2)}$ (1) where $\mu_{1}$ labels one of the muons and $m_{\mu\mu}$ is the reconstructed $B$ candidate mass333As the $B$ is a (pseudo)-scalar particle, this variable is uniformely distributed for signal candidates while is peaked at zero for $b\bar{b}\rightarrow\mu^{+}\mu^{-}X$ background candidates. In fact, muons from semi-leptonic decays are mostly emitted in the direction of the $b$’s and, therefore, lie in a plane formed by the $B$ momentum and the beam axis.; 3. 3. the $B$ isolation [20] $I_{B}=\frac{p_{\rm T}(B)}{p_{\rm T}(B)+\sum_{i}p_{{\rm T},i}},$ (2) where $p_{\rm T}(B)$ is the $B$ transverse momentum with respect to the beam line and the sum is over all the tracks, excluding the muon candidates, that satisfy $\sqrt{\delta\eta^{2}+\delta\phi^{2}}~{}<~{}1.0$, where $\delta\eta$ and $\delta\phi$ denote respectively the difference in pseudorapidity and azimuthal angle between the track and the $B$ candidate. The BDT output is found to be independent of the invariant mass for both signal and background and is defined such that the signal is uniformly distributed between zero and one and the background peaks at zero. The BDT range is then divided in four bins of equal width. The BDT is trained using simulated samples ($B^{0}_{(s)}\rightarrow\mu^{+}\mu^{-}$ for signals and $b\bar{b}\rightarrow\mu^{+}\mu^{-}X$ for background where $X$ is any other set of particles) and the PDF obtained from data as explained below. ### 5.1 Combinatorial background PDFs The BDT and invariant mass shapes for the combinatorial background inside the signal regions are determined from data by interpolating the number of expected events using the invariant mass sidebands for each BDT bin. The boundaries of the signal regions are defined as $m_{B^{0}}\pm 60$ ${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ and $m_{B^{0}_{s}}\pm 60$ ${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ and the mass sidebands as $[m_{B^{0}}-600{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}},m_{B^{0}}-60{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}]$ and $[m_{B^{0}_{s}}+60{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}},m_{B^{0}_{s}}+600{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}]$. Figure 1 shows the invariant mass distribution for events that lie in each BDT output bin. In each case the fit model used to estimate the expected number of combinatorial background events in the signal regions is superimposed. Aside from combinatorial background, the low-mass sideband is potentially polluted by two other contributions: cascading $b\rightarrow c\mu\nu\rightarrow\mu\mu X$ decays below 4900${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ and peaking background from $B^{0}_{(s)}\rightarrow h^{+}h^{{}^{\prime}-}$ candidates with the two hadrons misidentified as muons above 5000${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. To avoid these contaminations, the number of expected combinatorial background events is obtained by fitting a single exponential function to the events in the reduced low-mass sideband [4900, 5000] ${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ and in the full high-mass sideband. As a cross-check, two other models, a single exponential function and the sum of two exponential functions, have been used to fit the events in different ranges of sidebands providing consistent background estimates inside the signal regions. Figure 1: Distribution of the $\mu^{+}\mu^{-}$ invariant mass for events in each BDT output bin. The curve shows the model used to fit the sidebands and extract the expected number of combinatorial background events in the $B^{0}_{s}$ and $B^{0}$ signal regions, delimited by the vertical dotted orange and dashed green lines respectively. Only events in the region in which the line is solid have been considered in the fit. ### 5.2 Peaking background PDFs The peaking backgrounds due to $B^{0}_{(s)}\rightarrow h^{+}h^{{}^{\prime}-}$ events in which both hadrons are misidentified as muons have been evaluated from data and simulated events to be $N_{B^{0}_{s}}=1.0\pm 0.4$ events and $N_{B^{0}}=5.0\pm 0.9$ events within the two mass windows and in the whole BDT output range. The mass line shape of the peaking background is obtained from a simulated sample of doubly-misidentified $B^{0}_{(s)}\rightarrow h^{+}h^{{}^{\prime}-}$ events and normalized to the number of events expected in the two search windows from data, $N_{B^{0}_{s}}$ and $N_{B^{0}}$. The BDT PDF of the peaking background is assumed to be the same as for the signal. ### 5.3 Signal PDFs The BDT PDF for signal events is determined using an inclusive $B^{0}_{(s)}\rightarrow h^{+}h^{{}^{\prime}-}$ sample. Only events which are triggered independently on the signal candidates have been considered (TIS events). The number of $B^{0}_{(s)}\rightarrow h^{+}h^{{}^{\prime}-}$ signal events in each BDT output bin is determined by fitting the $hh^{\prime}$ invariant mass distribution under the $\mu\mu$ mass hypothesis [21]. Figure 2 shows the fit to the mass distribution of the full sample and for the three highest BDT output bins for $B^{0}_{(s)}\rightarrow h^{+}h^{{}^{\prime}-}$ TIS events. The $B^{0}_{(s)}\rightarrow h^{+}h^{{}^{\prime}-}$ exclusive decays, the combinatorial background and the physical background components are drawn under the fit to the data; the physical background is due to the partial reconstruction of three-body $B$ meson decays. In order to cross-check this result, two other fits have been performed on the same dataset. The signal line shape is parametrized either by a single or a double Crystal Ball function [15], the combinatorial background by an exponential function and the physical background by an ARGUS function [22]. In addition, exclusive $B^{0}_{(s)}\rightarrow\pi^{-}K^{+},\pi^{-}\pi^{+},K^{-}K^{+}$ channels, selected using the $K-\pi$ separation capability of the RICH system, are used to cross-check the calibration of the BDT output both using the $\pi^{-}K^{+},\pi^{-}\pi^{+},K^{-}K^{+}$ inclusive yields without separating $B$ and $B^{0}_{s}$ and using the $B^{0}\rightarrow K^{+}\pi^{-}$ exclusive channel alone. The maximum spread in the fractional yield obtained among the different models has been used as a systematic uncertainty in the signal BDT PDF. The BDT PDFs for signals and combinatorial background are shown in Fig. 3. Figure 2: Invariant mass distributions of $B^{0}_{(s)}\rightarrow h^{+}h^{{}^{\prime}-}$ candidates in the $\mu^{+}\mu^{-}$ mass hypothesis for the whole sample (top left) and for the samples in the three highest bins of the BDT output (top right, bottom left, bottom right). The $B^{0}_{(s)}\rightarrow h^{+}h^{{}^{\prime}-}$ exclusive decays, the combinatorial background and the physical background components are drawn under the fit to the data (solid blue line). Figure 3: BDT probability distribution functions of signal events (solid squares) and combinatorial background (open circles): the PDF for the signal is obtained from the inclusive sample of TIS $B^{0}_{(s)}\rightarrow h^{+}h^{{}^{\prime}-}$ events, the PDF for the combinatorial background is obtained from the events in the mass sidebands. The invariant mass shape for the signal is parametrized as a Crystal Ball function. The mean value is determined using the $B^{0}\rightarrow K^{+}\pi^{-}$ and $B^{0}_{s}\rightarrow K^{+}K^{-}$ exclusive channels and the transition point of the radiative tail is obtained from simulated events [11]. The central values are $\displaystyle m_{B^{0}_{s}}$ $\displaystyle=$ $\displaystyle 5358.0\pm 1.0{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}},$ $\displaystyle m_{B^{0}}$ $\displaystyle=$ $\displaystyle 5272.0\pm 1.0{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}.$ The measured values of $m_{B^{0}}$ and $m_{B^{0}_{s}}$ are $7-8$ ${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ below the PDG values [19] due to the fact that the momentum scale is uncalibrated in the dataset used in this analysis. The mass resolutions are extracted from data with a linear interpolation between the measured resolution of charmonium and bottomonium resonances decaying into two muons: ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$, $\psi{(2S)}$, $\mathchar 28935\relax{(1S)}$, $\mathchar 28935\relax{(2S)}$ and $\mathchar 28935\relax{(3S)}$. The mass line shapes for quarkonium resonances are shown in Fig. 4. Each resonance is fitted with two Crystal Ball functions with common mean value and common resolution but different parameterization of the tails. The background is fitted with an exponential function. The results of the interpolation at the $m_{B^{0}_{s}}$ and $m_{B^{0}}$ masses are $\displaystyle\sigma(m_{B^{0}_{s}})$ $\displaystyle=$ $\displaystyle 24.6\pm 0.2_{\rm(stat)}\pm 1.0_{\rm(syst)}{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}},$ $\displaystyle\sigma(m_{B^{0}})$ $\displaystyle=$ $\displaystyle 24.3\pm 0.2_{\rm(stat)}\pm 1.0_{\rm(syst)}{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}.$ This result has been checked using both the fits to the $B^{0}_{(s)}\rightarrow h^{+}h^{{}^{\prime}-}$ inclusive decay line shape and the $B^{0}\rightarrow K^{+}\pi^{-}$ exclusive decay. The results are in agreement within the uncertainties. Figure 4: Di-muon invariant mass spectrum in the ranges (2.9 – 3.9) ${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ (left) and (9–11) ${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ (right). ## 6 Normalization To estimate the signal branching fraction, the number of observed signal events is normalized to the number of events of a channel with a well known branching fraction. Three complementary normalization channels are used: $B^{+}\rightarrow J/\psi(\mu^{+}\mu^{-})K^{+}$, $B^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}(\mu^{+}\mu^{-})\phi(K^{+}K^{-})$ and $B^{0}\rightarrow K^{+}\pi^{-}$. The first two channels have similar trigger and muon identification efficiencies to the signal but different number of particles in the final state. The third channel has a similar topology but is selected by different trigger lines. The numbers of $B^{0}_{s}\rightarrow\mu^{+}\mu^{-}$ and $B^{0}\rightarrow\mu^{+}\mu^{-}$ candidates are translated into a branching fractions (${\cal B}$) using the equation ${\cal B}={\cal B}_{\rm norm}\times\frac{\rm\epsilon_{norm}^{REC}\epsilon_{norm}^{SEL\|REC}\epsilon_{norm}^{TRIG\|SEL}}{\rm\epsilon_{sig}^{REC}\epsilon_{sig}^{SEL\|REC}\epsilon_{sig}^{TRIG\|SEL}}\times\frac{f_{\rm norm}}{f_{d(s)}}\times\frac{N_{B^{0}_{(s)}\rightarrow\mu^{+}\mu^{-}}}{N_{\rm norm}}=\alpha^{\rm norm}_{B^{0}_{(s)}\rightarrow\mu^{+}\mu^{-}}\times N_{B^{0}_{(s)}\rightarrow\mu^{+}\mu^{-}},$ (3) where $f_{d(s)}$ and $f_{\rm norm}$ are the probabilities that a $b$ quark fragments into a $B^{0}_{(s)}$ and into the $b$ hadron involved for the chosen normalization mode. LHCb has measured $f_{s}/f_{d}=0.267^{+0.021}_{-0.020}$ [23]. ${\cal B}_{\rm norm}$ is the branching fraction and $N_{\rm norm}$ is the number of selected events of the normalization channel. The efficiency is the product of three factors: $\epsilon^{\rm REC}$ is the reconstruction efficiency of all the final state particles of the decay including the geometric acceptance of the detector; $\epsilon^{\rm SEL\|REC}$ is the selection efficiency for reconstructed events; $\epsilon^{\rm TRIG\|SEL}$ is the trigger efficiency for reconstructed and selected events. The subscript ($\rm sig,norm$) indicates whether the efficiency refers to the signal or the normalization channel. Finally, $\alpha^{\rm norm}_{B^{0}_{(s)}\rightarrow\mu^{+}\mu^{-}}$ is the normalization factor (or single event sensitivity) and $N_{B^{0}_{(s)}\rightarrow\mu^{+}\mu^{-}}$ the number of observed signal events. For each normalization channel $N_{\rm norm}$ is obtained from a fit to the invariant mass distribution. The invariant mass distributions for reconstructed $B^{+}\rightarrow J/\psi K^{+}$ and $B^{0}_{s}\rightarrow J/\psi\phi$ candidates are shown in Fig. 5, while the $B^{0}\rightarrow K^{+}\pi^{-}$ yield is obtained from the full $B^{0}_{(s)}\rightarrow h^{+}h^{{}^{\prime}-}$ fit as shown in the top left of Fig. 2. Figure 5: Invariant mass distributions of the $B^{+}\rightarrow J/\psi K^{+}$ (left) and $B^{0}_{s}\rightarrow J/\psi\phi$ (right) candidates used in the normalization procedure. The numbers used to calculate the normalization factors are summarized in Table 1. A weighted average of the three normalization channels, assuming the tracking and trigger uncertainties to be correlated between the two $J/\psi$ normalization channels and the uncertainty on $f_{d}/f_{s}$ to be correlated between the $B^{+}\rightarrow J/\psi K^{+}$ and $B^{0}\rightarrow K^{+}\pi^{-}$, gives $\displaystyle\alpha^{\rm norm}_{B^{0}_{s}\rightarrow\mu^{+}\mu^{-}}=(8.38\pm 0.74)\times 10^{-10}\,,$ $\displaystyle\alpha^{\rm norm}_{B^{0}\rightarrow\mu^{+}\mu^{-}}=(2.20\pm 0.11)\times 10^{-10}\,.$ These normalization factors are used to determine the limits. Table 1: Summary of the quantities and their uncertainties required to calculate the normalization factors ($\alpha^{\rm norm}_{B^{0}_{(s)}\rightarrow\mu^{+}\mu^{-}}$) for the three normalization channels considered. The branching fractions are taken from Refs. [19, 24]. The trigger efficiency and the number of $B^{0}\rightarrow K^{+}\pi^{-}$ candidates correspond to TIS events. \| $\cal B$ \| $\frac{\rm\epsilon_{norm}^{REC}\epsilon_{norm}^{SEL\|REC}}{\rm\epsilon_{sig}^{REC}\epsilon_{sig}^{SEL\|REC}}$ \| $\frac{\rm\epsilon_{norm}^{TRIG\|SEL}\rule{0.0pt}{7.71552pt}}{\rm\epsilon_{sig}^{TRIG\|SEL}\rule[-4.33998pt]{0.0pt}{0.0pt}}$ \| $N_{\rm norm}$ \| $\alpha^{\rm norm}_{B^{0}\rightarrow\mu^{+}\mu^{-}}$ \| $\alpha^{\rm norm}_{B^{0}_{s}\rightarrow\mu^{+}\mu^{-}}$ ---\|---\|---\|---\|---\|---\|--- \| $(\times 10^{-5})$ \| \| \| \| \| \| \| $(\times 10^{-10})$ \| $(\times 10^{-9})$ $B^{+}\rightarrow J/\psi K^{+}$ \| $6.01\,\pm$ \| $\,0.21$ \| $0.48\,\pm$ \| $\,0.014$ \| $0.95\,\pm$ \| $\,0.01$ \| $124\,518\,\pm$ \| $\,2\,025$ \| $2.23\,\pm$ \| $\,0.11$ \| $0.83\,\pm$ \| $\,0.08$ $B^{0}_{s}\rightarrow J/\psi\phi$ \| $3.4\,\pm$ \| $\,0.9$ \| $0.24\,\pm$ \| $\,0.014$ \| $0.95\,\pm$ \| $\,0.01$ \| $6\,940\,\pm$ \| $\,93$ \| $2.96\,\pm$ \| $\,0.84$ \| $1.11\,\pm$ \| $\,0.30$ $B^{0}\rightarrow K^{+}\pi^{-}$ \| $1.94\,\pm$ \| $\,0.06$ \| $0.86\,\pm$ \| $\,0.02$ \| $0.049\,\pm$ \| $\,0.004$ \| $4\,146\,\pm$ \| $\,608$ \| $1.98\,\pm$ \| $\,0.34$ \| $0.74\,\pm$ \| $\,0.14$ ## 7 Results The results for $B^{0}_{s}\rightarrow\mu^{+}\mu^{-}$ and $B^{0}\rightarrow\mu^{+}\mu^{-}$ are summarized in Table 2 and Table 3 respectively and in each of the bins the expected number of combinatorial background, peaking background, signal events, with the SM prediction assumed, is shown together with the observations on the data. The uncertainties in the signal and background PDFs and normalization factors are used to compute the uncertainties on the background and signal predictions. The two dimensional (mass, BDT) distribution of selected events can be seen in Fig. 6. The distribution of the invariant mass in the four BDT bins is shown in Fig. 7 for $B^{0}_{s}\rightarrow\mu^{+}\mu^{-}$ and in Fig. 8 for $B^{0}\rightarrow\mu^{+}\mu^{-}$ selected candidates. Figure 6: Distribution of selected di-muon events in the invariant mass–BDT plane. The orange short-dashed (green long-dashed) lines indicate the $\pm 60{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ search window around the mean $B^{0}_{s}$ ($B^{0}$) mass. Figure 7: Distribution of selected di-muon events in the $B^{0}_{s}\rightarrow\mu^{+}\mu^{-}$ mass window for the four BDT output bins. The black dots are data, the light grey histogram shows the contribution of the combinatorial background, the black filled histogram shows the contribution of the $B^{0}_{(s)}\rightarrow h^{+}h^{{}^{\prime}-}$ background and the dark grey filled histogram the contribution of $B^{0}_{s}\rightarrow\mu^{+}\mu^{-}$ signal events according to the SM rate. The hatched area depicts the uncertainty on the sum of the expected contributions. Figure 8: Distribution of selected di-muon events in the $B^{0}\rightarrow\mu^{+}\mu^{-}$ mass window for the four BDT output bins. The black dots are data, the light grey histogram shows the contribution of the combinatorial background, the black filled histogram shows the contribution of the $B^{0}_{(s)}\rightarrow h^{+}h^{{}^{\prime}-}$ background and the dark grey filled histogram shows the cross-feed of $B^{0}_{s}\rightarrow\mu^{+}\mu^{-}$ events in the $B^{0}$ mass window assuming the the SM rate. The hatched area depicts the uncertainty on the sum of the expected contributions. The compatibility of the distribution of events inside the search window in the invariant mass–BDT plane with a given branching fraction hypothesis is evaluated using the $\textrm{CL}_{\textrm{s}}$ method [16]. This method provides three estimators: $\textrm{CL}_{\textrm{s+b}}$, a measure of the compatibility of the observed distribution with the signal and background hypotheses, $\textrm{CL}_{\textrm{b}}$, a measure of the compatibility with the background-only hypothesis and $\textrm{CL}_{\textrm{s}}$, a measure of the compatibility of the observed distribution with the signal and background hypotheses normalized to the background-only hypothesis. The expected $\textrm{CL}_{\textrm{s}}$ values are shown in Fig. 9 for $B^{0}_{s}\rightarrow\mu^{+}\mu^{-}$ and for $B^{0}\rightarrow\mu^{+}\mu^{-}$ as dashed black lines under the hypothesis that background and SM events are observed. The shaded areas cover the region of $\pm 1\sigma$ of compatible observations. The observed values of $\textrm{CL}_{\textrm{s}}$ as a function of the assumed branching ratio is shown as dotted blue lines on both plots. The expected limits and the measured limits for $B^{0}_{s}\rightarrow\mu^{+}\mu^{-}$ and $B^{0}\rightarrow\mu^{+}\mu^{-}$ at 90 % and 95 % CL are shown in Table 4 and Table 5, respectively. For the $B^{0}_{s}\rightarrow\mu^{+}\mu^{-}$ decay, the expected limits are computed allowing the presence of $B^{0}_{s}\rightarrow\mu^{+}\mu^{-}$ events according to the SM branching fraction. For the $B^{0}\rightarrow\mu^{+}\mu^{-}$ decay the expected limit is computed in the background-only hypothesis and also allowing the presence of $B^{0}\rightarrow\mu^{+}\mu^{-}$ events with the SM rate: the two results are identical. In the determination of the limits, the cross-feed of $B^{0}_{s}\rightarrow\mu^{+}\mu^{-}$ ($B^{0}\rightarrow\mu^{+}\mu^{-}$) events in the $B^{0}$ ($B^{0}_{s}$) mass window has been taken into account assuming the SM rates. The observed $\textrm{CL}_{\textrm{b}}$ values are shown in the same tables. The comparison of the observed distribution of events with the expected background distribution results in a p-value $(1-\textrm{CL}_{\textrm{b}})$ of 5 % for the $B^{0}_{s}\rightarrow\mu^{+}\mu^{-}$ and 32 % for the $B^{0}\rightarrow\mu^{+}\mu^{-}$ decay. For the $B^{0}_{s}\rightarrow\mu^{+}\mu^{-}$ decay, the probability that the observed events are compatible with the sum of expected background events and signal events according to the SM rate is measured by $1-$$\textrm{CL}_{\textrm{s+b}}$ and it is 33%. The result obtained in 2011 with 0.37 fb-1 has been combined with the published result based on $\sim 37$ pb-1[11]. The expected and observed limits for 90 % and 95 % CL for the combined results are shown in Table 4 for the $B^{0}_{s}\rightarrow\mu^{+}\mu^{-}$ decay and in Table 5 for the $B^{0}\\!\rightarrow\mu^{+}\mu^{-}$ decay. Figure 9: $\textrm{CL}_{\textrm{s}}$ as a function of the assumed $\cal B$. Expected (observed) values are shown by dashed black (dotted blue) lines. The expected $\textrm{CL}_{\textrm{s}}$ values have been computed assuming a signal yield corresponding to the SM branching fractions. The green (grey) shaded areas cover the region of $\pm 1\sigma$ of compatible observations. The measured upper limits at 90% and 95% CL are also shown. Left: $B^{0}_{s}\rightarrow\mu^{+}\mu^{-}$, right: $B^{0}\rightarrow\mu^{+}\mu^{-}$. Table 2: Expected combinatorial background events, expected peaking ($B^{0}_{(s)}\rightarrow h^{+}h^{{}^{\prime}-}$) background events, expected signal events assuming the SM branching fraction prediction, and observed events in the $B^{0}_{s}\rightarrow\mu^{+}\mu^{-}$ search window. \| \| \| BDT ---\|---\|---\|--- \| \| \| 0\. – 0.25 \| 0.25 – 0.5 \| 0.5 – 0.75 \| 0.75 – 1. Invariant mass [$\,{\rm MeV}/c^{2}$ ] \| 5298 – 5318 \| Expected comb. bkg \| $575.5^{+6.5}_{-6.0}$ \| $6.96^{+0.63}_{-0.57}$ \| $1.19^{+0.39}_{-0.35}$ \| $0.111^{+0.083}_{-0.066}$ Expected peak. bkg \| $0.126^{+0.037}_{-0.030}$ \| $0.124^{+0.037}_{-0.030}$ \| $0.124^{+0.037}_{-0.030}$ \| $0.127^{+0.038}_{-0.031}$ Expected signal \| $0.059^{+0.023}_{-0.022}$ \| $0.0329^{+0.0128}_{-0.0095}$ \| $0.0415^{+0.0120}_{-0.0085}$ \| $0.0411^{+0.0135}_{-0.0099}$ Observed \| $533$ \| $10$ \| $1$ \| $0$ 5318 – 5338 \| Expected comb. bkg \| $566.8^{+6.3}_{-5.8}$ \| $6.90^{+0.61}_{-0.55}$ \| $1.16^{+0.38}_{-0.34}$ \| $0.109^{+0.079}_{-0.063}$ Expected peak. bkg \| $0.052^{+0.023}_{-0.018}$ \| $0.054^{+0.026}_{-0.019}$ \| $0.052^{+0.024}_{-0.018}$ \| $0.051^{+0.023}_{-0.018}$ Expected signal \| $0.205^{+0.073}_{-0.074}$ \| $0.114^{+0.040}_{-0.031}$ \| $0.142^{+0.036}_{-0.025}$ \| $0.142^{+0.042}_{-0.031}$ Observed \| $525$ \| $9$ \| $0$ \| $1$ 5338 – 5358 \| Expected comb. bkg \| $558.2^{+6.1}_{-5.6}$ \| $6.84^{+0.59}_{-0.54}$ \| $1.14^{+0.37}_{-0.33}$ \| $0.106^{+0.075}_{-0.060}$ Expected peak. bkg \| $0.024^{+0.028}_{-0.012}$ \| $0.025^{+0.026}_{-0.012}$ \| $0.024^{+0.027}_{-0.012}$ \| $0.025^{+0.025}_{-0.012}$ Expected signal \| $0.38^{+0.14}_{-0.14}$ \| $0.213^{+0.075}_{-0.058}$ \| $0.267^{+0.065}_{-0.047}$ \| $0.265^{+0.077}_{-0.058}$ Observed \| $561$ \| $6$ \| $2$ \| $1$ 5358 – 5378 \| Expected comb. bkg \| $549.8^{+6.0}_{-5.4}$ \| $6.77^{+0.57}_{-0.52}$ \| $1.11^{+0.36}_{-0.32}$ \| $0.103^{+0.073}_{-0.057}$ Expected peak. bkg \| $0.0145^{+0.0220}_{-0.0091}$ \| $0.0151^{+0.0230}_{-0.0091}$ \| $0.0153^{+0.0232}_{-0.0098}$ \| $0.015^{+0.023}_{-0.010}$ Expected signal \| $0.38^{+0.14}_{-0.14}$ \| $0.213^{+0.075}_{-0.057}$ \| $0.267^{+0.065}_{-0.047}$ \| $0.265^{+0.077}_{-0.057}$ Observed \| $515$ \| $7$ \| $0$ \| $0$ 5378 – 5398 \| Expected comb. bkg \| $541.5^{+5.8}_{-5.3}$ \| $6.71^{+0.55}_{-0.51}$ \| $1.09^{+0.34}_{-0.31}$ \| $0.101^{+0.070}_{-0.054}$ Expected peak. bkg \| $0.0115^{+0.0175}_{-0.0086}$ \| $0.0116^{+0.0177}_{-0.0090}$ \| $0.0118^{+0.0179}_{-0.0090}$ \| $0.0118^{+0.0179}_{-0.0088}$ Expected signal \| $0.204^{+0.073}_{-0.074}$ \| $0.114^{+0.040}_{-0.031}$ \| $0.142^{+0.036}_{-0.026}$ \| $0.141^{+0.042}_{-0.031}$ Observed \| $547$ \| $10$ \| $1$ \| $1$ 5398 – 5418 \| Expected comb. bkg \| $533.4^{+5.7}_{-5.2}$ \| $6.65^{+0.53}_{-0.49}$ \| $1.07^{+0.34}_{-0.30}$ \| $0.098^{+0.068}_{-0.051}$ Expected peak. bkg \| $0.0089^{+0.0136}_{-0.0065}$ \| $0.0088^{+0.0133}_{-0.0066}$ \| $0.0091^{+0.0138}_{-0.0070}$ \| $0.0090^{+0.0137}_{-0.0065}$ Expected signal \| $0.058^{+0.024}_{-0.021}$ \| $0.0323^{+0.0128}_{-0.0093}$ \| $0.0407^{+0.0120}_{-0.0087}$ \| $0.0402^{+0.0137}_{-0.0097}$ Observed \| $501$ \| $4$ \| $1$ \| $0$ Table 3: Expected combinatorial background events, expected peaking ($B^{0}_{(s)}\rightarrow h^{+}h^{{}^{\prime}-}$) background events, expected $B^{0}\\!\rightarrow\mu^{+}\mu^{-}$ signal events assuming the SM branching fraction, expected cross-feed events from $B^{0}_{s}\rightarrow\mu^{+}\mu^{-}$ assuming the SM branching fraction and observed events in the $B^{0}\rightarrow\mu^{+}\mu^{-}$ search window. \| \| \| BDT ---\|---\|---\|--- \| \| \| 0\. – 0.25 \| 0.25 – 0.5 \| 0.5 – 0.75 \| 0.75 – 1. Invariant mass [$\,{\rm MeV}/c^{2}$ ] \| 5212 – 5232 \| Expected comb. bkg \| $614.2^{+7.5}_{-7.0}$ \| $7.23^{+0.77}_{-0.68}$ \| $1.31^{+0.46}_{-0.40}$ \| $0.123^{+0.107}_{-0.072}$ Expected peak. bkg \| $0.203^{+0.038}_{-0.034}$ \| $0.206^{+0.038}_{-0.034}$ \| $0.203^{+0.037}_{-0.034}$ \| $0.205^{+0.038}_{-0.034}$ Cross-feed \| $0.0056^{+0.0021}_{-0.0020}$ \| $0.00312^{+0.00119}_{-0.00087}$ \| $0.00391^{+0.00107}_{-0.00078}$ \| $0.00387^{+0.00122}_{-0.00092}$ Expected signal \| $0.0070^{+0.0027}_{-0.0026}$ \| $0.0039^{+0.0015}_{-0.0011}$ \| $0.0049^{+0.0014}_{-0.0010}$ \| $0.0048^{+0.0016}_{-0.0012}$ \| Observed \| $554$ \| $6$ \| $0$ \| $2$ 5232 – 5252 \| Expected comb. bkg \| $605.0^{+7.2}_{-6.8}$ \| $7.17^{+0.74}_{-0.65}$ \| $1.29^{+0.44}_{-0.39}$ \| $0.121^{+0.102}_{-0.072}$ Expected peak. bkg \| $0.281^{+0.056}_{-0.049}$ \| $0.279^{+0.056}_{-0.049}$ \| $0.280^{+0.056}_{-0.049}$ \| $0.280^{+0.058}_{-0.050}$ Cross-feed \| $0.0071^{+0.0027}_{-0.0026}$ \| $0.0039^{+0.0015}_{-0.0011}$ \| $0.00496^{+0.00134}_{-0.00099}$ \| $0.0049^{+0.0016}_{-0.0012}$ Expected signal \| $0.0241^{+0.0086}_{-0.0087}$ \| $0.0135^{+0.0048}_{-0.0037}$ \| $0.0169^{+0.0042}_{-0.0031}$ \| $0.0167^{+0.0050}_{-0.0037}$ \| Observed \| $556$ \| $4$ \| $2$ \| $1$ 5252 – 5272 \| Expected comb. bkg \| $595.9^{+7.0}_{-6.5}$ \| $7.10^{+0.71}_{-0.63}$ \| $1.26^{+0.42}_{-0.37}$ \| $0.119^{+0.097}_{-0.072}$ Expected peak. bkg \| $0.323^{+0.075}_{-0.061}$ \| $0.326^{+0.074}_{-0.061}$ \| $0.324^{+0.072}_{-0.060}$ \| $0.325^{+0.075}_{-0.062}$ Cross-feed \| $0.0097^{+0.0036}_{-0.0035}$ \| $0.0054^{+0.0021}_{-0.0015}$ \| $0.0068^{+0.0018}_{-0.0013}$ \| $0.0067^{+0.0021}_{-0.0016}$ Expected signal \| $0.045^{+0.016}_{-0.016}$ \| $0.0252^{+0.0088}_{-0.0067}$ \| $0.0317^{+0.0077}_{-0.0057}$ \| $0.0313^{+0.0093}_{-0.0068}$ \| Observed \| $588$ \| $11$ \| $1$ \| $0$ 5272 – 5292 \| Expected comb. bkg \| $586.9^{+6.7}_{-6.3}$ \| $7.04^{+0.68}_{-0.60}$ \| $1.23^{+0.41}_{-0.36}$ \| $0.117^{+0.092}_{-0.071}$ Expected peak. bkg \| $0.252^{+0.058}_{-0.047}$ \| $0.252^{+0.056}_{-0.046}$ \| $0.253^{+0.059}_{-0.048}$ \| $0.250^{+0.056}_{-0.046}$ Cross-feed \| $0.0154^{+0.0058}_{-0.0055}$ \| $0.0086^{+0.0033}_{-0.0024}$ \| $0.0108^{+0.0029}_{-0.0021}$ \| $0.0106^{+0.0033}_{-0.0025}$ Expected signal \| $0.045^{+0.016}_{-0.016}$ \| $0.0251^{+0.0089}_{-0.0067}$ \| $0.0317^{+0.0077}_{-0.0057}$ \| $0.0313^{+0.0092}_{-0.0069}$ \| Observed \| $616$ \| $5$ \| $2$ \| $1$ 5292 – 5312 \| Expected comb. bkg \| $578.1^{+6.5}_{-6.1}$ \| $6.98^{+0.66}_{-0.58}$ \| $1.20^{+0.39}_{-0.35}$ \| $0.114^{+0.087}_{-0.067}$ Expected peak. bkg \| $0.124^{+0.023}_{-0.021}$ \| $0.124^{+0.023}_{-0.021}$ \| $0.123^{+0.023}_{-0.021}$ \| $0.124^{+0.023}_{-0.021}$ Cross-feed \| $0.038^{+0.015}_{-0.014}$ \| $0.0214^{+0.0086}_{-0.0061}$ \| $0.0270^{+0.0080}_{-0.0056}$ \| $0.0266^{+0.0089}_{-0.0064}$ Expected signal \| $0.0241^{+0.0086}_{-0.0087}$ \| $0.0134^{+0.0048}_{-0.0036}$ \| $0.0169^{+0.0042}_{-0.0030}$ \| $0.0167^{+0.0050}_{-0.0037}$ \| Observed \| $549$ \| $7$ \| $0$ \| $0$ 5312 – 5332 \| Expected comb. bkg \| $569.3^{+6.3}_{-5.9}$ \| $6.92^{+0.63}_{-0.57}$ \| $1.18^{+0.38}_{-0.34}$ \| $0.111^{+0.083}_{-0.064}$ Expected peak. bkg \| $0.047^{+0.023}_{-0.012}$ \| $0.047^{+0.022}_{-0.012}$ \| $0.047^{+0.021}_{-0.012}$ \| $0.047^{+0.021}_{-0.012}$ Cross-feed \| $0.149^{+0.055}_{-0.054}$ \| $0.083^{+0.031}_{-0.022}$ \| $0.104^{+0.027}_{-0.019}$ \| $0.103^{+0.031}_{-0.023}$ Expected signal \| $0.0068^{+0.0028}_{-0.0026}$ \| $0.0038^{+0.0015}_{-0.0011}$ \| $0.0048^{+0.0014}_{-0.0010}$ \| $0.0048^{+0.0016}_{-0.0012}$ \| Observed \| $509$ \| $10$ \| $1$ \| $1$ Table 4: Expected and observed limits on the $B^{0}_{s}\rightarrow\mu^{+}\mu^{-}$ branching fraction for the 2011 data and for the combination of 2010 and 2011 data. The expected limits are computed allowing the presence of $B^{0}_{s}\rightarrow\mu^{+}\mu^{-}$ events according to the SM branching fraction. \| \| at 90% CL \| at 95% CL \| $\textrm{CL}_{\textrm{b}}$ ---\|---\|---\|---\|--- 2011 \| expected limit \| $1.1\times 10^{-8}$ \| $1.4\times 10^{-8}$ \| \| observed limit \| $1.3\times 10^{-8}$ \| $1.6\times 10^{-8}$ \| 0.95 2010+2011 \| expected limit \| $1.0\times 10^{-8}$ \| $1.3\times 10^{-8}$ \| \| observed limit \| $1.2\times 10^{-8}$ \| $1.4\times 10^{-8}$ \| 0.93 Table 5: Expected and observed limits on the $B^{0}\rightarrow\mu^{+}\mu^{-}$ branching fraction for 2011 data and for the combination of 2010 and 2011 data. The expected limits are computed in the background only hypothesis. \| \| at 90% CL \| at 95% CL \| $\textrm{CL}_{\textrm{b}}$ ---\|---\|---\|---\|--- 2011 \| expected limit \| $2.5\times 10^{-9}$ \| $3.2\times 10^{-9}$ \| \| observed limit \| $3.0\times 10^{-9}$ \| $3.6\times 10^{-9}$ \| 0.68 2010+2011 \| expected limit \| $2.4\times 10^{-9}$ \| $3.0\times 10^{-9}$ \| \| observed limit \| $2.6\times 10^{-9}$ \| $3.2\times 10^{-9}$ \| 0.61 ## 8 Conclusions With 0.37$\mbox{\,fb}^{-1}$ of integrated luminosity, a search for the rare decays $B^{0}_{s}\rightarrow\mu^{+}\mu^{-}$ and $B^{0}\rightarrow\mu^{+}\mu^{-}$ has been performed and sensitivities better than the existing limits have been obtained. The observed events in the $B^{0}_{s}$ and in the $B^{0}$ mass windows are compatible with the background expectations at 5% and 32% confidence level, respectively. For the $B^{0}_{s}\rightarrow\mu^{+}\mu^{-}$ decay, the probability that the observed events are compatible with the sum of expected background events and signal events according to the SM rate is 33%. The upper limits for the branching fractions are evaluated to be $\displaystyle{\cal B}(B^{0}_{s}\\!\rightarrow\mu^{+}\mu^{-})$ $\displaystyle<$ $\displaystyle 1.3\,(1.6)\times 10^{-8}~{}{\rm at}~{}90\,\%\,(95\,\%)~{}{\rm CL},$ $\displaystyle{\cal B}(B^{0}\\!\rightarrow\mu^{+}\mu^{-})$ $\displaystyle<$ $\displaystyle 3.0\,(3.6)\times 10^{-9}~{}{\rm at}~{}90\,\%\,(95\,\%)~{}{\rm CL}.$ The ${\cal B}(B^{0}_{s}\rightarrow\mu^{+}\mu^{-})$ and ${\cal B}(B^{0}\rightarrow\mu^{+}\mu^{-})$ upper limits have been combined with those published previously by LHCb [11] and the results are $\displaystyle{\cal B}(B^{0}_{s}\\!\rightarrow\mu^{+}\mu^{-})(2010+2011)$ $\displaystyle<$ $\displaystyle 1.2\,(1.4)\times 10^{-8}~{}{\rm at}~{}90\,\%\,(95\,\%)~{}{\rm CL},$ $\displaystyle{\cal B}(B^{0}\\!\rightarrow\mu^{+}\mu^{-})(2010+2011)$ $\displaystyle<$ $\displaystyle 2.6\,(3.2)\times 10^{-9}~{}{\rm at}~{}90\,\%\,(95\,\%)~{}{\rm CL}.$ The above 90% (95%) CL upper limits are still about 3.8 (4.4) times the SM branching fractions for the $B^{0}_{s}$ and 26 (32) times for the $B^{0}$. These results represent the best upper limits to date. ## Acknowledgements We express our gratitude to our colleagues in the CERN accelerator departments for the excellent performance of the LHC. We thank the technical and administrative staff at CERN and at the LHCb institutes, and acknowledge support from the National Agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil); CERN; NSFC (China); CNRS/IN2P3 (France); BMBF, DFG, HGF and MPG (Germany); SFI (Ireland); INFN (Italy); FOM and NWO (The Netherlands); SCSR (Poland); ANCS (Romania); MinES of Russia and Rosatom (Russia); MICINN, XuntaGal and GENCAT (Spain); SNSF and SER (Switzerland); NAS Ukraine (Ukraine); STFC (United Kingdom); NSF (USA). We also acknowledge the support received from the ERC under FP7 and the Region Auvergne. ## References * [1] A. J. Buras, M. V. Carlucci, S. Gori, and G. Isidori, Higgs-mediated FCNCs: Natural Flavour Conservation vs. Minimal Flavour Violation, JHEP 1010 (2010) 009, [arXiv:1005.5310] * [2] A. J. Buras, Minimal Flavour Violation and Beyond: Towards a Flavour Code for Short Distance Dynamics, Acta Phys.Polon. B41 (2010) 2487–2561, [arXiv:1012.1447] * [3] S. Choudhury and N. Gaur, Dileptonic decay of B(s) meson in SUSY models with large tan Beta, Phys.Lett. B451 (1999) 86–92, [arXiv:hep-ph/9810307] * [4] C. Hamzaoui, M. Pospelov, and M. Toharia, Higgs Mediated FCNC in Supersymmetric Models with Large $\tan\beta$, Phys.Rev. D59 (1999) 095005, [arXiv:hep-ph/9807350] * [5] K. Babu and C. F. Kolda, Higgs Mediated $B^{0}\rightarrow\mu^{+}\mu^{-}$ in Minimal Supersymmetry, Phys.Rev.Lett. 84 (2000) 228–231, [arXiv:hep-ph/9909476] * [6] L. J. Hall, R. Rattazzi, and U. Sarid, The Top Quark Mass in Supersymmetric $SO$(10) Unification, Phys.Rev. D50 (1994) 7048–7065, [arXiv:hep-ph/9306309] * [7] C.-S. Huang, W. Liao, and Q.-S. Yan, The Promising process to distinguish supersymmetric models with large $\tan\beta$ from the standard model: $B\rightarrow X(s)\mu^{+}\mu^{-}$, Phys.Rev. D59 (1999) 011701, [arXiv:hep-ph/9803460] * [8] DØ collaboration, V. M. Abazov et al., Search for the Rare Decay $B^{0}_{s}\rightarrow\mu^{+}\mu^{-}$, Phys.Lett. B693 (2010) 539–544, [arXiv:1006.3469] * [9] CDF collaboration, T. Aaltonen et al., Search for $B^{0}_{s}\rightarrow\mu^{+}\mu^{-}$ and $B^{0}\rightarrow\mu^{+}\mu^{-}$ Decays with CDF II, Phys.Rev.Lett. 107 (2011) 191801, [arXiv:1107.2304] * [10] CMS Collaboration, S. Chatrchyan et al., Search for $B^{0}_{s}\rightarrow\mu^{+}\mu^{-}$ and $B^{0}\rightarrow\mu^{+}\mu^{-}$ decays in $pp$ Collisions at 7 TeV, Phys.Rev.Lett. 107 (2011) 191802, [arXiv:1107.5834] * [11] LHCb collaboration, R. Aaij et al., Search for the Rare Decays $B^{0}_{s}\rightarrow\mu^{+}\mu^{-}$ and $B^{0}\rightarrow\mu^{+}\mu^{-}$, Phys.Lett. B699 (2011) 330–340, [arXiv:1103.2465] * [12] LHCb collaboration, A. Alves et al., The LHCb Detector at the LHC, JINST 3 (2008) S08005. and references therein * [13] LHCb collaboration, R. Aaij et al., Measurement of $\sigma(pp\rightarrow b\overline{b}X)$ at $\sqrt{s}=7\mathrm{\,Te\kern-1.00006ptV}$ in the Forward Region, Phys.Lett. B694 (2010) 209–216, [arXiv:1009.2731] * [14] P. Speckmayer, A. Hocker, J. Stelzer, and H. Voss, The Toolkit for Multivariate Data Analysis TMVA 4, J.Phys.Conf.Ser. 219 (2010) 032057 * [15] T. Skwarnicki, A Study of the Radiative Cascade Transitions Between the $\Upsilon$ and $\Upsilon^{\prime}$ Resonances. PhD thesis, DESY F31-86-02, 1986. Appendix E * [16] A. Read, Presentation of Search Results: The CLs Technique, J.Phys. G28 (2002) 2693–2704 * [17] S. Kullback and R. A. Leibler, On Information and Sufficiency, Ann.Math.Statist. 22 (1951) 79–86 * [18] S. Kullback, Letter to Editor: the Kullback-Leibler Distance, The American Statistician 41 (1987) 340. The use of the Kullback-Leibler distance at LHCb is described in M. Needham, Clone Track Identification Using the Kullback-Leibler Distance, LHCb-2008-002. * [19] Particle Data Group, K. Nakamura et al., Review of particle physics, J.Phys. G37 (2010) 075021 * [20] CDF collaboration, A. Abulencia et al., Search for $B_{s}\rightarrow\mu^{+}\mu^{-}$ and $B_{d}\rightarrow\mu^{+}\mu^{-}$ Decays in $p\bar{p}$ Collisions with CDF II, Phys.Rev.Lett. 95 (2005) 221805, [arXiv:hep-ex/0508036] * [21] A. Carbone et al., Invariant Mass Line Shape of $B\rightarrow h^{+}h^{-}$ Decays at LHCb, LHCb-PUB-2009-031 * [22] ARGUS collaboration, H. Albrecht et al., Search for Hadronic $b\rightarrow u$ Decays, Phys.Lett. B241 (1990) 278–282 * [23] LHCb Collaboration, R. Aaij et al., Determination of $f_{s}/f_{d}$ for 7 TeV $pp$ Collisions and a Measurement of the Branching Fraction of the Decay $B\rightarrow D^{-}K^{+}$, Phys.Rev.Lett. 107 (2011) 211801, [arXiv:1106.4435] * [24] Belle collaboration, R. Louvot, $\Upsilon$(5S) Results at Belle, arXiv:0905.4345	arxiv-papers	2011-12-07T15:40:33	2024-09-04T02:49:25.067414	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "LHCb Collaboration: R. Aaij, C. Abellan Beteta, B. Adeva, M. Adinolfi,\n C. Adrover, A. Affolder, Z. Ajaltouni, J. Albrecht, F. Alessio, M. Alexander,\n G. Alkhazov, P. Alvarez Cartelle, A. A. Alves Jr, S. Amato, Y. Amhis, J.\n Anderson, R. B. Appleby, O. Aquines Gutierrez, F. Archilli, L. Arrabito, A.\n Artamonov, M. Artuso, E. Aslanides, G. Auriemma, S. Bachmann, J. J. Back, D.\n S. Bailey, V. Balagura, W. Baldini, R. J. Barlow, C. Barschel, S. Barsuk, W.\n Barter, A. Bates, C. Bauer, Th. Bauer, A. Bay, I. Bediaga, S. Belogurov, K.\n Belous, I. Belyaev, E. Ben-Haim, M. Benayoun, G. Bencivenni, S. Benson, J.\n Benton, R. Bernet, M.-O. Bettler, M. van Beuzekom, A. Bien, S. Bifani, T.\n Bird, A. Bizzeti, P. M. Bj{\\o}rnstad, T. Blake, F. Blanc, C. Blanks, J.\n Blouw, S. Blusk, A. Bobrov, V. Bocci, A. Bondar, N. Bondar, W. Bonivento, S.\n Borghi, A. Borgia, T. J. V. Bowcock, C. Bozzi, T. Brambach, J. van den Brand,\n J. Bressieux, D. Brett, M. Britsch, T. Britton, N. H. Brook, H. Brown, A.\n B\\\"uchler-Germann, I. Burducea, A. Bursche, J. Buytaert, S. Cadeddu, O.\n Callot, M. Calvi, M. Calvo Gomez, A. Camboni, P. Campana, A. Carbone, G.\n Carboni, R. Cardinale, A. Cardini, L. Carson, K. Carvalho Akiba, G. Casse, M.\n Cattaneo, Ch. Cauet, M. Charles, Ph. Charpentier, N. Chiapolini, K. Ciba, X.\n Cid Vidal, G. Ciezarek, P. E. L. Clarke, M. Clemencic, H. V. Cliff, J.\n Closier, C. Coca, V. Coco, J. Cogan, P. Collins, A. Comerma-Montells, F.\n Constantin, G. Conti, A. Contu, A. Cook, M. Coombes, G. Corti, G. A. Cowan,\n R. Currie, B. D'Almagne, C. D'Ambrosio, P. David, P. N. Y. David, I. De\n Bonis, S. De Capua, M. De Cian, F. De Lorenzi, J. M. De Miranda, L. De Paula,\n P. De Simone, D. Decamp, M. Deckenhoff, H. Degaudenzi, M. Deissenroth, L. Del\n Buono, C. Deplano, D. Derkach, O. Deschamps, F. Dettori, J. Dickens, H.\n Dijkstra, P. Diniz Batista, F. Domingo Bonal, S. Donleavy, F. Dordei, P.\n Dornan, A. Dosil Su\\'arez, D. Dossett, A. Dovbnya, F. Dupertuis, R.\n Dzhelyadin, A. Dziurda, S. Easo, U. Egede, V. Egorychev, S. Eidelman, D. van\n Eijk, F. Eisele, S. Eisenhardt, R. Ekelhof, L. Eklund, Ch. Elsasser, D.\n Elsby, D. Esperante Pereira, L. Est\\'eve, A. Falabella, E. Fanchini, C.\n F\\\"arber, G. Fardell, C. Farinelli, S. Farry, V. Fave, V. Fernandez Albor, M.\n Ferro-Luzzi, S. Filippov, C. Fitzpatrick, M. Fontana, F. Fontanelli, R.\n Forty, M. Frank, C. Frei, M. Frosini, S. Furcas, A. Gallas Torreira, D.\n Galli, M. Gandelman, P. Gandini, Y. Gao, J-C. Garnier, J. Garofoli, J. Garra\n Tico, L. Garrido, D. Gascon, C. Gaspar, N. Gauvin, M. Gersabeck, T. Gershon,\n Ph. Ghez, V. Gibson, V. V. Gligorov, C. G\\\"obel, D. Golubkov, A. Golutvin, A.\n Gomes, H. Gordon, M. Grabalosa G\\'andara, R. Graciani Diaz, L. A. Granado\n Cardoso, E. Graug\\'es, G. Graziani, A. Grecu, E. Greening, S. Gregson, B.\n Gui, E. Gushchin, Yu. Guz, T. Gys, G. Haefeli, C. Haen, S. C. Haines, T.\n Hampson, S. Hansmann-Menzemer, R. Harji, N. Harnew, J. Harrison, P. F.\n Harrison, J. He, V. Heijne, K. Hennessy, P. Henrard, J. A. Hernando Morata,\n E. van Herwijnen, E. Hicks, K. Holubyev, P. Hopchev, W. Hulsbergen, P. Hunt,\n T. Huse, R. S. Huston, D. Hutchcroft, D. Hynds, V. Iakovenko, P. Ilten, J.\n Imong, R. Jacobsson, A. Jaeger, M. Jahjah Hussein, E. Jans, F. Jansen, P.\n Jaton, B. Jean-Marie, F. Jing, M. John, D. Johnson, C. R. Jones, B. Jost, M.\n Kaballo, S. Kandybei, M. Karacson, T. M. Karbach, J. Keaveney, I. R. Kenyon,\n U. Kerzel, T. Ketel, A. Keune, B. Khanji, Y. M. Kim, M. Knecht, P.\n Koppenburg, A. Kozlinskiy, L. Kravchuk, K. Kreplin, M. Kreps, G. Krocker, P.\n Krokovny, F. Kruse, K. Kruzelecki, M. Kucharczyk, T. Kvaratskheliya, V. N. La\n Thi, D. Lacarrere, G. Lafferty, A. Lai, D. Lambert, R. W. Lambert, E.\n Lanciotti, G. Lanfranchi, C. Langenbruch, T. Latham, C. Lazzeroni, R. Le Gac,\n J. van Leerdam, J.-P. Lees, R. Lef\\'evre, A. Leflat, J. Lefran\\c{c}ois, O.\n Leroy, T. Lesiak, L. Li, L. Li Gioi, M. Lieng, M. Liles, R. Lindner, C. Linn,\n B. Liu, G. Liu, J. H. Lopes, E. Lopez Asamar, N. Lopez-March, H. Lu, J.\n Luisier, A. Mac Raighne, F. Machefert, I. V. Machikhiliyan, F. Maciuc, O.\n Maev, J. Magnin, S. Malde, R. M. D. Mamunur, G. Manca, G. Mancinelli, N.\n Mangiafave, U. Marconi, R. M\\\"arki, J. Marks, G. Martellotti, A. Martens, L.\n Martin, A. Mart\\'in S\\'anchez, D. Martinez Santos, A. Massafferri, Z. Mathe,\n C. Matteuzzi, M. Matveev, E. Maurice, B. Maynard, A. Mazurov, G. McGregor, R.\n McNulty, C. Mclean, M. Meissner, M. Merk, J. Merkel, R. Messi, S.\n Miglioranzi, D. A. Milanes, M.-N. Minard, J. Molina Rodriguez, S. Monteil, D.\n Moran, P. Morawski, R. Mountain, I. Mous, F. Muheim, K. M\\\"uller, R. Muresan,\n B. Muryn, B. Muster, M. Musy, J. Mylroie-Smith, P. Naik, T. Nakada, R.\n Nandakumar, I. Nasteva, M. Nedos, M. Needham, N. Neufeld, C. Nguyen-Mau, M.\n Nicol, V. Niess, N. Nikitin, A. Nomerotski, A. Novoselov, A.\n Oblakowska-Mucha, V. Obraztsov, S. Oggero, S. Ogilvy, O. Okhrimenko, R.\n Oldeman, M. Orlandea, J. M. Otalora Goicochea, P. Owen, K. Pal, J. Palacios,\n A. Palano, M. Palutan, J. Panman, A. Papanestis, M. Pappagallo, C. Parkes, C.\n J. Parkinson, G. Passaleva, G. D. Patel, M. Patel, S. K. Paterson, G. N.\n Patrick, C. Patrignani, C. Pavel-Nicorescu, A. Pazos Alvarez, A. Pellegrino,\n G. Penso, M. Pepe Altarelli, S. Perazzini, D. L. Perego, E. Perez Trigo, A.\n P\\'erez-Calero Yzquierdo, P. Perret, M. Perrin-Terrin, G. Pessina, A.\n Petrella, A. Petrolini, A. Phan, E. Picatoste Olloqui, B. Pie Valls, B.\n Pietrzyk, T. Pila\\v{r}, D. Pinci, R. Plackett, S. Playfer, M. Plo Casasus, G.\n Polok, A. Poluektov, E. Polycarpo, D. Popov, B. Popovici, C. Potterat, A.\n Powell, T. du Pree, J. Prisciandaro, V. Pugatch, A. Puig Navarro, W. Qian, J.\n H. Rademacker, B. Rakotomiaramanana, M. S. Rangel, I. Raniuk, G. Raven, S.\n Redford, M. M. Reid, A. C. dos Reis, S. Ricciardi, K. Rinnert, D. A. Roa\n Romero, P. Robbe, E. Rodrigues, F. Rodrigues, P. Rodriguez Perez, G. J.\n Rogers, S. Roiser, V. Romanovsky, M. Rosello, J. Rouvinet, T. Ruf, H. Ruiz,\n G. Sabatino, J. J. Saborido Silva, N. Sagidova, P. Sail, B. Saitta, C.\n Salzmann, M. Sannino, R. Santacesaria, C. Santamarina Rios, R. Santinelli, E.\n Santovetti, M. Sapunov, A. Sarti, C. Satriano, A. Satta, M. Savrie, D.\n Savrina, P. Schaack, M. Schiller, S. Schleich, M. Schlupp, M. Schmelling, B.\n Schmidt, O. Schneider, A. Schopper, M.-H. Schune, R. Schwemmer, B. Sciascia,\n A. Sciubba, M. Seco, A. Semennikov, K. Senderowska, I. Sepp, N. Serra, J.\n Serrano, P. Seyfert, B. Shao, M. Shapkin, I. Shapoval, P. Shatalov, Y.\n Shcheglov, T. Shears, L. Shekhtman, O. Shevchenko, V. Shevchenko, A. Shires,\n R. Silva Coutinho, T. Skwarnicki, A. C. Smith, N. A. Smith, E. Smith, K.\n Sobczak, F. J. P. Soler, A. Solomin, F. Soomro, B. Souza De Paula, B. Spaan,\n A. Sparkes, P. Spradlin, F. Stagni, S. Stahl, O. Steinkamp, S. Stoica, S.\n Stone, B. Storaci, M. Straticiuc, U. Straumann, V. K. Subbiah, S. Swientek,\n M. Szczekowski, P. Szczypka, T. Szumlak, S. T'Jampens, E. Teodorescu, F.\n Teubert, C. Thomas, E. Thomas, J. van Tilburg, V. Tisserand, M. Tobin, S.\n Topp-Joergensen, N. Torr, E. Tournefier, M. T. Tran, A. Tsaregorodtsev, N.\n Tuning, M. Ubeda Garcia, A. Ukleja, P. Urquijo, U. Uwer, V. Vagnoni, G.\n Valenti, R. Vazquez Gomez, P. Vazquez Regueiro, S. Vecchi, J. J. Velthuis, M.\n Veltri, B. Viaud, I. Videau, X. Vilasis-Cardona, J. Visniakov, A. Vollhardt,\n D. Volyanskyy, D. Voong, A. Vorobyev, H. Voss, S. Wandernoth, J. Wang, D. R.\n Ward, N. K. Watson, A. D. Webber, D. Websdale, M. Whitehead, D. Wiedner, L.\n Wiggers, G. Wilkinson, M. P. Williams, M. Williams, F. F. Wilson, J. Wishahi,\n M. Witek, W. Witzeling, S. A. Wotton, K. Wyllie, Y. Xie, F. Xing, Z. Xing, Z.\n Yang, R. Young, O. Yushchenko, M. Zavertyaev, F. Zhang, L. Zhang, W. C.\n Zhang, Y. Zhang, A. Zhelezov, L. Zhong, E. Zverev, A. Zvyagin", "submitter": "Gaia Lanfranchi", "url": "https://arxiv.org/abs/1112.1600" }
1112.1818	# Bubble tree of a class of conformal mappings and applications to Willmore functional Jingyi Chen & Yuxiang Li ###### Abstract. We develop a bubble tree construction and prove compactness results for $W^{2,2}$ branched conformal immersions of closed Riemann surfaces, with varying conformal structures whose limit may degenerate, in a compact Riemannian manifold with uniformly bounded areas and Willmore energies. The compactness property is applied to construct Willmore type surfaces in compact Riemannian manifolds. This includes (a) existence of a Willmore 2-sphere in ${\mathbb{S}}^{n}$ with at least 2 nonremovable singular points (b) existence of minimizers of the Willmore functional with prescribed area in a compact manifold $N$ provided (i) the area is small when genus is 0 and (ii) the area is close to that of the area minimizing surface of Schoen-Yau and Sacks- Uhlenbeck in the homotopy class of an incompressible map from a surface of positive genus to $N$ and $\pi_{2}(N)$ is trivial (c) existence of smooth minimizers of the Willmore functional if a Douglas type condition is satisfied. The first author is partially supported by NSERC. The second author was partially supported by NSERC during his visit to UBC in the spring of 2011 where most part of this work was done ## 1\. Introduction Let $\Sigma$ be a smooth Riemann surface and $\ f:\Sigma\rightarrow\mathbb{R}^{n}\ $ be a smooth immersion. The Willmore functional of $f$ is defined by $W(f)=\frac{1}{4}\int_{\Sigma}\|H_{f}\|^{2}d\mu_{f}$ where $H_{f}=\Delta_{g_{f}}f$ denotes the mean curvature vector of $f$, and $\Delta_{g_{f}}$ is the Laplace operator in the induced metric $g_{f}$ and $d\mu_{f}$ the induced area element on $\Sigma$. For a sequence of immersions $f_{k}$ of a compact surfaces $\Sigma$ in ${\mathbb{R}}^{n}$ with uniformly bounded areas $\mu(f_{k})$ and Willmore functionals $W(f_{k})$, a subsequence of $\Sigma_{k}=f_{k}(\Sigma)$ converges, as Radon measures, to a two dimensional integral varifold, by Allard’s integral compactness theorem. The second fundamental forms $A_{f_{k}}$ are uniformly bounded in the $L^{2}$-norm as $\int_{\Sigma}\|A_{f_{k}}\|^{2}d\mu_{f_{k}}=4W(f_{k})-4\pi\chi(\Sigma)$ from the Gauss equation and the Gauss-Bonnet formula. In general $\\|f_{k}\\|_{W^{2,2}}$ are not uniformly bounded: we can find diffeomorphisms $\phi_{k}$ from $\Sigma$ to $\Sigma$ such that $f_{k}=f\circ\phi_{k}$ diverge in $C^{0}$, while a uniform bound on $\\|f_{k}\\|_{W^{2,2}}$ would imply sequential convergence in $C^{0}$ (in fact $C^{\alpha},0<\alpha<1$) norm by the Rellich-Kondrachov embedding theorem. A recent advance in understanding the limit process is given in [13], where each $f_{k}$ is a conformal immersion from a Riemann surface $(\Sigma,h_{k})$ into $\mathbb{R}^{n}$ and $h_{k}$ is the smooth metric of constant curvature: (1.1) $\begin{array}[]{l}h_{k}\mbox{ has Gauss curvature }\pm 1,\mbox{ or }(\Sigma,h_{k})=\mathbb{C}/\\{1,a+bi\\}\mbox{ with }\\\ -\frac{1}{2}<a\leq\frac{1}{2},\,\,\,\,b\geq 0,a^{2}+b^{2}\geq 1\mbox{ and }a\geq 0\mbox{ whenever }a^{2}+b^{2}=1.\end{array}$ There are two reasons to use conformal immersions. One is that the conformal diffeomorphism group of $(\Sigma,h_{k})$ is rather small comparing with the group of diffeomorphisms. Secondly, if we set $g_{f_{k}}=e^{2u_{k}}g_{euc}$ in an isothermal coordinate system, then we can estimate $\\|u_{k}\\|_{L^{\infty}}$ from the compensated compactness property of $K_{f_{k}}e^{2u_{k}}$. Thus it is possible to get an upper bound of $\\|f_{k}\\|_{W^{2,2}}$ via the equation $\Delta_{h_{k}}f_{k}=H_{f_{k}}$. When the conformal structures determined by $f_{k}$ do not go to the boundary of the moduli space, convergence of $f_{k}$ is treated in [13]: if the conformal classes induced by $f_{k}$ converge in the moduli space, then there exist Mobius transformations $\sigma_{k}$, such that $\sigma_{k}\circ f_{k}$ converge locally in weak $W^{2,2}$ sense on $\Sigma$ minus finitely many concentration points. The weak limit $f_{0}$ is a $W^{2,2}$ branched conformal immersion. The $W^{2,2}$ conformal immersions and $W^{2,2}$ branched conformal immersions are as follows: ###### Definition 1. Let $(\Sigma,h)$ be a connected Riemann surface with $h$ satisfies (1.1). A map $f\in W^{2,2}(\Sigma,h,\mathbb{R}^{n})$ is called a conformal immersion of $(\Sigma,h)$, if $df\otimes df=e^{2u}h\,\,\,\,\hbox{with}\,\,\,\,\\|u\\|_{L^{\infty}(\Sigma)}<+\infty.$ We denote the set of all such immersions by $W^{2,2}_{conf}(\Sigma,h,\mathbb{R}^{n})$. It can be shown that for $f\in W^{2,2}_{conf}(\Sigma,\mathbb{R}^{n})$ the corresponding $u$ is continuous (see Appendix). When $f\in W^{2,2}_{loc}(\Sigma,h,\mathbb{R}^{n})$ with $df\otimes df=e^{2u}h$ and $u\in L^{\infty}_{loc}(\Sigma)$, we say $f\in W^{2,2}_{conf,loc}(\Sigma,h,\mathbb{R}^{n})$. ###### Definition 2. We say $f$ is a $W^{2,2}$ branched conformal immersion of $(\Sigma,h)$ with possible branch points $x_{1},\dots,x_{m}$, if $f\in W^{2,2}_{conf,loc}(\Sigma\backslash\\{x_{1},\dots,x_{m}\\},h,\mathbb{R}^{n})$ and $\int_{\Sigma\backslash\\{x_{1},\dots,x_{m}\\}}(1+\|A_{f}\|^{2})d\mu_{f}<+\infty.$ The set of $W^{2,2}$ branched conformal immersions is denoted by $W^{2,2}_{b,c}(\Sigma,h,\mathbb{R}^{n})$. We say $f\in\widetilde{W}^{2,2}(\Sigma,\mathbb{R}^{n})$, if we can find a smooth metric $h$ satisfying (1.1) over $\Sigma$, such that $f\in W^{2,2}_{b,c}(\Sigma,h,\mathbb{R}^{n})$. The first part of the paper is a study of a sequence of $W^{2,2}$ branched conformal immersions and the main goal is to establish compactness in Hausdorff distance for such immersions with uniformly bounded areas and Willmore functionals (cf. Theorem 1). Our compactness result holds not only when $h_{k}$ converges, but also when the conformal classes $c_{k}$ of $h_{k}$ diverge in the moduli space $\mathcal{M}_{g}$. Bubbles develop near points where the Willmore energy concentrates, and if $c_{k}$ go to a point in the boundary $\overline{\mathcal{M}}_{g}\backslash\mathcal{M}_{g}$ additional complication arises as the topology of the limit may be different from that of $\Sigma$ and stratified surfaces are used as possible limits. The main idea to deal with degenerating conformal structures in the limit process is as follows. First, pull the immersions $f_{k}$ to the immersions from components of $\Sigma_{0}$, by composing $f_{k}$ with diffeomorphisms from the regular parts of the limit $\Sigma_{0}$ of $(\Sigma,h_{k})$ in $\overline{\mathcal{M}_{p}}$. Then we study the limit of $f_{k}$ and construct bubble trees at the energy concentration points and collars, and investigate behavior between bubbles. In particular, we will prove that there is no loss of measure in the limit and there is no neck between the bubbles. Then the limit $f_{0}$ of $f_{k}$ is a union of conformal maps from some components $\overline{\Sigma_{0}^{1}},\dots,\overline{\Sigma_{0}^{m}}$ of $\Sigma_{0}$ (we delete those components whose images are points) and finitely many 2-spheres $S_{1},\dots,S_{l}$ into $\mathbb{R}^{n}$. “no neck” means that we can glue $\overline{\Sigma_{0}^{i}}$’s and $S_{j}$’s to a stratified surface $\Sigma_{\infty}$ (see definition below), and $f_{0}$ is a continuous map from $\Sigma_{\infty}$ into $\mathbb{R}^{n}$. Then we will apply a result of Hélein [10] and a removable singularity theorem in [13] to show that for a sequence of branched conformal immersions with uniformly bounded measures and Willmore functionals, the limit we get in section 2 is in fact a branched conformal immersion of a stratified surface. We point out that the “no loss of measure” and “no neck” phenomenon must occur whenever the following two equations hold: (1.2) $-\Delta f_{k}=\frac{1}{2}\|\nabla f_{k}\|^{2}H_{k},\,\,\,\,\mbox{with}\,\,\,\,\sup_{k}\int\|\nabla f_{k}\|^{2}(1+\|H_{k}\|^{2})<\infty,$ (1.3) $\partial f_{k}\otimes\partial f_{k}=0\,\,\,(\hbox{weakly conformal})$ where $\Delta,\nabla,\partial=\partial/\partial z$ are the operators in $h_{k}$ and its conformal structure $c_{k}$. In section 2, we study the blowup behavior of a sequence satisfies (1.2) and (1.3). The equation (1.2) looks similar to the equation of harmonic maps $-\Delta u=A(u)(du,du).$ In fact, the arguments in section 2 are originated from the “energy identity” and “no neck” arguments of harmonic maps [2, 5, 16, 18, 19, 23, 24, 25, 28]. When conformal structures go to the boundary of ${\mathcal{M}}_{g}$, non- trivial necks exist for harmonic map ( [2, 3, 23, 34]); in our case, however, there is no non-trivial neck due to conformality although (1.2) is much weaker than the harmonic map equation. ###### Definition 3. Let $(\Sigma,d)$ be a connected compact metric space. We say $\Sigma$ is a stratified surface with singular set $P$ if $P\subset\Sigma$ is finite set such that 1\. $(\Sigma\backslash P,d)$ is a smooth Riemann surface without boundary (possibly disconnected) and $d$ is a smooth metric $h=d\|_{\Sigma\backslash P}$, and 2\. For each $p\in P$, there is $\delta$, such that $B_{\delta}(p)\cap P=\\{p\\}$ and $B_{\delta}(p)\backslash\\{p\\}=\bigcup\limits_{i=1}^{m(p)}\Omega_{i}$, where $1<m(p)<+\infty$, and each $\Omega_{i}$ is topologically a disk with its center deleted. Moreover, on each $\Omega_{i}$, $h$ can be extended to be a smooth metric on the disk. The genus of $\Sigma$ is defined by $g(\Sigma)=\frac{2-\chi(\Sigma)+\sum\limits_{p\in P}(m(p)-1)}{2}.$ When $g(\Sigma)=0$, $\Sigma$ is called a stratified sphere. A stratified surface with singular set $P=\emptyset$ is a smooth Riemann surface. Figure 1. Stratified torus For a stratified surface $\Sigma$ with singular set $P$, we can write $\Sigma\backslash P=\bigcup_{i}\Sigma^{i}$ where $\Sigma^{i}$’s are the disjoint connected components of $\Sigma$, and each $\Sigma^{i}$ is a punctured Riemann surface when there are more than one components. The topological closure of $\Sigma^{i}$ is denoted by $\Sigma_{i}$, so as a point- set $\Sigma=\bigcup_{i}\Sigma_{i}$. By 2 in the above definition, each component $\Sigma^{i}$ can be extended to a closed Riemann surface $\overline{\Sigma^{i}}$ by adding finitely many points. To illustrate the difference of these notations, take, for example, the stratified torus on the left in Figure 1: $P$ contains two points, $\Sigma^{1}$ is the “torus” with two points deleted and $\Sigma^{2}$ is a 2-sphere with one point removed, $\Sigma_{1}$ is the “torus” and $\Sigma_{2}$ is the 2-sphere, while $\overline{\Sigma^{1}}$ is a Riemann sphere (adding 3 points at the punctures) and $\overline{\Sigma^{2}}$ is also a Riemann sphere (adding 1 point at the puncture). When $\Sigma$ is a stratified surface we define $f\in W^{2,2}_{b,c}(\Sigma,\mathbb{R}^{n})$ if $f$ is a $W^{2,2}$ non-trivial branched conformal immersion on each $\overline{\Sigma^{i}}$. We now state the main result in the first part of the paper: ###### Theorem 1. Suppose that $\\{f_{k}\\}$ is a sequence of $W^{2,2}$ branched conformal immersions of $(\Sigma,h_{k})$ in $\mathbb{R}^{n}$, where $h_{k}$ satisfies (1.1). If $f_{k}(\Sigma)\cap B_{R_{0}}\neq\emptyset$ for some fixed $R_{0}$ and $\sup_{k}\left(\mu(f_{k})+W(f_{k})\right)<+\infty$ then either $f_{k}$ converges to a point, or there is a stratified surface $\Sigma_{\infty}$ with $g(\Sigma_{\infty})\leq g(\Sigma)$, a map $f_{0}\in W^{2,2}_{b,c}(\Sigma_{\infty},\mathbb{R}^{n})$, such that a subsequence of $f_{k}(\Sigma)$ converges to $f_{0}(\Sigma_{\infty})$ in Hausdorff distance with $\mu(f_{0})=\lim_{k\rightarrow+\infty}\mu(f_{k})\,\,\,\,and\,\,\,\,W(f_{0})\leq\lim_{k\rightarrow+\infty}W(f_{k}).$ Moreover, if $y_{1},\dots,y_{m}\in f_{k}(\Sigma)$ for all $k$, then $y_{1},\dots,y_{m}\in f_{0}(\Sigma_{\infty}).$ In fact, we will prove that $f_{k}$ converges to $f_{0}$ in the sense of bubble tree. For each $k$, we can find a domain $U_{k}$ of $\Sigma$ and a domain $V_{k}$ of $\Sigma_{\infty}$, such that 1) $V_{k}\subset V_{k+1}$, and $P=\Sigma_{\infty}\backslash\bigcup_{k}V_{k}$ is a finite set which contains all singular points of $\Sigma_{\infty}$. Moreover, $\Sigma_{\infty}\backslash V_{k}$ is a union of topological disks and $H^{1}_{1}(\Sigma_{\infty}\backslash V_{k})\rightarrow 0$, where $H^{1}_{1}$ is the Hausdorff measure: $H_{1}^{1}(S)=\inf\left\\{\sum_{i=1}^{\infty}\mbox{\rm diam}(U_{i}):S\subset\bigcup_{i=1}^{\infty}U_{i},\,\,\,\,\mbox{\rm diam}(U_{i})<1\right\\}.$ 2) $\Sigma\backslash U_{k}$ is a smooth surface with boundary, possibly disconnected, $H^{1}_{1}(f_{k}(\Sigma\backslash U_{k}))\rightarrow 0$. Moreover, $f_{k}(\Sigma\backslash U_{k})$ converge to $P$ in Hausdorff distance. 3) There is a sequence of diffeomorphisms $\phi_{k}:V_{k}\rightarrow U_{k}$, such that for any $\Omega\subset\subset\Sigma_{\infty}\backslash P$, $f_{k}\circ\phi_{k}$ converge in $W^{2,2}(\Omega,\mathbb{R}^{n})$ weakly. In Theorem 1, the singular points of $f_{0}(\Sigma_{\infty})$ arise in three ways: (a) the limit point to which a sequence of closed geodesics that are not null-homotopic in $f_{k}(\Sigma)$ pinches, (b) a bubble point of $f_{k}$, so belonging to a 2-sphere (the bubble), (c) a point where both (a) and (b) happen. In the second part of the paper, we apply Theorem 1 to obtain several existence results of Willmore surfaces in compact Riemannian manifolds. Here we note that Theorem 1 is applicable for surfaces immersed in a compact Riemannian manifold $N$. To see this, for $\Sigma$ immersed in $N$ which is isometrically embedded in $\mathbb{R}^{n}$, direct calculation shows that the Willmore functional of $\Sigma$ in $\mathbb{R}^{n}$ is dominated by its Willmore functional in $N$ together with the area $\mu(\Sigma)$, see Lemma 4.1. We first consider 2-spheres immersed in the round unit sphere ${\mathbb{S}}^{n},n\geq 3$. Fix at least two distinct points $y_{1},\dots,y_{m},m\geq 2$ on ${\mathbb{S}}^{n}$. Define $\beta^{n}_{0}(y_{1},\dots,y_{m})=\inf\left\\{W_{n}(f):f\in W^{2,2}_{conf}(S^{2},{\mathbb{S}}^{n}),y_{1},\dots,y_{m}\in f({S}^{2})\right\\}$ where $W_{n}(f)=\int_{S^{2}}\left(1+\frac{1}{4}\left\|H_{f}\right\|^{2}\right)d\mu_{f}$ and $H_{f}$ is the mean curvature vector of $f(S^{2})$ in ${\mathbb{S}}^{n}$. We show ###### Theorem 2. If $\beta^{n}_{0}(y_{1},\dots,y_{m})<8\pi$, then there is a $W^{2,2}$ conformal immersion of $S^{2}$ in ${\mathbb{S}}^{n}$ without self- intersections realizing $\beta^{n}_{0}(y_{1},\dots,y_{m})$. For any $\epsilon>0$, there exists a Willmore sphere in ${\mathbb{S}}^{n}$ with $W_{n}(f)<4\pi+\epsilon$, which has at least 2 nonremovable singularities. By results in [15], [27], a singular point of a Willmore surface with density $\theta^{2}<2$ in $\mathbb{R}^{n}$ can be removed if its residue is 0. Kuwert and Schätzle also point out that the removability can not be true generally, for example, 0 is the true singular point of an inverted half catenoid ([15], P. 337). The second statement in Theorem 2 provides examples of embedded Willmore surface which has a nonremovable singular point with density $\theta^{2}=1$, and it is an application of the first statement with five points prescribed in ${\mathbb{S}}^{n}$. We then consider minimizers of the Willmore functional subject to area constraint. A fundamental existence result for incompressible minimal surfaces due to Schoen-Yau [30] and Sacks-Uhlenbeck [29] asserts: If $\varphi$ induces an injection from the fundamental groups to $\Sigma$ and $N$, then there is a branched minimal immersion $f:\Sigma\rightarrow N$ so that $f$ induces the same map between fundamental groups as $\varphi$ and $f$ has least area among all such maps. We denote the area of the minimizer by $a_{\varphi}$. ###### Theorem 3. Let $N$ be a compact Riemannian manifold and $\Sigma$ be a closed surface of genus $g$. Then (1) For $\beta_{0}(N,a)=\inf\\{W(f):\mu(f)=a>0,f\in W^{2,2}_{conf}(S^{2},N)\\}$, $\lim_{a\to 0}\beta_{0}(N,a)=4\pi$, and there is an embedding realizing $\beta_{0}(N,a)$ for all sufficiently small $a$. (2) Suppose $\varphi:\Sigma\to N$ induces an injection $\varphi_{\\#}:\pi_{1}(\Sigma)\to\pi_{1}(N)$ and $\pi_{2}(N)=0$. Let $\beta_{g}(N,a,\varphi)=\inf\\{W(f):f\in\widetilde{W}^{2,2}(\Sigma,N),\mu(f)=a,f\hbox{ is homotopic to}\,\,\,\varphi\\}$. Then there is $\delta>0$, such that for any $a\in[a_{\varphi},a_{\varphi}+\delta)$, there is a branched conformal immersion $f$ of $(\Sigma,h)$ attaining $\beta_{g}(N,a,\varphi)$. Moreover, when $\dim N=3$, $f$ is an immersion for small $\delta$. For $\beta_{0}(N,a)$, Lamm and Metzger showed in [17] that if it is attained by a surface with positive mean curvature in the sufficiently small geodesic ball around a point $p$, then the scalar curvature of $N$ must have a critical point at $p$. When $N$ has negative sectional curvature, the area of an immersed surface is dominated by the Willmore functional. We now describe a sufficient condition of Douglas type for existence. Let $S(g)$ be the set of connected stratified Riemann surfaces $\Sigma=\bigcup_{i}\Sigma_{i}$ satisfying (a) genus of $\Sigma_{i}<g$ if $g>0$ and (b) $i>1$ if $g=0$. Note that a surface in $S(g)$ has genus at most $g$ and smooth surfaces of genus $g$ are not in $S(g)$. Isometrically embed $N$ into ${\mathbb{R}}^{n}$. Define $\alpha^{}(g)=\inf\\{W(f):f\in W^{2,2}_{b,c}(\Sigma,\mathbb{R}^{n}),\Sigma\in{S}(g)\\}$ $\alpha(g)=\inf\\{W(f):f\in W^{2,2}_{b,c}(\Sigma,{\mathbb{R}}^{n}),\hbox{$\Sigma$ is a smooth surface of genus $g$}\\}.$ Similarly, for $0<a<\infty$, define $\displaystyle\gamma^{}(g,a)$ $\displaystyle=$ $\displaystyle\inf\\{W(f,\Sigma,\mathbb{R}^{n}):f\in W^{2,2}_{b,c}(\Sigma,\mathbb{R}^{n}),f(\Sigma)\subset N,\Sigma\in S(g),\mu(f(\Sigma))\leq a\\}$ $\displaystyle\gamma(g,a)$ $\displaystyle=$ $\displaystyle\inf\\{W(f,\Sigma,\mathbb{R}^{n}):f\in W^{2,2}_{b,c}(\Sigma,\mathbb{R}^{n}),f(\Sigma)\subset N,\Sigma\in{\mathcal{M}}_{g},\mu(f(\Sigma))\leq a\\}.$ ###### Theorem 4. Let $N$ be a compact Riemannian manifold. If $0<\alpha(g)<\alpha^{}(g)$ and $N$ has negative sectional curvature, then there is a $W^{2,2}$ branched conformal immersion $f$ from a smooth closed Riemann surface of genus $g$ with $W(f)=\alpha(g)$. If $0<\gamma(g,a)<\gamma^{}(g,a)$ then there is a $W^{2,2}$ branched conformal immersion $f$ from a smooth closed Riemann surface of genus $g$ with $W(f)=\gamma(g)$. ## 2\. blowup analysis - energy identity and absence of neck Let $(\Sigma,h)$ be a smooth Riemann surface which may not be compact, where $h$ is the metric compatible with the complex structure of $\Sigma$. For given $p>1$ and $R>0$, let $\mathcal{F}^{p}(\Sigma,h,R)$ be the set of mappings $f:\Sigma\rightarrow\mathbb{R}^{n}$ which satisfy 1. (1) $f\in W^{2,p}_{loc}(\Sigma,h)$; 2. (2) $f(\Sigma)$ is contained in the closed ball centered at the origin with radius $R$ in ${\mathbb{R}}^{n}$; 3. (3) $\Delta_{h}f=F(f)$ with $\|F(f)\|\leq\beta\,\|\nabla_{h}f\|^{2}$ a.e. on $\Sigma$, where $\beta$ is a nonnegative measurable function on $\Sigma$ with $\int_{\Sigma}\beta^{2}\|\nabla_{h}f\|^{2}d\mu_{h}<+\infty.$ When $f\in\mathcal{F}^{p}(\Sigma,h,R)$, we introduce a notation by $H(f)=\left\\{\begin{array}[]{ll}2\frac{\Delta_{h}f}{\|\nabla_{h}f\|^{2}},&\hbox{if $\|\nabla_{h}f\|\neq 0$}\\\ 0,&\hbox{if $\|\nabla_{h}f\|=0$}.\end{array}\right.$ The Willmore functional of $f$ is defined to be $W(f)=\frac{1}{4}\int_{\Sigma}\|H(f)\|^{2}\|\nabla_{h}f\|^{2}d\mu_{h}.$ That $W(f)<\infty$ for $f\in\mathcal{F}^{p}(\Sigma,h,R)$ follows from (3) as $\Delta_{h}f=F(f)=\frac{1}{2}H(f)\|\nabla_{h}f\|^{2}.$ We denote by $\mathcal{F}^{p}_{conf}(\Sigma,h,R)$ the set of $f\in\mathcal{F}^{p}(\Sigma,h,R)$ and $f$ is weakly conformal a.e., i.e. $\partial f\otimes\partial f=0$ almost everywhere on $\Sigma$, where $\partial f=\frac{\partial f}{\partial z}dz$ in a local complex coordinate system on $\Sigma$. Note that when $f$ is a smooth conformal immersion $H(f),\frac{1}{2}\|\nabla_{h}f\|^{2}d\mu_{h},W(f)$ are the mean curvature vector, the area element and the Willmore functional of $f(\Sigma)$, respectively. By the Kondrachov embedding theorem, functions in ${\mathcal{F}}^{p}(\Sigma,h,R)$ are also in $W^{1,2}$. The right hand side of the equation $\Delta_{h}f=F(f)$ is not necessarily in $L^{2}$ under the assumption (3). We point out that $H(f)$, $\mathcal{F}^{p},\mathcal{F}^{p}_{conf}$ are conformal invariant, in the sense that if $h^{\prime}=e^{2u}h$ for some smooth function $u$ on $\Sigma$, we always have $H_{h}(f)=H_{h^{\prime}}(f),\,\,\,\,\mathcal{F}^{p}(\Sigma,h,R)=\mathcal{F}^{p}(\Sigma,h^{\prime},R),\,\,\,\,\mathcal{F}^{p}_{conf}(\Sigma,h,R)=\mathcal{F}^{p}_{conf}(\Sigma,h^{\prime},R).$ Thus we may select preferred metrics $h$, e.g. the ones with constant curvature. In this section, we will study regularity, compactness and the blowup behavior of a sequence $\\{f_{k}\\}\subset\mathcal{F}^{p}$. ### 2.1. $\epsilon$-regularity, removable singularity and weak limit In this subsection, we will show that some well-known results for harmonic maps still hold for mappings in $\mathcal{F}^{p}$. Let $D$ be the unit 2-disk centered at 0. For simplicity, write $\mathcal{F}^{p}(D,dx^{2}+dy^{2},R)$ as $\mathcal{F}^{p}(D,R)$. ###### Proposition 2.1. ($\epsilon$-regularity) There is an $\epsilon_{0}$ such that for any $f\in\mathcal{F}^{p}(D,R)$, $1<p<2$, if $W(f)<\epsilon_{0}^{2}$, then $\\|\nabla f\\|_{W^{1,p}(D_{\frac{1}{2}})}\leq C\,\\|\nabla f\\|_{L^{2}(D)}.$ ###### Proof. Set $\bar{f}=\frac{1}{\|D\|}\int_{D}fd\sigma$ and let $\eta$ be a cut-off function which is 1 in $D_{1/2}$, 0 in $D\backslash D_{3/4}$ and $0\leq\eta\leq 1$. Then for the equation $\Delta\left(\eta(f-\bar{f})\right)=(f-\bar{f})\Delta\eta+2\nabla\eta\nabla f+\frac{1}{2}\eta H(f)\|\nabla f\|^{2}:=\phi$ we have $\displaystyle\|\phi\|$ $\displaystyle\leq$ $\displaystyle C_{1}\left(\|f-\bar{f}\|+\|\nabla f\|\right)+\frac{1}{2}\eta\left\|H(f)\right\|\|\nabla f\|^{2}$ $\displaystyle\leq$ $\displaystyle C_{1}\left(\|f-\bar{f}\|+\|\nabla f\|\right)+C_{2}\left\|H(f)\right\|\|\nabla f\|\left(\|\nabla\left(\eta(f-\bar{f})\right)\|+\|f-\bar{f}\|\right)$ since $\begin{array}[]{lll}\frac{1}{2}\eta\left\|H(f)\right\|\|\nabla f\|^{2}&=&\frac{1}{2}\eta\left\|H(f)\right\|\nabla(f-\bar{f})\nabla f\\\\[8.61108pt] &=&\frac{1}{2}\left\|H(f)\right\|\nabla\left(\eta(f-\bar{f})\right)\nabla f-\frac{1}{2}\left\|H(f)\right\|(f-\bar{f})\nabla\eta\nabla f\\\\[8.61108pt] &\leq&C_{2}\left\|H(f)\right\|\|\nabla f\|\left(\|\nabla\left(\eta(f-\bar{f})\right)\|+\|f-\bar{f}\|\right).\end{array}$ By the $L^{p}$ estimates for elliptic equations, $\displaystyle\left\\|\eta(f-\bar{f})\right\\|_{W^{2,p}(D)}$ $\displaystyle\leq$ $\displaystyle C_{3}\left(\left\\|f-\bar{f}\right\\|_{L^{p}(D)}+\left\\|\nabla f\right\\|_{L^{p}(D)}\right.$ $\displaystyle\left.+\left\\|H(f)\|\nabla f\|\left(\|\nabla\left(\eta(f-\bar{f})\right)\|+\|f-\bar{f}\|\right)\right\\|_{L^{p}(D)}\right).$ For $1<p<2$, the Hölder inequality and the Sobolev inequality imply $\begin{array}[]{lll}&&\left\\|H(f)\|\nabla f\|\left(\left\|\nabla\left(\eta(f-\bar{f})\right)\right\|+\left\|f-\bar{f}\right\|\right)\right\\|_{L^{p}(D)}\\\ &&\leq\\|H(f)\nabla f\\|_{L^{2}(D)}\left(\\|\nabla\left(\eta(f-\bar{f})\right)\\|_{L^{\frac{2p}{2-p}}(D)}+\\|f-\bar{f}\\|_{L^{\frac{2p}{2-p}}(D)}\right)\\\\[8.61108pt] &&\leq\epsilon_{0}\,C_{4}\,\\|\eta(f-\bar{f})\\|_{W^{2,p}(D)}+\epsilon_{0}\,C_{5}\,\\|f-\bar{f}\\|_{W^{1,p}(D)}\end{array}$ since $W(f)<\epsilon_{0}$. Applying the Poincaré inequality and noting $1<p<2$, we get $\\|f-\bar{f}\\|_{L^{p}(D)}+\\|\nabla f\\|_{L^{p}(D)}+\epsilon_{0}\,C_{5}\,\\|f-\bar{f}\\|_{W^{1,p}(D)}\leq C_{6}\,\\|\nabla f\\|_{L^{2}(D)}.$ Choose $\epsilon_{0}$ so that $C_{3}C_{4}\,\epsilon_{0}<1/2$, then we get $\\|\eta(f-\bar{f})\\|_{W^{2,p}(D)}<C_{7}\,\\|\nabla f\\|_{L^{2}(D)}$ which completes the proof. $\hfill\Box$ ###### Proposition 2.2. (Gap constant) Let $\Sigma$ be a closed surface. There is a constant ${\epsilon}_{1}$ which depends on $\Sigma$ and $R$, such that for any $f\in\mathcal{F}^{p}(\Sigma,h,R)$ where $1<p<2$, if $W(f)<{\epsilon}_{1}^{2}$, then $f$ is constant. ###### Proof. Let $\bar{f}=\frac{1}{\|\Sigma\|}\int_{\Sigma}f$. It follows from the equation $\Delta_{h}(f-\bar{f})=\frac{1}{2}H(f)\|\nabla f\|^{2}$ that $\begin{array}[]{lll}\displaystyle\int_{\Sigma}\|\nabla(f-\bar{f})\|^{2}&\leq&\displaystyle\frac{1}{2}\int_{\Sigma}\|f-\bar{f}\|\,\|{H(f)}\|\,\|\nabla f\|^{2}\\\\[8.61108pt] &\leq&\displaystyle\left(\int_{\Sigma}{H(f)}^{2}\|\nabla f\|^{2}\right)^{\frac{1}{2}}\left(\int_{\Sigma}\|f-\bar{f}\|^{\frac{2p}{2p-2}}\right)^{\frac{2p-2}{2p}}\left(\int_{\Sigma}\|\nabla f\|^{\frac{2p}{2-p}}\right)^{\frac{2-p}{2p}}\\\\[8.61108pt] &\leq&\displaystyle C_{1}W(f)^{\frac{1}{2}}\\|\nabla f\\|_{L^{2}(\Sigma)}\\|\nabla f\\|_{L^{\frac{2p}{2-p}}(\Sigma)}\end{array}$ where we used the Sobolev inequality, the Poincaré inequality and $1<p<2$. Then we get $\\|\nabla f\\|_{L^{2}(\Sigma)}\leq C_{1}W(f)^{\frac{1}{2}}\\|\nabla f\\|_{L^{\frac{2p}{2-p}}(\Sigma)}.$ Using the Poincaré inequality and $1<p<2$ again, we have $\\|f-\bar{f}\\|_{L^{p}(\Sigma)}\leq C_{2}\\|\nabla f\\|_{L^{2}(\Sigma)}\leq C_{1}C_{2}\,W(f)^{\frac{1}{2}}\\|\nabla f\\|_{L^{\frac{2p}{2-p}}(\Sigma)}.$ Since $\left\\|\frac{1}{2}H(f)\|\nabla f\|^{2}\right\\|_{L^{p}(\Sigma)}\leq\left(\int_{\Sigma}\frac{1}{4}{H(f)}^{2}\|\nabla f\|^{2}\right)^{\frac{1}{2}}\left(\int_{\Sigma}\|\nabla f\|^{\frac{2p}{2-p}}\right)^{\frac{2-p}{2p}}=W(f)^{\frac{1}{2}}\\|\nabla f\\|_{L^{\frac{2p}{2-p}}(\Sigma)},$ it follows from the $L^{p}$ estimates for elliptic equations that $\displaystyle\left\\|f-\bar{f}\right\\|_{W^{2,p}(\Sigma)}$ $\displaystyle\leq$ $\displaystyle C_{3}\,\left(\left\\|\frac{1}{2}H(f)\|\nabla f\|^{2}\right\\|_{L^{p}(\Sigma)}+\left\\|f-\bar{f}\right\\|_{L^{p}(\Sigma)}\right)$ $\displaystyle\leq$ $\displaystyle C_{3}(1+C_{1}C_{2})W(f)^{\frac{1}{2}}\left\\|\nabla f\right\\|_{L^{\frac{2p}{2-p}}(\Sigma)}$ $\displaystyle\leq$ $\displaystyle C_{4}W(f)^{\frac{1}{2}}\left\\|f-\bar{f}\right\\|_{W^{2,p}(\Sigma)}$ where the Sobolev inequality was used in the last step. By choosing $\epsilon_{1}<1/C_{4}$ we immediately have $f=\overline{f}$. $\hfill\Box$ We now derive a key estimate for later applications. Set $E(f,Q(t))=\int_{Q(t)}\|\nabla f\|^{2},\,\,\hbox{where}\,\,Q(t)=S^{1}\times[-t,t]$ and denote $\mathcal{F}^{p}(Q(t),dt^{2}+d\theta^{2},R)$ by $\mathcal{F}^{p}(Q(t),R)$. We will prove the following energy decay estimate: ###### Proposition 2.3. (Decay estimate) Let $f\in\mathcal{F}^{p}_{conf}(Q(T),R)$ with $T\gg 1,1<p<2$. Then there is a constant $\epsilon_{2}<\epsilon_{0}$, where $\epsilon_{0}$ is the constant in Proposition 2.1, such that if $\sup_{t\in[-T,T-1]}W(f,S^{1}\times[t,t+1])<\epsilon^{2}\leq\epsilon_{2}^{2}$ then $\int_{Q(t)}\|\nabla f\|^{2}<{C}E(f,Q(T))e^{-(1-C\epsilon)(T-t)}$ for some positive constant $C$ independent of $T$ and $f$. ###### Proof. Define $f^{}(t)=\frac{1}{2\pi}\int_{0}^{2\pi}f(t,\theta)d\theta.$ We have $\displaystyle\int_{Q(t)}\left\|\frac{\partial f^{}}{\partial t}\right\|^{2}$ $\displaystyle=$ $\displaystyle\int_{-t}^{t}\int_{0}^{2\pi}\left(\frac{1}{2\pi}\int_{0}^{2\pi}\frac{\partial f}{\partial t}d\theta\right)^{2}d\theta dt$ $\displaystyle\leq$ $\displaystyle\displaystyle\frac{1}{2\pi}\int_{-t}^{t}\left(\int_{0}^{2\pi}\left\|\frac{\partial f}{\partial t}\right\|^{2}d\theta\int_{0}^{2\pi}d\theta\right)dt$ $\displaystyle=$ $\displaystyle\int_{-t}^{t}\int_{0}^{2\pi}\left\|\frac{\partial f}{\partial t}\right\|^{2}dtd\theta$ $\displaystyle=$ $\displaystyle\int_{Q(t)}\left\|\frac{\partial f}{\partial t}\right\|^{2}dtd\theta.$ Then (2.2) $\begin{array}[]{lll}\displaystyle\int_{Q(t)}\nabla(f-f^{})\nabla f&=&\displaystyle\int_{Q(t)}\|\nabla f\|^{2}-\displaystyle\int_{Q(t)}\frac{\partial f}{\partial t}\frac{\partial f^{}}{\partial t}\\\\[8.61108pt] &\geq&\displaystyle\int_{Q(t)}\|\nabla f\|^{2}-\frac{1}{2}\left(\displaystyle\int_{Q(t)}\left\|\frac{\partial f}{\partial t}\right\|^{2}+\displaystyle\int_{Q(t)}\left\|\frac{\partial f^{}}{\partial t}\right\|^{2}\right)\\\\[8.61108pt] &\geq&\displaystyle\int_{Q(t)}\|\nabla f\|^{2}-\int_{Q(t)}\left\|\frac{\partial f}{\partial t}\right\|^{2}\\\\[8.61108pt] &=&\displaystyle\frac{1}{2}\int_{Q(t)}\|\nabla f\|^{2}\end{array}$ where in the last step we used the fact that $\|f_{t}\|^{2}=\|f_{\theta}\|^{2}$ a.e. as $f$ is conformal a.e. On the other hand, (2.3) $\begin{array}[]{lll}\displaystyle\int_{Q(t)}\nabla(f-f^{})\nabla f&=&-\displaystyle\int_{Q(t)}(f-f^{})\Delta f-\displaystyle\int_{\partial Q(t)}\frac{\partial f}{\partial t}(f-f^{})\\\\[8.61108pt] &\leq&\displaystyle\int_{Q(t)}\|f-f^{}\|\|\nabla f\|^{2}\frac{\|{H(f)}\|}{2}+\left\|\displaystyle\int_{\partial Q(t)}\frac{\partial f}{\partial t}(f-f^{})\right\|.\end{array}$ Let $m\in[t,t+1)$ be an integer. Then for each $i=-m,-m+1,\dots,m-1$, by (2.1) and the hypothesis in the proposition $\sup_{t\in[-T,T-1]}\frac{1}{4}\int_{S^{1}\times[t,t+1]}\|H(f)\|^{2}<\epsilon^{2}\leq\epsilon_{0}^{2}$ it follows from Proposition 2.1 that (2.4) $\\|f-f^{}\\|_{L^{\infty}(S^{1}\times[i,i+1])}\leq C\,\\|\nabla f\\|_{L^{2}(S^{1}\times[i-1,i+2])}.$ In fact, to see (2.4), denote the average of $f$ over $S\times[i-1,i+2]$ by $\overline{f}$ and observe that from Proposition 2.1 $\\|\nabla(f-\overline{f})\\|_{W^{1,p}(S^{1}\times[i,i+1])}=\\|\nabla f\\|_{W^{1,p}(S^{1}\times[i,i+1])}\leq C\\|\nabla f\\|_{L^{2}(S^{1}\times[i-1,i+2])}$ and from the Poincaré inequality $\\|f-\overline{f}\\|_{L^{2}(S^{1}\times[i,i+1])}\leq\\|f-\overline{f}\\|_{L^{2}(S^{1}\times[i-1,i+2])}\leq C\\|\nabla f\\|_{L^{2}(S^{1}\times[i-1,i+2])}$ hence $\\|f-\overline{f}\\|_{W^{2,p}(S^{1}\times[i,i+1])}\leq C\\|\nabla f\\|_{L^{2}(S^{1}\times[i-1,i+2])}.$ The Sobolev embedding theorem then implies $\\|f-\overline{f}\\|_{C^{0}(S^{1}\times[i,i+1])}\leq C\\|f-\overline{f}\\|_{W^{2,p}(S^{1}\times[i,i+1])}\leq C\\|\nabla f\\|_{L^{2}(S^{1}\times[i-1,i+2])}.$ Therefore for any $t\in[i,i+1]$ $\left\|f^{}(t)-\overline{f}\right\|=\left\|\frac{1}{2\pi}\int^{2\pi}_{0}\left(f(t,\theta)-\overline{f}\right)d\theta\right\|\leq C\\|\nabla f\\|_{L^{2}(S^{1}\times[i-1,i+2])}.$ It follows $\\|f-f^{}\\|_{C^{0}(S^{1}\times[i,i+1])}\leq\\|f-\bar{f}\\|_{C^{0}(S^{1}\times[i,i+1])}+\\|f^{}-\bar{f}\\|_{C^{0}(S^{1}\times[i,i+1])}\leq C\\|\nabla f\\|_{L^{2}(S^{1}\times[i-1,i+2])}.$ Clearly, by the mean value theorem, $f-f^{}$ equals 0 somewhere in $S^{1}\times[i,i+1]$, thus $\\|f-f^{}\\|_{L^{\infty}(S^{1}\times[i,i+1])}=\\|f-f^{}\\|_{C^{0}(S^{1}\times[i,i+1])}\leq C\\|\nabla f\\|_{L^{2}(S^{1}\times[i-1,i+2])}$ which shows (2.4) holds. Then $\begin{array}[]{l}\displaystyle\int_{S^{1}\times[i,i+1]}\|f-f^{}\|\|\nabla f\|^{2}\frac{\|{H(f)}\|}{2}\\\\[8.61108pt] \begin{array}[]{lll}&\leq&\displaystyle\left\\|f-f^{}\right\\|_{L^{\infty}(S^{1}\times[i,i+1])}\times\left(W(f_{k},S^{1}\times[i,i+1])\int_{S^{1}\times[i,i+1]}\|\nabla f\|^{2}\right)^{\frac{1}{2}}\\\\[8.61108pt] &\leq&\displaystyle C\epsilon\,\left(\int_{S^{1}\times[i-1,i+2]}\|\nabla f\|^{2}\int_{S^{1}\times[i,i+1]}\|\nabla f\|^{2}\right)^{\frac{1}{2}}\\\\[8.61108pt] &\leq&\displaystyle C\epsilon\int_{S^{1}\times[i-1,i+2]}\|\nabla f\|^{2}.\end{array}\end{array}$ Then (2.5) $\begin{array}[]{lll}\displaystyle{\int}_{Q(t)}\|f-f^{}\|\|\nabla f\|^{2}{H(f)}&\leq&\displaystyle\sum\limits_{i=-m}^{m-1}\int_{S^{1}\times[i,i+1]}\|f-f^{}\|\|\nabla f\|^{2}{H(f)}\\\\[8.61108pt] &\leq&\displaystyle C\epsilon\sum\limits_{i=-m}^{m-1}\int_{S^{1}\times[i-1,i+2]}\|\nabla f\|^{2}\\\\[8.61108pt] &\leq&\displaystyle 3C\epsilon\int_{Q(t)}\|\nabla f\|^{2}\\\\[8.61108pt] &&\displaystyle+C\epsilon\left(\int_{S^{1}\times[-m-1,-m]}\|\nabla f\|^{2}+\int_{S^{1}\times[m,m+1]}\|\nabla f\|^{2}\right)\\\\[8.61108pt] &\leq&3C\epsilon\displaystyle\int_{Q(t+2)}\|\nabla f\|^{2}.\end{array}$ From (2.2), (2.3), (2.5), we have (2.6) $\frac{1}{2}\int_{Q(t)}\|\nabla f\|^{2}\leq\epsilon^{\prime}\int_{Q(t+2)}\|\nabla f\|^{2}+\left\|\int_{\partial Q(t)}(f-f^{})\frac{\partial f}{\partial t}\right\|$ where $\epsilon^{\prime}=3C\epsilon/2$. Moreover, (2.7) $\begin{array}[]{lll}\displaystyle\left\|\int_{S^{1}\times\\{t\\}}\frac{\partial f}{\partial t}(f-f^{})\right\|&\leq&\displaystyle\left(\int_{0}^{2\pi}(f(\theta,t)-f^{}(t))^{2}d\theta\right)^{\frac{1}{2}}\left(\int_{0}^{2\pi}\left\|\frac{\partial f}{\partial t}(\theta,t)\right\|^{2}d\theta\right)^{\frac{1}{2}}\\\\[8.61108pt] &\leq&\displaystyle\left(\int_{0}^{2\pi}\left\|\frac{\partial f}{\partial\theta}(\theta,t)\right\|^{2}d\theta\right)^{\frac{1}{2}}\left(\int_{0}^{2\pi}\left\|\frac{\partial f}{\partial t}(\theta,t)\right\|^{2}d\theta\right)^{\frac{1}{2}}\\\\[8.61108pt] &=&\displaystyle{\frac{1}{2}}\displaystyle{\int}_{S^{1}\times\\{t\\}}\|\nabla f\|^{2}d\theta\end{array}$ here we used the Poincaré inequality on $S^{1}$ and the fact that $\|\frac{\partial f}{\partial t}\|^{2}=\|\frac{\partial f}{\partial\theta}\|^{2}$ a.e. Let $\varphi(t)=\frac{1}{2}\int_{Q(t)}\|\nabla f\|^{2}.$ By (2.6) and (2.7), we have $\varphi(t)\leq\varphi^{\prime}(t)+\epsilon^{\prime}\varphi(t+2).$ Then $-(e^{-t}\varphi(t))^{\prime}\leq\epsilon^{\prime}\varphi(t+2)e^{-t},$ and integrating the inequality from $t$ to $T-2$ leads to (2.8) $\begin{array}[]{lll}\displaystyle e^{-t}\varphi(t)&\leq&\displaystyle e^{-T+2}\varphi(T-2)+\epsilon^{\prime}\int_{t}^{T-2}\varphi(s+2)e^{-s}ds\\\\[8.61108pt] &=&\displaystyle e^{-T+2}\varphi(T-2)+\epsilon^{\prime}\int_{t+2}^{T}\varphi(s)e^{-s+2}ds\\\\[8.61108pt] &=&\displaystyle e^{-T+2}\varphi(T-2)+\epsilon^{\prime}e^{2}\int_{t+2}^{T-2}\varphi(s)e^{-s}ds+\epsilon^{\prime}e^{2}\int_{T-2}^{T}\varphi(s)e^{-s}ds\\\\[8.61108pt] &\leq&\displaystyle e^{-T+2}\varphi(T)+\epsilon^{\prime}e^{2}\int_{t}^{T-2}\varphi(s)e^{-s}ds+\epsilon^{\prime}e^{2}\varphi(T)\left(e^{-T+2}-e^{-T}\right)\end{array}$ as $\varphi(t)$ is increasing in $t$. Let $F(t)=\int_{t}^{T-2}\varphi(s)e^{-s}ds$ and $\epsilon_{2}=\epsilon^{\prime}e^{2}<1$. Now (2.8) leads to $-F^{\prime}(t)\leq 2\,\varphi(T)e^{-T+2}+\epsilon_{2}F(t)$ or equivalently $\left(e^{\epsilon_{2}t}F(t)\right)^{\prime}+2\varphi(T)e^{-T+2}e^{\epsilon_{2}t}\geq 0.$ Integrating over $[t,T-2]$ and noting $F(T-2)=0$, we have (2.9) $F(t)\leq\frac{2\varphi(T)}{\epsilon_{2}}e^{2-T}\left(e^{\epsilon_{2}(T-2)}-e^{\epsilon_{2}t}\right)e^{-\epsilon_{2}t}.$ Substitute (2.9) into (2.8): $\displaystyle\varphi(t)$ $\displaystyle\leq$ $\displaystyle e^{2-T+t}\varphi(T)+2\varphi(T)e^{\epsilon_{2}(T-t-2)}e^{2-T+t}+\epsilon_{2}\varphi(T)e^{t-T+2}$ $\displaystyle\leq$ $\displaystyle C\varphi(T)e^{(1-\epsilon_{2})(T-t)}$ $\displaystyle=$ $\displaystyle C\varphi(T)e^{(1-C\epsilon)(T-t)}$ for some positive constant $C$ independent of $T$ and $f$. $\hfill\Box$ ###### Proposition 2.4. (Removability of point singularity) Let $f\in\mathcal{F}^{p}_{conf}(D\backslash\\{0\\},R)$, where $1<p<2$. If $\int_{D}\|\nabla f\|^{2}<+\infty$, then $f\in\mathcal{F}^{p^{\prime}}_{conf}(D,R)$ for any $p^{\prime}\in(1,\frac{4}{3})\cap(1,p]$. ###### Proof. We may assume that $W(f)<\epsilon^{2}<\epsilon_{2}^{2}$, otherwise, we can replace $f$ with $f(\lambda x)$ for some $\lambda<1$. Let $\phi:\mathbb{R}^{1}\times S^{1}\rightarrow\mathbb{R}^{2}$ be the conformal mapping given by $r=e^{-t},\theta=\theta$. Then $f^{\prime}=f(\phi)$ is a map from $[0,+\infty)\times S^{1}$ into $\mathbb{R}^{n}$. By translating $S^{1}\times[t-1,t+1]\subset S^{1}\times[0,2t]$ to $S^{1}\times[-1,1]\subset S^{1}\times[-t,t]$, from Proposition 2.3 we conclude $\int_{S^{1}\times[t-1,t+1]}\|\nabla f^{\prime}\|^{2}\leq C_{1}e^{-\delta t},\,\,\,\,\hbox{where}\,\,\,\,\delta=1-C\epsilon.$ Then for any $r_{k}=e^{-k}$, we have $t_{k}=k$ and (2.10) $\int_{D_{r_{k-1}}\backslash D_{r_{k+1}}}\|\nabla f\|^{2}<C_{1}r_{k}^{-\delta}.$ Set $f_{k}(x)=f(r_{k}x)$. Applying Proposition 2.1 and (2.10), we get $\\|\nabla f_{k}\\|_{W^{1,p}(D_{1}\backslash D_{e^{-1}})}\leq C_{2}\,\\|\nabla f_{k}\\|_{L^{2}(D_{e}\backslash D_{e^{-2}})}\leq C_{3}\,r_{k}^{\frac{\delta}{2}}.$ By the Sobolev inequality, we have $\left(\int_{D_{1}\backslash D_{e^{-1}}}\|\nabla f_{k}\|^{q}\right)^{\frac{1}{q}}\leq C_{4}\,\\|\nabla f_{k}\\|_{W^{1,p}(D_{1}\backslash D_{e^{-1}})}\leq C_{5}\,r_{k}^{-\frac{\delta}{2}},\,\,\,\,where\,\,\,\,q\leq\frac{2p}{2-p}.$ Then $\int_{D_{1}\backslash D_{e^{-1}}}\|\nabla f_{k}\|^{q}\leq C_{6}\,e^{-qk\frac{\delta}{2}}.$ Since $r_{k}^{2-q}\int_{D_{1}\backslash D_{e^{-1}}}\|\nabla f_{k}\|^{q}=\int_{D_{r_{k}}\backslash D_{r_{k+1}}}\|\nabla f\|^{q},$ we have $\int_{D_{r_{k}}\backslash D_{r_{k+1}}}\|\nabla f\|^{q}\leq C_{6}\,e^{-qk\frac{\delta}{2}+(q-2)k}=C_{6}\,e^{k(-2+q(1-\frac{\delta}{2}))}.$ When $q<4$, we can choose $\epsilon$ suitably such that $q(1-\frac{\delta}{2})<2$, which yields $\int_{D}\|\nabla f\|^{q}\leq C_{6}\sum_{k}2^{-qk\frac{\delta}{2}+(q-2)k}<C_{7}<\infty.$ For any $p^{\prime}\in(1,\frac{4}{3})$, set $q=\frac{2p^{\prime}}{2-p^{\prime}}$, so $q\in(2,4)$. We have $\int_{D}{H(f)}^{p^{\prime}}\|\nabla f\|^{2p^{\prime}}\leq\left(\int_{D}{H(f)}^{2}\|\nabla f\|^{2}\right)^{\frac{p^{\prime}}{2}}\left(\int_{D}\|\nabla f\|^{q}\right)^{\frac{p^{\prime}}{q}}<C_{8}.$ Therefore, $F(f)\in L^{p^{\prime}}(D)$ with $p^{\prime}>1$ and then there exists $v$ which solves the equation $-\Delta v=F(f),\,\,\,\,v\|_{\partial D}=0,$ and $v\in W^{2,p^{\prime}}(D)$. Obviously, $f-v$ is a harmonic function on $D\backslash\\{0\\}$ with $\\|\nabla(f-v)\\|_{L^{2}(D)}+\\|f-v\\|_{L^{2}(D)}<+\infty.$ Then $f-v$ is smooth on $D$. Now $f\in{\mathcal{F}}^{p^{\prime}}_{conf}(D,R)$ is evident for $p^{\prime}\leq p$ and $1<p^{\prime}<\frac{4}{3}$. $\hfill\Box$ We now consider weak compactness property of a bounded sequence in ${\mathcal{F}}^{p}(D,R)$. Let $\\{f_{k}\\}\subset\mathcal{F}^{p}(D,R)$. The blowup set of $\\{f_{k}\\}$ is defined to be $\mathcal{C}(\\{f_{k}\\})=\left\\{z\in D:\lim_{r\rightarrow 0}\varliminf_{k\rightarrow+\infty}W(f_{k},D_{r}(z))>\epsilon_{2}^{2}\right\\}.$ Then for any $z\in D\backslash\mathcal{C}(\\{f_{k}\\})$, we can find $r$ and a subsequence of $\\{f_{k}\\}$ which is still denoted by $\\{f_{k}\\}$ for simplicity, such that $\lim_{k\rightarrow+\infty}W(f_{k},D_{r}(z))<\epsilon_{0}^{2}.$ Then we get from Proposition 2.1 that $\\|f_{k}\\|_{W^{2,p}(D_{r/2}(z))}<C\\|\nabla f_{k}\\|_{L^{2}(D_{r})}$. Thus we may assume $f_{k}$ converges weakly in $W^{2,p}(D\backslash\mathcal{C}(\\{f_{k}\\}))$. ###### Corollary 2.5. Let $\\{f_{k}\\}\subset\mathcal{F}^{p}_{conf}(D,R)$ with $\sup_{k}\\{E(f_{k},D)+W(f_{k},D)\\}<\Lambda<\infty$ and $f_{0}$ be the weak limit of $f_{k}$ in $W^{2,p}_{loc}(D\backslash\mathcal{C}(\\{f_{k}\\}))$. If $p\in(1,\frac{4}{3})$, then $f_{0}\in\mathcal{F}^{p}_{conf}(D,R)$ and (2.11) $W(f_{0},D)\leq\varliminf_{k\rightarrow+\infty}W(f_{k},D).$ ###### Proof. Set $\Delta f_{k}=F_{k},k\in{\mathbb{N}}$. For any $\Omega\subset\subset D\backslash\mathcal{C}(f_{k})$, we have $\\|f_{k}\\|_{W^{2,p}(\Omega)}<C(\Omega)$. Then by the Hölder inequality and the Sobolev inequality $\\|F_{k}\\|_{L^{p}(\Omega)}\leq\left\\|\frac{1}{2}H(f_{k})\nabla f_{k}\right\\|_{L^{2}(\Omega)}\left\\|\nabla f_{k}\right\\|_{L^{\frac{2p}{2-p}}(\Omega)}\leq C\Lambda^{\frac{1}{2}}\\|\nabla f_{k}\\|_{W^{1,p}(\Omega)}<C^{\prime}(\Omega,\Lambda).$ We may assume, by selecting subsequences if necessary, that $F_{k}\rightharpoonup F_{0}\,\,\,\,\hbox{in}\,\,\,\,L^{p}(\Omega)\,\,\,\,\hbox{and}\,\,\,\,\|H(f_{k})\|\|\nabla f_{k}\|\rightharpoonup\alpha\,\,\,\,\hbox{in}\,\,\,\,L^{2}.$ Since we may also assume $\|\nabla f_{k}\|\rightarrow\|\nabla f_{0}\|$ in $L^{2}(\Omega)$ because $f_{k}\rightharpoonup f_{0}$ in $W^{2,p}(\Omega)$, we have $\|H(f_{k})\|\|\nabla f_{k}\|^{2}\rightharpoonup\alpha\|\nabla f_{0}\|$ in the sense of measures in $\Omega$. Define $\beta_{0}=\left\\{\begin{array}[]{ll}\frac{\alpha}{\|\nabla f_{0}\|}&\hbox{ when}\,\|\nabla f_{0}\|\neq 0\\\ 0&\,\,\hbox{otherwise.}\end{array}\right.$ Clearly, $\beta_{0}\|\nabla f_{0}\|^{2}=\alpha\|\nabla f_{0}\|$. Let $F_{k}^{+}=\max\\{F_{k},0\\}$ and $F_{k}^{-}=-\min\\{F_{k},0\\}$. Then $F_{k}=F_{k}^{+}-F_{k}^{-}$ and $\|F_{k}\|=F_{k}^{+}+F_{k}^{-}$. We may assume that $F_{k}^{+}\rightharpoonup F_{0}^{1}\,\,\,\,\hbox{and}\,\,\,\,F_{k}^{-}\rightharpoonup F_{0}^{2}\,\,\,\,\hbox{in}\,\,\,\,L^{p}(\Omega).$ Obviously $F_{0}=F_{0}^{1}-F_{0}^{2}$. Then for any nonnegative function $\varphi\in C_{0}^{\infty}(\Omega)$, $\int_{\Omega}\varphi\|F_{0}\|\leq\int_{\Omega}\varphi(F_{0}^{1}+F_{0}^{2})=\lim_{k\rightarrow+\infty}\int_{\Omega}\varphi\|F_{k}\|\leq\lim_{k\rightarrow+\infty}\int_{\Omega}\frac{1}{2}\varphi\|H(f_{k})\|\|\nabla f_{k}\|^{2}=\int_{\Omega}\frac{1}{2}\varphi\beta_{0}\|\nabla f_{0}\|^{2}.$ Hence we conclude $\|F_{0}\|\leq\frac{1}{2}\beta_{0}\|\nabla f_{0}\|^{2},\,\,\,\,\hbox{a.e.}\,z\in D.$ Then, we have $\int_{\Omega}\beta_{0}^{2}\|\nabla f_{0}\|^{2}\leq\int_{\Omega}\alpha^{2}\leq\varliminf_{k\rightarrow+\infty}\int_{\Omega}\|H_{k}(f_{k})\|^{2}\|\nabla f_{k}\|^{2}.$ Moreover, as $f_{k}$ converge in $L^{2}(\Omega)$, it follows from $\partial f_{k}\otimes\partial f_{k}=0$ a.e. in $D$ that $\partial f_{0}\otimes\partial f_{0}=0$ a.e. in $D$ as $\Omega$ is arbitrary. Since $\sup_{k}\\{E(f_{k})+W(f_{k})\\}<\infty$, there are at most finitely many points in ${\mathcal{C}}(f_{k})$. Then we conclude that $f_{0}\in\mathcal{F}^{p}_{conf}(D,R)$ if $p\in(1,\frac{4}{3})$ by removing the point singularity across $\mathcal{C}(f_{k})$ ensured by Proposition 2.4. Furthermore, we have $H(f_{0})\leq\beta_{0}$ whenever $\|\nabla f_{0}\|\neq 0$, hence we get (2.11). $\hfill\Box$ ### 2.2. A criterion for absence of bubbles along cylinders Let $f_{k}\in\mathcal{F}^{p}_{conf}(Q(T_{k}),R)$, with $\sup_{k}\\{E(f_{k})+W(f_{k})\\}<\Lambda<\infty.$ Given a sequence $t_{k}\in(-T_{k},T_{k})$ with (2.12) $T_{k}-t_{k}\rightarrow+\infty\,\,\,\hbox{and}\,\,\,t_{k}-(-T_{k})\rightarrow+\infty,$ we say the limit $f_{0}$ of a subsequence of $f_{k}(\theta,t+t_{k})$, as in Corollary 2.5, is nontrivial if $E(f_{0})>0$. When $f_{0}$ is nontrivial, it is a bubble of $\\{f_{k}\\}$. ###### Proposition 2.6. Let $f_{k}\in{\mathcal{F}}^{p}_{conf}(S^{1}\times(-T_{k},T_{k}),R)$ with $\sup_{k}\\{E(f_{k})+W(f_{k})\\}=\Lambda<\infty.$ Let $\epsilon_{2}$ be the constant in Proposition 2.3. If $\lim_{T\rightarrow+\infty}\varliminf_{k\rightarrow+\infty}\sup_{t\in[-T_{k}+T,T_{k}-T]}W(f_{k},S^{1}\times[t,t+1])<\epsilon_{2}^{2},$ then we have the following 1. (1) $\\{f_{k}\\}$ has no bubble; 2. (2) there is no energy loss, i.e. (2.13) $\lim_{T\rightarrow+\infty}\lim_{k\rightarrow+\infty}\int_{S^{1}\times[-T_{k}+T,T_{k}-T]}\|\nabla f_{k}\|^{2}=0,$ 3. (3) there is no neck, i.e. (2.14) $\lim_{t\rightarrow+\infty}\lim_{k\rightarrow+\infty}f_{k}(\theta,-T_{k}+t)=\lim_{t\rightarrow+\infty}\lim_{k\rightarrow+\infty}f_{k}(\theta,T_{k}-t).$ ###### Proof. We may assume that $f_{k}(\theta,-T_{k}+t)$ and $f_{k}(\theta,T_{k}-t)$ converge to $f_{0}^{+}(\theta,t)$ and $f_{0}^{-}(\theta,t)$ weakly in $W^{2,p}_{loc}(S^{1}\times[0,+\infty))$, respectively. Then $f_{0}^{+}(\phi)$ and $f_{0}^{-}(\phi)\in\mathcal{F}^{p}(D\backslash\\{0\\},R)$ with $E(f^{\pm}_{0}(\phi))+W(f^{\pm}_{0}(\phi))\leq\Lambda$ where $\phi$ is the conformal diffeomorphism between $D\backslash\\{0\\}$ with $S^{1}\times(0,+\infty)$. By removability of point singularities asserted in Proposition 2.4, they are in $\mathcal{F}^{p^{\prime}}(D,R)$ for some $p^{\prime}>1$. It then follows from the compact embedding $W^{2,p^{\prime}}\subset L^{2}$: $\lim_{T\to\infty}\int_{S^{1}\times[T,T+1]}\left(\|\nabla f_{0}^{+}\|^{2}+\|\nabla f_{0}^{-}\|^{2}\right)=0.$ Define $f_{k}^{}(t)=\frac{1}{2\pi}\int_{0}^{2\pi}f_{k}(t,\theta)d\theta$. It is easy to check that $\lim_{t\rightarrow+\infty}\lim_{k\rightarrow+\infty}\left\|\int_{\partial Q(T_{k}-t)}(f_{k}-f_{k}^{})\frac{\partial f_{k}}{\partial t}\right\|=0.$ In fact, this can be seen as follows: $\sup_{S^{1}\times\\{T_{k}-T\\}}\|f_{k}-f_{k}^{}\|\leq C\,\underset{S^{1}\times\\{T_{k}-T\\}}{\hbox{osc}}f_{k}$ which will converge to $C\hbox{osc}_{S^{1}\times\\{-T\\}}f_{0}^{+}$ as $k\to\infty$. By removability of singularity, $\lim_{T\to\infty}\hbox{osc}_{S^{1}\times\\{-T\\}}f_{0}^{+}=0.$ By the Sobolev trace embedding, $\int_{S^{1}\times\\{T_{k}-T\\}}\|\nabla f_{k}\|\leq C\\|\nabla f_{k}\\|_{W^{1,p}(S^{1}\times[T_{k}-T-1,T_{k}-T+1])}$ By $\epsilon$-regularity, $\left\\|\nabla f_{k}\right\\|_{W^{1,p}(S^{1}\times[T_{k}-T-1,T_{k}-T+1])}\leq C\\|\nabla f_{k}\\|_{L^{2}(S^{1}\times[T_{k}-T-2,T_{k}-T+2])}<C.$ Then (2.13) follows from (2.6). Let $m_{k}$ be the integer in $[T_{k}-T,T_{k}-T+1)$. For $0\leq i\leq m_{k}-2$, applying Proposition 2.3 on $S^{1}\times[i-m_{k},m_{k}]$ (by shifting the center circle to $S^{1}\times\\{i\\}$, and the same below), we have $\int_{S^{1}\times[i-2,i+2]}\|\nabla f_{k}\|^{2}<CE(f_{k},Q(T_{k}-T))e^{-\delta(m_{k}-i)},\,\,\,\,\delta=1-C\epsilon_{2}.$ Then from (2.4) $\underset{S^{1}\times[i-1,i+1]}{\hbox{osc}}f_{k}\leq C\sqrt{E(f_{k},Q(T_{k}-T))}e^{-\frac{\delta}{2}(m_{k}-i)}.$ When $-m_{k}+2\leq i\leq 0$, applying Proposition 2.3 on $S^{1}\times[-m_{k},m_{k}+i]$, we get $\int_{S^{1}\times[i-2,i+2]}\|\nabla f_{k}\|^{2}<CE(f_{k},Q(T_{k}-T))e^{-\frac{\delta}{2}(m_{k}-\|i\|)},$ then we obtain $\underset{S^{1}\times[i-1,i+1]}{\hbox{osc}}f_{k}\leq C\sqrt{E(f_{k},Q(T_{k}-T))}e^{-\frac{\delta}{2}(m_{k}-\|i\|)}.$ Hence, $\underset{Q(T_{k}-T)}{\hbox{osc}}f_{k}\leq 2C\sqrt{E(f_{k},Q(T_{k}-T))}\sum_{i=1}^{m_{k}}e^{-\frac{\delta}{2}(m_{k}-i)}\leq C^{\prime}\sqrt{E(f_{k},Q(T_{k}-T))}.$ Then (2.14) can be deduced from (2.13). $\hfill\Box$ ### 2.3. Bubble trees for a sequence of maps from the disk $D$ Let $f_{k}\in\mathcal{F}^{p}_{conf}(D,R)$ with $\sup_{k}\left\\{E(f_{k},D)+W(f_{k},D)\right\\}=\Lambda<\infty.$ We assume $0$ is the only blowup point of $\\{f_{k}\\}$, i.e. the only point such that $\lim_{r\rightarrow 0}\varliminf_{k\rightarrow+\infty}W(f_{k},D_{r}(z))\geq\epsilon_{2}^{2}.$ We assume that $f_{k}$ converges to $f_{\infty}$ weakly in $W^{2,p}_{loc}(D\backslash\\{0\\})$. The construction of the bubble tree at $0$ will be divided into the following steps: Step 1. Construct the first level of the bubble tree. There exists a sequence of points $z_{k}\in D$ and a sequence of radii $r_{k}\to 0$ such that (2.15) $W(f_{k},D_{r_{k}}(z_{k}))=\frac{\epsilon_{2}^{2}}{2}$ and $W(f_{k},D_{r}(z))<{\epsilon_{2}^{2}/2}$ for any $r<r_{k}$ and $D_{r}(z)\subset D$. It is easy to check that $z_{k}\rightarrow 0$ as $0$ is the only blowup point of $\\{f_{k}\\}$. We set $f_{k}^{\prime}(z)=f_{k}(z_{k}+r_{k}z)$. Since $\mathcal{C}(\\{f_{k}^{\prime}\\})=\emptyset$, $f_{k}^{\prime}(z)$ converge weakly in $W^{2,p}_{loc}(\mathbb{C})$. We denote the limit by $f^{F}$. Note that it may be a trivial mapping. Let $(r,\theta)$ be the polar coordinates centered at $z_{k}$, and set $T_{k}=-\ln r_{k}$. Let $\phi_{k}:S^{1}\times[0,T_{k}]\rightarrow\mathbb{R}^{2}$ be the conformal mapping given by $\phi_{k}(t,\theta)=(e^{-t},\theta).$ Then $\phi_{k}^{}(dx^{1}\otimes dx^{1}+dx^{2}\otimes dx^{2})=\frac{1}{r^{2}}(dt^{2}+d\theta^{2}).$ Thus $f_{k}\circ\phi_{k}\in\mathcal{F}^{p}_{conf}(S^{1}\times[0,T_{k}],R)$. We will also denote $f_{k}\circ\phi_{k}$ by $f_{k}$ for simplicity of notations. ###### Lemma 2.7. There exists a subsequence of $\\{f_{k}\\}$ and $0=d_{k}^{0}<d_{k}^{1}<\cdots<d_{k}^{l}=T_{k}$ with $l<{\Lambda/\epsilon_{2}^{2}}+1$, such that (2.16) $\lim_{k\rightarrow+\infty}d_{k}^{j}-d_{k}^{j-1}=\infty,$ (2.17) $W(f_{k},S^{1}\times[d_{k}^{j},d_{k}^{j}+1])\geq\epsilon_{2}^{2},\,\,\,\,j\neq 0,l$ and (2.18) $\lim_{T\rightarrow+\infty}\varliminf_{k\rightarrow+\infty}\sup_{t\in[d_{k}^{j-1}+T,d_{k}^{j}-T]}W(f_{k},S^{1}\times[t,t+1])\leq\epsilon_{2}^{2},\,\,\,\,j=1,...,l.$ ###### Proof. Suppose $(m-1)\epsilon_{2}^{2}<W(f_{k},S^{1}\times[0,T_{k}])\leq\epsilon_{2}^{2}\,m,$ where $m$ is a positive integer. We prove the lemma by induction on $m$. When $m=1$, the lemma is obvious by taking $d^{0}_{k}=0,d^{1}_{k}=T_{k}$ and (2.17) is vacuous. Assuming the lemma is true for $m-1$, we will prove it also true for $m$. First of all, if (2.19) $\lim_{T\rightarrow+\infty}\varliminf_{k\rightarrow+\infty}\sup_{t\in[T,T_{k}-T]}W(f_{k},S^{1}\times[t,t+1])\leq\epsilon_{2}^{2},$ then the lemma follows since $[d^{j-1}_{k}+T,d^{j}_{k}-T]\subset[T,T_{k}-T]$. If (2.19) does not hold, we can find $t_{k}$ such that $t_{k}\rightarrow+\infty,\,\,\,\,T_{k}-t_{k}\rightarrow+\infty,$ and $W(f_{k},S^{1}\times[t_{k},t_{k}+1])\geq\epsilon_{2}^{2}.$ Then $W(f_{k},S^{1}\times[0,t_{k}])\leq\epsilon_{2}^{2}\,(m-1)\,\,\,\,\hbox{and}\,\,\,\,W(f_{k},S^{1}\times[t_{k}+1,T_{k}])\leq\epsilon_{2}^{2}\,(m-1).$ Using the induction hypothesis on $[0,t_{k}]$ and $[t_{k}+1,T_{k}]$, we can find $0=\bar{d}_{k}^{0}<\bar{d}_{k}^{1}<\cdots<\bar{d}_{k}^{\bar{l}}=t_{k},\,\,\,\,\hbox{and}\,\,\,\,t_{k}+1=\widehat{d}_{k}^{0}<\widehat{d}_{k}^{1}<\cdots<\widehat{d}_{k}^{\hat{l}}=T_{k},$ such that $\bar{d}_{k}^{i}-\bar{d}_{k}^{i-1}\rightarrow+\infty,\,\,\,\,\,\,\,\,\widehat{d}_{k}^{i}-\widehat{d}_{k}^{i-1}\rightarrow+\infty,$ $W(f_{k},S^{1}\times[\bar{d}_{k}^{j},\bar{d}_{k}^{j}+1])\geq\epsilon_{2}^{2},\,\,\,\,W(f_{k},S^{1}\times[\widehat{d}_{k}^{j},\widehat{d}_{k}^{j}+1])\geq\epsilon_{2}^{2},$ and $\lim_{T\rightarrow+\infty}\varliminf_{k\rightarrow+\infty}\sup_{t\in[\bar{d}_{k}^{j-1}+T,\bar{d}_{k}^{j}-T]}W(f_{k},S^{1}\times[t,t+1])\leq\epsilon_{2}^{2},$ $\lim_{T\rightarrow+\infty}\varliminf_{k\rightarrow+\infty}\sup_{t\in[\widehat{d}_{k}^{j-1}+T,\widehat{d}_{k}^{j}-T]}W(f_{k},S^{1}\times[t,t+1])\leq\epsilon_{2}^{2}.$ Put $d_{k}^{i}=\left\\{\begin{array}[]{ll}\bar{d}_{k}^{i}&i\leq\bar{l},\\\ \widehat{d}_{k}^{i-\bar{l}}&i>\bar{l}.\end{array}\right.$ The induction is complete. $\hfill\Box$ We now start to construct the bubble tree at the first level. In Lemma 2.7, if $l=1$, in view of Proposition 2.6, we do not do anything as there is no bubble developing in $S^{1}\times[0,T_{k}]$ when $k\to\infty$. If $l>1$, we set $f_{k}^{i}(\theta,t)=f_{k}(\theta,d_{k}^{i}+t)$. We may assume $\\{f_{k}^{i}\\}$ converges weakly in $W^{2,p}$ to a bubble $f_{\infty}^{i}$ in any compact set outside the blowup points of $\\{f_{k}^{i}\\}$. By Proposition 2.6, there is no other bubble of $f_{k}$ between $f_{\infty}^{i}$ and $f_{\infty}^{i+1}$ and $f_{\infty}^{i}\cup f_{\infty}^{i+1}$ is connected. Clearly, $\\{f_{k}^{0}\\}$ and $\\{f_{k}^{l}\\}$ have no blowup points. Moreover $f_{\infty}^{0}$ is just a part of $f_{\infty}$ and $f_{\infty}^{l}$ is just a part of $f^{F}$. Removing the point singularity by Proposition 2.4, $f_{\infty}^{1}$, $\cdots$, $f_{\infty}^{l-1}$ and $f^{F}$ can be considered as conformal mappings from $S^{2}$ into $\mathbb{R}^{n}$. Figure 2. Bubble tree: First level (dots denote concentration points) For a stratified sphere, we can define a dual graph as following: 1) Associate one vertex for each component of the stratified sphere; 2) Vertices are connected by edges if the corresponding components meet at a point. Let $S_{1}$ be the stratified sphere with $l$ components whose dual graph is a tree, i.e. no loops. We define $F^{1}$ to the continuous map from $S_{1}$ into $\mathbb{R}^{n}$, such that $F^{1}$ is $f_{\infty}^{i}$ on the $i$-th component when $i<l$ and $f^{F}$ on the $l$-th component. We call $F^{1}$ the first level of bubble tree of $\\{f_{k}\\}$. We define $E(F^{1})$ and $W(F^{1})$ by $E(F^{1})=\sum_{i=1}^{l-1}\int_{S^{1}\times\mathbb{R}}\|\nabla f_{\infty}^{i}\|^{2}+\int_{S^{2}}\|\nabla f^{F}\|,\,\,\,\,W(F^{1})=\sum_{i=1}^{l-1}W(f_{\infty}^{i})+W(f^{F}).$ Then $\lim_{\delta\rightarrow 0}\lim_{k\rightarrow+\infty}\int_{D_{\delta}}\|\nabla f_{k}\|^{2}=E(F^{1})+\sum_{i}\sum_{p\in\mathcal{C}(\\{f_{k}^{i}\\})}\lim_{r\rightarrow 0}\lim_{k\rightarrow+\infty}\int_{B_{r}(p)}\|\nabla f_{k}^{i}\|^{2}$ and $\lim_{\delta\rightarrow 0}\lim_{k\rightarrow+\infty}W(f_{k},D_{\delta})\geq W(F^{1})+\sum_{i}\sum_{p\in\mathcal{C}(\\{f_{k}^{i}\\})}\lim_{r\rightarrow 0}\lim_{k\rightarrow+\infty}W(f_{k}^{i},B_{r}(p)).$ Step 2. We consider the convergence of $\\{f_{k}^{i}\\}$ near its blow up points. For each $p\in\mathcal{C}(\\{f_{k}^{i}\\})$, we find a small $r$ such that $B_{r}(p)\subset S^{1}\times\mathbb{R}$ contains only one blowup point. Then for each $p$, using the arguments in Step 1, we will get the first level of bubble tree of $\\{f_{k}^{i}\\}$, which is a map $F_{p}$ from a stratified sphere $S_{p}$ into $\mathbb{R}^{n}$. Each $S_{p}$ is attached to $S_{1}$ at $p$. Taking union over $p\in\mathcal{C}(\\{f_{k}^{i}\\})$ gives us a continuous map $F^{2}$ from $S_{2}$, which is a union of stratified spheres, into $\mathbb{R}^{n}$. We call $F^{2}$ the second level of the bubble tree of $\\{f_{k}\\}$. Figure 3. Bubble tree: Second level Step 3. In the same way, we can build the third and higher levels of the bubble tree. Since each step will take away at least $\epsilon_{2}^{2}$ from the Willmore functional, the construction will stop after finite many steps. In the end we get a stratified surface $S$ which is the union of all levels and a mapping $F$ from $S$ into $\mathbb{R}^{n}$. We shrink all the components of $S$ on which $F$ is trivial into points, i.e. deleting the ghost bubbles, then we get a new stratified surface $S^{\prime}$ and a continuous map $F^{\prime}$ from $S^{\prime}$ into $\mathbb{R}^{n}$, such that $F^{\prime}$ is nontrivial on each component of $S^{\prime}$. We call $F^{\prime}$ is the bubble tree of $\\{f_{k}\\}$ at $0$. Moreover, we have $\lim_{\delta\rightarrow 0}\lim_{k\rightarrow+\infty}\int_{D_{\delta}}\|\nabla f_{k}\|^{2}=E(F^{\prime}),$ and $W(F^{\prime})\leq\lim_{\delta\rightarrow 0}\lim_{k\rightarrow+\infty}W(f_{k},D_{\delta}).$ ### 2.4. Bubble trees for a sequence of maps from cylinders $Q(T_{k})$ with $T_{k}\rightarrow+\infty$ In this subsection, we show that in the situation that blowup occurs along a long cylinder we can divide the long cylinder into smaller ones and apply the results in the subsection 2.3 on each. Let $f_{k}\in\mathcal{F}^{p}_{conf}(Q(T_{k}),R)$ with $T_{k}\rightarrow+\infty$. In light of Proposition 2.6, we only need to consider the case that the following happens: (2.20) $\lim_{T\rightarrow+\infty}\varliminf_{k\rightarrow+\infty}\sup_{t\in[-T_{k}+T,T_{k}-T]}W(f_{k},S^{1}\times[t,t+1])\geq\epsilon_{2}^{2}$ since otherwise there will be no bubbles, no necks and no energy loss. When (2.20) holds, there exist $t_{k}>0$ such that $T_{k}-t_{k}\to+\infty$ as $k\to\infty$ and $W(f_{k},S^{1}\times[t_{k},t_{k}+1])\geq\epsilon^{2}_{2}.$ Case I. If $\\{t_{k}\\}$ contains a bounded subsequence, we choose this subsequence and apply the bubble tree construction in subsection 2.3 at the blowup points. Case II. If $\\{t_{k}\\}$ does not contain any bounded subsequences, then by Lemma 2.7, we can find (by translations) $-T_{k}=d_{k}^{0}<d_{k}^{1}<\cdots<d_{k}^{l}=T_{k}$ which satisfy (2.16), (2.17) and (2.18). Recall that $l$ is independent of $k$. We may assume $f_{k}^{i}(t,\theta)=f_{k}(d_{k}^{i}+t,\theta)$ converge weakly to $f_{\infty}^{i}$ in $W^{2,p}$ outside the blowup points ${\mathcal{C}}(\\{f^{i}_{k}\\})$ of $\\{f_{k}^{i}\\}$. Let $\Sigma_{\infty}^{1}$ be the stratified surface with $l-1$ components whose dual graph is a tree. Then we get a continuous map $F^{1}$ from $\Sigma^{1}_{\infty}$ into $\mathbb{R}^{n}$, and $F^{1}$ is $f_{\infty}^{i}$ on the $i$-th component. Moreover, we have $\lim_{T\rightarrow+\infty}\lim_{k\rightarrow+\infty}\int\limits_{S^{1}\times[-T_{k}+T,T_{k}-T]}\|\nabla f_{k}\|^{2}=E(F^{1})+\sum_{i=1}^{l-1}\sum_{p\in\mathcal{C}(\\{f_{k}^{i}\\})}\lim_{r\rightarrow 0}\lim_{k\rightarrow+\infty}\int_{B_{r}(p)}\|\nabla f_{k}^{i}\|^{2}$ and $\lim_{T\rightarrow+\infty}\lim_{k\rightarrow+\infty}W(f_{k},Q(T_{k}-T))\geq W(F^{1})+\sum_{i=1}^{l-1}\sum_{p\in\mathcal{C}(\\{f_{k}^{i}\\})}\lim_{r\rightarrow 0}\lim_{k\rightarrow+\infty}W(f_{k}^{i},B_{r}(p)).$ The first level of the bubble tree of $\\{f_{k}\\}$ is $F^{1}$ together with the bubble tree in Case I. Then we repeat this process to construct the second level of the bubble tree at $\bigcup^{l}_{i=1}{\mathcal{C}}(\\{f^{i}_{k}\\})$, and similarly the third level and so on. The construction stops in finite steps. ### 2.5. Convergence in Hausdorff distance The main aim of this subsection is to prove the following: ###### Theorem 2.8. Assume that $(\Sigma,h_{k})$ are a sequence of close Riemann surface of genus $g$, where $h_{k}$ satisfies (1.1). Suppose that $f_{k}\in\mathcal{F}^{p}_{conf}(\Sigma,h_{k},R)$ with $p\in(1,\frac{4}{3})$ and (2.21) $\sup_{p}\\{E(f_{k})+W(f_{k})\\}<\Lambda<\infty.$ Then either $f_{k}$ converges to a point, or there is a stratified surface $\Sigma_{\infty}$ with $g(\Sigma_{\infty})\leq g$, an $f_{0}\in\mathcal{F}^{p}_{conf}(\Sigma_{\infty},R)$, such that a subsequence of $f_{k}(\Sigma_{k})$ converges to $f_{0}(\Sigma_{\infty})$ in the Hausdorff distance with $E(f_{0})=\lim_{k\rightarrow+\infty}E(f_{k}),\,\,\,\,and\,\,\,\,W(f_{0})\leq\lim_{k\rightarrow+\infty}W(f_{k}).$ Remark. Here $f_{0}\in\mathcal{F}^{p}_{conf}(\Sigma_{\infty},R)$ means that $f_{0}\in C^{0}(\Sigma_{\infty},\mathbb{R}^{n})$, and for any component $\Sigma^{i}_{\infty}$ of $\Sigma_{\infty}$, $F$ is nontrivial on $\Sigma^{i}_{\infty}$ and $F\|_{\Sigma^{i}_{\infty}}\in\mathcal{F}^{p}(\overline{\Sigma^{i}_{\infty}},h_{i},R)$. Proof of Theorem 2.8: The proof will be divided into three cases according to the genus of $\Sigma$. Spherical case. When $\Sigma$ is a sphere, as there is only one conformal structure on a 2-sphere, we may let $h_{k}\equiv h$. Let $\mathcal{C}(\\{f_{k}\\})=\\{p_{1},\dots,p_{m}\\}.$ We can choose $\delta$, such that $B_{\delta}(p_{i})\cap B_{\delta}(p_{j})=\emptyset$. Using isothermal coordinates, each $B_{\delta}(p_{i})$ with metric $h$ is conformal to a Euclidean disk, the results can be deduced from subsection 2.3 directly. Toric case. Suppose that $(\Sigma,h)$ is induced by lattice $\\{1,a+bi\\}$ in $\mathbb{C}$, where $-\frac{1}{2}<a\leq\frac{1}{2}$, $b>0$, $a^{2}+b^{2}\geq 1$, and $a\geq 0$ whenever $a^{2}+b^{2}=1$. Then the conformal map $f$ from $(\Sigma,h)$ into $\mathbb{R}^{n}$ can be lifted to a conformal map $\widetilde{f}$ from $\mathbb{C}$ into $\mathbb{R}^{n}$ which satisfies $\widetilde{f}(z+a+bi)=\widetilde{f}(z).$ Let $\begin{array}[]{lll}\Pi:&\mathbb{C}\rightarrow S^{1}\times\mathbb{R}\\\ &a+bi\rightarrow(\pi a,\pi b),\end{array}$ be the conformal covering map. Then $(\Sigma,h)$ is conformal to $(S^{1}\times\mathbb{R})/G$, where $G\cong\mathbb{Z}$ is the transformation group of $S^{1}\times\mathbb{R}$ generalized by $(\theta,t)\rightarrow(\theta+2\pi a,t+2\pi b).$ Then $f$ can be lifted to a map $f^{\prime}:S^{1}\times\mathbb{R}^{n}$, which satisfies $f^{\prime}(\Pi)=\widetilde{f}$. Now we assume $(\Sigma_{k},h_{k})=S^{1}\times\mathbb{R}/G_{k}$, where $G_{k}$ is generalized by $(\theta,t)\rightarrow(\theta+\theta_{k},t+b_{k}),\,\,\,\,where\,\,\,\,b_{k}\geq\sqrt{\pi^{2}-\theta_{k}^{2}},\,\,\,\,and\,\,\,\,\theta_{k}\in[-\frac{\pi}{2},\frac{\pi}{2}].$ In the moduli space $\mathcal{M}_{1}$ of genus 1 surfaces, $(\Sigma,h_{k})$ diverges if and only if $b_{k}\rightarrow+\infty$. For $f_{k}\in\mathcal{F}^{p}_{conf}(\Sigma,h_{k},R)$ with (2.21), we lift each $f_{k}$ to a mapping $f_{k}^{\prime}:S^{1}\times\mathbb{R}\rightarrow\mathbb{R}^{n}$ which satisfies $f_{k}^{\prime}(\theta,t)=f_{k}^{\prime}(\theta+\theta_{k},t+a_{k}).$ After translations, we may assume that $f_{k}^{\prime}(\theta,t+\frac{a_{k}}{2})$ and $f_{k}^{\prime}(\theta,t-\frac{a_{k}}{2})$ have no blowup points as $k\to\infty$. Then $f_{k}^{\prime}$ satisfies the conditions in subsection 2.4 for $T_{k}=a_{k}/2$. Since $f_{k}^{\prime}(\theta,-T_{k}+t)=f_{k}^{\prime}(\theta+\theta_{k},T_{k}+t)$, the weak limit of $f_{k}^{\prime}(\theta,-T_{k}+t)$ in $W^{2,p}_{loc}(S^{1}\times[0,+\infty))$ and the weak limit of $f_{k}^{\prime}(\theta,T_{k}+t)$ in $W^{2,p}_{loc}(S^{1}\times(-\infty,0])$ are just the two parts of a conformal map from $S^{1}\times\mathbb{R}$ into $\mathbb{R}^{n}$. So the Hausdorff limit of $f_{k}(\Sigma)$ is the image of a continuous map $F$ from a stratified surface $S$ of genus 1 into $\mathbb{R}^{n}$ with $E(F)=\lim_{k\rightarrow+\infty}E(f_{k}),\,\,\,\,W(F)\leq\lim_{k\rightarrow+\infty}W(f_{k}).$ Hyperbolic case. For the hyperbolic case, we first briefly review the compactness of moduli space. Let $\Sigma_{0}$ be a stable surface in $\overline{\mathcal{M}}_{g}$ with nodal points $\mathcal{N}=\\{a_{1},\dots,a_{m^{\prime}}\\}$. Geometrically, $\Sigma_{0}$ is obtained by pinching $m^{\prime}$ non null homotopy curves in a surface with genus $g>1$ to points $a_{1},\dots,a_{m^{\prime}}$, thus $\Sigma_{0}\backslash\mathcal{N}$ can be divided to finite components $\Sigma_{0}^{1},\dots,\Sigma_{0}^{s}$. For each $\Sigma_{0}^{i}$, we can extend $\Sigma_{0}^{i}$ to a smooth closed Riemann surface $\overline{\Sigma_{0}^{i}}$ by adding a point at each puncture. Moreover, the complex structure of $\Sigma_{0}^{i}$ can be extended smoothly to a complex structure of $\overline{\Sigma_{0}^{i}}$. We say $h_{0}$ determines a hyperbolic structure on $\Sigma_{0}$ if $h_{0}$ is a smooth complete metric on $\Sigma_{0}\backslash\mathcal{N}$ with finite volume and Gauss curvature $-1$. We define a neighborhood around each nodal point $a_{j}$ in $\Sigma_{0}$ by $\Sigma_{0}(a_{j},\delta)=\left\\{p\in\Sigma_{0}:\,\,\,\,\hbox{injrad}_{\Sigma_{0}\backslash\mathcal{N}}^{h_{0}}(p)<\delta,\,\,\,\,\forall p\in\Sigma_{0}(a_{j},\delta)\backslash\\{a_{j}\\}\right\\}\bigcup\,\\{a_{j}\\}.$ Let $h_{0}^{i}$ be the metric on $\overline{\Sigma_{0}^{i}}$ which has Gauss curvature $\pm 1$ or curvature 0, and is conformal to $h_{0}$ on $\Sigma_{0}^{i}$. Now, we let $\Sigma_{k}$ be a sequence of compact Riemann surfaces of fixed genus $g$ with hyperbolic structures $h_{k}$, such that $\Sigma_{k}\rightarrow\Sigma_{0}$ in the moduli space $\overline{\mathcal{M}_{g}}$. By Proposition 5.1 in [11], there exists a maximal collection $\Gamma_{k}=\\{\gamma_{k}^{1},\ldots,\gamma_{k}^{m^{\prime}}\\}$ of pairwise disjoint, simple closed geodesics in $\Sigma_{k}$ with $\ell^{j}_{k}=L(\gamma_{k}^{j})\to 0$, such that after passing to a subsequence the following holds: * (1) There are maps $\varphi_{k}\in C^{0}(\Sigma_{k},\Sigma_{0})$, such that $\varphi_{k}:\Sigma_{k}\backslash\Gamma_{k}\to\Sigma_{0}\backslash\mathcal{N}$ is diffeomorphic and $\varphi_{k}(\gamma_{k}^{j})=a_{j}$ for $j=1,\ldots,m^{\prime}$. * (2) For the inverse diffeomorphisms $\psi_{k}:\Sigma_{0}\backslash\mathcal{N}\to\Sigma_{k}\backslash\Gamma_{k}$, we have $\psi_{k}^{\ast}(h_{k})\to h_{0}$ in $C^{\infty}_{loc}(\Sigma_{0}\backslash\mathcal{N})$, where $h_{0}$ determine a hyperbolic structure over $\Sigma_{0}\backslash\mathcal{N}$. * (3) Let $c_{k}$ be the complex structure over $\Sigma_{k}$, and $c_{0}$ be the complex structure over $\Sigma_{0}\backslash\mathcal{N}$. Then $\psi_{k}^{}(c_{k})\rightarrow c_{0}\,\,\,\,in\,\,\,\,C^{\infty}_{loc}(\Sigma_{0}\backslash\mathcal{N}).$ Moreover, we have the following Collar Lemma [9, 12, 20, 26]: ###### Lemma 2.9. For each $\gamma_{k}^{j}$ as above, there is a collar $U_{k}^{j}$ containing $\gamma_{k}^{j}$, which is isometric to the cylinder $Q_{k}^{j}=Q(\frac{\pi^{2}}{l_{k}^{j}})$ with metric (2.22) $h_{k}^{j}=\left(\frac{1}{2\pi\sin(\frac{l_{k}^{j}}{2\pi}t+\theta_{k})}\right)^{2}(dt^{2}+d\theta^{2}),$ where $\theta_{k}=\arctan(\sinh(\frac{l_{k}^{j}}{2}))+\frac{\pi}{2}$. Moreover, for any $(\theta,t)\in S^{1}\times(-\frac{\pi^{2}}{l_{k}^{j}},\frac{\pi^{2}}{l_{k}^{j}})$, we have (2.23) $\sinh(\it{injrad}_{\Sigma_{k}}(\theta,t))\sin(\frac{l_{k}^{j}t}{2\pi}+\theta_{k})=\sinh\frac{l_{k}^{j}}{2}.$ Let $\phi_{k}^{j}$ be the isometry between $Q_{k}^{j}$ and $U_{k}^{j}$. Then $\varphi_{k}\circ\phi_{k}^{j}(\theta,\frac{\pi^{2}}{l_{k}^{j}}+t)\bigcup\varphi_{k}\circ\phi_{k}^{j}(\theta,-\frac{\pi^{2}}{l_{k}^{j}}+t)$ converges in $C^{\infty}_{loc}(S^{1}\times(-\infty,0)\cup S^{1}\times(0,\infty))$ to an isometry from $S^{1}\times(-\infty,0)\cup S^{1}\times(0,+\infty)$ to $\Sigma_{0}(a_{j},1)\backslash\\{a_{j}\\}$. The Collar Lemma can be found in [12], [9] and [11]. We also need the following local existence and compactness of conformal diffeomorphisms. ###### Theorem 2.10. [4] Let $h_{k},h_{0}$ be smooth Riemannian metrics on a surface $M$, such that $h_{k}\to h_{0}$ in $C^{s,\alpha}(M)$, where $s\in\mathbb{N}$, $\alpha\in(0,1)$. Then for each point $z\in M$ there exist neighborhoods $U_{k},U_{0}$ of $z$ and smooth conformal diffeomorphisms $\vartheta_{k}:D\to U_{k},\vartheta_{0}:D\rightarrow U$, such that $\vartheta_{k}\to\vartheta_{0}$ in $C^{s+1,\alpha}(\overline{D},M)$. Proof of Theorem 2.8 (continued): For a sequence $f_{k}\in\mathcal{F}^{p}_{conf}(\Sigma,h_{k},R)$ satisfying the energy bound (2.21), let $\widetilde{f}_{k}=f_{k}\circ\psi_{k}$ which is a mapping from $\Sigma_{0}\backslash\mathcal{N}$ to $\mathbb{R}^{n}$. It is easy to check that $\widetilde{f}_{k}\in\mathcal{F}^{p}_{conf}(\Sigma_{0}\backslash\mathcal{N},\psi_{k}^{\ast}(h_{k}),R)$. First, we show $\widetilde{f}_{k}$ converge in $W^{2,p}_{loc}(\Sigma_{0}\backslash(\mathcal{N}\cup\mathcal{C}(\\{f_{k}\\})))$. Given a point $z\in\Sigma_{0}\backslash(\mathcal{N}\cup\mathcal{C}(\\{\widetilde{f}_{k}\\}))$, we choose $U_{k},U,\vartheta_{k},\vartheta$ as in Theorem 2.10 and $U_{k}$, $U\subset\Sigma_{0}\backslash(\mathcal{N}\cup\mathcal{C}(\\{\widetilde{f}_{k}\\}))$. Let $\widehat{f}_{k}=\widetilde{f}_{k}\circ\varphi_{k}$ and note that $\widehat{f}_{k}\in\mathcal{F}^{p}_{conf}(D,R)$. We can assume that $\widehat{f}_{k}$ converge to $\widehat{f}_{\infty}$ in $W^{2,p}_{loc}(D_{3/4})$ with $\partial\widehat{f}_{\infty}\otimes\partial\widehat{f}_{\infty}=0$. Let $V=\vartheta(D_{1/2})$. Since $\vartheta_{k}$ converge to $\vartheta$, $\vartheta_{k}^{-1}(V)\subset D_{3/4}$ for sufficiently large $k$, $\widetilde{f}_{k}=\widehat{f}_{k}(\vartheta_{k}^{-1})$ converge to $\widetilde{f}_{\infty}=\widehat{f}_{\infty}(\vartheta_{0}^{-1})$ weakly in $W^{2,p}(V,h_{0})$. Then $\widetilde{f}_{\infty}\in\mathcal{F}^{p}_{conf}(V,h_{0},R)$. Moreover, for any nonnegative function $\varphi$ with support in $V$, from Fatou’s lemma (2.24) $\lim_{k\rightarrow+\infty}\int_{V}\varphi\|H(\widetilde{f}_{k})\|^{2}\|\nabla\widetilde{f}_{k}\|^{2}=\lim_{k\rightarrow+\infty}\int_{D}\varphi(\vartheta_{k})\|H(\widehat{f}_{k})\|^{2}\|\nabla\widehat{f}_{k}\|^{2}\geq\int_{D}\varphi(\vartheta_{0})\|H(\widehat{f}_{0})\|^{2}\|\nabla\widehat{f}_{0}\|^{2}.$ We may thus assume $\widetilde{f}_{k}$ converge weakly to $\widetilde{f}_{\infty}$ in $W^{2,p}_{loc}(\Sigma_{0}\backslash(\mathcal{N}\cup\,\mathcal{C}(\\{f_{k}\\})))$. Then $\widetilde{f}_{\infty}\|_{\Sigma_{0}^{i}}\in W^{2,p}_{loc}(\Sigma_{0}^{i},h_{0}^{i})$. So for $p\in(1,{4/3})$, $\widetilde{f}_{\infty}\|_{\Sigma_{0}^{i}}$ can be extended to a map in $\mathcal{F}^{p}_{conf}(\overline{\Sigma_{0}^{i}},h_{0}^{i},R)$. Further, $\lim_{k\rightarrow+\infty}E(f_{k})=E(\widetilde{f}_{\infty})+\sum_{z\in\mathcal{C}(\\{\widetilde{f}_{k}\\})}\lim_{r\rightarrow 0}\lim_{k\rightarrow+\infty}E(\widetilde{f}_{k},B_{r}(z,h_{0}))+\sum_{j}\lim_{\delta\rightarrow 0}\lim_{k\rightarrow+\infty}E(\widetilde{f}_{k},\Sigma_{0}(a_{j},\delta))$ and from (2.24) $\lim_{k\rightarrow+\infty}W(\widetilde{f}_{k})\geq W(\widetilde{f}_{\infty})+\sum_{z\in\mathcal{C}(\\{\widetilde{f}_{k}\\})}\lim_{r\rightarrow 0}\lim_{k\rightarrow+\infty}W(\widetilde{f}_{k},B_{r}(z,h_{0}))+\sum_{j}\lim_{\delta\rightarrow 0}\lim_{k\rightarrow+\infty}W(\widetilde{f}_{k},\Sigma_{0}(a_{j},\delta)).$ Next, we construct bubble trees at a point $z\in\mathcal{C}(\\{f_{k}\\})\backslash\mathcal{N}$. We have a bubble tree $F$ of $\widehat{f}_{k}$ at $z$. We define it to be a bubble tree of $\widetilde{f}_{k}$ at $z$. By the arguments in subsection 2.3, we have $\lim_{r\rightarrow 0}\lim_{k\rightarrow+\infty}E(\widetilde{f}_{k},B_{r}(z,h_{0}))=\lim_{r\rightarrow 0}\lim_{k\rightarrow+\infty}E(\widehat{f}_{k},D_{r})=E(F),$ and $\lim_{r\rightarrow 0}\lim_{k\rightarrow+\infty}W(\widetilde{f}_{k},B_{r}(z,h_{0}))=\lim_{r\rightarrow 0}\lim_{k\rightarrow+\infty}W(\widehat{f}_{k},D_{r})\geq W(F).$ Lastly, we consider the convergence of $f_{k}$ at the collars. Set $\check{f}_{k}^{j}=f_{k}\circ\phi_{k}^{j}$ and $T_{k}^{j}={\pi^{2}/l_{k}^{j}}-T$. We may choose $T$ to be sufficiently large such that the two sequences $\check{f}_{k}^{j}(T_{k}^{j}-t,\theta)$ and $\check{f}_{k}^{j}(-T_{k}^{j}+t,\theta)$ have no blowup points in $[0,T]$ and $[-T,0]$ respectively (otherwise, return to the previous case as in Case I, subsection 2.4). Then $\check{f}_{k}^{j}$ satisfies the conditions in subsection 2.4. We get a bubble tree $F^{j}$. So the convergence of $\check{f}_{k}^{j}$ is clear. Since $\check{f}_{k}^{j}=f_{k}\circ\phi_{k}^{j}=f_{k}\circ\psi_{k}\circ(\varphi_{k}\circ\phi_{k}^{j})=\widetilde{f}_{k}(\varphi_{k}\circ\phi_{k}^{j}),$ we have $\displaystyle\check{f}_{k}^{j}(T_{k}^{j}-t,\theta)$ $\displaystyle=$ $\displaystyle\widetilde{f}_{k}(\varphi_{k}\circ\phi_{k}^{j}(T_{k}^{j}-t,\theta)),$ $\displaystyle\check{f}_{k}^{j}(t-T_{k}^{j},\theta)$ $\displaystyle=$ $\displaystyle\widetilde{f}_{k}(\varphi_{k}\circ\phi_{k}^{j}(t-T_{k}^{j},\theta)).$ By the convergence statement of the Collar Lemma, $\varphi_{k}\circ\phi^{j}_{k}(t+T-\pi^{2}/l^{j}_{k},\theta)$ and $\varphi_{k}\circ\phi_{k}^{j}({\pi^{2}/l_{k}^{j}}-T-t,\theta)$ converge in $C^{\infty}_{loc}(S^{1}\times(0,\infty))$ to an isometry from $S^{1}\times(-\infty,0)\cup S^{1}\times(0,+\infty)$ to $\Sigma_{0}(a_{j},1)\setminus\\{a_{j}\\}$. We conclude that the image of the limit of $\check{f}_{k}^{j}(T_{k}^{j}-t,\theta)$ and that of $\check{f}_{k}^{j}(-T_{k}^{j}+t,\theta)$ are both contained in the image of $\widetilde{f}_{\infty}$. Moreover, $\lim_{\delta\rightarrow 0}\lim_{k\rightarrow+\infty}E(\widetilde{f}_{k},\Sigma_{0}(a_{j},\delta))=\lim_{T\rightarrow+\infty}\lim_{k\rightarrow+\infty}E(\check{f},Q(T_{k}-T))=E(F^{j}),$ and $\lim_{\delta\rightarrow 0}\lim_{k\rightarrow+\infty}W(\widetilde{f}_{k},\Sigma_{0}(a_{j},\delta))=\lim_{T\rightarrow+\infty}\lim_{k\rightarrow+\infty}W(\check{f},Q(T_{k}-T))=E(F^{j}).$ Thus, we complete the proof. $\hfill\Box$ ###### Remark 2.11. When $\Sigma_{0}\in\mathcal{M}_{p}$, i.e. $\mathcal{N}=\emptyset$, $\psi_{k}$ is just a smooth diffeomorphism sequence from $\Sigma$ to $\Sigma$. In this case, $g(\Sigma_{\infty})=g(\Sigma)$, and $\Sigma_{\infty}=\Sigma_{\infty}^{\prime}\cup S_{1}\cup S_{2}\cdots\cup S_{m},$ where $\Sigma_{\infty}^{\prime}$ is a smooth Riemann surface of genus $p$, and each $S_{i}$ is a sphere. Figure 4. $\Sigma_{0}$ (limit of $(\Sigma,h_{k}))$ and $\Sigma_{\infty}$ (some components of $\Sigma_{0}$ do not appear in $\Sigma_{\infty}$) We now generalize Theorem 2.8 to surfaces with marked points. Let us briefly review the compactification of the moduli space of surfaces with marked points. Let $\overline{\mathcal{M}}_{g,m}$ be the moduli space of closed Riemann surfaces of genus $g$ with $m$ marked points. Let $(\Sigma_{0},x_{0,1},\dots,x_{0,m})\in\partial\overline{\mathcal{M}}_{g,m}$ with nodal points $\mathcal{N}=\\{a_{1},\dots,a_{m^{\prime}}\\}$. Geometrically, $\Sigma_{0}$ is obtained by pinching some homotopically nontrivial closed curves which do not pass any of $x_{0,1},\dots,x_{0,m}$ into the points in ${\mathcal{N}}$, and $\Sigma\backslash\mathcal{N}$ can be divided to connected components $\Sigma_{0}^{1}$, $\cdots$, $\Sigma_{0}^{s}$. For each $\Sigma_{0}^{i}$, we can extend $\Sigma_{0}^{i}$ to a smooth closed Riemann surface $\overline{\Sigma_{0}^{i}}$ by adding a point at each puncture. Moreover, the complex structure of $\Sigma_{0}^{i}$ can be extended smoothly to a complex structure of $\overline{\Sigma^{i}_{0}}$. We say $h$ is a hyperbolic structure on $(\Sigma,x_{1},\dots,x_{m})\in\mathcal{M}_{g,m}$ if $h$ is a smooth complete metric on $\Sigma\backslash\\{x_{1},\dots,x_{m}\\}$ with curvature $-1$ and finite volume. We say $h_{0}$ is a hyperbolic structure on $(\Sigma_{0},x_{0,1},\dots,x_{0,m})\in\overline{\mathcal{M}}_{g,m}\backslash\mathcal{M}_{g,m}$ if $h_{0}$ is a smooth complete metric on $\Sigma\backslash\\{a_{1},\dots,a_{m^{\prime}},x_{0,1},...,x_{0,m}\\}$ with curvature $-1$ and finite volume. For a surface $\Sigma$ with hyperbolic structure $h$ and with marked points $x_{1},\dots,x_{m}$, we define $\Sigma^{}=\Sigma\backslash\\{x_{1},\dots,x_{m}\\}$, and $h^{}$ to be the hyperbolic structure on $(\Sigma,x_{1},\dots,x_{m})$ which is conformal to $h$ on $\Sigma^{}$. Let $\\{(\Sigma_{k},x_{k,1},\dots,x_{k,m}\\}$ be a sequence of marked surfaces in $\mathcal{M}_{g,m}$ with hyperbolic structures $h_{k}$ and $(\Sigma_{k},x_{k,1},\dots,x_{k,m})\rightarrow(\Sigma_{0},x_{0,1},\dots,x_{0,m})\,\,\,\,in\,\,\,\,\overline{\mathcal{M}}_{g,m}.$ By Proposition 5.1 in [11] again, there exists a maximal collection $\Gamma_{k}=\\{\gamma_{k}^{1},\ldots,\gamma_{k}^{m^{\prime}}\\}$ of pairwise disjoint, simple closed geodesics in $\Sigma_{k}$ with $\ell^{j}_{k}=L(\gamma_{k}^{j})\to 0$ as $k\to\infty$, such that, after passing to a subsequence if necessary, the following holds: * (1) There are maps $\varphi_{k}\in C^{0}(\Sigma_{k},\Sigma_{0})$, such that $\varphi_{k}:\Sigma_{k}\backslash\Gamma_{k}\to\Sigma_{0}\backslash\mathcal{N}$ is diffeomorphic and $\varphi_{k}(\gamma_{k}^{i})=a_{i}$ for $i=1,\dots,m^{\prime}$, and $\varphi_{k}(x_{k,j})=x_{0,j}$ for $j=1,\dots,m$. * (2) For the inverse diffeomorphisms $\psi_{k}:\Sigma_{0}\backslash\mathcal{N}\to\Sigma_{k}\backslash\Gamma_{k}$, we have $\psi_{k}^{\ast}(h_{k})\to h$ in $C^{\infty}_{loc}(\Sigma_{0}^{}\backslash\mathcal{N})$. (3) Let $c_{k}$ be the complex structure over $\Sigma_{k}$, and $c_{0}$ be the complex structure over $\Sigma_{0}\backslash\mathcal{N}$. Then $\psi_{}(c_{k})\rightarrow c_{0}\,\,\,\,in\,\,\,\,C^{\infty}_{loc}(\Sigma_{0}\backslash\mathcal{N}).$ Moreover, the Collar Lemma also holds for the moduli space with marked points. ###### Theorem 2.12. In addition to the assumptions in Theorem 2.8, we assume for $m\geq 2$ $y_{1},\dots,y_{m}\in f_{k}(\Sigma).$ Then there is a stratified surface $\Sigma_{\infty}$ with $g(\Sigma_{\infty})\leq g$, and an $f_{0}\in\mathcal{F}^{p}_{conf}(\Sigma_{\infty},R)$ with $y_{1},\dots,y_{m}\in f_{0}(\Sigma_{\infty}),$ such that a subsequence of $f_{k}(\Sigma_{k})$ converges to $f_{0}(\Sigma_{\infty})$ in Hausdorff distance with $E(f_{0})=\lim_{k\rightarrow+\infty}E(f_{k})\,\,\,\,and\,\,\,\,W(f_{0})\leq\lim_{k\rightarrow+\infty}W(f_{k}).$ ###### Proof. Let $\widetilde{f}_{k}=f_{k}\circ\psi_{k}$. In view of Theorem 2.8, we only need to consider convergence of $\\{\widetilde{f}_{k}\\}$ near $x_{0,j},j=1,\dots,m$. Choose a complex coordinate $\\{U,(x,y)\\}$ on $\Sigma_{0}$ compatible with $c_{0}$, with $x_{0,j}=(0,0)$. Let $c_{k}^{\prime}=\psi_{k}^{\ast}(c_{k})$. We set $e_{1}=\frac{\partial}{\partial x},\,\,\,\,e_{2}=c_{k}^{\prime}(e_{1}),$ and $h_{k}^{\prime}$ to be the metric on $U$ defined by $h_{k}^{\prime}(e_{1},e_{1})=h_{k}^{\prime}(e_{2},e_{2})=1,\,\,\,\,h_{k}^{\prime}(e_{1},e_{2})=0.$ Then $h_{k}^{\prime}$ is compatible with $c_{k}^{\prime}$, and converges smoothly to a metric which is compatible with $c_{0}$ in $U$. Then we consider the weak convergence of $\\{\widetilde{f}_{k}\\}$ in $U\backslash\mathcal{C}(\\{\widetilde{f}_{k}\\})$, using the arguments in subsection 2.3. It remains to check that each marked point $y_{i}$ is on the image of $f_{\infty}$ or one of the bubbles. If $x_{0,j}$ is not a blow up point of $\\{\widetilde{f}_{k}\\}$, it is obvious that $y_{j}\in f_{\infty}(U)$. Now assume $x_{0,j}$ is the only blow-up point in $D$. We take $U_{k}$, $U_{0}$, $\vartheta_{k},\vartheta_{0},\widehat{f}_{k},V$ as in the proof of Theorem 2.8 for $z=x_{0,j}$. We will prove it by induction on the number of the levels of the bubble tree. We take $z_{k}$, $r_{k}$, $\phi_{k}$ and $d_{k}$ as in subsection 2.3 for $\widehat{f}_{k}$. If ${\|z_{k}\|/r_{k}}<L$ for some fixed $L$, then we may assume $-{z_{k}/r_{k}}\rightarrow z_{\infty}$, by selecting a subsequence if necessary. Recalling that $\widehat{f}_{k}(0)\equiv y_{j}$, we get $y_{j}=\widehat{f}^{F}(z_{\infty})$. Let $(r,\theta)$ be the polar coordinates centered at $z_{k}$, $T_{k}=-\ln r_{k}$ and $\phi_{k}:[0,T_{k}]\times S^{1}\rightarrow\mathbb{R}^{2}$ be the conformal mapping given by $\phi_{k}(t,\theta)=(e^{-t},\theta)$. We set $\phi_{k}^{-1}(0)=(t_{k},\theta_{k})$. Then ${\|z_{k}\|/r_{k}}\rightarrow+\infty$ means that $t_{k}\in[0,T_{k}]$ and $T_{k}-t_{k}\rightarrow+\infty$. Thus we may assume $t_{k}\in[d_{k}^{i},d_{k}^{i+1}]$ for some $i$, where $d_{k}^{i}$ are defined in Lemma 2.7. Then, if $t_{k}-d_{k}^{i}\rightarrow+\infty$ and $d_{k}^{i+1}-t_{k}\rightarrow+\infty$, we have $y_{j}=f_{\infty}^{i}(+\infty)=f_{\infty}^{i+1}(-\infty).$ If at least one of $t_{k}-d_{k}^{i}$ and $d_{k}^{i+1}-t_{k}$ is bounded above for all large $k$, then we repeat the above argument at the second level of the bubble tree, and proceed in this way for the finitely many levels of the bubble tree if necessary, and we conclude that $y^{j}$ is on one of bubbles of $\widetilde{f}_{k}^{i}$ or $\widetilde{f}_{k}^{i+1}$. Finally, as $m\geq 2$ and all $y_{i}\in f_{0}(\Sigma_{\infty})$, $f_{k}$ cannot converge to a single point. $\hfill\Box$ ## 3\. Branched conformal immersions and proof of Theorem 1 For a branched conformal immersion, we have the following result: ###### Theorem 3.1. [13] Suppose that $f\in W^{2,2}_{conf,loc}(D\backslash\\{0\\},\mathbb{R}^{n})$ satisfies $\int_{D}\|A_{f}\|^{2}\,d\mu_{g}<\infty\quad\mbox{ and }\quad\mu_{g}(D)<\infty,$ where $g_{ij}=e^{2u}\delta_{ij}$ is the induced metric. Then $f\in W^{2,2}(D,\mathbb{R}^{n})$ and we have $\displaystyle u(z)$ $\displaystyle=$ $\displaystyle\lambda\log\|z\|+\omega(z)\quad\mbox{ where }\lambda\geq 0,\quad\lambda\in\mathbb{Z},\quad\omega\in C^{0}\cap W^{1,2}(D),$ $\displaystyle-\Delta u$ $\displaystyle=$ $\displaystyle-2\lambda\pi\delta_{0}+K_{g}e^{2u}\quad\mbox{ in }D.$ The density of $f(D_{\sigma})$ as varifolds at $f(0)$ is given by $\lambda+1$ for any small $\sigma>0$. The classical Gauss-Bonnet formula is generalized in [6] for smooth branched surface. Following arguments in [6], we provide a version for $W^{2,2}$ branched conformal immersions. ###### Lemma 3.2. Let $(\Sigma,g)$ be a smooth closed compact Riemann surface. Then for any $f\in W_{b,c}^{2,2}(\Sigma,g,\mathbb{R}^{n})$, there holds (3.1) $\int_{\Sigma}K_{f}d\mu_{f}=2\pi\chi(\Sigma)+2\pi b,$ where $b$ is the number of branch points counted with multiplicities and at each branch point $p$ the branching order is $\lambda=\theta^{2}(p)-1$. ###### Proof. Without loss of generality, we only prove the case that $f$ has only one branch point $p$. Let $g_{f}=e^{2u}g$ be the metric induced by $f$ and $K_{f}$ be its Gauss curvature. In [13], we proved that the equation $-\Delta_{g}u=K_{f}e^{2u}-K_{g}$ holds weakly in $\Sigma\backslash\\{p\\}$: for any smooth $\varphi$ with support in $\Sigma\backslash\\{p\\}$, it holds $\int_{\Sigma}\nabla_{g}u\nabla_{g}\varphi\,d\mu_{g}=\int_{\Sigma}\varphi K_{f}e^{2u}d\mu_{g}-\int_{\Sigma}\varphi K_{g}du_{g}.$ Take a complex coordinate chart $\\{U;z\\}$ around $p$ with $p=0$. For any small $\epsilon>0$, we choose a function $\varphi_{\epsilon}(z)=\varphi_{\epsilon}(\|z\|)$ between 0 and 1 with $\|\varphi_{\epsilon}^{\prime}\|<{C/\epsilon}$ and equals 1 outside $D_{\epsilon}$ and $0$ in $D_{\epsilon/2}$. Then we have $\int_{D_{\epsilon}}\frac{\partial u}{\partial r}\varphi_{\epsilon}^{\prime}dx=\int_{\Sigma}\varphi_{\epsilon}K_{f}e^{2u}d\mu_{g}-\int_{\Sigma}\varphi_{\epsilon}K_{g}du_{g}.$ By Theorem 3.1, $u=\lambda\log\|z\|+\omega$, we get $\int_{D_{\epsilon}}\frac{\partial u}{\partial r}\varphi_{\epsilon}^{\prime}dx=\int_{D_{\epsilon}}\frac{\partial\omega}{\partial r}\varphi_{\epsilon}^{\prime}dx+2\pi\lambda\left(\varphi_{\epsilon}(\epsilon)-\varphi_{\epsilon}(0)\right)=\int_{D_{\epsilon}}\frac{\partial\omega}{\partial r}\varphi_{\epsilon}^{\prime}dx+2\pi\lambda.$ Since $\int_{D_{\epsilon}}\left\|\frac{\partial\omega}{\partial r}\varphi_{\epsilon}^{\prime}\right\|\leq C\left(\int_{D_{\epsilon}\backslash D_{\epsilon/2}}\left\|\frac{\partial\omega}{\partial r}\right\|^{2}\right)^{1/2}\left(\int_{D_{\epsilon}\backslash D_{\epsilon/2}}\frac{1}{r^{2}}\right)^{1/2}\leq C\\|\nabla\omega\\|_{L^{2}(D_{\epsilon})}\rightarrow 0,\,\,\,\,\hbox{as}\,\,\,\,\epsilon\rightarrow 0,$ we conclude, by applying the classical Gauss-Bonnet theorem on $(\Sigma,g)$, that $\int_{\Sigma}K_{f}d\mu_{f}=\lim_{\epsilon\rightarrow 0}\int_{\Sigma}\varphi_{\epsilon}K_{f}d\mu_{f}=\lim_{\epsilon\rightarrow 0}\int_{\Sigma}\varphi_{\epsilon}K_{g}d\mu_{g}+\lim_{\epsilon\rightarrow 0}\int_{\Sigma}\frac{\partial u}{\partial r}\varphi_{\epsilon}^{\prime}d\mu_{g}=2\pi\chi(\Sigma)+2\lambda\pi$ and complete the proof. $\hfill\Box$ ###### Remark 3.3. Since $\int_{\Sigma}K_{f}d\mu_{f}\leq W(f)$, it follows from Lemma 3.2 $b\leq\frac{1}{2\pi}W(f)-\chi(\Sigma).$ Moreover, $\int_{\Sigma}\|A_{f}\|^{2}d\mu_{f}=4W(f)-2\int K_{f}\leq 4W(f)-2\pi\chi(\Sigma).$ Then $\sup_{k}W(f_{k})<+\infty$ implies that $\sup_{k}b_{k}<+\infty$ and $\sup_{k}\int_{\Sigma}\|A_{f_{k}}\|^{2}d\mu_{f_{k}}<+\infty$. To study the convergence conformal immersions, we recall an important result of Hélein. ###### Theorem 3.4. [10] Let $f_{k}\in W^{2,2}_{conf}(D,\mathbb{R}^{n})$ be a sequence of conformal immersions with induced metrics $(g_{k})_{ij}=e^{2u_{k}}\delta_{ij}$, and assume $\int_{D}\|A_{f_{k}}\|^{2}\,d\mu_{g_{k}}\leq\gamma<\gamma_{n}=\begin{cases}8\pi&\mbox{ for }n=3,\\\ 4\pi&\mbox{ for }n\geq 4.\end{cases}$ Assume also that $\mu_{g_{k}}(D)\leq C$ and $f_{k}(0)=0$. Then $f_{k}$ is bounded in $W^{2,2}_{loc}(D,\mathbb{R}^{n})$, and there is a subsequence such that one of the following two alternatives holds: (a) $u_{k}$ is bounded in $L^{\infty}_{loc}(D)$ and $f_{k}$ converges weakly in $W^{2,2}_{loc}(D,\mathbb{R}^{n})$ to a conformal immersion $f\in W^{2,2}_{conf,loc}(D,\mathbb{R}^{n})$. * (b) $u_{k}\to-\infty$ and $f_{k}\to 0$ locally uniformly on $D$. Hélein first proved the above result for $\gamma_{n}={8\pi/3}$ [10, Theorem 5.1.1]. In [13] $\gamma_{n}$ is shown to be optimal. Before proving Theorem 1, we recall a monotonicity formula (for more details, see [15, 31]). Let $\mu$ be a 2-dimensional integral varifold with square integrable weak mean curvature $H_{\mu}\in L^{2}(\mu)$. Then we have $g_{x_{0}}(\varrho)\leq g_{x_{0}}(\sigma),\,\,\,\,\hbox{when}\,\,\,\,\varrho<\sigma,$ where $g_{x_{0}}(r)=\frac{\mu(B_{r}(x_{0}))}{\pi r^{2}}+\frac{1}{4\pi}W(\mu,B_{r}(x_{0}))+\frac{1}{2\pi r^{2}}\int_{B_{r}(x_{0})}\langle x-x_{0},H\rangle\,d\mu.$ When $\Sigma$ is compact and connected, if we let $\sigma\rightarrow+\infty$, and $\varrho\rightarrow 0$, then the area density of $\Sigma$ at $x_{0}$ satisfies (3.2) $\theta^{2}(\mu,x_{0})\leq\frac{1}{4\pi}W(f).$ If we only let $\varrho\rightarrow 0$, then we get (3.3) $\theta^{2}(\mu,x_{0})\leq\frac{\mu(B_{\sigma}(x_{0}))}{\pi\sigma^{2}}+CW(f,B_{\sigma}(x_{0}))+C\left(\frac{\mu(B_{\sigma}(x_{0}))}{\pi\sigma^{2}}\right)^{\frac{1}{2}}W(f,B_{\sigma}(x_{0}))^{\frac{1}{2}}.$ Another useful consequence (cf. [31], [15]) is the following: (3.4) $\Big{(}\frac{\mu(\Sigma)}{W(f)}\Big{)}^{\frac{1}{2}}\leq{\rm diam\,}f(\Sigma)\leq C\Big{(}\mu(\Sigma)\,W(f)\Big{)}^{\frac{1}{2}}.$ Proof of Theorem 1. Consider a branched conformal immersion $f_{k}\in W^{2,2}_{b,c}(\Sigma,h_{k},\mathbb{R}^{n})$, where $h_{k}$ satisfies (1.1). The following equation clearly holds on $\Sigma$ away from the finitely many branch points; the singularities at the branch points can be removed by Theorem 3.1, thus it holds on entire $\Sigma$: $\Delta_{h_{k}}f_{k}=\frac{1}{2}H_{f_{k}}\|\nabla_{h_{k}}f_{k}\|^{2}.$ By Remark 3.3, the number of branch points and $\\|A_{f_{k}}\\|_{L^{2}}$ are both bounded from above. By (3.4), $\mbox{\rm diam}f_{k}(\Sigma)\leq{R}$ for some $R>0$. Then $f_{k}\in\mathcal{F}^{2}_{conf}(\Sigma,h_{k},R+R_{0})$. We only need to prove that $f_{0}$ in Theorem 2.8 and Theorem 2.12 is also branched conformal immersion. In fact, we only need to prove the following: If $f_{k}$ are branched conformal immersions from $D$ into $\mathbb{R}^{n}$ with a uniform upper bound on the number of branch points, then the limit $f_{0}$ is either a point or a branched conformal immersion. Let $P$ be the limit set of the branch points, and $\mathcal{S}(\\{f_{k}\\})=\\{z\in D:\lim_{r\rightarrow 0}\varliminf_{k\rightarrow+\infty}\int_{D_{r}(z)}\|A_{k}\|^{2}\geq\hat{\epsilon}^{2}\\},$ where $\hat{\epsilon}\leq\min\\{\sqrt{4\pi},4\epsilon_{0}\\}$. By Theorem 3.4, $f_{k}$ will converge weakly in $W^{2,2}_{loc}(D\backslash(\mathcal{S}\cup P))$ to either a conformal immersion or a point. If the limit is not a single point, by Theorem 3.1, the limit can be extended across the finite set ${\mathcal{S}}\cup P$ to a branched conformal immersion of $D$, hence $\Sigma$, in $\mathbb{R}^{n}$. $\hfill\Box$ ## 4\. Willmore functional for surfaces in compact manifolds Let $N$ be a compact Riemannian manifold without boundary. We embed $N$ into $\mathbb{R}^{n}$ isometrically so that any immersion of $\Sigma$ in $N$ can be regarded as an immersion in $\mathbb{R}^{n}$. Let $A_{\Sigma,N},A_{\Sigma,\mathbb{R}^{n}}$ and $A_{N,\mathbb{R}^{n}}$ be the second fundamental forms of $\Sigma$ in $N$, in $\mathbb{R}^{n}$ and $N$ in $\mathbb{R}^{n}$ respectively. The $L^{2}$ integrals of these quantities can be related as in the following simple lemma. ###### Lemma 4.1. For any $f\in W^{2,2}_{b,c}(\Sigma,h,N)$, we have (4.1) $\int_{\Sigma}\|H_{f,\Sigma,\mathbb{R}^{n}}\|^{2}d\mu_{f}\leq C\mu(f)+\int_{\Sigma}\|H_{f,\Sigma,N}\|^{2}d\mu_{f},$ and (4.2) $\int_{\Sigma}\|A_{f,\Sigma,\mathbb{R}^{n}}\|^{2}d\mu_{f}\leq C\int_{\Sigma}(1+\|H_{f,\Sigma,N}\|^{2})d\mu_{f}+C^{\prime},$ where $C$ only depends on $N$ and $C^{\prime}$ only depends on the Euler characteristic of $\Sigma$. ###### Proof. Let $e_{1},\dots,e_{n}$ be an orthonormal frame of $T_{f_{k}(p)}\mathbb{R}^{n}$ with $e_{1}$, $e_{2}\in Tf(\Sigma)$ and $e_{1},\dots,e_{k}\in TN$. Then for $f_{i}=\frac{\partial f}{\partial x^{i}}$ we have $\nabla^{N}_{f_{i}}f_{j}=\sum_{l=1}^{k}\langle f_{ij},e_{l}\rangle e_{l}$ and $A_{\Sigma,N}(f_{i},f_{j})=\sum_{m=3}^{k}\langle\nabla^{N}_{f_{i}}f_{j},e_{m}\rangle e_{m}=\sum_{m=3}^{k}\langle f_{ij},e_{m}\rangle e_{m}.$ Thus if $F=i\circ f$ where $i:N\to\mathbb{R}^{n}$ is the isometric embedding, we have $A_{\Sigma,\mathbb{R}^{n}}(F_{i},F_{j})=\sum_{m=3}^{n}\langle F_{ij},e_{m}\rangle e_{m}=A_{\Sigma,N}(f_{i},f_{j})+A_{N,\mathbb{R}^{n}}(F_{i},F_{j}).$ Hence, we have $H_{\Sigma,\mathbb{R}^{n}}(f)=H_{\Sigma,N}(f)+g^{ij}A_{N,\mathbb{R}^{n}}(F_{i},F_{j}).$ Noting that $H_{\Sigma,N}(f)\perp A_{N,\mathbb{R}^{n}}$, we get $\left\|H_{\Sigma,\mathbb{R}^{n}}(f)\right\|^{2}=\left\|H_{\Sigma,N}(f)\right\|^{2}+\left\|g^{ij}A_{N,\mathbb{R}^{n}}(F_{i},F_{j})\right\|^{2}\leq\left\|H_{\Sigma,\mathbb{R}^{n}}\right\|^{2}+\\|A_{N,\mathbb{R}^{n}}\\|_{L^{\infty}}$ where $\\|A_{N,\mathbb{R}^{n}}\\|_{L^{\infty}}$ is bounded since $N$ is compact. Thus, integrating over $\Sigma$ yields (4.1), and by (4.1) and Remark 3.3, we get $\int_{\Sigma}\|A_{\Sigma,\mathbb{R}^{n}}(f)\|^{2}d\mu_{f}<C\left(1+\mu(f)+\int_{\Sigma}\|H_{\Sigma,N}(f)\|^{2}d\mu_{f}\right).$ $\hfill\Box$ ### 4.1. Willmore sphere passing through fixed points In this subsection, we let $W_{n}(f)=\int_{S^{2}}\left(1+\frac{1}{4}\left\|H_{f}\right\|^{2}\right)d\mu_{f}$ where $f$ is a $W^{2,2}$ conformal immersion of $S^{2}$ in the round unit sphere ${\mathbb{S}}^{n}$ for some $n>2$. We consider the existence of minimizers of $\beta_{0}^{n}(y_{1},\dots,y_{m})=\inf\\{W_{n}(f):{y_{1},\dots,y_{m}\in f(S^{2})}\\}$ where $y_{1},\dots,y_{m}$ are fixed distinct points in ${\mathbb{S}}^{n}$. When $m\geq 2$, $\beta^{n}_{0}(y_{1},\dots,y_{m})$ is positive by the conformality of the functions $f$. ###### Proposition 4.2. When $m\geq 2$, any $\beta_{0}^{n}(y_{1},\dots,y_{m})<8\pi$ is attained by a $W^{2,2}$-conformally embedded $S^{2}$ in ${\mathbb{S}}^{n}$. ###### Proof. Let $\\{f_{k}\\}$ be a minimizing sequence of $\beta_{0}^{n}(y_{1},\dots,y_{m})$. We can consider $f_{k}$ as conformal map from $S^{2}$ into $\mathbb{R}^{n}$. By Theorem 1, $f_{k}$ will converge to a mapping $f_{0}$ which is a $W^{2,2}$ branched conformal immersion from a stratified sphere $\Sigma_{\infty}$ into ${\mathbb{S}}^{n}$ with $y_{1},\dots,y_{m}\in f_{0}(\Sigma_{\infty}),\,\,\,\,W_{n}(f_{0})\leq\beta_{0}^{n}(y_{1},\dots,y_{m})<8\pi.$ Composing with a stereographic projection $\Pi$ from ${\mathbb{S}}^{n}$ minus a point not on $f_{0}(S^{2})$ into $\mathbb{R}^{n}$, we see $W_{n}(f_{0})=W(\Pi\circ f_{0})$ and $\theta^{2}_{f_{0}(p)}=\theta^{2}_{\Pi\circ f(p)}$. Now, by (3.2) we have $\theta^{2}_{f_{0}(p)}\leq\frac{1}{4\pi}W_{n}(f_{0}).$ By Theorem 3.1 $\lambda(p)+1=\theta^{2}_{f(p)}\leq\frac{1}{4\pi}W_{n}(f_{0})<2$ thus $\lambda(p)=0$ which means $f_{0}$ has no branched points. Moreover, that the area density of $\Sigma_{\infty}$ is one everywhere implies that $\Sigma_{\infty}$ has only 1 component and $f_{0}$ has no intersection points. Thus $\Sigma_{\infty}=S^{2}$, and $f_{0}$ is an (Lipschitz) embedding. $\hfill\Box$ ###### Corollary 4.3. For any $\epsilon>0$, there is a Willmore sphere $f:S^{2}\to{\mathbb{S}}^{n}$ with $W_{n}(f)<4\pi+\epsilon$, which has at least 2 nonremovable singular points. ###### Proof. Take five distinct points $y_{1},\dots,y_{5}\in{\mathbb{S}}^{n}$, such that there is no round 2-sphere passing through all of them. Recall the Willmore functional $W_{n}$ of a round 2-sphere is $4\pi$. We can choose the five points to be very closed to a round 2-sphere, such that there is a 2-sphere $\Sigma$ which is not round and contains $y_{1},\dots,y_{5}$ with $W_{n}(\Sigma)<4\pi+\epsilon.$ Then we can find a $W^{2,2}$ conformal embedding $f:S^{2}\rightarrow{\mathbb{S}}^{n}$, such that $f(S^{2})$ passes through $y_{1},\dots,y_{5}$, and attains $\beta_{0}^{n}(y_{1},\dots,y_{5})$, by Proposition 4.2. Choose a point $P\in{\mathbb{S}}^{n}\backslash\Sigma$ as the north pole. Let $\Pi$ be the stereographic projection from ${\mathbb{S}}^{n}\backslash\\{P\\}$ to $\mathbb{R}^{n}$, and denote $\widetilde{y}_{i}=\Pi(y_{i})$ and $\widetilde{f}=\Pi(f)$. By the conformal invariance of the Willmore functional, we have $W_{n}(f)=\frac{1}{4}\int_{S^{2}}\|H_{\widetilde{f}}\|^{2}d\mu_{\widetilde{f}}.$ Then $\widetilde{f}$ attains $\inf\left\\{\frac{1}{4}\int_{S^{2}}\|H_{\varphi}\|^{2}d\mu_{\varphi}:\varphi\in W^{2,2}_{conf}(S^{2},\mathbb{R}^{n}),\,\,\,\,\widetilde{y}_{1},\dots,\widetilde{y}_{5}\in\varphi(S^{2})\right\\}.$ Then by results in [27], $\widetilde{f}(S^{2})$ is smooth on $\widetilde{f}(S^{2})\backslash\\{\widetilde{y}_{1},\dots,\widetilde{y}_{5}\\}$. However, the Gap Lemma in [14, Theorem 2.7] tells us that there is an $\epsilon>0$, such that any closed smooth Willmore sphere with Willmore functional $4\pi+\epsilon$ is a round sphere. Therefore, at least one of $\widetilde{y}_{1},\dots,\widetilde{y}_{5}$ is a nonremovable singular point. However, a Willmore sphere cannot have only one singular point, by Lemma 4.2 in [15] (which is true in $\mathbb{R}^{n}$), therefore $\widetilde{f}$ has at least 2 singular points. $\hfill\Box$ ### 4.2. Minimizing Willmore functional subject to area constraint In this subsection, $N$ stands for a compact closed submanifold of $\mathbb{R}^{n}$ with induced metric. We say $f\in W^{2,2}_{conf}(\Sigma,h,N)$ if $f\in W^{2,2}_{conf}(\Sigma,h,\mathbb{R}^{n})$ and $f(\Sigma)\subset N$. For $f\in W^{2,2}_{conf}(\Sigma,h,N)$, we define $W(f)=W(f,\Sigma,N)=\frac{1}{4}\int_{\Sigma}\|H_{f,\Sigma,N}\|^{2}d\mu_{f}.$ First, we consider the case of genus zero. Set $\beta_{0}(N,a)=\inf\\{W(f):\mu(f)=a,\,\,\,\,f\in W^{2,2}_{conf}(S^{2},N)\\}.$ ###### Proposition 4.4. We have $\lim_{a\rightarrow 0}\beta_{0}(N,a)=4\pi.$ Moreover, when $a$ is sufficiently small, there is an embedding $f\in W^{2,2}_{conf}(S^{2},N)$, such that $\mu(f)=a,\,\,\,\,and\,\,\,\,W(f)=\beta_{0}(N,a).$ ###### Proof. First, we show that (4.3) $\limsup_{a\rightarrow 0}\beta_{0}(N,a)\leq 4\pi.$ Take a point $p\in N$ and a normal coordinate neighborhood $U$ around $p$. Let $S_{r}=\\{(x^{1},x^{2},x^{3},0,\dots,0)\in T_{p}N:(x^{1})^{2}+(x^{2})^{2}+(x^{3})^{2}=r^{2}\\}.$ It is easy to check that $\lim_{r\rightarrow 0}W(\exp_{p}(S_{r}),N)=4\pi.$ For any $a$ which is sufficiently small, we can find $r=r(a)$ such that $\mu(\exp_{p}(S_{r}))=a$ and $r\rightarrow 0$ as $a\rightarrow 0$. Then (4.3) follows from $\beta_{0}(N,a)\leq W(\exp_{p}(S_{r}))$. Next, we prove that $\beta_{0}(N,a)$ can be attained by an embedded 2-sphere. Let $f_{k}\in W^{2,2}_{conf}(S^{2},N)$ be a minimizing sequence of $\beta_{0}(N,a)$. By Lemma 4.1 and (4.3), when $a$ is sufficiently small and $k$ is sufficiently large $W(f_{k},S^{2},\mathbb{R}^{n})\leq W(f_{k},S^{2},N)+C\mu(f_{k})<4\pi+\epsilon(a,k)+Ca$ where $\epsilon(a,k)\to 0$ as $a\to 0$ and $k\to\infty$. By Theorem 1, $\\{f_{k}\\}$ has a limit $f_{0}$, which is a branched conformal immersion from a stratified sphere $S$ into $N$ with $\mu(f_{0})=a\,\,\,\,\hbox{and}\,\,\,\,W(f_{0})\leq\beta_{0}(N,a).$ Then by (3.2), for any $p\in S$ it holds $\theta^{2}(f_{0}(p))<2.$ Thus $S$ is a 2-sphere and $f_{0}$ has no branch points and no self- intersection points. Hence $f_{0}$ is an embedding. Therefore $f_{0}$ is a minimizer for $\beta_{0}(N,a)$: $W(f_{0})=\beta_{0}(N,a).$ Finally, we prove $\varliminf_{a\rightarrow 0}\beta_{0}(N,a)\geq 4\pi.$ By Lemma 4.1, $W(f_{0},S^{2},\mathbb{R}^{n})\leq W(f_{0},S^{2},N)+Ca.$ It is well-known that $W(f_{0},S^{2},\mathbb{R}^{n})\geq 4\pi$, which completes the proof. $\hfill\Box$ We now consider the case of genus larger than 0. Recall a result of Schoen-Yau [30] and Sacks-Uhlenbeck [29]: If $\varphi:\Sigma\to N$ induces an injection from the fundamental groups to $\Sigma$ and $N$, then there is a branched minimal immersion $f:\Sigma\rightarrow N$ so that $f$ induces the same map between fundamental groups as $\varphi$ and $f$ has least area among all such maps. If $\pi_{2}(N)=0$ then $f$ is minimizing in its homotopy class. We denote the area of the branched minimal immersion $f_{\varphi}$ by $a_{\varphi}$. Let $g>0$ be the genus of the closed Riemann surface $\Sigma$ and $\phi:\Sigma\to N$ be a continuous map. Define $\beta_{g}(N,a,\phi)=\inf\left\\{W(f):f\in\widetilde{W}^{2,2}(\Sigma,N),\,\,\mu(f)=a,\,\,f\sim\phi\right\\},$ where $f\sim\phi$ means that $f$ is homotopic to $\phi$. ###### Proposition 4.5. Let $\Sigma$ be a closed Riemann surface with genus $g>0$ and $N$ be a compact Riemannian manifold with $\pi_{2}(N)=0$. Let $\varphi:\Sigma\to N$ be a map which induces an injective $\varphi_{\\#}:\pi_{1}(\Sigma)\to\pi_{1}(N)$. Then we can find an $\delta>0$, such that for any $a\in[a_{\varphi},a_{\varphi}+\delta)$, there is a branched conformal immersion $f_{0}$ of a smooth Riemann surface $(\Sigma,h)$ of genus $g$ in $\mathbb{R}^{n}$, such that $\mu(f_{0})=a$ and $W(f_{0})=\beta_{g}(N,a,\varphi)$ and $f_{0}$ is homotopic to $\varphi$. Moreover, when $\dim N=3$, we can choose $\delta$ to be small such that $f_{0}$ is an immersion. ###### Proof. The proof will be divided into several steps. Step 1. We prove that $\lim_{a\rightarrow a_{0}}\beta_{g}(N,a,\varphi)=0.$ Let $F\in C^{\infty}(\Sigma\times[0,1],\mathbb{R}^{n})$, such that $F(\cdot,t)$ is an immersion for each $t$ and $F(\cdot,0)=f_{\varphi},\,\,\,\,\mu(F(\cdot,1))\geq a_{\varphi}.$ As $F(\cdot,t)\sim\varphi$ and $f_{\varphi}$ is a minimal surface, $\lim_{a\rightarrow a_{\varphi}}\beta_{p}(N,a,\varphi)\leq\lim_{t\rightarrow 0}W(F(\cdot,t))=W(f_{\varphi})=0.$ Step 2. Smooth convergence of conformal structures. We take a minimizing sequence $\\{f_{k}\\}$ of $\beta_{g}(N,a,f)$. Recall that $f_{k}$ are $W^{2,2}$ branched conformal immersions from $(\Sigma,h_{k})$ into $\mathbb{R}^{n}$, where $h_{k}$ are the smooth metrics with curvature 0 or $-1$. Because $\pi_{2}(N)=0$ and $f_{k}\sim\varphi$ for each $k$, $f_{k}$ induces the same injective action on the fundamental groups as $\varphi$ does; hence the conformal structures of $h_{k}$ stay in a compact set of the moduli space for both the cases $g>1$ and $g=1$, therefore, after passing to a subsequence if necessary, $\Sigma_{k}=(\Sigma,h_{k})$ converges to a Riemann surface $(\Sigma,h_{0})$ in $\mathcal{M}_{g}$ (cf. [30]). The results in [30] applies as $f_{k}$ belong to $W^{1,2}\cap C^{0}$. Step 3. We prove that $\\{f_{k}\\}$ has no bubbles, i.e. the limit $f_{0}$ is a map defined on $\Sigma$. By Remark 2.11, $f_{0}$ is defined on $\Sigma_{\infty}=\Sigma_{0}\cup S_{1}\cup S_{2}\cdots\cup S_{m}$, where $S_{i}$ are all 2-spheres and $\Sigma_{0}$ is a smooth surface of genus $g$. We prove $m=0$. Assume $m\geq 1$. By Theorem 1, $\mu(f_{0})=a$ and $W(f_{0})\leq\beta_{p}(N,a,\varphi)$. Further, $f_{k}(\Sigma)$ converge to $f_{0}(\Sigma_{\infty})$ in Hausdorff distance and $f_{0}\|_{S_{j}}$ is homotopic to a constant map for each $j=1,\dots,m$ as $\pi_{2}(N)=0$. We conclude that $f_{0}\|_{\Sigma_{0}}$ is homotopic to $\varphi$. Consequently, $\mu(f_{0}(\Sigma_{0}))\geq a_{\varphi}$. Then we get $\mu(f_{0},S_{i})\leq\mu(\Sigma)-\mu(\Sigma_{0})\leq a-a_{\varphi}\,\,\,\,\hbox{and}\,\,\,\,W(f_{0},S_{i},N)\leq\beta_{g}(N,a,\varphi).$ By Lemma 4.1 and Step 1, $W(f_{0},S_{i},\mathbb{R}^{n})\leq C(a-a_{\varphi})+\beta_{g}(N,a,\varphi)\to 0\,\,\,\,\hbox{as}\,\,\,\,a\to a_{\varphi}.$ This, however, contradicts Proposition 2.2 when the Willmore functional of $S_{i}$ goes below the gap constant. Step 4. We consider the case of $\dim N=3$. We will use the result that there are no branch points for minimal surfaces [8, 22] to prove that $f_{0}$ has no branch points when $\delta$ is sufficiently small. If the claimed result is not true, then there is a sequence of numbers $a_{k}>a_{\varphi}$ with $a_{k}\rightarrow a_{\varphi}$ and a sequence of $W^{2,2}$ branched conformal immersions ${f}_{0,k}$ of $(\Sigma,h_{k})$ in $N$ with $\mu(f_{0,k})=a_{k}$, $W(f_{0,k},\Sigma,N)=\beta_{g}(N,a_{k},\varphi)$ by the first part of the proposition, and each $f_{0,k}$ has at least a branch point $p_{k}$. By Step 1, $W(f_{0,k},\Sigma,N)\rightarrow 0$. As in Step 2, $(\Sigma,h_{k})$ converge to a smooth surface $(\Sigma,h_{0})$ in $\mathcal{M}_{g}$. For simplicity, we will still denote ${f}_{0,k}\circ\psi_{k}$ (see Remark 2.11) by ${f}_{0,k}$ which is a branched conformal immersion from $(\Sigma,\psi^{}_{k}(h_{k}))$ into $\mathbb{R}^{n}$. By Theorem 2.8, we may set ${f}_{0,0}$ to be the limit of ${f}_{0,k}$ with $\mu({f}_{0,0})=a$ and $W({f}_{0,0})=0$. Arguing as in Step 3, $\\{{f}_{0,k}\\}$ has no bubbles, and ${f}_{0,0}\in W^{2,2}_{b,c}(\Sigma,h_{0},\mathbb{R}^{n})$ for some smooth $h_{0}$. Moreover, ${f}_{0,0}$ is a minimal surface in $N$. By the result of Gulliver and Osserman, ${f}_{0,0}$ is a smooth immersion of $\Sigma$ in $N$. Since $p_{k}$ is a branch point, by Theorem 3.1, the area density $\theta^{2}_{f_{0,k}(p_{k})}(f_{0,k}(U))\geq 2,$ where $U$ is a neighborhood of $p_{k}\rightarrow p$ in $\Sigma$ for sufficiently large $k$. As $f_{0,0}$ is immersive, we can take $U$ small so that ${f}_{0,0}$ is an embedding on $U$ and $\mu({f}_{0,0}(U))<\epsilon^{\prime}$. Further, by the monotonicity formula for minimal surfaces, for small $r$ and geodesic balls $B^{N}_{r}(f_{0,0}(p))$ in $N$, it holds $\mu({f}_{0,0}(U)\cap B^{N}_{r}({f}_{0,0}(p)))\leq(1+\epsilon^{\prime})\pi r^{2}.$ From the expansion of metric in normal coordinates, for small $r$ and the Euclidean ball $B_{r}(f_{0,0}(p))$ in $\mathbb{R}^{n}$ we have $\mu(f_{0,0}(U)\cap B_{r}(f_{0,0}(p)))\leq\mu(f_{0,0}(U)\cap B^{N}_{r+cr^{2}}(f_{0,0}(p)))\leq(1+\epsilon^{\prime})\pi r^{2}+O(r^{3})$ where $c$ depends on $N$. In light of Lemma 4.1, $W({f}_{0,k},U,\mathbb{R}^{n})<\epsilon_{0}^{2}$ if we choose $\epsilon^{\prime}$ to be very small and $k$ large enough. Then $\\{{f}_{0,k}\\}$ has no blowup points in $U$ by the $\epsilon$-regularity. Then we have $\mu({f}_{0,k}(U)\cap B_{r}({f}_{0,k}(p_{k})))\rightarrow\mu({f}_{0,0}(U)\cap B_{r}({f}_{0,0}(p)))\,\,\,\,\,\,\,\,\hbox{as $k\to\infty$}.$ By Lemma 4.1, $W({f}_{0,k},U,\mathbb{R}^{n})\leq C\epsilon^{\prime}+W(f_{0,k},U,N).$ Then by (3.3), $\displaystyle\theta^{2}_{f_{0,k}(p)}(f_{0,k}(U))$ $\displaystyle\leq$ $\displaystyle\frac{\mu(f_{0,k}(U)\cap B_{r}(f_{0,k}(p_{k})))}{\pi r^{2}}+W(f_{0,k},U,\mathbb{R}^{n})+CW(f_{0,k},U,\mathbb{R}^{n})^{\frac{1}{2}}.$ Hence, $\displaystyle 2$ $\displaystyle\leq$ $\displaystyle\lim_{U\to p}\lim_{k\to\infty}\left(\frac{\mu(f_{0,k}(U)\cap B_{r}(f_{0,k}(p_{k})))}{\pi r^{2}}+W(f_{0,k},U,\mathbb{R}^{n})+CW(f_{0,k},U,\mathbb{R}^{n})^{\frac{1}{2}}\right)$ $\displaystyle\leq$ $\displaystyle 1+\epsilon^{\prime}.$ This is impossible for $\epsilon^{\prime}$ small. $\hfill\Box$ ### 4.3. Minimizing Willmore functional of surfaces with a Douglas type condition In this subsection, we consider a sufficient condition of Douglas type as in the minimal surface theory for existence of minimizers of the Willmore functional. First, we assume $N$ to be a compact Riemannian manifold with negative sectional curvatures. In negatively curved $N$, surface area is bounded by the Willmore functional and the genus of the surface. ###### Lemma 4.6. Let $N$ be a compact Riemannian manifold with $K\leq-c<0$. Then for any $f\in\widetilde{W}^{2,2}(\Sigma,N)$, $\mu(f)\leq c^{-1}\left(W(f,\Sigma,N)-2\pi\chi(\Sigma)\right).$ Especially, when $g(\Sigma)=0$ or $1$, $\mu(f)\leq c^{-1}W(f,\Sigma,N).$ ###### Proof. From the Gauss equation: $R^{\Sigma}(X,Y,X,Y)=R^{N}(X,Y,X,Y)+\langle A(X,X),A(Y,Y)\rangle-\langle A(X,Y),A(X,Y)\rangle.$ we have $K_{\Sigma}\leq K_{f_{}(T\Sigma)}+\frac{1}{4}\|H_{f,\Sigma,N}\|^{2}.$ Then from the generalized Gauss-Bonnet formula - Lemma 3.2, we have $2\pi\chi(\Sigma)+2\pi b\leq-c\mu_{f}(\Sigma)+W(f,\Sigma,N)$ where $b$ is the number of branch points, in turn $c\mu_{f}(\Sigma)\leq W(f,\Sigma,N)-2\pi\chi(\Sigma).$ When $g(\Sigma)\leq 1$ the Euler number $\chi(\Sigma)$ is nonnegative, in this case $c\mu_{f}(\Sigma)\leq W(f,\Sigma,N).$ Dividing by $c$ yields the desired area bounds. $\hfill\Box$ Recall that any connected stratified surface $\Sigma$ can be written as union of finitely many connected 2-dimensional components: $\Sigma=\bigcup_{i}\Sigma_{i}$. Denote the genus of $\Sigma$ and $\Sigma_{i}$ by $g(\Sigma)$ and $g(\Sigma_{i})$, accordingly. We introduce a subset $S(g)$ of all stratified surfaces as follows. 1. (1) If $g>0$, $S(g)=\left\\{\Sigma:\hbox{$\Sigma=\bigcup_{i}\Sigma_{i}$ with $g(\Sigma_{i})<g$ for all $i$}\right\\}.$ 2. (2) If $g=0$, $S(0)=\left\\{\Sigma:\hbox{$\Sigma=\bigcup_{i}\Sigma_{i}$ with $g(\Sigma)=0$ and $i\geq 2$}\right\\}.$ Note that any $\Sigma\in S(g)$ with $g(\Sigma)=g$ must be singular, in the sense that it has more than one components. Especially, $S(g)\cap{\mathcal{M}}_{g}=\emptyset$. However, when $g\geq 1$, $S(g)$ contains smooth surfaces of genus $\leq g-1$. Define $\displaystyle\alpha^{}(g)$ $\displaystyle=$ $\displaystyle\inf\\{W(f,\Sigma,\mathbb{R}^{n}):f\in W^{2,2}_{b,c}(\Sigma,\mathbb{R}^{n}),f(\Sigma)\subset N,\Sigma\in S(g)\\}$ $\displaystyle\alpha(g)$ $\displaystyle=$ $\displaystyle\inf\\{W(f,\Sigma,\mathbb{R}^{n}):f\in W^{2,2}_{b,c}(\Sigma,\mathbb{R}^{n}),f(\Sigma)\subset N,\Sigma\in{\mathcal{M}}_{g}\\}.$ We now state a sufficient condition, similar to the Douglas condition for minimal surfaces, for existence of minimizers for the Willmore functional. ###### Proposition 4.7. Let $N$ be a compact Riemannian manifold with negative sectional curvatures. If $0<\alpha(g)<\alpha^{}(g)$, then there is a $W^{2,2}$ branched conformal immersion $f$ from a smooth surface of genus $g$ into $N$ which minimizes the Willmore functional among all such maps. ###### Proof. Let $f_{k}:(\Sigma,h_{k})\rightarrow N\hookrightarrow\mathbb{R}^{n}$ be a minimizing sequence of $\alpha(g)$. By Lemma 4.6, the areas $\mu(f_{k}(\Sigma))$ are uniformly bounded as well by Lemma 4.6. Since $\alpha(g)$ is positive, $f_{k}$ cannot converge to a point. Then from Theorem 1, there exists a subsequence of $\\{f_{k}\\}$, still denoted by $\\{f_{k}\\}$, a limit map $f_{0}\in W^{2,2}_{b,c}(\Sigma_{\infty},\mathbb{R}^{n})$ from a stratified Riemann surface $\Sigma_{\infty}$ with $g(\Sigma_{\infty})\leq g$ into $N\hookrightarrow\mathbb{R}^{n}$, and $W(f_{0},\Sigma_{\infty},\mathbb{R}^{n})\leq\lim_{k\to\infty}W(f_{k},\Sigma,\mathbb{R}^{n})=\alpha(g).$ We write $\Sigma_{\infty}=\bigcup_{i=1}^{m}\Sigma_{i}$. If $g(\Sigma_{\infty})=g$, we consider two cases. Case 1: $g(\Sigma_{i})=g$ for some $i=1,...,m$. In this case, $W(f_{0}\|_{\Sigma_{1}},\Sigma_{1},\mathbb{R}^{n})\leq W(f_{0},\Sigma_{\infty},\mathbb{R}^{n})=\alpha(g).$ So $f_{0}(\Sigma_{i})$ is a smooth genus $g$ surface attains $\alpha(g)$. Case 2: $g(\Sigma_{i})<g$ for all $i=1,...,m$. Thus $\Sigma_{\infty}\in S(g)$, and in turn $\alpha^{}(g)\leq W(f_{0},\Sigma_{\infty},\mathbb{R}^{n})\leq\alpha(g)<\alpha^{}(g).$ This contradiction rules out Case 2. If $g(\Sigma_{\infty})<g$ then $\Sigma_{g}\in S(g)$. Therefore $\alpha^{}(g)\leq W(f_{0},\Sigma_{\infty},\mathbb{R}^{n})\leq\alpha(g)<\alpha^{}(g)$ and this is impossible. $\hfill\Box$ Instead of the curvature assumption on $N$, we set, for $0<a<\infty$, $\displaystyle\gamma^{}(g,a)$ $\displaystyle=$ $\displaystyle\inf\\{W(f,\Sigma,\mathbb{R}^{n}):f\in W^{2,2}_{b,c}(\Sigma,\mathbb{R}^{n}),f(\Sigma)\subset N,\Sigma\in S(g),\mu(f(\Sigma))\leq a\\}$ $\displaystyle\gamma(g,a)$ $\displaystyle=$ $\displaystyle\inf\\{W(f,\Sigma,\mathbb{R}^{n}):f\in W^{2,2}_{b,c}(\Sigma,\mathbb{R}^{n}),f(\Sigma)\subset N,\Sigma\in{\mathcal{M}}_{g},\mu(f(\Sigma))\leq a\\}.$ Since there is no loss in measures in the limit process, as asserted in Theorem 1, the same proof above allows us to conclude ###### Proposition 4.8. Let $N$ be a compact Riemannian manifold. If $0<\gamma(g,a)<\gamma^{}(g,a)$, then there is a $W^{2,2}$ branched conformal immersion $f$ from a smooth surface of genus $g$ into $N$ which minimizes the Willmore functional among all such maps. ## 5\. Appendix Wente’s inequality [1, 7, 33] states that if $u\in W^{1,2}_{0}(D)$ solves the equation $-\Delta u=\nabla a\,\nabla^{\bot}b,$ then we have (5.1) $\\|u\\|_{L^{\infty}(D)}\leq\frac{1}{2\pi}\\|\nabla a\\|_{L^{2}(D)}\\|\nabla b\\|_{L^{2}(D)}.$ and (5.2) $\\|\nabla u\\|_{L^{2}(D)}\leq\frac{3}{16\pi}\\|\nabla a\\|_{L^{2}(D)}\\|\nabla b\\|_{L^{2}(D)}.$ ###### Lemma 5.1. Let $u\in W^{1,2}_{0}(D)$ be the unique solution to the equation $-\Delta u=\nabla a\nabla^{\bot}b,$ where $a,b\in W^{1,2}(D)$. Then $u\in C^{0}(D)$. ###### Proof. Let $a_{k}\in C^{\infty}(\bar{D})$ with $a_{k}\rightarrow a$ in $W^{1,2}(D)$. There exist solutions $u_{k}\in W^{2,2}(D)\cap C^{0,\alpha}(D)$ to the Dirichlet problem $\displaystyle-\Delta u_{k}$ $\displaystyle=$ $\displaystyle\nabla a_{k}\nabla^{\bot}b,\,\,\,\,\,\,\,\,\,\,\,\,\mbox{in $D$ }$ $\displaystyle u_{k}$ $\displaystyle=$ $\displaystyle 0,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\mbox{on $\partial D$.}$ We have $-\Delta(u_{k}-u_{m})=\nabla(a_{k}-a_{m})\nabla^{\bot}b.$ Then by (5.1) $\displaystyle\\|u_{k}-u_{m}\\|_{C^{0}(D)}$ $\displaystyle=$ $\displaystyle\\|u_{k}-u_{m}\\|_{L^{\infty}(D)}$ $\displaystyle\leq$ $\displaystyle C\,\\|a_{k}-a_{m}\\|_{L^{2}(D)}\\|b\\|_{L^{2}(D)}.$ Hence $u_{k}$ converge in $C^{0}(D)$ to a continuous function $u_{0}$ which vanished on $\partial D$. By (5.2), we may assume $u_{k}$ converges to $u_{0}$ in $W^{1,2}(D)$ as well. For any smooth $\varphi$, $\int_{D}\varphi\nabla a\nabla^{\bot}b=\lim_{k\rightarrow+\infty}\int_{D}\varphi\nabla a_{k}\nabla^{\bot}b=\lim_{k\rightarrow+\infty}\int_{D}\nabla u_{k}\nabla\varphi=\int_{D}\nabla u_{0}\nabla\varphi.$ Hence $-\Delta u_{0}=\nabla a\nabla^{\bot}b$, and it follows from the uniqueness of solution to the Dirichlet problem that $u=u_{0}$, so $u\in C^{0}(D)$. $\hfill\Box$ Now, we are ready to prove: ###### Proposition 5.2. If $f\in W^{2,2}_{conf}(D,\mathbb{R}^{n})$ with $df\otimes df=e^{2u}g_{euclid}$ and $\\|u\\|_{L^{\infty}}<+\infty$, then $u\in C^{0}(D)$. ###### Proof. In a complex coordinate $z=x+iy$ on $D$, since $f$ is conformal and the induced metric is $e^{2u}(dx^{2}+dy^{2})$, we have $f_{x}\cdot f_{x}=f_{y}\cdot f_{y}=e^{2u}$ and we can take $a=e^{-u}\,f_{x},b=e^{-u}\,f_{y}$ as a local orthonormal frame for the tangent bundle of $f(D)$. Straight computation shows $a_{x}=e^{-u}f_{xx}+(e^{-u})_{x}f_{x}=e^{-u}f_{xx}-e^{-3u}(f_{xx}\cdot f_{x})f_{x}$ which leads to $a_{x}\cdot a_{x}\leq Ce^{-2u}f_{xx}\cdot f_{xx}\leq Ce^{2\\|u\\|_{L^{\infty}}}\left\|f_{xx}\right\|^{2}$ and similarly $a_{y}\cdot a_{y}\leq Ce^{2\\|u\\|_{L^{\infty}}}\left\|f_{yy}\right\|^{2}.$ Therefore, $a\in W^{1,2}(D)$ and similarly $b\in W^{1,2}(D)$. On the other hand, we can check $K_{f}e^{2u}=\nabla a\cdot\nabla^{\perp}b,\,\,\,\,-\Delta u=K_{f}e^{2u}.$ Let $v$ be the harmonic function on $D$ which agrees with $u$ on $\partial D$. Then $\displaystyle\Delta(u-v)$ $\displaystyle=$ $\displaystyle\nabla a\cdot\nabla^{\perp}b\,\,\,\,\,\,\,\,\,\hbox{in $D$}$ $\displaystyle u-v$ $\displaystyle=$ $\displaystyle 0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\hbox{on $\partial D$}$ and Lemma 5.1 implies $u-v\in C^{0}(D)$, hence $u\in C^{0}(D)$. $\hfill\Box$ ## References * [1] S. Baraket: Estimations of the best constant involving the $L^{\infty}$ norm in Wente’s inequality, Ann. Fac. Sci. Toulouse Math. (6) 5 (1996), 373 C385. * [2] J. Chen and G. Tian: Compactification of moduli space of harmonic mappings, _Comment. Math. Helv._ 74 (1999), 201-237. * [3] L. Chen, Y. Li, Y. Wang: The Refined Analysis on the Convergence Behavior of Harmonic Map Sequence from Cylinders, _J. Geom. Anal._ , to appear. * [4] D. DeTurck and J. Kazdan: Some regularity theorems in Riemannian geometry, _Ann. Sci. École Norm. Sup. (4)_ 14 (1981), 249-260. * [5] W. Ding and G. Tian: Energy identity for a class of approximate harmonic maps from surfaces, _Comm. Anal. Geom._ 3 (1995), 543-554. * [6] J. Eschenburg and R. Tribuzy: Branch points of conformal mappings of surfaces, _Math. Ann._ 279 (1988), 621-633. * [7] Y. Ge: Estimations of the best constant involving the $L^{2}$ norm in Wente’s inequality and compact $H$-surfaces in Euclidean space, ESAIM Control Optim. Calc. Var. 3 (1998), 263–300. * [8] R. Gulliver: Regularity of minimizing surfaces of prescribed mean curvature, _Ann. of Math. (2)_ 97 (1973), 275-305. * [9] N. Halpern: A proof of the collar lemma, _Bull. London Math. Soc._ 13 (1981), 141-144. * [10] F. Hélein: Harmonic maps, conservation laws and moving frames. Translated from the 1996 French original. With a foreword by James Eells. Second edition. Cambridge Tracts in Mathematics, 150. Cambridge University Press, Cambridge, 2002. * [11] C. Hummel: Gromov’s compactness theorem for pseudo-holomorphic curves, _Progress in Mathematics_ 151, Birkhäuser Verlag, Basel (1997). * [12] L. Keen: Collars on Riemann surfaces, Discontinuous groups and Riemann surfaces (Proc. Conf., Univ. Maryland, College Park, Md., 1973), 263268. Ann. of Math. Studies, No. 79, Princeton Univ. Press, Princeton, N.J., 1974. * [13] E. Kuwert and Y. Li: $W^{2,2}$-conformal immersions of a closed Riemann surface into $\mathbb{R}^{n}$, _arXiv:1007.3967_. * [14] E. Kuwert and R. Schätzle: The Willmore flow with small initial energy, _J. Differential Geom._ , 57 (2001), 409-441. * [15] E. Kuwert and R. Schätzle: Removability of point singularities of Willmore surfaces, _Ann. of Math._ 160 (2004), 315-357. * [16] T. Lamm: Energy identity for approximations of harmonic maps from surfaces, _Trans. Amer. Math. Soc._ 362 (2010), 4077-4097. * [17] T. Lamm and J. Metzger: Small surfaces of Willmore type in Riemannian manifolds, _Int. Math. Res. Not._ (2010), no. 3786-3813. * [18] Y. Li and Y. Wang: A weak energy identity and the length of necks for a Sacks-Uhlenbeck -harmonic map sequence, _Adv. Math._ 225 (2010), 1134-1184 . * [19] F. Lin and C. Wang: Energy identity of harmonic map flows from surfaces at finite singular time, _Calc. Var. Partial Differential Equations_ , 6 (1998), 369-380. * [20] J.P. Matelski: A compactness theorem for Fuchsian groups of the second kind, _Duke Math. J._ 43 (1976), 829-840. * [21] S. Müller and V. Šverák: On surfaces of finite total curvature, _J. Differential Geom._ 42 (1995), 229-258. * [22] R. Osserman: A proof of the regularity everywhere of the classical solution to Plateau’s problem, _Ann. of Math. (2)_ 91 (1970) 550-569. * [23] T. H. Parker: Bubble tree convergence for harmonic maps, _J. Differential Geom._ 44 (1996), 595-633. * [24] J. Qing: On singularities of the heat flow for harmonic maps from surfaces into spheres, _Comm. Anal. Geom._ , 3 (1995), 297-315. * [25] J. Qing and G. Tian: Bubbling of the heat flow for harmonic maps from surfaces, _Comm. Pure. Apple. Math._ , 50 (1997), 295-310. * [26] B. Randol: Cylinders in Riemann surfaces, _Comment. Math. Helv._ 54 (1979), 1-5. * [27] T. Riviére: Analysis aspects of Willmore surfaces, Invent. Math. 174 (2008), 1-45. * [28] J. Sacks and K. Uhlenbeck: The existence of minimal immersions of 2-spheres, _Ann. of Math._ , 113 (1981), 1-24. * [29] J. Sacks and K. Uhlenbeck: Minimal immersions of closed Riemann surfaces, _Trans. Amer. Math. Soc._ 271 (1982), no. 2, 639-652. * [30] R. Schoen and S.T. Yau, Existence of incompressible minimal surfaces and the topology of three-dimensional manifolds with nonnegative scalar curvature, _Ann. of Math. (2)_ 110 (1979), no. 1, 127-142. * [31] L. Simon: Existence of surfaces minimizing the Willmore functional, _Comm. Anal. Geom._ 1 (1993), 281-326. * [32] R. Schätzle: The Willmore boundary problem, _Cal. Var. P.D.E._ , 37 (2010), 275-302. * [33] P. Topping: The optimal constant in Wente’s $L^{\infty}$ estimate, Comment. Math. Helv. 72 (1997), 316 C328. * [34] M. Zhu: Harmonic maps from degenerating Riemann surfaces, _Math. Z._ 264 (2010), no. 1, 63-85. Jingyi Chen \| Yuxiang Li ---\|--- Department of Mathematics \| Department of Mathematical Sciences University of British Columbia \| Tsinghua University Vancouver,B.C., V6T1Z2, Canada \| Beijing 100084, P.R. China jychen@math.ubc.ca \| yxli@math.tsinghua.edu.cn	arxiv-papers	2011-12-08T12:30:36	2024-09-04T02:49:25.089036	{ "license": "Public Domain", "authors": "Jingyi Chen and Yuxiang Li", "submitter": "Yuxiang Li", "url": "https://arxiv.org/abs/1112.1818" }
1112.2082	2011-204 09.12.2011 Underlying Event measurements with ALICE The ALICE Collaboration††thanks: See Appendix A for the list of collaboration members The ALICE Collaboration We present measurements of Underlying Event observables in pp collisions at $\sqrt{s}=0.9$ and 7 TeV. The analysis is performed as a function of the highest charged-particle transverse momentum $p_{\mathrm{T,LT}}$ in the event. Different regions are defined with respect to the azimuthal direction of the leading (highest transverse momentum) track: Toward, Transverse and Away. The Toward and Away regions collect the fragmentation products of the hardest partonic interaction. The Transverse region is expected to be most sensitive to the Underlying Event activity. The study is performed with charged particles above three different $p_{\mathrm{T}}$ thresholds: 0.15, 0.5 and 1.0 ${\rm GeV}/c$. In the Transverse region we observe an increase in the multiplicity of a factor 2-3 between the lower and higher collision energies, depending on the track $p_{\mathrm{T}}$ threshold considered. Data are compared to Pythia 6.4, Pythia 8.1 and Phojet. On average, all models considered underestimate the multiplicity and summed $p_{\mathrm{T}}$ in the Transverse region by about 10-30%. ## 1 Introduction The detailed characterization of hadronic collisions is of great interest for the understanding of the underlying physics. The production of particles can be classified according to the energy scale of the process involved. At high transverse momentum transfers ($p_{\mathrm{T}}\gtrsim 2\,\mathrm{\mbox{${\rm GeV}/c$}}$) perturbative Quantum Chromodynamics (pQCD) is the appropriate theoretical framework to describe partonic interactions. This approach can be used to quantify parton yields and correlations, whereas the transition from partons to hadrons is a non-perturbative process that has to be treated using phenomenological approaches. Moreover, the bulk of particles produced in high- energy hadronic collisions originate from low-momentum transfer processes. For momenta of the order of the QCD scale, $\mathcal{O}$(100 MeV), a perturbative treatment is no longer feasible. Furthermore, at the center-of-mass energies of the Large Hadron Collider (LHC), at momentum transfers of a few GeV/$c$, the calculated QCD cross-sections for 2-to-2 parton scatterings exceed the total hadronic cross-section [1]. This result indicates that Multiple Partonic Interactions (MPI) occur in this regime. The overall event dynamics cannot be derived fully from first principles and must be modeled using phenomenological calculations. Measurements at different center-of-mass energies are required to test and constrain these models. In this paper, we present an analysis of the bulk particle production in pp collisions at the LHC by measuring the so-called Underlying Event (UE) activity [2]. The UE is defined as the sum of all the processes that build up the final hadronic state in a collision excluding the hardest leading order partonic interaction. This includes fragmentation of beam remnants, multiple parton interactions and initial- and final-state radiation (ISR/FSR) associated to each interaction. Ideally, we would like to study the correlation between the UE and perturbative QCD interactions by isolating the two leading partons with topological cuts and measuring the remaining event activity as a function of the transferred momentum scale ($Q^{2}$). Experimentally, one can identify the products of the hard scattering, usually the leading jet, and study the region azimuthally perpendicular to it as a function of the jet energy. Results of such an analysis have been published by the CDF [2, 3, 4, 5] and STAR [6] collaborations for pp collisions at $\sqrt{s}=1.8$ and 0.2 TeV, respectively. Alternatively, the energy scale is given by the leading charged-particle transverse momentum, circumventing uncertainties related to the jet reconstruction procedure at low $p_{\mathrm{T}}$. It is clear that this is only an approximation to the original outgoing parton momentum, the exact relation depends on the details of the fragmentation mechanism. The same strategy based on the leading charged particle has recently been applied by the ATLAS [7] and CMS [8] collaborations. In the present paper we consider only charged primary particles111Primary particles are defined as prompt particles produced in the collision and their decay products (strong and electromagnetic decays), except products of weak decays of strange particles such as $K^{0}_{S}$ and $\Lambda$., due to the limited calorimetric acceptance of the ALICE detector systems in azimuth. Distributions are measured for particles in the pseudorapidity range $\|\eta\|<0.8$ with $p_{\mathrm{T}}>p_{\rm T,min}$, where $p_{\rm T,min}=$ 0.15, 0.5 and 1.0 ${\rm GeV}/c$, and are studied as a function of the leading particle transverse momentum. Many Monte Carlo (MC) generators for the simulation of pp collisions are available; see [9] for a recent review discussing for example Pythia [10], Phojet [11], Sherpa [9] and Herwig [12]. These provide different descriptions of the UE associated with high energy hadron collisions. A general strategy is to combine a perturbative QCD treatment of the hard scattering with a phenomenological approach to soft processes. This is the case for the two models used in our analysis: Pythia and Phojet. In Pythia the simulation starts with a hard LO QCD process of the type 2 $\to$ 2\. Multi-jet topologies are generated with the parton shower formalism and hadronization is implemented through the Lund string fragmentation model [13]. Each collision is characterized by a different impact parameter $b$. Small $b$ values correspond to a large overlap of the two incoming hadrons and to an increased probability for MPIs. At small $p_{\mathrm{T}}$ values color screening effects need to be taken into account. Therefore a cut-off $p_{\rm T,0}$ is introduced, which damps the QCD cross-section for $p_{\mathrm{T}}\ll p_{\rm T,0}$. This cut-off is one of the main tunable model parameters. In Pythia version 6.4 [10] MPI and ISR have a common transverse momentum evolution scale (called interleaved evolution [14]). Version 8.1 [15] is a natural extension of version 6.4, where the FSR evolution is interleaved with MPI and ISR and parton rescatterings [16] are considered. In addition initial- state partonic fluctuations are introduced, leading to a different amount of color-screening in each event. Phojet is a two-component event generator, where the soft regime is described by the Dual Parton Model (DPM) [17] and the high-$p_{\mathrm{T}}$ particle production by perturbative QCD. The transition between the two regimes happens at a $p_{\mathrm{T}}$ cut-off value of 3 ${\rm GeV}/c$. A high-energy hadronic collision is described by the exchange of effective Pomerons. Multiple-Pomeron exchanges, required by unitarization, naturally introduce MPI in the model. UE observables allow one to study the interplay of the soft part of the event with particles produced in the hard scattering and are therefore good candidates for Monte Carlo tuning. A better understanding of the processes contributing to the global event activity will help to improve the predictive power of such models. Further, a good description of the UE is needed to understand backgrounds to other observables, e.g., in the reconstruction of high-$p_{\mathrm{T}}$ jets. The paper is organized in the following way: the ALICE sub-systems used in the analysis are described in Section 2 and the data samples in Section 3. Section 4 is dedicated to the event and track selection. Section 5 introduces the analysis strategy. In Sections 6 and 7 we focus on the data correction procedure and systematic uncertainties, respectively. Final results are presented in Section 8 and in Section 9 we draw conclusions. ## 2 ALICE detector Optimized for the high particle densities encountered in heavy-ion collisions, the ALICE detector is also well suited for the study of pp interactions. Its high granularity and particle identification capabilities can be exploited for precise measurements of global event properties [18, 19, 20, 21, 22, 23, 24]. The central barrel covers the polar angle range $45^{\circ}-135^{\circ}$ ($\|\eta\|$ $<$ $1$) and full azimuth. It is contained in the L3 solenoidal magnet which provides a nominal uniform magnetic field of $0.5\,{\rm T}$. In this section we describe only the trigger and tracking detectors used in the analysis, while a detailed discussion of all ALICE sub-systems can be found in [25]. The V0A and V0C counters consist of scintillators with a pseudorapidity coverage of $-3.7<\eta<-1.7$ and $2.8<\eta<5.1$, respectively. They are used as trigger detectors and to reject beam–gas interactions. Tracks are reconstructed combining information from the two main tracking detectors in the ALICE central barrel: the Inner Tracking System (ITS) and the Time Projection Chamber (TPC). The ITS is the innermost detector of the central barrel and consists of six layers of silicon sensors. The first two layers, closely surrounding the beam pipe, are equipped with high granularity Silicon Pixel Detectors (SPD). They cover the pseudorapidity ranges $\|\eta\|<2.0$ and $\|\eta\|<1.4$ respectively. The position resolution is $12\,{\rm\mu m}$ in $r\phi$ and about $100\,{\rm\mu m}$ along the beam direction. The next two layers are composed of Silicon Drift Detectors (SDD). The SDD is an intrinsically 2-dimensional sensor. The position along the beam direction is measured via collection anodes and the associated resolution is about $50\,{\rm\mu m}$. The $r\phi$ coordinate is given by a drift time measurement with a spatial resolution of about $60\,{\rm\mu m}$. Due to drift field non-uniformities, which were not corrected for in the 2010 data, a systematic uncertainty of $300\,{\rm\mu m}$ is assigned to the SDD points. Finally, the two outer layers are made of double-sided Silicon micro-Strip Detectors (SSD) with a position resolution of $20\,{\rm\mu m}$ in $r\phi$ and about $800\,{\rm\mu m}$ along the beam direction. The material budget of all six layers including support and services amounts to 7.7% of a radiation length. The main tracking device of ALICE is the Time Projection Chamber that covers the pseudorapidity range of about $\|\eta\|<0.9$ for tracks traversing the maximum radius. In order to avoid border effects, the fiducial region has been restricted in this analysis to $\|\eta\|<0.8$. The position resolution along the $r\phi$ coordinate varies from $1100\,{\rm\mu m}$ at the inner radius to $800\,{\rm\mu m}$ at the outer. The resolution along the beam axis ranges from $1250\,{\rm\mu m}$ to $1100\,{\rm\mu m}$. For the evaluation of the detector performance we use events generated with the Pythia 6.4 [10] Monte Carlo with tune Perugia-0 [26] passed through a full detector simulation based on Geant3 [27]. The same reconstruction algorithms are used for simulated and real data. ## 3 Data samples The analysis uses two data sets which were taken at the center-of-mass energies of $\sqrt{s}=$ 0.9 and 7 TeV. In May 2010, ALICE recorded about 6 million good quality minimum-bias events at $\sqrt{s}=$ 0.9 TeV. The luminosity was of the order of 1026 cm-2 s-1 and, thus, the probability for pile-up events in the same bunch crossing was negligible. The $\sqrt{s}=$ 7 TeV sample of about 25 million events was collected in April 2010 with a luminosity of $10^{27}$ cm-2 s-1. In this case the mean number of interactions per bunch crossing $\mu$ ranges from 0.005 to 0.04. A set of high pile-up probability runs ($\mu=0.2-2$) was analysed in order to study our pile-up rejection procedure and determine its related uncertainty. Those runs are excluded from the analysis. Corrected data are compared to three Monte Carlo models: Pythia 6.4 (tune Perugia-0), Pythia 8.1 (tune 1 [15]) and Phojet 1.12. Collision energy: 0.9 TeV --- \| Events \| % of all Offline trigger \| 5,515,184 \| 100.0 Reconstructed vertex \| 4,482,976 \| 81.3 Leading track $p_{\mathrm{T}}>0.15$ ${\rm GeV}/c$ \| 4,043,580 \| 73.3 Leading track $p_{\mathrm{T}}>0.5$ ${\rm GeV}/c$ \| 3,013,612 \| 54.6 Leading track $p_{\mathrm{T}}>1.0$ ${\rm GeV}/c$ \| 1,281,269 \| 23.2 Collision energy: 7 TeV \| Events \| % of all Offline trigger \| 25,137,512 \| 100.0 Reconstructed vertex \| 22,698,200 \| 90.3 Leading track $p_{\mathrm{T}}>0.15$ ${\rm GeV}/c$ \| 21,002,568 \| 83.6 Leading track $p_{\mathrm{T}}>0.5$ ${\rm GeV}/c$ \| 17,159,249 \| 68.3 Leading track $p_{\mathrm{T}}>1.0$ ${\rm GeV}/c$ \| 9,873,085 \| 39.3 Table 1: Events remaining after each event selection step. ## 4 Event and track selection ### 4.1 Trigger and offline event selection Events are recorded if either of the three triggering systems, V0A, V0C or SPD, has a signal. The arrival time of particles in the V0A and V0C are used to reject beam–gas interactions that occur outside the nominal interaction region. A more detailed description of the online trigger can be found in [20]. An additional offline selection is made following the same criteria but considering reconstructed information instead of online trigger signals. For each event a reconstructed vertex is required. The vertex reconstruction procedure is based on tracks as well as signals in the SPD. Only vertices within $\pm 10$ cm of the nominal interaction point along the beam axis are considered. Moreover, we require at least one track with $p_{\mathrm{T}}>$ $p_{\rm T,min}=$ 0.15, 0.5 or 1.0 ${\rm GeV}/c$ in the acceptance $\|\eta\|<0.8$. A pile-up rejection procedure is applied to the set of data taken at $\sqrt{s}=$ 7 TeV: events with more than one distinct reconstructed primary vertex are rejected. This cut has a negligible effect on simulated events without pile-up: only 0.06% of the events are removed. We have compared a selection of high pile-up probability runs (see Section 3) with a sample of low pile-up probability runs. The UE distributions differ by 20-25% between the two samples. After the above mentioned rejection procedure, the difference is reduced to less than 2%. Therefore, in the runs considered in the analysis, the effect of pile-up is negligible. No explicit rejection of cosmic-ray events is applied since cosmic particles are efficiently suppressed by our track selection cuts [23]. This is further confirmed by the absence of a sharp enhanced correlation at $\Delta\phi=\pi$ from the leading track which would be caused by almost straight high-$p_{\mathrm{T}}$ tracks crossing the detector. Table 1 summarizes the percentage of events remaining after each event selection step. We do not explicitly select non-diffractive events, although the above mentioned event selection significantly reduces the amount of diffraction in the sample. Simulated events show that the event selections reduce the fraction of diffractive events from 18-33% to 11-16% (Pythia 6.4 and Phojet at 0.9 and 7 TeV). We do not correct for this contribution. Selection criteria \| Value ---\|--- Detectors required \| ITS,TPC Minimum number of TPC clusters \| 70 Maximum $\chi^{2}$ per TPC cluster \| 4 Minimum number of ITS clusters \| 3 Minimum number of SPD or $1^{st}$ layer SDD clusters \| 1 Maximum $DCA_{Z}$ \| 2 cm Maximum $DCA_{XY}(p_{\mathrm{T}})$ \| 7$\sigma$ Table 2: Track selection criteria. ### 4.2 Track cuts The track cuts are optimized to minimize the contamination from secondary tracks. For this purpose a track must have at least 3 ITS clusters, one of which has to be in the first 3 layers. Moreover, we require at least 70 (out of a maximum of 159) clusters in the TPC drift volume. The quality of the track fitting measured in terms of the $\chi^{2}$ per space point is required to be lower than 4 (each space point having 2 degrees of freedom). We require the distance of closest approach of the track to the primary vertex along the beam axis (DCAZ) to be smaller than 2 cm. In the transverse direction we apply a $p_{\mathrm{T}}$ dependent DCAXY cut, corresponding to 7 standard deviations of its inclusive probability distribution. These cuts are summarized in Table 2. ## 5 Analysis strategy The Underlying Event activity is characterized by the following observables [2]: * • average charged particle density vs. leading track transverse momentum $p_{\rm T,LT}$: $\frac{1}{\Delta\eta\cdot\Delta\Phi}\frac{1}{N_{\rm ev}(p_{\rm T,LT})}N_{\rm ch}(p_{\rm T,LT})$ (1) * • average summed $p_{\mathrm{T}}$ density vs. leading track $p_{\rm T,LT}$: $\frac{1}{\Delta\eta\cdot\Delta\Phi}\frac{1}{N_{\rm ev}(p_{\rm T,LT})}\sum p_{\mathrm{T}}(p_{\rm T,LT})$ (2) * • $\Delta\phi$-correlation between tracks and the leading track: $\frac{1}{\Delta\eta}\frac{1}{N_{\rm ev}(p_{\rm T,LT})}\frac{{\rm d}N_{\rm ch}}{{\rm d}\Delta\phi}$ (3) (in bins of leading track $p_{\rm T,LT}$). $N_{\rm ev}$ is the total number of events selected and $N_{\rm ev}(p_{\rm T,LT})$ is the number of events in a given leading-track transverse-momentum bin. The first two variables are evaluated in three distinct regions. These regions, illustrated in Fig. 1, are defined with respect to the leading track azimuthal angle: * • Toward: $\|\Delta\phi\|<1/3$ $\pi$ * • Transverse: $1/3$ $\pi<\|\Delta\phi\|<2/3$ $\pi$ * • Away: $\|\Delta\phi\|>2/3$ $\pi$ where $\Delta\phi=\phi_{LT}-\phi$ is defined in $\pm\pi$. In Eq. (1)-(3) the normalization factor $\Delta\Phi$ is equal to $2/3\pi$, which is the size of each region. $\Delta\eta=$ 1.6 corresponds to the acceptance in pseudorapidity. The leading track is not included in the final distributions. Figure 1: Definition of the regions Toward, Transverse and Away w.r.t. leading track direction. ## 6 Corrections We correct for the following detector effects: vertex reconstruction efficiency, tracking efficiency, contamination from secondary particles and leading-track misidentification bias. The various corrections are explained in more detail in the following subsections. We do not correct for the trigger efficiency since its value is basically 100% for events which have at least one particle with $p_{\mathrm{T}}>$ 0.15 ${\rm GeV}/c$ in the range $\|\eta\|<0.8$. In Table 3 we summarize the maximum effect of each correction on the measured final observables at the two collision energies for $p_{\rm T,min}=0.5\,\mathrm{\mbox{${\rm GeV}/c$}}$. #### Vertex reconstruction The correction for finite vertex reconstruction efficiency is performed as a function of the measured multiplicity. Its value is smaller than 0.7% and 0.3% at $\sqrt{s}=$ 0.9 and $\sqrt{s}=7\,\mathrm{TeV}$, respectively. #### Tracking efficiency The tracking efficiency depends on the track level observables $\eta$ and $p_{\mathrm{T}}$. The projections of the tracking efficiency on the $p_{\mathrm{T}}$ and $\eta$ axes are shown in Fig. 2. In the pseudorapidity projection we observe a dip of about 1% at $\eta=0$ due to the central TPC cathode. The slight asymmetry between positive and negative $\eta$ is due to a different number of active SPD and SDD modules in the two halves of the detector. The number of active modules also differs between the data-taking periods at the two collision energies. Moreover, the efficiency decreases by 5% in the range 1-3 ${\rm GeV}/c$. This is due to the fact that above about 1 ${\rm GeV}/c$ tracks are almost straight and can be contained completely in the dead areas between TPC sectors. Therefore, at high $p_{\mathrm{T}}$ the efficiency is dominated by geometry and has a constant value of about 80% at both collision energies. To avoid statistical fluctuations, the estimated efficiency is fitted with a constant for $p_{\mathrm{T}}>5\,\mathrm{\mbox{${\rm GeV}/c$}}$ (not shown in the figure). #### Contamination from secondaries We correct for secondary tracks that pass the track selection cuts. Secondary tracks are predominantly produced by weak decays of strange particles (e.g. $K^{0}_{S}$ and $\Lambda$), photon conversions or hadronic interactions in the detector material, and decays of charged pions. The relevant track level observables for the contamination correction are transverse momentum and pseudorapidity. The correction is determined from detector simulations and is found to be 15-20% for tracks with $p_{\mathrm{T}}<$ 0.5 ${\rm GeV}/c$ and saturates at about 2% for higher transverse momenta (see Fig. 3). Correction \| $\sqrt{s}=0.9$ TeV \| $\sqrt{s}=7$ TeV ---\|---\|--- Leading track misidentification \| $<5$% \| $<8$% Contamination \| $<3$% \| $<3$% Efficiency \| $<19$% \| $<19$% Vertex reconstruction \| $<0.7$% \| $<0.3$% Table 3: Maximum effect of corrections on final observables for $p_{\rm T,min}=0.5\,\mathrm{\mbox{${\rm GeV}/c$}}$. We multiply the contamination estimate by a data-driven coefficient to take into account the low strangeness yield in the Monte Carlo compared to data [24]. The coefficient is derived from a fit of the discrepancy between data and Monte Carlo strangeness yields in the tails of the DCAXY distribution which are predominantly populated by secondaries. The factor has a maximum value of 1.07 for tracks with $p_{\mathrm{T}}<$ 0.5 ${\rm GeV}/c$ and is equal to 1 for $p_{\mathrm{T}}>$ 1.5 ${\rm GeV}/c$. This factor is included in the Contamination entry in Table 3. #### Leading-track misidentification Experimentally, the real leading track can escape detection because of tracking inefficiency and the detector’s finite acceptance. In these cases another track (i.e. the sub-leading or sub-sub-leading etc.) will be selected as the leading one, thus biasing the analysis in two possible ways. Firstly, the sub-leading track will have a different transverse momentum than the leading one. We refer to this as leading-track $p_{\mathrm{T}}$ bin migration. It has been verified with Monte Carlo that this effect is negligible due to the weak dependence of the final distributions on $p_{\rm T,LT}$. Secondly, the reconstructed leading track might have a significantly different orientation with respect to the real one, resulting in a rotation of the overall event topology. The largest bias occurs when the misidentified leading track falls in the Transverse region defined by the real leading track. We correct for leading-track misidentification with a data-driven procedure. Starting from the measured distributions, for each event the track loss due to inefficiency is applied a second time to the data (having been applied the first time naturally by the detector) by rejecting tracks randomly. If the leading track is considered reconstructed it is used as before to define the different regions. Otherwise the sub-leading track is used. Since the tracking inefficiency is quite small (about 20%) applying it on the reconstructed data a second time does not alter the results significantly. To verify this statement we compared our results with a two step procedure. In this case the inefficiency is applied two times on measured data, half of its value at a time. The correction factor obtained in this way is compatible with the one step procedure. Furthermore, the data-driven procedure has been tested on simulated data where the true leading particle is known. We observed a discrepancy between the two methods, especially at low leading-track $p_{\mathrm{T}}$ values, which is taken into account in the systematic error. The maximum leading-track misidentification correction is 8% on the final distributions. Figure 2: Tracking efficiency vs. track $p_{\mathrm{T}}$ (left, $\|\eta\|<0.8$) and $\eta$ (right, $p_{\mathrm{T}}>0.5\,\mathrm{\mbox{${\rm GeV}/c$}}$) from a Pythia 6.4 and Geant3 simulation. #### Two-track effects By comparing simulated events corrected for single-particle efficiencies with the input Monte Carlo, we observe a 0.5% discrepancy around $\Delta\phi=0$. This effect is called non-closure in Monte Carlo (it will be discussed further in Section 7) and in this case is related to small two-track resolution effects. Data are corrected for this discrepancy. ## 7 Systematic uncertainties In Tables 4, 5 and 6 we summarize the systematic uncertainties evaluated in the analysis for the three track thresholds: $p_{\mathrm{T}}>0.15$, 0.5 and 1.0 ${\rm GeV}/c$. Each uncertainty is explained in more detail in the following subsections. Uncertainties which are constant as a function of leading-track $p_{\mathrm{T}}$ are listed in Table 4. Leading-track $p_{\mathrm{T}}$ dependent uncertainties are summarized in Tables 5 and 6 for $\sqrt{s}=0.9\,\mathrm{TeV}$ and $7\,\mathrm{TeV}$, respectively. Positive and negative uncertainties are propagated separately, resulting in asymmetric final uncertainties. #### Particle composition The tracking efficiency and contamination corrections depend slightly on the particle species mainly due to their decay length and absorption in the material. To assess the effect of an incorrect description of the particle abundances in the Monte Carlo, we varied the relative yields of pions, protons, kaons, and other particles by 30% relative to the default Monte Carlo predictions. The maximum variation of the final values is 0.9% and represents the systematic uncertainty related to the particle composition (see Table 4). Moreover, we have compared our assessment of the underestimation of strangeness yields with a direct measurement from the ALICE collaboration [24]. Based on the discrepancy between the two estimates, we assign a systematic uncertainty of 0-2.3% depending on the $p_{\mathrm{T}}$ threshold and collision energy, see Tables 5 and 6. Figure 3: Contamination correction: correction factor vs. track $p_{\mathrm{T}}$ (left, $\|\eta\|<0.8$) and $\eta$ (right, $p_{\mathrm{T}}>0.5\,\mathrm{\mbox{${\rm GeV}/c$}}$) from a Pythia 6.4 and Geant3 simulation. #### ITS and TPC efficiency The tracking efficiency depends on the level of precision of the description of the ITS and TPC detectors in the simulation and the modeling of their response. After detector alignment with survey methods, cosmic-ray events and pp collision events [28], the uncertainty on the efficiency due to the ITS description is estimated to be below 2% and affects only tracks with $p_{\mathrm{T}}<$ 0.3 ${\rm GeV}/c$. The uncertainty due to the TPC reaches 4.5% at very low $p_{\mathrm{T}}$ and is smaller than 1.2% for tracks with $p_{\mathrm{T}}>$ 0.5 ${\rm GeV}/c$. The resulting maximum uncertainty on the final distributions is below 1.9%. Moreover, an uncertainty of 1% is included to account for uncertainties in the MC description of the matching between TPC and ITS tracks (see Table 4). \| $\sqrt{s}=0.9$ TeV ---\|--- \| $p_{\mathrm{T}}>0.15\,\mbox{${\rm GeV}/c$}$ \| $p_{\mathrm{T}}>0.5\,\mbox{${\rm GeV}/c$}$ \| $p_{\mathrm{T}}>1.0\,\mbox{${\rm GeV}/c$}$ Particle composition \| $\pm$ 0.9% \| $\pm$ 0.7% \| $\pm$ 0.4% ITS efficiency \| $\pm$ 0.6% \| – \| – TPC efficiency \| $\pm$ 1.9% \| $\pm$ 0.8% \| $\pm$ 0.4% Track cuts \| ${}^{+\ 3.0\%}_{-\ 1.1\%}$ \| ${}^{+\ 2.0\%}_{-\ 1.1\%}$ \| ${}^{+\ 0.9\%}_{-\ 1.5\%}$ ITS/TPC matching \| $\pm$ 1.0% \| $\pm$ 1.0% \| $\pm$ 1.0% MC dependence \| $+$ 1.1% , $+$ 1.1% , $+$ 1.6% \| $+$ 0.9% \| $+$ 0.9% , $+$ 0.9% , $+$ 1.3% Material budget \| $\pm$ 0.6% \| $\pm$ 0.2% \| $\pm$ 0.2% \| $\sqrt{s}=7$ TeV \| $p_{\mathrm{T}}>0.15\,\mbox{${\rm GeV}/c$}$ \| $p_{\mathrm{T}}>0.5\,\mbox{${\rm GeV}/c$}$ \| $p_{\mathrm{T}}>1.0\,\mbox{${\rm GeV}/c$}$ Particle composition \| $\pm$ 0.9% \| $\pm$ 0.7% \| $\pm$ 0.5% ITS efficiency \| $\pm$ 0.5% \| – \| – TPC efficiency \| $\pm$ 1.8% \| $\pm$ 0.8% \| $\pm$ 0.5% Track cuts \| ${}^{+\ 2.1\%}_{-\ 2.3\%}$ \| ${}^{+\ 1.6\%}_{-\ 3.2\%}$ \| ${}^{+\ 2.5\%}_{-\ 3.5\%}$ ITS/TPC matching \| $\pm$ 1.0% \| $\pm$ 1.0% \| $\pm$ 1.0% MC dependence \| $+$ 0.8% , $+$ 0.8% , $+$ 1.2% \| $+$ 0.8% \| $+$ 1.0% Material budget \| $\pm$ 0.6% \| $\pm$ 0.2% \| $\pm$ 0.2% Table 4: Constant systematic uncertainties at both collision energies. When more than one number is quoted, separated by a comma, the first value refers to the number density distribution, the second to the summed $p_{\mathrm{T}}$ and the third to the azimuthal correlation. Some of the uncertainties are quoted asymmetrically. #### Track cuts By applying the efficiency and contamination corrections we correct for those particles which are lost due to detector effects and for secondary tracks which have not been removed by the selection cuts. These corrections rely on detector simulations and therefore, one needs to estimate the systematic uncertainty introduced in the correction procedure by one particular choice of track cuts. To do so, we repeat the analysis with different values of the track cuts, both for simulated and real data. The variation of the final distributions with different track cuts is a measure of the systematic uncertainty. The overall effect, considering all final distributions, is smaller than 3.5% at both collision energies (see Table 4). #### Misidentification bias The uncertainty on the leading-track misidentification correction is estimated from the discrepancy between the data-driven correction used in the analysis and that based on simulations. The effect influences only the first two leading-track $p_{\mathrm{T}}$ bins at both collision energies. The maximum uncertainty ($\sim 18\%$) affects the first leading-track $p_{\mathrm{T}}$ bin for the track $p_{\mathrm{T}}$ cut-off of 0.15 GeV/$c$. In all other bins this uncertainty is of the order of few percent. As summarized in Tables 5 and 6, the uncertainty has slightly different values for the various UE distributions. #### Vertex-reconstruction efficiency The analysis accepts reconstructed vertices with at least one contributing track. We repeat the analysis requiring at least two contributing tracks. The systematic uncertainty related to the vertex reconstruction efficiency is given by the maximum variation in the final distributions between the cases of one and two contributing tracks. Its value is 2.4% for $p_{\rm T,min}=$ 0.15 GeV/$c$ and below 1% for the other cut-off values (see Tables 5 and 6). The effect is only visible in the first leading-track $p_{\mathrm{T}}$ bin. #### Non-closure in Monte Carlo By correcting a Monte Carlo prediction after full detector simulation with corrections extracted from the same generator, we expect to obtain the input Monte Carlo prediction within the statistical uncertainty. This consideration holds true only if each correction is evaluated with respect to all the variables to which the given correction is sensitive. Any statistically significant difference between input and corrected distributions is referred to as non-closure in Monte Carlo. The overall non-closure effect is sizable ($\sim 17\%$) in the first leading- track $p_{\mathrm{T}}$ bin and is 0.6-5.3% in all other bins at both collision energies. #### Monte-Carlo dependence The difference in final distributions when correcting the data with Pythia 6.4 or Phojet generators is of the order of 1% and equally affects all the leading-track $p_{\mathrm{T}}$ bins. #### Material budget The material budget has been measured by reconstructing photon conversions which allows a precise $\gamma$-ray tomography of the ALICE detector. For the detector regions important for this analysis the remaining uncertainty on the extracted material budget is less than 7%. Varying the material density in the detector simulation, the effect on the observables presented is determined to be 0.2-0.6% depending on the $p_{\mathrm{T}}$ threshold considered. \| \| $\sqrt{s}=0.9$ TeV ---\|---\|--- \| \| Number density \| $p_{\rm T,LT}$ \| $p_{\mathrm{T}}>0.15\,\mbox{${\rm GeV}/c$}$ \| $p_{\mathrm{T}}>0.5\,\mbox{${\rm GeV}/c$}$ \| $p_{\mathrm{T}}>1.0\,\mbox{${\rm GeV}/c$}$ Lead. track misid. \| $1^{st}$ bin \| $+$ (17.8, 16.3, 16.3)% \| $+$ (4.6, 3.5, 3.5)% \| $+$ (4.2, 2.9, 1.7)% \| $2^{nd}$ bin \| $+$ 2.9% \| $+$ 1.3% \| – MC non closure \| $1^{st}$ bin \| $-$ 17.2% \| $-$ 3.6% \| $-$ 1.2% \| $2^{nd}$ bin \| $-$ 3.2% \| $-$ 0.8% \| $-$ 1.2% \| others \| $-$ 0.6% \| $-$ 0.8% \| $-$ 1.2% Strangeness \| $1^{st}$ bin \| $\pm$ 1.9% \| $\pm$ 0.2% \| – \| others \| $\pm$ 1.0% \| $\pm$ 0.2% \| – Vertex reco. \| $1^{st}$ bin \| $-$ 2.4% \| $-$ 0.7% \| $-$ 0.5% \| \| Summed $p_{\mathrm{T}}$ \| $p_{\rm T,LT}$ \| $p_{\mathrm{T}}>0.15\,\mbox{${\rm GeV}/c$}$ \| $p_{\mathrm{T}}>0.5\,\mbox{${\rm GeV}/c$}$ \| $p_{\mathrm{T}}>1.0\,\mbox{${\rm GeV}/c$}$ Lead. track misid. \| $1^{st}$ bin \| $+$ (20.0, 18.1, 18.1)% \| $+$ (5.3, 4.1, 4.1)% \| $+$ (4.8, 3.4, 3.4)% \| $2^{nd}$ bin \| $+$ 3.7% \| $+$ 1.6% \| – MC non closure \| $1^{st}$ bin \| $-$ 17.0% \| $-$ 2.8% \| $-$ 1.1% \| $2^{nd}$ bin \| $-$ 3.0% \| $-$ 1.0% \| $-$ 1.1% \| others \| $-$ 0.7% \| $-$ 1.0% \| $-$ 1.1% Strangeness \| $1^{st}$ bin \| $\pm$ 1.9% \| $\pm$ 0.2% \| – \| others \| $\pm$ 1.0% \| $\pm$ 0.2% \| – Vertex reco. \| $1^{st}$ bin \| $-$ 2.4% \| $-$ 0.7% \| $-$ 0.5% \| \| Azimuthal correlation \| $p_{\rm T,LT}$ \| $p_{\mathrm{T}}>0.15\,\mbox{${\rm GeV}/c$}$ \| $p_{\mathrm{T}}>0.5\,\mbox{${\rm GeV}/c$}$ \| $p_{\mathrm{T}}>1.0\,\mbox{${\rm GeV}/c$}$ Lead. track misid. \| $1^{st}$ bin \| $+$ 12.0% \| $+$ 3.9% \| $+$ 2.5% \| $2^{nd}$ bin \| $+$ 2.6% \| $+$ 1.1% \| – MC non closure \| $1^{st}$ bin \| $-$ 17.1% \| $-$ 3.3% \| $-$ 1.6% \| $2^{nd}$ bin \| $-$ 3.5% \| $-$ 3.0% \| $-$ 1.6% \| others \| $-$ 2.4% \| $-$ 3.0% \| $-$ 1.6% Strangeness \| $1^{st}$ bin \| $\pm$ 1.9% \| $\pm$ 0.2% \| – \| others \| $\pm$ 1.0% \| $\pm$ 0.2% \| – Vertex reco. \| $1^{st}$ bin \| $-$ 2.4% \| $-$ 0.4% \| – \| others \| $-$ 0.5% \| $-$ 0.4% \| – Table 5: Systematic uncertainties vs. leading track $p_{\mathrm{T}}$ at $\sqrt{s}=0.9\,\mathrm{TeV}$. When more than one number is quoted, separated by a comma, the first value refers to the Toward, the second to the Transverse and the third to the Away region. The second column denotes the leading track $p_{\mathrm{T}}$ bin for which the uncertainty applies. The numbering starts for each case from the first bin above the track $p_{\mathrm{T}}$ threshold. \| \| $\sqrt{s}=7$ TeV ---\|---\|--- \| \| Number density \| $p_{\rm T,LT}$ \| $p_{\mathrm{T}}>0.15\,\mbox{${\rm GeV}/c$}$ \| $p_{\mathrm{T}}>0.5\,\mbox{${\rm GeV}/c$}$ \| $p_{\mathrm{T}}>1.0\,\mbox{${\rm GeV}/c$}$ Lead. track misid. \| $1^{st}$ bin \| $+$ (17.9, 16.3, 16.3)% \| $+$ (4.0, 3.2, 3.2)% \| $+$ (2.5, 1.2, 1.2)% \| $2^{nd}$ bin \| $+$ 2.7% \| – \| $+$ 0.7% MC non closure \| $1^{st}$ bin \| $-$ 16.8% \| $-$ 2.6% \| $-$ 1.9% \| $2^{nd}$ bin \| $-$ 2.9% \| $-$ 1.4% \| $-$ 1.9% \| others \| $-$ 0.6% \| $-$ 1.0% \| $-$ 1.9% Strangeness \| $1^{st}$ bin \| $\pm$ 1.8% \| $\pm$ 2.3% \| – \| others \| $\pm$ 1.0% \| $\pm$ 2.3% \| – Vertex reco. \| $1^{st}$ bin \| $-$ 2.4% \| $-$ 0.7% \| $-$ 0.5% \| \| Summed $p_{\mathrm{T}}$ \| $p_{\rm T,LT}$ \| $p_{\mathrm{T}}>0.15\,\mbox{${\rm GeV}/c$}$ \| $p_{\mathrm{T}}>0.5\,\mbox{${\rm GeV}/c$}$ \| $p_{\mathrm{T}}>1.0\,\mbox{${\rm GeV}/c$}$ Lead. track misid. \| $1^{st}$ bin \| $+$ (20.0, 17.9, 17.9)% \| $+$ (4.9, 3.8, 3.8)% \| $+$ (3.4, 1.9, 1.9)% \| $2^{nd}$ bin \| $+$ 3.4% \| $+$ 0.8% \| $+$ 1.1% MC non closure \| $1^{st}$ bin \| $-$ 16.7% \| $-$ 2.7% \| $-$ 1.5% \| $2^{nd}$ bin \| $-$ 2.6% \| $-$ 1.2% \| $-$ 1.5% \| others \| $-$ 0.8% \| $-$ 1.0% \| $-$ 1.5% Strangeness \| $1^{st}$ bin \| $\pm$ 1.8% \| $\pm$ 2.3% \| – \| others \| $\pm$ 1.0% \| $\pm$ 2.3% \| – Vertex reco. \| $1^{st}$ bin \| $-$ 2.4% \| $-$ 0.7% \| $-$ 0.5% \| \| Azimuthal correlation \| $p_{\rm T,LT}$ \| $p_{\mathrm{T}}>0.15\,\mbox{${\rm GeV}/c$}$ \| $p_{\mathrm{T}}>0.5\,\mbox{${\rm GeV}/c$}$ \| $p_{\mathrm{T}}>1.0\,\mbox{${\rm GeV}/c$}$ Lead. track misid. \| $1^{st}$ bin \| $+$ 16.8% \| $+$ 3.4% \| $+$ 0.9% \| $2^{nd}$ bin \| $+$ 2.5% \| – \| – MC non closure \| $1^{st}$ bin \| $-$ 25.3% \| $-$ 4.3% \| $-$ 1.2% \| $2^{nd}$ bin \| $-$ 5.3% \| $-$ 2.1% \| $-$ 1.2% \| others \| $-$ 2.1% \| $-$ 2.1% \| $-$ 1.2% Strangeness \| $1^{st}$ bin \| $\pm$ 1.8% \| $\pm$ 2.3% \| – \| others \| $\pm$ 1.0% \| $\pm$ 2.3% \| – Vertex reco. \| $1^{st}$ bin \| $-$ 2.4% \| $-$ 0.4% \| – \| others \| $-$ 0.5% \| $-$ 0.4% \| – Table 6: Systematic uncertainties vs. leading track $p_{\mathrm{T}}$ at $\sqrt{s}=7\,\mathrm{TeV}$. When more than one number is quoted, separated by a comma, the first value refers to the Toward, the second to the Transverse and the third to the Away region. The second column denotes the leading track $p_{\mathrm{T}}$ bin for which the uncertainty applies. The numbering starts for each case from the first bin above the track $p_{\mathrm{T}}$ threshold. ## 8 Results In this section we present and discuss the corrected results for the three UE distributions in all regions at the two collision energies. The upper part of each plot shows the relevant measured distribution (black points) compared to a set of Monte Carlo predictions (coloured curves). Shaded bands represent the systematic uncertainty only. Error bars along the $x$ axis indicate the bin width. The lower part shows the ratio between Monte Carlo and data. In this case the shaded band is the sum in quadrature of statistical and systematic uncertainties. The overall agreement of data and simulations is of the order of 10-30% and we were not able to identify a preferred model that can reproduce all measured observables. In general, all three generators underestimate the event activity in the Transverse region. Nevertheless, an agreement of the order of 20% has to be considered a success, considering the complexity of the system under study. Even though an exhaustive comparison of data with the latest models available is beyond the scope of this paper, in the next sections we will indicate some general trends observed in the comparison with the chosen models. In the following discussion we define the leading track $p_{\mathrm{T}}$ range from 4 to 10 ${\rm GeV}/c$ at $\sqrt{s}=0.9\,\mathrm{TeV}$ and from 10 to 25 ${\rm GeV}/c$ at $\sqrt{s}=7\,\mathrm{TeV}$ as the plateau. \| $\sqrt{s}=0.9$ TeV ---\|--- \| Number density \| Summed $p_{\mathrm{T}}$ \| Slope $(\mbox{${\rm GeV}/c$})^{-1}$ \| Mean \| Slope \| Mean $(\mbox{${\rm GeV}/c$})$ $p_{\mathrm{T}}>$ 0.15 GeV/c \| 0.00 $\pm$ 0.02 \| 1.00 $\pm$ 0.04 \| 0.00 $\pm$ 0.01 \| 0.62 $\pm$ 0.02 $p_{\mathrm{T}}>$ 0.5 GeV/c \| 0.00 $\pm$ 0.01 \| 0.45 $\pm$ 0.02 \| 0.01 $\pm$ 0.01 \| 0.45 $\pm$ 0.02 $p_{\mathrm{T}}>$ 1.0 GeV/c \| 0.003 $\pm$ 0.003 \| 0.16 $\pm$ 0.01 \| 0.006 $\pm$ 0.005 \| 0.24 $\pm$ 0.01 \| $\sqrt{s}=7$ TeV \| Number density \| Summed $p_{\mathrm{T}}$ \| Slope $(\mbox{${\rm GeV}/c$})^{-1}$ \| Mean \| Slope \| Mean $(\mbox{${\rm GeV}/c$})$ $p_{\mathrm{T}}>$ 0.15 GeV/c \| 0.00 $\pm$ 0.01 \| 1.82 $\pm$ 0.06 \| 0.01 $\pm$ 0.01 \| 1.43 $\pm$ 0.05 $p_{\mathrm{T}}>$ 0.5 GeV/c \| 0.005 $\pm$ 0.007 \| 0.95 $\pm$ 0.03 \| 0.01 $\pm$ 0.01 \| 1.15 $\pm$ 0.04 $p_{\mathrm{T}}>$ 1.0 GeV/c \| 0.001 $\pm$ 0.003 \| 0.41 $\pm$ 0.01 \| 0.008 $\pm$ 0.006 \| 0.76 $\pm$ 0.03 \| $\sqrt{s}=1.8$ TeV (CDF) \| Number density (at leading charged jet $p_{\mathrm{T}}=20\,\mathrm{\mbox{${\rm GeV}/c$}}$) $p_{\mathrm{T}}>$ 0.5 GeV/c \| 0.60 Table 7: Saturation values in the Transverse region for the two collision energies. The result from CDF is also given, for details see text. ### 8.1 Number density In Fig. 4-6 we show the multiplicity density as a function of leading track $p_{\mathrm{T}}$ in the three regions: Toward, Transverse and Away. Toward and Away regions are expected to collect the fragmentation products of the two back-to-back outgoing partons from the elementary hard scattering. We observe that the multiplicity density in these regions increases monotonically with the $p_{\rm T,LT}$ scale. In the Transverse region, after a monotonic increase at low leading track $p_{\mathrm{T}}$, the distribution tends to flatten out. The same behaviour is observed at both collision energies and all values of $p_{\rm T,min}$. The rise with $p_{\rm T,LT}$ has been interpreted as evidence for an impact parameter dependence in the hadronic collision [29]. More central collisions have an increased probability for MPI, leading to a larger transverse multiplicity. Nevertheless, we must be aware of a trivial effect also contributing to the low $p_{\rm T,LT}$ region. For instance for any probability distribution, the maximum value per randomized sample averaged over many samples rises steadily with the sample size $M$. In our case, the conditional probability density $\mathcal{P}(p_{\rm T,LT}\|M)$ shifts towards larger $p_{\rm T,LT}$ with increasing $M$. Using Bayes’ theorem one expects the conditional probability density $\mathcal{P}(M\|p_{\rm T,LT})$ to shift towards larger $M$ with rising $p_{\rm T,LT}$: $\mathcal{P}(M\|p_{\rm T,LT})\sim\mathcal{P}(p_{\rm T,LT}\|M)\mathcal{P}(M).$ (4) The saturation of the distribution at higher values of $p_{\rm T,LT}$ indicates the onset of the event-by-event partitioning into azimuthal regions containing the particles from the hard scattering and the UE region. The bulk particle production becomes independent of the hard scale. The plateau range is fitted with a line. The fit slopes, consistent with zero, and mean values for the three $p_{\mathrm{T}}$ thresholds are reported in Table 7. In the fit, potential correlations of the systematic uncertainties in different $p_{\mathrm{T}}$ bins are neglected. ATLAS has published a UE measurement where the hard scale is given by the leading track $p_{\mathrm{T}}$, with a $p_{\mathrm{T}}$ threshold for particles of 0.5 ${\rm GeV}/c$ and an acceptance of $\|\eta\|<2.5$ [7]. Given the different acceptance with respect to our measurement, the results in the Toward and Away regions are not comparable. On the other hand the mean values of the Transverse plateaus from the two measurements are in good agreement, indicating an independence of the UE activity on the pseudorapidity range. The CDF collaboration measured the UE as a function of charged particle jet $p_{\mathrm{T}}$ at a collision energy of 1.8 TeV[2]. The particle $p_{\mathrm{T}}$ threshold is 0.5 ${\rm GeV}/c$ and the acceptance $\|\eta\|<1$. In the Transverse region CDF measures 3.8 charged particles per unit pseudorapidity above the $p_{\mathrm{T}}$ threshold at leading-jet $p_{\mathrm{T}}=20\,\mathrm{\mbox{${\rm GeV}/c$}}$. This number needs to be divided by $2\pi$ in order to be compared with the average number of particles in the plateau from Table 7 at the same threshold value. The scaled CDF result is 0.60, also shown in Table 7 for comparison. As expected it falls between our two measurements at $\sqrt{s}=0.9\,\mathrm{\mbox{${\rm TeV}$}}$ and $\sqrt{s}=7\,\mathrm{\mbox{${\rm TeV}$}}$. The values do not scale linearly with the collision energy, in particular the increase is higher from 0.9 to 1.8 ${\rm TeV}$ than from 1.8 to 7 ${\rm TeV}$. Interpolating between our measurements assuming a logarithmic dependence on $\sqrt{s}$ results in 0.62 charged particles per unit area at 1.8 ${\rm TeV}$, consistent with the CDF result. For illustration, Figure 7 presents the number density in the plateau of the Transverse region for $p_{\mathrm{T}}>0.5\,\mathrm{\mbox{${\rm GeV}/c$}}$ (our measurement as well as the value measured by CDF at 1.8 TeV) compared with $dN_{\rm ch}/d\eta\|_{\eta=0}$ of charged particles with $p_{\mathrm{T}}>$ 0.5 ${\rm GeV}/c$ in minimum-bias events [30] (scaled by $1/2\pi$).222These data are for events that have at least one charged particle in $\|\eta\|<2.5$. The UE activity in the plateau region is more than a factor 2 larger than the $dN_{\rm ch}/d\eta$. Both can be fitted with a logarithmic dependence on $s$ ($a+b\ln{s}$). The relative increase from 0.9 to 7 TeV for the UE is larger than that for the $dN_{\rm ch}/d\eta$: about 110% compared to about 80%, respectively. In Fig. 8 (left) we show the ratio between the number density distribution at $\sqrt{s}=7\,\mathrm{\mbox{${\rm TeV}$}}$ and $\sqrt{s}=0.9\,\mathrm{\mbox{${\rm TeV}$}}$. Most of the systematic uncertainties are expected to be correlated between the two energies, therefore we consider only statistical uncertainties. The ratio saturates for leading track $p_{\mathrm{T}}>4\,\mathrm{\mbox{${\rm GeV}/c$}}$. The results of a constant fit in the range $4<p_{\mathrm{T,LT}}<10\,\mathrm{\mbox{${\rm GeV}/c$}}$ are reported in Table 8. The measured scaling factor for a $p_{\mathrm{T}}$ threshold of 0.5 ${\rm GeV}/c$ is in agreement with the observations of ATLAS [7, 31] and CMS [32]. For the track threshold $p_{\mathrm{T}}>0.15\,\mathrm{\mbox{${\rm GeV}/c$}}$ all models underestimate the charged multiplicity in the Transverse and Away regions. In particular at $\sqrt{s}=7\,\mathrm{\mbox{${\rm TeV}$}}$ PHOJET predictions largely underestimate the measurement in the Transverse region (up to $\sim 50\%$), the discrepancy being more pronounced with increasing $p_{\mathrm{T}}$ cut-off value. Pythia 8 correctly describes the Toward region at both collision energies and Phojet only at $\sqrt{s}=0.9\,\mathrm{\mbox{${\rm TeV}$}}$. For track $p_{\mathrm{T}}>$ 1 ${\rm GeV}/c$, Pythia 8 systematically overestimates the event activity in the jet fragmentation regions (Toward and Away). \| Number density \| Summed $p_{\mathrm{T}}$ ---\|---\|--- $p_{\mathrm{T}}>0.15\,\mathrm{\mbox{${\rm GeV}/c$}}$ \| 1.76 $\pm$ 0.02 \| 2.00 $\pm$ 0.03 $p_{\mathrm{T}}>0.5\,\mathrm{\mbox{${\rm GeV}/c$}}$ \| 1.97 $\pm$ 0.03 \| 2.16 $\pm$ 0.03 $p_{\mathrm{T}}>1.0\,\mathrm{\mbox{${\rm GeV}/c$}}$ \| 2.32 $\pm$ 0.04 \| 2.48 $\pm$ 0.05 Table 8: Constant fit in $4<p_{\mathrm{T,LT}}<10\,\mathrm{GeV/\textit{c}}$ to the ratio between $\sqrt{s}=0.9\,\mathrm{TeV}$ and $\sqrt{s}=7\,\mathrm{TeV}$ for number density (left) and summed $p_{\mathrm{T}}$ (right) distributions in the Transverse region. The shown uncertainties are based on statistical and systematic uncertainties summed in quadrature. ### 8.2 Summed $p_{\mathrm{T}}$ In Fig. 9-11 we show the summed $p_{\mathrm{T}}$ density as a function of leading track $p_{\mathrm{T}}$ in the three topological regions. The shape of the distributions follows a trend similar to that discussed above for the number density. The general trend of Pythia 8 is to overestimate the fragmentation in the Toward region at all $p_{\mathrm{T}}$ cut-off values. Also in this case at $\sqrt{s}=7\,\mathrm{\mbox{${\rm TeV}$}}$ PHOJET largely underestimates the measurement in the Transverse region (up to $\sim 50\%$), especially at higher values of $p_{\mathrm{T}}$ cut-off. Other systematic trends are not very pronounced. In Table 7 we report the mean value of a linear fit in the plateau range. Our results agree with the ATLAS measurement in the Transverse plateau. In Fig. 8 (right) we show the ratio between the distribution at $\sqrt{s}=7\,\mathrm{\mbox{${\rm TeV}$}}$ and $\sqrt{s}=0.9\,\mathrm{\mbox{${\rm TeV}$}}$, considering as before only statistical errors. The results of a constant fit in the range $4<p_{\mathrm{T,LT}}<10\,\mathrm{\mbox{${\rm GeV}/c$}}$ are reported in Table 8. Also in this case the scaling factor is in agreement with ATLAS and CMS results. The summed $p_{\mathrm{T}}$ density in the Transverse region can be interpreted as a measurement of the UE activity in a given leading track $p_{\mathrm{T}}$ bin. Therefore, its value in the plateau can be used, for example, to correct jet spectra. ### 8.3 Azimuthal correlation In Fig. 12-22 azimuthal correlations between tracks and the leading track are shown in different ranges of leading track $p_{\mathrm{T}}$. The range $1/3\pi<\|\Delta\phi\|<2/3\pi$ corresponds to the Transverse region. The regions $-1/3\pi<\Delta\phi<1/3\pi$ (Toward) and $2/3\pi<\|\Delta\phi\|<\pi$ (Away) collect the fragmentation products of the leading and sub-leading jets. In general, all Monte Carlo simulations considered fail to reproduce the shape of the measured distributions. Pythia 8 provides the best prediction for the Transverse activity in all leading track $p_{\mathrm{T}}$ ranges considered. Unfortunately the same model significantly overestimates the jet fragmentation regions. ## 9 Conclusions We have characterized the Underlying Event in pp collisions at $\sqrt{s}=$ 0.9 and 7 ${\rm TeV}$ by measuring the number density, the summed $p_{\mathrm{T}}$ distribution and the azimuthal correlation of charged particles with respect to the leading particle. The analysis is based on about $6\cdot 10^{6}$ minimum bias events at $\sqrt{s}=$ 0.9 ${\rm TeV}$ and $25\cdot 10^{6}$ events at $\sqrt{s}=$ 7 ${\rm TeV}$ collected during the data taking periods from April to July 2010. Measured data have been corrected for detector related effects; in particular we applied a data-driven correction to account for the misidentification of the leading track. The fully corrected final distributions are compared with Pythia 6.4, Pythia 8 and Phojet, showing that pre-LHC tunes have difficulties describing the data. These results are an important ingredient in the required retuning of those generators. Among the presented distributions, the Transverse region is particularly sensitive to the Underlying Event. We find that the ratio between the distributions at $\sqrt{s}=$ 0.9 and 7 ${\rm TeV}$ in this region saturates at a value of about 2 for track $p_{\mathrm{T}}>$ 0.5 ${\rm GeV}/c$. The summed $p_{\mathrm{T}}$ distribution rises slightly faster as a function of $\sqrt{s}$ than the number density distribution, indicating that the available energy tends to increase the particle’s transverse momentum in addition to the multiplicity. This is in qualitative agreement with an increased relative contribution of hard processes to the Underlying Event with increasing $\sqrt{s}$. Moreover, the average number of particles at large $p_{\rm T,LT}$ in the Transverse region seems to scale logarithmically with the collision energy. In general our results are in good qualitative and quantitative agreement with measurements from other LHC experiments (ATLAS and CMS) and show similar trends to that of the Tevatron (CDF). Our results show that the activity in the Transverse region increases logarithmically and faster than $dN_{\rm ch}/d\eta$ in minimum-bias events. Models aiming to correctly reproduce these minimum-bias and underlying event distributions need a precise description of the interplay of the hard process, the associated initial and final-state radiation and multiple parton interactions. ## Number Density - track $p_{\mathrm{T}}>0.15\,\mbox{${\rm GeV}/c$}$ Figure 4: Number density in Toward (top), Transverse (middle) and Away (bottom) regions at $\sqrt{s}=0.9$ TeV (left) and $\sqrt{s}=7$ TeV (right). Right and left vertical scales differ by a factor 2. Shaded area in upper plots: systematic uncertainties. Shaded areas in bottom plots: sum in quadrature of statistical and systematic uncertainties. Horizontal error bars: bin width. ## Number Density - track $p_{\mathrm{T}}>0.5\,\mbox{${\rm GeV}/c$}$ Figure 5: Number density in Toward (top), Transverse (middle) and Away (bottom) regions at $\sqrt{s}=0.9$ TeV (left) and $\sqrt{s}=7$ TeV (right). Right and left vertical scales differ by a factor 2. Shaded area in upper plots: systematic uncertainties. Shaded areas in bottom plots: sum in quadrature of statistical and systematic uncertainties. Horizontal error bars: bin width. ## Number Density - track $p_{\mathrm{T}}>1.0\,\mbox{${\rm GeV}/c$}$ Figure 6: Number density in Toward (top), Transverse (middle) and Away (bottom) regions at $\sqrt{s}=0.9$ TeV (left) and $\sqrt{s}=7$ TeV (right). Right and left vertical scales differ by a factor 2. Shaded area in upper plots: systematic uncertainties. Shaded areas in bottom plots: sum in quadrature of statistical and systematic uncertainties. Horizontal error bars: bin width. Figure 7: Comparison of number density in the plateau of the Transverse region (see Table 8) and $dN_{\rm ch}/d\eta$ in minimum-bias events (scaled by $1/2\pi$) [30]. Both are for charged particles with $p_{\mathrm{T}}>0.5\,\mathrm{\mbox{${\rm GeV}/c$}}$. For this plot, statistical and systematic uncertainties have been summed in quadrature. The lines show fits with the functional form $a+b\ln{s}$. Figure 8: Ratio between $\sqrt{s}=0.9\,\mathrm{TeV}$ and $\sqrt{s}=7\,\mathrm{TeV}$ for number density (left) and summed $p_{\mathrm{T}}$ (right) distributions in the Transverse region. Statistical uncertainties only. ## Summed $p_{\mathrm{T}}$ \- track $p_{\mathrm{T}}>0.15\,\mbox{${\rm GeV}/c$}$ Figure 9: Summed $p_{\mathrm{T}}$ in Toward (top), Transverse (middle) and Away (bottom) regions at $\sqrt{s}=0.9$ TeV (left) and $\sqrt{s}=7$ TeV (right). Right and left vertical scales differ by a factor 4 (2) in the top (middle and bottom) panel. Shaded area in upper plots: systematic uncertainties. Shaded areas in bottom plots: sum in quadrature of statistical and systematic uncertainties. Horizontal error bars: bin width. ## Summed $p_{\mathrm{T}}$ \- track $p_{\mathrm{T}}>0.5\,\mbox{${\rm GeV}/c$}$ Figure 10: Summed $p_{\mathrm{T}}$ in Toward (top), Transverse (middle) and Away (bottom) regions at $\sqrt{s}=0.9$ TeV (left) and $\sqrt{s}=7$ TeV (right). Right and left vertical scales differ by a factor 4 (2) in the top (middle and bottom) panel. Shaded area in upper plots: systematic uncertainties. Shaded areas in bottom plots: sum in quadrature of statistical and systematic uncertainties. Horizontal error bars: bin width. ## Summed $p_{\mathrm{T}}$ \- track $p_{\mathrm{T}}>1.0\,\mbox{${\rm GeV}/c$}$ Figure 11: Summed $p_{\mathrm{T}}$ in Toward (top), Transverse (middle) and Away (bottom) regions at $\sqrt{s}=0.9$ TeV (left) and $\sqrt{s}=7$ TeV (right). Right and left vertical scales differ by a factor 4 (3) in the top (middle and bottom) panel. Shaded area in upper plots: systematic uncertainties. Shaded areas in bottom plots: sum in quadrature of statistical and systematic uncertainties. Horizontal error bars: bin width. ## Azimuthal correlations - track $p_{\mathrm{T}}>0.15\,\mbox{${\rm GeV}/c$}$ Figure 12: Azimuthal correlation at $\sqrt{s}=0.9$ TeV (left) and $\sqrt{s}=7$ TeV (right). Leading-track: 0.5 $<p_{T,LT}<$ 1.5 GeV/$c$. For visualization purposes the $\Delta\phi$ axis is not centered around 0. Shaded area in upper plots: systematic uncertainties. Shaded areas in bottom plots: sum in quadrature of statistical and systematic uncertainties. Horizontal error bars: bin width. Figure 13: Azimuthal correlation at $\sqrt{s}=0.9$ TeV (left) and $\sqrt{s}=7$ TeV (right). Leading-track: 2.0 $<p_{T,LT}<$ 4.0 GeV/$c$. For visualization purposes the $\Delta\phi$ axis is not centered around 0. Shaded area in upper plots: systematic uncertainties. Shaded areas in bottom plots: sum in quadrature of statistical and systematic uncertainties. Horizontal error bars: bin width. Figure 14: Azimuthal correlation at $\sqrt{s}=0.9$ TeV (left) and $\sqrt{s}=7$ TeV (right). Leading-track: 4.0 $<p_{T,LT}<$ 6.0 GeV/$c$. For visualization purposes the $\Delta\phi$ axis is not centered around 0. Shaded area in upper plots: systematic uncertainties. Shaded areas in bottom plots: sum in quadrature of statistical and systematic uncertainties. Horizontal error bars: bin width. Figure 15: Azimuthal correlation at $\sqrt{s}=0.9$ TeV (left) and $\sqrt{s}=7$ TeV (right). Leading-track: 6.0 $<p_{T,LT}<$ 10.0 GeV/$c$. For visualization purposes the $\Delta\phi$ axis is not centered around 0. Shaded area in upper plots: systematic uncertainties. Shaded areas in bottom plots: sum in quadrature of statistical and systematic uncertainties. Horizontal error bars: bin width. ## Azimuthal correlations - track $p_{\mathrm{T}}>0.5\,\mbox{${\rm GeV}/c$}$ Figure 16: Azimuthal correlation at $\sqrt{s}=0.9$ TeV (left) and $\sqrt{s}=7$ TeV (right). Leading-track: 0.5 $<p_{T,LT}<$ 1.5 GeV/$c$. For visualization purposes the $\Delta\phi$ axis is not centered around 0. Shaded area in upper plots: systematic uncertainties. Shaded areas in bottom plots: sum in quadrature of statistical and systematic uncertainties. Horizontal error bars: bin width. Figure 17: Azimuthal correlation at $\sqrt{s}=0.9$ TeV (left) and $\sqrt{s}=7$ TeV (right). Leading-track: 2.0 $<p_{T,LT}<$ 4.0 GeV/$c$. For visualization purposes the $\Delta\phi$ axis is not centered around 0. Shaded area in upper plots: systematic uncertainties. Shaded areas in bottom plots: sum in quadrature of statistical and systematic uncertainties. Horizontal error bars: bin width. Figure 18: Azimuthal correlation at $\sqrt{s}=0.9$ TeV (left) and $\sqrt{s}=7$ TeV (right). Leading-track: 4.0 $<p_{T,LT}<$ 6.0 GeV/$c$. For visualization purposes the $\Delta\phi$ axis is not centered around 0. Shaded area in upper plots: systematic uncertainties. Shaded areas in bottom plots: sum in quadrature of statistical and systematic uncertainties. Horizontal error bars: bin width. Figure 19: Azimuthal correlation at $\sqrt{s}=0.9$ TeV (left) and $\sqrt{s}=7$ TeV (right). Leading-track: 6.0 $<p_{T,LT}<$ 10.0 GeV/$c$. For visualization purposes the $\Delta\phi$ axis is not centered around 0. Shaded area in upper plots: systematic uncertainties. Shaded areas in bottom plots: sum in quadrature of statistical and systematic uncertainties. Horizontal error bars: bin width. ## Azimuthal correlations - track $p_{\mathrm{T}}>1.0\,\mbox{${\rm GeV}/c$}$ Figure 20: Azimuthal correlation at $\sqrt{s}=0.9$ TeV (left) and $\sqrt{s}=7$ TeV (right). Leading-track: 2.0 $<p_{T,LT}<$ 4.0 GeV/$c$. For visualization purposes the $\Delta\phi$ axis is not centered around 0. Shaded area in upper plots: systematic uncertainties. Shaded areas in bottom plots: sum in quadrature of statistical and systematic uncertainties. Horizontal error bars: bin width. Figure 21: Azimuthal correlation at $\sqrt{s}=0.9$ TeV (left) and $\sqrt{s}=7$ TeV (right). Leading-track: 4.0 $<p_{T,LT}<$ 6.0 GeV/$c$. For visualization purposes the $\Delta\phi$ axis is not centered around 0. Shaded area in upper plots: systematic uncertainties. Shaded areas in bottom plots: sum in quadrature of statistical and systematic uncertainties. Horizontal error bars: bin width. Figure 22: Azimuthal correlation at $\sqrt{s}=0.9$ TeV (left) and $\sqrt{s}=7$ TeV (right). Leading-track: 6.0 $<p_{T,LT}<$ 10.0 GeV/$c$. For visualization purposes the $\Delta\phi$ axis is not centered around 0. Shaded area in upper plots: systematic uncertainties. Shaded areas in bottom plots: sum in quadrature of statistical and systematic uncertainties. Horizontal error bars: bin width. ## 10 Acknowledgements The ALICE collaboration would like to thank all its engineers and technicians for their invaluable contributions to the construction of the experiment and the CERN accelerator teams for the outstanding performance of the LHC complex. The ALICE collaboration acknowledges the following funding agencies for their support in building and running the ALICE detector: Calouste Gulbenkian Foundation from Lisbon and Swiss Fonds Kidagan, Armenia; Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq), Financiadora de Estudos e Projetos (FINEP), Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP); National Natural Science Foundation of China (NSFC), the Chinese Ministry of Education (CMOE) and the Ministry of Science and Technology of China (MSTC); Ministry of Education and Youth of the Czech Republic; Danish Natural Science Research Council, the Carlsberg Foundation and the Danish National Research Foundation; The European Research Council under the European Community’s Seventh Framework Programme; Helsinki Institute of Physics and the Academy of Finland; French CNRS-IN2P3, the ‘Region Pays de Loire’, ‘Region Alsace’, ‘Region Auvergne’ and CEA, France; German BMBF and the Helmholtz Association; General Secretariat for Research and Technology, Ministry of Development, Greece; Hungarian OTKA and National Office for Research and Technology (NKTH); Department of Atomic Energy and Department of Science and Technology of the Government of India; Istituto Nazionale di Fisica Nucleare (INFN) of Italy; MEXT Grant-in-Aid for Specially Promoted Research, Japan; Joint Institute for Nuclear Research, Dubna; National Research Foundation of Korea (NRF); CONACYT, DGAPA, México, ALFA-EC and the HELEN Program (High-Energy physics Latin-American–European Network); Stichting voor Fundamenteel Onderzoek der Materie (FOM) and the Nederlandse Organisatie voor Wetenschappelijk Onderzoek (NWO), Netherlands; Research Council of Norway (NFR); Polish Ministry of Science and Higher Education; National Authority for Scientific Research - NASR (Autoritatea Naţională pentru Cercetare Ştiinţifică - ANCS); Federal Agency of Science of the Ministry of Education and Science of Russian Federation, International Science and Technology Center, Russian Academy of Sciences, Russian Federal Agency of Atomic Energy, Russian Federal Agency for Science and Innovations and CERN-INTAS; Ministry of Education of Slovakia; Department of Science and Technology, South Africa; CIEMAT, EELA, Ministerio de Educación y Ciencia of Spain, Xunta de Galicia (Consellería de Educación), CEADEN, Cubaenergía, Cuba, and IAEA (International Atomic Energy Agency); Swedish Reseach Council (VR) and Knut $\&$ Alice Wallenberg Foundation (KAW); Ukraine Ministry of Education and Science; United Kingdom Science and Technology Facilities Council (STFC); The United States Department of Energy, the United States National Science Foundation, the State of Texas, and the State of Ohio. ## References * [1] T. Sjostrand and P. Z. Skands, Multiple interactions and the structure of beam remnants, JHEP 03 (2004) 053, [hep-ph/0402078]. * [2] CDF Collaboration, A. A. Affolder et. al., Charged jet evolution and the underlying event in $p\bar{p}$ collisions at 1.8 TeV, Phys. Rev. D65 (2002) 092002. * [3] CDF Collaboration, D. E. Acosta et. al., The underlying event in hard interactions at the Tevatron $\bar{p}p$ collider, Phys. Rev. D70 (2004) 072002, [hep-ex/0404004]. * [4] CDF Collaboration, R. Field, Min-bias and the underlying event in Run 2 at CDF, Acta Phys. Polon. B36 (2005) 167–178. * [5] CDF Collaboration, T. Aaltonen et. al., Studying the Underlying Event in Drell-Yan and High Transverse Momentum Jet Production at the Tevatron, Phys. Rev. D82 (2010) 034001, [hep-ex/1003.3146]. * [6] H. Caines, Underlying Event Studies at RHIC, nucl-ex/0910.5203. eConf C090726. * [7] ATLAS Collaboration, G. Aad et. al., Measurement of underlying event characteristics using charged particles in pp collisions at $\sqrt{s}=900GeV$ and 7 TeV with the ATLAS detector, Phys. Rev. D 83 (2011) 112001, [hep-ex/1012.0791]. * [8] CMS Collaboration, V. Khachatryan et. al., First Measurement of the Underlying Event Activity at the LHC with $\sqrt{s}$ = 0.9 TeV, Eur. Phys. J. C70 (2010) 555–572, [hep-ex/1006.2083]. * [9] A. Buckley et. al., General-purpose event generators for LHC physics, Phys. Rept. 504 (2011) 145–233, [hep-ph/1101.2599]. * [10] T. Sjostrand, S. Mrenna, and P. Z. Skands, PYTHIA 6.4 Physics and Manual, JHEP 05 (2006) 026, [hep-ph/0603175]. * [11] S. Roesler, R. Engel, and J. Ranft, The Monte Carlo event generator DPMJET-III, Lisbon 2000 - Advanced Monte Carlo (2000) 1033–1038, [hep-ph/0012252]. * [12] M. Bahr et. al., Herwig++ Physics and Manual, Eur. Phys. J. C58 (2008) 639–707, [hep-ph/0803.0883]. * [13] B. Andersson, G. Gustafson, G. Ingelman, and T. Sjostrand, Parton fragmentation and string dynamics, Physics Reports 97 (1983), no. 2-3 31 – 145. * [14] T. Sjostrand and P. Z. Skands, Transverse-momentum-ordered showers and interleaved multiple interactions, Eur. Phys. J. C39 (2005) 129–154, [hep-ph/0408302]. * [15] T. Sjostrand, S. Mrenna, and P. Z. Skands, A Brief Introduction to PYTHIA 8.1, Comput. Phys. Commun. 178 (2008) 852–867, [hep-ph/0710.3820]. * [16] R. Corke and T. Sjostrand, Multiparton Interactions and Rescattering, JHEP 01 (2010) 035, [hep-ph/0911.1909]. * [17] A. Capella, U. Sukhatme, C.-I. Tan, and J. Tran Thanh Van, Dual parton model, Phys. Rept. 236 (1994) 225–329. * [18] ALICE Collaboration, K. Aamodt et. al., First proton-proton collisions at the LHC as observed with the ALICE detector: Measurement of the charged particle pseudorapidity density at s*(1/2) = 900-GeV, Eur. Phys. J. C65 (2010) 111–125, [hep-ex/0911.5430]. [19] ALICE Collaboration, K. Aamodt et. al., Charged-particle multiplicity measurement in proton-proton collisions at sqrt(s) = 0.9 and 2.36 TeV with ALICE at LHC, Eur. Phys. J. C68 (2010) 89–108, [hep-ex/1004.3034]. * [20] ALICE Collaboration, K. Aamodt et. al., Charged-particle multiplicity measurement in proton-proton collisions at sqrt(s) = 7 TeV with ALICE at LHC, Eur. Phys. J. C68 (2010) 345–354, [hep-ex/1004.3514]. * [21] ALICE Collaboration, A. K. Aamodt et. al., Midrapidity antiproton-to-proton ratio in pp collisions at $\sqrt{s}=0.9$ and $7$ TeV measured by the ALICE experiment, Phys. Rev. Lett. 105 (2010) 072002, [hep-ex/1006.5432]. * [22] ALICE Collaboration, K. Aamodt et. al., Two-pion Bose-Einstein correlations in central Pb-Pb collisions at $sqrt(s_{N}N)$ = 2.76 TeV, Phys. Lett. B696 (2011) 328–337, [nucl-ex/1012.4035]. * [23] ALICE Collaboration, K. Aamodt et. al., Transverse momentum spectra of charged particles in proton-proton collisions at $\sqrt{s}=900$ GeV with ALICE at the LHC, Phys. Lett. B693 (2010) 53–68, [hep-ex/1007.0719]. * [24] ALICE Collaboration, K. Aamodt et. al., Strange particle production in proton-proton collisions at sqrt(s) = 0.9 TeV with ALICE at the LHC, Eur.Phys.J. C71 (2011) 1594, [hep-ex/1012.3257]. * [25] ALICE Collaboration, K. Aamodt et. al., The ALICE experiment at the CERN LHC, JINST 3 (2008) S08002. * [26] P. Z. Skands, Tuning Monte Carlo Generators: The Perugia Tunes, Phys. Rev. D82 (2010) 074018, [hep-ph/1005.3457]. * [27] R. Brun, F. Bruyant, M. Maire, A. C. McPherson, and P. Zanarini, GEANT3, . CERN-DD-EE-84-1. * [28] ALICE Collaboration, K. Aamodt et. al., Alignment of the ALICE Inner Tracking System with cosmic-ray tracks, JINST 5 (2010) P03003. * [29] T. Sjöstrand and M. van Zijl, A multiple-interaction model for the event structure in hadron collisions, Phys. Rev. D 36 (Oct, 1987) 2019–2041. * [30] ATLAS Collaboration Collaboration, G. Aad et. al., Charged-particle multiplicities in pp interactions measured with the ATLAS detector at the LHC, New J.Phys. 13 (2011) 053033, [arXiv:1012.5104]. Long author list - awaiting processing. * [31] ATLAS Collaboration, G. Aad et. al., Measurement of underlying event characteristics using charged particles in pp collisions at sqrts =900 gev and 7 tev with the atlas detector in a limited phase space, ATLAS-CONF-2011-009 (Feb, 2011). * [32] CMS Collaboration, S. Chatrchyan et. al., Measurement of the Underlying Event Activity at the LHC with sqrt(s)= 7 TeV and Comparison with sqrt(s) = 0.9 TeV, hep-ex/1107.0330. CERN-PH-EP-2011-059. ## Appendix A The ALICE Collaboration B. Abelevorg1234&A. Abrahantes Quintanaorg1197&D. Adamováorg1283&A.M. Adareorg1260&M.M. Aggarwalorg1157&G. Aglieri Rinellaorg1192&A.G. Agocsorg1143&A. Agostinelliorg1132&S. Aguilar Salazarorg1247&Z. Ahammedorg1225&N. Ahmadorg1106&A. Ahmad Masoodiorg1106&S.U. Ahnorg1160,org1215&A. Akindinovorg1250&D. Aleksandrovorg1252&B. Alessandroorg1313&R. Alfaro Molinaorg1247&A. Aliciorg1133,org1192,org1335&A. Alkinorg1220&E. Almaráz Aviñaorg1247&T. Altorg1184&V. Altiniorg1114,org1192&S. Altinpinarorg1121&I. Altsybeevorg1306&C. Andreiorg1140&A. Andronicorg1176&V. Anguelovorg1200&C. Ansonorg1162&T. Antičićorg1334&F. Antinoriorg1271&P. Antonioliorg1133&L. Aphecetcheorg1258&H. Appelshäuserorg1185&N. Arbororg1194&S. Arcelliorg1132&A. Arendorg1185&N. Armestoorg1294&R. Arnaldiorg1313&T. Aronssonorg1260&I.C. Arseneorg1176&M. Arslandokorg1185&A. Asryanorg1306&A. Augustinusorg1192&R. Averbeckorg1176&T.C. Awesorg1264&J. Äystöorg1212&M.D. Azmiorg1106&M. Bachorg1184&A. Badalàorg1155&Y.W. Baekorg1160,org1215&R. Bailhacheorg1185&R. Balaorg1313&R. Baldini Ferroliorg1335&A. Baldisseriorg1288&A. Balditorg1160&F. Baltasar Dos Santos Pedrosaorg1192&J. Bánorg1230&R.C. Baralorg1127&R. Barberaorg1154&F. Barileorg1114&G.G. Barnaföldiorg1143&L.S. Barnbyorg1130&V. Barretorg1160&J. Bartkeorg1168&M. Basileorg1132&N. Bastidorg1160&B. Bathenorg1256&G. Batigneorg1258&B. Batyunyaorg1182&C. Baumannorg1185&I.G. Beardenorg1165&H. Beckorg1185&I. Belikovorg1308&F. Belliniorg1132&R. Bellwiedorg1205&E. Belmont- Morenoorg1247&S. Beoleorg1312&I. Berceanuorg1140&A. Bercuciorg1140&Y. Berdnikovorg1189&D. Berenyiorg1143&C. Bergmannorg1256&D. Berzanoorg1313&L. Betevorg1192&A. Bhasinorg1209&A.K. Bhatiorg1157&N. Bianchiorg1187&L. Bianchiorg1312&C. Bianchinorg1270&J. Bielčíkorg1274&J. Bielčíkováorg1283&A. Bilandzicorg1109&F. Blancoorg1242&F. Blancoorg1205&D. Blauorg1252&C. Blumeorg1185&M. Boccioliorg1192&N. Bockorg1162&A. Bogdanovorg1251&H. Bøggildorg1165&M. Bogolyubskyorg1277&L. Boldizsárorg1143&M. Bombaraorg1229&J. Bookorg1185&H. Borelorg1288&A. Borissovorg1179&C. Bortolinorg1270,Dipartimento di Fisica dell’Universita, Udine, Italy&S. Boseorg1224&F. Bossúorg1192,org1312&M. Botjeorg1109&S. Böttgerorg27399&B. Boyerorg1266&P. Braun-Munzingerorg1176&M. Bregantorg1258&T. Breitnerorg27399&M. Brozorg1136&R. Brunorg1192&E. Brunaorg1260,org1312,org1313&G.E. Brunoorg1114&D. Budnikovorg1298&H. Bueschingorg1185&S. Bufalinoorg1312,org1313&K. Bugaievorg1220&O. Buschorg1200&Z. Butheleziorg1152&D. Caffarriorg1270&X. Caiorg1329&H. Cainesorg1260&E. Calvo Villarorg1338&P. Cameriniorg1315&V. Canoa Romanorg1244,org1279&G. Cara Romeoorg1133&F. Carenaorg1192&W. Carenaorg1192&N. Carlin Filhoorg1296&F. Carminatiorg1192&C.A. Carrillo Montoyaorg1192&A. Casanova Díazorg1187&M. Caselleorg1192&J. Castillo Castellanosorg1288&J.F. Castillo Hernandezorg1176&E.A.R. Casulaorg1145&V. Catanescuorg1140&C. Cavicchioliorg1192&J. Cepilaorg1274&P. Cerelloorg1313&B. Changorg1212,org1301&S. Chapelandorg1192&J.L. Charvetorg1288&S. Chattopadhyayorg1224&S. Chattopadhyayorg1225&M. Cherneyorg1170&C. Cheshkovorg1192,org1239&B. Cheynisorg1239&E. Chiavassaorg1313&V. Chibante Barrosoorg1192&D.D. Chinellatoorg1149&P. Chochulaorg1192&M. Chojnackiorg1320&P. Christakoglouorg1109,org1320&C.H. Christensenorg1165&P. Christiansenorg1237&T. Chujoorg1318&S.U. Chungorg1281&C. Cicaloorg1146&L. Cifarelliorg1132,org1192&F. Cindoloorg1133&J. Cleymansorg1152&F. Coccettiorg1335&J.-P. Coffinorg1308&F. Colamariaorg1114&D. Colellaorg1114&G. Conesa Balbastreorg1194&Z. Conesa del Valleorg1192,org1308&P. Constantinorg1200&G. Continorg1315&J.G. Contrerasorg1244&T.M. Cormierorg1179&Y. Corrales Moralesorg1312&P. Corteseorg1103&I. Cortés Maldonadoorg1279&M.R. Cosentinoorg1125,org1149&F. Costaorg1192&M.E. Cotalloorg1242&E. Crescioorg1244&P. Crochetorg1160&E. Cruz Alanizorg1247&E. Cuautleorg1246&L. Cunqueiroorg1187&A. Daineseorg1270,org1271&H.H. Dalsgaardorg1165&A. Danuorg1139&I. Dasorg1224,org1266&K. Dasorg1224&D. Dasorg1224&A. Dashorg1127,org1149&S. Dashorg1254,org1313&S. Deorg1225&A. De Azevedo Moregulaorg1187&G.O.V. de Barrosorg1296&A. De Caroorg1290,org1335&G. de Cataldoorg1115&J. de Cuvelandorg1184&A. De Falcoorg1145&D. De Gruttolaorg1290&H. Delagrangeorg1258&E. Del Castillo Sanchezorg1192&A. Delofforg1322&V. Demanovorg1298&N. De Marcoorg1313&E. Dénesorg1143&S. De Pasqualeorg1290&A. Deppmanorg1296&G. D Erasmoorg1114&R. de Rooijorg1320&D. Di Bariorg1114&T. Dietelorg1256&C. Di Giglioorg1114&S. Di Libertoorg1286&A. Di Mauroorg1192&P. Di Nezzaorg1187&R. Diviàorg1192&Ø. Djuvslandorg1121&A. Dobrinorg1179,org1237&T. Dobrowolskiorg1322&I. Domínguezorg1246&B. Dönigusorg1176&O. Dordicorg1268&O. Drigaorg1258&A.K. Dubeyorg1225&L. Ducrouxorg1239&P. Dupieuxorg1160&M.R. Dutta Majumdarorg1225&A.K. Dutta Majumdarorg1224&D. Eliaorg1115&D. Emschermannorg1256&H. Engelorg27399&H.A. Erdalorg1122&B. Espagnonorg1266&M. Estienneorg1258&S. Esumiorg1318&D. Evansorg1130&G. Eyyubovaorg1268&D. Fabrisorg1270,org1271&J. Faivreorg1194&D. Falchieriorg1132&A. Fantoniorg1187&M. Faselorg1176&R. Fearickorg1152&A. Fedunovorg1182&D. Fehlkerorg1121&L. Feldkamporg1256&D. Feleaorg1139&G. Feofilovorg1306&A. Fernández Téllezorg1279&A. Ferrettiorg1312&R. Ferrettiorg1103&J. Figielorg1168&M.A.S. Figueredoorg1296&S. Filchaginorg1298&R. Finiorg1115&D. Finogeevorg1249&F.M. Fiondaorg1114&E.M. Fioreorg1114&M. Florisorg1192&S. Foertschorg1152&P. Fokaorg1176&S. Fokinorg1252&E. Fragiacomoorg1316&M. Fragkiadakisorg1112&U. Frankenfeldorg1176&U. Fuchsorg1192&C. Furgetorg1194&M. Fusco Girardorg1290&J.J. Gaardhøjeorg1165&M. Gagliardiorg1312&A. Gagoorg1338&M. Gallioorg1312&D.R. Gangadharanorg1162&P. Ganotiorg1264&C. Garabatosorg1176&E. Garcia-Solisorg17347&I. Garishviliorg1234&J. Gerhardorg1184&M. Germainorg1258&C. Geunaorg1288&A. Gheataorg1192&M. Gheataorg1192&B. Ghidiniorg1114&P. Ghoshorg1225&P. Gianottiorg1187&M.R. Girardorg1323&P. Giubellinoorg1192&E. Gladysz-Dziadusorg1168&P. Glässelorg1200&R. Gomezorg1173&E.G. Ferreiroorg1294&L.H. González-Truebaorg1247&P. González- Zamoraorg1242&S. Gorbunovorg1184&A. Goswamiorg1207&S. Gotovacorg1304&V. Grabskiorg1247&L.K. Graczykowskiorg1323&R. Grajcarekorg1200&A. Grelliorg1320&C. Grigorasorg1192&A. Grigorasorg1192&V. Grigorievorg1251&A. Grigoryanorg1332&S. Grigoryanorg1182&B. Grinyovorg1220&N. Grionorg1316&P. Grosorg1237&J.F. Grosse-Oetringhausorg1192&J.-Y. Grossiordorg1239&R. Grossoorg1192&F. Guberorg1249&R. Guernaneorg1194&C. Guerra Gutierrezorg1338&B. Guerzoniorg1132&M. Guilbaudorg1239&K. Gulbrandsenorg1165&T. Gunjiorg1310&A. Guptaorg1209&R. Guptaorg1209&H. Gutbrodorg1176&Ø. Haalandorg1121&C. Hadjidakisorg1266&M. Haiducorg1139&H. Hamagakiorg1310&G. Hamarorg1143&B.H. Hanorg1300&L.D. Hanrattyorg1130&A. Hansenorg1165&Z. Harmanovaorg1229&J.W. Harrisorg1260&M. Hartigorg1185&D. Haseganorg1139&D. Hatzifotiadouorg1133&A. Hayrapetyanorg1192,org1332&S.T. Heckelorg1185&M. Heideorg1256&H. Helstruporg1122&A. Herghelegiuorg1140&G. Herrera Corralorg1244&N. Herrmannorg1200&K.F. Hetlandorg1122&B. Hicksorg1260&P.T. Hilleorg1260&B. Hippolyteorg1308&T. Horaguchiorg1318&Y. Horiorg1310&P. Hristovorg1192&I. Hřivnáčováorg1266&M. Huangorg1121&S. Huberorg1176&T.J. Humanicorg1162&D.S. Hwangorg1300&R. Ichouorg1160&R. Ilkaevorg1298&I. Ilkivorg1322&M. Inabaorg1318&E. Incaniorg1145&P.G. Innocentiorg1192&G.M. Innocentiorg1312&M. Ippolitovorg1252&M. Irfanorg1106&C. Ivanorg1176&M. Ivanovorg1176&V. Ivanovorg1189&A. Ivanovorg1306&O. Ivanytskyiorg1220&A. Jachołkowskiorg1192&P. M. Jacobsorg1125&L. Jancurováorg1182&H.J. Jangorg20954&S. Jangalorg1308&R. Janikorg1136&M.A. Janikorg1323&P.H.S.Y. Jayarathnaorg1205&S. Jenaorg1254&R.T. Jimenez Bustamanteorg1246&L. Jirdenorg1192&P.G. Jonesorg1130&W. Jungorg1215&H. Jungorg1215&A. Juskoorg1130&A.B. Kaidalovorg1250&V. Kakoyanorg1332&S. Kalcherorg1184&P. Kaliňákorg1230&M. Kaliskyorg1256&T. Kalliokoskiorg1212&A. Kalweitorg1177&K. Kanakiorg1121&J.H. Kangorg1301&V. Kaplinorg1251&A. Karasu Uysalorg1192,org15649&O. Karavichevorg1249&T. Karavichevaorg1249&E. Karpechevorg1249&A. Kazantsevorg1252&U. Kebschullorg1199,org27399&R. Keidelorg1327&P. Khanorg1224&M.M. Khanorg1106&S.A. Khanorg1225&A. Khanzadeevorg1189&Y. Kharlovorg1277&B. Kilengorg1122&J.H. Kimorg1300&D.J. Kimorg1212&D.W. Kimorg1215&J.S. Kimorg1215&M. Kimorg1301&S.H. Kimorg1215&S. Kimorg1300&B. Kimorg1301&T. Kimorg1301&S. Kirschorg1184,org1192&I. Kiselorg1184&S. Kiselevorg1250&A. Kisielorg1192,org1323&J.L. Klayorg1292&J. Kleinorg1200&C. Klein-Bösingorg1256&M. Kliemantorg1185&A. Klugeorg1192&M.L. Knichelorg1176&K. Kochorg1200&M.K. Köhlerorg1176&A. Kolojvariorg1306&V. Kondratievorg1306&N. Kondratyevaorg1251&A. Konevskikhorg1249&A. Korneevorg1298&C. Kottachchi Kankanamge Donorg1179&R. Kourorg1130&M. Kowalskiorg1168&S. Koxorg1194&G. Koyithatta Meethaleveeduorg1254&J. Kralorg1212&I. Králikorg1230&F. Kramerorg1185&I. Krausorg1176&T. Krawutschkeorg1200,org1227&M. Kretzorg1184&M. Krivdaorg1130,org1230&F. Krizekorg1212&M. Krusorg1274&E. Kryshenorg1189&M. Krzewickiorg1109,org1176&Y. Kucheriaevorg1252&C. Kuhnorg1308&P.G. Kuijerorg1109&P. Kurashviliorg1322&A.B. Kurepinorg1249&A. Kurepinorg1249&A. Kuryakinorg1298&S. Kushpilorg1283&V. Kushpilorg1283&H. Kvaernoorg1268&M.J. Kweonorg1200&Y. Kwonorg1301&P. Ladrón de Guevaraorg1246&I. Lakomovorg1266,org1306&R. Langoyorg1121&C. Laraorg27399&A. Lardeuxorg1258&P. La Roccaorg1154&C. Lazzeroniorg1130&R. Leaorg1315&Y. Le Bornecorg1266&S.C. Leeorg1215&K.S. Leeorg1215&F. Lefèvreorg1258&J. Lehnertorg1185&L. Leistamorg1192&M. Lenhardtorg1258&V. Lentiorg1115&H. Leónorg1247&I. León Monzónorg1173&H. León Vargasorg1185&P. Lévaiorg1143&X. Liorg1118&J. Lienorg1121&R. Lietavaorg1130&S. Lindalorg1268&V. Lindenstruthorg1184&C. Lippmannorg1176,org1192&M.A. Lisaorg1162&L. Liuorg1121&P.I. Loenneorg1121&V.R. Logginsorg1179&V. Loginovorg1251&S. Lohnorg1192&D. Lohnerorg1200&C. Loizidesorg1125&K.K. Looorg1212&X. Lopezorg1160&E. López Torresorg1197&G. Løvhøidenorg1268&X.-G. Luorg1200&P. Luettigorg1185&M. Lunardonorg1270&J. Luoorg1329&G. Luparelloorg1320&L. Luquinorg1258&C. Luzziorg1192&R. Maorg1260&K. Maorg1329&D.M. Madagodahettige- Donorg1205&A. Maevskayaorg1249&M. Magerorg1177,org1192&D.P. Mahapatraorg1127&A. Maireorg1308&M. Malaevorg1189&I. Maldonado Cervantesorg1246&L. Malininaorg1182,M.V.Lomonosov Moscow State University, D.V.Skobeltsyn Institute of Nuclear Physics, Moscow, Russia&D. Mal’Kevichorg1250&P. Malzacherorg1176&A. Mamonovorg1298&L. Manceauorg1313&L. Mangotraorg1209&V. Mankoorg1252&F. Mansoorg1160&V. Manzariorg1115&Y. Maoorg1194,org1329&M. Marchisoneorg1160,org1312&J. Marešorg1275&G.V. Margagliottiorg1315,org1316&A. Margottiorg1133&A. Marínorg1176&C. Markertorg17361&I. Martashviliorg1222&P. Martinengoorg1192&M.I. Martínezorg1279&A. Martínez Davalosorg1247&G. Martínez Garcíaorg1258&Y. Martynovorg1220&A. Masorg1258&S. Masciocchiorg1176&M. Maseraorg1312&A. Masoniorg1146&L. Massacrierorg1239&M. Mastromarcoorg1115&A. Mastroserioorg1114,org1192&Z.L. Matthewsorg1130&A. Matyjaorg1258&D. Mayaniorg1246&C. Mayerorg1168&J. Mazerorg1222&M.A. Mazzoniorg1286&F. Meddiorg1285&A. Menchaca-Rochaorg1247&J. Mercado Pérezorg1200&M. Meresorg1136&Y. Miakeorg1318&A. Michalonorg1308&J. Midoriorg1203&L. Milanoorg1312&J. Milosevicorg1268,Institute of Nuclear Sciences, Belgrade, Serbia&A. Mischkeorg1320&A.N. Mishraorg1207&D. Miśkowiecorg1176,org1192&C. Mituorg1139&J. Mlynarzorg1179&A.K. Mohantyorg1192&B. Mohantyorg1225&L. Molnarorg1192&L. Montaño Zetinaorg1244&M. Montenoorg1313&E. Montesorg1242&T. Moonorg1301&M. Morandoorg1270&D.A. Moreira De Godoyorg1296&S. Morettoorg1270&A. Morschorg1192&V. Mucciforaorg1187&E. Mudnicorg1304&S. Muhuriorg1225&H. Müllerorg1192&M.G. Munhozorg1296&L. Musaorg1192&A. Mussoorg1313&B.K. Nandiorg1254&R. Naniaorg1133&E. Nappiorg1115&C. Nattrassorg1222&N.P. Naumovorg1298&S. Navinorg1130&T.K. Nayakorg1225&S. Nazarenkoorg1298&G. Nazarovorg1298&A. Nedosekinorg1250&M. Nicassioorg1114&B.S. Nielsenorg1165&T. Niidaorg1318&S. Nikolaevorg1252&V. Nikolicorg1334&V. Nikulinorg1189&S. Nikulinorg1252&B.S. Nilsenorg1170&M.S. Nilssonorg1268&F. Noferiniorg1133,org1335&P. Nomokonovorg1182&G. Noorenorg1320&N. Novitzkyorg1212&A. Nyaninorg1252&A. Nyathaorg1254&C. Nygaardorg1165&J. Nystrandorg1121&H. Obayashiorg1203&A. Ochirovorg1306&H. Oeschlerorg1177,org1192&S.K. Ohorg1215&S. Ohorg1260&J. Oleniaczorg1323&C. Oppedisanoorg1313&A. Ortiz Velasquezorg1246&G. Ortonaorg1192,org1312&A. Oskarssonorg1237&P. Ostrowskiorg1323&I. Otterlundorg1237&J. Otwinowskiorg1176&K. Oyamaorg1200&K. Ozawaorg1310&Y. Pachmayerorg1200&M. Pachrorg1274&F. Padillaorg1312&P. Paganoorg1290&G. Paićorg1246&F. Painkeorg1184&C. Pajaresorg1294&S. Palorg1288&S.K. Palorg1225&A. Palahaorg1130&A. Palmeriorg1155&V. Papikyanorg1332&G.S. Pappalardoorg1155&W.J. Parkorg1176&A. Passfeldorg1256&B. Pastirčákorg1230&D.I. Patalakhaorg1277&V. Paticchioorg1115&A. Pavlinovorg1179&T. Pawlakorg1323&T. Peitzmannorg1320&M. Peralesorg17347&E. Pereira De Oliveira Filhoorg1296&D. Peresunkoorg1252&C.E. Pérez Laraorg1109&E. Perez Lezamaorg1246&D. Periniorg1192&D. Perrinoorg1114&W. Perytorg1323&A. Pesciorg1133&V. Peskovorg1192,org1246&Y. Pestovorg1262&V. Petráčekorg1274&M. Petranorg1274&M. Petrisorg1140&P. Petrovorg1130&M. Petroviciorg1140&C. Pettaorg1154&S. Pianoorg1316&A. Piccottiorg1313&M. Piknaorg1136&P. Pillotorg1258&O. Pinazzaorg1192&L. Pinskyorg1205&N. Pitzorg1185&F. Piuzorg1192&D.B. Piyarathnaorg1205&M. Płoskońorg1125&J. Plutaorg1323&T. Pocheptsovorg1182,org1268&S. Pochybovaorg1143&P.L.M. Podesta- Lermaorg1173&M.G. Poghosyanorg1192,org1312&K. Polákorg1275&B. Polichtchoukorg1277&A. Poporg1140&S. Porteboeuf-Houssaisorg1160&V. Pospíšilorg1274&B. Potukuchiorg1209&S.K. Prasadorg1179&R. Preghenellaorg1133,org1335&F. Prinoorg1313&C.A. Pruneauorg1179&I. Pshenichnovorg1249&S. Puchaginorg1298&G. Pudduorg1145&A. Pulvirentiorg1154,org1192&V. Puninorg1298&M. Putišorg1229&J. Putschkeorg1179,org1260&E. Quercighorg1192&H. Qvigstadorg1268&A. Rachevskiorg1316&A. Rademakersorg1192&S. Radomskiorg1200&T.S. Räihäorg1212&J. Rakorg1212&A. Rakotozafindrabeorg1288&L. Ramelloorg1103&A. Ramírez Reyesorg1244&S. Raniwalaorg1207&R. Raniwalaorg1207&S.S. Räsänenorg1212&B.T. Rascanuorg1185&D. Ratheeorg1157&K.F. Readorg1222&J.S. Realorg1194&K. Redlichorg1322,org23333&P. Reicheltorg1185&M. Reicherorg1320&R. Renfordtorg1185&A.R. Reolonorg1187&A. Reshetinorg1249&F. Rettigorg1184&J.-P. Revolorg1192&K. Reygersorg1200&L. Riccatiorg1313&R.A. Ricciorg1232&M. Richterorg1268&P. Riedlerorg1192&W. Rieglerorg1192&F. Riggiorg1154,org1155&M. Rodríguez Cahuantziorg1279&D. Rohrorg1184&D. Röhrichorg1121&R. Romitaorg1176&F. Ronchettiorg1187&P. Rosnetorg1160&S. Rosseggerorg1192&A. Rossiorg1270&F. Roukoutakisorg1112&P. Royorg1224&C. Royorg1308&A.J. Rubio Monteroorg1242&R. Ruiorg1315&E. Ryabinkinorg1252&A. Rybickiorg1168&S. Sadovskyorg1277&K. Šafaříkorg1192&P.K. Sahuorg1127&J. Sainiorg1225&H. Sakaguchiorg1203&S. Sakaiorg1125&D. Sakataorg1318&C.A. Salgadoorg1294&S. Sambyalorg1209&V. Samsonovorg1189&X. Sanchez Castroorg1246,org1308&L. Šándororg1230&A. Sandovalorg1247&M. Sanoorg1318&S. Sanoorg1310&R. Santoorg1256&R. Santoroorg1115,org1192&J. Sarkamoorg1212&E. Scapparoneorg1133&F. Scarlassaraorg1270&R.P. Scharenbergorg1325&C. Schiauaorg1140&R. Schickerorg1200&C. Schmidtorg1176&H.R. Schmidtorg1176,org21360&S. Schreinerorg1192&S. Schuchmannorg1185&J. Schukraftorg1192&Y. Schutzorg1192,org1258&K. Schwarzorg1176&K. Schwedaorg1176,org1200&G. Scioliorg1132&E. Scomparinorg1313&R. Scottorg1222&P.A. Scottorg1130&G. Segatoorg1270&I. Selyuzhenkovorg1176&S. Senyukovorg1103,org1308&J. Seoorg1281&S. Serciorg1145&E. Serradillaorg1242,org1247&A. Sevcencoorg1139&I. Sguraorg1115&A. Shabetaiorg1258&G. Shabratovaorg1182&R. Shahoyanorg1192&N. Sharmaorg1157&S. Sharmaorg1209&K. Shigakiorg1203&M. Shimomuraorg1318&K. Shtejerorg1197&Y. Sibiriakorg1252&M. Sicilianoorg1312&E. Sickingorg1192&S. Siddhantaorg1146&T. Siemiarczukorg1322&D. Silvermyrorg1264&G. Simonettiorg1114,org1192&R. Singarajuorg1225&R. Singhorg1209&S. Singhaorg1225&B.C. Sinhaorg1225&T. Sinhaorg1224&B. Sitarorg1136&M. Sittaorg1103&T.B. Skaaliorg1268&K. Skjerdalorg1121&R. Smakalorg1274&N. Smirnovorg1260&R. Snellingsorg1320&C. Søgaardorg1165&R. Soltzorg1234&H. Sonorg1300&J. Songorg1281&M. Songorg1301&C. Soosorg1192&F. Soramelorg1270&I. Sputowskaorg1168&M. Spyropoulou- Stassinakiorg1112&B.K. Srivastavaorg1325&J. Stachelorg1200&I. Stanorg1139&I. Stanorg1139&G. Stefanekorg1322&G. Stefaniniorg1192&T. Steinbeckorg1184&M. Steinpreisorg1162&E. Stenlundorg1237&G. Steynorg1152&D. Stoccoorg1258&M. Stolpovskiyorg1277&K. Strabykinorg1298&P. Strmenorg1136&A.A.P. Suaideorg1296&M.A. Subieta Vásquezorg1312&T. Sugitateorg1203&C. Suireorg1266&M. Sukhorukovorg1298&R. Sultanovorg1250&M. Šumberaorg1283&T. Susaorg1334&A. Szanto de Toledoorg1296&I. Szarkaorg1136&A. Szostakorg1121&C. Tagridisorg1112&J. Takahashiorg1149&J.D. Tapia Takakiorg1266&A. Tauroorg1192&G. Tejeda Muñozorg1279&A. Telescaorg1192&C. Terrevoliorg1114&J. Thäderorg1176&J.H. Thomasorg1176&D. Thomasorg1320&R. Tieulentorg1239&A.R. Timminsorg1205&D. Tlustyorg1274&A. Toiaorg1184,org1192&H. Toriiorg1203,org1310&L. Toscanoorg1313&F. Toselloorg1313&T. Traczykorg1323&D. Truesdaleorg1162&W.H. Trzaskaorg1212&T. Tsujiorg1310&A. Tumkinorg1298&R. Turrisiorg1271&T.S. Tveterorg1268&J. Uleryorg1185&K. Ullalandorg1121&J. Ulrichorg1199,org27399&A. Urasorg1239&J. Urbánorg1229&G.M. Urciuoliorg1286&G.L. Usaiorg1145&M. Vajzerorg1274,org1283&M. Valaorg1182,org1230&L. Valencia Palomoorg1266&S. Valleroorg1200&N. van der Kolkorg1109&P. Vande Vyvreorg1192&M. van Leeuwenorg1320&L. Vannucciorg1232&A. Vargasorg1279&R. Varmaorg1254&M. Vasileiouorg1112&A. Vasilievorg1252&V. Vecherninorg1306&M. Veldhoenorg1320&M. Venaruzzoorg1315&E. Vercellinorg1312&S. Vergaraorg1279&D.C. Vernekohlorg1256&R. Vernetorg14939&M. Verweijorg1320&L. Vickovicorg1304&G. Viestiorg1270&O. Vikhlyantsevorg1298&Z. Vilakaziorg1152&O. Villalobos Baillieorg1130&L. Vinogradovorg1306&Y. Vinogradovorg1298&A. Vinogradovorg1252&T. Virgiliorg1290&Y.P. Viyogiorg1225&A. Vodopyanovorg1182&S. Voloshinorg1179&K. Voloshinorg1250&G. Volpeorg1114,org1192&B. von Hallerorg1192&D. Vranicorg1176&G. Øvrebekkorg1121&J. Vrlákováorg1229&B. Vulpescuorg1160&A. Vyushinorg1298&B. Wagnerorg1121&V. Wagnerorg1274&R. Wanorg1308,org1329&Y. Wangorg1200&M. Wangorg1329&D. Wangorg1329&Y. Wangorg1329&K. Watanabeorg1318&J.P. Wesselsorg1192,org1256&U. Westerhofforg1256&J. Wiechulaorg1200,org21360&J. Wikneorg1268&M. Wildeorg1256&G. Wilkorg1322&A. Wilkorg1256&M.C.S. Williamsorg1133&B. Windelbandorg1200&L. Xaplanteris Karampatsosorg17361&H. Yangorg1288&S. Yangorg1121&S. Yanoorg1203&S. Yasnopolskiyorg1252&J. Yiorg1281&Z. Yinorg1329&H. Yokoyamaorg1318&I.-K. Yooorg1281&J. Yoonorg1301&W. Yuorg1185&X. Yuanorg1329&I. Yushmanovorg1252&C. Zachorg1274&C. Zampolliorg1133,org1192&S. Zaporozhetsorg1182&A. Zarochentsevorg1306&P. Závadaorg1275&N. Zaviyalovorg1298&H. Zbroszczykorg1323&P. Zelnicekorg1192,org27399&I.S. Zguraorg1139&M. Zhalovorg1189&X. Zhangorg1160,org1329&F. Zhouorg1329&Y. Zhouorg1320&D. Zhouorg1329&X. Zhuorg1329&A. Zichichiorg1132,org1335&A. Zimmermannorg1200&G. Zinovjevorg1220&Y. Zoccaratoorg1239&M. Zynovyevorg1220 ## Affiliation notes 0Deceased Dipartimento di Fisica dell’Universita, Udine, ItalyAlso at: Dipartimento di Fisica dell’Universita, Udine, Italy M.V.Lomonosov Moscow State University, D.V.Skobeltsyn Institute of Nuclear Physics, Moscow, RussiaAlso at: M.V.Lomonosov Moscow State University, D.V.Skobeltsyn Institute of Nuclear Physics, Moscow, Russia Institute of Nuclear Sciences, Belgrade, SerbiaAlso at: ”Vinča” Institute of Nuclear Sciences, Belgrade, Serbia ## Collaboration Institutes org1279Benemérita Universidad Autónoma de Puebla, Puebla, Mexico org1220Bogolyubov Institute for Theoretical Physics, Kiev, Ukraine org1262Budker Institute for Nuclear Physics, Novosibirsk, Russia org1292California Polytechnic State University, San Luis Obispo, California, United States org14939Centre de Calcul de l’IN2P3, Villeurbanne, France org1197Centro de Aplicaciones Tecnológicas y Desarrollo Nuclear (CEADEN), Havana, Cuba org1242Centro de Investigaciones Energéticas Medioambientales y Tecnológicas (CIEMAT), Madrid, Spain org1244Centro de Investigación y de Estudios Avanzados (CINVESTAV), Mexico City and Mérida, Mexico org1335Centro Fermi – Centro Studi e Ricerche e Museo Storico della Fisica “Enrico Fermi”, Rome, Italy org17347Chicago State University, Chicago, United States org1118China Institute of Atomic Energy, Beijing, China org1288Commissariat à l’Energie Atomique, IRFU, Saclay, France org1294Departamento de Física de Partículas and IGFAE, Universidad de Santiago de Compostela, Santiago de Compostela, Spain org1106Department of Physics Aligarh Muslim University, Aligarh, India org1121Department of Physics and Technology, University of Bergen, Bergen, Norway org1162Department of Physics, Ohio State University, Columbus, Ohio, United States org1300Department of Physics, Sejong University, Seoul, South Korea org1268Department of Physics, University of Oslo, Oslo, Norway org1132Dipartimento di Fisica dell’Università and Sezione INFN, Bologna, Italy org1315Dipartimento di Fisica dell’Università and Sezione INFN, Trieste, Italy org1145Dipartimento di Fisica dell’Università and Sezione INFN, Cagliari, Italy org1270Dipartimento di Fisica dell’Università and Sezione INFN, Padova, Italy org1285Dipartimento di Fisica dell’Università ‘La Sapienza’ and Sezione INFN, Rome, Italy org1154Dipartimento di Fisica e Astronomia dell’Università and Sezione INFN, Catania, Italy org1290Dipartimento di Fisica ‘E.R. Caianiello’ dell’Università and Gruppo Collegato INFN, Salerno, Italy org1312Dipartimento di Fisica Sperimentale dell’Università and Sezione INFN, Turin, Italy org1103Dipartimento di Scienze e Tecnologie Avanzate dell’Università del Piemonte Orientale and Gruppo Collegato INFN, Alessandria, Italy org1114Dipartimento Interateneo di Fisica ‘M. Merlin’ and Sezione INFN, Bari, Italy org1237Division of Experimental High Energy Physics, University of Lund, Lund, Sweden org1192European Organization for Nuclear Research (CERN), Geneva, Switzerland org1227Fachhochschule Köln, Köln, Germany org1122Faculty of Engineering, Bergen University College, Bergen, Norway org1136Faculty of Mathematics, Physics and Informatics, Comenius University, Bratislava, Slovakia org1274Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague, Prague, Czech Republic org1229Faculty of Science, P.J. Šafárik University, Košice, Slovakia org1184Frankfurt Institute for Advanced Studies, Johann Wolfgang Goethe- Universität Frankfurt, Frankfurt, Germany org1215Gangneung-Wonju National University, Gangneung, South Korea org1212Helsinki Institute of Physics (HIP) and University of Jyväskylä, Jyväskylä, Finland org1203Hiroshima University, Hiroshima, Japan org1329Hua-Zhong Normal University, Wuhan, China org1254Indian Institute of Technology, Mumbai, India org1266Institut de Physique Nucléaire d’Orsay (IPNO), Université Paris-Sud, CNRS-IN2P3, Orsay, France org1277Institute for High Energy Physics, Protvino, Russia org1249Institute for Nuclear Research, Academy of Sciences, Moscow, Russia org1320Nikhef, National Institute for Subatomic Physics and Institute for Subatomic Physics of Utrecht University, Utrecht, Netherlands org1250Institute for Theoretical and Experimental Physics, Moscow, Russia org1230Institute of Experimental Physics, Slovak Academy of Sciences, Košice, Slovakia org1127Institute of Physics, Bhubaneswar, India org1275Institute of Physics, Academy of Sciences of the Czech Republic, Prague, Czech Republic org1139Institute of Space Sciences (ISS), Bucharest, Romania org27399Institut für Informatik, Johann Wolfgang Goethe-Universität Frankfurt, Frankfurt, Germany org1185Institut für Kernphysik, Johann Wolfgang Goethe-Universität Frankfurt, Frankfurt, Germany org1177Institut für Kernphysik, Technische Universität Darmstadt, Darmstadt, Germany org1256Institut für Kernphysik, Westfälische Wilhelms-Universität Münster, Münster, Germany org1246Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México, Mexico City, Mexico org1247Instituto de Física, Universidad Nacional Autónoma de México, Mexico City, Mexico org23333Institut of Theoretical Physics, University of Wroclaw org1308Institut Pluridisciplinaire Hubert Curien (IPHC), Université de Strasbourg, CNRS-IN2P3, Strasbourg, France org1182Joint Institute for Nuclear Research (JINR), Dubna, Russia org1143KFKI Research Institute for Particle and Nuclear Physics, Hungarian Academy of Sciences, Budapest, Hungary org18995Kharkiv Institute of Physics and Technology (KIPT), National Academy of Sciences of Ukraine (NASU), Kharkov, Ukraine org1199Kirchhoff-Institut für Physik, Ruprecht-Karls-Universität Heidelberg, Heidelberg, Germany org20954Korea Institute of Science and Technology Information org1160Laboratoire de Physique Corpusculaire (LPC), Clermont Université, Université Blaise Pascal, CNRS–IN2P3, Clermont-Ferrand, France org1194Laboratoire de Physique Subatomique et de Cosmologie (LPSC), Université Joseph Fourier, CNRS-IN2P3, Institut Polytechnique de Grenoble, Grenoble, France org1187Laboratori Nazionali di Frascati, INFN, Frascati, Italy org1232Laboratori Nazionali di Legnaro, INFN, Legnaro, Italy org1125Lawrence Berkeley National Laboratory, Berkeley, California, United States org1234Lawrence Livermore National Laboratory, Livermore, California, United States org1251Moscow Engineering Physics Institute, Moscow, Russia org1140National Institute for Physics and Nuclear Engineering, Bucharest, Romania org1165Niels Bohr Institute, University of Copenhagen, Copenhagen, Denmark org1109Nikhef, National Institute for Subatomic Physics, Amsterdam, Netherlands org1283Nuclear Physics Institute, Academy of Sciences of the Czech Republic, Řež u Prahy, Czech Republic org1264Oak Ridge National Laboratory, Oak Ridge, Tennessee, United States org1189Petersburg Nuclear Physics Institute, Gatchina, Russia org1170Physics Department, Creighton University, Omaha, Nebraska, United States org1157Physics Department, Panjab University, Chandigarh, India org1112Physics Department, University of Athens, Athens, Greece org1152Physics Department, University of Cape Town, iThemba LABS, Cape Town, South Africa org1209Physics Department, University of Jammu, Jammu, India org1207Physics Department, University of Rajasthan, Jaipur, India org1200Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg, Germany org1325Purdue University, West Lafayette, Indiana, United States org1281Pusan National University, Pusan, South Korea org1176Research Division and ExtreMe Matter Institute EMMI, GSI Helmholtzzentrum für Schwerionenforschung, Darmstadt, Germany org1334Rudjer Bošković Institute, Zagreb, Croatia org1298Russian Federal Nuclear Center (VNIIEF), Sarov, Russia org1252Russian Research Centre Kurchatov Institute, Moscow, Russia org1224Saha Institute of Nuclear Physics, Kolkata, India org1130School of Physics and Astronomy, University of Birmingham, Birmingham, United Kingdom org1338Sección Física, Departamento de Ciencias, Pontificia Universidad Católica del Perú, Lima, Peru org1146Sezione INFN, Cagliari, Italy org1115Sezione INFN, Bari, Italy org1313Sezione INFN, Turin, Italy org1133Sezione INFN, Bologna, Italy org1155Sezione INFN, Catania, Italy org1316Sezione INFN, Trieste, Italy org1286Sezione INFN, Rome, Italy org1271Sezione INFN, Padova, Italy org1322Soltan Institute for Nuclear Studies, Warsaw, Poland org1258SUBATECH, Ecole des Mines de Nantes, Université de Nantes, CNRS-IN2P3, Nantes, France org1304Technical University of Split FESB, Split, Croatia org1168The Henryk Niewodniczanski Institute of Nuclear Physics, Polish Academy of Sciences, Cracow, Poland org17361The University of Texas at Austin, Physics Department, Austin, TX, United States org1173Universidad Autónoma de Sinaloa, Culiacán, Mexico org1296Universidade de São Paulo (USP), São Paulo, Brazil org1149Universidade Estadual de Campinas (UNICAMP), Campinas, Brazil org1239Université de Lyon, Université Lyon 1, CNRS/IN2P3, IPN-Lyon, Villeurbanne, France org1205University of Houston, Houston, Texas, United States org20371University of Technology and Austrian Academy of Sciences, Vienna, Austria org1222University of Tennessee, Knoxville, Tennessee, United States org1310University of Tokyo, Tokyo, Japan org1318University of Tsukuba, Tsukuba, Japan org21360Eberhard Karls Universität Tübingen, Tübingen, Germany org1225Variable Energy Cyclotron Centre, Kolkata, India org1306V. Fock Institute for Physics, St. Petersburg State University, St. Petersburg, Russia org1323Warsaw University of Technology, Warsaw, Poland org1179Wayne State University, Detroit, Michigan, United States org1260Yale University, New Haven, Connecticut, United States org1332Yerevan Physics Institute, Yerevan, Armenia org15649Yildiz Technical University, Istanbul, Turkey org1301Yonsei University, Seoul, South Korea org1327Zentrum für Technologietransfer und Telekommunikation (ZTT), Fachhochschule Worms, Worms, Germany	arxiv-papers	2011-12-09T12:18:32	2024-09-04T02:49:25.126448	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/", "authors": "ALICE collaboration", "submitter": "Alice Publications", "url": "https://arxiv.org/abs/1112.2082" }
1112.2160	On the role of enrichment and statical admissibility of recovered fields in a-posteriori error estimation for enriched finite element methods Octavio A. González-Estrada1, Juan José Ródenas2, Stéphane P.A. Bordas1, Marc Duflot3, Pierre Kerfriden1, Eugenio Giner2 1Institute of Mechanics and Advanced Materials. Cardiff School of Engineering, Cardiff University, The Parade, Cardiff CF24 3AA Wales, UK. 2Centro de Investigación de Tecnología de Vehículos(CITV), Universitad Politècnica de València, E-46022-Valencia, Spain. 3CENAERO, Rue des Frères Wright 29, B-6041 Gosselies, Belgium ###### Abstract Purpose – This paper aims at assessing the effect of (1) the statical admissibility of the recovered solution; (2) the ability of the recovered solution to represent the singular solution; on the accuracy, local and global effectivity of recovery-based error estimators for enriched finite element methods (e.g. the extended finite element method, XFEM). Design/methodology/approach – We study the performance of two recovery techniques. The first is a recently developed superconvergent patch recovery procedure with equilibration and enrichment (SPR-CX). The second is known as the extended moving least squares recovery (XMLS), which enriches the recovered solutions but does not enforce equilibrium constraints. Both are extended recovery techniques as the polynomial basis used in the recovery process is enriched with singular terms for a better description of the singular nature of the solution. Findings – Numerical results comparing the convergence and the effectivity index of both techniques with those obtained without the enrichment enhancement clearly show the need for the use of extended recovery techniques in Zienkiewicz-Zhu type error estimators for this class of problems. The results also reveal significant improvements in the effectivities yielded by statically admissible recovered solutions. Originality/value – This work shows that both extended recovery procedures and statical admissibility are key to an accurate assessment of the quality of enriched finite element approximations. Keywords extended finite element method; error estimation; linear elastic fracture mechanics; statical admissibility; extended recovery Paper Type Research Paper ## 1 Introduction Engineering structures, in particular in aerospace engineering, are intended to operate with flawless components, especially for safety critical parts. However, there is always a possibility that cracking will occur during operation, risking catastrophic failure and associated casualties. The mission of Damage Tolerance Assessment (DTA) is to assess the influence of these defects, cracks and damage on the ability of a structure to perform safely and reliably during its service life. An important goal of DTA is to estimate the fatigue life of a structure, i.e. the time during which it remains safe given a pre-existing flaw. Damage tolerance assessment relies on the ability to accurately predict crack paths and growth rates in complex structures. Since the simulation of three- dimensional crack growth is either not supported by commercial software, or requires significant effort and time for the analysts and is generally not coupled to robust error indicators, reliable, quality-controlled, industrial damage tolerance assessment is still a major challenge in engineering practice. The extended finite element method (XFEM) (Moës et al., 1999) is now one of many successful numerical methods to solve fracture mechanics problems. The particular advantage of the XFEM, which relies on the partition of unity (PU) property (Melenk and Babuška, 1996) of finite element shape functions, is its ability to model cracks without the mesh conforming to their geometry. This allows the crack to split the background mesh arbitrarily, thereby leading to significantly increased freedom in simulating crack growth. This feat is achieved by adding new degrees of freedom to (i) describe the discontinuity of the displacement field across the crack faces, within a given element, and (ii) reproduce the asymptotic fields around the crack tip. Thanks to the advances made in the XFEM during the last years, the method is now considered to be a robust and accurate means of analysing fracture problems, has been implemented in commercial codes (ABAQUS, 2009, CENAERO, 2011) and is industrially in use for damage tolerance assessment of complex three dimensional structures (Bordas and Moran, 2006, Wyart et al., 2007, Duflot, 2007). While XFEM allows to model cracks without (re)meshing the crack faces as they evolve and yields exceptionally accurate stress intensity factors (SIF) for 2D problems, Bordas and Moran (2006), Wyart et al. (2007), Duflot (2007) show that for realistic 3D structures, a “very fine” mesh is required to accurately capture the complex three-dimensional stress field and obtain satisfactory stress intensity factors (SIFs), the main drivers of linear elastic crack propagation. Consequently, based on the mesh used for the stress analysis, a new mesh offering sufficient refinement throughout the whole potential path of the crack must be constructed _a priori_ , i.e. before the crack path is known. Practically, this is done by running preliminary analyses on coarse meshes to obtain an approximative crack path and heuristically refining the mesh around this path to increase accuracy. Typically, this refinement does not rely on sound error measures, thus the heuristically chosen mesh is in general inadequately suited and can cause large inaccuracies in the crack growth path, especially around holes, which can lead to non-conservative estimates of the safe life of the structure. Thus, it is clear that although XFEM simplifies the treatment of cracks compared to the standard FEM by lifting the burden of a geometry conforming mesh, it still requires iterations and associated user intervention and it employs heuristics which are detrimental to the robustness and accuracy of the simulation process. It would be desirable to minimise the changes to the mesh topology, thus user intervention, while ensuring the stress fields and SIFs are accurately computed at each crack growth step. This paper is one more step in attempting to control the discretization error committed by enriched FE approximations, decrease human intervention in damage tolerance assessment of complex industrial structures and enhance confidence in the results by providing enriched FEMs with sound error estimators which will guarantee a predetermined accuracy level and suppress recourse to manual iterations and heuristics. The error assessment procedures used in finite element analysis are well known and can be usually classified into different families (Ainsworth and Oden, 2000): residual based error estimators, recovery based error estimators, dual techniques, etc. Residual based error estimators were substantially improved with the introduction of the _residual equilibration_ by Ainsworth and Oden (2000). These error estimators have a strong mathematical basis and have been frequently used to obtain lower and upper bounds of the error (Ainsworth and Oden, 2000, Díez et al., 2004). Recovery based error estimates were first introduced by Zienkiewicz and Zhu (1987) and are often preferred by practitioners because they are robust and simple to use and provide an enhanced solution. Further improvements were made with the introduction of new recovery processes such as the superconvergent patch recovery (SPR) technique proposed by Zienkiewicz and Zhu (1992a, b). Dual techniques based on the evaluation of two different fields, one compatible in displacements and another equilibrated in stresses, have also been used by Pereira et al. (1999) to obtain bounds of the error. Herein we are going to focus on recovery based techniques which follow the ideas of the Zienkiewicz-Zhu (ZZ) error estimator proposed by Zienkiewicz and Zhu (1987). The literature on error estimation techniques for mesh based partition of unity methods is still scarce. One of the first steps in that direction was made in the context of the Generalized Finite Element Method (GFEM) by Strouboulis et al. (2001). The authors proposed a recovery-based error estimator which provides good results for h-adapted meshes. In a later work two new a posteriori error estimators for GFEM were presented (Strouboulis et al., 2006). The first one was based on patch residual indicators and provided an accurate theoretical upper bound estimate, but its computed version severely underestimated the exact error. The second one was an error estimator based on a recovered displacement field and its performance was closely related to the quality of the GFEM solution. This recovery technique constructs a recovered solution on patches using a basis enriched with handbook functions. In order to obtain a recovered stress field that improves the accuracy of the stresses obtained by XFEM, Xiao and Karihaloo (2006) proposed a moving least squares fitting adapted to the XFEM framework which considers the use of statically admissible basis functions. Nevertheless, the recovered stress field was not used to obtain an error indicator. Pannachet et al. (2009) worked on error estimation for mesh adaptivity in XFEM for cohesive crack problems. Two error estimates were used, one based on the error in the energy norm and another that considers the error in a local quantity of interest. The error estimation was based on solving a series of local problems with prescribed homogeneous boundary conditions. Panetier et al. (2010) presented an extension to enriched approximations of the constitutive relation error (CRE) technique already available to evaluate error bounds in FEM (Ladevèze and Pelle, 2005). This procedure has been used to obtain local error bounds on quantities of interest for XFEM problems. Bordas and Duflot (2007) and Bordas et al. (2008) proposed a recovery based error estimator for 2D and 3D XFEM approximations for cracks known as the extended moving least squares (XMLS). This method intrinsically enriched an MLS formulation to include information about the singular fields near the crack tip. Additionally, it used a diffraction method to introduce the discontinuity in the recovered field. This error estimator provided accurate results with effectivity indices close to unity (optimal value) for 2D and 3D fracture mechanics problems. Later, Duflot and Bordas (2008) proposed a global derivative recovery formulation extended to XFEM problems. The recovered solution was sought in a space spanned by the near tip strain fields obtained from differentiating the Westergaard asymptotic expansion. Although the results provided by this technique were not as accurate as those in Bordas and Duflot (2007), Bordas et al. (2008), they were deemed by the authors to require less computational power. Ródenas et al. (2008) presented a modification of the superconvergent patch recovery (SPR) technique tailored to the XFEM framework called $\textrm{SPR}_{\textrm{XFEM}}$. This technique was based on three key ingredients: (1) the use of a _singular_ +_smooth_ stress field decomposition procedure around the crack tip similar to that described by Ródenas et al. (2006) for FEM; (2) direct calculation of recovered stresses at integration points using the partition of unity, and (3) use of different stress interpolation polynomials at each side of the crack when a crack intersects a patch. In order to obtain an equilibrated smooth recovered stress field, a simplified version of the SPR-C technique presented by Ródenas et al. (2007) was used. This simplified SPR-C imposed the fulfilment of the boundary equilibrium equation at boundary nodes but did not impose the satisfaction of the internal equilibrium equations. The numerical results presented by Bordas and Duflot (2007), Bordas et al. (2008) and Ródenas et al. (2008) showed promising accuracy for both the XMLS and the $\textrm{SPR}_{\textrm{XFEM}}$ techniques. It is apparent in those papers that $\textrm{SPR}_{\textrm{XFEM}}$ led to effectivity indices remarkably close to unity. Yet, these papers do not shed any significant light on the respective roles played by two important ingredients in those error estimators, namely: 1. 1. The statistical admissibility of the recovered solution; 2. 2. The enrichment of the recovered solution. Moreover, the differences in the test cases analysed and in the quality measures considered in each of the papers makes it difficult to objectively compare the merits of both methods. The aim of this paper is to assess the role of statistical admissibility and enrichment of the recovered solution in recovery based error estimation of enriched finite element approximation for linear elastic fracture. To do so, we perform a systematic study of the results obtained when considering the different features of two error estimation techniques: XMLS and SPR-CX. SPR-CX is an enhanced version of the $\textrm{SPR}_{\textrm{XFEM}}$ presented by Ródenas et al. (2008) for 2D problems. Advantages and disadvantages of each method are also provided. The outline of the paper is as follows. In Section 2, the XFEM is briefly presented. Section 3 deals with error estimation and quality assessment of the solution. In Sections 4 and 5 we introduce the error estimators used in XFEM approximations: the SPR-CX and the XMLS, respectively. Some numerical examples analysing both techniques and the effect of the enrichment functions in the recovery process are presented in Section 6. Finally some concluding remarks are provided in Section 7. ## 2 Reference problem and XFEM solution Let us consider a 2D linear elastic fracture mechanics (LEFM) problem on a bounded domain $\Omega\subset\mathbb{R}^{2}$. The unknown displacement field $\bm{\mathrm{u}}$ is the solution of the boundary value problem $\displaystyle\nabla\cdot\boldsymbol{\upsigma}(\bm{\mathrm{u}})+\bm{\mathrm{b}}$ $\displaystyle=\mathbf{0}$ $\displaystyle\textrm{in }\Omega$ (1) $\displaystyle\boldsymbol{\upsigma}(\bm{\mathrm{u}})\cdot\bm{\mathrm{n}}$ $\displaystyle=\bm{\mathrm{t}}$ $\displaystyle\textrm{on }\Gamma_{N}$ (2) $\displaystyle\boldsymbol{\upsigma}(\bm{\mathrm{u}})\cdot\bm{\mathrm{n}}$ $\displaystyle=\mathbf{0}$ $\displaystyle\textrm{on }\Gamma_{C}$ (3) $\displaystyle\bm{\mathrm{u}}$ $\displaystyle=\bar{\bm{\mathrm{u}}}$ $\displaystyle\textrm{on }\Gamma_{D}$ (4) where $\Gamma_{N}$ and $\Gamma_{D}$ are the Neumann and Dirichlet boundaries and $\Gamma_{C}$ represents the crack faces, with $\partial\Omega=\Gamma_{N}\cup\Gamma_{D}\cup\Gamma_{C}$ and $\Gamma_{N}\cap\Gamma_{D}\cap\Gamma_{C}=\varnothing$. $\bm{\mathrm{b}}$ are the body forces per unit volume, $\bm{\mathrm{t}}$ the tractions applied on $\Gamma_{N}$ (being $\bm{\mathrm{n}}$ the normal vector to the boundary) and $\bar{\bm{\mathrm{u}}}$ the prescribed displacements on $\Gamma_{D}$. The weak form of the problem reads: Find $\bm{\mathrm{u}}\in V$ such that: $\forall\bm{\mathrm{v}}\in V\qquad a(\bm{\mathrm{u}},\bm{\mathrm{v}})=l(\bm{\mathrm{v}}),$ (5) where $V$ is the standard test space for elasticity problems such that $V=\\{\bm{\mathrm{v}}\;\|\;\bm{\mathrm{v}}\in H^{1}(\Omega),\bm{\mathrm{v}}\|_{\Gamma_{D}}(\bm{\mathrm{x}})=\mathbf{0},\bm{\mathrm{v}}\;\textrm{discontinuous on }\Gamma_{C}\\}$, and $\displaystyle a(\bm{\mathrm{u}},\bm{\mathrm{v}})$ $\displaystyle:=\int_{\Omega}\boldsymbol{\upsigma}(\bm{\mathrm{u}}):\bm{\mathrm{\epsilon}}(\bm{\mathrm{v}})d\Omega=\int_{\Omega}\boldsymbol{\upsigma}(\bm{\mathrm{u}}):\boldsymbol{\mathsf{S}}:\boldsymbol{\upsigma}(\bm{\mathrm{v}})d\Omega$ (6) $\displaystyle l(\bm{\mathrm{v}})$ $\displaystyle:=\int_{\Omega}\bm{\mathrm{b}}\cdot\bm{\mathrm{v}}d\Omega+\int_{\Gamma_{N}}\bm{\mathrm{t}}\cdot\bm{\mathrm{v}}d\Gamma,$ (7) where $\boldsymbol{\mathsf{S}}$ is the compliance tensor, $\boldsymbol{\upsigma}$ and $\bm{\mathrm{\epsilon}}$ represent the stress and strain operators. LEFM problems are denoted by the singularity present at the crack tip. The following expressions represent the first term of the asymptotic expansion which describes the displacements and stresses for combined loading modes I and II in 2D. These expressions can be found in the literature (Szabó and Babuška, 1991, Ródenas et al., 2008) and are reproduced here for completeness: $\displaystyle u_{1}(r,\phi)$ $\displaystyle=\frac{K_{\rm I}}{2\mu}\sqrt{\frac{r}{2\pi}}\cos\frac{\phi}{2}\left(\kappa-\cos\phi\right)+\frac{K_{\rm II}}{2\mu}\sqrt{\frac{r}{2\pi}}\sin\frac{\phi}{2}\left(2+\kappa+\cos\phi\right)$ (8) $\displaystyle u_{2}(r,\phi)$ $\displaystyle=\frac{K_{\rm I}}{2\mu}\sqrt{\frac{r}{2\pi}}\sin\frac{\phi}{2}\left(\kappa-\cos\phi\right)+\frac{K_{\rm II}}{2\mu}\sqrt{\frac{r}{2\pi}}\cos\frac{\phi}{2}\left(2-\kappa-\cos\phi\right)$ $\displaystyle\sigma_{11}(r,\phi)$ $\displaystyle=\frac{K_{\rm I}}{\sqrt{2\pi r}}\cos\frac{\phi}{2}\left(1-\sin\frac{\phi}{2}\sin\frac{3\phi}{2}\right)-\frac{K_{\rm II}}{\sqrt{2\pi r}}\sin\frac{\phi}{2}\left(2+\cos\frac{\phi}{2}\cos\frac{3\phi}{2}\right)$ (9) $\displaystyle\sigma_{22}(r,\phi)$ $\displaystyle=\frac{K_{\rm I}}{\sqrt{2\pi r}}\cos\frac{\phi}{2}\left(1+\sin\frac{\phi}{2}\sin\frac{3\phi}{2}\right)+\frac{K_{\rm II}}{\sqrt{2\pi r}}\sin\frac{\phi}{2}\cos\frac{\phi}{2}\cos\frac{3\phi}{2}$ $\displaystyle\sigma_{12}(r,\phi)$ $\displaystyle=\frac{K_{\rm I}}{\sqrt{2\pi r}}\sin\frac{\phi}{2}\cos\frac{\phi}{2}\cos\frac{3\phi}{2}+\frac{K_{\rm II}}{\sqrt{2\pi r}}\cos\frac{\phi}{2}\left(1-\sin\frac{\phi}{2}\sin\frac{3\phi}{2}\right)$ where $r$ and $\phi$ are the crack tip polar coordinates, $K_{\rm I}$ and $K_{\rm II}$ are the stress intensity factors for modes I and II, $\mu$ is the shear modulus, and $\kappa$ the Kolosov’s constant, defined in terms of the parameters of material $E$ (Young’s modulus) and $\upsilon$ (Poisson’s ratio), according to the expressions: $\mu=\frac{E}{2\left(1+\upsilon\right)},\qquad\kappa=\left\\{\begin{array}[]{l c r}{\displaystyle 3-4\upsilon\qquad\textrm{plane strain}}\\\ \vskip 6.0pt plus 2.0pt minus 2.0pt\cr{\displaystyle\frac{3-\upsilon}{1+\upsilon}\,\qquad\textrm{plane stress}}\end{array}\right.$ This type of problem is difficult to model using a standard FEM approximation as the mesh needs to explicitly conform to the crack geometry. With the XFEM the discontinuity of the displacement field along the crack faces is introduced by adding degrees of freedom to the nodes of the elements intersected by the crack. This tackles the problem of adjusting the mesh to the geometry of the crack (Moës et al., 1999, Stolarska et al., 2001). Additionally, to describe the solution around the crack tip, the numerical model introduces a basis that spans the near tip asymptotic fields. The following expression is generally used to interpolate the displacements at a point of coordinates $\bm{\mathrm{x}}$ accounting for the presence of a crack tip in a 2D XFEM approximation: $\bm{\mathrm{u}}_{h}(\bm{\mathrm{x}})=\sum_{i\in\mathcal{I}}N_{i}(\bm{\mathrm{x}})\textbf{a}_{i}+\sum_{j\in\mathcal{J}}N_{j}(\bm{\mathrm{x}})H(\bm{\mathrm{x}})\textbf{b}_{j}+\sum_{m\in\mathcal{M}}N_{m}(\bm{\mathrm{x}})\left(\sum_{\ell=1}^{4}F_{\ell}(\bm{\mathrm{x}})\textbf{c}_{m}^{\ell}\right)$ (10) where $N_{i}$ are the shape functions associated with node $i$, $\textbf{a}_{i}$ represent the conventional nodal degrees of freedom, $\textbf{b}_{j}$ are the coefficients associated with the discontinuous enrichment functions, and $\bm{\mathrm{c}}_{m}$ those associated with the functions spanning the asymptotic field. In the above equation, $\mathcal{I}$ is the set of all the nodes in the mesh, $\mathcal{M}$ is the subset of nodes enriched with crack tip functions, and $\mathcal{J}$ is the subset of nodes enriched with the discontinuous enrichment (see Figure 1). In (10), the Heaviside function $H$, with unitary modulus and a change of sign on the crack face, describes the displacement discontinuity if the finite element is intersected by the crack. The $F_{\ell}$ are the set of branch functions used to represent the asymptotic expansion of the displacement field around the crack tip seen in (8). For the 2D case, the following functions are used (Belytschko and Black, 1999): $\left\\{F_{\ell}\left(r,\phi\right)\right\\}\equiv\sqrt{r}\left\\{\sin\frac{\phi}{2},\cos\frac{\phi}{2},\sin\frac{\phi}{2}\sin\phi,\cos\frac{\phi}{2}\sin\phi\right\\}$ (11) Figure 1: Classification of nodes in XFEM. Fixed enrichment area of radius $r_{e}$ The main features of the XFEM implementation considered to evaluate the numerical results is described in detail in Ródenas et al. (2008) and can be summarized as follows: * • Use of bilinear quadrilaterals. * • Decomposition of elements intersected by the crack into integration subdomains that do not contain the crack. Alternatives which do not required this subdivision are proposed by Ventura (2006), Natarajan et al. (2010). * • Use of a quasi-polar integration with a $5\times 5$ quadrature rule in triangular subdomains for elements containing the crack tip. * • No correction for blending elements. Methods to address blending errors are proposed by Chessa et al. (2003), Gracie et al. (2008), Fries (2008), Tarancón et al. (2009). ### 2.1 Evaluation of stress intensity factors The stress intensity factors (SIFs) in LEFM represent the amplitude of the singular stress fields and are key quantities of interest to simulate crack growth in LEFM. Several post–processing methods, following local or global (energy) approaches, are commonly used to extract SIFs (Banks-Sills, 1991) or to calculate the energy release rate $G$. Energy or global methods are considered to be the most accurate and efficient methods (Banks-Sills, 1991, Li et al., 1985). Global methods based on the equivalent domain integral (EDI methods) are specially well-suited for FEM and XFEM analyses as they are easy to implement and can use information far from the singularity. In this paper, the interaction integral as described in Shih and Asaro (1988), Yau et al. (1980) has been used to extract the SIFs. This technique provides $K_{\rm I}$ and $K_{\rm II}$ for problems under mixed mode loading conditions using auxiliary fields. Details on the implementation of the interaction integral can be found for example in Moës et al. (1999), Ródenas et al. (2008), Giner et al. (2005). ## 3 Error estimation in the energy norm The approximate nature of the FEM and XFEM approximations implies a discretization error which can be quantified using the error in the energy norm for the solution $\lVert\bm{\mathrm{e}}\rVert=\lVert\bm{\mathrm{u}}-\bm{\mathrm{u}}^{h}\rVert$. To obtain an estimate of the discretization error $\lVert\bm{\mathrm{e}}_{es}\rVert$, in the context of elasticity problems solved using the FEM, the expression for the ZZ estimator is defined as (Zienkiewicz and Zhu, 1987) (in matrix form): $\lVert\bm{\mathrm{e}}\rVert^{2}\approx\lVert\bm{\mathrm{e}}_{es}\rVert^{2}=\int_{\Omega}\left(\bm{\mathrm{\sigma}}^{}-\bm{\mathrm{\sigma}}^{h}\right)^{T}\bm{\mathrm{D}}^{-1}\left(\bm{\mathrm{\sigma}}^{}-\bm{\mathrm{\sigma}}^{h}\right)d\Omega$ (12) or alternatively for the strains $\lVert\bm{\mathrm{e}}_{es}\rVert^{2}=\int_{\Omega}\left(\bm{\mathrm{\varepsilon}}^{}-\bm{\mathrm{\varepsilon}}^{h}\right)^{T}\bm{\mathrm{D}}\left(\bm{\mathrm{\varepsilon}}^{}-\bm{\mathrm{\varepsilon}}^{h}\right)d\Omega,$ where the domain $\Omega$ refers to the full domain of the problem or a local subdomain (element), $\bm{\mathrm{\sigma}}^{h}$ represents the stress field provided by the FEM, $\bm{\mathrm{\sigma}}^{}$ is the recovered stress field, which is a better approximation to the exact solution than $\bm{\mathrm{\sigma}}^{h}$ and $\bm{\mathrm{D}}$ is the elasticity matrix of the constitutive relation $\bm{\mathrm{\sigma}}=\bm{\mathrm{D}}\bm{\mathrm{\varepsilon}}$. The recovered stress field $\bm{\mathrm{\sigma}}^{}$ is usually interpolated in each element using the shape functions $\bm{\mathrm{N}}$ of the underlying FE approximation and the values of the recovered stress field calculated at the nodes $\bar{\bm{\mathrm{\sigma}}}^{}$ $\bm{\mathrm{\sigma}}^{}(\bm{\mathrm{x}})=\sum_{i=1}^{n_{e}}N_{i}(\bm{\mathrm{x}})\bar{\bm{\mathrm{\sigma}}}^{}_{i}(\bm{\mathrm{x}}_{i}),$ (13) where $n_{e}$ is the number of nodes in the element under consideration and $\bar{\bm{\mathrm{\sigma}}}^{}_{i}(\bm{\mathrm{x}}_{i})$ are the stresses provided by the least squares technique at node $i$. The components of $\bar{\bm{\mathrm{\sigma}}}^{}_{i}$ are obtained using a polynomial expansion, $\bar{\sigma}_{i,j}=\bm{\mathrm{p}}\bm{\mathrm{a}}$ (with $j=xx,yy,xy$), defined over a set of contiguous elements connected to node $i$ called patch, where $\bm{\mathrm{p}}$ is the polynomial basis and $\bm{\mathrm{a}}$ are the unknown coefficients. The ZZ error estimator is asymptotically exact if the recovered solution used in the error estimation converges at a higher rate than the finite element solution (Zienkiewicz and Zhu, 1992b). In this paper, we are interested in the role played by statical admissibility and enrichment of the recovered solution in estimating the error committed by XFEM. To do so, we study the performance of two recovery-based error estimators, which exhibit different features, that have been recently developed for XFEM: • The SPR-CX derived from the error estimator developed by Ródenas et al. (2008) and summarized in Section 4 (two other versions of the SPR-CX technique have been also considered); * • The XMLS proposed by Bordas and Duflot (2007) and summarized in Section 5. Both estimators provide a recovered stress field in order to evaluate the estimated error in the energy norm by means of the expression shown in (12). ## 4 SPR-CX error estimator The SPR-CX error estimator is an enhancement of the error estimator first introduced by Ródenas et al. (2008), which incorporates the ideas proposed in Ródenas et al. (2007) to guarantee the exact satisfaction of the equilibrium locally on patches. In Ródenas et al. (2008) a set of key ideas are proposed to modify the standard SPR by Zienkiewicz and Zhu (1992a), allowing its use for singular problems. The recovered stresses $\bm{\mathrm{\sigma}}^{}$ are directly evaluated at an integration point $\bm{\mathrm{x}}$ through the use of a partition of unity procedure, properly weighting the stress interpolation polynomials obtained from the different patches formed at the vertex nodes of the element containing $\bm{\mathrm{x}}$: $\bm{\mathrm{\sigma}}^{}(\bm{\mathrm{x}})=\sum_{i=1}^{n_{v}}N_{i}(\bm{\mathrm{x}})\bm{\mathrm{\sigma}}^{}_{i}(\bm{\mathrm{x}}),$ (14) where $N_{i}$ are the shape functions associated to the vertex nodes $n_{v}$. One major modification is the introduction of a splitting procedure to perform the recovery. For singular problems the exact stress field $\bm{\mathrm{\sigma}}$ is decomposed into two stress fields, a smooth field $\bm{\mathrm{\sigma}}_{smo}$ and a singular field $\bm{\mathrm{\sigma}}_{sing}$: $\bm{\mathrm{\sigma}}=\bm{\mathrm{\sigma}}_{smo}+\bm{\mathrm{\sigma}}_{sing}.$ (15) Then, the recovered field $\bm{\mathrm{\sigma}}^{}$ required to compute the error estimate given in (12) can be expressed as the contribution of two recovered stress fields, one smooth $\bm{\mathrm{\sigma}}^{}_{smo}$ and one singular $\bm{\mathrm{\sigma}}^{}_{sing}$: $\bm{\mathrm{\sigma}}^{}=\bm{\mathrm{\sigma}}^{}_{smo}+\bm{\mathrm{\sigma}}^{}_{sing}.$ (16) For the recovery of the singular part, the expressions in (9) which describe the asymptotic fields near the crack tip are used. To evaluate $\bm{\mathrm{\sigma}}^{}_{sing}$ from (9) we first obtain estimated values of the stress intensity factors $K_{\rm I}$ and $K_{\rm II}$ using the interaction integral as indicated in Section 2.1. The recovered part $\bm{\mathrm{\sigma}}^{}_{sing}$ is an equilibrated field as it satisfies the internal equilibrium equations. Once the field $\bm{\mathrm{\sigma}}^{}_{sing}$ has been evaluated, an FE approximation to the smooth part $\bm{\mathrm{\sigma}}^{h}_{smo}$ can be obtained subtracting $\bm{\mathrm{\sigma}}^{}_{sing}$ from the raw FE field: $\bm{\mathrm{\sigma}}^{h}_{smo}=\bm{\mathrm{\sigma}}^{h}-\bm{\mathrm{\sigma}}^{}_{sing}.$ (17) Then, the field $\bm{\mathrm{\sigma}}^{}_{smo}$ is evaluated applying an SPR-C recovery procedure over the field $\bm{\mathrm{\sigma}}^{h}_{smo}$. For patches intersected by the crack, the recovery technique uses different stress interpolation polynomials on each side of the crack. This way it can represent the discontinuity of the recovered stress field along the crack faces, which is not the case for SPR that smoothes out the discontinuity (see, e.g. Bordas and Duflot (2007)). In order to obtain an equilibrated recovered stress field $\bm{\mathrm{\sigma}}^{}_{smo}$, the SPR-CX enforces the fulfilment of the equilibrium equations locally on each patch. The constraint equations are introduced via Lagrange multipliers into the linear system used to solve for the coefficients of the polynomial expansion of the recovered stresses on each patch. These include the satisfaction of the: * • Internal equilibrium equations. * • Boundary equilibrium equation: A point collocation approach is used to impose the satisfaction of a second order approximation to the tractions along the Neumann boundary. * • Compatibility equation: This additional constraint is also imposed to further increase the accuracy of the recovered stress field. To evaluate the recovered field, quadratic polynomials have been used in the patches along the boundary and crack faces, and linear polynomials for the remaining patches. As more information about the solution is available along the boundary, polynomials one degree higher are useful to improve the quality of the recovered stress field. The enforcement of equilibrium equations provides an equilibrated recovered stress field locally on patches. However, the process used to obtain a continuous field $\bm{\mathrm{\sigma}}^{}$ shown in (14) introduces a small lack of equilibrium as explained in Ródenas et al. (2010a). The reader is referred to Ródenas et al. (2010a), Díez et al. (2007) for details. ## 5 XMLS error estimator In the XMLS the solution is recovered through the use of the _moving least squares_ (MLS) technique, developed by mathematicians to build and fit surfaces. The XMLS technique extends the work of Tabbara et al. (1994) for FEM to enriched approximations. The general idea of the XMLS is to use the displacement solution provided by XFEM to obtain a recovered strain field (Bordas and Duflot, 2007, Bordas et al., 2008). The smoothed strains are recovered from the derivative of the MLS-smoothed XFEM displacement field: $\displaystyle\bm{\mathrm{u}}^{}(\bm{\mathrm{x}})$ $\displaystyle=\sum_{i\in\mathcal{N}_{x}}\psi_{i}(\bm{\mathrm{x}})\bm{\mathrm{u}}_{i}^{h}$ (18) $\displaystyle\bm{\mathrm{\varepsilon}}^{}(\bm{\mathrm{x}})$ $\displaystyle=\sum_{i\in\mathcal{N}_{x}}\nabla_{s}(\psi_{i}(\bm{\mathrm{x}})\bm{\mathrm{u}}_{i}^{h}),$ (19) where the $\bm{\mathrm{u}}_{i}^{h}$ are the raw nodal XFEM displacements and $\psi_{i}(\bm{\mathrm{x}})$ are the MLS shape function values associated with a node $i$ at a point $\bm{\mathrm{x}}$. $\mathcal{N}_{x}$ is the set of $n_{x}$ nodes in the domain of influence of point $\bm{\mathrm{x}}$, $\nabla_{s}$ is the symmetric gradient operator, $\bm{\mathrm{u}}^{}$ and $\bm{\mathrm{\varepsilon}}^{}$ are the recovered displacement and strain fields respectively. At each point $\bm{\mathrm{x}}_{i}$ the MLS shape functions $\psi_{i}$ are evaluated using weighting functions $\omega_{i}$ and an enriched basis $\bm{\mathrm{p}}(\bm{\mathrm{x}}_{i})$. The total $n_{x}$ non-zero MLS shape functions at point $\bm{\mathrm{x}}$ are evaluated as: $(\psi_{i}(\bm{\mathrm{x}}))_{1\leq i\leq n_{x}}=(\bm{\mathrm{A}}^{-1}(\bm{\mathrm{x}})\bm{\mathrm{p}}(\bm{\mathrm{x}}))^{T}\bm{\mathrm{p}}(\bm{\mathrm{x}}_{i})\omega_{i}(\bm{\mathrm{x}})$ (20) where $\bm{\mathrm{A}}(\bm{\mathrm{x}})=\sum_{i=1}^{n_{x}}\omega_{i}(\bm{\mathrm{x}})\bm{\mathrm{p}}(\bm{\mathrm{x}}_{i})\bm{\mathrm{p}}(\bm{\mathrm{x}}_{i})^{T}$ is a matrix to be inverted at every point $\bm{\mathrm{x}}$ (see Bordas and Duflot (2007), Bordas et al. (2008) for further details). For each supporting point $\bm{\mathrm{x}}_{i}$, the weighting function $\omega_{i}$ is defined such that: $\omega_{i}(s)=f_{4}(s)=\begin{cases}1-6s^{2}+8s^{3}-3s^{4}&\text{if }\left\|s\right\|\leq 1\\\ 0&\text{if }\left\|s\right\|>1\end{cases}$ (21) where $s$ is the normalized distance between the supporting point $\bm{\mathrm{x}}_{i}$ and a point $\bm{\mathrm{x}}$ in the computational domain. In order to describe the discontinuity, the distance $s$ in the weight function defined for each supporting point is modified using the _diffraction criterion_ (Belytschko et al., 1996). The basic idea of this criterion is depicted in Figure 2. The weight function is continuous except across the crack faces since the points at the other side of the crack are not considered as part of the support. Near the crack tip the weight of a node $i$ over a point of coordinates $\bm{\mathrm{x}}$ diminishes as the crack hides the point. When the point $\bm{\mathrm{x}}$ is hidden by the crack the following expression is used: $s=\frac{\left\\|\bm{\mathrm{x}}-\bm{\mathrm{x}}_{C}\right\\|+\left\\|\bm{\mathrm{x}}_{C}-\bm{\mathrm{x}}_{i}\right\\|}{d_{i}}$ (22) where $d_{i}$ is the radius of the support. Figure 2: Diffraction criteria to introduce the discontinuity in the XMLS approximation. The MLS shape functions can reproduce any function in their basis. The basis $\bm{\mathrm{p}}$ used is a linear basis enriched with the functions that describe the first order asymptotic expansion at the crack tip as indicated in (11): $\bm{\mathrm{p}}=\left[1,x,y,\left[F_{1}(r,\phi),F_{2}(r,\phi),F_{3}(r,\phi),F_{4}(r,\phi)\right]\right]$ (23) Note that although the enriched basis can reproduce the singular behaviour of the solution around the crack tip, the resulting recovered field not necessarily satisfies the equilibrium equations. ## 6 Numerical results In this section, numerical experiments are performed to verify the behaviour of both XFEM recovery based error estimators considered in this paper. Babuška et al. (1994a, b, 1997) proposed a robustness patch test for quality assessment of error estimators. However, the use of this test is not within the scope of this paper and furthermore, to the authors’ knowledge, it has not been used in the context of XFEM. Therefore, the more traditional approach of using benchmark problems is considered here to analyse the response of the different error estimators. The accuracy of the error estimators is evaluated both locally and globally. This evaluation has been based on the effectivity of the error in the energy norm, which is quantified using the _effectivity index_ $\theta$ defined as: $\theta=\frac{\lVert\textbf{e}_{es}\rVert}{\lVert\textbf{e}\rVert}.$ (24) When the enhanced or recovered solution is close to the analytical solution the effectivity approaches the theoretical value of 1, which indicates that it is a good error estimator, i.e. the approximate error is close to the exact error. To assess the quality of the estimator at a local level, the local effectivity $D$, inspired on the _robustness index_ found in Babuška et al. (1994a), is used. For each element $e$, $D$ represents the variation of the effectivity index in this element, $\theta^{e}$, with respect to the theoretical value (the error estimator can be considered to be of good quality if it yields $D$ values close to zero). $D$ is defined according to the following expression, where superscripts e indicate the element $e$: $\begin{array}[]{c}{\displaystyle D=\theta^{e}-1\qquad{\rm if}\qquad\theta^{e}\geq 1}\\\ {\displaystyle D=1-\frac{1}{\theta^{e}}\qquad{\rm if}\qquad\theta^{e}<1}\end{array}\qquad\qquad{\rm with}\qquad\theta^{e}=\frac{\left\\|\textbf{e}_{es}^{e}\right\\|}{\left\\|\textbf{e}^{e}\right\\|}.$ (25) To evaluate the overall quality of the error estimator we use the global effectivity index $\theta$, the mean value $m\left(\|D\|\right)$ and the standard deviation $\sigma\left(D\right)$ of the local effectivity. A good quality error estimator yields values of $\theta$ close to one and values of $m\left(\|D\|\right)$ and $\sigma\left(D\right)$ close to zero. The techniques can be used in practical applications. However, in order to properly compare their performance we have used an academic problem with exact solution. In the analysis we solve the Westergaard problem (Gdoutos, 1993) as it is one of the few problems in LEFM under mixed mode with an analytical solution. In Giner et al. (2005), Ródenas et al. (2008) we can find explicit expressions for the stress fields in terms of the spatial coordinates. In the next subsection we show a description of the Westergaard problem and XFEM model, taken from Ródenas et al. (2008) and reproduced here for completeness. ### 6.1 Westergaard problem and XFEM model The Westergaard problem corresponds to an infinite plate loaded at infinity with biaxial tractions $\sigma_{x\infty}=\sigma_{y\infty}=\sigma_{\infty}$ and shear traction $\tau_{\infty}$, presenting a crack of length $2a$ as shown in Figure 3. Combining the externally applied loads we can obtain different loading conditions: pure mode I, II or mixed mode. Figure 3: Westergaard problem. Infinite plate with a crack of length $2a$ under uniform tractions $\sigma_{\infty}$ (biaxial) and $\tau_{\infty}$. Finite portion of the domain $\Omega_{0}$, modelled with FE. In the numerical model only a finite portion of the domain ($a=1$ and $b=4$ in Figure 3) is considered. The projection of the stress distribution corresponding to the analytical Westergaard solution for modes I and II, given by the expressions below, is applied to its boundary: $\begin{array}[]{r@{\hspace{1ex}}c@{\hspace{1ex}}l}{\sigma_{x}^{I}}(x,y)\hskip 4.30554pt&{=}\hfil\hskip 4.30554pt&{\displaystyle\frac{\sigma_{\infty}}{\sqrt{\left\|t\right\|}}\bigg{[}\left(x\cos\frac{\phi}{2}-y\sin\frac{\phi}{2}\right)+y\frac{a^{2}}{\left\|t\right\|^{2}}\left(m\sin\frac{\phi}{2}-n\cos\frac{\phi}{2}\right)\bigg{]}}\\\ \vskip 6.0pt plus 2.0pt minus 2.0pt\cr{\sigma_{y}^{I}}(x,y)\hskip 4.30554pt&{=}\hfil\hskip 4.30554pt&{\displaystyle\frac{\sigma_{\infty}}{\sqrt{\left\|t\right\|}}\bigg{[}\left(x\cos\frac{\phi}{2}-y\sin\frac{\phi}{2}\right)-y\frac{a^{2}}{\left\|t\right\|^{2}}\left(m\sin\frac{\phi}{2}-n\cos\frac{\phi}{2}\right)\bigg{]}}\\\ \vskip 6.0pt plus 2.0pt minus 2.0pt\cr{\tau_{xy}^{I}}(x,y)\hskip 4.30554pt&{=}\hfil\hskip 4.30554pt&{\displaystyle y\frac{a^{2}\sigma_{\infty}}{\left\|t\right\|^{2}\sqrt{\left\|t\right\|}}\left(m\cos\frac{\phi}{2}+n\sin\frac{\phi}{2}\right)}\end{array}$ (26) $\begin{array}[]{r@{\hspace{1ex}}c@{\hspace{1ex}}l}{\sigma_{x}^{II}}(x,y)\hskip 4.30554pt&{=}\hfil\hskip 4.30554pt&{\displaystyle\frac{\tau_{\infty}}{\sqrt{\left\|t\right\|}}\bigg{[}2\left(y\cos\frac{\phi}{2}+x\sin\frac{\phi}{2}\right)-y\frac{a^{2}}{\left\|t\right\|^{2}}\left(m\cos\frac{\phi}{2}+n\sin\frac{\phi}{2}\right)\bigg{]}}\\\ \vskip 6.0pt plus 2.0pt minus 2.0pt\cr{\sigma_{y}^{II}}(x,y)\hskip 4.30554pt&{=}\hfil\hskip 4.30554pt&{\displaystyle y\frac{a^{2}\tau_{\infty}}{\left\|t\right\|^{2}\sqrt{\left\|t\right\|}}\left(m\cos\frac{\phi}{2}+n\sin\frac{\phi}{2}\right)}\\\ \vskip 6.0pt plus 2.0pt minus 2.0pt\cr{\tau_{xy}^{II}}(x,y)\hskip 4.30554pt&{=}\hfil\hskip 4.30554pt&{\displaystyle\frac{\tau_{\infty}}{\sqrt{\left\|t\right\|}}\bigg{[}\left(x\cos\frac{\phi}{2}-y\sin\frac{\phi}{2}\right)+y\frac{a^{2}}{\left\|t\right\|^{2}}\left(m\sin\frac{\phi}{2}-n\cos\frac{\phi}{2}\right)\bigg{]}}\end{array}$ (27) where the stress fields are expressed as a function of $x$ and $y$, with origin at the centre of the crack. The parameters $t$, $m$, $n$ and $\phi$ are defined as $\begin{split}t&=(x+iy)^{2}-a^{2}=(x^{2}-y^{2}-a^{2})+i(2xy)=m+in\\\ m&=\textrm{Re}(t)=\textrm{Re}(z^{2}-a^{2})=x^{2}-y^{2}-a^{2}\\\ n&=\textrm{Im}(t)=(z^{2}-a^{2})=2xy\\\ \phi&=\textrm{Arg}(\bar{t})=\textrm{Arg}(m-in)\qquad\textrm{with }\phi\in\left[-\pi,\pi\right],\;i^{2}=-1\end{split}$ (28) For the problem analysed, the exact value of the SIF is given as $K_{{\rm I},ex}=\sigma\sqrt{\pi a}\qquad\qquad K_{{\rm II},ex}=\tau\sqrt{\pi a}$ (29) Three different loading configurations corresponding to the _pure mode I_ ($\sigma_{\infty}=100,\;\tau_{\infty}=0$), _pure mode II_ ($\sigma_{\infty}=0,\;\tau_{\infty}=100$), and _mixed mode_ ($\sigma_{\infty}=30,\;\tau_{\infty}=90$) cases of the Westergaard problem are considered. The geometric models and boundary conditions are shown in Figure 4. Figure 4: Model for an infinite plate with a crack subjected to biaxial tractions $\sigma_{\infty},\;\tau_{\infty}$ in the infinite. Bilinear elements are used in the models, with a _singular_ +_smooth_ decomposition area of radius $\rho=0.5$ equal to the radius $r_{e}$ used for the fixed enrichment area (note that the splitting radius $\rho$ should be greater or equal to the enrichment radius $r_{e}$, Ródenas et al. (2008)). The radius for the Plateau function used to extract the SIF is $r_{q}=0.9$. Young’s modulus is $E=10^{7}$ and Poisson’s ratio $\upsilon=0.333$. For the numerical integration of standard elements we use a $2\times 2$ Gaussian quadrature rule. For split elements we use 7 Gauss points in each triangular subdomain, and a $5\times 5$ quasipolar integration (Béchet et al., 2005) in the subdomains of the element containing the crack tip. Regarding global error estimation, the evolution of global parameters in sequences of uniformly refined structured (Figure 5) and unstructured (Figure 6) meshes is studied. In the first case, the mesh sequence is defined so that the crack tip always coincided with a node. For a more general scenario, this condition has not been applied for the unstructured meshes. Figure 5: Sequence of structured meshes. Figure 6: Sequence of unstructured meshes. ### 6.2 Mode I and structured meshes The first set of results presented in this study are for the Westergaard problem under mode I load conditions. Figure 7 shows the values for the local effectivity index $D$ for the first mesh in the sequence analysed. In the figure, the size of the enrichment area is denoted by a circle. It can be observed that far from the enrichment area both techniques yield similar results, however, the XMLS technique exhibits a higher overestimation of the error close to the singularity. Figure 7: Mode I, structured mesh 1. Local effectivity index $D$ (ideal value $D=0$). The circles denote the enriched zones around the crack tips. The superiority of the statically admissible recovery (SPR-CX) compared to the standard XMLS is clear. The same behaviour is also observed for more refined meshes in the sequence. Figure 8 shows a zoom in the enriched area of a finer mesh where, as before, a higher overestimation of the error can be observed for the results obtained with the XMLS error estimator. In this example and in order to study the evolution of the accuracy of the recovered stress field when considering different features in the recovery process, we consider two additional versions: • The first considers a recovery procedure which enforces both internal and boundary equilibrium, but does not include the singular+smooth splitting technique (SPR-C). This approach is very similar in form to conventional SPR- based recovery techniques widely used in FEM. It is well-known that this type of recovery process will produce unreliable results when the stress field contains a singularity because the polynomial representation of the recovered stresses is not able to describe the singular field (e.g. Bordas and Duflot (2007)). * • The second version of the error estimator performs the splitting but does not equilibrate the recovered field (SPR-X). As previously commented, $\bm{\mathrm{\sigma}}^{}_{sing}$ is an equilibrated field but $\bm{\mathrm{\sigma}}^{}_{smo}$ is not equilibrated. The aim of this second version is to assess the influence of enforcing the equilibrium constraints. Table 1 summarizes the main features of the different recovery procedures we considered. \| Singular \| Equilibrated sin- \| Locally equili- ---\|---\|---\|--- \| functions \| gular functions \| brated field SPR-X \| yes \| yes \| no SPR-C \| no \| not applicable \| yes SPR-CX \| yes \| yes \| yes XMLS \| yes \| no \| no Table 1: Comparison of features for the different recovery procedures Figure 8: Mode I, structured mesh 4. Local effectivity index $D$ (ideal value $D=0$). The circles denote the enriched zones around the crack tips. The results show the need for the inclusion of the near-tip fields in the recovery process (SPR-X is far superior to SPR-C). It is also clear that enforcing statical admissibility (SPR-CX) greatly improves the accuracy along the crack faces and the Neumann boundaries. Also notice that XMLS leads to slightly improved effectivities compared to SPR-CX around the crack faces, between the enriched zone and the left boundary. SPR-C is clearly not able to reproduce the singular fields, which is shown by large values of $D$ inside the enriched region. The performance of SPR-X in Figure 8 is poor, especially along the Neumann boundaries, where the equilibrium equations are not enforced, as they are in the SPR-CX. Particularly interesting are the results for the SPR-C technique, where considerable overestimation of the exact error is observed _in the whole enriched region_. This is due to the inability of polynomial functions to reproduce the near-tip fields, even relatively far from the crack tip. The above compared the relative performance of the methods locally. It is also useful to have a measure of the global accuracy of the different error estimators, Figure 9(left) shows the convergence of the estimated error in the energy norm $\lVert\textbf{e}_{es}\rVert$ (see equation (12)) evaluated with the proposed techniques, the convergence of the exact error $\lVert\textbf{e}\rVert$111The exact error measures the error of the raw XFEM compared to the exact solution. The theoretical convergence rate is $\mathcal{O}(h^{p})$, $h$ being the element size and $p$ the polynomial degree (1 for linear elements,…), or $\mathcal{O}(-\text{dof}^{p/2})$ if we consider the number of degrees of freedom. is shown for comparison. It can be observed that SPR-CX leads to the best approximation to the exact error in the energy norm. The best approach is that which captures as closely as possible the exact error. Non-singular formulations such as SPR-C greatly overestimate the error, especially close to the singular point. Only SPR-C suffers from the characteristic suboptimal convergence rate (factor 1/2, associated with the strength of the singularity) In order to evaluate the quality of the recovered field $\bm{\mathrm{\sigma}}^{}$ obtained with each of the recovery techniques, the convergence in energy norm of the approximate error on the recovered field $\lVert\textbf{e}^{}\rVert=\lVert\bm{\mathrm{u}}-\bm{\mathrm{u}}^{}\rVert$ is compared for the error estimators in Figure 9(right). It can be seen that the XMLS, the SPR-CX and the SPR-X provide convergence rates higher than the convergence rate for the exact error $\lVert\textbf{e}\rVert$, which is an indicator of the asymptotic exactness of the error estimators based on these recovery techniques (Zienkiewicz and Zhu, 1992b). Figure 9: Mode I, structured meshes: Convergence of the estimated error in the energy norm $\\|\mathbf{e}_{es}\\|$ (left). All methods which include the near- tip fields do converge with the optimal convergence rate of 1/2. The best method is SPR-CX. XMLS performs worse than SPR-X for coarse meshes, but it becomes equivalent to SPR-CX for fine meshes whilst the SPR-X error increases slightly. Convergence of the error for the recovered field $\\|\mathbf{e}_{es}^{}\\|$ (right). For the recovered field, SPR-CX is the best method closely followed by SPR-X. XMLS errors are half an order of magnitude larger than SPR-CX/SPR-X, but superior to the non-enriched recovery method SPR-C. For all methods, except SPR-C, the error on the recovered solution converges faster than the error on the raw solution, which shows that the estimators are asymptotically exact. Figure 10 shows the evolution of the parameters $\theta$, $m\left(\|D\|\right)$ y $\sigma\left(D\right)$ with respect to the number of degrees of freedom for the error estimators. It can be verified that although the SPR-C technique includes the satisfaction of the equilibrium equations, the effectivity of the error estimator does not converge to the theoretical value ($\theta=1$). This is due to the absence of the near-tip fields in the recovered solution. The influence of introducing singular functions in the recovery process can be observed in the results provided by SPR-X. In this case, the convergence is obtained both locally and globally. This shows the importance of introducing a description of the singular field in the recovery process, i.e. of using _extended recovery techniques_ in XFEM. Figure 10: Global indicators $\theta$, $m\left(\|D\|\right)$ and $\sigma\left(D\right)$ for mode I and structured meshes. Regarding the enforcement of the equilibrium equations, we can see that an equilibrated formulation (SPR-CX) leads to better effectivities compared to other non-equilibrated configurations (SPR-X, XMLS), which is an indication of the advantages associated with equilibrated recoveries in this context. Still in Figure 10, the results for the SPR-CX and XMLS show that the values of the global effectivity index are close to (and less than) unity for the SPR-CX estimator, whilst the XMLS is a lot less effective and shows effectivities larger than 1. The next step in our analysis is the verification of the convergence of the mean value and standard deviation of the local effectivity index $D$. From a practical perspective, one would want the local effectivity to be equally good, on average, everywhere inside the domain; one would also want this property to improve with mesh refinement. In other words, the average local effectivity $m$ and its standard deviation $\sigma$ should decrease with mesh refinement: $m$ and $\sigma$ measure the average behaviour of the method and how far the results deviate from the mean, i.e. how spatially consistent they are. The idea is to spot areas where there may be compensation between overestimated and underestimated areas that could produce an apparently ’accurate’ global estimation. It can be confirmed that, although the curves for $m(\|D\|)$ and $\sigma(D)$ for both estimators tend to zero with mesh refinement, the SPR-CX is clearly superior to the XMLS. These results can be explained by the fact that the SPR-CX enforces the fulfilment of the equilibrium equations in patches, and evaluates the singular part of the recovered field using the equilibrated expression that represents the first term of the asymptotic expansion, whereas the XMLS enrichment uses only a set of singular functions that would be able to reproduce this term but are not necessarily equilibrated. Although the results for the SPR-CX proved to be more accurate, the XMLS-type techniques will prove useful to obtain error bounds. There is an increasing interest in evaluating upper and lower bounds of the error for XFEM approximations. Some work has been already done to obtain upper bounds using recovery techniques as indicated in Ródenas et al. (2010a), where the authors showed that an upper bound can be obtained if the recovered field is continuous and equilibrated. However, they demonstrated that due to the use of the conjoint polynomials process to enforce the continuity of $\bm{\mathrm{\sigma}}^{}$, some residuals in the equilibrium equations appear, and the recovered field is only nearly equilibrated. Then, correction terms have to be evaluated to obtain the upper bound. An equilibrated version of the XMLS could provide a recovered stress field which would be continuous and equilibrated, thus facilitating the evaluation of the upper bound. Initial results for this class of techniques can be found in Ródenas et al. (2009, 2010b). In this first example we have shown the effect of using the SPR-C and SPR-X recovery techniques in the error estimation. Considering that these two techniques are special cases of the SPR-CX with inferior results, for further examples we will focus only on the SPR-CX and XMLS techniques. ### 6.3 Mode II and structured meshes Figure 11 presents the results considering mode II loading conditions for the local effectivity index $D$ on the fourth mesh of the sequence. Similarly to the results for mode I, the XMLS estimator presents a higher overestimation of the error near the enrichment area. This same behaviour is observed for the whole set of structured meshes analysed under pure mode II. Figure 11: Mode II, structured mesh 4. Local effectivity index $D$ (ideal value $D=0$). SPR-CX leads to much better local effectivities than XMLS. It is also remarkable that the effectivities are clearly worse in the enriched region (circle) for both the XMLS and the SPR-CX, and that this effect is more pronounced in the former method. Note the slightly worse results obtained by SPR-CX around the crack faces between the boundary of the enriched region and the left boundary, as in mode I. As for the mode I case, the evolution of global accuracy parameters in mode II exhibits the same behaviour seen for mode I loading conditions. Figure 12 shows the results for the convergence of the error in the energy norm. Figure 13 shows the evolution of global indicators $\theta$, $m\left(\|D\|\right)$ and $\sigma\left(D\right)$. Again, SPR-CX provides the best results for the Westergaard problem under pure mode II, using structured meshes. Figure 12: Mode II, structured meshes: Convergence of the estimated error in the energy norm $\\|\mathbf{e}_{es}\\|$ (left). The convergence rates are very similar to those obtained in the mode I case, and, SPR-CX is still the most accurate method, i.e. that for which the estimated error is closest to the exact error. Convergence of the error in the recovered field $\\|\mathbf{e}_{es}^{}\\|$ (right). It can be noticed that the SPR-CX error still converges faster than the exact error, but with a lower convergence rate (0.59 versus 0.73) than in the mode I case. There is no such difference for the XMLS results, which converge at practically the same rate (0.71 versus 0.72). The error level difference between SPR-CX and XMLS is about half an order of magnitude, as in the mode I case. Figure 13: Global indicators $\theta$, $m\left(\|D\|\right)$ and $\sigma\left(D\right)$ for mode II and structured meshes. The results are qualitatively and quantitatively similar to those obtained in mode I. ### 6.4 Mixed mode and unstructured meshes Considering a more general problem, Figure 14 shows the local effectivity index $D$ for the fourth mesh in a sequence of unstructured meshes, having a number of degrees of freedom (dof) similar to the mesh represented previously for load modes I and II. The XMLS recovered field presents higher overestimation of the error around the crack tip which corroborates the results found in previous load cases. Figure 14: Mixed mode, unstructured mesh 4. Local effectivity index $D$ (ideal value $D=0$). Note the improved results compared to the structured case with $D$ values ranging from -1.5 to 1.5 as opposed to -4 to 4. Figures 15 and 16 represent the evolution of global parameters for unstructured meshes and mixed mode. Once more, the best results for the error estimates are obtained using the SPR-CX technique. Figure 15: Mixed mode, unstructured meshes: Convergence of the estimated error in the energy norm $\\|\mathbf{e}_{es}\\|$ (left). Convergence of the error for the recovered field $\\|\mathbf{e}_{es}^{}\\|$ (right). The results are almost identical to the structured mesh case, except for the faster convergence obtained for SPR-CX in the unstructured compared to the structured case (0.77 versus 0.59), keeping in mind that optimal convergence rates are only formally obtained for structured meshes. Figure 16: Global indicators $\theta$, $m\left(\|D\|\right)$ and $\sigma\left(D\right)$ for mixed mode and unstructured meshes. Notice that the XMLS performs more closely to SPR-CX using those indicators, for unstructured than for structured meshes, especially when measuring the mean value of the effectivities $m\left(\|D\|\right)$. ## 7 Conclusions The aim of this paper was to assess the accuracy gains provided by 1. 1. Including relevant enrichment functions in the recovery process; 2. 2. Enforcing statical admissibility of the recovered solution. We focused on two recovery-based error estimators already available for LEFM problems using the XFEM. The first technique called SPR-CX is an enhancement of the SPR-based error estimator presented by Ródenas et al. (2008), where the stress field is split into two parts (singular and smooth) and equilibrium equations are enforced locally on patches. The second technique is the XMLS proposed by Bordas and Duflot (2007), Bordas et al. (2008) which enriches the basis of MLS shape functions, and uses a diffraction criterion, in order to capture the discontinuity along the crack faces and the singularity at the crack tip. To analyse the behaviour of the ZZ error estimator using both techniques and to assess the quality of the recovered stress field, we have evaluated the effectivity index considering problems with an exact solution. Convergence of the estimated error in the energy norm and other local error indicators are also evaluated. To analyse the influence of the special features introduced in the recovery process we have also considered two additional configurations: SPR-C and SPR-X. The results indicate that both techniques, SPR-CX and XMLS, provide error estimates that converge to the exact value and can be considered as asymptotically exact. Both techniques could be effectively used to estimate the error in XFEM approximations, while other conventional recovery procedures which do not include the enrichment functions in the recovery process have proved not to converge to the exact error. This shows (albeit only numerically) the need for the use of extended recovery techniques for accuracy assessment in the XFEM context. Better results are systematically obtained when using the SPR-CX to recover the stress field, specially in the areas close to the singular point. For all the different test cases analysed, the XMLS produced higher values of the effectivity index in the enriched area, where the SPR-CX proved to be more accurate. This can be ascribed to the fact that the SPR-CX technique recovers the singular part of the solution using the known equilibrated exact expressions for the asymptotic fields around the crack tip and enforces the fulfilment of the equilibrium equations on patches. Further work currently in progress will include the development of an equilibrated XMLS formulation, which could provide a continuous and globally equilibrated recovered stress field which can then be used to obtain upper bounds of the error in energy norm. The aim of our project is to tackle practical engineering problems such as those presented in Bordas and Moran (2006), Wyart et al. (2007) where the accuracy of the stress intensity factor is the target. The superiority of SPR- CX may then be particularly relevant. To minimise the error on the stress intensity factors we will target this error directly, through goal-oriented error estimates. This will be reported in a forthcoming publication. However, although the SPR-CX results are superior in general, an enhanced version of the XMLS technique presented in this paper where we enforce equilibrium conditions could result useful to evaluate upper error bounds when considering goal-oriented error estimates, as it directly produces a continuous equilibrated recovered stress field. Our next step will be the comparison of available error estimators in three dimensional settings in terms of accuracy versus computational cost to minimise the error on the crack path and damage tolerance of the structure. ## 8 Acknowledgements Stéphane Bordas would like to thank the support of the Royal Academy of Engineering and of the Leverhulme Trust for his Senior Research Fellowship entitled “Towards the next generation surgical simulators” as well as the support of EPSRC under grant EP/G042705/1 Increased Reliability for Industrially Relevant Automatic Crack Growth Simulation with the eXtended Finite Element Method. This work has been carried out within the framework of the research projects DPI 2007-66773-C02-01, DPI2010-20542 and DPI2010-20990 of the Ministerio de Ciencia e Innovación (Spain). Funding from FEDER, Universitat Politècnica de València and Generalitat Valenciana is also acknowledged. ## References ABAQUS (2009) ABAQUS (2009). ABAQUS/standard user’s manual, v.6.9. Hibbitt, Karlsson & Sorensen, Inc., Pawtucket, Rhode Island. * Ainsworth and Oden (2000) Ainsworth, M. and Oden, J. T. (2000). _A posteriori_ Error Estimation in Finite Element Analysis. John Wiley & Sons, Chichester. * Babuška et al. (1994a) Babuška, I., Strouboulis, T., and Upadhyay, C. S. (1994a). A model study of the quality of _a posteriori_ error estimators for linear elliptic problems. Error estimation in the interior of patchwise uniform grids of triangles. Computer Methods in Applied Mechanics and Engineering, 114(3-4):307–378. * Babuška et al. (1997) Babuška, I., Strouboulis, T., and Upadhyay, C. S. (1997). A model study of the quality of _a posteriori_ error estimators for finite element solutions of linear elliptic problems, with particular reference to the behaviour near the boundary. International Journal for Numerical Methods in Engineering, 40(14):2521–2577. * Babuška et al. (1994b) Babuška, I., Strouboulis, T., Upadhyay, C. S., Gangaraj, J., and Copps, K. (1994b). Validation of _a posteriori_ error estimators by numerical approach. International Journal for Numerical Methods in Engineering, 37(7):1073–1123. * Banks-Sills (1991) Banks-Sills, L. (1991). Application of the finite element method to linear elastic fracture mechanics. Applied Mechanics Review, 44(10):447–461. * Béchet et al. (2005) Béchet, E., Minnebo, H., Moës, N., and Burgardt, B. (2005). Improved implementation and robustness study of the X-FEM method for stress analysis around cracks. International Journal for Numerical Methods in Engineering, 64(8):1033–1056. * Belytschko and Black (1999) Belytschko, T. and Black, T. (1999). Elastic crack growth in finite elements with minimal remeshing. International Journal for Numerical Methods in Engineering, 45(5):601–620. * Belytschko et al. (1996) Belytschko, T., Krongauz, Y., Fleming, M., Organ, D., and Liu, W. K. (1996). Smoothing and accelerated computations in the element free Galerkin method. Journal of Computational and Applied Mathematics, 74(1-2):111–126. * Bordas and Duflot (2007) Bordas, S. and Duflot, M. (2007). Derivative recovery and _a posteriori_ error estimate for extended finite elements. Computer Methods in Applied Mechanics and Engineering, 196(35-36):3381–3399. * Bordas et al. (2008) Bordas, S., Duflot, M., and Le, P. (2008). A simple error estimator for extended finite elements. Communications in Numerical Methods in Engineering, 24(11):961–971. * Bordas and Moran (2006) Bordas, S. and Moran, B. (2006). Enriched finite elements and level sets for damage tolerance assessment of complex structures. Engineering Fracture Mechanics, 73(9):1176–1201. * CENAERO (2011) CENAERO (2011). Morfeo. CENAERO, Gosselies, Belgium. * Chessa et al. (2003) Chessa, J., Wang, H., and Belytschko, T. (2003). On the construction of blending elements for local partition of unity enriched finite elements. International Journal of Numerical Methods, 57(7):1015–1038. * Duflot (2007) Duflot, M. (2007). A study of the representation of cracks with level sets. International Journal for Numerical Methods in Engineering, 70(11):1261–1302. * Duflot and Bordas (2008) Duflot, M. and Bordas, S. (2008). _A posteriori_ error estimation for extended finite element by an extended global recovery. International Journal for Numerical Methods in Engineering, 76(8):1123–1138. * Díez et al. (2004) Díez, P., Parés, N., and Huerta, A. (2004). Accurate upper and lower error bounds by solving flux-free local problems in stars. Revue européenne des éléments finis, 13(5-6-7):497. * Díez et al. (2007) Díez, P., Ródenas, J. J., and Zienkiewicz, O. C. (2007). Equilibrated patch recovery error estimates: simple and accurate upper bounds of the error. International Journal for Numerical Methods in Engineering, 69(10):2075–2098. * Fries (2008) Fries, T. (2008). A corrected XFEM approximation without problems in blending elements. International Journal for Numerical Methods in Engineering, 75(5):503–532. * Gdoutos (1993) Gdoutos, E. E. (1993). Fracture Mechanics: and Introduction. Solid Mechanics and its Applications. Kluwer Academic Publishers, Dordrecht. * Giner et al. (2005) Giner, E., Fuenmayor, F. J., Baeza, L., and Tarancón, J. E. (2005). Error estimation for the finite element evaluation of ${G}_{\text{I}}$ and ${G}_{\text{II}}$ in mixed-mode linear elastic fracture mechanics. Finite Elements in Analysis and Design, 41(11-12):1079–1104. * Gracie et al. (2008) Gracie, R., Wang, H., and Belytschko, T. (2008). Blending in the extended finite element method by discontinuous galerkin and assumed strain methods. International Journal for Numerical Methods in Engineering, 74(11):1645–1669. * Ladevèze and Pelle (2005) Ladevèze, P. and Pelle, J. P. (2005). Mastering calculations in linear and nonlinear mechanics. Mechanical engineering series. Springer Science. * Li et al. (1985) Li, F. Z., Shih, C. F., and Needleman, A. (1985). A comparison of methods for calculating energy release rates. Engineering Fracture Mechanics, 21(2):405–421. * Melenk and Babuška (1996) Melenk, J. M. and Babuška, I. (1996). The partition of unity finite element method: Basic theory and applications. Computer Methods in Applied Mechanics and Engineering, 139(1-4):289–314. * Moës et al. (1999) Moës, N., Dolbow, J., and Belytschko, T. (1999). A finite element method for crack growth without remeshing. International Journal for Numerical Methods in Engineering, 46(1):131–150. * Natarajan et al. (2010) Natarajan, S., Mahapatra, D. R., and Bordas, S. P. A. (2010). Integrating strong and weak discontinuities without integration subcells and example applications in an XFEM/GFEM framework. International Journal for Numerical Methods in Engineering, 83(3):269–294. * Panetier et al. (2010) Panetier, J., Ladevèze, P., and Chamoin, L. (2010). Strict and effective bounds in goal-oriented error estimation applied to fracture mechanics problems solved with XFEM. International Journal for Numerical Methods in Engineering, 81(6):671–700. * Pannachet et al. (2009) Pannachet, T., Sluys, L. J., and Askes, H. (2009). Error estimation and adaptivity for discontinuous failure. International Journal for Numerical Methods in Engineering, 78(5):528–563. * Pereira et al. (1999) Pereira, O. J. B. A., de Almeida, J. P. M., and Maunder, E. A. W. (1999). Adaptive methods for hybrid equilibrium finite element models. Computer Methods in Applied Mechanics and Engineering, 176(1-4):19–39. * Ródenas et al. (2006) Ródenas, J. J., Giner, E., Tarancón, J. E., and González, O. A. (2006). A recovery error estimator for singular problems using _singular+smooth_ field splitting. In Topping, B. H. V., Montero, G., and Montenegro, R., editors, Fifth International Conference on Engineering Computational Technology, Stirling, Scotland. Civil-Comp Press. * Ródenas et al. (2010a) Ródenas, J. J., González-Estrada, O. A., Díez, P., and Fuenmayor, F. J. (2010a). Accurate recovery-based upper error bounds for the extended finite element framework. Computer Methods in Applied Mechanics and Engineering, 199(37-40):2607–2621. * Ródenas et al. (2009) Ródenas, J. J., González-Estrada, O. A., Fuenmayor, F. J., and Chinesta, F. (2009). Accurate upper bound of the error in the finite element method using an equilibrated moving least squares recovery technique. In Bouillard, P. and Díez, P., editors, ADMOS 2009. IV International Conference on Adaptive Modeling and Simulation, Barcelona, Spain. CINME. * Ródenas et al. (2010b) Ródenas, J. J., González-Estrada, O. A., Fuenmayor, F. J., and Chinesta, F. (2010b). Upper bounds of the error in X-FEM based on a moving least squares (MLS) recovery technique. In Khalili, N., Valliappan, S., Li, Q., and Russell, A., editors, 9th. World Congress on Computational Mechanics (WCCM9). 4th. Asian Pacific Congress on Computational Methods (APCOM2010). Centre for Infrastructure Engineering and Safety. * Ródenas et al. (2008) Ródenas, J. J., González-Estrada, O. A., Tarancón, J. E., and Fuenmayor, F. J. (2008). A recovery-type error estimator for the extended finite element method based on singular+smooth stress field splitting. International Journal for Numerical Methods in Engineering, 76(4):545–571. * Ródenas et al. (2007) Ródenas, J. J., Tur, M., Fuenmayor, F. J., and Vercher, A. (2007). Improvement of the superconvergent patch recovery technique by the use of constraint equations: the SPR-C technique. International Journal for Numerical Methods in Engineering, 70(6):705–727. * Shih and Asaro (1988) Shih, C. and Asaro, R. (1988). Elastic-plastic analysis of cracks on bimaterial interfaces: Part I \- small scale yielding. Journal of Applied Mechanics, 8:537–545. * Stolarska et al. (2001) Stolarska, M., Chopp, D. L., Moës, N., and Belytschko, T. (2001). Modelling crack growth by level sets and the extended finite element method. International Journal for Numerical Methods in Engineering, 51(8):943–960. * Strouboulis et al. (2001) Strouboulis, T., Copps, K., and Babuška, I. (2001). The generalized finite element method. Computer Methods in Applied Mechanics and Engineering, 190(32-33):4081–4193. * Strouboulis et al. (2006) Strouboulis, T., Zhang, L., Wang, D., and Babuška, I. (2006). A posteriori error estimation for generalized finite element methods. Computer Methods in Applied Mechanics and Engineering, 195(9-12):852–879. * Szabó and Babuška (1991) Szabó, B. A. and Babuška, I. (1991). Finite Element Analysis. John Wiley & Sons, New York. * Tabbara et al. (1994) Tabbara, M., Blacker, T., and Belytschko, T. (1994). Finite element derivative recovery by moving least square interpolants. Computer Methods in Applied Mechanics and Engineering, 117(1-2):211–223. * Tarancón et al. (2009) Tarancón, J. E., Vercher, A., Giner, E., and Fuenmayor, F. J. (2009). Enhanced blending elements for XFEM applied to linear elastic fracture mechanics. International Journal for Numerical Methods in Engineering, 77(1):126–148. * Ventura (2006) Ventura, G. (2006). On the elimination of quadrature subcells for discontinuous functions in the eXtended Finite-Element Method. International Journal for Numerical Methods in Engineering, 66(5):761–795. * Wyart et al. (2007) Wyart, E., Coulon, D., Duflot, M., Pardoen, T., Remacle, J. ., and Lani, F. (2007). A substructured FE-shell/XFE-3D method for crack analysis in thin-walled structures. International Journal for Numerical Methods in Engineering, 72(7):757–779. * Xiao and Karihaloo (2006) Xiao, Q. Z. and Karihaloo, B. L. (2006). Improving the accuracy of XFEM crack tip fields using higher order quadrature and statically admissible stress recovery. International Journal for Numerical Methods in Engineering, 66(9):1378–1410. * Yau et al. (1980) Yau, J., Wang, S., and Corten, H. (1980). A mixed-mode crack analysis of isotropic solids using conservation laws of elasticity. Journal of Applied Mechanics, 47(2):335–341. * Zienkiewicz and Zhu (1987) Zienkiewicz, O. C. and Zhu, J. Z. (1987). A simple error estimation and adaptive procedure for practical engineering analysis. International Journal for Numerical Methods in Engineering, 24:337–357. * Zienkiewicz and Zhu (1992a) Zienkiewicz, O. C. and Zhu, J. Z. (1992a). The superconvergent patch recovery and _a posteriori_ error estimates. Part I: The recovery technique. International Journal for Numerical Methods in Engineering, 33(7):1331–1364. * Zienkiewicz and Zhu (1992b) Zienkiewicz, O. C. and Zhu, J. Z. (1992b). The superconvergent patch recovery and _a posteriori_ error estimates. Part II: Error estimates and adaptivity. International Journal for Numerical Methods in Engineering, 33(7):1365–1382. Corresponding author Octavio Andrés González-Estrada can be contacted at: ocgones@upv.es	arxiv-papers	2011-12-09T17:41:45	2024-09-04T02:49:25.142079	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Octavio A. Gonz\\'alez-Estrada, Juan Jos\\'e R\\'odenas, St\\'ephane P.A.\n Bordas, Marc Duflot, Pierre Kerfriden, Eugenio Giner", "submitter": "Octavio Andres Gonzalez Estrada", "url": "https://arxiv.org/abs/1112.2160" }
1112.2196	# Ultra-high-Q wedge-resonator on a silicon chip Hansuek Lee, Tong Chen, Jiang Li, Ki Youl Yang, Seokmin Jeon, Oskar Painter, and Kerry J. Vahala T. J. Watson Laboratory of Applied Physics, California Institute of Technology, Pasadena, California 91125, USA Ultra-high-Q optical resonators are being studied across a wide range of research subjects including quantum information, nonlinear optics, cavity optomechanics, and telecommunications vahala_nature ; review_vahala ; review_vahala2 ; review_ilchenko ; review_ilchenko2 ; review_comb ; cqed . Here, we demonstrate a new, resonator on-a-chip with a record Q factor of 875 million, surpassing even microtoroids toroid . Significantly, these devices avoid a highly specialized processing step that has made it difficult to integrate microtoroids with other photonic devices and to also precisely control their size. Thus, these devices not only set a new benchmark for Q factor on a chip, but also provide, for the first time, full compatibility of this important device class with conventional semiconductor processing. This feature will greatly expand the possible kinds of system on a chip functions enabled by ultra-high-Q devices. Figure 1: Micrographs and mode renderings of the wedge resonator from top and side views. (a) An optical micrograph shows a top view of a $1\,$mm diameter wedge resonator. (b) A scanning electron micrograph shows the side-view of a resonator. The insets here give slightly magnified micrographs of resonators in which the wedge angle is $12$ degrees (upper inset) and $27\,$ degrees (lower inset). (c) A rendering shows calculated fundamental mode intensity profiles in 10 degree and 30 degree wedge angle resonators at two wavelengths. As a guide, the center-of-motion of the mode is provided to illustrate how the wedge profile introduces normal dispersion that is larger for smaller wedge angles. Long photon storage time (high Q factor) in microcavities relies critically upon use of low absorption dielectrics and creation of very smooth (low scattering) dielectric interfaces. For chip-compatible devices, silica has by far the lowest intrinsic material loss. Microtoroid resonators combine this low material loss with a reflow technique in which surface tension is used to smooth lithographic and etch-related blemishes toroid . At the same time, reflow smoothing makes it very challenging to fabricate larger diameter UHQ resonators and likewise to leverage the full range of integration tools and devices available on silicon. The devices reported here attain ultra-high-Q performance using only conventional semiconductor processing methods on a silicon wafer. Moreover, the best Q performance occurs for diameters greater than $500$ microns, a size range that is difficult to access for microtoroids on account of the limitations of the reflow process. Microcombs will benefit from such a combination of UHQ and larger diameter resonators (microwave-rate free-spectral-range) to create combs that are both efficient in turn-on power and that can be self-referenced review_comb . Moreover, integrated reference cavities and ring gyroscopes are two other applications that can benefit from larger ($1-50\,$mm diameter) UHQ resonators. Fabrication control of the free- spectral range to $1:20,000$ is also demonstrated here, opening the possibility of precision repetition rate control in microcombs or precision spectral placement of modes in certain nonlinear oscillators SBS_carmon ; SBS_ivan . Earlier work considered the Q factor in a wedge-shaped resonator fabricated of silica on a silicon wafer. Q factors as high as $50$ million were obtained PRA . That approach isolated the mode from the lithographic blemishes near the outer rim of the resonator by using a shallow wedge angle. In the current work, we have boosted the optical Q by about $20$X beyond these earlier results through a combination of process improvements. These improvements make it unnecessary to isolate the mode from the resonator rim. Indeed, the highest Q factor demonstrated uses the largest wedge angles. A top-view optical micrograph is provided in figure $1$ to illustrate the basic geometry. The process flow begins with thermal oxide on silicon, followed by lithography and oxide etching with buffered hydrofluoric acid. In the insets to figure $1$, scanning electron micrographs of devices featuring $12$-degree and $27$-degree wedge angles are imaged. Empirically, the angle can be controlled through adjustment of the photoresist adhesion using commercially available adhesion promoters. The oxide disk structures function as an etch mask for an isotropic dry etch of the silicon using XeF2. During the dry etch, the silicon undercut is set so as to reduce coupling of the optical mode to the silicon support pillar. This value is typically set to about $100$ microns for $1\,$mm diameter structures and over $150$ microns for $7.5\,$mm diameter disks, however, smaller undercuts are possible while preserving ultra-high-Q performance. Further information on the processing is given in the Methods section. To measure intrinsic Q factor, devices were coupled to SMF-$28$ optical fiber using a fiber taper taper ; ideality_PRL and spectral lineshape data were obtained by tuning an external cavity semiconductor laser across the resonance while monitoring transmission on an oscilloscope. To accurately calibrate the laser scan in this measurement, a portion of the laser output was also monitored after transmission through a calibrated Mach-Zehnder interferometer having a free spectral range of $7.75$ MHz. The inset in figure $2$ shows a spectral scan obtained on a device having a record Q factor of $875$ million. In these measurements, the taper coupling was applied on the upper surface of the resonator near the center of the wedge region. Modeling shows that the fundamental mode has its largest field amplitude in this region. Moreover, this mode is expected to feature the lowest overall scattering loss resulting from the three, dielectric-air interfaces as well as from the silicon support pillar. An additional test that can be performed to verify the fundamental mode is to measure the mode index by monitoring the free-spectral-range (FSR). The fundamental mode features the largest mode index and hence smallest FSR. Figure 2: Data showing the measured Q factor plotted versus resonator diameter with oxide thickness as a parameter. The solid lines show the predicted Q factor from a model that accounts for surface roughness induced scattering loss and also material loss. The rms roughness is measured using an AFM (see Methods section for values) and the fitted bulk material loss corresponds to a Q value of $2.5$ billion. The red data points correspond to a wedge angle of $27$ degrees. All other data are obtained using a wedge angle of approximately $10$ degrees. The inset shows a spectral scan for the case of a record Q factor of $875$ million. The sinusoidal curve accompanying the spectrum is a calibration scan performed using a fiber interferometer. The typical coupled power in all measurements was maintained around $1$ microWatt to minimize thermal effects. However, there was little or no evidence of thermal effects in the optical spectrum. Typically, these appear as an asymmetry in the lineshape and also a scan-direction dependent (to higher or lower frequency) spectral linewidth. As a further check that thermal effects were negligible, ring down measurements ring_down were also performed on a range of devices for comparison to the spectral-based Q measurement. For these, the laser was tuned into resonance with the cavity and a lithium niobate modulator was used to abruptly switch off the input. The output cavity decay rate was then monitored to ascertain the cavity lifetime. Ring-down data and spectral linewidths were consistently in good agreement. This insensitivity to thermal effects is a result of the larger mode volumes of these devices in comparison to earlier work on microtoroids (for which thermal effects must be carefully monitored). The mode volumes in the present devices are typically $100-1000$X larger. Figure 3: Plot of measured free spectral range (FSR) versus the target design- value resonator diameter on a lithographic mask. The plot shows one device at each size and five different sizes. The rms variance is $2.4\,$MHz (relative variance of less than $1:4,500$). The inset shows the FSR data measured on four devices having the same target FSR. An improved variance of $0.45\,$MHz is obtained (a relative variance of $1:20,000$). Measurements showing the effects of oxide thickness and device diameter on Q factor are presented in the figure $2$ main panel. Four, oxide thicknesses are shown ($2$, $4$, $7.5$ and $10$ microns) over diameters ranging from $0.2\,$mm to $7.5\,$mm. All data points, with the exception of the red points, correspond to a wedge angle of approximately $10$ degrees. The upper most (highest Q at a given diameter) data correspond to a wedge angle of $27$ degrees. The solid curves are a model of optical loss caused by surface scattering on the upper, wedge, and lower oxide-air interfaces and by bulk- oxide loss. In the model, the surface roughness was measured independently on each of these surfaces using an atomic force microscope (AFM) (r.m.s. roughness values are given in the Methods section). The bulk optical loss of the thermal silica corresponds to a Q value of $2.5$ billion by fit to the data. The data corresponding to the $10$ degree wedge angle show that Q increases for thicker oxides and also larger diameters. Using the model, this trend can be understood to result from loss that is caused primarily by scattering at the oxide-air interfaces. Specifically, both thicker oxides and larger diameter structures feature a reduced field amplitude at the dielectric-air interface, leading to reduced scattered power. A slight, overall boost to the Q factor is possible by increasing the wedge angle. In this case, the mode experiences reduced upper and lower surface scattering as compared to the smaller angle case. A record Q factor of $875$ million for any chip based resonators is obtained under these conditions. In general, there is reasonably good agreement between the model and the data, except in the case of the thinner oxides. For these thinner structures, there is a tendency for stress-induced buckling to occur at larger radii. This is believed to create the discrepancy with the model. Figure 4: Data plot showing the effect of etch time on appearance of the “foot” region in etching of a $10$ micron thick silica layer. The foot region is a separate etch front produced by wet etch of silica that is empirically observed to adversely affect the optical Q factor. The data show that by control of the etch time the “foot” region can be eliminated. The upper-left inset is an image of the foot region and the lower right inset shows the foot region eliminated by increase of the wet etch time. The ability to lithographically define ultra-high-Q resonators as opposed to relying upon the reflow process enables a multi-order-of-magnitude improvement in precision control of resonator diameter and FSR. This feature is especially important in microcombs and also certain nonlinear sources SBS_carmon ; SBS_ivan . As a preliminary test of the practical limits of FSR control, two studies were conducted. In the first, a series of resonator diameters were set in a CAD file used to create a photo mask. A plot of the measured FSR (fundamental mode) versus CAD file target diameter is provided in figure $3$ (main panel). The variance from ideal linear behavior is $2.4\,$MHz, giving a relative variance of better than $1:4,500$ (FSR $\approx$ $11$ GHz). The inset to figure $3$ shows that for separate devices having the same target CAD file diameter, the variance is further improved to a value of $0.45\,$MHz or $1:20,000$. The Q factor for these new resonators is not only higher in an absolute sense than what has been possible with microtoroids, but it also accesses an important regime of resonator FSR that has not been possible using microtoroids. To date, the smallest FSR achieved with the toroid reflow process has been $86\,$GHz (D $=750\,\mu m$) and the corresponding Q factor was $20$ million kippenberg_PRL_2008 . The present structures attain their best Q factors for FSRs that are complementary to microtoroids (FSRs less than $100\,$GHz). This range has become increasingly important in applications like microcombs where self-referencing is important. Specifically, low turn-on power and microwave-rate repetition are conflicting requirements in these devices on account of the inverse dependence of threshold power on FSR. However, such increases can be compensated using ultra-high-Q because turn-on power depends inverse quadratically on Q OPO_vahala . The ability to manipulate normal dispersion through the wedge angle (see figure 1) can be shown to provide control over the zero dispersion point in spectral regions where silica exhibits anomalous dispersion. Ultra-high-Q performance in large area resonators is also important in rotation sensing rotation and for on- chip frequency references freq_ref ; freq_ref2 . In the former case, the larger resonator area enhances the Sagnac effect. In the latter, the larger mode volume lowers the impact of thermal fluctuations on the frequency noise of the resonator noise . The precision control of FSR is important to determine repetition rate in microcombs, and also in applications such as stimulated Brillouin lasers where a precise match of FSR to the Brillouin shift is a prerequisite for oscillation. Application of these devices to low turn-on power, microwave-rate microcombs and to high-efficiency SBS lasers will be reported elsewhere. Finally, an upper bound to the material loss of thermal silica was established in this work. The value of $2.5$ billion bodes well for further application of thermal silica to photonic devices. Methods Disks were fabricated on ($100$) prime grade float zone silicon wafers. Photo- resist was patterned using a GCA $6300$ stepper on thermally grown oxide of thickness in the range of $2-10$ microns. Post exposure bake followed in order to cure the surface roughness of photo-resist pattern which acted as an etch mask during immersion in buffered hydrofluoric solution (Transene, buffer-HF improved). Careful examination of the wet etch revealed that the vertex formed by the lower oxide and upper surface contains an etch front that is distinct from that associated with the upper surface (see “foot” region in figure $4$ inset). This region has a roughness level that is higher than any other surface and is a principle contributor to Q degradation. By extending the etch time beyond what is necessary to reach the silicon substrate, this foot region can be eliminated as shown in figure $4$. With elimination of the foot etch front, the isotropic and uniform etching characteristic of buffered hydrofluoric solution results in oxide disks and waveguides having very smooth wedge-profiles which enhance Q factors. After the conventional cleaning process to remove photo-resist and organics, silicon was isotropically etched by xenon difluoride to create an air-cladding whispering gallery resonator. An atomic force microscope was used to measure the surface roughness of the three, silica-air dielectric surfaces. For the lower surface, the resonators were detached by first etching the silicon pillar to a few microns in diameter and then removing the resonator using tape. The r.m.s. roughness values on $10$-degree wedge-angle devices are: $0.15\,$nm (upper), $0.46\,$nm (wedge), $0.70\,$nm (lower); and for $27$-degree wedge-angle devices are: $0.15\,$ nm (upper), $0.75\,$nm (wedge), $0.70\,$nm (lower). The correlation length is approximately a few hundred nm. The difference in the wedge surface roughness obtained for the large and small wedge angle cases is not presently understood. Acknowledgments We gratefully acknowledge the Defense Advanced Research Projects Agency under the iPhod and Orchid programs and also the Kavli Nanoscience Institute at Caltech. H. L. thanks the Center for the Physics of Information. ## References * (1) Vahala, K. J. Optical microcavities. _Nature_ 424, 839–846 (2003). * (2) Kippenberg, T. J. & Vahala, K. J. Cavity optomechanics: Back-action at the mesoscale. _Science_ 321, 1172–1176 (2008). * (3) Kippenberg, T. J. & Vahala, K. J. Cavity opto-mechanics. _Optics Express_ 15, 17172–17205 (2007). * (4) Matsko, A. B. & Ilchenko, V. S. Optical resonators with whispering-gallery modes-part I: basics. _IEEE J. Sel. Top. Quant._ 12, 3–14 (2006). * (5) Ilchenko, V. S. & Matsko, A. B. Optical resonators with whispering-gallery modes-part II: applications. _IEEE J. Sel. Top. Quant._ 12, 15–32 (2006). * (6) Kippenberg, T. J., Holzwarth, R. & Diddams, S. A. Microresonator-based optical frequency combs. _Science_ 332, 555–559 (2011). * (7) Aoki, T. _et al._ Observation of strong coupling between one atom and a monolithic microresonator. _Nature_ 442, 671–674 (2006). * (8) Armani, D. K., Kippenberg, T. J., Spillane, S. M. & Vahala, K. J. Ultra-high-Q toroid microcavity on a chip. _Nature_ 421, 925–929 (2003). * (9) Tomes, M. & Carmon, T. Photonic micro-electromechanical systems vibrating at X-band (11-GHz) rates. _Phys. Rev. Lett._ 102, 113601 (2009). * (10) Grudinin, I. S., Yu, N. & Maleki, L. Brillouin lasing with a CaF2 whispering gallery mode resonator. _Phys. Rev. Lett._ 102, 043902 (2009). * (11) Kippenberg, T. J., Kalkman, J., Polman, A. & Vahala, K. J. Demonstration of an erbium-doped microdisk laser on a silicon chip. _Phys. Rev. A_ 74, 051802 (2006). * (12) Cai, M., Painter, O. J. & Vahala, K. J. Observation of critical coupling in a fiber taper to silica-microsphere whispering gallery mode system. _Phys. Rev. Lett._ 74, 051802 (2006). * (13) Spillane, S. M., Kippenberg, T. J., Painter, O. J. & Vahala, K. J. Ideality in a fiber-taper-coupled microresonator system for application to cavity quantum electrodynamics. _Phys. Rev. Lett._ 91, 043902 (2003). * (14) Vernooy, D. W., Ilchenko, V. S., Mabuchi, H., Streed, E. W. & Kimble, H. J. High-Q measurements of fused-silica microspheres in the near infrared. _Optics Letters_ 23, 247–249 (1998). * (15) Del’Haye, P., Arcizet, O., Schliesser, A., Holzwarth, R. & Kippenberg, T. J. Full stabilization of a microresonator frequency comb. _Phys. Rev. Lett._ 101, 053903 (2008). * (16) Kippenberg, T. J., Spillane, S. M. & Vahala, K. J. Kerr-nonlinearity optical parametric oscillation in an ultrahigh-Q toroid microcavity. _Phys. Rev. Lett._ 93, 083904 (2004). * (17) Ciminelli, C., Dell’Olio, F., Campanella, C. & Armenise, M. Photonic technologies for angular velocity sensing. _Adv. Opt. Photon._ 2, 370–404 (2010). * (18) Matsko, A. B., Savchenkov, A. A., Yu, N. & Maleki, L. Whispering-gallery-mode resonators as frequency references. I. fundamental limitations. _J. Opt. Soc. Am. B_ 24, 1324–1335 (2007). * (19) Savchenkov, A. A., Matsko, A. B., Ilchenko, V. S., Yu, N. & Maleki, L. Whispering-gallery-mode resonators as frequency references. II. stabilization. _J. Opt. Soc. Am. B_ 24, 2988–2997 (2007). * (20) Gorodetsky, M. L. & Grudinin, I. S. Fundamental thermal fluctuations in microspheres. _J. Opt. Soc. Am. B_ 21, 697–705 (2004).	arxiv-papers	2011-12-09T20:10:40	2024-09-04T02:49:25.153919	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Hansuek Lee, Tong Chen, Jiang Li, Ki Youl Yang, Seokmin Jeon, Oskar\n Painter, and Kerry J. Vahala", "submitter": "Tong Chen", "url": "https://arxiv.org/abs/1112.2196" }
1112.2343	# Spatial confinement effects on quantum harmonic oscillator I: Nonlinear coherent state approach M. Bagheri Harouni, R. Roknizadeh, M. H. Naderi Quantum Optics Group, Department of physics, University of Isfahan, Isfahan, Iran m-baghreri@phys.ui.ac.ir rokni@sci.ui.ac.ir mhnaderi@phys.ui.ac.ir , , ###### Abstract In this paper we study some basic quantum confinement effects through investigation of a deformed harmonic oscillator algebra. We show that spatial confinement effects on a quantum harmonic oscillator can be represented by a deformation function within the framework of nonlinear coherent states theory. We construct the coherent states associated with the spatially confined quantum harmonic oscillator in a one-dimensional infinite well and examine some of their quantum statistical properties, including sub-poissonian statistics and quadrature squeezing. ## 1 Introduction The harmonic oscillator is one of the models most extensively used in both classical and quantum mechanics. The usefulness and simplicity make this model a subject of lots of studies. One of the most important aspects of quantum harmonic oscillator (QHO) is its dynamic algebra i.e. Weyl-Heisenberg algebra. This algebra appears in many areas of modern theoretical physics, as an example we notice that the one-dimensional quantum harmonic oscillator was successfully used in second quantization formalism [1]. Due to the relevance of Weyl-Heisenberg algebra, some efforts have been devoted to studying possible deformations of the QHO algebra [2]. A deformed algebra is a nontrivial generalization of a given algebra through the introduction of one or more deformation parameters, such that, in a certain limit of parameters the non-deformed algebra is recovered. A particular deformation of Heisenberg algebra has led to the notion of $f$-oscillator [3]. An $f$-oscillator is a non-harmonic system, that from mathematical point of view its dynamical variables (creation and annihilation operators) are constructed from a non canonical transformation through $\hat{A}=\hat{a}f(\hat{n})\hskip 28.45274pt,\hskip 28.45274pt\hat{A}^{{\dagger}}=f(\hat{n})\hat{a}^{{\dagger}},$ (1) where $\hat{a}$ and $\hat{a}^{{\dagger}}$ are the usual (non-deformed) harmonic oscillator operators with $[\hat{a},\hat{a}^{{\dagger}}]=1$ and $\hat{n}=\hat{a}^{{\dagger}}\hat{a}$. The function $f(\hat{n})$ is called deformation function which depends on the number of excitation quanta and some physical parameters. The presence of the operator-valued deformation function causes the Heisenberg algebra of the standard QHO to transform into a deformed Heisenberg algebra. The nonlinearity in $f$-oscillators means dependence of the oscillation frequency on the intensity [4]. On the other hand, in contrast to the standard QHO, $f$-oscillators have not equal spaced energy spectrum. For example, if we confine a simple QHO inside an infinite well, due to the spatial confinement, the energy levels constitute a spectrum that is not equal spaced. Therefore, in this case it is reasonable to investigate the corresponding $f$-oscillator. The confined QHO can be used to describe confinement effects on physical properties of confined systems. Physical size and shape of the materials strongly affect the nature, dynamics of the electronic excitations, lattice vibrations, and dynamics of carriers. For example, in the mesoscopic systems, the dimension of system is comparable with the coherence length of carriers and this leads to some new phenomena that they do not appear in a bulk semiconductor, such as quantum interference between carrier’s motion [5]. Recent progress in growth techniques and development of micromachinig technology in designing mesoscopic systems and nanostructures, have led to intensive theoretical [6] and experimental investigations [7] on electronic and optical properties of those systems. The most important point about the nanoscale structures is that the quantum confinement effects play the center- stone role. One can even say, in general, that recent success in nanofabrication technique have resulted in great interest in various artificial physical systems (quantum dots, quantum wires and quantum wells) with new phenomena driven by the quantum confinement. A number of recent experiments have demonstrated that isolated semiconductor quantum dots are capable of emitting light [8]. It becomes possible to combine high-Q optical microcavities with quantum dot emitters as the active medium [9]. Furthermore, there are many theoretical attempts for understanding the optical and electronic properties of nanostructures especially semiconductor quantum dots [10]. On the other hand, a nanostructure such as quantum dot, is a system that carrier’s motion is confined inside a small region, and during the interaction with other systems, the generated excitations such as phonons, excitons and plasmons are confined in small region. In order to describe the physical properties of these excitations one can consider them as harmonic oscillator. As another application of deformed algebra we can refer to the notion of parastatistics [11]. The concept of parastatistics has found many application in fractal statistics and anyon theory [12]. In addition to the anyon theory, the parastatistics has found many interesting application in supersymmetry and non-commutative quantum mechanics [13]. The construction of generalized deformed oscillators corresponding to well- known potentials and study of the correspondence between the properties of the conventional potential picture and the algebraic one has been done [14]. Recently, the generalized deformed algebra and its associated generalized operators have been considered [15]. By looking at the classical correspondence of the Hamiltonian, the potential energy and the effective mass function is obtained. In this contribution we derive the generalized operators associated with a definite potential by comparing the physical properties of system and physical results of generalized algebra. One of the most interesting features of the QHO is the construction of its coherent states as the eigenfunctions of the annihilation operator. As is well known [3], one can introduce nonlinear coherent states (NLCSs) or $f$-coherent states as the right-hand eigenstates of the deformed annihilation operator $\hat{A}$. It has been shown [16] that these families of generalized coherent states exhibit various non-classical properties. Due to these properties and their applications, generation of these states is a very important issue in the context of quantum optics. The $f$-coherent states may appear as stationary states of the center-of-mass motion of a trapped ion [17]. Furthermore, a theoretical scheme for generation of these states in a coherently pumped micromaser within the frame-work of intensity-dependent Jaynes-Cummings model has been proposed [18]. One of the most important questions is the physical meaning of the deformation in the NLCSs theory. It has been shown [19] that there is a close connection between the deformation function appeared in the algebraic structure of NLCSs and the non-commutative geometry of the configuration space. Furthermore, it has been shown recently [20], that a two-mode QHO confined on the surface of a sphere, can be interpreted as a single mode deformed oscillator, whose quantum statistics depends on the curvature of sphere. Motivated by the above-mentioned studies, in the present contribution we are intended to investigate the spatial confinement effects on physical properties of a standard QHO. It will be shown that the spatial confinement leads to deformation of standard QHO. We consider a QHO confined in a one-dimensional infinite well without periodic boundary conditions, and we find its energy levels, as well as associated ladder operators. We show that the ladder operators can be interpreted as a special kind of the so-called $f$-deformed creation and annihilation operators [3]. This paper is organized as follows: In section 2, we review some physical properties of $f$-oscillator and its coherent states. In section 3 we consider the spatially confined QHO in a one-dimensional infinite well and construct its associated coherent states. We shall also examine some of their quantum statistical properties, including sub-Poissonian statistics and quadrature squeezing. Finally, we summarize our conclusions in section 4. ## 2 $f$-oscillator and nonlinear coherent states In this section, we review the basics of the $f$-deformed quantum oscillator and the associated coherent states known in the literature as nonlinear coherent states. In the first step, to investigate one of the sources of deformation we consider an eigenvalue problem for a given quantum physical system and we focus our attention on the properties of creation and annihilation operators, that allow to make transition between the states of discrete spectrum of the system Hamiltonian [21]. As usual, we expand the Hamiltonian in its eigenvectors $\hat{H}=\sum_{i=0}^{\infty}E_{i}\|i\rangle\langle i\|\>,$ (2) where we have choosed $E_{0}=0$. We introduce the creation (raising) and annihilation (lowering) operators as follows $\hat{a}^{{\dagger}}=\sum_{i=0}^{\infty}\sqrt{E_{i+1}}\|i+1\rangle\langle i\|,\hskip 56.9055pt\hat{a}=\sum_{i=0}^{\infty}\sqrt{E_{i}}\|i-1\rangle\langle i\|\>,$ (3) so that $\hat{a}\|0\rangle=0$. These ladder operators satisfy the following commutation relation $[\hat{a},\hat{a}^{{\dagger}}]=\sum_{i=1}^{\infty}(E_{i+1}-E_{i})\|i\rangle\langle i\|\>.$ (4) Obviously if the energy spectrum is equally spaced that is, it should be linear in quantum numbers, as in the case of ordinary QHO, then $E_{i+1}-E_{i}=c$, where $c$ is a constant and in this condition the commutator of $\hat{a}$ and $\hat{a}^{{\dagger}}$ becomes a constant (a rescaled Weyl-Heisenberg algebra). On the other hand, if the energy spectrum is not equally spaced, the ladder operators of the system satisfy a deformed Heisenberg algebra, i.e. their commutator depends on the quantum numbers that appear in the energy spectrum. This is one of the most important properties of the quantum $f$-oscillators [3]. An $f$-oscillator is a non-harmonic system characterized by a Hamiltonian of the harmonic oscillator form $\hat{H}_{D}=\frac{\Omega}{2}(\hat{A}\hat{A}^{{\dagger}}+\hat{A}^{{\dagger}}\hat{A})\hskip 28.45274pt(\hbar=1)\>,$ (5) ($\hat{A}=\hat{a}f(\hat{n})$) with a specific frequency $\Omega$ and deformed boson creation and annihilation operators defined in (1). The deformed operators obey the commutation relation $[\hat{A}\>,\>\hat{A}^{{\dagger}}]=(\hat{n}+1)f^{2}(\hat{n}+1)-\hat{n}f^{2}(\hat{n})\>.$ (6) The $f$-deformed Hamiltonian $\hat{H}_{D}$ is diagonal on the eingenstates $\|n\rangle$ in the Fock space and its eigenvalues are $E_{n}=\frac{\Omega}{2}[(n+1)f^{2}(n+1)+nf^{2}(n)].$ (7) In the limit $f\rightarrow 1$, the ordinary expression $E_{n}=\Omega(n+\frac{1}{2})$ and the usual (non-deformed) commutation relation $[\hat{a}\>,\>\hat{a}^{{\dagger}}]=1$ are recovered. Furthermore, by using the Heisenberg equation of motion with Hamiltonian (5) $i\frac{d\hat{A}}{dt}=[\hat{A}\>,\>\hat{H}_{D}],$ (8) we obtain the following solution for the $f$-deformed operators $\hat{A}$ and $\hat{A}^{{\dagger}}$ $\hat{A}(t)=e^{-i\Omega G(\hat{n})t}\hat{A}(0),\hskip 28.45274pt\hat{A}^{{\dagger}}(t)=\hat{A}^{{\dagger}}(0)e^{i\Omega G(\hat{n})t},$ (9) where $G(\hat{n})=\frac{1}{2}\left((\hat{n}+2)f^{2}(\hat{n}+2)-\hat{n}f^{2}(\hat{n})\right).$ (10) In this sense, the $f$-deformed oscillator can be interpreted as a nonlinear oscillator whose frequency of vibrations depends explicitly on its number of excitation quanta [4]. It is interesting to point out that recent studies have revealed strictly physical relationship between the nonlinearity concept resulting from the $f$-deformation and some nonlinear optical effects, e.g., Kerr nonlinearity, in the context of atom-field interaction [22]. The nonlinear transformation of the creation and annihilation operators leads naturally to the notion of nonlinear coherent states or $f$-coherent states. The nonlinear coherent states $\|\alpha\rangle_{f}$ are defined as the right- hand eigenstates of the deformed operator $\hat{A}\|\alpha\rangle_{f}=\alpha\|\alpha\rangle_{f}\>.$ (11) From Eq.(11) one can obtain an explicit form of the nonlinear coherent states in a number state representation $\|\alpha\rangle_{f}=C\sum_{n=0}^{\infty}\alpha^{n}d_{n}\|n\rangle,$ (12) where the coefficients $d_{n}$’s and normalization constant $C$ are, respectively, given by $\displaystyle d_{0}$ $\displaystyle=$ $\displaystyle 1\hskip 14.22636pt,\hskip 14.22636ptd_{n}=\left(\sqrt{n!}[f(n)]!\right)^{-1}\hskip 14.22636pt,\hskip 14.22636pt[f(n)]!=\prod_{j=1}^{n}f(j),$ $\displaystyle C$ $\displaystyle=$ $\displaystyle\left(\sum_{n=0}^{\infty}d_{n}^{2}\|\alpha\|^{2n}\right)^{\frac{-1}{2}}.$ (13) In recent years the nonlinear coherent states have been paid much attentions because they exhibit nonclassical features [16] and many quantum optical states, such as squeezed states, phase states, negative binomial states and photon-added coherent states can be viewed as a sort of nonlinear coherent states [23]. ## 3 Quantum harmonic oscillator in a one dimensional infinite well ### 3.1 $f$-deformed oscillator description of confined QHO In this section we consider a quantum harmonic oscillator confined in a one dimensional infinite well. Many attempts have been done for solving this problem (see [24],[25], and references therein). In most of those works, authors tried to solve the problem numerically. But in our consideration we try to solve the problem analytically, to reveal the relationship between the confinement effect and given deformation function. We start from the Schrödinger equation ($\hbar=1$) $\left[-\frac{1}{2m}\frac{d^{2}}{dx^{2}}+\frac{1}{2}kx^{2}+V(x)\right]\psi(x)=E\psi(x),$ (14) where $V(x)=\left\\{\begin{array}[]{ll}0&\textrm{$-a\leq x\leq a$}\\\ \infty&\textrm{elsewhere}.\end{array}\right.$ According to the approach introduced in previous section, we can obtain raising and lowering operators from the spectrum of Schrödinger operator. On the other hand, by comparing the energy spectrum of particular system with energy spectrum of general $f$-deformed oscillator (7), one could obtain deformed raising and lowering operators. Hence, we need an analytical expression for energy spectrum of the system which explicitly shows dependence on special quantum numbers. The original problem, confined QHO (14), can be solved only by using the approximation methods. When applying perturbation theory, one is usually concern with a small perturbation of an exactly solvable Hamiltonian system. In the case of confined QHO we deal with three limits. Inside the well, for small values of position we have harmonic oscillator, for large values we have an infinite well and at the positions of the boundaries the two potentials have the same power. Hence the approximation method can not lead to acceptable results. Therefore, we model the original problem by a model potential that has mathematical behavior such as confined QHO. Instead of solving the Schrödinger equation for the QHO confined between infinite rectangular walls in positions $\pm a$, we propose to solve the eigenvalue problem for the potential $V(x)=\frac{1}{2}k\left(\frac{\tan(\delta x)}{\delta}\right)^{2}\>,$ (15) where $\delta=\frac{\pi}{2a}$, is a scaling factor depending on the width of the well. This potential models a QHO placed in the center of the rectangular infinite well [26]. The potential $V(x)$ (15) fulfills two asymptotic requirements: 1) $V(x)\rightarrow\frac{1}{2}kx^{2}$ when $a\rightarrow\infty$ (free harmonic oscillator limit). 2) $V(x)$ at equilibrium position has the same curvature as a free QHO, $\left[\frac{d^{2}V}{dx^{2}}\right]_{x=0}=k$. This model potential belongs to the exactly solvable trigonometric Pöschl- Teller potentials family [27]. Stationary coherent states for special kind of this potential have been considered [28]. Now we consider the following equation $\left[-\frac{1}{2m}\frac{d^{2}}{dx^{2}}+\frac{1}{2}k\left(\frac{\tan(\delta x)}{\delta}\right)^{2}-E\right]\psi(x)=0\;.$ (16) To solve analytically this equation, we use the factorization method [29]. By changing the variable and some mathematical manipulation, the corresponding energy eigenvalues are found as $E_{n}=\gamma(n+\frac{1}{2})^{2}+\sqrt{\gamma^{2}+\omega^{2}}(n+\frac{1}{2})+\frac{\gamma}{4}\>,$ (17) where $\gamma=\frac{4\pi^{2}}{32a^{2}m}$,and $\omega=\sqrt{\frac{k}{m}}$ is the frequency of the QHO. The first term in the energy spectrum can be interpreted as the energy of a free particle in a well, the second term denotes the energy spectrum of the QHO, and the last term shifts the energy spectrum by a constant amount. It is evident that if $a\rightarrow\infty$ then $\gamma\rightarrow 0$ and the energy spectrum (17) reduces to the spectrum of a free QHO. As is clear from (17), different energy levels are not equally spaced. Hence, confining a free QHO leads to deformation of its dynamical algebra and we can interpret the parameter $\gamma$ as the corresponding deformation parameter. In Table 1 the numerical results associated with the original potential, given in Ref. [24], are compared with the generated results from the model potential under consideration. As is seen, the results are in a good agreement when boundary size is of order of characteristic length of the harmonic oscillator. The original oscillator potential when approaches to the boundaries of the well becomes infinite suddenly, while the model potential is smooth and approaches to the infinity asymptotically. Therefore, the model potential (15) is more appropriate for the physical systems. If we normalize Eq.(17) to energy quanta of the simple harmonic oscillator and introduce the new variables $n+\frac{1}{2}=h$, $\sqrt{\frac{\gamma^{2}}{\omega^{2}}+1}=\eta$, and $\gamma^{\prime}=\frac{\gamma}{\omega}$ then it takes the following form $E_{l}=\gamma^{\prime}h^{2}+\eta h+\frac{\gamma^{\prime}}{4}.$ (18) By comparing this spectrum with the energy spectrum of an $f$-deformed oscillator, given by (7), we find the corresponding deformation function as $f(\hat{n})=\sqrt{\gamma^{\prime}\hat{n}+\eta}.$ (19) Furthermore, the ladder operators associated with the confined oscillator under consideration can be written in terms of the conventional (non-deformed) operators $\hat{a}$ , $\hat{a}^{{\dagger}}$ as follows $\hat{A}=\hat{a}\sqrt{\gamma^{\prime}\hat{n}+\eta}\hskip 28.45274pt,\hskip 28.45274pt\hat{A}^{{\dagger}}=\sqrt{\gamma^{\prime}\hat{n}+\eta}\,\,\hat{a}^{{\dagger}}.$ (20) These two operators satisfy the following commutation relation $[\hat{A},\hat{A}^{{\dagger}}]=\gamma^{\prime}(2\hat{n}+1)+\eta.$ (21) It is obvious that in the limiting case $a\rightarrow\infty$ ($\gamma^{\prime}\rightarrow 0$,$\eta\rightarrow 1$), the right hand side of the above commutation relation becomes independent of $\hat{n}$, and the deformed algebra reduces to a the conventional Weyl-Heisenberg algebra for a free QHO. Classically, harmonic oscillator is a particle that attached to an ideal spring, and can oscillate with specific amplitude. When that particle be confined, boundaries can affect particle’s motion if the boundaries position be in a smaller distance in comparison with a characteristic length that particle oscillates within it. This characteristic length for the QHO is given by $\frac{\hbar}{m\omega}$ $(\hbar=1)$ , and if $2a\leq\frac{1}{m\omega}$, then the presence of the boundaries affects the behavior of QHO, otherwise it behaves like a free QHO. Therefore, one can interpret $l_{0}=\frac{1}{m\omega}$ as a scale length where the deformation effects become relevant. ### 3.2 Coherent states of confined oscillator Now, we focus our attention on the coherent states associated with the QHO under consideration. As usual, we define coherent states as the right-hand eigenstates of the deformed annihilation operator $\hat{A}\|\beta\rangle_{f}=\beta\|\beta\rangle_{f}.$ (22) From (22) and using the NLCS formalism introduced in (11)-(2) the explicit form of the corresponding NLCS of the confined QHO is written as $\|\beta\rangle_{f}=\mathcal{N}\sum_{n}\frac{\beta^{n}}{\sqrt{n!(\gamma^{\prime}n+\eta)!}}\|n\rangle,$ (23) where $\mathcal{N}=\left(\sum_{n}\frac{\|\beta\|^{2n}}{[f(n)!]^{2}n!}\right)^{-\frac{1}{2}}$ is the normalization factor, $\beta$ is a complex number, and the deformation function $f(n)$ is given by Eq.(19). The ensemble of states $\|\beta\rangle_{f}$ labeled by the single complex number $\beta$ is called a set of coherent states if the following conditions are satisfied [30]: * • normalizability $_{f}\langle\beta\|\beta\rangle_{f}=1,$ (24) * • continuity in the label $\beta$ $\|\beta-\beta^{\prime}\|\rightarrow 0\hskip 14.22636pt\Rightarrow\hskip 14.22636pt\\|\;\|\beta\rangle_{f}-\|\beta^{\prime}\rangle_{f}\\|\rightarrow 0,$ (25) * • resolution of the identity $\int_{c}d^{2}\beta\|\beta\rangle_{f}{}_{f}\langle\beta\|w(\|\beta\|^{2})=\hat{I},$ (26) where $w(\|\beta\|^{2})$ is a proper measure that ensures the completeness and the integration is restricted to the part of the complex plane where normalization converges. The first two conditions can be proved easily. For the third condition, we choose the normalization constant as $\mathcal{N}^{2}=\frac{\|\beta\|^{\eta}}{I_{\eta}^{\gamma^{\prime}}(2\|\beta\|)},$ (27) where $I_{\eta}^{\gamma^{\prime}}(x)=\sum_{s=0}^{\infty}\frac{1}{s!(\gamma^{\prime}s+\eta)!}(\frac{x}{2})^{2s+\eta},$ (28) is similar to the modified Bessel function of the first kind of the order $\eta$ with the series expansion $I_{\eta}(x)=\sum_{s=0}^{\infty}\frac{1}{s!(s+\eta)!}(\frac{x}{2})^{2s+\eta}$. Resolution of the identity of the deformed coherent states $\|\beta\rangle_{f}$ can be written as $\displaystyle\int d^{2}\beta\|\beta\rangle_{f}\langle\beta\|w(\|\beta\|)=$ $\displaystyle\pi\sum_{n}\|n\rangle\langle n\|\frac{1}{n!(\gamma^{\prime}n+\eta)!}\int_{0}^{\infty}d\|\beta\|\|\beta\|\|\beta\|^{2n}$ $\displaystyle\times\frac{\|\beta\|^{\eta}}{I_{\eta}^{\gamma^{\prime}}(2\|\beta\|)}w(\|\beta\|).$ Now we introduce the new variable $\|\beta\|^{2}=x$ and the measure $w(\sqrt{x})=\frac{8}{\pi}I_{\eta}^{\gamma^{\prime}}(2\sqrt{x})K_{m}(2\sqrt{x})x^{\frac{l}{2}},$ (30) where $K_{m}(x)$ is the modified Bessel function of the second kind of the order $m$, $m=(\gamma^{\prime}-1)n+\alpha$, and $l=(\gamma^{\prime}-1)n+1$. Using the integral relation $\int_{0}^{\infty}K_{\nu}(t)t^{\mu-1}dt=2^{\mu-2}\Gamma\left(\frac{\mu-\nu}{2}\right)\Gamma\left(\frac{\mu+\nu}{2}\right)$ [31], we obtain $\int d^{2}\beta\|\beta\rangle_{f}{}_{f}\langle\beta\|w(\|\beta\|)=\sum_{n}\|n\rangle\langle n\|=\hat{I}.$ (31) We therefore conclude that the states $\|\beta\rangle_{f}$ qualify as coherent states in the sense described by the conditions (24)-(26). We now proceed to examine some nonclassical properties of the nonlinear coherent states $\|\beta\rangle_{f}$. As an important quantity, we consider the variance of the number operator $\hat{n}$. Since for the coherent states the variance of number operator is equal to its average, deviation from the Poissonian statistics can be measured with the Mandel parameter [32] $M=\frac{(\Delta n)^{2}-\langle\hat{n}\rangle}{\langle\hat{n}\rangle}.$ (32) This parameter vanishes for the Poisson distribution, is positive for the super-Poissonian distribution (bunching effect), and is negative for the sub- Poissonian distribution (antibunchig effect). Fig. 1 shows the size dependence of the Mandel parameter for different values of $\|\beta\|^{2}$. As is seen, the Mandel parameter exhibits the sub-Poissonian statistics and with further increasing values of $a$ it is finally stabilized at an asymptotical zero value corresponding to the Poissonian statistics. In addition, the smaller the parameter $\|\beta\|^{2}$ is, the more rapidly the Mandel parameter tends to the Poissonian statistics. As another important nonclassical property we examine the quadrature squeezing. For this purpose we first consider the conventional quadrature operators $\hat{X}_{a}$ and $\hat{Y}_{a}$ defined in terms of nondeformed operators $\hat{a}$ and $\hat{a}^{\dagger}$ as [33] $\hat{X}_{a}=\frac{1}{2}(\hat{a}e^{i\phi}+\hat{a}^{{\dagger}}e^{-i\phi})\hskip 28.45274pt\hat{Y}_{a}=\frac{1}{2i}(\hat{a}e^{i\phi}-\hat{a}^{{\dagger}}e^{-i\phi}).$ (33) In this equation, $\phi$ is the phase of quadrature operators which can effectivly affect the squeezing properties. The commutation relation for $\hat{a}$ and $\hat{a}^{{\dagger}}$ leads to the following uncertainty relation $(\Delta\hat{X}_{a})^{2}(\Delta\hat{Y}_{a})^{2}\geq\frac{1}{4}\|\langle[\hat{X}_{a},\hat{Y}_{a}]\rangle\|^{2}=\frac{1}{16}.$ (34) For the vacuum state $\|0\rangle$, we have $(\Delta\hat{X}_{a})^{2}=(\Delta\hat{Y}_{a})^{2}=\frac{1}{4}$ and hence $(\Delta\hat{X}_{a})^{2}(\Delta\hat{Y}_{a})^{2}=\frac{1}{16}$. A given quantum state of the QHO is said to be squeezed when the variance of one of the quadrature components $\hat{X}_{a}$ and $\hat{Y}_{a}$ satisfies the relation $(\Delta\hat{O}_{a})^{2}<(\Delta\hat{O}_{a})^{2}_{vacuum}=\frac{1}{4}\hskip 14.22636pt(\hat{O}_{a}=\hat{X}_{a}\hskip 8.5359ptor\hskip 8.5359pt\hat{Y}_{a}).$ (35) The degree of quadrature squeezing can be measured by the squeezing parameter $s_{\hat{O}}$ defined by $s_{\hat{O}}=4(\Delta\hat{O}_{a})^{2}-1.$ (36) Then, the condition for squeezing in the quadrature component can be simply written as $s_{\hat{O}}<0$. In Fig. 2 we have plotted the parameter $s_{\hat{X}_{a}}$ corresponding to the squeezing of $\hat{X}_{a}$ with respect to the phase angle $\phi$ for three different values of $a$. As is seen, the state $\|\beta\rangle_{f}$ exhibits squeezing for different values of the confinement size, and when $a_{l}=\frac{a}{l_{0}}=2.5$, the quadrature $\hat{X}_{a}$ exhibits squeezing for all values of the phase angle $\phi$. Fig. 3 shows the plot of $s_{\hat{X}_{a}}$ versus the dimensionless parameter $a_{l}=\frac{a}{l_{0}}$ for different values of the phase $\phi$. As is seen, with the increasing value of $a_{l}\;(\frac{a}{l_{0}})$, the quadrature component tends to the zero according to the vacuum fluctuation. Let us also consider the deformed quadrature operators $\hat{X}_{A}$ and $\hat{Y}_{A}$ defined in terms of the deformed operators $\hat{A}$ and $\hat{A}^{{\dagger}}$ as $\hat{X}_{A}=\frac{1}{2}(\hat{A}e^{i\phi}+\hat{A}^{{\dagger}}e^{-i\phi}),\hskip 28.45274pt\hat{Y}_{A}=\frac{1}{2i}(\hat{A}e^{i\phi}-\hat{A}^{{\dagger}}e^{-i\phi}).$ (37) By considering the commutation relation (6) for the deformed operators $\hat{A}$ and $\hat{A}^{{\dagger}}$, the squeezing condition for the deformed quadrature operators $\hat{O}_{A}$ ($=\hat{X}_{A}$, $\hat{Y}_{A}$)can be written as $S=4(\Delta\hat{O}_{A})^{2}-\langle(\hat{n}+1)f^{2}(\hat{n}+1)\rangle+\langle\hat{n}f^{2}(\hat{n})\rangle<0.$ (38) In Fig. 4 we have plotted the parameter $S_{\hat{X}_{A}}$ versus the dimensionless parameter $\frac{a}{l_{0}}$ for three different values of $\|\beta\|^{2}$. As is seen, the deformed quadrature operator exhibits squeezing for all values of $a$. Furthermore, with the increasing value of $\|\beta\|^{2}$ the squeezing of the quadrature $\hat{X}_{A}$ is enhanced. ## 4 Conclusion In this paper, we have considered the relation between the spatial confinement effects and a special kind of $f$-deformed algebra. We have found that the confined simple harmonic oscillator can be interpreted as an $f$-oscillator, and we have obtained the corresponding deformation function. By constructing the associated NLCSs, we have examined the effects of confinement size on non- classical statistical properties of those states. The result show that the stronger confinement leads to the strengthening of non-classical properties. We hope that our approach may be used in description of phonons in the strong excitation regimes, photons in a microcavity and different elementary excitations in confined systems. The work on this direction is in progress. Acknowledgment The authors wish to thank the Office of Graduate Studies of the University of Isfahan and Iranian Nanotechnology initiative for their support. Table 1: Calculated energy levels of the confined QHO in a one dimensional infinite well by using our model potential in comparison with the numerical results given in Ref.[24] state \| boundary size \| model potential \| numerical results ---\|---\|---\|--- 0 \| a=0.5 \| 4.98495312 \| 4.95112932 0 \| 1 \| 1.41089325 \| 1.29845983 0 \| 2 \| 0.67745392 \| 0.53746120 0 \| 3 \| 0.57321464 \| 0.50039108 0 \| 4 \| 0.54003728 \| 0.50000049 1 \| a=0.5 \| 19.88966157 \| 19.77453417 1 \| 1 \| 5.46638033 \| 5.07558201 1 \| 2 \| 2.34078691 \| 1.76481643 1 \| 3 \| 1.85672176 \| 1.50608152 1 \| 4 \| 1.69721813 \| 1.50001461 2 \| a=0.5 \| 44.66397441 \| 44.45207382 2 \| 1 \| 11.98926850 \| 11.25882578 2 \| 2 \| 4.62097017 \| 3.39978824 2 \| 3 \| 3.41438455 \| 2.54112725 2 \| 4 \| 3.00861155 \| 2.50020117 3 \| a=0.5 \| 79.30789166 \| 78.99692115 3 \| 1 \| 20.97955777 \| 19.89969649 3 \| 2 \| 7.51800371 \| 5.58463907 3 \| 3 \| 5.24620303 \| 3.66421964 3 \| 4 \| 4.47421754 \| 3.50169153 4 \| a=0.5 \| 123.82141330 \| 123.41071050 4 \| 1 \| 32.43724814 \| 31.00525450 4 \| 2 \| 11.03188752 \| 8.36887442 4 \| 3 \| 7.35217718 \| 4.95418047 4 \| 4 \| 6.09403610 \| 4.50964099 ## References ## References * [1] W. Greiner, J. Reinhardt, Field Quantization (Springer-Verlag, 1996). S. Weinberg, The Quantum Theory of Fields, Vol.1 (Cambridge University Press, 1995). * [2] A. J. Macfarlane, J. Phys. A: Math. Gen. 22, 4581 (1989); L. C. Biedenharn, J. Phys. A: Math. Gen. 22, L873 (1989). * [3] V. I. Man’ko, G. Marmo, E. C. G. Sudarshan, F. Zaccaria, Phys. Scr. 55, 528 (1997). * [4] V. I. Man’ko, R. Vilela Mendes, J. Phys. A: Math. Gen. 31, 6037 (1998); V. I. Man’ko, G. Marmo, S. Solimeno, F. Zaccaria, Int. J. Mod. Phys. A 8, 3577 (1993); V. I. Man’ko, G. Marmo, S. Solimeno, F. Zaccaria, Phys. Lett. A 176, 173 (1993). * [5] D. K. Ferry, S. M. Goodnick, Transport in Nanostructures, (Cambridge University Press, 1999). B. L. Altshuler, P. A. Lee, R. A. Web, Mesoscopic Phenomena in Solids, (North- Holland, Amsterdam, 1991). * [6] H. Haug, S. W. Koch, Quantum Theory of the Optical and Electronic Properties of Semiconductors, 4th ed. (World Scientific, Singapore, 2004). * [7] P. Michler, Single Quantum Dots: Fundamental, Application, and New Concepts, Topics in applied physics (Springer, 2003). * [8] P. Michler, A. Imamoğlu, M. D. Mason, P. J. Carson, G. F. Strouse, S. K. Buratto, Nature (London)406, 968 (2000); P. Michler, A. Kiraz, C. Becher, W. V. Shoenfeld, P. M. Petroff, L. Zhang, E. Hu, A. Imamoğlu, Science 290, 2282 (2000). * [9] P. Michler, A. Kiraz, L. Zhang, C. Becher, E. Hu, A. Imamoğlu, Appl. Phys. Lett. 77, 184 (2000). * [10] T. Feldtmann, L. Schneebeli, M. Kira, S. W. Koch, Phys. Rev. B 73, 155319 (2006); C. Gies, J. Wiersing, M. Lorke, F. Jahnke, Phys. Rev A 75, 013803 (2007); K. Ahn, J. Förstner, A. Knorr, Phys. Rev. B 71, 153309 (2005). * [11] H. S. Green, Phys. Rev. 90, 270 (1953). * [12] F. Wilczek, Phys. Rev. Lett. 48, 1144 (1982). * [13] M. S. Plyushchay, Ann. Phys. 245, 339 (1996); M. S. Plyushchay, Nucl. Phys. B 491, 619 (1997); M. S. Plyushchay, Int. J. Mod. Phys. A 15, 3679 (2000); P. A. Horvathy, M. S. Plyushchay and M. Valenzuela, Nucl. Phys. B 768, 247 (2007). * [14] C. Daskaloyannis, K. Ypsilantis, J. Phys. A: Math. Gen. 25, 4157 (1992); C. Daskaloyannis, J. Phys. A: Math. Gen. 25, 2261 (1992). * [15] S. S. Mizrahi, J. P. Camargo Lima, V. V. Dodonov, J. Phys. A: Math. Gen. 37, 3707 (2004). * [16] V. I. Man’ko, G. Marmo, E. C. G. Sudarshan, and F. Zaccaria, in: N. M. Atakishiev (Ed), Proc. IV Wigner Symp. (Guadalajara, Mexico, July 1995), World Scientific, Singapore, 1996, p.421; S. Mancini, Phys. Lett. A 233, 291 (1997); S. Sivakumar, J. Phys. A: Math. Gen. 33, 2289 (2000); B. Roy, Phys. Lett. A 249, 25 (1998); H. C. Fu, R. Sasaki, J. Phys. A: Math. Gen. 29, 5637 (1996); R. Roknizadeh, M. K. Tavassoly, J. Phys. A: Math. Gen. 37, 5649 (2004); M. H. Naderi, M. Soltanolkotabi, R. Roknizadeh, J. Phys. A: Math. Gen. 37, 3225 (2004); A-S. F. Obada, G. M. Abd Al-Kader, J. Opt. B: Quantum Semiclass. Opt. 7, S635 (2005). * [17] R. L. deMatos Filho, W. Vogel, Phys. Rev. A 54, 4560 (1996). * [18] M. H. Naderi, M. Soltanolkotabi, R. Roknizadeh, Eur. Phys. J. D 32, 397 (2005). * [19] M. H. Naderi, Ph.D. thesis, University of Isfahan, Iran,2004 (unpublished). * [20] A. Mahdifar, R. Roknizadeh, M. H. Naderi, J. Phys. A: Math. Gen. 39, 7003 (2006). * [21] A. Wünsche, J. Opt. B: Quantum SemiClass. Opt. 4, 359 (2002). * [22] M. H. Naderi, M. Soltanolkotabi, R. Roknizadeh, J. Phys. Soc. Japan 73, 2413 (2004). * [23] S. Sivakumar, Phys, Lett. A 250, 257 (1998); V. I. Man’ko, A. Wünshe, Quantum Semiclass. Opt. 9, 381 (1997); G. N. Jones, J. Haight, C. T. Lee, Quantum Semiclass. Opt. 9, 411 (1997); V. V. Dodonov, Y. A. Korennoy, V. I. Man’ko, Y. A. Moukhin, Quantum Semiclass. Opt. 8, 413 (1996); Nai-le Liu, Zhi-hu Sun, Hong-yi Fan, J. Phys. A: Math. Gen. 33, 1933 (2000);B. Roy, P. Roy, J. Opt. B: Quantum Semiclass. Opt. 1, 341 (1999). * [24] G. Campoy, N. Aquino, V. D. Granados, J. Phys. A: Math. Gen. 35, 4903 (2002). * [25] V. C. Aguilera-Navarros, E. Ley Koo, A. H. Zimerman, J. Phys. A: Math. Gen. 13, 3585 (1980). * [26] C. Zicovich-Wilson, J. H. Planelles, W. Jackolski, Int. J. Quan. Chem. 50, 429 (1994). * [27] G. Pöschl and E. Teller, Z. Phys. 83, 143 (1933). * [28] J.-P. Antoine, J.-P. Gazeau, P. Monceau, J. R. Klauder and K. A. Penson, J. Math. Phys. 42, 2349 (2001). * [29] L. Infeld, T. E. Hull, Rev. Mod. Phys. 23, 21 (1951). * [30] J. R. Klauder, B. Skagerstam, Coherent states: Application in Physics and Mathematical Physics (World Scientific, Singapore, 1985); S. T. Ali, J-P. Antoine, J-P. Gazeau, Coherent states, Wavalets, and their Generalization (New York: Springer, 2000). * [31] G. N. Watson, Theory of Bessel Functions, Second ed. (Cambridge University Press, 1966) p.388. * [32] L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, 1995). * [33] M. O. Scully, M. S. Zubairy, Quantum Optics(Cambridge University Press, 1997). Figure 1: Plots of the Mandel parameter versus the dimensionless parameter $a_{l}=\frac{a}{l_{0}}$. For $\|\beta\|^{2}=0.5$ (dashed curve), for $\|\beta\|^{2}=1$ (longdashed curve), for $\|\beta\|^{2}=1.5$ (solid curve) and for $\|\beta\|^{2}=4.0$ (bold curve). Figure 2: Plot of $s_{\hat{X}_{a}}$ versus $\phi$ for $\|\beta\|^{2}=4$. The dashed, longdashed and solid curves respectively relate to $a=2.5$, $a=1$, $a=0.5$ (the values of $a$ are renormalized to $l_{0}$). Figure 3: Plots of $s_{\hat{X}_{a}}$ versus the dimensionless parameter $a_{l}=\frac{a}{l_{0}}$ for different phases and $\|\beta\|^{2}=1$. Dashed curve, solid curve and bold curve ,respectively, correspond to $\phi=100$, $\phi=110$ and $\phi=90$. Figure 4: Plots of deformed squeezing parameter $S_{X_{A}}$ versus the dimensionless parameter $a_{l}=\frac{a}{l_{0}}$. The dashed curve, longdashed curv, solid curve and bold curve are respectively, correspond to $\|\beta\|^{2}=1$, $\|\beta\|^{2}=1.5$ $\|\beta\|^{2}=2.5$ and $\|\beta\|^{2}=4$.	arxiv-papers	2011-12-11T10:52:14	2024-09-04T02:49:25.163343	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "M. Bagheri Harouni, R. Roknizadeh, M. H. Naderi", "submitter": "Malek Bagheri", "url": "https://arxiv.org/abs/1112.2343" }
1112.2344	# Nonlinear coherent state of an exciton in a wide quantum dot M. Bagheri Harouni, R. Roknizadeh, M. H. Naderi Quantum Optics Group, Physics Department, University of Isfahan m-baghreri@phys.ui.ac.ir rokni@sci.ui.ac.ir mhnaderi@phys.ui.ac.ir , , ###### Abstract In this paper, we derive the dynamical algebra of a particle confined in an infinite spherical well by using the $f$-deformed oscillator approach. We consider an exciton with definite angular momentum in a wide quantum dot interacting with two laser beams. We show that under the weak confinement condition, and quantization of the center-of-mass motion of exciton, the stationary state of it can be considered as a special kind of nonlinear coherent states which exhibits the quadrature squeezing. ## 1 Introduction The conventional coherent states of the quantum harmonic oscillator, defined by Glauber [1] as the right-hand eigenstates of non-hermitian annihilation operator $\hat{a}([\hat{a},\hat{a}^{{\dagger}}]=1)$, have found many interesting applications in different areas of physics such as quantum optics, condensed matter physics, statistical physics and atomic physics [2]. These states play an important role in the quantum theory of coherence, are considered as the most classical ones among the pure quantum states, and laser light can be supposed as a physical realization of them. Due to the vast application of these states, there have been many attempts to generalize them [3]. Among the all generalizations, nonlinear coherent states (NLCS) [4] have been paid attention in recent years because they exhibit nonclassical features such as quadrature squeezing and sub-poissonian statistics [5]. These states are defined as the right-hand eigenstates of a deformed operator $\hat{A}$ $\hat{A}=\hat{a}f(\hat{n})\hskip 56.9055pt\hat{A}\|\alpha,f\rangle=\alpha\|\alpha,f\rangle,$ (1) where the deformation function $f(\hat{n})$ is an operator-valued function of the number operator $\hat{n}$. From (1) one can obtain an explicit form of NLCS in the number state representation $\displaystyle\|\alpha,f\rangle=\mathcal{N}_{f}\sum_{n}\frac{\alpha^{n}}{\sqrt{n!}f(n)!}\|n\rangle,$ $\displaystyle\mathcal{N}_{f}=\left[\sum_{n}\frac{\|\alpha\|^{2n}}{[f(n)!]^{2}n!}\right]^{-\frac{1}{2}}.$ (2) A class of NLCS can be realized physically as the stationary state of the center-of-mass motion of a laser driven trapped ion [6, 7]. Furthermore, it has been proposed a theoretical scheme to show the possibility of generating various families of NLCS [8] of the radiation field in a lossless coherently pumped micromaser within the frame work of the intensity-dependent Jaynes- Cummings model. Recently, the influences of the spatial confinement [9] and the curvature of physical space [10] on the algebraic structure of the coherent states of the quantum harmonic oscillator have been investigated within the frame work of nonlinear coherent states approach. It has been shown that if a quantum harmonic oscillator be confined within a small region of order of its characteristic length [9] or its physical space to be a sphere [10], then it can be regarded as a deformed oscillator, i.e., an oscillator that its creation and annihilation operators are deformed operator $\hat{A}$ and $\hat{A}^{{\dagger}}$ given by Eq.(1). On the other hand, we can consider nanostructures as systems whose physical properties are related to the confinement effects. Thus, we expect that it is possible to realize some natural deformations in these systems [9, 11]. In addition, in nanostructures different kinds of quantum states can be prepared. One of the most applicable of these states is exciton state. Exciton is an elementary excitation in semiconductors interacting with light, electron in conduction band which is bounded to hole in valance band that can easily move through the sample. In one of the nano size systems, quantum dot (QD), due to the confinement in three dimensions, energy bands reduce to quasi energy levels. Therefore, in order to describe the interaction of QD with light we can consider it as a few level atom [12]. These Exciton states can be used in quantum information processes. It has been shown that excitons in coupled QDs are ideal for preparation of entangled state in solid-state systems [13]. Entanglement of the exciton states in a single QD or in a QD molecule has been demonstrated experimentally [14]. Entanglement of the coherent states of the excitons in a system of two coupled QDs has been considered [15]. Recently, coherent exciton states of excitonic nano-crystal-molecules has been considered [16]. Theoretical approach for generating Dick states of excitons in optically driven QD has been proposed in Ref.[17]. In a QD, the effects of exciton-phonon interaction, exciton-impurity interaction and exciton-exciton interaction play an important role. These effects are the main sources for the decoherence of exciton states [18]. Furthermore, these effects cause the exciton has the spontaneous recombination or scattered to other exciton modes [19, 20]. In this paper we propose a theoretical scheme for generating excitonic NLCS. We will show that under certain conditions the quantized motion of wave packet of center-of-mass of exciton can be consider as a special kind of NLCSs. Our scheme is based on the interaction of a quantum dot with two laser beams. By using the approach considered in Ref.[6], we propose a theoretical scheme for generation of NLCS of an exciton in a wide QD. In section 2, we consider different confinement regimes in a QD, and the explicit forms of the creation and annihilation operators for a particle confined in an infinite well are derived by using the deformed quantum oscillator approach. In section 3, we consider an exciton in a wide QD which interacts with two laser beams. We shall show that under the weak confinement condition, the stationary state of the exciton center-of-mass motion can be considered as a NLCS. ## 2 Algebraic approach for a particle in an infinite spherical well In nanostructures and confined systems, there are three different confinement regimes. The criteria for this classification is based on the comparison between excitation Bohr radius and the spatial dimensions of the system under consideration. In the case of a QD, these regimes are defined as follows [21]. We first introduce three quantities $\Delta E_{c}$, $\Delta E_{v}$ and $V_{exc}$ which, respectively denote: the electron energy due to the confinement, the hole energy due to the confinement and Coulomb energy between correlated electron-hole (exciton). 1) $V_{exc}>\Delta E_{c}-\Delta E_{v}$: In this case, the exciton energy is much greater than the confinement energies of electron and hole. If we show the system size by $L$ and the exciton Bohr radius by $a$, then in this regime $L>a$. This regime corresponds to the weak confinement (in some literature the weak confinement is characterized by the situation in which the electron and the hole are not in the same matter, for example, hole be in QD and excited electron in host matter. In this paper, by the weak confinement regime we mean $L>a$ and the excitations in the same matter). In this regime due to the confinement, the center-of-mass motion of the exciton is quantized and the confinement do not affect electron and hole separately. Hence, the confinement affect the exciton motion as a whole [22]. 2) $V_{exc}<\Delta E_{c}\;,\Delta E_{v}$: This regime, in contrast to the previous one, is associated with the cases where $L<a$. In this regime the exciton is completely localized, and the confinement affects both the electron and the hole independently and their states become quantized in conduction and valance bands. This regime is called strong confinement. 3) $\Delta E_{c}>V_{exc}\;,\Delta E_{v}$: This condition is equivalent to the situation $a_{c}<a<a_{v}$, where $a_{c}$ and $a_{v}$ are, respectively, the Bohr radii of electron and hole. Here, due to the different effective masses of electron and hole, the hole which has heavier effective mass is localized and the electron motion will be quantized. This regime is called intermediate confinement. In the first case (weak confinement), in a wide QD, an exciton can move due to its center-of-mass momentum, and because of the presence of the barriers, its center-of-mass motion is quantized. Therefore, it moves as a whole between energy levels of an infinite well. We consider a wide spherical QD whose energy levels are equivalent to the energy levels of a spherical well $E_{nl}=\frac{\hbar^{2}}{2M}\frac{\alpha_{nl}^{2}}{R^{2}},$ (3) where $\alpha_{nl}$ is the n’th zero of the first kind Bessel function of order $l$, $j_{l}(x)$. In this energy spectrum according to the azimuthal symmetry around $z$ axis, we have a degenerate spectrum. As mentioned before, in the weak confinement regime, the Coulomb potential plays an essential role and its spectrum is given by $E_{k}^{b}=\frac{\mu e^{4}}{2\hbar^{2}\varepsilon^{2}}\frac{1}{k^{2}}\hskip 14.22636pt,\hskip 14.22636pt\mu=\frac{m_{e}m_{h}}{m_{e}+m_{h}},$ (4) where superscript $b$ shows binding energy related to the Coulomb interaction and $\varepsilon$ shows dielectric constant of the system. As is usual, we interpret the Coulomb part as an exciton and another degree of freedom (motion between energy levels of the well) as the exciton center-of-mass motion. Therefore, in a wide QD an exciton has two different kinds of degrees of freedom: internal degrees of freedom due to the Coulomb potential and external degrees of freedom related to the quantum confinement. Here we consider the lowest exciton state, $1s$ exciton, because this exciton state has the largest oscillator strength among other exciton state. Then the energy of the exciton in a wide QD can be written as $E_{nlm}=E_{g}-E_{1}^{b}+\frac{\hbar^{2}}{2MR^{2}}\alpha_{nl}^{2},$ (5) where $E_{g}$ is the energy gap of QD, $E_{1}^{b}=E_{k}^{b}\|_{k=1}$ is the exciton binding energy, $M=m_{e}+m_{h}$ is the total mass of exciton, and $R$ is the radius of QD. Due to the relation of quantum numbers $l$ and $m$ with the angular momentum and the selection rules for optical transitions, we can fix $l$ and $m$ (by choosing a certain condition), and hence the energy of exciton depends only on a single quantum number $E_{n}=E_{g}-E_{1}^{b}+\frac{\hbar^{2}}{2MR^{2}}\alpha_{nl}^{2}.$ (6) Therefore, we can prepare the conditions under which the exciton center-of- mass motion has a one-dimensional degree of freedom. Due to the quantization of the exciton center-of-mass motion, we can describe the exciton motion between the energy levels by the action of a special kind of ladder operators. In order to find these operators we use the $f$-deformed oscillator approach [4]. As mentioned elsewhere [9], if the energy spectrum of the system is equally spaced, such as harmonic oscillator, its creation and annihilation operators satisfy the ordinary Weyl-Heisenberg algebra, otherwise we can interpret them as the generators of a generalized Weyl-Heisenberg algebra. The energy spectrum of a particle with mass $M$ confined in an infinite spherical well can be written as (3). According to the conservation of angular momentum, we assume that particle has been prepared with definite angular momentum (for example by measuring its angular momentum). Then $l$ becomes completely determined, i.e., in the energy spectrum the number $l$ is a constant. By determining the number $l$ and considering the rotational symmetry of the system around the $z$ axis, the angular part of the spectrum becomes completely determined, and the radius part is described by (3). Now we use a factorization method and write the Hamiltonian of the center-of-mass motion of the system as follows $\hat{H}=\frac{1}{2}(\hat{A}\hat{A}^{{\dagger}}+\hat{A}^{{\dagger}}\hat{A}),$ (7) where $\hat{A}$ and $\hat{A}^{{\dagger}}$ are defined through the relation (1). Therefore the spectrum of $\hat{H}$, after straightforward calculation, is obtained as $E_{n}=\frac{1}{2}[(n+1)f^{2}(n+1)+nf^{2}(n)].$ (8) By comparing (8) with Eq.(3) we arrive at the following expression for the corresponding deformation function $f_{1}(\hat{n})$ $f_{1}(n)=\sqrt{\frac{\hbar^{2}}{MR^{2}}\frac{(-1)^{n}}{n}\sum_{i=1}^{n}(-1)^{i}\alpha_{i-1l}^{2}}.$ (9) Then, the ladder operators associated with the radial motion of a confined particle in a spherical infinite well is given by $\displaystyle\hat{A}=\hat{a}\sqrt{\frac{\hbar^{2}}{MR^{2}}\frac{(-1)^{n}}{n}\sum_{i=1}^{n}(-1)^{i}\alpha_{i-1l}^{2}},$ (10) $\displaystyle\hat{A}^{{\dagger}}=\sqrt{\frac{\hbar^{2}}{MR^{2}}\frac{(-1)^{n}}{n}\sum_{i=1}^{n}(-1)^{i}\alpha_{i-1l}^{2}}\,\,\hat{a}^{{\dagger}}.$ These two deformed operators obey the following commutation relation $[\,\hat{A}\,,\,\hat{A}^{{\dagger}}]=-nf_{1}^{2}(n)+\frac{\hbar^{2}}{MR^{2}}\alpha_{nl}^{2}.$ (11) As is usual in the $f$-deformation approach, for a particular limit of the corresponding deformation parameter, the deformed algebra should be reduced to the conventional oscillator algebra. However, in this treatment we note that there is no thing in common between the harmonic oscillator potential and an infinite spherical well. Only in the limit $R\rightarrow\infty$, the system reduces to a free particle which has continuous spectrum. As a result, in this section we conclude that the radial motion of a particle confined in a three-dimensional infinite spherical well can be interpreted by an $f$-deformed Weyl-Heisenberg algebra. ## 3 Exciton dynamics in QD Now we consider the formation of an exciton and its dynamics in a wide QD during the exciton lifetime. As mentioned before, in this situation the center-of-mass motion of the exciton is quantized. The exciton is created during the interaction of a QD with light, and because of the angular momentum conservation, the exciton has a well-defined angular momentum. The exciton is a quasiparticle composed of an electron and a hole and thus the exciton spin state can be in a singlet state or a triplet state. According to the optical transition selection rules, the triplet state is optically inactive and is called dark exciton [23]. By adding spin and angular momentum of absorbed photons, the angular momentum of the exciton state can be determined. Hence, the exciton behaves like a particle in a spherical well with the definite angular momentum. According to the previous section, the center-of-mass motion of the exciton in the QD and the barriers of QD can be described by an oscillator-like Hamiltonian expressed in terms of the $f$-deformed annihilation and creation operators given by Eq.(10) $H_{well}=\frac{1}{2}(\hat{A}\hat{A}^{{\dagger}}+\hat{A}^{{\dagger}}\hat{A}),$ (12) where we interpret the operator $\hat{A}$ $(\hat{A}^{{\dagger}})$ as the operator whose action causes the transition of exciton center-of-mass motion to a lower (an upper) energy state. In fact the Hamiltonian (12) is related to the external degree of freedom of exciton. On the other hand, one can imagine QD as a two-level system with the ground state $\|g\rangle$ and the excited state $\|e\rangle$ (associated with the presence of exciton). Thus, for the internal degree of freedom we can consider the following Hamiltonian $H_{ex}=\hbar\omega_{ex}\hat{S}_{22},$ (13) where $\hat{S}_{22}=\|e\rangle\langle e\|-\|g\rangle\langle g\|$ and $\hbar\omega_{ex}=E_{g}-E_{1}^{b}$ is the exciton energy. We consider a single exciton of frequency $\omega_{ex}$ confined in a wide QD interacting with two laser fields, respectively, tuned to the internal degree of freedom of the frequency $\omega_{ex}$ and to the non-equal spaced energy levels of the infinite well. It is necessary that the second laser has special conditions, because it should give rise to the transitions between energy levels whose frequencies depend on intensity. The interacting system can be described by the Hamiltonian $\hat{H}=\hat{H}_{0}+\hat{H}_{int},$ (14) where $\hat{H}_{0}=\hat{H}_{well}+\hat{H}_{ex}$ and $H_{int}=g[E_{0}e^{-i(k_{0}\hat{x}-\omega_{ex}t)}+E_{1}e^{-i(k_{1}\hat{x}-(\omega_{ex}-\omega_{\overline{n}})t)}]\hat{S}_{12}+H.c.,$ (15) in which $g$ is the coupling constant, $k_{0}$ and $k_{1}$ are the wave vectors of the laser fields, $\hat{S}_{12}=\|g\rangle\langle e\|$ is the exciton annihilation operator, and $\omega_{\overline{n}}$ is the frequency of exciton transition between energy levels of QD due to the spatial confinement. Here, we consider transition between specific side-band levels hence, we show the frequency transition with definite dependence to $n$. We show this by a c-number quantity $\overline{n}$. The exciton has a finite lifetime that in systems with small dimension, is increased [24]. The interaction with phonons is the main reason of damping of the exciton [25]. Phonons in bulk matter have a continuous spectrum while in a confined system such as QD their spectrum becomes discrete. Hence in a QD, the resonant interaction between the exciton and phonons decreases and in this system the exciton lifetime will increase. Therefore during the lifetime of an exciton, its dynamics is under influence of a bath reservoir, and its damping play an important role. We assume that during the presence of the exciton in QD, it interacts with the reservoir and hence we can consider its steady state. We consider an exciton in dark state. Experimental preparation methods of such exciton has been described in [23]. In this situation lifetime of exciton will increase and exciton has not spontaneously recombination radiation. However, its interaction with phonons causes a finite lifetime for it. The operator of the center-of-mass motion position $\hat{x}$ of the exciton in a spherical QD may be defined as $\hat{x}=\frac{\kappa}{k_{ex}}(\hat{A}+\hat{A}^{{\dagger}}),$ (16) where $\kappa$ being a parameter similar to the Lamb-Dick parameter in ion trapped systems and is defined as the ratio of QD radius to the wavelength of the driving laser (because of the spatial confinement of exciton, its wave function width is determined by the barriers of QD), and we assume $k_{0}\simeq k_{1}\simeq k_{ex}$ ($k_{ex}$ is the wavevector of the exciton). The operators $\hat{A}$ and $\hat{A}^{{\dagger}}$ are defined in Eq.(10). The interaction Hamiltonian (15) can be written as $H_{int}=\hbar e^{i\omega_{ex}t}\Omega_{1}\left[\frac{\Omega_{0}}{\Omega_{1}}+e^{-i\omega_{\overline{n}}t}\right]e^{i\kappa(\hat{A}+\hat{A}^{{\dagger}})}\hat{S}_{12}+H.c.,$ (17) where $\Omega_{0}=\frac{gE_{0}}{\hbar}$ and $\Omega_{1}=\frac{gE_{1}}{\hbar}$ are the Rabi frequencies of the lasers, respectively, tuned to the electronic transition of QD (internal degree of freedom) and the first center-of-mass motion transition of exciton. Since the external degree of freedom is definite, then $\omega_{\overline{n}}$ depends on a special value of $n$ such that it can be consider as a c-number quantity. The frequency $\omega_{\overline{n}}$ is depend on the number of quanta for each transition and hence the laser tuned to the center-of-mass motion must be so strong that causes transition. This allows us to treat the interaction of the confined exciton in a wide QD with two lasers separately, by using a nonlinear Jaynes- Cummings Hamiltonian [26] for each coupling. The interaction Hamiltonian in the interaction picture can be written as $H_{I}=\hbar\Omega_{1}\hat{S}_{12}\left[\frac{\Omega_{0}}{\Omega_{1}}+e^{i\omega_{\overline{n}}t}\right]\exp[i\kappa(e^{-i\omega_{\hat{n}}t}\hat{A}+\hat{A}^{{\dagger}}e^{i\omega_{\hat{n}}t})]+H.c.,$ (18) where $\omega_{\hat{n}}=\frac{1}{2\hbar}[(\hat{n}+2)f_{1}(\hat{n}+2)-\hat{n}f_{1}(\hat{n})]$. By using the vibrational rotating wave approximation [6], applying the disentangling formula introduced in [27], and using the fact that in the present case the Lamb-Dick parameter is small, the interaction Hamiltonian (18) is simplified to $H_{I}^{(1)}=\hbar\Omega_{1}\hat{S}_{12}\left[F_{0}(\hat{n},\kappa)\frac{\Omega_{0}}{\Omega_{1}}+i\kappa F_{1}(\hat{n},\kappa)\hat{a}\right]+H.c.,$ (19) where the function $F_{i}(\hat{n},\kappa)\;(i=0,1)$ is defined by $\displaystyle F_{i}(\hat{n},\kappa)$ $\displaystyle=$ $\displaystyle e^{-\frac{\kappa^{2}}{2}((n+1+i)f_{1}^{2}(n+1+i)-(n+i)f_{1}^{2}(n+i))}\times$ $\displaystyle\sum_{l=0}^{n}\frac{\left(i\kappa\right)^{2l}}{l!(l+i)!}\frac{f_{1}(\hat{n})f_{1}(\hat{n}+i)}{[f_{1}(\hat{n}-l)!]^{2}}(\hat{a}^{{\dagger}})^{l}\hat{a}^{l}.$ It should be noted that this function in the limit $f_{1}(\hat{n})\rightarrow 1$ (which is equivalent to the harmonic confinement) is proportional to the associated Laguerre polynomials $F_{i}(\hat{n},\kappa)\|_{f_{1}(\hat{n})\rightarrow 1}=\frac{e^{-\frac{\kappa^{2}}{2}}}{\hat{n}+i}L_{\hat{n}}^{i}\left(\kappa^{2}\right).$ (21) Now we write the function $F_{i}(\hat{n},\kappa)$ (3) $F_{i}(\hat{n},\kappa)=\frac{e^{-\frac{\kappa^{2}}{2}((n+1+i)f_{1}^{2}(n+1+i)-(n+i)f_{1}^{2}(n+i))}}{\hat{n}+i}f_{1}(\hat{n})!f_{1}(\hat{n}+1)!L_{f,\hat{n}}^{i}\left(\kappa^{2}\right),$ (22) where the function $L_{f,\hat{n}}^{i}(x)$ is defined as $L_{f,\hat{n}}^{i}(x)=\sum_{l=0}^{n}\frac{1}{[f_{1}(\hat{n}-l)!]^{2}}\frac{(\hat{n}+i)!}{(\hat{n}-l)!l!(l+i)!}(-x)^{l}.$ (23) This function is similar to the associated Laguerre function. The time evolution of the system under consideration is characterized by the master equation $\frac{d\hat{\rho}}{dt}=-\frac{i}{\hbar}[\hat{H}_{I}^{(1)},\hat{\rho}]+\mathfrak{L}\hat{\rho},$ (24) where $\mathfrak{L}\hat{\rho}$ defines damping of the system due to the different kinds of interactions which lead to annihilation of exciton. We assume a bosonic reservoir that causes damping of exciton system. Due to the properties of dark exciton, the rate of spontaneous recombination and hence spontaneous emission is decrease. On the other hand, interactions of exciton- phonon and exciton-impurities cause the exciton to be damped. In fact in low temperatures it is possible to ignore the phonon effects and by assuming a pure system we neglect the impurity effects. Hence we can write $\mathfrak{L}\hat{\rho}=\frac{\Gamma}{2}(2\hat{b}\hat{\rho}\hat{b}^{{\dagger}}-\hat{b}^{{\dagger}}\hat{b}\hat{\rho}-\hat{\rho}\hat{b}^{{\dagger}}\hat{b}),$ (25) where $\Gamma$ is the energy relaxation rate, $\hat{b}$ and $\hat{b}^{{\dagger}}$ are the annihilation and creation operators of the reservoir. Due to the confinement and dark state properties, spontaneous recombination of exciton decreases and hence the lifetime of exciton becomes so long that we can consider the stationary solution of Eq.(24). We assume a finite lifetime for exciton, and during this time we neglect damping effects. The stationary solution of the master equation (24) in the time scales of our interest is $\hat{\rho}=\|e\rangle\|\psi\rangle\langle\psi\|\langle e\|,$ (26) where $\|e\rangle$ is the electronic excited state correspond to the presence of exciton and $\|\psi\rangle$ is the center-of-mass motion state of the exciton, which can be considered as a right-hand eigenstate of the deformed operator $\hat{A}=\frac{F_{1}(\hat{n},\kappa)}{F_{0}(\hat{n},\kappa)}\hat{a}$ $\frac{F_{1}(\hat{n},\kappa)}{F_{0}(\hat{n},\kappa)}\hat{a}\|\psi\rangle=\frac{i\Omega_{0}}{\Omega_{1}\kappa}\|\psi\rangle.$ (27) According to Eq.(22) the corresponding deformation function reads as $\displaystyle f(\hat{n})$ $\displaystyle=$ $\displaystyle\frac{F_{1}(\hat{n}-1,\kappa)}{F_{0}(\hat{n}-1,\kappa)}$ $\displaystyle=$ $\displaystyle\frac{f_{1}(\hat{n})L_{f,\hat{n}-1}^{1}(\kappa^{2})}{nL_{f,\hat{n}-1}^{0}(\kappa^{2})}e^{-\frac{\kappa^{2}}{2}\left((n+1)f_{1}^{2}(n+1)-(n-1)f_{1}^{2}(n-1)\right)}.$ Hence, we can express the state $\|\psi\rangle$ in the Fock space representation as $\|\psi\rangle=\mathcal{N}_{f}\sum_{n}\frac{\chi^{n}}{\sqrt{n!}f(n)!}\|n\rangle,$ (29) where $\chi=\frac{i\Omega_{0}}{\kappa\Omega_{1}}$. According to the definition (1), it is evident that the state $\|\psi\rangle$ can be regarded as a special kind of NLCS. As is seen from equation (27), the eigenvalues of the deformed operator $\hat{A}$ depends on some physical parameters such as the Rabi frequencies, the parameter $\kappa$ and radius of QD. As is clear from equation (3), the deformation function $f(\hat{n})$ depends on the quantum number $\hat{n}$ and physical parameters such as QD radius and $\kappa$ which characterizes the confinement regime. In the limit $f_{1}(\hat{n})\rightarrow 1$, (harmonic confinement), which corresponds, for example, to a QD in lens shape [28], the function $L_{f,\hat{n}}^{i}$ reduces to the ordinary associated Laguerre polynomials, its argument tends to $\kappa^{2}$ and therefore, the deformation function (3) takes the following form $f(\hat{n})=e^{-\kappa^{2}}L_{\hat{n}}^{1}(\kappa^{2})[(\hat{n}+1)L_{\hat{n}}^{0}(\kappa^{2})]^{-1}.$ (30) This is the deformation function that appears in the center-of-mass motion of a trapped ion confined in a harmonic trap [6]. In order to investigate the nonclassical behavior of the NLCS $\|\psi\rangle$ we consider the quadrature squeezing of the center-of-mass motion. For this purpose, we define the deformed quadratures operators as follows $\hat{X}_{1}=\frac{1}{2}(\hat{A}e^{i\phi}+\hat{A}^{{\dagger}}e^{-i\phi}),\hskip 28.45274pt\hat{X}_{2}=\frac{1}{2i}(\hat{A}e^{i\phi}-\hat{A}^{{\dagger}}e^{-i\phi}).$ (31) In the limiting case $f(\hat{n})\rightarrow 1$, these two operators reduce to the conventional (non-deformed) quadrature operators [29]. The commutation relation of $\hat{X}_{1}$ and $\hat{X}_{2}$ is $[\hat{X}_{1},\hat{X}_{2}]=\frac{i}{2}[(\hat{n}+1)f^{2}(\hat{n}+1)-\hat{n}f^{2}(\hat{n})].$ (32) The variances $\langle(\Delta\hat{X}_{i})^{2}\rangle\equiv\langle\hat{X}_{i}^{2}\rangle-\langle\hat{X}_{i}\rangle^{2}(i=1,2)$ satisfy the uncertainty relation $\langle(\Delta\hat{X}_{1})^{2}\rangle\langle(\Delta\hat{X}_{2})^{2}\rangle\geq\frac{1}{16}(\langle(\hat{n}+1)f^{2}(\hat{n}+1)-\hat{n}f^{2}(\hat{n})\rangle)$ (33) A quantum state is said to be squeezed when one of the quadratures components $\hat{X}_{1}$ and $\hat{X}_{2}$ satisfies the relation $\langle(\Delta\hat{X}_{i})^{2}\rangle<\frac{1}{4}\langle(\hat{n}+1)f^{2}(\hat{n}+1)-\hat{n}f^{2}(\hat{n})\rangle\hskip 28.45274pti=1\;or\;2$ (34) The degree of squeezing can be measured by the squeezing parameter $s_{i}(i=1,2)$ defined by $s_{i}=4\langle(\Delta\hat{X}_{i})^{2}\rangle-\frac{1}{4}\langle(\hat{n}+1)f^{2}(\hat{n}+1)-\hat{n}f^{2}(\hat{n})\rangle.$ (35) Then the condition for squeezing in the quadrature component can be simply written as $s_{i}<0$. In Fig.(1) we plot the squeezing parameter $s_{1}$ versus the parameter $\frac{R}{a_{B}}$ defined as the ratio of the QD radius to the Bohr radius of exciton for two different values of ratio $\frac{\Omega_{1}}{\Omega_{0}}$. As is clear from Fig.(1) for small values of the parameter $\frac{R}{a_{B}}$ the state shows quadrature squeezing and by increasing this parameter the quadrature squeezing disappears. ## 4 Conclusion In this paper, we first considered a particle confined in a spherical infinite well and we found the explicit forms of its creation and annihilation operators by using the $f$-deformed oscillator approach. Then we considered an exciton in a wide QD interacts with two laser beams. We showed that under the weak confinement condition, the exciton is influenced as a whole and its center-of-mass motion will be quantized. Within the framework of the $f$-deformed oscillator approach, we found that under certain circumstances of exciton-laser interaction the stationary state of the exciton center-of-mass is a nonlinear coherent state which exhibits the quadrature squeezing. Acknowledgment The authors wish to thank the Office of Graduate Studies of the University of Isfahan and Iranian Nanotechnology initiative for their support. ## References ## References * [1] R. J. Glauber, Phys. Rev. 130 (1963) 2529; R. J. Glauber, Phys. Rev. 131 (1963) 2766; R. J. Glauber, Phys. Rev. Lett. 10, 84 (1963). * [2] J. R. Klauder and B. S. Skagerstam, Coherent states, Applications in Physics and Mathematical Physics (Singapore: World Scientific, 1985). * [3] P. A. Perelomov, Generalized Coherent States and Their Applications, (Berlin: Springer, 1986). S. T. Ali, J-P Antoine and J-P Gazeau, Coherent States, Wavelets and Their Generalization, (New York: Springer, 2000). * [4] V. I. Man’ko, G. Marmo, E. C. G. Sudarshan and F. Zaccaria, Phys. Scr. 55, 528 (1997). * [5] V. I. Man’ko, G. Marmo, E. C. G. Sudarshan and F. Zaccaria, in: N. M. Atakishiev(Ed.), Proc. IV Wigner Symp. (Guadalajara, Mexico, July 1995), World Scientific, Singapore, 1996, P. 427; S. Mancini, Phys. Lett. A 233, 291 (1997); S. Sivakumar, J. Phys. A: Math. Gen. 33, 2289 (2000); B. Roy, Phys. Lett. A 249, 25 (1998); H. C. Fu and R. Sasaki, J. Phys. A: Math. Gen. 29, 5637 (1996); R. Roknizadeh and M. K. Tavassoli, J. Phys. A: Math. Gen. 37, 5649 (2004); M. H. Naderi, M. Soltaolkotabi and R. Roknizadeh, J. Phys. A: Math. Gen. 37, 3225 (2004). * [6] R. L. deMatos Filho and W. Vogel, Phys. Rev. A 54, 4560 (1996). * [7] V. I. Man’ko, G. Marmo, A. Porzio, S. Solimeno and F. Zaccaria, Phys. Rev. A 62, 053407 (2000). * [8] M. H. Naderi, M. Soltanolkotabi and R. Roknizadeh, Eur. Phys. J. D. 32, 397 (2005). * [9] M. Bagheri Harouni, R. Roknizadeh and M. H. Naderi, * [10] A. Mahdifar, R. Roknizadeh and M. H. Naderi, J. Phys. A: Math. Gen. 39, 7003 (2006). * [11] Y. X. Liu, C. P. Sun, S. X. Yu and D. L. Zhou, Phys. Rev. A 63,023802 (2001). * [12] S. Schmitt-Rink, D. A. B. Miller and D. S. Chemla, Phys. Rev. B 35, 8113 (1987). * [13] L. Quiroga and N. F. Johnson, Phys. Rev. Lett. 83, 2270 (1999). * [14] G. Chen, N. H. Bonadeo, D. G. Steel, D. Gammon, D. S. Katzer, D. Park and L. J. Sham, Science 289, 1906 (2000); M. Bayer, P. Hawrylak, K. Hinzer, S. Fafard, D. Korkushiski, Z. R. Wasilewski, O. Stern and A. Forchel, Science 291, 451 (2001). * [15] Y. X. Liu, S. K. Özdemir, M. Koashi and N. Imoto, Phys. Rev. A 65, 042326 (2002). * [16] S-K Hong, S. W. Nam and K-H Yeon, Phys. Rev. B 76, 115330 (2007). * [17] X. Zou, K. Pahlke and W. Mathis, Phys. Rev. A 68, 034306 (2003). * [18] Ka-Di Zhu, Z-J Wu, X-Z Yuam and Zhen, Phys. Rev. B 71, 135312 (2005). * [19] F. Tassone and Y. Yamamoto, Phys. Rev. B 59, 1003 (1999). * [20] I. V. Bondarev, S. A. Maksimenko, G. Ya. Slepyan, I. L. Krestnikov and A. Hoffmann, Phys. Rev. B 68, 073310 (2003). * [21] E. Hanamura, Phys. Rev B 37, 1273 (1988). * [22] A. D’Andrea and R. Del Sole, Phys. Rev B 41, 1413 (1990); S. Jazirl, G. Bastard and R. Bennaceur, Semicond. Sci. Tech. 8, 670 (1993). * [23] M. Nirmal, D. J. Norris, M. Kuno, M. G. Bawendi, Al. L. Efros and M. Rosen, Phys. Rev. Lett. 75, 3728 (1995). * [24] M. Sugawara, Phys. Rev. B 51, 10743(1995); M. Califano, A. Franceschetti, and A. Zunger, Phys. Rev. B 75, 115401 (2007). * [25] P. Machikowski and L. Jacak, Phys. Rev. B 71, 115309 (2005). * [26] W. Vogel and R. L. de Matos Filho, Phys. Rev. A 52,4214(1995); D. M. Meekhof, C. Monroe, B. E. King, W. M. Itano and D. J. Wineland, Phys. Rev. Lett. 76, 1796(1996). * [27] R. P. Feynman, Phys. Rev. 84, 108 (1951). * [28] A. Wojs, P. Hawrylak, S. Fafard and L. Jacak, Phys. Rev. B 54, 5604 (1996). * [29] M. O. Scully and M. S. Zubairy, Quantum optics, (Cambridge University Press, Cambridge, (1997)). Figure 1: Plots of squeezing versus ration $\frac{r}{a_{B}}$. Solid line correspond to $\frac{\Omega_{0}}{\Omega_{1}}=0.5$, dash line correspond to $\frac{\Omega_{0}}{\Omega_{1}}=0.2$. In both plots Lamb-Dick parameter is chosen as $\kappa=0.3$.	arxiv-papers	2011-12-11T11:05:03	2024-09-04T02:49:25.171054	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "M. Bagheri Harouni, R. Roknizadeh, M. H. Naderi", "submitter": "Malek Bagheri", "url": "https://arxiv.org/abs/1112.2344" }
1112.2346	# $Q$-deformed description of excitons and associated physical results M. Bagheri Harouni, R. Roknizadeh, M. H. Naderi Quantum Optics Group, Physics Department, University of Isfahan, Iran m-baghreri@phys.ui.ac.ir rokni@sci.ui.ac.ir mhnaderi@phys.ui.ac.ir , , ###### Abstract We consider excitons in a quantum dot as $q$-deformed systems. Interaction of some excitonic systems with one cavity mode is considered. Dynamics of the system is obtained by diagonalizing total Hamiltonian and emission spectrum of quantum dot is derived. Physical consequences of $q$-deformed exciton on emission spectrum of quantum dot is given. It is shown that when the exciton system deviates from Bose statistics, emission spectra will become multi peak. With our investigation we try to find the origin of the $q$-deformation of exciton. The optical response of excitons, which affected by the nonlinear nature of $q$-deformed systems, up to the second order of approximation is calculated and absorption spectra of the system is given. ###### pacs: 73.20.La, 71.35.Cc, 03.65.Fd ††: J. Phys. B: At. Mol. Phys. ## 1 Introduction Exciton is an elementary excitation of a semiconductor which consists of a pair of two correlated fermions, the electron and the hole. Analogous to the Hydrogen atom, it is characterized by a binding energy $E_{b}$ and a Bohr radius $a_{B}$. Because an exciton is composed of two fermions, it is a composite boson. Particularly, in a bulk semiconductor, when the excitation of system is dilute, i.e. $n_{ex}a_{B}^{3}\ll 1$, where $n_{ex}$ is the exciton density, a bosonic description of system is convenient [1]. Also Bose-Einstein condensation of excitons, which is an essential characteristic of boson systems has been considered theoretically [2]. When density of excitons increase, the above condition is violated. In this situation the statistics of excitons deviates from Bose statistics. In low dimensional semiconductor systems such as quantum well (QW), quantum wire and quantum dot (QD), due to the small dimensions and loss of translational symmetry, exciton excitation differs from exciton in bulk materials. In semiconductor nanostructures the size of the system strongly affects exciton properties. For example, in the case of quantum well it is shown [3], if the well width is larger (smaller) than the Bohr radius of exciton, the spectrum of quantum well has properties similar to the situation in which excitons are boson (fermion). Hence, the size of the system directly affects the quantum statistics of excitons in that system. Recently similar results has been obtained for QD [4]. In Ref.[4] the effects of different statistics of excitons on emission spectra of a QD is investigated, and the origin of different statistics of excitons is considered. The same results have also been obtained for the quantum well. If the size of QD is smaller (larger) than the exciton Bohr radius, excitons behave like fermion (boson). Real statistics of excitons in the interaction is considered in [5] and references therein. As mentioned before in high density regime, exciton statistics deviates from Bose statistics. This is due to the increase of mutual forces between the excitations of the system and then the Pauli exclusion principle plays a dominant role [6]. Appearance of Bose statistics of exciton-biexciton system and Pauli exclusion effects in superlattice has been considered experimentally [7]. Bosons and fermions are the only two kinds of particles realized in nature. The conditions mentioned for excitons (in one regime they are like bosons and in another one like fermions) are property of a special kind of statistics called intermediate statistics [8]. Bose and Fermi statistics are two limiting cases of this statistics. Properties of this statistics have been considered by many authors [9]$-$[11]. Operator realization of intermediate statistics is similar to $q$-deformed operators [12]. Bosonic $q$-deformed operators [13] are a generalization of the Heisenberg algebra obtained by introducing a deformation parameter $q$. Deviation of this parameter from 1 shows deviation of algebra from the Heisenberg algebra. It is shown that it is possible to describe correlated fermion pairs with $q$-deformed bosons [14]. Therefore it is reasonable to consider an exciton system as a $q$-deformed system. We assume the creation and annihilation operators of excitons obey a $q$-deformed algebra. A $q$-deformed description of Frenkel exciton has been considered recently [15]. The algebra generated by $q$-deformed operators are given by $\displaystyle[\hat{b}_{q},\hat{b}_{q}^{\dagger}]_{q}$ $\displaystyle=$ $\displaystyle\hat{b}_{q}\hat{b}_{q}^{\dagger}-q^{-1}\hat{b}_{q}^{\dagger}\hat{b}_{q}=q^{\hat{n}},$ (1) $\displaystyle[\hat{n},\hat{b}_{q}^{\dagger}]$ $\displaystyle=$ $\displaystyle\hat{b}_{q}^{\dagger},\hskip 42.67912pt[\hat{n},\hat{b}_{q}]=-\hat{b}_{q}.$ where $\hat{n}=\hat{b}^{\dagger}\hat{b}$ is the usual particle number operator. Representation of this algebra is given in [16]. In the case of excitons, $q$-parameter can depend on excitation number and physical size of system. In this paper we consider the interaction of light with a QD embedded in a microcavity. By considering excitons in QD as $q$-deformed bosons (case of $q$-deformed fermion is straightforward) we study the emission spectrum of the system. As is clear, the commutator (1) explicitly depends on the number of excitons. Hence, this is a system in which light interacts with a nonlinear active medium. Therefore, we shall obtain the linear and nonlinear response of a $q$-deformed exciton system. Knowledge of interaction of light with a nonlinear medium ($q$-deformed excitons) and its optical response is important for the interpretation of experimental results such as [30]. On the other hand, we compared the obtained results with some experimental ones and in this manner we investigate the physical origin of $q$-deformation of excitons. In section 2 we derive the spectrum of QD when one exciton mode interacts with a single mode cavity-field. In section 3 we consider the interaction of two exciton modes with a single cavity mode. In section 4 the nonlinear response of QD is derived up to second order of approximation. Finally we summarize our conclusions in section 5. ## 2 Model Hamiltonian We consider a QD embedded in a microcavity which interacts with a single mode cavity-field. We assume the excitations in QD have an intermediate statistics [4], and their creation and annihilation operators obey $q$-deformed algebra. We can express the $q$-deformed operators in terms of ordinary boson operators by the following maps $\hat{b}_{q}=\hat{b}\sqrt{\frac{q^{\hat{n}}-q^{-\hat{n}}}{\hat{n}(q-q^{-1})}},\hskip 42.67912pt\hat{b}_{q}^{\dagger}=\sqrt{\frac{q^{\hat{n}}-q^{-\hat{n}}}{\hat{n}(q-q^{-1})}}\hat{b}^{\dagger},$ (2) where $\hat{b}$ and $\hat{b}^{\dagger}$ are the ordinary boson operators and $\hat{n}=\hat{b}^{\dagger}\hat{b}$. Ordinary commutator of $q$-deformed exciton operators is $[\hat{b}_{q},\hat{b}_{q}^{\dagger}]=\frac{q}{q+1}[q^{n}+q^{-(n+1)}]\equiv k(\hat{n}).$ (3) Deviation of this commutator from ordinary boson algebra ($[\hat{b},\hat{b}^{\dagger}]=1$) relates to deviation of $q$-parameter from 1. It is clear, this generalized commutator depends on the number of excitations. It seems that by using this algebra we can consider some nonlinear phenomena in the system related to the population of excitons. For example, biexciton effects can be considered in this manner as an effective approach. So that the deformation parameter $q$ can represent some physical parameters such as the ratio of size of system to the Bohr radius of exciton. Interaction of QD with single mode cavity-field in rotating wave approximation can be described by the following Hamiltonian $\hat{H}=\hbar\omega\hat{a}^{\dagger}\hat{a}+\hbar\omega_{ex}\hat{b}_{q}^{\dagger}\hat{b}_{q}+\hbar g(\hat{a}\hat{b}_{q}^{\dagger}+\hat{a}^{\dagger}\hat{b}_{q}),$ (4) where $\hat{a}$ and $\hat{a}^{\dagger}$ are creation and annihilation operators of cavity field and $[\hat{a}_{i},\hat{a}_{j}^{\dagger}]=\delta_{ij}$. We shall consider a phenomenological damping for the system which relates to both subsystems: photon and exciton. As is clear from the Hamiltonian (4), the exciton number is not a constant of motion. Because of the dependence of exciton operator, $\hat{b}_{q}$, on the exciton number, resulting equations of motion become a nontrivial set of coupled equations. On the other hand, since the total number of excitation (exciton and photon) is conserved we can diagonalize the Hamiltonian in the subspace of a definite excitation. To consider this dynamics we propose an approach based on diagonalization of the Hamiltonian by using the polariton transformation [17]. This procedure depends on some unitary transformations which diagonalize the model Hamiltonian. As is usual in this procedure [18], new operators have the same commutation relation as the original operators (free operators). Here, there are two distinct sets of operators, the cavity mode operators which obey the usual boson commutation relation and exciton operators that are $q$-deformed boson. Therefore, with the presence of these two different statistics, mixed operators (polariton operators) do not have specific statistics. They can be considered as ordinary boson operators or $q$-deformed operators. We consider both situations and we study the physical results associated with each situation in the resonance fluorescence spectrum of QD. ### 2.1 Boson polaritons In order to solve the dynamical system, we perform the following transformation $\hat{p}_{k}=u_{k}\hat{b}_{q}+v_{k}\hat{a}.$ (5) Due to the presence of $q$-deformed operator $\hat{b}_{q}$, we call this transformation a polariton-like transformation. As mentioned before, $\hat{b}_{q}$ depends on the number of excitons explicitly and this causes the Hopfield coefficients $u_{k}$ and $v_{k}$ will depend on the number of excitons. Hence, the transformation (5) can be considered as a nonlinear polariton transformation. This kind of transformation has been considered recently for the case of Bogoliubov transformation [19, 20]. We assume polariton-like operators obey the usual boson commutation relations $[\hat{p}_{k},\hat{p}^{\dagger}_{k^{\prime}}]=\delta_{kk^{\prime}}\Rightarrow[\hat{p}_{k},\hat{p}^{\dagger}_{k}]=\|u_{k}\|^{2}k(\hat{n})+\|v_{k}\|^{2}=1,$ (6) where the operator valued function $k(\hat{n})$ was introduced by Eq.(3). We choose unknown coefficients $u_{k}$ and $v_{k}$ so that the Hamiltonian (4) becomes diagonal in terms of the polariton-like operators $\hat{H}=\hbar\sum_{k}\Omega_{k}\hat{p}_{k}^{\dagger}\hat{p}_{k},$ (7) where $\Omega_{k}$ is the polariton spectrum and $k$ refers to different polariton branches. By taking into account a phenomenological damping for exciton and photon systems separately, the unknown parameters satisfy following set of equations $[\omega_{ex}k(\hat{n})-\Omega_{k}-i\gamma_{ex}]u_{k}+v_{k}g=0,\hskip 31.2982ptu_{k}gk(\hat{n})+(\omega-\Omega_{k}-i\gamma_{ph})v_{k}=0.$ (8) In this set of equations, $\gamma_{ex}$ and $\gamma_{ph}$ are the exciton and photon damping constants, respectively. From these equations the polariton spectrum can be obtained as $\displaystyle\Omega_{k}$ $\displaystyle=$ $\displaystyle\frac{\omega_{ex}k(\hat{n})+\omega-i(\gamma_{ex}+\gamma_{ph})}{2}$ (9) $\displaystyle\pm$ $\displaystyle\frac{1}{2}\sqrt{[\omega_{ex}k(\hat{n})-\omega-i(\gamma_{ex}-\gamma_{ph})]^{2}+4g^{2}k(\hat{n}))}.$ It is apparent that $q$-deformed description of excitons causes the splitting between these energy eigenvalues be increased in compare to the case of bosonic description of exciton. Using the set of equations (8) and the polariton spectrum (9) we find the coefficients for two polariton branches $\displaystyle u_{k}=\sqrt{\frac{\omega-i\gamma_{ph}-\Omega_{k}}{k(\hat{n})[\omega-2\Omega_{k}+\omega_{ex}k(\hat{n})-i(\gamma_{ex}+\gamma_{ph})]}},$ (10) $\displaystyle v_{k}=-\sqrt{\frac{\omega_{ex}k(\hat{n})-i\gamma_{ex}-\Omega_{k}}{\omega-2\Omega_{k}+\omega_{ex}k(\hat{n})-i(\gamma_{ex}+\gamma_{ph})}}.$ By employing these coefficients all necessary parameters for the polariton Hamiltonian are determined. Now we can consider the dynamics of polariton operators. The time evolution of polariton operators is governed by the polariton Hamiltonian (7) $\hat{\dot{p}}_{k}=\frac{-i}{\hbar}[\hat{p}_{k},\hat{H}]=-i\Omega_{k}\hat{p}_{k}.$ (11) Let us consider damping effects by taking into account a phenomenological damping term and noise operator in the dynamical equations of polariton operators. Hence, the time evolution of polariton operator is given by $\hat{\dot{p}}_{k}=-i\Omega_{k}\hat{p}_{k}-\Gamma_{k}\hat{p}_{k}+\hat{F}_{\hat{p}_{k}}(t),$ (12) where $\hat{F}_{\hat{p}_{k}}(t)$ is the Langevin noise operator which depends on the reservoir variables and $\Gamma_{k}$ is the damping constant of $k$th polariton branch given by $\Gamma_{k}=\frac{\gamma_{ex}+\gamma_{ph}}{2}$. Correlation functions of the noise operators determine physical properties of the system. The Langevin noise operator are such that their expectation values $\langle\hat{F}_{x}\rangle$ vanishes, but their second order moments do not [21]. They are intimately linked up with the global dissipation and in a Markovian environment they take the form $\langle\hat{F}^{\dagger}_{\hat{p}_{k}}(t)\hat{F}_{\hat{p}_{k}}(t^{\prime})\rangle=2\Gamma_{k}\delta(t-t^{\prime}).$ (13) With neglecting the phonon effects by decreasing the temperature, other sources of damping like spontaneous recombination of exciton and photon loss are considered as Markovian procedures. It follows, on solving Eq.(12), that $\hat{p}_{k}(t)=\hat{p}_{k}(0)e^{(-i\Omega_{k}-\Gamma_{k})t}+\int_{0}^{t}e^{(-i\Omega_{k}-\Gamma_{k})(t-t^{\prime})}\hat{F}_{\hat{p}_{k}}(t^{\prime})dt^{\prime}.$ (14) In this equation we set initial time equal zero. The power spectrum of the scattered light for statistical stationary fields is given by [22] $S(r,\omega)=\frac{1}{\pi}Re\int_{0}^{\infty}\langle\hat{E}^{-}(r,t)\hat{E}^{+}(r,t+\tau)\rangle e^{i\omega\tau}d\tau,$ (15) where $\hat{E}^{\pm}$ are the positive and negative frequency parts of the electric field operator. Expressing field operators in terms of creation and annihilation operators we have $S(r,\omega)=\frac{A(r)}{\pi}Re\int_{0}^{\infty}\langle\hat{a}^{\dagger}(0)\hat{a}(\tau)\rangle e^{i\omega\tau}d\tau.$ (16) Here, we set $t=0$, and $A(r)$ depends on mode function of the cavity-field. Now we can express, the field and exciton creation and annihilation operators in terms of polariton ones: $\hat{a}=v_{1}^{\ast}\hat{p}_{1}+v_{2}^{\ast}\hat{p}_{2},\hskip 42.67912pt\hat{b}_{q}=k(\hat{n})(u_{1}^{\ast}\hat{p}_{1}+u_{2}^{\ast}\hat{p}_{2}),$ (17) and at the time $t$ we have $\hat{a}(t)=v_{1}^{\ast}\hat{p}_{1}(t)+v_{2}^{\ast}\hat{p}_{2}(t).$ (18) Now to calculate the resonance fluorescence spectrum we have to determine the initial state of system. we assume at $t=0$, the cavity-field is in a coherent state $\|\alpha\rangle$, and the exciton subsystem in its vacuum state. Under this condition, by using Eq.(14) the resonance fluorescence spectrum is obtained as $S(r,\omega)=\frac{A(r)\|\alpha\|^{2}}{\pi}\left[\|v_{1}\|^{2}\frac{\Gamma_{1}}{(\omega-\Omega_{1})^{2}+\Gamma_{1}^{2}}+\|v_{2}\|^{2}\frac{\Gamma_{2}}{(\omega-\Omega_{2})^{2}+\Gamma_{2}^{2}}\right].$ (19) In deriving this result we implicitly assume that at $t=0$ the noise operator and polariton operators are uncorrelated. Fig.(1) shows the plot of $S(r,\omega)$ versus $\omega$ for different values of deformation parameter q. Material parameters are chosen as $\omega=1.75\;eV$, $\omega_{ex}=1.75\;eV$, $g=200\;\mu eV$, $\gamma_{ex}=20\;\mu eV$, $\gamma_{ph}=40\;\mu eV$ [23], $n=100$ and $\|\alpha\|^{2}=9$ . As is clear when $q=1$, spectrum has similar variation as experimental results [23]. This figure shows that when $q=1$ (nondeformed case) the power spectrum of the fluorescence light is a double peak centered at $\omega=\Omega_{1}$ and $\omega=\Omega_{2}$. By increasing deviation of q from 1, it is apparent from the different plots in this figure that splitting between two peaks increases and the height of one of peaks decreases. This result has been reported in resonance fluorescence of excitons when the biexcitonic interaction is taken into account. It has been shown [3] that biexcitonic effects are a red shift of the transition frequencies, emergence of sidebands due to the switch-on forbidden transitions and asymmetry of the emission spectrum. The binding energy of biexciton in QD causes a shift in the spectrum of the system. In the present model the splitting of spectrum (Rabi splitting) depends on the $q$-parameter. Hence, changing this parameter affects the spectrum. Then as a one reason of deviation of excitons from ideal Bose system we can consider Coulomb interaction between them. On the other hand, $q$-deformed exciton operators depend on the total number of exciton, and biexciton interaction occurs when there are more than one exciton. This similarity makes this clue that the $q$-deformation can be consider as an effective approach to take into account the biexciton effects. As mentioned before, the $q$-parameter can depend on the size of sample. The plotted resonance fluorescence spectrum in Fig.(1) makes clear some differences of optical properties of different size QD. For large values of q, compare with 1, spectrum will reduce to one peak. This case is a characteristic of the weak coupling regime. ### 2.2 $Q$-deformed polaritons In this subsection we assume that the polariton operators are $q$-deformed operators. According to the $q$-deformed nature of the exciton system we assume the following algebra for polariton operators $[\hat{p}_{k},\hat{p}^{\dagger}_{k}]_{s}=\hat{p}_{k}\hat{p}^{\dagger}_{k}-s^{-1}\hat{p}^{\dagger}_{k}\hat{p}_{k}=s^{\hat{n}_{k}},$ (20) where $s$ denotes the deformation parameter corresponding to the polariton system and $\hat{n}_{k}$ shows the number operator for $k$th polariton branch. Ordinary commutator for these operators is $[\hat{p}_{k},\hat{p}^{\dagger}_{k}]=\|u_{k}\|^{2}k(\hat{n})+\|v_{k}\|^{2}=\frac{s}{s+1}[s^{\hat{n}_{k}}+s^{-(\hat{n}_{k}+1)}]=M(\hat{n}_{k}).$ (21) Using the same approach of the previous subsection we obtain the following set of equations for the coefficients of transformation $\displaystyle[(\omega_{ex}k(\hat{n})-i\gamma_{ex}-\Omega^{\prime}_{k}M(n_{k})]u_{k}+v_{k}g$ $\displaystyle=$ $\displaystyle 0,$ $\displaystyle u_{k}gk(\hat{n})+[\omega-i\gamma_{ph}-\Omega^{\prime}_{k}M(n_{k})]v_{k}$ $\displaystyle=$ $\displaystyle 0.$ (22) From this set of equations we derive the deformed polariton spectrum as $\displaystyle\Omega^{\prime}_{k}$ $\displaystyle=$ $\displaystyle\frac{\omega_{ex}k(\hat{n})+\omega-i(\gamma_{ex}+\gamma_{ph})}{2M(n_{k})}$ (23) $\displaystyle\pm$ $\displaystyle\frac{\sqrt{[\omega_{ex}k(\hat{n})-\omega-i(\gamma_{ex}-\gamma_{ph})]^{2}+4g^{2}k(\hat{n}))}}{2M(n_{k})}.$ and the transformation coefficients read as $\displaystyle u_{k}=-\sqrt{\frac{M(n_{k})[\omega-i\gamma_{ph}-\Omega^{\prime}_{k}M(n_{k})]}{k(\hat{n})[\omega-2\Omega^{\prime}_{k}M(n_{k})+\omega_{ex}k(\hat{n})-i(\gamma_{ex}+\gamma_{ph})]}},$ (24) $\displaystyle v_{k}=\sqrt{\frac{M(n_{k})[\omega_{ex}k(\hat{n})-i\gamma_{ex}-\Omega^{\prime}_{k}M(n_{k})]}{\omega-2\Omega^{\prime}_{k}M(n_{k})+\omega_{ex}k(\hat{n})-i(\gamma_{ex}+\gamma_{ph})}}.$ By determining all the variables, polariton Hamiltonian (diagonal Hamiltonian) will be determined. By applying the same procedure as before we derive the resonance fluorescence spectrum in this case as follows $S(r,\omega)=\frac{A(r)\|\alpha\|^{2}(\|v_{1}\|^{2}+\|v_{2}\|^{2})}{\pi}\sum_{i=1,2}\|v_{i}\|^{2}\frac{\Gamma_{i}}{(\omega-\Omega^{\prime}_{i}M(n_{k}))^{2}+\Gamma_{i}^{2}}.$ (25) Fig. (2) shows the plot of $S(r,\omega)$ versus $\omega$ for different values of polariton deformation parameter $s$. This figure shows that changes of s-parameter (deformation parameter of polariton) does not cause any shift in transition frequencies, but causes strengths of peaks increase. ## 3 Interaction of light with two exciton modes We now consider the interaction of one cavity mode with QD when two exciton modes are coupled to the field mode. As before, we assume exciton system is expressed by the $q$-deformed operators. The total Hamiltonian of the system under consideration can be written as follows $\hat{H}=\hbar\omega\hat{a}^{\dagger}\hat{a}+\hbar\sum_{i=1,2}\omega_{{ex}_{i}}\hat{b}_{q_{i}}^{\dagger}\hat{b}_{q_{i}}+\hbar g\sum_{i=1,2}(\hat{a}\hat{b}_{q_{i}}^{\dagger}+\hat{a}^{\dagger}\hat{b}_{q_{i}}).$ (26) We assume both excitons have the same coupling constant with the cavity mode. We solve this system as before by diagonalizing the Hamiltonian. For this purpose we perform the following transformation $\hat{p}_{k}=u_{k}\hat{b}_{q_{1}}+x_{k}\hat{b}_{q_{2}}+v_{k}\hat{a}.$ (27) We consider the situation in which the polariton operators obey the nondeformed Bose statistics $[\hat{p}_{k},\hat{p}^{\dagger}_{k}]=\|u_{k}\|^{2}k(\hat{n}_{1})+\|x_{k}\|^{2}k(\hat{n}_{2})+\|v_{1}\|^{2}=1,$ (28) where $\hat{n}_{i}$ ($i=1,2$) represents the number operator for each excitonic mode. As is clear in this case there are three polariton branches. Assuming the transformation (27) diagonalizes the Hamiltonian (26), this polariton Hamiltonian takes the following form $\hat{H}=\hbar\sum_{k}\Omega_{k}\hat{p}^{\dagger}_{k}\hat{p}_{k},$ (29) where summation is over all polariton branches. The following equation determines the polariton spectrum $(c-\Omega_{k})[(d-\Omega_{k})(\omega-i\gamma_{ph}-\Omega_{k})-g^{2}k(\hat{n}_{1})]-g^{2}k(\hat{n}_{2})(d-\Omega_{k})=0,$ (30) where $c=\omega_{ex_{1}}k(\hat{n}_{1})-i\gamma_{ex_{1}}$ and $d=\omega_{ex_{2}}k(\hat{n}_{2})-i\gamma_{ex_{2}}$. By deriving the polariton spectrum the transformation parameters are obtained as $\displaystyle u_{k}=\frac{g[(d-\Omega_{k})(\omega-i\gamma_{ph}-\Omega_{k})-g^{2}k(\hat{n}_{2})]}{A},$ (31) $\displaystyle x_{k}=\frac{g^{3}k(\hat{n}_{1})}{A}$ $\displaystyle v_{k}=-\frac{(c-\Omega_{k})[(d-\Omega_{k})(\omega-i\gamma_{ph}-\Omega_{k})-g^{2}k(\hat{n}_{2})]}{A},$ where the parameter $A$ is given by $\displaystyle A$ $\displaystyle=$ $\displaystyle([g^{2}k(\hat{n}_{1})+(c-\Omega_{k})^{2}][(d-\Omega_{k})(\omega-i\gamma_{ph}-\Omega_{k})-g^{2}k(\hat{n}_{2})]^{2}$ (32) $\displaystyle+$ $\displaystyle g^{6}k^{2}(\hat{n}_{1})k(\hat{n}_{2}))^{\frac{1}{2}}.$ In this manner, all the parameters which appear in the polariton Hamiltonian are determined. By repeating the approach of previous section the resonance fluorescence spectrum of system with different initial conditions can be determined. If we assume at $t=0$ the cavity mode is in the coherent state $\|\alpha\rangle$ and QD in vacuum state $\|0\rangle$, the resonance fluorescence spectrum is given by $S(r,\omega)=\frac{\|\alpha\|^{2}A(r)}{\pi}\sum_{k}\frac{\|v_{k}\|^{2}\Gamma_{k}}{\Gamma_{k}^{2}+(\omega-\Omega_{k})^{2}}.$ (33) To show complex structure (multi-peak structure) of this spectrum Fig. (3) presents the spectra on a logarithmic scale. For the sake of clarity, we have powered some peaks compare to other ones in this figure. In the case of $q=1$ (nondeformed exciton) the spectrum has two peaks. Increasing the $q$-parameter causes that splitting between peaks be increased and spectrum becomes multi- peaks. Multi-peaks structure in emission of exciton such as Mollow triplet was predicted when excitons obey statistics different from Bose statistics [3, 4]. When, $q$-parameter is changed, the energy and intensities of emission change. Effects of exciton number on absorption spectrum of QD is considered . Due to the relation of absorption spectrum and resonance fluorescence, similar result is obtain in [24]. ## 4 Nonlinear response of excitons in $q$-deformed regime In previous sections we considered some physical results of $q$-deformed description of excitons. The $q$-deformed description can be served as a nonlinear description of excitons. It is well-known that different kinds of nonlinearity in an exciton system lead to different orders of nonlinear response of the system [25, 26]. Therefore, we try to obtain optical response of a driven quantum dot, which its optical excitations are considered as $q$-deformed systems. For this purpose we will calculate the coefficient absorption of a QD in this regime. In this section we neglect all damping effects and we consider the Hamiltonian of the system as follows $\hat{H}=\hbar\omega\hat{a}^{\dagger}\hat{a}+\hbar\omega_{ex}\hat{b}_{q}^{\dagger}\hat{b}_{q}+\hbar g(\hat{a}\hat{b}_{q}^{\dagger}+\hat{a}^{\dagger}\hat{b}_{q}).$ (34) In the electron picture, the induced dipole moment by transition of an electron is described by $\hat{\mu}=\hat{a}^{\dagger}_{v}\hat{a}_{c}+\hat{a}^{\dagger}_{c}\hat{a}_{v}$ [27]. The operator $\hat{a}^{\dagger}_{v}\;(\hat{a}_{v})$ is the creation (annihilation) operator for an electron in the valance band (level in the case of QD), and $\hat{a}^{\dagger}_{c}\;(\hat{a}_{c})$ is the creation (annihilation) operator for an electron in the conduction band. Hence, creation of an exciton is denoted by $\hat{a}^{\dagger}_{c}\hat{a}_{v}=\hat{b}_{q}^{\dagger}$. Therefore we can write the dipole operator of QD as $\hat{\mu}=\hat{b}^{\dagger}_{q}+\hat{b}_{q}$. The macroscopic polarization is expectation value of polarization operator. The optical response function represents the reaction of the system to an external classic field $E(t)$ coupled to the variables of system [28], i.g., the dipole operator. Hence, we consider an external field as a pump source and we treat the reaction of QD to it. Then the total Hamiltonian of system is then given by $\hat{H}=\hbar\omega\hat{a}^{\dagger}\hat{a}+\hbar\omega_{ex}\hat{b}_{q}^{\dagger}\hat{b}_{q}+\hbar g(\hat{a}\hat{b}_{q}^{\dagger}+\hat{a}^{\dagger}\hat{b}_{q})-[\vec{d}_{vc}\cdot\vec{E}(t)\hat{b}_{q}+\vec{d}_{cv}\cdot\vec{E}(t)\hat{b}^{\dagger}_{q}],$ (35) where $\vec{d}_{vc}$ denotes the dipole matrix element. The Hamiltonian in the interaction picture has the form $\displaystyle\hat{H}_{int}$ $\displaystyle=$ $\displaystyle\hbar g\left[\hat{a}\hat{b}^{\dagger}_{q}e^{-i[\omega-\omega_{ex}k(\hat{n})]t}+\hat{a}^{\dagger}e^{i[\omega-\omega_{ex}k(\hat{n})]t}\hat{b}_{q}\right]$ $\displaystyle-$ $\displaystyle\left[\vec{d}_{cv}\cdot\vec{E}(t)e^{-i\omega_{ex}k(\hat{n})t}\hat{b}_{q}+\vec{d}_{vc}\cdot\vec{E}(t)\hat{b}^{\dagger}_{q}e^{i\omega_{ex}k(\hat{n})t}\right].$ The observable of interest for the optical response is the time-dependent dipole density $\mu(t)=\langle\hat{b}_{q}(t)\rangle+h.c.=Tr_{ex}(\hat{b}_{q}\rho_{ex}(t))+h.c.$, where $Tr_{ex}$ means trace over the exciton system and $\rho_{ex}(t)=Tr_{f}\rho(t)$, which $\rho(t)$ is the total time dependent density matrix of the system and $\rho_{ex}(t)$ is the time dependent density matrix of exciton system. The total time dependent density matrix is given by $\hat{\rho}(t)=\hat{U}(t,t_{0})\hat{\rho}(t_{0})\hat{U}^{-1}(t,t_{0}),$ (37) where $U(t,t_{0})=\hat{T}\exp[-\frac{i}{\hbar}\int_{t_{0}}^{t}\hat{H}_{int}(t^{\prime})dt^{\prime}]$ is the time ordered evolution operator and $\hat{\rho}(t_{0})$ is the total density matrix of system at initial time. We assume that the quantum field and exciton system are both in vacuum state. Therefore, the time dependent density matrix of excitons is given by $\hat{\rho}_{ex}(t)=\sum_{n}\langle n\|\hat{U}(t,t_{0})(\|0\rangle_{f}\|0\rangle_{ex})(_{ex}\langle 0\|_{f}\langle 0\|)\hat{U}^{-1}(t,t_{0})\|n\rangle,$ (38) where summation is carried on field state and the matrix elements of the time evolution operator are in the basis of field states. By using the Feynman disentanglement theorem [29] the matrix elements of the time evolution operator $\hat{U}(t,t_{0})$ can be evaluated. We can write Hamiltonian in (4) as $\hat{H}_{int}=\hat{H}_{1}(t)+\hat{H}_{2}(t)$, where $\displaystyle\hat{H}_{1}(t)=\hbar g\left[\hat{a}\hat{b}^{\dagger}_{q}e^{-i[\omega-\omega_{ex}k(\hat{n})]t}+\hat{a}^{\dagger}e^{i[\omega-\omega_{ex}k(\hat{n})]t}\hat{b}_{q}\right]$ (39) $\displaystyle\hat{H}_{2}(t)=-\left[\vec{d}_{cv}\cdot\vec{E}(t)e^{-i\omega_{ex}k(\hat{n})t}\hat{b}_{q}+\vec{d}_{vc}\cdot\vec{E}(t)\hat{b}^{\dagger}_{q}e^{i\omega_{ex}k(\hat{n})t}\right].$ As is clear $\hat{H}_{2}(t)$ depends only on exciton operators. The time evolution operator can be written as $\displaystyle\hat{U}(t,t_{0})$ $\displaystyle=$ $\displaystyle\hat{T}\exp[-\frac{i}{\hbar}\int_{t_{0}}^{t}(\hat{H}_{1}(t^{\prime})+\hat{H}_{2}(t))dt^{\prime}]$ (40) $\displaystyle=$ $\displaystyle\hat{T}\exp[-\frac{i}{\hbar}\int_{t_{0}}^{t}\hat{H}_{2}(t^{\prime})dt^{\prime}]\exp[-\frac{i}{\hbar}\int_{t_{0}}^{t}\hat{H}_{1}(s)ds].$ In this equation we use Feynman notation [29]. These two exponential terms are not disentangle from each other. They are correlated and in doing integration, we have to take into account ordering of operators. In calculation of matrix element of this operator in the basis of field states, second exponential can be considered as a ordinary c-number function of $t^{\prime}$, because it is independent of field operators: $\langle i\|\hat{U}(t,t_{0})\|j\rangle=\langle i\|\hat{T}\exp[-\frac{i}{\hbar}\int_{t_{0}}^{t}\hat{H}_{1}(t^{\prime})dt^{\prime}]\|j\rangle\exp[-\frac{i}{\hbar}\int_{t_{0}}^{t}\hat{H}_{2}(s)ds].$ (41) On the other hand we consider all the exciton operators in $\hat{H}_{1}(t)$ as ordinary c-number functions, and we can write $\displaystyle\langle i\|\hat{U}(t,t_{0})\|j\rangle$ $\displaystyle=$ $\displaystyle\langle i\|\hat{T}\exp[-\frac{i}{\hbar}\int_{t_{0}}^{t}(\hat{a}^{\dagger}_{t^{\prime}}e^{i\omega t^{\prime}}B(t^{\prime})+\hat{a}_{t^{\prime}}e^{-i\omega t^{\prime}}B^{\ast}(t^{\prime}))dt^{\prime}]\|j\rangle$ (42) $\displaystyle\times$ $\displaystyle\exp[-\frac{i}{\hbar}\int_{t_{0}}^{t}\hat{H}_{2}(s)ds],$ where $B(t)$ is a ordinary function corresponding to exciton operators. As is clear this matrix element is a function of exciton operators. The influence of exciton system is completely contained in this operator functional and factored term in (42). By using Feynman theorem, above matrix element can written as $\langle i\|\exp[-ig\int_{t_{0}}^{t}\hat{a}^{\dagger}_{t^{\prime}}e^{i\omega t^{\prime}}B(t^{\prime})dt^{\prime}]\exp[-ig\int_{t_{0}}^{t}\hat{a}^{\prime}_{t^{\prime}}e^{-i\omega t^{\prime}}B^{\ast}(t^{\prime})dt^{\prime}]\|j\rangle,$ (43) where in this equation $\hat{a}^{\prime}_{t^{\prime}}=\hat{V}^{-1}(t)\hat{a}_{t}\hat{V}(t)$, and $\hat{V}(t)=\exp[-ig\hat{a}^{\dagger}\int_{t_{0}}^{t}B(t^{\prime})e^{i\omega t^{\prime}}].$ (44) In this manner the density matrix of exciton system takes the following form $\hat{\rho}_{ex}(t)=\sum_{n}\frac{1}{n!}S_{1}(\hat{n},\hat{b}_{q},\hat{b}^{\dagger}_{q})\|0\rangle_{ex}\langle 0\|S_{2}(\hat{n},\hat{b}_{q},\hat{b}^{\dagger}_{q}),$ (45) where $\displaystyle S_{1}(\hat{n},\hat{b}_{q},\hat{b}^{\dagger}_{q})$ $\displaystyle=$ $\displaystyle\left[-g\hat{b}^{\dagger}_{q}\frac{e^{i\omega_{ex}k(\hat{n}_{ex})(t-t_{0})}}{\omega_{ex}k(\hat{n}_{ex})}\right]^{n}e^{-\frac{g^{2}}{2}L(\hat{n}_{ex})}f(\hat{b}_{q},\hat{b}_{q}^{\dagger}),$ $\displaystyle S_{2}(\hat{n},\hat{b}_{q},\hat{b}^{\dagger}_{q})$ $\displaystyle=$ $\displaystyle\left[-g\hat{b}_{q}\frac{e^{-i\omega_{ex}k(\hat{n}_{ex}+1)(t-t_{0})}}{\omega_{ex}k(\hat{n}_{ex}+1)}\right]^{n}e^{-\frac{g^{2}}{2}L(\hat{n}_{ex})}f^{-1}(\hat{b}_{q},\hat{b}_{q}^{\dagger}),$ $\displaystyle L(\hat{n}_{ex})$ $\displaystyle=$ $\displaystyle\hat{b}_{q}^{\dagger}\hat{b}_{q}[\frac{e^{-i\omega_{ex}[k(\hat{n}_{ex}+1)-k(\hat{n}_{ex}-1)](t-t_{0})}}{[\omega-\omega_{ex}k(\hat{n}_{ex}-1)][\omega-\omega_{ex}k(\hat{n}_{ex}+1)]}$ (46) $\displaystyle+$ $\displaystyle\frac{e^{-i\omega_{ex}[k(\hat{n}_{ex}+2)-k(\hat{n}_{ex}))](t-t_{0})}}{[\omega-\omega_{ex}k(\hat{n}_{ex}+2)][\omega-\omega_{ex}k(\hat{n}_{ex})]}],$ (47) and $\displaystyle f(\hat{b}_{q},\hat{b}_{q}^{\dagger})=\hat{T}\exp$ $\displaystyle[$ $\displaystyle\frac{i}{\hbar}\int_{t_{0}}^{t}dt^{\prime}(\vec{d}_{cv}\cdot\vec{E}(t^{\prime})\hat{b}_{q}e^{-i\omega_{ex}k(\hat{n}_{ex}+1)t^{\prime}}$ (48) $\displaystyle+$ $\displaystyle\vec{d}_{vc}\cdot\vec{E}(t^{\prime})\hat{b}^{\dagger}_{q}e^{i\omega_{ex}k(\hat{n}_{ex})t^{\prime}})],$ By expanding the function $f(\hat{b}_{q},\hat{b}^{\dagger}_{q})$ up to second order in $E(t)$ and using (45) we obtain the time-dependent dipole density as follows $\displaystyle p(t)=\sum_{n}\frac{g^{2n}}{n!}h_{1}(n)!\sqrt{f_{q}(n)!}e^{-\frac{g^{2}}{2}L(n)}\times$ $\displaystyle[\frac{i}{\hbar}h_{0}(n+1)!e^{-\frac{g^{2}}{2}L(1)}\sqrt{f_{q}(n+1)!f_{q}(n+1)}\int_{t_{0}}^{t}\vec{d}_{cv}\cdot\vec{E}(t^{\prime})e^{i\omega_{ex}t^{\prime}}dt^{\prime}$ $\displaystyle-\frac{i}{\hbar}\sqrt{f_{q}(n)}h_{0}(n)!e^{-\frac{g^{2}}{2}L(0)}\sqrt{f_{q}(n)!f_{q}(n)}\int_{t_{0}}^{t}\vec{d}_{cv}\cdot\vec{E}(t^{\prime})e^{i\omega_{ex}k(n-1)t^{\prime}}dt^{\prime}$ $\displaystyle+\frac{i}{2\hbar^{3}}\sqrt{f_{q}(n+1)}h_{0}(n+2)!e^{-\frac{g^{2}}{2}L(0)}\sqrt{f_{q}(n+2)!f_{q}(n+2)}\times$ $\displaystyle\int_{t_{0}}^{t}\int_{t_{0}}^{t}\int_{t_{0}}^{t}\vec{d}_{vc}\cdot\vec{E}(t^{\prime})\vec{d}_{cv}\cdot\vec{E}(r)\vec{d}_{cv}\cdot\vec{E}(s)e^{i\omega_{ex}[s+r-k(n+1)t^{\prime}]}dt^{\prime}drds$ $\displaystyle+\frac{i}{2\hbar^{3}}\sqrt{f_{q}(n)}h_{0}(n)!e^{-\frac{g^{2}}{2}L(0)}\sqrt{f_{q}(n)!f_{q}(n)}\times$ $\displaystyle\int_{t_{0}}^{t}\int_{t_{0}}^{t}\int_{t_{0}}^{t}\vec{d}_{cv}\cdot\vec{E}(t^{\prime})\vec{d}_{vc}\cdot\vec{E}(r)\vec{d}_{cv}\cdot\vec{E}(s)e^{i\omega_{ex}[k(n-1)t^{\prime}-(r-s)]}dt^{\prime}drds$ $\displaystyle-\frac{i}{2\hbar^{3}}\sqrt{f_{q}(n+1)}h_{0}(n+1)!e^{-\frac{g^{2}}{2}L(1)}\sqrt{f_{q}(n+1)!f_{q}(n+1)}\int_{t_{0}}^{t}\int_{t_{0}}^{t}\int_{t_{0}}^{t}\times$ $\displaystyle\vec{d}_{vc}\cdot\vec{E}(r)\vec{d}_{cv}\cdot\vec{E}(s)\vec{d}_{cv}\cdot\vec{E}(t^{\prime})e^{-i\omega_{ex}[k(n+1)(r-s)-t^{\prime}]}dt^{\prime}drds$ $\displaystyle-\frac{i}{2\hbar^{3}}\sqrt{f_{q}(n)}h_{0}(n+1)!e^{-\frac{g^{2}}{2}L(1)}\sqrt{f_{q}(n+1)!f_{q}(n+1)}\int_{t_{0}}^{t}\int_{t_{0}}^{t}\int_{t_{0}}^{t}\times$ $\displaystyle\vec{d}_{cv}\cdot\vec{E}(r)\vec{d}_{vc}\cdot\vec{E}(s)\vec{d}_{cv}\cdot\vec{E}(t^{\prime})e^{i\omega_{ex}[k(n+1)(r-s)+t^{\prime}]}dt^{\prime}drds],$ (49) where $h_{i}(n)=\frac{e^{(-1)^{i}i\omega_{ex}k(n+i)(t-t_{0})}}{\omega_{ex}k(n+i)}$ and $f_{q}(n)=\sqrt{\frac{q_{n}-q^{-n}}{q-q^{-1}}}$. These equation shows that in this conditions second order response function is equal zero. Now we can calculate linear and nonlinear electric susceptibility of this exciton system from this equation. Generalized linear and nonlinear absorption spectra of this system is shown in figures (4)-(6) for different values of $q$-parameter. In these plots, $1s$-exciton is considered. In these figures we choose $\hbar=e=1$, $g=200\;\mu ev$ and $\omega_{ex}=1574mev$. Fig.(4) shows plots of linear absorption spectra and Fig.(6) shows plots of nonlinear spectra. On the other hand, 3-dimension plot of linear absorption coefficients is given in figure (5). It is clear that changes of $q$-parameter strongly affects absorption spectra of the system. These figures show in the presence of $q$-values absorption of probe beam shows a complex structure: a multiple-like absorption pattern appears with one strong peak and some side bands. Presence of these side bands is a signature of the optical generation of an nonlinear exciton (an exciton which expresses with $q$-deformed operator). Negative part of the absorption spectrum demonstrates gain of the probe beam. Due to the resonance interaction of pump with exciton transition, the gain effect comes from the coherent energy exchange between the pump and probe beams through the QD nonlinearity. The obtained absorption spectra are very similar to experimental results [30]. In Ref. [30] absorption spectra of a driven charged QD is derived experimentally. Charged QD is a nonlinear medium and is similar to our model. Then It can be consider as a experimental test of our model. ## 5 Conclusion $Q$-deformed description of excitons in a QD and its physical consequences was considered. We showed that increasing the $q$-parameter will lead to increase of splitting between peaks in the spectrum and asymmetry of spectrum. Similar effects were observe when biexciton effects taken into account. In experiments of QD it is shown [23] the same results are obtained in different temperatures. Then we can associate this physical parameter as source of $q$-deformation. The temperature dependence of emission energy of system can be attributed to the change in the refractive index of its active medium with temperature. We have derived the optical response of QD with $q$-deformed exciton. As mentioned before $q$-deformed description of excitons will lead to dependence of optical response on $q$ parameter. Hence, due to the wide range of $q$ parameter and its effects on optical response we can consider some parameters like temperature and interaction between the excitons which affects the optical response of QD as sources of $q$-deformation of excitons. As mentioned, the relation of quantum statistics of excitons in the QD and the size of QD has been considered. Then we can consider the ratio of exciton Bohr radius to dimension of system and exciton population as two main sources of $q$-deformation. $Q$-deformed operator depends on total number of associated particles of system. Therefore we can interpret $q$-deformed operator as an operator which consists of effects of other excitations of system implicitly. Then it is reasonable to consider this description as an effective description which takes into account some nonlinearity in exciton system. As we saw, in the case of interaction of light with two excitons, when $q=1$ this system showed a two peaks spectrum. While by increasing deviation of exciton from Bose statistics, spectrum becomes multi peak. Due to the nonlinear nature of $q$-deformed exciton we showed that different orders of nonlinear response function of this system can be calculated. From coincidence of obtained results and experimental results, we can conclude that $q$-deformed description of excitons can be a considerable model for excitons. With comparing the obtained results in this paper with experimental ones we can investigate the origin of this description of excitons. As pointed out the ratio of system dimension to the Bohr radius of exciton is one of the sources of deviation of excitons from usual boson. The obtained results are very similar to the effects of the exciton-exciton interaction [3],[31] which is relates to exciton population and biexciton binding energy. On the other hand, it is shown that [1] exciton density is another source of their deviation from ordinary bosons. To sum up we attribute the origin of $q$-deformation of the excitons to their density, their mutual interactions, confinement size and other parameters which cause fluctuation of optical response of the system. $Q$-deformed description of an active medium causes that the optical properties of system depend on the $q$-parameter. Then, it is seem that parameters which can affect optical properties of the active medium (like refractive index) their effects can be considered by this formulation. The $q$-parameter can be considered as a variation parameter which its values can be obtained from comparison of theoretical and experimental results. Acknowledgment The authors wish to thank the Office of Graduate Studies of the University of Isfahan and Iranian Nanotechnology initiative for their support. ## References ## References * [1] A. S. Davydov, Theory of Molecular Excitons, (Plenum Press, NewYork, 1971). * [2] E. Hanamura and H. Haug, Phys. Rep. 33, 209 (1977). * [3] Y. Yamamoto, F. Tassone and H. Cao, Semiconductor Cavity Electrodynamics, (Springer-Verlag Berlin Heidelberg, 2000). * [4] F. P. Laussy, M. M. Glazov, A. Kavokin, D. M. Whittaker and G. Malpuech, Phys. Rev. B 73, 115343 (2006). * [5] B. Laikhtman, J. Phys: Condens. Matter 19, 295214 (2007). * [6] M. Combescot, O. Betbeder-Matibet and F. Dubin, Phys. Rev. A 76, 033601 (2007). * [7] H. Ichida, M. Nakayama and J. Lumin, J. Lumin. 94-95, 379 (2001). * [8] A. Khare, Fractional Statistics and Quantum Theory, (World Scientific, Singapore, 1997). * [9] M. P. Blencowe and N. C-Koshnick, J. Math. Phys. 42, 5713 (2001). * [10] W. S. Dai and M. Xie, Physica A 331, 497 (2204). * [11] Y. Shen, W. S. Dai and M. Xie, Phys. Rev. A 75, 042111(2007). * [12] S. Chaturvedi, V. Srinivasan, Phys. Rev. A 44, 8024 (1991). * [13] A. J. Macfarlane, J. Phys. A: Math. Gen. 22, 4581 (1989); L. C. Biedenharn, J. Phys. A: Math. Gen. 22, L873 (1989). * [14] D. Bonatsos, J. Phys. A: Math. Gen. 25, L101 (1992). * [15] Y. X. Liu, C. P. Sun, S. X. Yu and D. L. Zhou, Phys. Rev. A 63, 023802 (2001). * [16] G. Rideau, Lett. Math. Phys. 24, 147(1992). * [17] J. J. Hopfield, Phys. Rev. 112, 1555 (1958). * [18] U. Fano. Phys. Rev. 103, 1202 (1956). * [19] J. Katriel, Phys. Lett. A 307, 1(2003). * [20] M. H. Naderi, R. Roknizadeh and M. Soltanolkotabi, Prog. Theor. Phys. 112, 797 (2004); ibid 112, 811 (2004). * [21] M. Lax, Phys. Rev. 145, 110 (1966). * [22] M. O. Scully, M. S. Zubairy, Quantum Optics, (Cambridge University Press 1997). * [23] E. Peter, P. Senellart, D. Martrou, A. Lemaître, J. Hours, J. M. Gérard, and J. Bloch, Phys. Rev. Lett. 95, 067401 (2005). * [24] A. Franceschetti and Y. Zhang, Phys. Rev. Lett. 100, 136805 (2008). * [25] V. M. Axt, A. Stahl, Z. Phys. B 93, 205 (1994). * [26] Th. Östreich, K. Schönhammer, L. J. Sham, Phys. Rev. Lett. 74, 4698 (1995). * [27] H. Haug, S. W. Koch, Quantum Theory of The Optical and Electronic Properties of Semiconductor, 4rd edition (World Scientific, Singapore, 2004). * [28] S. Mukamel, Principles of Nonlinear Optical Spectroscopy, (Oxford, NewYork, 1995). * [29] R. P. Feynman, Phys. Rev. 84, 108 (1951). * [30] X. Xu et al., arxiv:0803,0734 (Cond-Mat. mes-hall). * [31] U. Hohenester and E. Molinari, Phys. Stat. Sol. (b) 221, 19 (2000). Figure 1: Plots of $S(\omega)$ versus $\omega$. Parameter are choose as $\omega=1.75\;eV$, $\omega_{ex}=1.75\;eV$, $g=200\;\mu eV$, $\gamma_{ex}=20\;\mu eV$, $\gamma_{ph}=40\;\mu eV$, $n=1$ and $\|\alpha\|^{2}=9$. Solid plot corresponds to $q=1$, nondeformed case. Dotted one corresponds to $q=1.01$, and for dash line $q$ is equal $1.015$. Figure 2: Plots of $S(\omega)$ versus $\omega$. Parameter are choose as $\omega=1.75\;eV$, $\omega_{ex}=1.75\;eV$, $g=200\;\mu eV$, $\gamma_{ex}=20\;\mu eV$, $\gamma_{ph}=40\;\mu eV$, $n=1$ and $\|\alpha\|^{2}=9$. In all figures we have $q=1$. Solid line corresponds to case $s=1$. In dotted one we have $s=1.007$ and for dash line $s=1.01$. Figure 3: Plots of $S(\omega)$ versus $\omega$. Parameter are choose as $\omega=1.75\;eV$, $\omega_{ex_{1}}=1.75\;eV$, $\omega_{ex_{2}}=1.77\;eV$, $g=200\;\mu eV$, $\gamma_{ex_{1}}=\gamma_{ex_{2}}=200\;\mu eV$, $\gamma_{ph}=45\;\mu eV$, $n_{1}=1$, $n_{2}=1$ and $\|\alpha\|^{2}=9$. Dotted line corresponds to nondeformed case $q_{1},q_{2}=1$. For solid line $q_{1},q_{2}=1.04$. In the case of dashed line $q_{1},q_{2}=1.08$. Figure 4: Plots of spectrum absorption versus $\omega$. We consider 1s-exciton and Parameter are choose as $\hbar=e=1$, $g=200\;\mu ev$ and $\omega_{ex}=1574\;mev$. Solid plot corresponds to nondeformed case $q=1$. For dotted one $q=1.01$ and in dash one $q=0.99$. Figure 5: 3D-Plots of spectrum absorption versus $\omega$ and deformation parameter $q$. Physical parameter are the same as Fig.(4). Figure 6: Plots of nonlinear spectrum absorption versus $\omega$. We consider 1s-exciton and Parameters are choose as $\hbar=e=1$, $g=200\;\mu ev$ and $\omega_{ex}=1574\;mev$ Solid plot corresponds to nondeformed case $q=1$. In dotted plot $q=1.01$. In dash plot $q=0.99$.	arxiv-papers	2011-12-11T11:12:54	2024-09-04T02:49:25.179194	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "M. Bagheri Harouni, R. Roknizadeh, M. H. Naderi", "submitter": "Malek Bagheri", "url": "https://arxiv.org/abs/1112.2346" }
1112.2531	# Double-relativistic-electron-shell laser proton acceleration Yongsheng Huang http://www.anianet.com/adward huangyongs@gmail.com China Institute of Atomic Energy, Beijing 102413, China. Naiyan Wang Xiuzhang Tang Yijin Shi China Institute of Atomic Energy, Beijing 102413, China. Yan Xueqing Institute of Heavy Ion Physics, Peking University, Beijing 100871, China Zhang Shan Beijing Normal University, Beijing 100875, China ###### Abstract A new laser-proton acceleration structure combined by two relativistic electron shells, a suprathermal electron shell and a thermal electron cloud is proposed for $a\gtrapprox 80\sigma_{0}$, where a is the normalized laser field and $\sigma_{0}$ is the normalized plasma surface density. In the new region, a uniform energy distribution of several GeV and a monoenergetic hundreds-of- MeV proton beam have been obtained for $a=39.5$. The first relativistic electron shell maintains opaque for incident laser pulse in the whole process. A monoenergetic electron beam has been generated with energy hundreds of MeV and charge of hundreds of pC. It is proposed a stirring solution for relativistic laser-particle acceleration. ###### pacs: 52.38.Kd,41.75.Jv,52.40.Kh,52.65.-y Laser-ion acceleration has been an international research focusMakoTajima ; Machnisms ; Esirkepov , however it is still a challenge to obtain mono- energetic proton beams larger than $100\mathrm{MeV}$. Although the field in the laser-plasma acceleration is three to four orders higher than that of the classic accelerators, it decreases to zero quickly in several pulse durations for target normal sheath acceleration (TNSA)Machnisms for $a\ll\sigma_{0}=\frac{n_{e}l_{e}}{n_{c}\lambda}$, where $a=eE_{l}/m\omega c$, $E_{l}$ is the electric field of the laser pulse, $\omega$ is the laser frequency, $e$ is the elementary charge, $m$ is the electron mass, $c$ is the light velocity, $n_{0}$ is the initial plasma density, $n_{c}$ is the critical density, $\lambda$ is the wave length, $n_{e}$ is the electron density, $l_{e}$ is the thickness of the electron shell. As a promising method to generate relativistic mono-energetic protons, radiation pressure acceleration (RPA) has attracted more attentionEsirkepov ; Henig ; YanxqPRL ; unlimitedRPA and becomes dominant in the interaction of the ultra-intense laser pulse with thin foils if $a\approx\sigma_{0}$. Even in the unlimited ion accelerationunlimitedRPA , only the ions trapped in the electron shell can obtain efficient acceleration, therefore, the total charge of the ion beam is quite limited due to the transverse expansionunlimitedRPA . Although in RPA region, the energy dispersion will become worse with time. Fortunately, for $a\gtrapprox 80\sigma_{0}$, in the relativistic case, a new acceleration region appears: double relativistic electron shells come into being. The ions between the two electron shells will be accelerated most efficiently and obtain a uniform energy distribution. The first electron shell is ultra-relativistic and is totally separated from the ions. The second electron shell comes into being in the potential well induced by the electron recirculation and will also be relativistic. It is in the ion region and follows the ion front and forms a potential well which traps energetic ions and accelerates them to be quasi-mono-energetic and relativistic. Following the second electron shell, a suprathermal electron beam comes into being and induces another potential well which can also trap lots of ions and accelerate them to obtain a monoenergetic relativistic one. On the whole, the maximum ion energy can reach several GeV and a relativistic monoenergetic ion beam with relative energy dispersion smaller then $5\%$ can be obtained. As a ultraintense laser pulse is shot on a ultra-thin plasma foil for $a\gtrapprox 80\sigma_{0}$, the electron shell is compressed to ultra-high density and pushed forward to be separated from the ion shell totally, and gains ultra-relativistic energy that can make sure it opaque for the laser pulse. In the whole process, the first electron shell keeps opaque for the incident laser pulse and is pushed by it continuously. According to Eq. (22)unlimitedRPA , it can be satisfied that the opaqueness condition of electron shell for laser in the acceleration: $a_{0}\leq\pi(\gamma_{e}+p_{e})\hat{n}_{e}\hat{l}_{e},$ (1) since $d\ln(p\hat{n}_{e}\hat{l}_{e})/dt>0$, as pointed by Bulanov, $p\propto t^{1/3}$ is the normalized electron momentum, and $n_{e}l_{e}=n_{0}l_{0}$ in the no transverse expansion case, where the electron density $n_{e},n_{0}$ are normalized by $n_{c}$ and $l_{e}$ is normalized by $\lambda$, $a_{0}=eE_{0}/m\omega_{0}c$, $E_{0}$ is the electric field of the laser pulse, $\omega_{0}$ is the laser frequency, $e$ is the elementary charge, $m$ is the electron mass, $c$ is the light velocity, $n_{0}$ is the initial plasma density, $n_{c}$ is the critical density, $\lambda$ is the wave length. In the ultra-relativistic case, the r.h.s. of Eq. (1) is approximate $2\pi p\hat{n}_{e}\hat{l}_{e}$. For $a=39.5$ and $\hat{n}_{e}\hat{l}_{e}=49\times 0.01=0.49$, $a\approx 81\hat{n}_{e}\hat{l}_{e}$. When the laser pulse interacts with the plasma shell, the electrons are compressed to high density and pushed forward to be separated from the ion shell. As shown by Figure 1 (c) and (d), at $t=25\mathrm{fs}$, the normalized momentum of the electron shell reaches $20-100$. With Eq. (1), the compressed high-density electron shell is opaque for the laser pulse as shown by Figure 1 (f). The electrons will be accelerated efficiently and continuously by the radiation pressure of the laser. Figure 1: (Color online) Simulation results by one-dimensional VORPAL at $t=25$fs: the first opaque relativistic electron shell forms and is totally separated from ions. (a) and (c): the phase space distribution of ions and electrons. The electron shell is totally separated from the ions and is relativistic and opaque for laser pulse. (b) and (d): the density distribution of the ions and electrons with normalized momentum. The ion energy distribution is uniform. The electron energy distribution contains several monoenergetic ones. (e): the longitudinal space charge separation field, which is similar to the field in a capacity, is uniform and is $9.95\times 10^{12}$V/m. (f): the potential and laser field. The electron shell is relativistic and opaque for the laser pulse. This high-density relativistic electron shell is called the first relativistic electron shell. Between the first electron shell and the ions, a uniform space-charge separation field forms and accelerates the ions at the ion front and drags the electrons as shown by Figure 1 (e). The ”capacity” field is decided by the surface density of the electron shell: $E_{cap}=\frac{en_{e}l_{e}}{\epsilon_{0}},$ (2) For $n_{e}=5.5\times 10^{22}\mathrm{/cm^{3}}$, $l_{e}=10\mathrm{nm}$, the stable field is $9.95\times 10^{12}\mathrm{V/m}$, which accelerates the ions at the rear of the ion shell continuously. Therefore the maximum ion energy is proportional to the acceleration length, $d_{acc}$, $E_{i}=E_{cap}d_{acc}(\mathrm{eV}),$ (3) before the electron shell breaks up. After several hundreds of femtoseconds, some electrons leak out from the electron shell continuously and move backward and round again and follow the ion front, however, they can not catch the ion front. Figure 2 (c), (e) and (f) shows the electron recirculation, the decrease of the separation field, and the formation of the potential well for electrons at $t=450\mathrm{fs}$ respectively. The normalized maximum electron momentum reaches about $500$ as shown by Figure 2 (d). Figure 2 (f) shows that the deepness of the potential well for electrons increases about to $0.3\mathrm{GeV}$. From Figure 2 (a) and (b), the maximum ion energy is about $500\mathrm{MeV}$. The acceleration length is about $65\mathrm{\mu m}$. A monoenergetic electron beam of $186\mathrm{MeV}$ and $132$pC is obtained as shown by Figure 2 (d). It is obvious that the first relativistic electron shell is still opaque for the laser pulse. Figure 2: (Color online) Simulation results by one-dimensional VORPAL at $t=450$fs: electron recirculation begins and generates a potential well for electrons, where $p_{i},p_{e}$ is momentum of ion and electron respectively, $M$ is the proton mass. (a) and (c): the phase space of ions and electrons respectively. The electron recirculation happens and some of them move in the opposite direction relativistically. (b) and (d): the energy distribution of ions and electrons respectively. The first electron shell is accelerated most efficiently and continuously. (e) the longitudinal field decreases due to the electron recirculation. However it is still uniform between the ion front and the first electron shell. (f): a potential hill for ions and a potential well for electrons forms due to the electron recirculation. The first relativistic electron shell maintains opaque for the laser pulse. As shown by Figure 3 (d), the recirculating electrons cumulate and drive up the electron potential. In front of and behind the accumulating electrons, two potential wells are forming for electrons. The potential well I still traps and accelerates the accumulating electrons to generate the second relativistic high-density electron shell. In the potential well II, some electrons at the end of the second relativistic electron shell drop into it and will be trapped and accelerated to form a suprathermal electron shell as shown by Figure 5 (d). At the same time and at the local position of the second relativistic electron shell, potential well III for ions traps lots of ions and accelerates them to be relativistic and monoenergetic. Figure 3: (Color online) Simulation results by one-dimensional VORPAL at $t=1.25$ps: the second relativistic electron shell forms and traps ions to be accelerated to relativistic. (a): the second relativistic electron shell forms in the potential well. It drives up the potential and forms two potential well I and II for electrons in front of and behind itself, a potential well III for ions at the local position of itself. It is shown clearly in (d). (b): the number density distribution of electrons. (c): the electron recirculation decreases the longitudinal field continuously. It is still uniform between the ion front and the first electron shell. Figure 4: (Color online) Simulation results by one-dimensional VORPAL at $t=3.075$ps: the suprathermal electron shell forms and traps ions to obtain a flap-top $3-47$MeV energy distribution. (a) and (c): the phase space of ions and electrons. The suprathermal electron shell forms in the potential well II and then potential V for electrons and potential IV are generated. (b) and (d) the energy distribution of ions and electrons. A monoenergy ion beam with energy of $627$MeV is obtained in the second relativistic electron shell. (e): the longitudinal field induced by the double electron shell. (f): the potential IV for ions is induced by the suprathermal electron shell and traps ions and improves the energy dispersion of the $3-47$MeV ion beam. The potential V can trap slow electrons and thermalize them to obtain thermal electron cloud. Potential III is nearly filled to be flat and the energy dispersion of the 627MeV monoenergetic ion beam will become worse. With time, the slow recirculating electrons can also be trapped in potential well V as shown in Figure 4(f) and the third suprathermal electron shell forms in the potential well II as shown by Figure 5. At the position of the shell, potential well IV traps the ions and accelerates them to obtain a quasi- monoenergetic distribution of $171\pm 10\mathrm{MeV}$. Behind the shell, a potential well for electrons traps them and a thermal electron cloud is generated. The ions between the double relativistic electron shell have a uniform distribution from $1\mathrm{GeV}$ to $2.18\mathrm{GeV}$. Trapped by the second relativistic electron shell, the maximum energy reach $981\mathrm{MeV}$. As shown in Figure 4 (f), the potential well III has been filled up nearly, then the energy dispersion will become worse. The ions with larger energy will coast down the following potential slope and get into the ion beam between the double relativistic electron shell. The ion number has a steep descent for the energy larger than $981\mathrm{MeV}$ and has a slow drop for the energy smaller than $981\mathrm{MeV}$. In the electron energy distribution, there is a monoenergetic one of $385\pm 10\mathrm{MeV}$ and $163\mathrm{pC}$, a ultra-relativistic one of $1\mathrm{GeV}$ and a Maxwellian one which contains the thermal electron cloud and the suprathermal electron shell. As shown by Figure 5(c) and (f), the ion front is between the double electron shell. The ions between the double electron shell coast down the potential slope and obtain relativistic energy as shown in Figure 5(h) and (b). Figure 5: (Color online) Simulation results by one-dimensional VORPAL at $t=4.025$ps: the thermal electron cloud and the suprathermal electron shell come into being. (a) and (d): the phase-space of ions and electrons respectively. The electrons contain four main parts: the double relativistic electron shells, the suprathermal electron shell, the thermal electron cloud. (b) and (e): the energy distribution of ions and electrons respectively. A monoenergetic ion beam with energy of $171\pm 10$MeV is obtained by the suprathermal electron shell. Trapped and accelerated by the second relativistic electron shell, the ion energy distribution drops down at $981$MeV. (c) and (f) the number density of ions and electrons respectively. (g) the longitudinal field. (h) the potential for ions and electrons. (i) the laser pulse field. The first electron shell maintains opaque for laser pulse. In conclusion, the double relativistic electron shells, the suprathermal electron shell and the thermal electron cloud induce a new region of laser particle acceleration. In the process, several potential wells for ions and electrons are generated. On the whole, the double relativistic electron shells induce two relativistic platforms of the ion energy distribution. The suprathermal electron shell traps and accelerates a monoenergetic ion beam with several hundreds of MeV, whose relative energy dispersion is near $5\%$. Together with the thermal electron cloud, a thermal Maxwellian ion beam has been obtained. ###### Acknowledgements. The authors would like to thank Dr. Hong-Yu Wang for useful discussion. The computation was carried out at the HSCC of Beijing Normal University. This work was supported by the Key Project of Chinese National Programs for Fundamental Research (973 Program) under contract No. $2011CB808104$ and the Chinese National Natural Science Foundation under contract No. $10834008$. ## References * (1) F. Mako and T. Tajima, Phys. Fluids 27, 1815 (1984). * (2) H. Schwoerer, S. Pfotenhauer, O. Jackel, et al., Nature 439, 445 (2006). electronacc * (3) T. Esirkepov, M. Borghesi, S. V. Bulanov, G. Mourou, and T. Tajima, Phys. Rev. Lett. 92, 175003 (2004). * (4) A. Henig, S. Steinke, M. Schn rer, et al., Phys. Rev. Lett. 103, 245003 (2009). * (5) H. Schwoerer, S. Pfotenhauer, O. Jackel, et al., Nature 439, 445 (2006). * (6) X. Q. Yan, C. Lin, Z.M. Sheng, et al., Phys. Rev. Lett. 100, 135003 (2008). * (7) S. V. Bulanov, E. Yu. Echkina, T. Zh. Esirkepov, et al., Phys. Rev. Lett. 104, 135003 (2010). * (8) X. Q. Yan, T. Tajima, M. Hegelich, et al., Appl. Phys. B, 98 711-721 (2010). * (9) Y. S. Huang, * (10) T. P. Yu, A. Pukhov, G. Shvets, and M. Chen, Phys. Rev. Lett. 105, 065002 (2010). * (11) A. Macchi, F. Cattani, T. V. Liseykina and F. Cornolti, Phys. Rev. Lett. 94, 165003 (2005).	arxiv-papers	2011-12-12T12:41:42	2024-09-04T02:49:25.190632	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Yongsheng Huang, Naiyan Wang, Xiuzhang Tang, Yijin Shi, Zhang Shan", "submitter": "Yongsheng Huang", "url": "https://arxiv.org/abs/1112.2531" }
1112.2570	# Minkowski momentum of an MHD wave Tadas K Nakamura CFAAS, Fukui Prefectural University, 910-1195 Fukui, Japan ###### Abstract The momentum of an MHD wave has been examined from the view point of the electromagnetic momentum expression derived by Minkowski. Basic calculations show that the Minkowski momentum is the sum of electromagnetic momentum and the momentum of the medium, as proposed in some of the past literature. The result has been explicitly confirmed by an example of an MHD wave, whose dynamics can be easily and precisely calculated from basic equations. The example of MHD wave also demonstrates the possiblility to construct a symmetric energy-momentum tensor based on the Minkowski momentum. ###### pacs: 03.50.De,42.20.Jb,52.35.Bj ## 1 Introduction The Minkowski-Abraham controversy has been discussed by a number of authors over a hundred years. Minkowski [1] proposed the electromagnetic momentum density in a dielectric medium must be $\mathbf{D}\times\mathbf{B}$, and Abraham [2, 3] proposed $\mathbf{E}\times\mathbf{H}$ for that (in the present letter symbols have conventional meanings, e.g., $\mathbf{E}$ = electric field, unless otherwise stated). There have been published numerous papers on this problem both theoretically and experimentally, but the final conclusion is still yet to come; papers are still beeing published in this century (see, e.g., [4] for a review). Several authors [5, 4, 6] pointed that the electromagnetic field inevitably affect the dynamics of the medium to change its energy-momentum, and therefore, the energy-momentum of an electromagnetic wave must include the contribution of the medium. In the present letter we show that the Minkowski momentum is the sum of electromagnetic momentum and the momentum of the medium. Feigel [5] obtained a similar result based on the Noether’s theorem using Lagrangian formulation. Compared to his elegant approach, the calculation here is rather a down-to-eath type, which is more closer to the Minkowski’s original derivation. This approach is less elegant, however, easier to understand its meaning intuitively. Perhaps the largest weak point of the Minkowski momentum is the fact that the four dimensional energy-momentum tensor does not become symmetric with this momentum, which means the violation of angular momentum conservation (see, e.g. [7]). Most of the past literature argued the legitimacy of the momentum part of the tensor in this point. Here, in contrast, we elucidate the possibility to alter the energy part to make a symmetric tensor; provided the momentum part of the Minkowski energy-momentum tensor includes the momentum of medium, the same should be true for the energy part. To treat it in a relativistically consistent way, the mass flux must be included in the energy flux even in the non-relativistic regime. The energy-momentum tensor with Minkowski momentum can become symmetric when the mass flux is taken into account. The consideration stated above is confirmed by an example of an MHD wave in a collisionless magnetized plasma. Usually the behavior of an ordinary medium is complicated and need to calculate microscopic states of molecules, which is difficult to solve exactly. A collisionless plasma is, in contrast, easy to calculate its response to the electromagnetic field from the classical basic equations (Maxwell equations and Newtonian mechanics). Here in this short letter we use the MHD approximation, however, if one wishes it is possible to derive an exact solution of the basic equation system to confirm the result. The result agree with the “frozen-in” of a magnetized plasma, which has been confirmed by a wide variety of experimental and observational facts. ## 2 Basics Microscopic Ampere’s equation in a medium is $-\varepsilon_{0}\frac{\partial\mathbf{E}}{\partial t}+\mu_{0}^{-1}\nabla\times\mathbf{B}=\mathbf{J}$ (1) Suppose there is no external current, and $\mathbf{J}$ consists of the polarization current $\mathbf{J}_{P}$ and magnetization current $\mathbf{J}_{M}$, which are generated in response to the electric field $\mathbf{E}$ and magnetic field $\mathbf{B}$ respectively. We introduce the polarization vector $\mathbf{P}$ and magnetization vector $\mathbf{M}$ such that $\frac{\partial}{\partial t}\mathbf{P=\mathbf{J}}_{P}\,,\;\;\nabla\times\mathbf{M}=\mathbf{J}_{M}$ (2) In this context, $\mathbf{P}$ and $\mathbf{M}$ should be understood as convenient mathematical expressions to represent the response of the medium to the electromagnetic field, rather than real physical entities. When averaged over a microscopically large but macroscopically small volume, $\bar{\mathbf{P}}$ and $\bar{\mathbf{M}}$ are assumed to have simple linear relations to the electromagnetic fields as $\bar{\mathbf{P}}=\chi_{P}\bar{\mathbf{E}}\,,\;\;\bar{\mathbf{M}}=\chi_{M}\bar{\mathbf{B}}\,.$ (3) The linear coefficients $\chi_{P}$ and $\chi_{M}$ are matrices in general because the medium may not be isotropic (as in our example of magnetized plasmas). The fields $\bar{\mathbf{D}}$ and $\bar{\mathbf{H}}$ are then defined as macroscopic quantities as $\bar{\mathbf{D}}=\varepsilon_{0}\bar{\mathbf{E}}+\bar{\mathbf{P}}=\varepsilon\bar{\mathbf{E}}\,,\;\;\bar{\mathbf{H}}=\mu_{0}^{-1}\bar{\mathbf{B}}+\bar{\mathbf{M}}=\mu^{-1}\bar{\mathbf{B}}$ (4) ## 3 Minkowski Momentum The momentum of microscopic electromagnetic field is $\varepsilon_{0}\mathbf{E}\times\mathbf{B}$ and its conservation law is $\frac{\partial}{\partial t}(\varepsilon_{0}\mathbf{E}\times\mathbf{B})+\nabla\cdot T+(\mathbf{J}_{P}+\mathbf{J}_{M})\times\mathbf{B}+(Q_{P}+Q_{M})\mathbf{E}=0\,,$ (5) where $T$ is the Maxwell stress tensor and we denote $\partial T_{ij}/\partial x_{i}=(\nabla\cdot T)_{j}$ in short. The polarization/magnetization charge $Q_{P}$ and $Q_{M}$ are the result of polarization/magnetization current ($\partial Q_{P,M}/\partial t=\nabla\mathbf{J}_{P,M}$). The charge due to magnetization current vanishes when averaged, $\bar{Q}_{M}=0$ since $\bar{\mathbf{J}}_{M}$ satisfies (2). The third and fourth term of (5) are the Lorentz and Coulomb force acting on the medium, and therefore, it can expressed by the momentum change of the medium. $(\mathbf{J}_{p}+\mathbf{J}_{M})\times\mathbf{B}+(Q_{P}+Q_{M})\mathbf{E}=\frac{\partial}{\partial t}\mathbf{g}+\nabla\cdot T_{M}\,,$ (6) where $\mathbf{g}$ and $T_{M}$ are the momentum density and stress tensor of the medium. It should be noted that the right hand side of the above expression has mathematical ambiguity. If we define new values of momentum/stress by $\mathbf{g}^{\prime}=\mathbf{g}+\mathbf{a}$ and $T^{\prime}=T+G$ with arbitrary vector $\mathbf{a}$ and tensor $G$ that satisfy $\partial\mathbf{a}/\partial t=\nabla G=0$, they also satisfy the above equation. Therefore, $\mathbf{g}$ and $T$ do not necessarily have to be the total momentum/stress of the medium. For example, the medium may contain a part that does not interact with the electromagnetic field, and such part causes this ambiguity. From (4) we obtain $\displaystyle\bar{\mathbf{J}}_{P}\times\bar{\mathbf{B}}$ $\displaystyle=$ $\displaystyle\frac{\partial}{\partial t}(\bar{\mathbf{P}}\times\bar{\mathbf{B}})+\chi_{P}\left[(\bar{\mathbf{E}}\cdot\nabla)\bar{\mathbf{E}}+\frac{1}{2}\nabla\bar{\mathbf{E}}^{2}+\bar{\mathbf{E}}(\nabla\bar{\mathbf{E}})\right]\,,$ (7) $\displaystyle\bar{\mathbf{J}}_{M}\times\bar{\mathbf{B}}$ $\displaystyle=$ $\displaystyle\chi_{M}\left[(\bar{\mathbf{B}}\bigtriangledown)\bar{\mathbf{B}}+\frac{1}{2}\nabla\bar{\mathbf{B}}^{2}\right]\,.$ Here we neglected cross terms of fluctuation in averaging as usually done in this kind of calculation, e.g., $\overline{\mathbf{P}\times\mathbf{B}}=\bar{\mathbf{P}}\times\bar{\mathbf{B}}$. Combining (5), (6) and (7) we obtain $\frac{\partial}{\partial t}(\varepsilon_{0}\bar{\mathbf{E}}\times\bar{\mathbf{B}}+\bar{\mathbf{g}})+\nabla\cdot(\bar{T}+\bar{T}_{M})=\frac{\partial}{\partial t}(\bar{\mathbf{D}}\times\bar{\mathbf{B}})+\nabla\cdot\bar{T}^{\prime}=0\,,$ (8) where $\bar{T}^{\prime}$ is the stress tensor in a dielectric medium defined as $\bar{T}^{\prime}_{ij}=\bar{E}_{i}\bar{D}_{j}+\mu_{0}^{-1}\bar{B}_{i}\bar{H}_{j}-\frac{1}{2}\delta_{ij}(\bar{\mathbf{E}}\bar{\cdot\mathbf{D}}+\bar{\mathbf{B}}\cdot\bar{\mathbf{H}})\,,$ (9) which is the sum of the fluxes of electromagnetic momentum and momentum carried by the medium. Now that we understand the Minkowski momentum includes the part of the medium then so should be for the energy. The energy is equivalent to mass in relativity, thus the energy flux of the medium must include mass, and then the flux may take the form of $\bar{\mathbf{D}}\times\bar{\mathbf{B}}$ to make the energy momentum symmetric. We will check this for the case of an Alfven wave in the following. ## 4 MHD wave Let us confirm the above discussion with an example of an MHD wave. Suppose a linearly polarized one dimensional ($\partial/\partial x=\partial/\partial y=0$) MHD wave (Alfven wave in this case) propagating in the $z$ direction, which is the direction of the background magnetic field: $B_{0}=B_{Z}$. The wave amplitude is small enough for linear approximation, and the plasma velocity is low enough for non-relativistic approximation. Also we assume “cold plasma limit”, which means the thermal energy of plasma particles is negligibly small. The wave has an electric field perpendicular to its propagation direction, and we take the $x$ axis in this electric field direction. Then the current is also in the $x$ direction, whereas the magnetic perturbation and the plasma velocity is in the $y$ direction (see Appendix). The magnetic field created by cyclotron motion of plasma particles is negligible in a cold plasma limit, and thus we treat the case with $\mu=\mu_{0}$ hereafter. The current is the polarization current due to the temporal change of the electric field, which is $J_{z}=\frac{\mu_{0}}{V_{A}^{2}}\frac{\partial}{\partial t}E_{x}$ (10) where $V_{A}$ is th Alfven speed defined by $V_{A}=B_{0}/\sqrt{\mu_{0}\rho}$ with $\rho$ being the mass density of the plasma. From (2) and the above expression we obtain $D_{x}=\varepsilon_{0}\left(1+\frac{c^{2}}{V_{A}^{2}}\right)E_{x}$ (11) The plasma frozen-in condition $\mathbf{E}+\mathbf{v}\times\mathbf{B}=0$ means that the plasma is moving in the $y$ direction with the $E\times B$ drift speed as $v_{y}=\frac{E_{x}}{B_{0}}\,.$ (12) Then the momentum carried by the plasma particles is $\rho v_{y}=\frac{\mu_{0}B_{0}}{V_{A}^{2}}E_{x}\,.$ (13) The $y$ component of the Minkowski momentum can be calculated from (11) and (13), which is $(\mathbf{D}\times\mathbf{B})_{y}=D_{x}B{}_{0}=\varepsilon_{0}E_{x}B_{0}+\rho v_{y}\,.$ (14) The above expression means the Minkowski momentum is the sum of electromagnetic momentum and momentum of the plasma particles as long as the frozen-in condition is satisfied. When multiplied by $c^{2}$, the first term of the right hand side of (14) becomes the electromagnetic energy flux in the $y$ direction. The second term becomes the relativistic energy comes from the rest mass, which is the predominant energy flux in the non-relativistic limit here; thermal or kinetic energy flux is negligible. The balance equation of the energy-momentum tensor is in a derivative form, and therefore, it has an ambiguity as discussed below (6) for the momentum. The same is true for the energy, and its conservation also holds when we add the mass density and mass flux ($\rho,\rho\mathbf{v})$, since $\partial\rho/\partial t+\nabla(\rho\mathbf{v})=0$. The energy momentum tensor becomes symmetric with these terms. ## 5 Concluding Remarks Momentum carried by an electromagnetic wave has been examined with an example of an MHD wave. It has been shown that the total momentum (electromagnetic momentum plus momentum of the medium) is expressed by $\mathbf{D}\times\mathbf{B}$ as proposed by Minkowski. This result is based on the very simple and basic two properties of an MHD plasma, the frozen-in condition (12) and polarization current (10), namely. It would not be exaggeration if one says the whole kingdom of MHD plasma physics would fall if these two basic properties were wrong. Here in this letter we examined a simplest case of an parallel (to the $\mathbf{B}$ field) propagating MHD wave, but similar calculations can be done for more complicated plasma waves to confirm the result here. A collisionless plasma contains a wide variety of wave phenomena, and the properties of waves can be precisely calculated at least in the linear limit. Calculation of the Minkowski momentum for various plasma waves would be a good exercise to understand the Abraham-Minkowski controversy. The drawback of the Minkowski momentum has been believed that the momentum fails to form a symmetric four dimensional energy-momentum tensor when coupled with the Poyinting flux; an asymmetric energy-momentum tensor means the violation of angular momentum conservation. This difficulty can be overcome when we include the mass flux as a part of energy flux, which is reasonable from the relativistic point of view. The energy-momentum tensor can be symmetric as we have examined with an MHD wave here. What we have shown in the present letter is that the Minkowski momentum can be self consistent description of the total momentum of an electromagnetic wave in a polarizable medium. This does not necessarily mean the Abraham momentum is wrong and inconsistent; it might be possible to give Abraham momentum another appropriate meaning to make it consistent. For example, Barnett [8] recently argued both Abraham and Minkowski momentum can be consistent when we interpret the former as kinetic momentum and latter as canonical momentum. It is out of our scope here to examine this argument, however, it should be noted the legitimacy of the Minkowski momentum does not automatically exclude the validity of the Abraham momentum. ## Appendix This appendix is to derive (10) and (12) in a shortest way for a physicist not familiar with plasma physics. For further information, see any textbook on plasma physics, e.g., [9, 10]. Note that many books derive Alfven waves from the MHD equations, which is different from the derivation here; of course the result is the same. Suppose a plasma consists of equal number of protons and electrons in a uniform magnetic field, which is in the $z$ direction of Cartesian coordinates. The plasma response to the electromagnetic field can be expressed by $\mathbf{P}$ only and we do not need the magnetization current for our calculation. Therefore we can set $\mathbf{H}=\mu_{0}^{-1}\mathbf{B}$ here. We assume an MHD wave described above is propagating in this plasma. Let us denote a vector in the $xy$ plane by a complex number as $A=A_{x}+iA_{y}$. Then the wave electric filed in the $x$ direction is denoted as $E(t)=\frac{E_{0}}{2}(e^{-i\omega t}+e^{\omega t})\,.$ (15) The equation of motion of a plasma particle in the $xy$ plane is written as $\frac{dv}{dt}=i\Omega v+\frac{e}{m}E(t)\,,$ (16) where $\Omega=eB/m$ is the gyro frequency. We include the sign of the charge in $\Omega$, thus $\Omega$ is positive/negative for a proton/electron. The above equation can be directly solved as $v=v_{0}e^{i\Omega t}+\frac{eE_{0}}{2m}\left(\frac{e^{-i\omega t}}{\Omega-\omega}+\frac{e^{i\omega t}}{\Omega+\omega}\right)\,,$ (17) where $v_{0}$ is the integration constant. Now we assume the wave frequency is much smaller than the gyro frequency ($\omega\ll\Omega$), which is true for most of MHD waves. Then the effect of the first term in (17) will be averaged out for MHD time scale; we do not pay attention to this term hereafter. The rest of the motion is called “drift” in plasma physics. The drift velocity $v_{d}$ can be expanded as $v_{d}=\frac{E_{0}}{B}\left(i\cos\omega t+\frac{\omega}{\Omega}\sin\omega t+\cdots\right)\,.$ (18) The first term of (18) is called $E\times B$ drift; protons and electrons drift in the same direction with the same speed with this drift. This term is pure imaginary, which means the drift is in the $y$ direction. This drift gives the predominant motion of the bulk plasma as in (12), however, it does not cause a current because both protons and electrons have the same drift velocity. What contribute to a current is the second term of (18), which is called the polarization drift. Since this term contains $\Omega^{-1}$ factor, protons and electrons moves in the opposite direction with different speed. The electron gyro frequency is much larger than that of protons, therefore, protons predominantly carry currents. The drift direction is the same as the electric field since it is pure real, and the drift speed is proportional to time derivative of the field because of the factor $\omega$ and $\cos\omega t\rightarrow\sin\omega t$. Multiplying the proton’s second term of (18) with the number density and charge, and replacing the factor $\omega$ and $\cos\omega t\rightarrow\sin\omega t$ by the time derivative, we obtain (18). From the Maxwell’s equation we have $\nabla\times\nabla\times\mathbf{E}=c^{-2}\partial^{2}\mathbf{E}/\partial^{2}t+\mu_{0}\mathbf{J}\,.$ (19) When we assume the wave propagation is in the $z$ direction ($\partial/\partial x=\partial/\partial y=0$) and use (10), we obtain the propagation equation of an MHD wave (Alfven wave) as $\left(\frac{1}{c^{2}}+\frac{1}{V_{A}^{2}}\right)\frac{\partial^{2}}{\partial t^{2}}E_{x}-\frac{\partial^{2}}{\partial z^{2}}E_{x}=0\,.$ (20) The Alfven speed $V_{A}$ is often much smaller than the speed of light in space and laboratory plasmas. We obtain an wave propagating with the Alfven speed $V_{A}$ when we ignore the $1/c^{2}$ term in the above expression. ## References ## References * [1] H. Minkowski. Math. Ann, 68:472, 1910. * [2] M. Abraham. Rend. Pal, 28:1, 1909. * [3] M. Abraham. Rend. Pal, 30:33, 1910. * [4] R.N.C. Pfeifer, T.A. Nieminen, N.R. Heckenberg, and H. Rubinsztein-Dunlop. Colloquium: Momentum of an electromagnetic wave in dielectric media. Reviews of Modern Physics, 79(4):1197, 2007. * [5] A. Feigel. Quantum vacuum contribution to the momentum of dielectric media. Physical review letters, 92(2):20404, 2004. * [6] R N C Pfeifer, T A Nieminen, N R Heckenberg, and H Rubinsztein-Dunlop. Constraining Validity of the Minkowski Energy-Momentum Tensor. Physical Review A, 79(2):023813, 2009. * [7] J. D. Jackson. Cassical Electrodynamics. Wiley & Sons, 1962. * [8] Stephen M Barnett. Resolution of the Abraham-Minkowski Dilemma. Physical Review Letters, 104(7):070401, 2010. * [9] F. Chen. Introduction to Plasma Physics. Plenum Press, 1974. * [10] R. Dendy. Plasma Dynamics. Oxford University Press, 1990.	arxiv-papers	2011-12-09T10:21:33	2024-09-04T02:49:25.197093	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Tadas K Nakamura", "submitter": "Tadas Nakamura", "url": "https://arxiv.org/abs/1112.2570" }
1112.2678	# Self-similar solutions of viscous and resistive ADAFs with thermal conduction Kazem Faghei ###### Abstract We have studied the effects of thermal conduction on the structure of viscous and resistive advection-dominated accretion flows (ADAFs). The importance of thermal conduction on hot accretion flow is confirmed by observations of hot gas that surrounds Sgr A∗ and a few other nearby galactic nuclei. In this research, thermal conduction is studied by a saturated form of it, as is appropriated for weakly-collisional systems. It is assumed the viscosity and the magnetic diffusivity are due to turbulence and dissipation in the flow. The viscosity also is due to angular momentum transport. Here, the magnetic diffusivity and the kinematic viscosity are not constant and vary by position and $\alpha$-prescription is used for them. The govern equations on system have been solved by the steady self-similar method. The solutions show the radial velocity is highly subsonic and the rotational velocity behaves sub- Keplerian. The rotational velocity for a specific value of the thermal conduction coefficient becomes zero. This amount of conductivity strongly depends on magnetic pressure fraction, magnetic Prandtl number, and viscosity parameter. Comparison of energy transport by thermal conduction with the other energy mechanisms implies that thermal conduction can be a significant energy mechanism in resistive and magnetized ADAFs. This property is confirmed by non-ideal magnetohydrodynamics (MHD) simulations. 00footnotetext: School of Physics, Damghan University, Damghan, Iran e-mail: kfaghei@du.ac.ir Received: 14 October 2011 / Accepted: 30 November 2011 Keywords accretion, accretion discs – conduction – magnetohydrodynamics: MHD ## 1 Introduction The observational features in active galactic nuclei (AGN) and X-ray binaries can be successfully explained by the standard geometrically thin, optically thick accretion disc model (Shakura & Sunyaev 1973). The motion of the matter in the standard thin model of the accretion disc is approximately Keplerian, and the energy released in the accreting gas is radiated away locally. In the past two decades, another type of accretion flow has been considered that the energy released due to heating processes in the flow may be trapped within accreting gas. As, only the small fraction of the energy released in the accretion flow is radiated away due to inefficient cooling, and most of the energy is stored in the accretion flow and advected to the central object. This type of accretion flow is called as advection-dominated accretion flow (ADAF). The models of ADAF have been studied by a number of researchers (e.g. Ichimaru 1977; Rees et al. 1982; Narayan & Yi 1994; Abramowicz et al. 1995; Blandford & Begelman 1999; Ogilvie 1999). The observations of black holes confirm existence of hot accretion flow that contrasted with classical cold and thin accretion disc model (Shakura & Sunyaev 1973). Hot accretion flows can be seen in supermassive black holes of galactic nuclei and during quiescent of accretion onto stellar-mass black holes in X-ray transients (e.g., Lasota et al. 1996; Esin et al. 1997, 2001; Narayan et al. 1998a; Menou et al. 1999; Di Matteo et al. 2000; see Narayan et al. 1998b; Melia & Falcke 2001; Narayan 2002; Narayan & Quataert 2005 for reviews). Chandra observations provide constraints on the density and temperature of gas at or near the the Bondi capture radius in Sgr A∗ and several nearby galactic nuclei (Loewenstein et al. 2001; Baganoff et al. 2003; Di Matteo et al. 2003; Ho et al. 2003). Tanaka & Menou (2006) exploited these constraints to calculate mean free path of the observed gas. They suggested accretion in such systems are under weakly collisional condition. Moreover, they suggested thermal conduction as a possible mechanism by which the sufficient extra heating is provided in hot accretion flows. In following, Johnson & Quataert (2007) studied the effects of electron thermal conduction on the properties of a spherical hot accretion flows. Their model is applicable for Sgr A∗ in the Galactic centre. Because, electron heat conduction is important for low accretion rate systems. They also found a supervirial temperature in the presence of thermal conduction. Similar to Tanaka & Menou (2006), they assumed a steady state model, but they solved their equations numerically. Abbassi et al. (2008) presented a set of self- similar solutions for ADAFs with a toroidal magnetic field in which the saturated thermal conduction has a important role in the energy transport in the radial direction. Since the observational evidences and magnetohydrodynamics (MHD) simulations have expressed the toroidal magnetic field and the magnetic diffusivity are important in accretion flows (see Faghei 2011a and references therein), Faghei (2011a) examined the self-similar solutions of viscous and resistive ADAFs in the presence of a toroidal magnetic field. But, he did not consider the effects of thermal conduction in his model. While, the recent studies of resistive accretion flows have represented that thermal conduction can play an importance role in such systems (e.g. Sharma et al. 2008; Ghanbari et al. 2009). Sharma et al. (2008) studied the effects of thermal conduction on magnetized spherical accretion flows using global axisymmetric MHD simulations. In their model, when the magnetic energy density becomes comparable to the gravitational potential energy density, the plasma due to resistivity is heated to roughly the virial temperature, the mean inflow becomes highly subsonic, and most of the energy released by accretion is transported to large radii by thermal conduction and the accretion rate became much smaller than Bondi accretion rate. Moreover, they found that for a larger values of conductive parameter, energy transport through thermal conduction becomes the dominant energy transport mechanism at small radii. Ghanbari et al. (2009) studied a two-dimensional advective accretion disc bathed in the poloidal magnetic field of a central accretor in the presence of thermal conduction. They did not consider toroidal component of magnetic field and for simplicity assumed the resistivity as a constant. They studied induction equation of magnetic field in a steady state that is not according to anti- dynamo theorem (e.g. Cowling 1981) and is useful only in particular systems where the magnetic dissipation time is much longer than the age of the system. On the other hand, this assumption implies that the flow is in balance between escape and creation of the magnetic field. In this paper, we adopt the presented solutions by Tanaka & Menou (2006) and Faghei (2011a). Thus, we will investigate the influences of thermal conduction on a viscous and resistive ADAF in the presence of a toroidal magnetic field. Moreover, it is assumed that magnetic diffusivity in the present model is not constant, and escaping and creating of magnetic field are unbalanced. From some aspects will be shown that the present model is in according with the observations and the resistive MHD simulations. The paper is organized as follows. In section 2, the basic equations of constructing a model for quasi- spherical magnetized advection dominated accretion flow with thermal conduction will be defined. In section 3, self- similar method for solving equations which govern the behaviour of the accreting gas was utilized. The summary of the model will appear in section 4. ## 2 Basic Equations We suppose a rotating and accreting gas around a Schwarzschild black hole of mass $M$. The flow is assumed to be in advection dominated stage, where viscous and resistive heating are balanced by the advection cooling and thermal conduction. We use a spherical coordinate ($r$, $\theta$, $\phi$) centred on the accreting object. Furthermore, the flow is assumed to be steady and axisymmetric ($\partial_{t}=\partial_{\phi}=0$), and the equations will be considered in the equatorial plane, $\theta=\pi/2$. Thus, all flow variables are a function of only $r$. For the sake of simplicity, the general relativistic effects are ignored and Newtonian gravity is used. The magnetic field in the present model has only a toroidal component. Under assumptions, the model is described by the following equations: The continuity equation with mass loss is $\frac{1}{r^{2}}\frac{d}{dr}(r^{2}\rho v_{r})=\dot{\rho},$ (1) where $\rho$ is density, $v_{r}$ is the radial infall velocity, and $\dot{\rho}$ the mass-loss rate per unit volume. The radial equation of momentum is $v_{r}\frac{dv_{r}}{dr}=r(\Omega^{2}-\Omega^{2}_{K})-\frac{1}{\rho}\frac{d}{dr}(\rho c^{2}_{s})-\frac{c^{2}_{A}}{r}-\frac{1}{2\rho}\frac{d}{dr}(\rho c^{2}_{A}),$ (2) where $\Omega$ is the angular velocity of the flow, $\Omega_{K}=\sqrt{GM/r^{3}}$ is the Keplerian angular velocity, $c_{s}$ the isothermal sound speed, which is defined as $c_{s}^{2}=p_{gas}/\rho$, $p_{gas}$ being the gas pressure, and $c_{A}$ is Alfven speed, which is defined as $c_{A}^{2}=B_{\varphi}^{2}/4\pi\rho=2p_{mag}/\rho$, $p_{mag}$ being the magnetic pressure. The angular momentum transfer equation is $\rho v_{r}\frac{d}{dr}(r^{2}\Omega)=\frac{1}{r^{2}}\frac{d}{dr}\left[\nu\rho r^{4}\frac{d\Omega}{\partial r}\right],$ (3) where the right-hand side of above equation describes the effects of viscous torques due to shear ($\nu$, here, is kinematic coefficient of viscosity). As noted in the introduction, we assume both of the kinematic coefficient of viscosity and the magnetic diffusivity due to turbulence in the accretion flow. Thus, it is reasonable to use these parameters in analogy to the $\alpha$-prescription of Shakura & Sunyaev (1973) for the turbulent, $\nu=P_{m}\eta=\alpha\frac{c_{s}^{2}}{\Omega_{K}},$ (4) where $P_{m}$ is the magnetic Prandtl number, which is assumed a constant of order of unity, $\eta$ is the magnetic diffusivity, and $\alpha$ is a free parameter less than unity. The energy equation becomes $\displaystyle\frac{v_{r}}{\gamma-1}\frac{d}{dr}(\rho c^{2}_{s})+\frac{\gamma}{\gamma-1}\frac{\rho c^{2}_{s}}{r^{2}}\frac{d}{dr}\left(r^{2}v_{r}\right)=$ $\displaystyle Q_{diss}-Q_{rad}+Q_{cond},$ (5) where $Q_{diss}=Q_{vis}+Q_{resis}$ is the dissipation rate by viscosity $Q_{vis}$ and resistivity $Q_{resis}$, $Q_{rad}$ represents the energy loss through radiative cooling, and $Q_{cond}$ is the energy transported by thermal conduction. For the right-hand side of the energy equation, we can write $Q_{adv}=Q_{diss}-Q_{rad}+Q_{cond}$ (6) where $Q_{adv}$ is the advective transport of energy. We exploit the advection factor, $f=1-Q_{rad}/Q_{diss}$, that describes the fraction of the dissipation energy which is stored in the accretion flow and advected into the central object rather than being radiated away. In general, the advection factor depends on the details of the heating and radiative cooling mechanisms and will vary by position (e.g. Watari 2006, 2007; Sinha et al. 2009). Here, we assume a constant $f$ for simplicity. Clearly, the flow in the case of $f=1$ is in the extreme limit of no radiative cooling and in the limit of efficient radiative cooling, we have $f=0$. As mentioned, the inner regions of hot accretion flows are collisionless and the electron mean free path due to Coulomb collision is larger than the radius. This property is described as saturation (Cowie & McKee 1977). The traditional equation of heat flux due to thermal conduction, $F_{cond}=-\kappa\nabla T$ which $\kappa$ being the thermal conduction coefficient, is not valid for such systems. Because this equation is suitable for the collisional plasma, in which mean free path of electron energy exchange is smaller than temperature scale height. Cowie & McKee (1977) derived the heat flux for the collisionless plasma as $F_{sat}=5\phi_{s}\rho c_{s}^{3}=5\phi_{s}p\sqrt{\frac{p}{\rho}},$ (7) where $\phi_{s}$ is a factor is less than unity and is called as saturation constant. Now, the viscous and resistive heating rates and the energy transport by thermal conduction are expressed as $Q_{vis}=\nu\rho r^{2}\left(\frac{\partial\Omega}{\partial r}\right)^{2}$ (8) $Q_{resis}=\frac{\eta}{4\pi}{\bf J}^{2}$ (9) $Q_{cond}=-\left\|\frac{1}{r^{2}}\frac{\partial}{\partial r}\left(r^{2}F_{sat}\right)\right\|$ (10) where ${\mathbf{J}}=\nabla\times{\mathbf{B}}$ is the current density, ${\mathbf{B}}$ being the magnetic field. We used a minus sign in equation (10). Because we want thermal conduction as energy transport mechanism outward. On the other hand, heat conduction flux will be behaved like a cooling mechanism in accretion flow. Finally, since we consider only the toroidal field, the induction equation with field escape can be written as $\frac{1}{r}\frac{d}{dr}\left[rv_{r}B_{\varphi}-\eta\frac{d}{dr}(rB_{\varphi})\right]=\dot{B}_{\varphi}.$ (11) where $B_{\varphi}$ is the toroidal component of magnetic field and $\dot{B}_{\varphi}$ is the field escaping/creating rate due to a magnetic instability or dynamo effect. This induction equation is rewritten as $\displaystyle\frac{1}{r}\frac{d}{dr}\left[\sqrt{4\pi\rho c^{2}_{A}}\left(rv_{r}-\frac{\alpha}{4\beta P_{m}}\frac{1}{r\rho\Omega_{K}}\frac{d}{dr}(r^{2}\rho c^{2}_{A})\right)\right]=$ $\displaystyle\dot{B}_{\varphi},$ (12) where $\beta$ is the degree of magnetic pressure to the gas pressure and can be defined by $\beta=\frac{p_{mag}}{p_{gas}}=\frac{1}{2}\left(\frac{c_{A}}{c_{s}}\right)^{2}.$ (13) In this paper, we apply the steady self-similar methods to solve the system equations. Thus, this parameter will be constant throughout the disc. While, Khesali & Faghei (2008, 2009) showed that it varies by position. In hot accretion flows, typical value of $\beta$ is in the range $0.01$-$1$ (e.g. De Villiers et al. 2003; Beckwith et al. 2008). Here, we will also consider the magnetically dominated case ($\beta>1$). Because, when thermal instability happens in an ADAF, the MHD numerical simulations imply that the thermal pressure rapidly decreases while the magnetic pressure increases due to the conservation of magnetic flux (Machida et al. 2006). This will result in large $\beta$ and forms a magnetically dominated accretion flow (Bu et al. 2009). ## 3 Self-Similar Solutions ### 3.1 Analysis The self-similar method is useful to understand of the physics of accretion flows. This method is familiar due to its wide range of applications in many research fields of astrophysics. Self-similar solution, although constituting only a limited part of problem, is often useful to understand the basic behaviour of the system. Thus, in order to seek similarity solutions for the above equations, we seek solutions in the following form: $v_{r}(r)=-c_{1}\alpha\sqrt{\frac{GM_{}}{r}}$ (14) $\Omega(r)=c_{2}\sqrt{\frac{GM_{}}{r^{3}}}$ (15) $c^{2}_{s}(r)=c_{3}\frac{GM_{}}{r}$ (16) $c^{2}_{A}(r)=\frac{B^{2}_{\varphi}}{4\pi\rho}=2\beta c_{3}\frac{GM_{}}{r}$ (17) where coefficients $c_{1}$, $c_{2}$, and $c_{3}$ are determined later. We assume a power-law relation for density $\rho(r)=\rho_{0}r^{s}$ (18) where $\rho_{0}$ and $s$ are constant. By using above self-similar quantities, the mass-loss rate and the field escaping/creating rate must have the following form: $\dot{\rho}(r)=\dot{\rho}_{0}r^{s-3/2}$ (19) $\dot{B}_{\varphi}(r)=\dot{B}_{0}r^{\frac{s-4}{2}}$ (20) where $\dot{\rho}_{0}$ and $\dot{B}_{0}$ are constant. Substituting the above solutions in the continuity, momentum, angular momentum, energy, and induction equations [(1)-(3), (5), and (11)], we can obtain the following relations: $\dot{\rho}_{0}=-\left(s+\frac{3}{2}\right)\alpha\rho_{0}c_{1}\sqrt{GM_{}},$ (21) $-\frac{1}{2}c^{2}_{1}\alpha^{2}=c^{2}_{2}-1-c_{3}\left[s-1+\beta(1+s)\right],$ (22) $c_{1}=3(s+2)c_{3},$ (23) $\displaystyle-\alpha c_{1}\left[\frac{2(s-1)+3\gamma}{\gamma-1}\right]=\alpha f\left[\frac{9}{2}c^{2}_{2}+\frac{\beta}{P_{m}}c_{3}(1+s)^{2}\right]$ $\displaystyle-10\phi_{s}c_{3}^{1/2}\left\|s+\frac{1}{2}\right\|,$ (24) $\dot{B}_{0}=-\frac{\alpha s}{2}GM_{}\sqrt{2\pi\rho_{0}\beta c_{3}}\left[2c_{1}+\frac{c_{3}}{P_{m}}(1+s)\right].$ (25) Above equations express for $s=-3/2$, there is no mass loss, while for $s>-3/2$ mass loss (wind) exists. After algebraic manipulations, we obtain an algebraic equation for $c_{3}$: $\displaystyle{\frac{81}{8}}\,{\alpha}^{3}(s+2)^{2}f{c_{{3}}}^{2}+c_{3}\Bigg{[}\frac{3\,\alpha(s+2)}{2}\times\frac{2(s-1)+3\gamma}{\gamma-1}$ $\displaystyle-\frac{9\alpha f}{4}\,\Bigg{(}\frac{2\beta}{9P_{m}}(s+1)^{2}+(s+1)\beta+s-1\Bigg{)}\Bigg{]}$ $\displaystyle-5\phi_{s}\left\|s+\frac{1}{2}\right\|\sqrt{c_{{3}}}-\frac{9}{4}\,f\alpha=0,$ (26) and the rest of the physical variables are $\dot{\rho}_{0}=-3\alpha\rho_{0}\sqrt{GM_{}}(s+2)\left(s+\frac{3}{2}\right)c_{3},$ (27) $c_{1}=3c_{3}(s+2),$ (28) $c_{2}^{2}=1-\frac{9\alpha^{2}}{2}(s+2)^{2}c_{3}^{2}+c_{3}\left[(s+1)\beta+s-1\right],$ (29) $\displaystyle\dot{B}_{0}=-3\alpha sGM_{}(s+2)^{3/2}c_{3}^{3/2}\sqrt{6\pi\rho_{0}\beta}\times$ $\displaystyle\left[1+\frac{s+1}{6P_{m}(s+2)}\right].$ (30) We can solve algebraic equation (26) numerically and clearly only real roots which correspond to positive $c_{1}$ are physically acceptable. Without thermal conduction, i.e. $\phi_{s}=0$, (26) and similarity solutions reduce to Faghei (2011a). But our main algebraic equation includes thermal conduction. Fig. 1 : Physical quantities of the flow as a function of saturation constant for $\gamma=1.3$, $\alpha=0.5$, $f=1$, and $P_{m}=100$. Solid, dotted, dashed, and dot-dashed lines represent $\beta=0.1,0.3,0.7$, and $1.5$. ### 3.2 Numerical Results Now, we consider the behaviour of solutions in the presence of thermal conduction. But in this paper, only case of no wind ($s=-3/2$) is considered that $\dot{\rho}_{0}=0$ and $\dot{B}_{\varphi}\propto r^{-11/4}$. In addition to introduced coefficients, we also define a new parameter of $c_{4}$ that is the right-hand side of equation (24) $c_{4}=\alpha f\left[\frac{9}{2}c^{2}_{2}+\frac{\beta}{P_{m}}c_{3}(1+s)^{2}\right]-10\phi_{s}c_{3}^{1/2}\left\|s+\frac{1}{2}\right\|.$ (31) Equations (5), (24), and (31) imply that the parameter $c_{4}$ is the advection transport of energy. The behaviour of coefficients $c_{i}$ as a function of $\phi_{s}$ are shown in Figures 1-3. Moreover, Figure 1 represents the profiles of physical quantities for several values of the magnetic pressure fraction, i. e. $\beta=0.1,0.3,0.7$, and $1.5$. The value of $\beta$ measures the strength of magnetic field, and a larger $\beta$ denotes a stronger magnetic field. Figure 2 represents the profiles of physical quantities for several values of magnetic Prandtl number, i. e. $P_{m}=\infty,1,2/3$, and $1/2$. The smaller values of $P_{m}$ denotes a stronger magnetic diffusivity, $\eta$. Figure 3 shows the profiles of physical quantities for several values of adiabatic index, i. e. $\gamma=1.2,1.25,1.3$, and $1.35$. Fig. 2 : Same as Fig. 1, but $\beta=1.0$, and solid, dotted, dashed, and dot- dashed lines represent $P_{m}=\infty,1,2/3$, and $1/2$. Fig. 3 : Same as Fig. 1, but $\beta=1.0$, and solid, dotted, dashed, and dot- dashed lines represent $\gamma=1.2,1.25,1.3$, and $1.35$. The solutions in Figures 1-3 imply that the radial infall velocity, $c_{1}$, and the sound speed, $c_{3}$, both decrease with the magnitude of conduction, while the squared angular velocity, $c_{2}^{2}$, increases. These properties are qualitatively consistent with dynamical analysis of Faghei (2011b). One and two dimensional simulations of hot accretion flows have also shown that thermal conduction reduces the flow temperature (Sharmal et al. 2008; Wu et al. 2010). The profiles of advection transport of energy, $c_{4}$, in Figures 1-3 imply that thermal conduction behaves as a cooling mechanism, resulting in a local decrease of the gas temperature relative to the original ADAF solution. At the same time, the gas adjust its angular velocity (which increases the level of viscous dissipation) and reduces its inflow speed. Figure 1 shows the radial velocity, sound speed, and advection transport of energy decrease by adding the magnetic pressure fraction, $\beta$. These properties are qualitatively consistent with results of Bu et al. (2009) and Faghei (2011a). Moreover, the angular velocity decreases with the magnitude of magnetic field that is in according with results of Khesali & Faghei (2009) and Faghei (2011a). Figure 2 shows the magnetic diffusivity has the opposite effects of thermal conduction on the physical variables. As, by adding the magnetic diffusivity, $P_{m}^{-1}$, the radial velocity, sound speed, and advection transport of energy increase, while the rotational velocity decrease. These results are similar to resistive ADAF models without thermal conduction (e. g. Faghei 2011a). Figure 3 represents the gas adiabatic index similar to magnetic diffusivity has the opposite effects of thermal conduction on the physical quantities. As, with the magnitude of $\gamma$, the inflow and sound speed increase, while the rotational velocity decreases. This is in accord with dynamical study of hot accretion flow (Faghei 2011b). Moreover, Figure 3 shows that the gas adiabatic index contributes with thermal conduction to reduce advection transport of energy. It can be due to decrease of rotational velocity by adding $\gamma$, which reduces the level of viscous dissipation. The studies of hot accretion flows (e. g. Shadmehri 2008) imply that the solution for a given set of the input parameters reaches to a non-rotating limit at a specific of $\phi_{s}$ which we denote it by $\phi_{s}^{c}$. With zero insertion of $c_{2}$ in equations (22)-(24), $\phi_{s}^{c}$ can be written as $\displaystyle\phi_{s}^{c}=\frac{1}{60}\left[\frac{\beta f}{P_{m}}-\frac{5/3-\gamma}{\gamma-1}\right]\times$ $\displaystyle\sqrt{2(\beta+5)\left[-1+\sqrt{1+\frac{18\alpha^{2}}{(\beta+5)^{2}}}\,\right]}.$ (32) We can not extend the studies beyond $\phi_{s}^{c}$, because the right-hand side of equation (29) becomes negative and a negative $c_{2}^{2}$ is clearly unphysical. As, the rotational velocity profiles in Figures 1-3 show, we have selected the input parameters that $c_{2}^{2}$ is positive. The behaviour of critical saturation constant, $\phi_{s}^{c}$, as a function of $\beta$ for several values of magnetic Prandtl number is shown in Figures 4. In left panel of Figure 4, the viscosity parameter value is $\alpha=0.1$, and in right panel is $\alpha=0.2$. The profiles of Figure 4 show the critical saturation constant highly depends on magnetic pressure fraction, $\beta$, magnetic Prandtl number, $P_{m}$, and viscosity parameter, $\alpha$. As, higher values of $\beta$ corresponds to larger $\phi_{s}^{c}$. Critical saturation constant also increases with higher value of $\alpha$. Since, magnetic diffusivity is proportional to $P_{m}^{-1}$, Figure 4 shows that the magnetic diffusivity similar to magnetic field increases $\phi_{s}^{c}$ value. As mentioned in the introduction, Sharma et al. (2008) by resistive MHD simulation studied a spherical accretion with thermal conduction. They found for even modest thermal conductivities, conduction is the significant mechanism of energy. Here, to compare conduction mechanism to others, we study the ratio of energy transport by thermal conduction, $Q_{cond}$, to the gas heating rate by viscosity $Q_{vis}$ and resistivity $Q_{resis}$. Such solutions are shown in Figure 5 for two cases of non-resistive (left-panel) and resistive (right-panel) flows. The solutions imply that thermal conduction is the significant energy mechanism in the flow. This result confirms simulation of Sharma et al. (2008). Moreover, the ratio of $Q_{cond}$ to $Q_{diss}$ increases slightly by adding the magnetic field and does not change for different values of magnetic Prandtl number. Because, a large fraction of $Q_{diss}$ is generated by viscous dissipation. ## 4 Summary and Discussion The collision timescale between ions and electrons in hot accretion flows is longer than the inflow timescale. Thus, the inflow plasma is collisionless, and transfer of energy by thermal conduction can be dynamically important. The low collisional rate of the gas is confirmed by direct observation, particularly in the case of the Galactic centre (Quataert 2004; Tanaka & Menou 2006) and in the intracluster medium of galaxy clusters (Sarazin 1986). In this paper, the structure of a magnetized ADAF in the presence of resistivity and thermal conduction is investigated. We assumed the magnetic field has a purely toroidal component. We adopted the presented solutions by Tanaka & Menou (2006) and Faghei (2011a). Thus, we assumed that angular momentum transport is due to viscous turbulence and the $\alpha$\- prescription is used for the kinematic coefficient of viscosity. We also assumed the flow does not have a good cooling efficiency and so a fraction of energy accretes along with matter on to the central object. In order to solve the equations that govern the structure behaviour of magnetized ADAF with thermal conduction, we have used steady self-similar solution. Fig. 4 : The critical saturation constant as a function of the ratio of magnetic pressure to gas pressure. Solid, dotted, dashed, and dot-dashed lines represent $P_{m}=10,5,1$, and $0.5$. The input parameters are set to $\gamma=5/3$, $f=1$, $s=-3/2$, and the viscous parameter $\alpha$ in left- panel is $0.1$ and in right-panel is $0.2$. The solutions showed the radial infall velocity and sound speed in the presence of thermal conduction both decrease, while angular velocity increase. These properties are consistent with dynamical study of hot accretion flow (e. g. Faghei 2011b) and from some aspects also are in accord with simulations of Sharma et al. (2008) and Wu et al. (2010). Moreover, the solutions represent the magnetic diffusivity and thermal conduction have the opposite effects on physical quantities. For a moderate thermal conduction, the solutions imply that thermal conduction can play an important role in energy mechanism of the system. This property is qualitatively consistent with non-ideal simulations of Sharma et al. (2008). In the present model, accretion flow is studied in one-dimensional approach and ignored from latitudinal dependence of physical quantities. There are some researches in two-dimensional approach that express the importance of such studies (Tanaka & Menou 2006; Ghanbari et al. 2009; Wu et al. 2010 ). Thus, latitudinal study of present model can be investigated in other research. Here, we used a saturated heat conduction flux. While, there are some studies with unsaturated heat flux (e. g. Shcherbakkov & baganoff 2010) that show a good agreement with observations. Thus, the study of the present model in a unsaturated case will be interesting. Fig. 5 : The ratio of energy transport by thermal conduction, $Q_{cond}$, to the gas heating rate by viscosity $Q_{vis}$ and resistivity $Q_{resis}$ as a function of saturation constant. Solid, dotted, dashed, and dot-dashed lines represent $\beta=0.0,1.0,3.0$, and $5.0$. The input parameters are set to $\gamma=4/3$, $f=1$, $s=-3/2$, $\alpha=0.5$, and the magnetic Prandtl number in left-panel is $\infty$ and in right-panel is $0.5$. ## Acknowledgements I wish to thank the anonymous referee for very useful comments that helped me to improve the initial version of the paper. I would also like to thank Roman V. Shcherbakov for his helpful comments. ## References * (1) Abbassi S., Ghanbari G., Najjar S., 2008, MNRAS, 388, 663 * (2) * (3) Abramowicz, M., Chen, X., Kato, S., Lasota, J. P., Regev, O., 1995, ApJ, 438, L37 * (4) * (5) Baganoff, F. K., et al. 2003, ApJ, 591, 891 * (6) * (7) Blandford, R. D., Begelman, M. C. 1999, MNRAS, 303, L1 * (8) * (9) Bu D., Yuan F., Xie F., 2009, MNRAS, 392, 325 * (10) * (11) Cowie L. L., McKee C. F., 1977, ApJ, 211, 135 * (12) * (13) Cowling, T. G., 1981, ARA&A, 19, 115 * (14) * (15) Di Matteo T., Allen S. W., Fabian A. C., Wilson A. S., Young, A. J. 2003, ApJ, 582, 133 * (16) * (17) Di Matteo T., Quataert, E., Allen S. W., Narayan R., Fabian A. C., 2000, MNRAS, 311, 507 * (18) * (19) Esin A. A., McClintock J.E., Narayan R. 1997, ApJ, 489, 865 * (20) * (21) Esin A. A., McClintock J. E., Drake J. J., Garcia M. R., Haswell C. A., Hynes R. I., Muno M. P., 2001, ApJ, 555, 483 * (22) * (23) Faghei, K., 2011a, J. Astrophys. Astr., accepted * (24) * (25) Faghei, K., 2011b, MNRAS, accepted * (26) * (27) Ghanbari J., Abbassi S., Ghasemnezhad M., 2009, MNRAS, 400, 422 * (28) * (29) Ho L. C., Terashima Y., Ulvestad J. S., 2003, ApJ, 589, 783 * (30) * (31) Ichimaru, S. 1977, ApJ, 214, 840 * (32) * (33) Johnson B. M., Quataert E., 2007, ApJ, 660, 1273 * (34) * (35) Khesali, A., Faghei, K., 2008, MNRAS, 389, 1218 * (36) * (37) Khesali, A., Faghei, K., 2009, MNRAS, 398, 1361 * (38) * (39) Lasota J.-P., Abramowicz M. A., Chen X., Krolik J., Narayan R., Yi I., 1996, ApJ, 462, 142L * (40) * (41) Loewenstein M., Mushotzky R. F., Angelini L., Arnaud K. A., Quataert, E., 2001, ApJ, 555, L21 * (42) * (43) Machida M., Nakamura K. E., Matsumoto R., 2006, PASJ, 58, 193 * (44) * (45) Melia F., Falcke H. 2001, ARA&A, 39, 309 * (46) * (47) Menou K., Esin, A. A., Narayan R., Garcia M. R., Lasota J.-P., McClintock J. E., 1999, ApJ, 520, 276 * (48) * (49) Narayan R. 2002, in Lighthouses of the Universe, ed. Gilfanov M., Sunyaev R. A., Churazov E. ( Berlin: Springer), 405 * (50) * (51) Narayan R., Mahadevan R., Grindlay J. E., Popham R. G., Gammie C., 1998a, ApJ, 492, 554 * (52) * (53) Narayan, R., Mahadevan, R., Quataert, E. 1998b, in Theory of Black Hole Accretion Disks, ed. Abramowicz M., Bjornsson G., Pringle J. (Cambridge: Cambridge Univ. Press), 148 * (54) * (55) Narayan R., Quataert E., 2005, Science, 307, 77 * (56) * (57) Narayan, R., Yi, I., 1994, ApJ, 428, L13 * (58) * (59) Ogilvie, G. I., 1999, MNRAS, 306, L9O * (60) * (61) Quataert E., 2004, ApJ, 613, 322 * (62) * (63) Rees, M. J., Begelman, M. C., Blandford, R. D., Phinney, E. S. 1982, Nature, 295, 17 * (64) * (65) Sarazin C. L., 1986, Rev. Mod. Phys., 58, 1 * (66) * (67) Shadmehri, M., 2008, Ap&SS, 317, 201 * (68) * (69) Shakura N. I., Sunyaev R. A., 1973, A&A, 24, 337 * (70) * (71) Sharma P., Quataert E., Stone J. M., 2008, MNRAS, 389, 1815 * (72) * (73) Shcherbakov R. V., Baganoff F. K., 2010, ApJ, 716, 504 * (74) * (75) Sinha M., Rajesh S. R., Mukhopadhyay B., 2009, RAA, 9, 1331 * (76) * (77) Tanaka T., Menou K., 2006, ApJ, 649, 345 * (78) * (79) Watarai, K. 2006, ApJ, 648, 523 * (80) * (81) Watarai, K. Y. 2007, PASJ, 59, 443 * (82) * (83) Wu M., Yuan F., Bu D., 2010, ScChG, 53S, 168 * (84)	arxiv-papers	2011-12-12T20:11:22	2024-09-04T02:49:25.205520	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Kazem Faghei", "submitter": "Kazem Faghei", "url": "https://arxiv.org/abs/1112.2678" }

Subsets and Splits

No community queries yet

The top public SQL queries from the community will appear here once available.